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label{eq:LnCdns}
\begin{aligned}
&{x\_{i,i+1}} - {r\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}{v^{(t\_i-1)}\_{i+1}}-4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}- \\
&\hspace\*{0.87in} 4{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}} {v^{(t\_i-1)}\_{i+1}} - 4{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}-\\
&\hspace\*{0.87in} 4{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}} - 4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}} - 4{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}-2{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}- \\
&\hspace\*{1.55in} 2{P^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-4{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}-4{P^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}-{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}{v^{(t\_i-1)}\_{i+1}} - 4{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}} \\
&~-2{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}- 2{P^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}\geq 0 \\
&({{v^{(t\_i-1)}\_{i+1}}})^3 - 4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i}}{v^{(t\_i-1)}\_{i+1}} \\
&~-2{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}}-4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-8{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}} \geq 0 \\
&{v^{(t\_i-1)}\_{i}}-8{Q^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}} \geq 0 \\
&{v^{(t\_i-1)}\_{i}}-8{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}} \geq 0
\end{aligned}$$
We note that most conditions in hold for typical values of the variables except the first one. But we observe global convergence in practice in the sequential manner even when *x**i*, *i* + 1 ≥ *r**i*, *i* + 1 may not hold.
Note that conditions in are the same as those in from Theorem [thm:single-bd] applied for the local subsystem {*i*, *i* + 1, *i* + 2} in place of {*i*, *j*, *k*}. Since conditions in hold for *i* = 1, the first local subsystem {1, 2, 3} is convergent at iteration step *t*1 following Theorem [thm:single-bd]. By Assumption [asm:cvgcpres], the convergence of node 2 is preserved in the next iteration step, and hence it can be treated as the fixed source node for the next local subsystem {2, 3, 4}. The convergence of this subsystem is then guaranteed at time *t*2 ≥ *t*1 + 1 by conditions in holding for *i* = 2. The overall result follows by the sequential application of Theorem [thm:single-bd].
Numerical Study
===============
In this section, we demonstrate the convergence properties of the ENDiCo-OPF algorithm with the help of numerical simulations. We also validate the optimality of our algorithm by comparing its results with those of a centralized solution. These simulations not only justify the convergence analysis of the method but also showcase the efficacy of the proposed real-time distributed controller to attain optimal power flow solutions. After attaining the optimal dispatch, the controller shares the computed boundary variables with it its neighbor instead of implementing and measuring the variables.
IEEE-123 Bus Test System
[Testsys]
Simulated System and Results
----------------------------
As a test system, we simulated a balanced IEEE-123 bus system with a maximum of 85 DERs (PVs) connected to the network (Fig. [Testsys]), where the DER/PV penetration can vary from 10% to 100%. As a cost function, we have simulated both (i) active power loss minimization (*f* = *r**i**j**l**i**j*), and (ii) voltage deviation (Δ*V*) minimization (*f* = (*v**j* − *v*ref)2 ) optimization problems.
### Residual and Objective Value Convergence
We have simulated the test system with 10%, 50%, and 100% DER/PV penetration cases for both active power loss minimization and Δ*V* minimization. The system converged after 42 iterations for all six cases (Fig. [convergence]). For the loss minimization objective, the maximum border residual goes below the tolerance value of 10− 3 after the 42nd iteration. The objective values for the loss minimization OPF is 26.5 kW, 19.6 kW, and 11.8 kW, for 10, 50, and 100% DER penetration, respectively. Similarly, for Δ*V* minimization, we can see that the maximum residual goes below the tolerance after the 42nd iteration as well. Thus the convergence is related to the network size, but not to the number of controllable variables.
0.75
0.78
[convergence]
0.49
0.49
[Validation]
0.49
0.49
[Voltageconv]
[!b]
[comptable]
| | | | | |
| --- | --- | --- | --- | --- |
| | Central | 26.4 | 19.6 | 11.78 |
| | ENDiCo-OPF | 26.5 | 19.6 | 11.80 |
| | Central | 0.5300 | 0.5038 | 0.4640 |
| | ENDiCo-OPF | 0.5306 | 0.5042 | 0.4642 |
[ht!]
0.672
0.672
0.672
0.672
0.672
0.672
[boundV]
### Validation of the Optimal Solution
Besides a faster convergence, we also present the efficacy of the distributed OPF controller in terms of the optimality gaps and feasibility. To this end, we have compared (a) the objective values, and (b) the nodal voltages with the centralized solution (see Fig. [Validation]). It can be observed in Table [comptable] that the value of the objective functions from centralized and distributed solutions matches for all the cases. For example, for the 100% DER penetration case, the line loss is 11.80 kW for proposed ENDiCo-OPF method, and the central solution is 11.78 kW. Similar comparisons can be found for other DER/PV penetration cases with different OPF objectives. This validates the solution quality of ENDiCo-OPF. Further, we can see in Fig. [Validation] that upon implementing ENDiCo-OPF, the difference in nodal voltages in the system and those from a centralized solution is in the order of 10− 4; this is true for both OPF objectives. This validates the feasibility of ENDiCo-OPF.
Numerical Experiments of Convergence
------------------------------------
In this section, we provide further simulated results on the convergence of the proposed ENDiCo-OPF method. Here we showcase the boundary variable convergence with respect to iterations, as well as their properties. We also compare the numerical convergence results with the theoretical analysis presented in Section [sec:cnvgcanal].
### Convergence at the Boundary
The convergence of the boundary variables (shared boundary voltage) for the simulated cases has been shown in Fig. [Voltageconv] and Fig.[boundV]; Fig. [Voltageconv] shows the boundary variables for 100% PV cases for 3 different locations that helps to visualize the convergence of the shared variable w.r.t. the distance from the root node (substation node). Specifically, from Fig. [boundV], we can see that after the initial values, the shared variables (shared nodal voltages) suddenly changes abruptly till 22*n**d* iterations. This location, i.e., iteration number, where this sudden changes happen depends on the distance of that shared node from the root node (substation node). For example, ‘Bus 3’ is 2 node distant from the substation, and thus this abrupt changes happen at the 2nd iteration (Fig. [Voltageconv]); similarly, ‘Bus 62’ is 11 node distant from the substation, and that changes happen at the 11*t**h* iteration as well. Along with these characteristics, the overall convergence properties of the shared variables are consistent with both objectives and for all the PV penetration cases as well (Fig. [boundV]). It corroborates with the statement that the convergence properties is not dependent on the OPF objective or the DER penetration percentage, but rather dependent on the system network. Instead of using a flat start with 1.02 pu for the controller, a measured voltage initialization would reduce the iteration number; however, we would like to mention that, this method is robust enough to initialize with any reasonable flat start values.
### Discussion on Convergence
In, we guaranteed the convergence of the proposed method under some sufficient conditions. Specifically, we showed (i) convergence of the local sub-problem for a given iteration step (Theorem [thm:subsyscvgce]), (ii) convergence of the local sub-problem over iteration steps (Theorem [thm:single-bd]), and (iii) convergence over time for a line network with multiple nodes (Theorem [thm:globalcvgnc]). At the same time, our numerical experiments demonstrated similar convergence behavior for more general (than line) networks.
The condition for the convergence of the local sub-problem in a single iteration step is expressed in equation. Generally for a stable electric power supply in a power distribution system, *v**i* and *P**i**j*, *Q**i**j* are in the order of 1 pu., and the corresponding line parameters, i.e., *r**i**j*, *x**i**j*, are both in the order of 10− 2 or less. This guarantees that condition is always satisfied for a practical power distribution system. In our simulated cases, line parameters are also in the order of ≤ 10− 2, thus satisfying the condition for Theorem [thm:subsyscvgce]. Further, this also satisfies most of the sufficient conditions for Theorem [thm:single-bd] and [thm:globalcvgnc], except the first conditions, i.e., *x**i**j* − *r**i**j* ≥ 0 of both of the theorems. We note that these are sufficient conditions, and that we can observe overall convergence for the cases where *x**i**j* − *r**i**j* < 0 as well. For the simulation cases, while other sufficient conditions hold true, we observed *x**i**j* − *r**i**j* < 0 for some of the lines, but still the controller converged. In addition, Fig. [Voltageconv] showcases the same result as Theorem [thm:globalcvgnc]. The node that is closer to the root node/substation node, i.e., the node with a strong voltage source, converges earlier than the node that is more distant from the root node. For instance, “Bus 3”, which is two nodes away from the substation, converges earlier than the “Bus 62” that is 11 nodes distant from the substation. Bus 3 converges around the 22nd iteration, whereas Bus 62 converges around 30th iteration for both loss and ΔV minimization optimization problems.
Comparison against Centralized OPF and an ADMM-based Distributed OPF Approach
-----------------------------------------------------------------------------
Table [table:timecomptable] compares the total solve time for the three different algorithms: a centralized OPF, the proposed distributed ENDiCo-OPF, and an ADMM-based distributed OPF. All three approaches are applied to both objective functions: loss minimization and voltage deviation minimization. The simulation was performed using a Core i7-8550U CPU @ 1.80GHz with 16GB of memory. All three algorithms use fmincon solver from MATLAB to solve the associated nonlinear optimization problems. Since all algorithms use the same compute system and same nonlinear solver, the simulation results provided are appropriate to demonstrate the relative improvements observed via the proposed distributed algorithm. The results show that the proposed distributed approach is significantly faster than both the centralized and ADMM-based distributed OPF methods. For example, for the loss minimization problem with 100% PV penetration, the solution time for ENDiCo-OPF is only 0.71 seconds, while the centralized OPF and ADMM-based distributed OPF take 15.8 seconds and 37.3 minutes, respectively. Moreover, the solution time for the centralized OPF increases with the increase in the number of controllable nodes (i.e. %PV penetration). The proposed ENDiCo-OPF method, however, scales well even for larger number of controllable nodes.
[!h] [table:timecomptable]
| | | | | |
| --- | --- | --- | --- | --- |
| | Centralized OPF | 4.2 sec | 10.6 sec | 15.8 sec |
| | ENDiCo-OPF | 0.67 sec | 0.71 sec | 0.71 sec |
| | ADMM-based OPF | 41.6 min | 36.5 min | 37.3 min |
| | Centralized OPF | 2.1 sec | 3.4 sec | 4.8 sec |
| | ENDiCo-OPF | 0.68 sec | 0.70 sec | 0.70 sec |
| | ADMM-based OPF | 120 min | 119 min | 121 min |
Additionally, compared to ADMM-based approach, the proposed ENDiCo-OPF method also reduces the required number of communication rounds/iterations by order of magnitudes. Besides the iteration counts, the developed method converges faster compared to the ADMM-based distributed OPF. Figure [fig:comparisonADMM] illustrates the convergence properties of the objective values for both the ADMM-based method and the ENDiCo-OPF method for 100% PV penetration cases. As can be observed, the ADMM-based method requires 7, 000 and 10, 00 iterations for loss minimization and ΔV minimization problems, respectively. It is worth noting that the proposed ENDiCo-OPF method requires only 42 iterations for both cases, which highlights the effectiveness of the developed method. These results demonstrate that the proposed ENDiCo-OPF method outperforms both centralized OPF and ADMM-based distributed OPF methods.
0.48
0.48
[fig:comparisonADMM]
Applicability and Extension to Real-world Setting
-------------------------------------------------
Although in this paper, we focus on a single-period optimization problem, the proposed distributed algorithm has been numerically demonstrated under different realistic test conditions, including for a large-scale single-phase system consisting of over 50, 000 variables, three-phase unbalanced systems, and simulations conducted under diverse communication conditions. We would also like to emphasize that the improvement in computational speed for single-period optimization, observed in this work, will help scale more complex versions of OPF problems, including multi-period and stochastic versions.
Another major challenge relate to optimization under fast varying conditions. The existing literature, employs online optimization approaches where the algorithm doesn’t wait to obtain optimal solution, but rather takes step towards the steepest decent direction. In our previous work, we have extended the proposed distributed approach to a setting similar to online optimization techniques. Our simulations show that the proposed approach is able to efficiently track the optimal solution, even for rapidly changing system conditions, modeled as fast varying load and PV injections.
A reliable communication system is crucial for the practical viability of the distributed OPF algorithms. The communication system is needed to exchange the boundary variables among distributed agents and arrive at a converged system-level optimal solution. Therefore, the convergence, speed and accuracy of distributed OPF methods depend upon the communication systems conditions. It is imperative to evaluate the impacts of communication system-specific attributes (such as, latency, bandwidth, reliability) on the convergence of distributed OPF algorithm. Related literature includes simplified analysis to numerically evaluate the effects of communication system-related challenges. In our prior work, we have used a cyber-power co-simulation platform, using HELICS, to evaluate several related concerns. Additional work is needed to both numerically and analytically evaluate the effects of communication system attributes on the convergence and performance of distributed optimization methods. One can also determine an optimal communication system design to meet the required performance for distributed optimization methods.
Conclusions
===========
The optimal coordination of growing DER penetrations requires computationally efficient models for distribution-level optimization. In this paper, we have developed a nonlinear distributed optimal power flow algorithm with convergence guarantees using network equivalence methods. Then we present sufficient conditions to guarantee the convergence of the proposed method. While our most general sufficient conditions for global convergence over time are presented for line networks, our numerical simulations demonstrate similar convergence behavior for more general, e.g., radial, networks. The numerical simulation on the IEEE 123 bus test system corroborates the theoretical analysis. The proposed distributed method is also validated by comparing the results with a centralized formulation. Developing similar sufficient conditions for global convergence of radial networks, or other general network topologies, will be of high interest.
---
1. YL is with University of Science and Technology of China, RS and AD are with EECS, Washington State University, Pullman and, BK is with Mathematics and Statistics, Washington State University, Vancouver. YL and BK acknowledge funding from NSF through grant 1819229. RS and AD acknowledge funding from DOE under contract DE-AC05-76RL01830. Corresponding author E-mail: [email protected].[↩](#fnref1)
Convergence Guarantees of a Distributed Network Equivalence Algorithm for Distribution-OPF
==========================================================================================
Optimization-based approaches have been proposed to handle the integration of distributed energy resources into the electric power distribution system. The added computational complexities of the resulting optimal power flow (OPF) problem have been managed by approximated or relaxed models; but they may lead to infeasible or inaccurate solutions. Other approaches based on decomposition-based methods require several message-passing rounds for relatively small systems, causing significant delays in decision-making. We propose a distributed algorithm with convergence guarantees called ENDiCo-OPF for nonlinear OPF. Our method is based on a previously developed decomposition-based optimization method that employs network equivalence. We derive a sufficient condition under which ENDiCo-OPF is guaranteed to converge for a single iteration step on a local subsystem. We then derive conditions that guarantee convergence of a local subsystem over time. Finally, we derive conditions under a suitable assumption that when satisfied in a time sequential manner guarantee global convergence of a *line network* in a sequential manner. We also present simulations using the IEEE-123 bus test system to demonstrate the algorithm’s effectiveness and provide additional insights into theoretical results.
Distributed optimization, optimal power flow, power distribution systems, Method of multipliers.
Nomenclature
============
[]
Set of all nodes
Set of all distribution lines
Set of all DER nodes in the system
Set of all load nodes in the system
Cost function for the OPF sub-problem
Function describing the sub-problem
Equality constraint function
Inequality constraint function
Lagrangian function
Line reactance for line (*i*, *j*)
Line resistance for line (*i*, *j*)
Voltage magnitude for node *i*
Squared voltage magnitude for node *i*
Squared magnitude of the current flow in line (*i*, *j*)
Sending end active power for line (*i*, *j*)
Sending end reactive power for line (*i*, *j*)
Active load demand at node *j*
Reactive load demand at node *j*
Active power output by DER at node *j*
Reactive power output by DER at node *j*
kVA rating of the DER at node *j*
Thermal rating for the line (*i*, *j*)
Vector of local variables at node *j*
Cost coefficient vector
Dual variable for equality constraints of sub-problems
Dual variable for inequality constraints of sub-problems
Auxiliary variable to convert inequality constraint to the equation
Convergence parameter
Introduction
============
The nature and the requirements of power systems, especially at the distribution level, are changing rapidly with the massive integration of controllable distributed energy resources (DERs). The continued proliferation of DERs that include photovoltaic (PV) systems, battery energy storage systems (BESS), and controllable loads such as electric vehicles (EVs) is drastically increasing the number of active nodes at the distribution level that needs to be controlled/managed optimally for efficient and resilient grid operations. This has motivated the development of optimal power flow (OPF) methods for power distribution systems that aim at optimizing a pre-specified network-level objective function subject to the distribution grid’s operating constraints, including limits on node voltages and line currents. The nonconvex power flow constraints in the distribution OPF (D-OPF) problem pose significant computational challenges that increase drastically with the size of the distribution systems. Existing methods manage the computational challenges using convex relaxation or linear approximation techniques. The primary drawbacks of the convex relaxed models are the possibilities of inexact/infeasible power flow solutions. The linear approximated models may lead to infeasible power flow solutions and high optimality gap depending upon the problem type.
Another approach to scale D-OPF for large feeders is to use distributed optimization methods that decompose the large-scale optimization problem into several smaller subproblems (solved by distributed computing nodes) with the method converging to the centralized optimal solution over multiple rounds of message passing. Even if nonlinear and nonconvex, the smaller subproblems are relatively easier to solve, and the message-passing algorithm typically involves simple arithmetic computations. Current approaches to model a distributed optimal power flow problem (D-OPF) include the use of traditional distributed optimization techniques such as Augmented Lagrangian & Method of Multipliers (ALMM), Alternating Direction Method of Multipliers (ADMM), Auxiliary Problem Principle (APP), Predictor-Corrector Proximal Multiplier method (PCPM), and Analytical Target Cascading (ATC). Along with computational advantages, the D-OPF methods can be used to coordinate the decisions of physically distributed agents that lead to a distributed coordination paradigm with added robustness to single-point failure and also lower the communication overheads. To this end, it is worth mentioning distributed online feedback-based voltage controllers used to solve the D-OPF problem in a distributed manner. In contrast with traditional distributed optimization methods, these controllers asymptotically arrive at the optimal solution over several steps of real-time decision-making instead of waiting to reach a convergence at the boundary. They generally take one step towards the optimal solution and then move on to the next time step of the system simulations/observation.
Unfortunately, generic distributed optimization algorithms such as ADMM do not guarantee convergence for a general nonconvex optimization problem and may take a many message-passing rounds to converge to a local optimal solution. Specific to the D-OPF problem, the existing methods require many message-passing rounds among the agents (on the order of 102–103) to converge for a single-step optimization, which is not preferred from both distributed computing and distributed coordination standpoints. A large number of communication rounds or message-passing events among distributed agents or distributed computing nodes increases the time-of-convergence (ToC) for distributed computing and leads to significant communication delays with decision-making when used for distributed coordination. Online distributed controllers based on traditional distributed optimization methods also take hundreds of iterations to converge or track the optimal solution for even a mid-size feeder. This raises further challenges to the performance of the algorithm for larger feeders, especially during the fast-varying phenomenon under slow communication channels. To address these challenges, we have recently developed a distributed algorithm for the optimization of radial distribution systems based on the equivalence of networks principle. The proposed approach solves the original non-convex OPF problem for power distribution systems using a novel decomposition technique that leverages the structure of the power flow problem in radial distribution systems. The use of problem structure in our distributed algorithm results in a significant reduction in the number of message-passing rounds needed to converge to an optimal solution by orders of magnitude ( ∼ 102). This results in significant advantages over generic application of distributed optimization techniques for distributed computing or distributed coordination in radial power distribution systems. However, our previous work requires solving a generic nonlinear optimization problem at each distributed node and does not provide any convergence guarantees.
The objective of this paper to develop a provable convergent distributed optimization algorithm to solve D-OPF problems in a radial power distribution system. Our decomposition approach is based on the structure of power flow problem in radial distribution systems, and employs method of multipliers to solve the distributed subproblems exchanging specific power flow. We present a comprehensive mathematical analysis on the convergence of the proposed approach and how it relates to the structural decomposition of the problem and problem-specific variables. Our analysis results in a relationship among power flow variables (which is trivially satisfied) under which the proposed distributed optimization approach shows linear convergence. The proposed approach is demonstrated by solving DER coordination for loss minimization and voltage deviation minimization problems.
Decomposition approaches based on the augmented Lagrangian method (ALM) and its variant, the alternating direction method of multipliers (ADMM), have been applied successfully to ACOPF problems. In a series of early papers, Baldick and coauthors applied a linearized ALM to a regional decomposition of ACOPF. Peng and Low applied ADMM to certain convex relaxations of ACOPF on radial networks. Computational efficiency of ADMM has been reported in practice for nonconvex ACOPF as well, with convergence guarantees studied under certain technical assumptions.
Standard sufficiency conditions for optimality in nonlinear optimization could be used to derive a set of conditions that guarantee convergence of local systems within a single time step. While the distributed nature stemming from the decomposition approach of our algorithm leads to its strong performances, the same setting poses considerable challenges to derive theoretical convergence guarantees over the entire network and also over time. As we employ a decomposition approach that solves local subsystems to optimality followed by communication rounds to achieve global convergence, we develop a similar strategy to derive guarantees for the same. To this end, we specify an additional condition on the convergence of voltage over time (Eqn. [eq:Deltacdn]) which when satisfied along with second order sufficient conditions for the local subsystem provide guarantees of its convergence over time. We then utilize the structure of the network to derive a set of conditions that guarantee convergence of a *line network* in a sequential fashion starting from the root node and propagating the convergence down the line in subsequent time steps.
The nature and the requirements of power systems, especially at the distribution level, are rapidly changing with the large-scale integration of controllable distributed energy resources (DERs). Optimal power flow (OPF) methods have emerged as a possible mechanism to coordinate DERs for specified grid services. However, the nonconvex power flow constraints in the distribution OPF (D-OPF) problem poses significant computational challenges that increase drastically with the size of the distribution systems. The OPF problems involve nonconvex power flow equations and are generally NP-hard problems, even for the radial power distribution networks. There is an extensive body of literature highlighting the computational challenges of OPF problems that worsens with the system size. Existing methods manage the computational challenges using convex relaxation or linear approximation techniques that may lead to infeasible power flow solutions or high optimality gap. The existing literature also includes successive linear programming and convex iterations techniques to address the scalability of D-OPF problems, while still achieving feasible and optimal power flow solutions. However, they have not been shown to scale beyond a medium size feeder.
[htp!]
[table:qualcomparisonopfs]
p0.5 in p1.9in p0.5 in p0.4in p0.6in p1.2in p0.8in & *Summary* & *Penalty Term* & *Workflow* & *Decomposition Method* & *Shared Variables* & *Global Variable*
& Augmented Lagrangian is solved (with added penalty term to minimize computed shared boundary variables) for the decomposed sub-problems & quadratic & Sequential & Dual decomposition & Voltage magnitude, and active and reactive power flow & Is needed
& Augmented Lagrangian is solved (with added penalty term to minimize computed shared boundary variables) for the decomposed sub-problems & quadratic & Parallel & Dual decomposition & Voltage magnitude &Not needed
& Augmented Lagrangian is solved for the decomposed sub-problems, however, primal and dual variable is updated by solving an additional quadratic consensus optimization problem for the boundaries using hessian and gradients & quadratic & Parallel & Dual decomposition & Voltage phasor (magnitude and angle), and active and reactive power flows &Not needed
& Augmented Lagrangian with added penalty terms is solved for the decomposed sub-problems & Linear & Parallel & Dual decomposition & Voltage magnitude and active and reactive power flows &Is needed
& At every micro-iterations, decomposed sub-problems approximate the upstream and the downstream areas as a voltage source and fixed loads, respectively. The updated values of voltage and power flows (obtained from solving micro-iterations) are shared at each macro-iteration. & N/A & Parallel & Primal decomposition & Voltage magnitude and active and reactive power flows &Not needed
On the other hand, decomposition approaches based on the Augmented Lagrangian Method (ALM) and its variant, the Alternating Direction Method of Multipliers (ADMM), have been applied successfully to scale ACOPF problems for large feeders. In a series of early papers, Baldick et al. applied a linearized ALM to a regional decomposition of ACOPF. Peng and Low applied ADMM to certain convex relaxations of ACOPF on radial networks. Computational efficiency of ADMM has been reported in practice for nonconvex ACOPF as well, with convergence guarantees studied under certain technical assumptions. Along with computational advantages, the distributed methods can be used to coordinate the decisions of physically distributed agents, provide added robustness to single-point failure, and reduce communication overheads. Unfortunately, generic distributed optimization algorithms such as ADMM do not guarantee convergence for a general nonconvex optimization problem and may take a many message-passing rounds to converge to a local optimal solution. Specific to the D-OPF problem, the existing methods require a large number of message-passing rounds among the agents (on the order of 102–103) to converge for a single-step optimization, which is not preferred from both distributed computing and distributed coordination standpoints. When used for distributed coordination, many communication/message-passing rounds among distributed agents increases the time-of-convergence (ToC) and results in significant delays with decision-making. Some of these challenges are mitigated by distributed online controllers; however, they also take several time-steps to track the optimal decisions.
To address these challenges, we recently proposed a distributed algorithm for the optimization of radial distribution systems based on the equivalence of networks principle. A qualitative comparison of the existing distributed algorithms with the proposed OPFs is included in Table [table:qualcomparisonopfs]. The proposed approach solves the original non-convex OPF problem for power distribution systems using a novel decomposition technique that leverages the structure of the power flow problem. The primary distinction from the dual-decomposition approaches, is that the proposed method exploits the physics, i.e., the unique upstream/downstream relation among the power flow variables observed in a radial power distribution system. The use of problem structure in our distributed algorithm results in a significant reduction in the number of message-passing rounds needed to converge to an optimal solution by orders of magnitude ( ∼ 102). This results in significant advantages over generic application of distributed optimization techniques for distributed computing or distributed coordination in radial power distribution systems. However, our previous work requires solving a generic nonlinear optimization problem at each distributed node and does not provide any convergence guarantees.
The objective of this paper to develop a distributed optimization algorithm with convergence guarantees to solve D-OPF problems in a radial power distribution system. Our decomposition approach is based on the structure of power flow problem in radial distribution systems and employs method of multipliers to solve the distributed subproblems exchanging specific power flow. Then, we present a comprehensive mathematical analysis on the convergence of the proposed approach and how it relates to the structural decomposition of the problem and problem-specific variables. Standard sufficiency conditions for optimality in nonlinear optimization could be used to derive a set of conditions that guarantee convergence of local systems within a single iteration step. While the distributed nature stemming from the decomposition approach of our algorithm leads to its strong performances, the same setting poses considerable challenges to derive theoretical convergence guarantees over the entire network and also over time. As we employ a decomposition approach that solves local subsystems to optimality followed by communication rounds to achieve global convergence, we develop a similar strategy to derive guarantees for the same. To this end, we specify an additional condition on the convergence of voltage over time (Eqn. [eq:Deltacdn]) which when satisfied along with second order sufficient conditions for the local subsystem provide guarantees of its convergence over time. We then utilize the structure of the network to derive a set of conditions that guarantee convergence of a *line network* in a sequential fashion starting from the root node and propagating the convergence down the line in subsequent iteration steps. Our analysis results in a relationship among power flow variables (which is trivially satisfied for a well-designed power distribution system) under which the proposed distributed optimization approach shows linear convergence. Finally, we validate the efficacy of the proposed approach by solving multiple distribution-level OPF problems. We also use simulations to provide additional insights into the convergence properties.
Our results present *sufficient* conditions that guarantee convergence when satisfied. But we note that our conditions are somewhat different in structure from typical convergence guarantees for optimization algorithms. Our conditions are specified on the values of variables in the problem over one or more iterations, rather than on ranges of values of problem parameters. At the same time, these conditions are satisfied by the typical values taken by the system variables in power systems. Furthermore, our computational experiments (see Section [sec:nmrclstud]) confirm the expected convergence behavior.
Modeling and Problem Formulation
================================
In this paper ( ⋅ )(*t*) represents the variable at iteration step *t*, $\underline{(.)}$ and $\overline{(.)}$ denote the minimum and maximum limit of any quantity, respectively, and ${\tilde{j}}= \sqrt{-1}$. We assume a radial single-phase power distribution network, where N and E denote the set of nodes and edges of the system. Here, edge *i**j* ∈ E identifies the distribution lines connecting the ordered pair of buses (*i*, *j*) and is weighted with the series impedance of the line, represented by *r**i**j* + *j̃**x**i**j*. The set of load buses and DER buses are denoted by N*L* and N*D*, respectively.
Network and DER Model
---------------------
Let node *i* be the unique parent node and node *k* is the children node for the controllable node *j*, i.e., *k* : *j* → *k* where, {*j**k*} ∈ E. We denote *v**j* and *l**i**j* as the squared magnitude of voltage and current flow at node *j* and in branch {*i**j*}, respectively. The network is modeled using the nonlinear branch flow equations shown in Eqn. . Here, *p**L**j* + *j̃**q**L**j* is the load connected at node *j* ∈ N*L*, *P**i**j*, *Q**i**j* ∈ R are the sending-end active and reactive power flows for the edge *i**j*, and *p**D**j* + *j̃**q**D**j* is the power output of the DER connected at node *j* ∈ N*D*.
C C [eqModelnonlin] Pij-rijlij-pLj+pDj= k:j k Pjk [eqModelnonlin1]
Qij-xijlij-qLj+qDj= k:j k Qjk [eqModelnonlin2]
vj=vi-2(rijPij+xijQij)+(rij2+xij2)lij[eqModelnonlin3]
vilij = Pij2+Qij2 [eqModelnonlin4]
The DERs are modeled as Photovoltaic modules (PVs) interfaced using smart inverters, capable of two-quadrant operation. Both the reactive and the active power output can be optimally controlled. For reactive power control, at the controllable node *j* ∈ N*D*, the real power generation by the DER, *p**D**j* is assumed to be known (measured), and the controllable reactive power generation, *q**D**j*, is modeled as the decision variable. With the rating of the DER connected at node *j* ∈ N*D* denoted as *S**D**R**j*, the limits on *q**D**j* are given by Eqn. . Similarly for the active power control (*p**D**j*), equation can be modeled as the DER assuming reactive power output to be zero; further, for both reactive and active power control, equation may be used to model such DERs.
C [eq:DGlim] - qDj [eq:DGlimq]
0pDj SDRj [eq:DGlimp]
pDj2+qDj2SDRj2[eq:DGlimpq]
Problem Formulation and Algorithm
---------------------------------
We recently developed a real-time distributed controller to solve OPF problems by controlling the reactive power outputs of the DERs for distribution systems, and the method is a variant of nodal-level extension of previously developed distributed OPF. In this section, we develop a similar approach for distributed OPF termed *ENDiCo-OPF* by decomposing the overall problem at each node that is solved using the decomposition method developed previously in . All controllable nodes in the system receive updated computed voltage and power flow quantities from their parent and children nodes, respectively, and in parallel calculate their optimum dispatches. Briefly, each node *j* ∈ N*D* solves a small-scale OPF problem defined by problem (**P1**) in Eqn. . The reactive power dispatch *q**D**j* is controlled to minimize some cost/objective function **f**. Note that the resulting nonlinear optimization problem is in five variables, {*P**i**j*, *Q**i**j*, *v**j*, *l**i**j*, *q**D**j*}, with four equality and five inequality constraints. Some examples of the cost function **f** include active power loss (**f** = *r**i**j**l**i**j*) and voltage deviation (**f** = (*v**j* − *v*ref)2), among others. The ENDiCO-OPF assumes the parent node voltage and the power flow to the children node to be constant and solves the problem (**P1**) locally for the reduced network. In this paper, we denote the sub-problem **(P1)** by the function ***F***, i.e., the argmin of this function corresponds to the optimal solution of the OPF problem **(P1)**. Steps of ENDiCo-OPF are presented in Algorithm [alg:ENDiCo-OPF]. We work under the following standard assumption.
[asm:nodeagent] All the nodes in the network have an agent that can measure its local power flow quantities (node voltages and line flows) and communicate with neighboring nodes.
We assume the small nonlinear optimization problem (P1) in is solved by a commercial solver, as we did in our earlier work . This assumption does not affect the convergence analysis as it is just a small subproblem. Alternatively, we could present a subroutine for this subproblem, based on the Augmented Lagrangian Multiplier (ALM) method for instance, which could make it easier to include the details of this step in the overall convergence analysis.
C [eq:P1]
(t)
[eq:P1obj]
Pij(t)-rijlij(t)-pLj(t)+pDj(t)= k:j k Pjk(t-1)
[eq:P1rl]
Qij(t)-xijlij(t)-qLj(t)+qDj(t)= k:j k Qjk(t-1) [eq:P1qDmxl]
vj(t)=vi(t-1)-2(rijPij(t)+xijQij(t))+(rij2+xij2)lij(t)
[eq:P1vjt]
lij(t) = [eq:P1lijt]
2 vj(t) 2[eq:P1vjtbds]
- qDj(t)
[eq:P1qDjtbds]
lij(t) (Iij)2 [eq:P1lijtub]
Here, $\underline{V} = 0.95$ and $\overline{V} = 1.05$ pu are the limits on bus voltages, and *I**i**j*rated is the thermal limit for the branch {*i**j*}.
[ht!] [alg:ENDiCo-OPF]
Calculate ∑*P**j**k*(*t* − 1) + *j̃**Q**j**k*(*t* − 1) from all the *P**j**k**i*(*t* − 1) + *j̃**Q**j**k**i*(*t* − 1), received from child nodes *k**i* ∈ N*j**k*
Approximate the upstream and downstream network of line (*i*, *j*) with fixed value of *v**i*(*t* − 1) and ∑*P**j**k*(*t* − 1) + *j̃**Q**j**k*(*t* − 1)
Solve optimization problem (P1) for iteration step (t), i.e., [st:optP1]
$$\label{eq:qDjStep3}
q\_{Dj}^\* = \underset{q\_{Dj}}{\arg\min} \hspace{0.2cm} \mathbf {F}^{(t)}(q\_{Dj})$$
Set Reactive power output *q**D**j*(*t*) = *q**D**j*\*
Calculate the node voltage *v**j*(*t*) at node *j* and complex power flow *P**i**j*(*t*) + *j̃**Q**i**j*(*t*) in the line (*i*, *j*)
Sends *v**j*(*t*) and *P**i**j*(*t*) + *j̃**Q**i**j*(*t*) to child node *k* and parent node *i*, respectively
Receives *v**i*(*t*) and *P**j**k**i*(*t*) + *j̃**Q**j**k**i*(*t*) from parent and child nodes, respectively
Move forward to the next iteration step (*t* + 1)
[algo]
For presenting the main results on convergence of ENDiCo-OPF, we first write the optimization model (P1) in Eqn. in standard form. We set the local variables of the bus *j* at iteration step *t* as
$$\label{eq:zj}
{\mathbf{z}\_j}= \begin{bmatrix} P\_{ij}^{(t)} & Q\_{ij}^{(t)} & v\_{j}^{(t)} & l\_{ij}^{(t)} & q\_{D\_j}^{(t)} \end{bmatrix}^T.$$
We also write the set of four equality constraints in Eqns. [eq:P1rl]–[eq:P1lijt] as A*p*(**z***j*) = 0 for *p* = 1–4 and the five inequality constraints in Eqns. [eq:P1vjtbds]–[eq:P1lijtub] as B*r*(**z***j*) ≤ 0 for *r* = 1–5. Finally, setting *R**i**j* = [0 0 0 *r**i**j* 0]*T* we rewrite the original system as follows.
$$\label{eq:P1'} \tag{P1'}
\begin{array}{llcll}
\min & f({\mathbf{z}\_j}) = R\_{ij}^T {\mathbf{z}\_j}\\
\text{s.t.} & {\mathscr{A}}\_p({\mathbf{z}\_j}) & = & 0 & p=1,\dots,4 \\
& {\mathscr{B}}\_r({\mathbf{z}\_j}) & \leq & 0 & r=1,\dots,5
\end{array}$$
Please note that only reactive power output of the DERs, *q**D**j*, are assumed to be the controllable variables in the OPF formulation for the convergence analysis. However, the analysis is extendable for the active power controls, *p**D**j*, as well.
Convergence Analysis
====================
We present theoretical guarantees for the convergence of the ENDiCo-OPF algorithm under certain standard assumptions on the network topology and structural properties of the distribution system. We use the method of Lagrangian multipliers from nonlinear optimization coupled with techniques from linear algebra to derive the convergence property of ENDiCo-OPF. We (i) first study the convergence of the simplest structure of a distributed network for a single iteration step (). As a first step toward generalizing this property to multiple iteration steps as well as to general networks, we present a slight modification of the ENDiCo-OPF algorithm using a new convergence parameter Δ (). We (ii) derive conditions that guarantee the convergence of the Δ-ENDiCo-OPF algorithm for a local system over iteration steps (). Then we (iii) generalize the problem to achieve a network-level convergence guarantee for line networks (). We introduce auxiliary variables to measure the difference of variable values between the adjacent iteration steps in the process of deriving guarantees for global convergence of ENDiCo-OPF over time for line networks.
Local System Convergence Guarantees
-----------------------------------
We first study the convergence of the ENDiCo-OPF algorithm at a local level on a *subsystem* under Assumption [asm:nodeagent], which refers to a system with only one communication layer that contains a middle node to receive voltages from a parent node as well as power flows from its children nodes (). The system consists of nodes {*i*, *j*, *k*} when there is a single child node, or nodes {*i*, *j*, *k*1, …, *k**l*} in general when there are *l* child nodes. We assume that the system conditions are changing at a slower rate than the decision variables.
0.99 [fig:3ndW1child]
0.99 [fig:3ndWmltchild]
[fig:single]
We introduce auxiliary variables *φ**r* to convert the inequality constraints in the system to equations.
$$\label{eq:P2} \tag{P2}
\begin{array}{llcll}
\min & f({\mathbf{z}\_j}) = R\_{ij}^T {\mathbf{z}\_j}\\
\text{s.t.} & {\mathscr{A}}\_p({\mathbf{z}\_j}) & = & 0 & p=1,\dots,4 \\
& {\mathscr{B}}\_r({\mathbf{z}\_j}) + \varphi\_r^2 & = & 0 & r=1,\dots,5
\end{array}$$
We apply second order sufficiency conditions that guarantee when **z***j*\* is a strict local minimum of the objective function *f* in the system ([eq:P2]) under standard assumptions.
[, Proposition 3.2.1] [prop:sufcdnopt] Assume *f* and A*p*, B*r* in the System ([eq:P2]) are twice continuously differentiable for all *p*, *r* and let ${\mathbf{z}\_j}^\* \in \mathbb{R}^n, {\boldsymbol{\lambda}}^\*\in \mathbb{R}^p$, and ${\boldsymbol{\mu}}^\*\in \mathbb{R}^r$ satisfy
$$\begin{aligned}
\nabla\_{{\mathbf{z}\_j}} L({\mathbf{z}\_j}^\*,{\boldsymbol{\lambda}}^\*,{\boldsymbol{\mu}}^\*)=0,~
\nabla\_{{\boldsymbol{\lambda}}} L({\mathbf{z}\_j}^\*, {\boldsymbol{\lambda}}^\*, {\boldsymbol{\mu}}^\*)=0, \hspace\*{0.5in}\\
\nabla\_{{\boldsymbol{\mu}}} L({\mathbf{z}\_j}^\*, {\boldsymbol{\lambda}}^\*, {\boldsymbol{\mu}}^\*)=0, \text{ and } \hspace\*{1.5in} \\
a^T\nabla\_{{\mathbf{z}\_j}{\mathbf{z}\_j}}L({\mathbf{z}\_j}^\*,{\boldsymbol{\lambda}}^\*,{\boldsymbol{\mu}}^\*)a>0
\text{ for all } a\neq 0 \text{ with } \hspace\*{0.42in} \\
\begin{bmatrix} \nabla{\mathscr{A}}({\mathbf{z}\_j}^\*) & \nabla{\mathscr{B}}({\mathbf{z}\_j}^\*) \end{bmatrix}^T a=0.
\hspace\*{1in}
\end{aligned}$$
Then **z***j*\* is a strict local minimum of *f*.
We now present the theorem that specifies a basic condition under which the ENDiCo-OPF algorithm is guaranteed to converge for a single iteration step on single subsystem networks as shown in. We show that this condition always holds for power systems operating under standard settings. To specify the result, we consider the augmented Lagrangian function for the System ([eq:P2]) for penalty parameter *c* > 0:
$$\label{eq:augLggnP2}
\small
\begin{aligned}
\hspace\*{-0.1in} L\_c({\mathbf{z}\_j}, {\boldsymbol{\varphi}}, {\boldsymbol{\lambda}}, {\boldsymbol{\mu}}) & = f({\mathbf{z}\_j})+ {\boldsymbol{\lambda}}{\mathscr{A}}({\mathbf{z}\_j}) + \frac{c}{2}\|{\mathscr{A}}({\mathbf{z}\_j})\|^2 ~+ \\
& ~~~~{\boldsymbol{\mu}}({\mathscr{B}}({\mathbf{z}\_j})+{\boldsymbol{\varphi}}^2)+\frac{c}{2}\|{\mathscr{B}}({\mathbf{z}\_j})+{\boldsymbol{\varphi}}^2\|^2 \,.
\end{aligned}$$
We first show that Lagrangian multipliers in Proposition [prop:sufcdnopt] do exist. Note that is a convex optimization problem, and hence the strict local minimum **z***j*\* in Proposition [prop:sufcdnopt] will also be the global optimal solution. We provide a proof similar in nature to the arguments presented by Andreani et al.
[lem:lgrngexist] There exist Lagrangian multipliers ${\boldsymbol{\lambda}}^\*, {\boldsymbol{\mu}}^\*$ that along with the system variables satisfy the conditions in Proposition [prop:sufcdnopt] for the System ([eq:P2]).
Let *γ* > 0 be such that **z***j*\* is the unique global solution of the following version of the problem in System ([eq:P2]) where **B**(**z***j*\*, *γ*) is the *γ*-ball centered at **z***j*\*.
$$\label{eq:P2zg} \tag{P2-$\gamma$}
\begin{array}{ll}
\min & f({\mathbf{z}\_j}) + (1/2) \|{\mathbf{z}\_j}-{\mathbf{z}\_j}^\*\|\_2^2 \\
\text{s.t.} & {\mathscr{A}}\_p({\mathbf{z}\_j})~~~~~~~~ = 0,~ p=1,\dots,4 \\
& {\mathscr{B}}\_r({\mathbf{z}\_j}) + \varphi\_r^2 ~= 0,~r=1,\dots,5 \\
& {\mathbf{z}\_j}\in \mathbf{B}({\mathbf{z}\_j}^\*,\gamma)
\end{array}$$
We consider the sequence of penalty subproblems associated with ([eq:P2zg]) for *k* ∈ N (natural numbers) with the penalty parameter *ρ**k* assumed to be large enough:
$$\label{eq:P2phi} \tag{P2-$\phi$}
\hspace\*{-0.15in}
\begin{array}{l}
\min\limits\_{{\mathbf{z}\_j}\in \mathbf{B}({\mathbf{z}\_j}^\*,\gamma)}~~\phi({\mathbf{z}\_j}) := f({\mathbf{z}\_j}) + (1/2) \|{\mathbf{z}\_j}-{\mathbf{z}\_j}^\*\|\_2^2 + \\
\hspace\*{0.5in} \rho\_k \left(\sum\_{p=1}^4 {\mathscr{A}}\_p({\mathbf{z}\_j}) + \sum\_{r=1}^5({\mathscr{B}}\_r({\mathbf{z}\_j}) + \varphi\_r^2) \right)
\end{array}$$
By the continuity of *ϕ*( ⋅ ) and the compactness of **B**(**z***j*\*, *γ*), as *ρ**k* → ∞, **z***j**k* is well-defined for all *k* ∈ N such that **z***j**k* is the global solution of ([eq:P2phi]). By the boundedness of {**z***j**k*}*k* ∈ N, we have that
$$\begin{aligned}
f({\mathbf{z}\_j}^k) + (1/2)\|{\mathbf{z}\_j}^k-{\mathbf{z}\_j}^\*\|^2\_2\leq \phi({\mathbf{z}\_j}^k)\leq \phi({\mathbf{z}\_j}^\*) = f({\mathbf{z}\_j}^\*)
\end{aligned}$$
such that the limit point of **z***j**k*, denoted by ${\overline{\mathbf{z}}\_j}$ satisfies
$$\begin{aligned}
f({\overline{\mathbf{z}}\_j}) + (1/2) \|{\overline{\mathbf{z}}\_j}- {\mathbf{z}\_j}^\*\|^2\_2 \leq f({\mathbf{z}\_j}^\*)
\end{aligned}$$
which can be fulfilled, due to the continuity and boundedness of *ϕ*( ⋅ ), only by making ${\mathscr{A}}({\overline{\mathbf{z}}\_j})=0$ and ${\mathscr{B}}({\overline{\mathbf{z}}\_j}) + \boldsymbol{\varphi}^2=0$. Hence, with ${\overline{\mathbf{z}}\_j}\in \mathbf{B}({\mathbf{z}\_j}^\*,\gamma)$, we get that ${\overline{\mathbf{z}}\_j}$ globally solves the problem ([eq:P2zg]), which further implies that ${\overline{\mathbf{z}}\_j}= {\mathbf{z}\_j}^\*$ due to the uniqueness of the global solution. Hence for some large enough *k* ∈ N, we have that **z***j**k* globally solves the unconstrained minimization problem with objective function *ϕ*( ⋅ ). Hence we get that ∇*ϕ*(**z***j**k*) = 0 for large *k*, which gives
$$\label{eq:gradphi}
\begin{array}{c}
\nabla f({\mathbf{z}\_j}^k) + {\mathbf{z}\_j}^k - {\mathbf{z}\_j}^\* + \sum\_{p=1}^4\lambda^k\_p \nabla{\mathscr{A}}\_p({\mathbf{z}\_j}^k) \hspace\*{0.5in} \\
+ \sum\_{r=1}^5\mu^k\_i(\nabla{\mathscr{B}}\_r({\mathbf{z}\_j}^k)+\varphi\_r^2)=0
\end{array}$$
with the the sequences of Lagrangian multipliers {*λ**p**k*}*k* ∈ N, and {*μ**r**k*}*k* ∈ N. We argue that both $\{{\boldsymbol{\lambda}}^k\}$ and $\{{\boldsymbol{\mu}}^k\}$ are bounded for *k* ∈ N. Assume that at least one sequence is unbounded, and let L*k* represent the maximum norm of ${\boldsymbol{\lambda}}^k\_p, p=1,\dots,4$ and ${\boldsymbol{\mu}}^k\_r, r= 1,\dots, 5$, Dividing ([eq:gradphi]) by L*k* and taking the limits along convergent subsequences, we get that L*k* → ∞ and
$$\begin{aligned}
\label{eq:zk}
{\boldsymbol{\lambda}}^\* \nabla{\mathscr{A}}({\mathbf{z}\_j}^\*) + {\boldsymbol{\mu}}^\*(\nabla{\mathscr{B}}({\mathbf{z}\_j}^\*)+\boldsymbol{\varphi}^{\*2})=0
\end{aligned}$$
where $\|{\boldsymbol{\lambda}}^\*\|, \|{\boldsymbol{\mu}}^\*\|\geq 0$. However, since ∇A and $\nabla{\mathscr{B}}+ \boldsymbol{\varphi}^{2}$ are all linearly independent at **z***j*\*, we get a contradiction. Hence we get the boundedness of the two sequences.
[thm:subsyscvgce] Assume *f* and A*p*, B*r* in the System ([eq:P2]) are twice continuously differentiable for all *p*, *r* and there exists a threshold penalty parameter *c̄* in the augmented Lagrangian (). If
*v**i*(*t* − 1) − 4*P**i**j*(*t*)*r**i**j*(*t*) − 4*Q**i**j*(*t*)*x**i**j*(*t*) > 0,
then the ENDiCo-OPF algorithm converges for a single iteration step *t* for all *c* > *c̄*.
Before presenting the proof, we note that the main condition in holds for typical values that the variables take in power systems. At the same time, this is a sufficient condition—local convergence could happen even when does not hold.
We obtain this result from the application of Proposition [prop:sufcdnopt]. We first consider the gradient of the Lagrangian function in with respect to **z***j*. We use the equations specifying the constraints in the System ([eq:P2]) to simplify the expression for the gradient, and for the sake of brevity, we suppress the subscript of **z***j* of ∇ for the terms on the right-hand side.
$$\label{eq:gradLggn}
\small
\hspace\*{-0.1in}
\begin{aligned}
\nabla\_{{\mathbf{z}\_j}} L\_c({\mathbf{z}\_j}, {\boldsymbol{\varphi}}, {\boldsymbol{\lambda}}, {\boldsymbol{\mu}}) = \nabla f({\mathbf{z}\_j}) + \nabla{\mathscr{A}}({\mathbf{z}\_j})({\boldsymbol{\lambda}}+c{\mathscr{A}}({\mathbf{z}\_j}))\\
+ \nabla{\mathscr{B}}({\mathbf{z}\_j})({\boldsymbol{\mu}}+c({\mathscr{B}}({\mathbf{z}\_j})+{\boldsymbol{\varphi}}^2))~~\\
=\nabla f({\mathbf{z}\_j})+{\boldsymbol{\lambda}}\nabla{\mathscr{A}}({\mathbf{z}\_j})+{\boldsymbol{\mu}}\nabla{\mathscr{B}}({\mathbf{z}\_j})
\end{aligned}$$
Lemma [lem:lgrngexist] ensures that there exist Lagrangian multipliers ${\boldsymbol{\lambda}}^\*, {\boldsymbol{\mu}}^\*$ and variables ${\mathbf{z}\_j}^\*, {\boldsymbol{\varphi}}^\*$ that satisfy $\nabla\_{{\mathbf{z}\_j}} L\_c({\mathbf{z}\_j}^\*, {\boldsymbol{\varphi}}^\*, {\boldsymbol{\lambda}}^\*, {\boldsymbol{\mu}}^\*) = 0$.
The main technical work is involved in specifying the structure of the Hessian such that Proposition [prop:sufcdnopt] will hold. To this end, we first note that the last sufficient condition specified in Proposition [prop:sufcdnopt] can be shown to hold by equivalently showing that the Hessian ∇**z***j***z***j*2*L**c* is positive definite. We assume the iteration step *t* is fixed, use the expression for **z***j* in, suppress the superscripts (*t*) of variables, let *v**i* represent *v**i*(*t* − 1), and let *d**j* = *λ*4\* to get the Hessian as
$$\begin{aligned}
\nabla^2\_{{\mathbf{z}\_j}{\mathbf{z}\_j}} L\_c({\mathbf{z}\_j}, {\boldsymbol{\varphi}}, {\boldsymbol{\lambda}}, {\boldsymbol{\mu}}) = d\_{j}
\begin{bmatrix}
2 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}
+ cM \label{eq:HsnwM}
\end{aligned}$$
where *M* =
$$\hspace\*{-0.08in}
\begin{array}{l}
\left[
\hspace\*{-0.07in}
\begin{array}{cc}
4{r\_{ij}}^2+4{P\_{ij}}^2+1 & 4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}\\
4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 4{x\_{ij}}^2+4{Q\_{ij}}^2+1 \\
2{r\_{ij}}& 2{x\_{ij}}\\
-{r\_{ij}}\hspace\*{-0.03in}-2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)\hspace\*{-0.03in}-\hspace\*{-0.03in}2{P\_{ij}}{v\_i}& \hspace\*{-0.03in}-{x\_{ij}}\hspace\*{-0.03in}-\hspace\*{-0.03in}2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)\hspace\*{-0.03in}-\hspace\*{-0.03in}2{Q\_{ij}}{v\_i}\\
0 & 1
\end{array}
\right.\\
\vspace\*{-0.02in} \\
\left.
\hspace\*{0.4in}
\begin{array}{ccc}
2{r\_{ij}}& -{r\_{ij}}-2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{P\_{ij}}{v\_i}& 0 \\
2{x\_{ij}}& -{x\_{ij}}-2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{Q\_{ij}}{v\_i}& 1\\
1 & -({r\_{ij}}^2+{x\_{ij}}^2) & 0 \\
-({r\_{ij}}^2+{x\_{ij}}^2) &{r\_{ij}}^2+{x\_{ij}}^2+({r\_{ij}}^2+{x\_{ij}}^2)^2+{v\_i}^2& -x \\
0 & -{x\_{ij}}& 1 \\
\end{array}
\right].
\end{array}$$
We simplify the expression in to obtain the following expression for the Hessian, which we investigate in detail.
$$\begin{aligned}
\nabla^2\_{{\mathbf{z}\_j}{\mathbf{z}\_j}} L\_c({\mathbf{z}\_j}, {\boldsymbol{\varphi}}, {\boldsymbol{\lambda}}, {\boldsymbol{\mu}}) = cM'\, \label{eq:Hsnzj}
\end{aligned}$$
where *M*ʹ =
$$\begin{array}{l}
\hspace\*{-0.08in}
\left[
\hspace\*{-0.05in}
\begin{array}{cc}
4{r\_{ij}}^2+4{P\_{ij}}^2+1+(2d\_{j}/c) & 4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}\\
4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 4{x\_{ij}}^2+4{Q\_{ij}}^2+1+(2d\_{j}/c) \\
2{r\_{ij}}& 2{x\_{ij}}\\
-{r\_{ij}}\hspace\*{-0.03in}-\hspace\*{-0.03in} 2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2) \hspace\*{-0.03in}-\hspace\*{-0.03in} 2{P\_{ij}}{v\_i}& \hspace\*{-0.03in}-\hspace\*{-0.03in}{x\_{ij}}\hspace\*{-0.03in}-\hspace\*{-0.03in} 2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2) \hspace\*{-0.03in}-\hspace\*{-0.03in} 2{Q\_{ij}}{v\_i}\\
0 & 1 \\
\end{array}
\right.\\
\nonumber \\
\left.
\hspace\*{0.3in}
\begin{array}{ccc}
2{r\_{ij}}& -{r\_{ij}}-2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{P\_{ij}}{v\_i}& 0 \\
2{x\_{ij}}& -{x\_{ij}}-2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{Q\_{ij}}{v\_i}& 1\\
1 & -({r\_{ij}}^2+{x\_{ij}}^2) & 0 \\
-({r\_{ij}}^2+{x\_{ij}}^2) & {r\_{ij}}^2+{x\_{ij}}^2+({r\_{ij}}^2+{x\_{ij}}^2)^2+{v\_i}^2& -x \\
0 & -{x\_{ij}}& 1 \\
\end{array}
\right]. \hspace\*{0.1in}
\end{array}$$
Given that *c* > 0 and ${\boldsymbol{\lambda}}> {\mathbf{0}}$, we can show that the Hessian in is positive definite if *M*ʹ is so. And we can show that the 5 × 5 matrix *M*ʹ is positive definite by checking that all its five upper left subdeterminants are positive.
1. 1 × 1 upper-left subdeterminant: We get that
4*r**i**j*2 + 4*P**i**j*2 + 1 + (2*d**j*/*c*) > 0
since *d**j* = *λ*4\* > 0 and *c* > 0, and from the observation that *r**i**j*2, *P**i**j*2 ≥ 0.
2. 2 × 2 upper-left subdeterminant:
$$\hspace\*{-0.18in}
\begin{array}{l}
\left\vert
\begin{matrix}
4{r\_{ij}}^2+4{P\_{ij}}^2+1+(2d\_{j}/c) & 4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}\\
4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 4{x\_{ij}}^2+4{Q\_{ij}}^2+1+(2d\_{j}/c)
\end{matrix} \right\vert \\
\vspace\*{-0.05in} \\
= 16({r\_{ij}}^2{Q\_{ij}}^2+{P\_{ij}}^2{x\_{ij}}^2)+4({r\_{ij}}^2+{P\_{ij}}^2+{x\_{ij}}^2+{Q\_{ij}}^2) + \\
~~\,(1/c)(8{x\_{ij}}^2d\_{j}+8{r\_{ij}}^2d\_{j}+8{P\_{ij}}^2d\_{j}+8{Q\_{ij}}^2d\_{j}+4d\_{j}+(4d\_{j}^2/c))\\
~~ -32{r\_{ij}}{Q\_{ij}}{P\_{ij}}{x\_{ij}}\\
\vspace\*{-0.1in}\\
\geq 4({r\_{ij}}^2+{P\_{ij}}^2+{x\_{ij}}^2+{Q\_{ij}}^2)+ \\
~\, (1/c)\Big(8{x\_{ij}}^2d\_{j}+8{r\_{ij}}^2d\_{j}+8{P\_{ij}}^2d\_{j}+8{Q\_{ij}}^2d\_{j}+4d\_{j}+(4d\_{j}^2/c)\Big)\\
\vspace\*{-0.15in}\\
> 0 \\
\vspace\*{-0.15in}\\
\end{array}$$
since (4*r**i**j**Q**i**j* − 4*P**i**j**x**i**j*)2 ≥ 0, *d**j* = *λ*4\* > 0, and *c* > 0.
3. 3 × 3 upper-left subdeterminant: Expanding along the third row, for instance, gives
$$\hspace\*{-0.1in}
\begin{array}{l}
\left\vert
\begin{matrix}
4{r\_{ij}}^2+4{P\_{ij}}^2+1+2d\_{j}/c & 4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 2{r\_{ij}}\\
4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 4{x\_{ij}}^2+4{Q\_{ij}}^2+1+2d\_{j}/c & 2{x\_{ij}}\\
2{r\_{ij}}& 2{x\_{ij}}& 1
\end{matrix}
\right\vert\\
\vspace\*{-0.1in}\\
= \left[16({Q\_{ij}}^2{r\_{ij}}^2+{P\_{ij}}^2{Q\_{ij}}^2)+4({r\_{ij}}^2+{P\_{ij}}^2+{Q\_{ij}}^2) \, + \right. \\
~~~ \left. (1/c)\Big(8{r\_{ij}}^2d\_{j}+8{P\_{ij}}^2d\_{j}+8{Q\_{ij}}^2d\_{j}+4d\_{j}+ (4d\_{j}^2/c)\Big)+1\right]\\
~~~ -\left(16{P\_{ij}}{Q\_{ij}}{r\_{ij}}{x\_{ij}}+16{P\_{ij}}^2{Q\_{ij}}^2\right)+ \\
~~~~ \Big(16{P\_{ij}}{Q\_{ij}}{r\_{ij}}{x\_{ij}}-16{Q\_{ij}}^2{r\_{ij}}^2-4{r\_{ij}}^2-(1/c)8{r\_{ij}}^2d\_{j}\Big)\\
\vspace\*{-0.12in}\\
= 4({P\_{ij}}^2+{Q\_{ij}}^2)+(1/c)\Big(8{P\_{ij}}^2d\_{j}+8{Q\_{ij}}^2d\_{j}+4d\_{j}+(4d\_{j}^2/c)\Big
) \\
~~~ +1\\
\vspace\*{-0.12in}\\
> 0
\end{array}$$
following the same observations as before.
4. 4 × 4 upper-left subdeterminant: Expanding along the fourth row gives
$$\begin{array}{l}
\hspace\*{-0.2in}
\left\vert
\begin{array}{c@{\hspace\*{0.08in}}c}
4{r\_{ij}}^2+4{P\_{ij}}^2+1+(2d\_{j}/c) & 4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}\\
4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 4{x\_{ij}}^2+4{Q\_{ij}}^2+1+(2d\_{j}/c) \\
2{r\_{ij}}& 2{x\_{ij}}\\
\hspace\*{-0.07in}-\hspace\*{-0.03in} {r\_{ij}}\hspace\*{-0.03in}-\hspace\*{-0.03in} 2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2) \hspace\*{-0.03in}-\hspace\*{-0.03in} 2{P\_{ij}}{v\_i}&
-{x\_{ij}}\hspace\*{-0.02in}-\hspace\*{-0.02in}2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)\hspace\*{-0.02in}-\hspace\*{-0.02in}2{Q\_{ij}}{v\_i}\end{array}
\right. \\
\vspace\*{-0.05in}\\
\hspace\*{0.38in}
\left.
\begin{array}{cc}
2{r\_{ij}}& -{r\_{ij}}-2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{P\_{ij}}{v\_i}\\
2{x\_{ij}}& -{x\_{ij}}-2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{Q\_{ij}}{v\_i}\\
1 & -({r\_{ij}}^2+{x\_{ij}}^2)\\
-({r\_{ij}}^2+{x\_{ij}}^2) & {r\_{ij}}^2+{x\_{ij}}^2+({r\_{ij}}^2+{x\_{ij}}^2)^2+{v\_i}^2
\end{array}
\hspace\*{-0.03in}\right\vert \\
\vspace\*{0.05in}\\
\hspace\*{-0.21in}
= \left[{r\_{ij}}+2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)+2{P\_{ij}}{v\_i}\right]\left[4{x\_{ij}}{P\_{ij}}{Q\_{ij}}-4{r\_{ij}}{Q\_{ij}}^2 \right. \\
\hspace\*{0.9in} -2{P\_{ij}}{v\_i}-r-(1/c)(2d\_{j}{r\_{ij}}+4d\_{j}{P\_{ij}}{v\_i}) \Big]\\
\hspace\*{-0.2in}+\left[-{x\_{ij}}-2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{Q\_{ij}}{v\_i}\right]\left[ 4{P\_{ij}}^2{x\_{ij}}-4{r\_{ij}}{P\_{ij}}{Q\_{ij}}\right.\\
\hspace\*{0.83in} +2{Q\_{ij}}{v\_i}+x+(1/c)(2d\_{j}{x\_{ij}}+4d\_{j}{Q\_{ij}}{v\_i}) \Big]\\
\hspace\*{-0.2in}+\left[{r\_{ij}}^2+{x\_{ij}}^2\right]\left[-16{P\_{ij}}{Q\_{ij}}{r\_{ij}}{v\_i}+4{r\_{ij}}^2{Q\_{ij}}^2+4{x\_{ij}}^2{P\_{ij}}^2-4{r\_{ij}}^2{P\_{ij}}^2 \right. \\
\hspace\*{0.55in} -4{x\_{ij}}^2{Q\_{ij}}^2+4{x\_{ij}}{Q\_{ij}}{v\_i}+4{P\_{ij}}{v\_i}{r\_{ij}}+{r\_{ij}}^2+{x\_{ij}}^2 \Big]\\
\hspace\*{-0.2in} -(1/c)\left[ 8d\_{j}{x\_{ij}}^2{P\_{ij}}^2+8d\_{j}{r\_{ij}}^2{P\_{ij}}^2+8d\_{j}{x\_{ij}}^2{Q\_{ij}}^2-8d\_{j}{x\_{ij}}{Q\_{ij}}{v\_i}+ \right. \\
\hspace\*{0.4in} 8d\_{j}{Q\_{ij}}^2r^2-8d\_{j}{P\_{ij}}{r\_{ij}}{v\_i}+(1/c)(4d\_{j}^2{x\_{ij}}^2+4d\_{j}^2{r\_{ij}}^2) \Big]\\
\hspace\*{-0.2in} +\left[ {r\_{ij}}^2+{x\_{ij}}^2+({r\_{ij}}^2+{x\_{ij}}^2)^2+{v\_i}^2 \right] \left[ 4{P\_{ij}}^2+4{Q\_{ij}}^2+1+ \right. \\
\hspace\*{0.8in} (1/c)\Big( 8d\_{j}{Q\_{ij}}^2+8d\_{j}{P\_{ij}}^2+4d\_{j}+(4d\_{j}^2/c) \Big) \Big]\\
\vspace\*{-0.1in} \\
\hspace\*{-0.25in} = 8{x\_{ij}}{P\_{ij}}{Q\_{ij}}{v\_i}+4{P\_{ij}}^2{r\_{ij}}^2+4{Q\_{ij}}^2{x\_{ij}}^2+ \\
\hspace\*{-0.25in} {v\_i}\Big({v\_i}-4{P\_{ij}}{r\_{ij}}-4{x\_{ij}}{Q\_{ij}}\Big)
+(1/c) \Big[8d\_{j}{P\_{ij}}^2{r\_{ij}}^2+8d\_{j}{Q\_{ij}}^2{r\_{ij}}^2 + \\
\hspace\*{-0.25in} 8d\_{j}{P\_{ij}}^2{x\_{ij}}^2+8d\_{j}{Q\_{ij}}^2{x\_{ij}}^2+2d\_{j}v^2\_i+2d\_{j}v\_i\Big({v\_i}\hspace\*{-0.03in}-\hspace\*{-0.03in} 4{P\_{ij}}{r\_{ij}}\hspace\*{-0.03in}-\hspace\*{-0.03in} 4{x\_{ij}}{Q\_{ij}}\Big) \\
\hspace\*{0.22in} +2d\_{j}{r\_{ij}}^2+2d\_{j}{x\_{ij}}^2 +(1/c)\Big(4d\_{j}^2{r\_{ij}}^2+4d\_{j}^2{x\_{ij}}^2+4d\_{j}^2{v\_i}^2\Big) \Big]\\
\vspace\*{-0.07in} \\
\hspace\*{-0.2in} > 0
\end{array}$$
following the assumption of the theorem in which guarantees that *v**i* − 4*P**i**j**r**i**j* − 4*x**i**j**Q**i**j* > 0.
5. 5 × 5 upper-left subdeterminant: Expanding along the fifth row gives
$$\begin{array}{l}
\hspace\*{-0.2in}
\left|
\hspace\*{-0.05in}
\begin{array}{c@{\hspace\*{0.12in}}c}
4{r\_{ij}}^2+4{P\_{ij}}^2+1+(2d\_{j}/c) & 4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}\\
4{r\_{ij}}{x\_{ij}}+4{P\_{ij}}{Q\_{ij}}& 4{x\_{ij}}^2+4{Q\_{ij}}^2+1+(2d\_{j}/c) \\
2{r\_{ij}}& 2{x\_{ij}}\\
\hspace\*{-0.02in}-\hspace\*{-0.02in}{r\_{ij}}\hspace\*{-0.02in}-\hspace\*{-0.02in}2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2) \hspace\*{-0.02in}-\hspace\*{-0.02in}2{P\_{ij}}{v\_i}& \hspace\*{-0.05in} -\hspace\*{-0.02in}{x\_{ij}}\hspace\*{-0.02in}-\hspace\*{-0.02in}2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)\hspace\*{-0.03in}-\hspace\*{-0.02in}2{Q\_{ij}}{v\_i}\\
0 & 1 \\
\end{array}
\right.\\
\nonumber \\
\left.
\hspace\*{0.1in}
\begin{array}{ccc}
2{r\_{ij}}& -{r\_{ij}}-2{r\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{P\_{ij}}{v\_i}& 0 \\
2{x\_{ij}}& -{x\_{ij}}-2{x\_{ij}}({r\_{ij}}^2+{x\_{ij}}^2)-2{Q\_{ij}}{v\_i}& 1\\
1 & -({r\_{ij}}^2+{x\_{ij}}^2) & 0 \\
-({r\_{ij}}^2+{x\_{ij}}^2) & {r\_{ij}}^2+{x\_{ij}}^2+({r\_{ij}}^2+{x\_{ij}}^2)^2+{v\_i}^2& -x \\
0 & -{x\_{ij}}& 1 \\
\end{array}
\hspace\*{-0.05in} \right| \\
\vspace\*{-0.04in}\\
\hspace\*{-0.17in} = -4{r\_{ij}}{x\_{ij}}{P\_{ij}}{Q\_{ij}}-4{r\_{ij}}^2{P\_{ij}}^2+4{r\_{ij}}{P\_{ij}}{v\_i}+2{Q\_{ij}}{x\_{ij}}{v\_i}-{v\_i}^2+ \\
\hspace\*{-0in} (1/c)(-2d\_{j}{r\_{ij}}^2-2d\_{j}{v\_i}^2+4d\_{j}{Q\_{ij}}{x\_{ij}}{v\_i}) -4{r\_{ij}}{x\_{ij}}{P\_{ij}}{Q\_{ij}}\\
\hspace\*{0in} -4{Q\_{ij}}^2{x\_{ij}}^2 + 2{Q\_{ij}}{v\_i}{x\_{ij}}+ (1/c)\left[-8d\_{j}{P\_{ij}}^2{x\_{ij}}^2-8d\_{j}{Q\_{ij}}^2{x\_{ij}}^2 \right.\\
\left. ~+4d\_{j}{Q\_{ij}}{x\_{ij}}{v\_i}-2d\_{j}{x\_{ij}}^2-(1/c)4d\_{j}^2{x\_{ij}}^2 \right]+8{x\_{ij}}{r\_{ij}}{P\_{ij}}{Q\_{ij}}\\
\hspace\*{0in}\left. +4{P\_{ij}}^2{r\_{ij}}^2 + 4{Q\_{ij}}^2{x\_{ij}}^2 - 4{P\_{ij}}{r\_{ij}}{v\_i}-4{Q\_{ij}}{x\_{ij}}{v\_i}+{v\_i}^2 \right. \\
\hspace\*{0in} +(1/c)\left[8d\_{j}{P\_{ij}}^2{r\_{ij}}^2 + 8d\_{j}{Q\_{ij}}^2{r\_{ij}}^2 + 8d\_{j}{P\_{ij}}^2{x\_{ij}}^2+8d\_{j}{Q\_{ij}}^2{x\_{ij}}^2\right. \\
\hspace\*{0.42in} \left. -8d\_{j}{P\_{ij}}{r\_{ij}}{v\_i}- 8d\_{j}{Q\_{ij}}{x\_{ij}}{v\_i}+ 4d\_{j}{v\_i}^2 + 2d\_{j}{r\_{ij}}^2 \right.\\
\hspace\*{0.42in} \left. + 2d\_{j}{x\_{ij}}^2 + (1/c)(4d\_{j}^2{r\_{ij}}^2+4d\_{j}^2{x\_{ij}}^2+4d\_{j}^2{v\_i}^2)\right] \\
\vspace\*{-0.04in}\\
\hspace\*{-0.2in} = (1/c)\left[ 8d\_{j}{P\_{ij}}^2{r\_{ij}}^2+8d\_{j}{Q\_{ij}}^2{r\_{ij}}^2+ 2d\_{j}{v\_i}\Big({v\_i}-4{P\_{ij}}{r\_{ij}}\Big)+ \right. \\
\hspace\*{0.3in} \left. (1/c)(4d\_{j}^2{r\_{ij}}^2+4d\_{j}^2{v\_i}^2) \right] \\
\hspace\*{-0.2in} > 0
\end{array}$$
as the assumption in gives that *v**i* − 4*P**i**j**r**i**j* > 0.
Hence we get that **z***j*\* satisfying the setting of Proposition [prop:sufcdnopt] is a strict local minimum of *f*. Thus we get the convergence of ENDiCo-OPF algorithm for a single iteration step when *f* is convex.
Modification of Algorithm: -ENDiCo-OPF
--------------------------------------
Developing similar convergence guarantees for local systems over time and further extending the same to global networks present non-trivial challenges. Motivated by the convergence behavior in practice of the original ENDiCO algorithm, we present a modification of the algorithm where a convergence parameter Δ(*t*) ≥ 1 is adaptively chosen in each iteration step *t*. We then derive conditions generalizing ones in Theorem [thm:subsyscvgce] that guarantee convergence of a local system over time when Δ(*t*) = 1.
Motivated by the definition of bi-Lipschitz functions in real analysis, we define the following condition for the convergence of voltage *v**j* over time as captured by the parameter Δ(*t*) ≥ 1:
$$\label{eq:Deltacdn}
\small
\frac{1}{{{\Delta}^{(t)}}} v\_j^{(t-1)} ~ \leq~ v\_j^{(t)} ~\leq~ {{\Delta}^{(t)}}v\_j^{(t-1)}.$$
Note that Δ(*t*) = 1 in the above condition implies *v**j*(*t*) = *v**j*(*t* − 1), which certifies convergence of *v**j* over time.
The only modification we make to the ENDiCo-OPF algorithm is the use of the scaling parameter Δ(*t*) that is suitably initialized by the user (Δ(0) > 1) and the addition of Step 8 presented in Algorithm [alg:DENDiCo]. Subsequently, the final step advancing the algorithm to the next iteration step is now numbered as Step 9.
Stop ${\Delta}^{(t+1)} = 1+\frac{{{\Delta}^{(t)}}-1}{2}$ Move forward to the next iteration step (*t* + 1) [alg:DENDiCo]
Local Convergence over Time
---------------------------
We now consider the convergence under Δ-ENDiCo-OPF of a local subsystem with a single child node, i.e., one consisting of nodes {*i*, *j*, *k*} as shown in. The system of equations ([eq:bdcase-3]) corresponds to System [eq:P1] but has subsystems for iteration steps *t* and (*t* + 1) (sub-equations ([eq:bdcase-3-t]) and ([eq:bdcase-3-tp1])) as well as a subsystem for determining *P**j**k*(*t*) and *Q**j**k*(*t*) ([eq:bdcase-3-PQ]) as well as the convergence condition ([eq:Deltacdn]) in sub-equation ([eq:bdcase-3-Dcdn]) apart from bounds ([eq:bdcase-3-bds]).
[eq:bdcase-3]
$$\begin{aligned}
\min~ &{r\_{ij}}{\ell^{(t)}\_{ij}}+{r\_{ij}}{\ell^{(t+1)}\_{ij}}+{r\_{jk}}{\ell^{(t)}\_{jk}}\\
\text{s.t.}~~\,& \nonumber \\
&\text{Constraints for iteration step $t$ :} \label{eq:bdcase-3-t}\\
& \hspace\*{-0.2in} \nonumber
\begin{array}{l}
\,-{r\_{ij}}{\ell^{(t)}\_{ij}} = P\_{jk}^{(t-1)} - {P^{(t)}\_{ij}} + p\_{Lj}^{(t)} - p\_{Dj}^{(t)}\\
\vspace\*{-0.1in} \\
-{x\_{ij}}{\ell^{(t)}\_{ij}} = Q\_{jk}^{(t-1)} - {Q^{(t)}\_{ij}} + q\_{Lj}^{(t)} - q\_{Dj}^{(t)}\\
\vspace\*{-0.1in} \\
\hspace\*{0.27in} {v^{(t)}\_{j}} = {v^{(t-1)}\_{i}} - 2({r\_{ij}}{P^{(t)}\_{ij}}+{x\_{ij}}{Q^{(t)}\_{ij}}) + ({r\_{ij}}^2+{x\_{ij}}^2){\ell^{(t)}\_{ij}}\\
\vspace\*{-0.1in} \\
\hspace\*{0.27in}{\ell^{(t)}\_{ij}} = \left( ({P^{(t)}\_{ij}})^2+({Q^{(t)}\_{ij}})^2 \right) / {v^{(t-1)}\_{i}}
\end{array} \\
\vspace\*{-0.2in} \nonumber \\
&\text{Solve for ${P^{(t)}\_{jk}},{Q^{(t)}\_{jk}}$ :} \label{eq:bdcase-3-PQ} \\
& \hspace\*{-0.2in} \nonumber
\begin{array}{l}
-{r\_{jk}}{\ell^{(t)}\_{jk}} = - {P^{(t)}\_{jk}} + p\_{Lk}^{(t)} - p\_{Dk}^{(t)}\\
\vspace\*{-0.1in} \\
-{x\_{jk}}{\ell^{(t)}\_{jk}} = - {Q^{(t)}\_{jk}} + q\_{Lk}^{(t)} - q\_{Dk}^{(t)}\\
\vspace\*{-0.1in} \\
\hspace\*{0.27in} {v^{(t)}\_{k}}= {v^{(t-1)}\_{j}} - 2({r\_{jk}}{P^{(t)}\_{jk}}+{x\_{jk}}{Q^{(t)}\_{jk}}) + ({r\_{jk}}^2+{x\_{jk}}^2){\ell^{(t)}\_{jk}}\\
\vspace\*{-0.1in} \\
\hspace\*{0.27in} {\ell^{(t)}\_{jk}} = ({P^{(t)}\_{jk}})^2+({Q^{(t)}\_{jk}})^2/{v^{(t-1)}\_{j}}
\end{array} \\
\vspace\*{-0.2in} \nonumber \\
&\text{Constraints for iteration step $t+1$ :} \label{eq:bdcase-3-tp1} \\
& \hspace\*{-0.3in} \nonumber
\begin{array}{l}
-{r\_{ij}}{\ell^{(t+1)}\_{ij}} = P\_{jk}^{(t)} - {P^{(t)}\_{ij}} + p\_{Lj}^{(t+1)} - p\_{Dj}^{(t)}\\
\vspace\*{-0.1in} \\
-{x\_{ij}}{\ell^{(t+1)}\_{ij}} = Q\_{jk}^{(t)} - {Q^{(t)}\_{ij}} + q\_{Lj}^{(t+1)} - q\_{Dj}^{(t+1)}\\
\vspace\*{-0.1in} \\
\hspace\*{0.15in} {v^{(t+1)}\_{j}}= {v^{(t)}\_{i}} - 2({r\_{ij}}{P^{(t+1)}\_{ij}}+{x\_{ij}}{Q^{(t+1)}\_{ij}}) + ({r\_{ij}}^2+{x\_{ij}}^2){\ell^{(t+1)}\_{ij}}\\
\vspace\*{-0.1in} \\
\hspace\*{0.27in} {\ell^{(t+1)}\_{ij}} = \left(({P^{(t+1)}\_{ij}})^2+({Q^{(t+1)}\_{ij}})^2\right)/ {v^{(t)}\_{i}}
\end{array} \end{aligned}$$
[eq:bdcase-32]
$$\begin{aligned}
&\text{Bounding ${v^{(t+1)}\_{j}}$ in terms of ${v^{(t)}\_{j}}$ :} \label{eq:bdcase-3-Dcdn}\\
& \left(1/{\Delta}^{(t+1)}\right) {v^{(t)}\_{j}} ~\leq~ {v^{(t+1)}\_{j}} ~\leq~ {\Delta}^{(t+1)} \, {v^{(t)}\_{j}} \nonumber \\
\vspace{-0.1in} \nonumber \\
&\text{Bounds on variables :} \label{eq:bdcase-3-bds} \\
&~~~~~~~\underline{V}^2\leq {v^{(t)}\_{i}},{v^{(t)}\_{j}}, {v^{(t)}\_{k}}, {v^{(t+1)}\_{j}}\leq \Bar{V}^2 \nonumber\\
&-\sqrt{S\_{DR\_j}^2-p\_{Dj}^2}\leq q\_{Dj}^{(t)}\leq \sqrt{S\_{DR\_j}^2-p\_{Dj}^2} \nonumber \\
&-\sqrt{S\_{DR\_k}^2-p\_{Dk}^2}\leq q\_{Dk}^{(t)}\leq \sqrt{S\_{DR\_k}^2-p\_{Dk}^2} \nonumber \\
&-\sqrt{S\_{DR\_j}^2-p\_{Dj}^2}\leq q\_{Dj}^{(t+1)}\leq \sqrt{S\_{DR\_j}^2-p\_{Dj}^2} \nonumber \\
& {\ell^{(t)}\_{ij}}\leq \left(I\_{ij}^{\text{rated}}\right)^2, \hspace\*{0.1in} {\ell^{(t)}\_{jk}}\leq \left(I\_{jk}^{\text{rated}}\right)^2, \hspace\*{0.1in} {\ell^{(t+1)}\_{ij}}\leq \left(I\_{ij}^{\text{rated}}\right)^2 \nonumber
\end{aligned}$$
[thm:single-bd] Assume that the objective function and all constraints in System ([eq:bdcase-3]) are twice differentiable. Then the local subsystem converges at iteration step *t* if the Δ-convergence condition ([eq:Deltacdn]) holds with Δ(*t*) = 1 as well as the following conditions hold.
$$\small
\label{eq:cdnsingle-bd}
\begin{aligned}
&{x\_{ij}} - {r\_{ij}}\geq 0 \\
&{v^{(t-1)}\_{i}}{v^{(t-1)}\_{j}}-4{P^{(t+1)}\_{ij}}{r\_{ij}}{v^{(t-1)}\_{j}}-4{P^{(t)}\_{jk}}{r\_{jk}}{v^{(t-1)}\_{i}}\geq 0 \\
&{v^{(t-1)}\_{i}} {v^{(t-1)}\_{j}} - 4{P^{(t)}\_{ij}}{r\_{ij}}{v^{(t-1)}\_{j}}-4{P^{(t)}\_{jk}}{r\_{jk}}{v^{(t-1)}\_{i}}\geq 0 \\
&{v^{(t-1)}\_{i}} - 4{P^{(t+1)}\_{ij}}{r\_{ij}} - 4{P^{(t)}\_{ij}}{r\_{ij}}\geq 0 \\
&{v^{(t-1)}\_{i}}-4{P^{(t+1)}\_{ij}}{r\_{ij}}-2{P^{(t)}\_{ij}}{r\_{ij}}-2{P^{(t)}\_{ij}}{x\_{ij}}\geq 0 \\
&{v^{(t-1)}\_{i}}-4{P^{(t)}\_{ij}}{r\_{ij}}-4{P^{(t)}\_{ij}}{x\_{ij}}\geq 0 \\
&{v^{(t-1)}\_{i}}-{P^{(t+1)}\_{ij}}{r\_{ij}}-{P^{(t)}\_{jk}}{r\_{jk}}\geq 0 \\
&{v^{(t-1)}\_{i}}{v^{(t-1)}\_{j}} - 4{P^{(t)}\_{jk}}{r\_{jk}}{v^{(t-1)}\_{i}} \\
&\hspace\*{0.7in} -2{P^{(t)}\_{ij}}{r\_{ij}}{v^{(t-1)}\_{j}}-2{P^{(t)}\_{ij}}{x\_{ij}}{v^{(t-1)}\_{j}}\geq 0 \\
&({{v^{(t-1)}\_{j}}})^3 - 4{P^{(t+1)}\_{ij}}{r\_{ij}}{v^{(t-1)}\_{i}}{v^{(t-1)}\_{j}} \\
&\hspace\*{0.52in} -2{P^{(t)}\_{jk}}{r\_{jk}}{v^{(t-1)}\_{i}}-4{P^{(t+1)}\_{ij}}{r\_{ij}}\geq 0 \\
& {v^{(t-1)}\_{i}}-8{P^{(t)}\_{ij}}{r\_{ij}} \geq 0 \\
& {v^{(t-1)}\_{i}}-8{Q^{(t)}\_{ij}}{x\_{ij}} \geq 0 \\
& {v^{(t-1)}\_{i}}-8{P^{(t)}\_{jk}}{r\_{jk}} \geq 0
\end{aligned}$$
We note that most conditions in hold for typical values of the variables except the first one: *x**i**j* ≥ *r**i**j* may not always hold. Similar to the condition specified in Theorem [thm:subsyscvgce] for local convergence, the above system also gives a sufficient condition. In practice, we observe convergence even when *x**i**j* ≥ *r**i**j* does not hold.
Analogous to the proof of Theorem [thm:subsyscvgce], we construct the augment Lagrangian function of System w.r.t. to the following variable vector (in place of the one in ):
$$\label{eq:zj3t}
\hspace\*{-0.1in}
\begin{aligned}
{\mathbf{z}\_j}= \left[ \begin{matrix}
{P^{(t)}\_{ij}} & {Q^{(t)}\_{ij}} & {v^{(t)}\_{j}} & {\ell^{(t)}\_{ij}} & q\_{Dj}^{(t)} & \ldots \hspace\*{0.65in}
\end{matrix}\right.\\
\left.\begin{matrix}
{P^{(t)}\_{jk}} & {Q^{(t)}\_{jk}} & {v^{(t)}\_{k}} & {\ell^{(t)}\_{jk}} & q\_{Dk}^{(t)} & \ldots \hspace\*{0.65in}
\end{matrix}\right.\\
\left.\begin{matrix}
{P^{(t+1)}\_{ij}} & {Q^{(t+1)}\_{ij}} & {v^{(t+1)}\_{j}} & {\ell^{(t+1)}\_{ij}} & q\_{Dj}^{(t+1)} &
\end{matrix} \right]^T
\end{aligned}$$
which gives the following first-order gradient:
$$\begin{aligned}
\hspace\*{-0.04in} \nabla\_{{\mathbf{z}\_j}} L \hspace\*{-0.02in} = \hspace\*{-0.05in}
\left[
\begin{matrix}
-1 & 0 & -2{r\_{ij}} & 2{P^{(t)}\_{ij}} & 1 & 1 & 0 \\
0 & -1 & -2{x\_{ij}} & 2{Q^{(t)}\_{ij}} & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 \\
{r\_{ij}} & {x\_{ij}} & {r\_{ij}}^2+{x\_{ij}}^2 & -{v^{(t)}\_{j}} & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & -2{r\_{jk}} \\
0 & 0 & 0 & 0 & 0 & -1 & -2{x\_{jk}} \\
0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & {r\_{jk}} & {x\_{jk}} & {r\_{jk}}^2+{x\_{jk}}^2 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right.
\end{aligned}$$
$$\begin{aligned}
\hspace\*{-0.05in}
\left.
\begin{matrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
2{P^{(t)}\_{jk}} & 0 & 0 & 0 & 0 & 0 & 0 \\
2{Q^{(t)}\_{jk}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1\\
-{v^{(t)}\_{k}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 2{r\_{ij}} & 2{P^{(t+1)}\_{ij}} & 0 & 0 \\
0 & 0 & -1 & -2{x\_{ij}} & 2{Q^{(t+1)}\_{ij}} & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & {r\_{ij}} & {x\_{ij}} & {r\_{ij}}^2+{x\_{ij}}^2 & -{v^{(t+1)}\_{j}} & -1/{{\Delta}^{(t)}}& -{{\Delta}^{(t)}}\\
0 & 0 & -1 & 0 & 0 & 0 & 0 \\
\end{matrix}
\,\right]
\end{aligned}$$
We then considered the positive definiteness of the Hessian ∇**z***j***z***j**L* to derive conditions under which all its upper left sub-determinants are positive. We used symbolic computation in Maple to simplify expressions for the higher order sub-determinants. These expressions turn out to be too tedious to present here in full, so we present them online. Examining expressions of the sub-determinants led to the conditions in, which when satisfied guarantee positive definiteness of the Hessian. Our approach was similar to the one we used in the proof of Theorem [thm:subsyscvgce] where we paired or grouped positive terms in the determinant expressions with negative ones so that their sum is guaranteed to be positive (when the condition in is satisfied).
While Δ(*t*) appears in the last two columns of the gradient ∇**z***j**L*, it does not appear in the Hessian ∇**z***j***z***j**L* and hence does not appear in. But these conditions holding along with the Δ-convergence condition ([eq:Deltacdn]) with Δ(*t*) = 1 guarantee the convergence over time of the local subsystem.
Global System Convergence for Line Systems
------------------------------------------
As the next generalization, we consider a *line* network with *n* ≥ 4 nodes. To keep notation simple, we label the nodes {1, …, *n*} where 1 is the source node and *n* is the final leaf node (note that *n* is also the number of nodes in the line). Not surprisingly, this system presents even more challenges to derive conditions that guarantee global convergence over time. Guided by the convergence behavior of the original ENDiCo algorithm in practice, we derive conditions under a suitable assumption that when satisfied in a time sequential manner guarantee global convergence of the line system in a sequential manner, i.e., starting with the first local subsystem {1, 2, 3}, moving to the next local subsystem {2, 3, 4}, and so on until the last local subsystem {*n* − 2, *n* − 1, *n*}.
[asm:cvgcpres] Given two adjacent and overlapping local subsystems {*i*, *i* + 1, *i* + 2} and {*i* + 1, *i* + 2, *i* + 3} from the line network {1, …, *n*} where the first subsystem is convergent at iteration step *t**i*, the convergence of the controllable node *i* + 1 is preserved in the next iteration step *t**i* + 1 ≥ *t**i* + 1 for the second subsystem.
We get the guarantee of global convergence of a line system under Assumption [asm:cvgcpres] by repeatedly applying Theorem [thm:single-bd] in sequence going from the source node to the final leaf node.
[thm:globalcvgnc] The global convergence of a line system {1, …, *n*} is guaranteed under Assumption [asm:cvgcpres] if the following conditions hold at each local subsystem {*i*, *i* + 1, *i* + 2} in a sequential manner with iteration steps *t**i* + 1 ≥ *t**i* + 1 for *i* = 1, …, *n* − 2.
$$\label{eq:LnDeltacdn}
\begin{aligned}
\left( 1/{\Delta}^{t\_i} \right) v\_{i+1}^{(t\_i-1)} ~ \leq~ v\_{i+1}^{(t\_i)} ~\leq~ {\Delta}^{t\_i} v\_{i+1}^{(t\_i-1)} \\
\text{ holds for } {\Delta}^{t\_i} = 1 \text{ for } i=1, \dots, n-2.
\end{aligned}$$
$$\label{eq:LnCdns}
\begin{aligned}
&{x\_{i,i+1}} - {r\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}{v^{(t\_i-1)}\_{i+1}}-4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}- \\
&\hspace\*{0.87in} 4{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}} {v^{(t\_i-1)}\_{i+1}} - 4{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}-\\
&\hspace\*{0.87in} 4{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}} - 4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}} - 4{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}-2{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}- \\
&\hspace\*{1.55in} 2{P^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-4{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}-4{P^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}-{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}{v^{(t\_i-1)}\_{i+1}} - 4{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}} \\
&~-2{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}- 2{P^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}}{v^{(t\_i-1)}\_{i+1}}\geq 0 \\
&({{v^{(t\_i-1)}\_{i+1}}})^3 - 4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}{v^{(t\_i-1)}\_{i}}{v^{(t\_i-1)}\_{i+1}} \\
&~-2{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}}{v^{(t\_i-1)}\_{i}}-4{P^{(t\_i+1)}\_{i,i+1}}{r\_{i,i+1}}\geq 0 \\
&{v^{(t\_i-1)}\_{i}}-8{P^{(t\_i)}\_{i,i+1}}{r\_{i,i+1}} \geq 0 \\
&{v^{(t\_i-1)}\_{i}}-8{Q^{(t\_i)}\_{i,i+1}}{x\_{i,i+1}} \geq 0 \\
&{v^{(t\_i-1)}\_{i}}-8{P^{(t\_i)}\_{i+1,i+2}}{r\_{i+1,i+2}} \geq 0
\end{aligned}$$
We note that most conditions in hold for typical values of the variables except the first one. But we observe global convergence in practice in the sequential manner even when *x**i*, *i* + 1 ≥ *r**i*, *i* + 1 may not hold.
Note that conditions in are the same as those in from Theorem [thm:single-bd] applied for the local subsystem {*i*, *i* + 1, *i* + 2} in place of {*i*, *j*, *k*}. Since conditions in hold for *i* = 1, the first local subsystem {1, 2, 3} is convergent at iteration step *t*1 following Theorem [thm:single-bd]. By Assumption [asm:cvgcpres], the convergence of node 2 is preserved in the next iteration step, and hence it can be treated as the fixed source node for the next local subsystem {2, 3, 4}. The convergence of this subsystem is then guaranteed at time *t*2 ≥ *t*1 + 1 by conditions in holding for *i* = 2. The overall result follows by the sequential application of Theorem [thm:single-bd].
Numerical Study
===============
In this section, we demonstrate the convergence properties of the ENDiCo-OPF algorithm with the help of numerical simulations. We also validate the optimality of our algorithm by comparing its results with those of a centralized solution. These simulations not only justify the convergence analysis of the method but also showcase the efficacy of the proposed real-time distributed controller to attain optimal power flow solutions. After attaining the optimal dispatch, the controller shares the computed boundary variables with it its neighbor instead of implementing and measuring the variables.
IEEE-123 Bus Test System
[Testsys]
Simulated System and Results
----------------------------
As a test system, we simulated a balanced IEEE-123 bus system with a maximum of 85 DERs (PVs) connected to the network (Fig. [Testsys]), where the DER/PV penetration can vary from 10% to 100%. As a cost function, we have simulated both (i) active power loss minimization (*f* = *r**i**j**l**i**j*), and (ii) voltage deviation (Δ*V*) minimization (*f* = (*v**j* − *v*ref)2 ) optimization problems.
### Residual and Objective Value Convergence
We have simulated the test system with 10%, 50%, and 100% DER/PV penetration cases for both active power loss minimization and Δ*V* minimization. The system converged after 42 iterations for all six cases (Fig. [convergence]). For the loss minimization objective, the maximum border residual goes below the tolerance value of 10− 3 after the 42nd iteration. The objective values for the loss minimization OPF is 26.5 kW, 19.6 kW, and 11.8 kW, for 10, 50, and 100% DER penetration, respectively. Similarly, for Δ*V* minimization, we can see that the maximum residual goes below the tolerance after the 42nd iteration as well. Thus the convergence is related to the network size, but not to the number of controllable variables.
0.75
0.78
[convergence]
0.49
0.49
[Validation]
0.49
0.49
[Voltageconv]
[!b]
[comptable]
| | | | | |
| --- | --- | --- | --- | --- |
| | Central | 26.4 | 19.6 | 11.78 |
| | ENDiCo-OPF | 26.5 | 19.6 | 11.80 |
| | Central | 0.5300 | 0.5038 | 0.4640 |
| | ENDiCo-OPF | 0.5306 | 0.5042 | 0.4642 |
[ht!]
0.672
0.672
0.672
0.672
0.672
0.672
[boundV]
### Validation of the Optimal Solution
Besides a faster convergence, we also present the efficacy of the distributed OPF controller in terms of the optimality gaps and feasibility. To this end, we have compared (a) the objective values, and (b) the nodal voltages with the centralized solution (see Fig. [Validation]). It can be observed in Table [comptable] that the value of the objective functions from centralized and distributed solutions matches for all the cases. For example, for the 100% DER penetration case, the line loss is 11.80 kW for proposed ENDiCo-OPF method, and the central solution is 11.78 kW. Similar comparisons can be found for other DER/PV penetration cases with different OPF objectives. This validates the solution quality of ENDiCo-OPF. Further, we can see in Fig. [Validation] that upon implementing ENDiCo-OPF, the difference in nodal voltages in the system and those from a centralized solution is in the order of 10− 4; this is true for both OPF objectives. This validates the feasibility of ENDiCo-OPF.
Numerical Experiments of Convergence
------------------------------------
In this section, we provide further simulated results on the convergence of the proposed ENDiCo-OPF method. Here we showcase the boundary variable convergence with respect to iterations, as well as their properties. We also compare the numerical convergence results with the theoretical analysis presented in Section [sec:cnvgcanal].
### Convergence at the Boundary
The convergence of the boundary variables (shared boundary voltage) for the simulated cases has been shown in Fig. [Voltageconv] and Fig.[boundV]; Fig. [Voltageconv] shows the boundary variables for 100% PV cases for 3 different locations that helps to visualize the convergence of the shared variable w.r.t. the distance from the root node (substation node). Specifically, from Fig. [boundV], we can see that after the initial values, the shared variables (shared nodal voltages) suddenly changes abruptly till 22*n**d* iterations. This location, i.e., iteration number, where this sudden changes happen depends on the distance of that shared node from the root node (substation node). For example, ‘Bus 3’ is 2 node distant from the substation, and thus this abrupt changes happen at the 2nd iteration (Fig. [Voltageconv]); similarly, ‘Bus 62’ is 11 node distant from the substation, and that changes happen at the 11*t**h* iteration as well. Along with these characteristics, the overall convergence properties of the shared variables are consistent with both objectives and for all the PV penetration cases as well (Fig. [boundV]). It corroborates with the statement that the convergence properties is not dependent on the OPF objective or the DER penetration percentage, but rather dependent on the system network. Instead of using a flat start with 1.02 pu for the controller, a measured voltage initialization would reduce the iteration number; however, we would like to mention that, this method is robust enough to initialize with any reasonable flat start values.
### Discussion on Convergence
In, we guaranteed the convergence of the proposed method under some sufficient conditions. Specifically, we showed (i) convergence of the local sub-problem for a given iteration step (Theorem [thm:subsyscvgce]), (ii) convergence of the local sub-problem over iteration steps (Theorem [thm:single-bd]), and (iii) convergence over time for a line network with multiple nodes (Theorem [thm:globalcvgnc]). At the same time, our numerical experiments demonstrated similar convergence behavior for more general (than line) networks.
The condition for the convergence of the local sub-problem in a single iteration step is expressed in equation. Generally for a stable electric power supply in a power distribution system, *v**i* and *P**i**j*, *Q**i**j* are in the order of 1 pu., and the corresponding line parameters, i.e., *r**i**j*, *x**i**j*, are both in the order of 10− 2 or less. This guarantees that condition is always satisfied for a practical power distribution system. In our simulated cases, line parameters are also in the order of ≤ 10− 2, thus satisfying the condition for Theorem [thm:subsyscvgce]. Further, this also satisfies most of the sufficient conditions for Theorem [thm:single-bd] and [thm:globalcvgnc], except the first conditions, i.e., *x**i**j* − *r**i**j* ≥ 0 of both of the theorems. We note that these are sufficient conditions, and that we can observe overall convergence for the cases where *x**i**j* − *r**i**j* < 0 as well. For the simulation cases, while other sufficient conditions hold true, we observed *x**i**j* − *r**i**j* < 0 for some of the lines, but still the controller converged. In addition, Fig. [Voltageconv] showcases the same result as Theorem [thm:globalcvgnc]. The node that is closer to the root node/substation node, i.e., the node with a strong voltage source, converges earlier than the node that is more distant from the root node. For instance, “Bus 3”, which is two nodes away from the substation, converges earlier than the “Bus 62” that is 11 nodes distant from the substation. Bus 3 converges around the 22nd iteration, whereas Bus 62 converges around 30th iteration for both loss and ΔV minimization optimization problems.
Comparison against Centralized OPF and an ADMM-based Distributed OPF Approach
-----------------------------------------------------------------------------
Table [table:timecomptable] compares the total solve time for the three different algorithms: a centralized OPF, the proposed distributed ENDiCo-OPF, and an ADMM-based distributed OPF. All three approaches are applied to both objective functions: loss minimization and voltage deviation minimization. The simulation was performed using a Core i7-8550U CPU @ 1.80GHz with 16GB of memory. All three algorithms use fmincon solver from MATLAB to solve the associated nonlinear optimization problems. Since all algorithms use the same compute system and same nonlinear solver, the simulation results provided are appropriate to demonstrate the relative improvements observed via the proposed distributed algorithm. The results show that the proposed distributed approach is significantly faster than both the centralized and ADMM-based distributed OPF methods. For example, for the loss minimization problem with 100% PV penetration, the solution time for ENDiCo-OPF is only 0.71 seconds, while the centralized OPF and ADMM-based distributed OPF take 15.8 seconds and 37.3 minutes, respectively. Moreover, the solution time for the centralized OPF increases with the increase in the number of controllable nodes (i.e. %PV penetration). The proposed ENDiCo-OPF method, however, scales well even for larger number of controllable nodes.
[!h] [table:timecomptable]
| | | | | |
| --- | --- | --- | --- | --- |
| | Centralized OPF | 4.2 sec | 10.6 sec | 15.8 sec |
| | ENDiCo-OPF | 0.67 sec | 0.71 sec | 0.71 sec |
| | ADMM-based OPF | 41.6 min | 36.5 min | 37.3 min |
| | Centralized OPF | 2.1 sec | 3.4 sec | 4.8 sec |
| | ENDiCo-OPF | 0.68 sec | 0.70 sec | 0.70 sec |
| | ADMM-based OPF | 120 min | 119 min | 121 min |
Additionally, compared to ADMM-based approach, the proposed ENDiCo-OPF method also reduces the required number of communication rounds/iterations by order of magnitudes. Besides the iteration counts, the developed method converges faster compared to the ADMM-based distributed OPF. Figure [fig:comparisonADMM] illustrates the convergence properties of the objective values for both the ADMM-based method and the ENDiCo-OPF method for 100% PV penetration cases. As can be observed, the ADMM-based method requires 7, 000 and 10, 00 iterations for loss minimization and ΔV minimization problems, respectively. It is worth noting that the proposed ENDiCo-OPF method requires only 42 iterations for both cases, which highlights the effectiveness of the developed method. These results demonstrate that the proposed ENDiCo-OPF method outperforms both centralized OPF and ADMM-based distributed OPF methods.
0.48
0.48
[fig:comparisonADMM]
Applicability and Extension to Real-world Setting
-------------------------------------------------
Although in this paper, we focus on a single-period optimization problem, the proposed distributed algorithm has been numerically demonstrated under different realistic test conditions, including for a large-scale single-phase system consisting of over 50, 000 variables, three-phase unbalanced systems, and simulations conducted under diverse communication conditions. We would also like to emphasize that the improvement in computational speed for single-period optimization, observed in this work, will help scale more complex versions of OPF problems, including multi-period and stochastic versions.
Another major challenge relate to optimization under fast varying conditions. The existing literature, employs online optimization approaches where the algorithm doesn’t wait to obtain optimal solution, but rather takes step towards the steepest decent direction. In our previous work, we have extended the proposed distributed approach to a setting similar to online optimization techniques. Our simulations show that the proposed approach is able to efficiently track the optimal solution, even for rapidly changing system conditions, modeled as fast varying load and PV injections.
A reliable communication system is crucial for the practical viability of the distributed OPF algorithms. The communication system is needed to exchange the boundary variables among distributed agents and arrive at a converged system-level optimal solution. Therefore, the convergence, speed and accuracy of distributed OPF methods depend upon the communication systems conditions. It is imperative to evaluate the impacts of communication system-specific attributes (such as, latency, bandwidth, reliability) on the convergence of distributed OPF algorithm. Related literature includes simplified analysis to numerically evaluate the effects of communication system-related challenges. In our prior work, we have used a cyber-power co-simulation platform, using HELICS, to evaluate several related concerns. Additional work is needed to both numerically and analytically evaluate the effects of communication system attributes on the convergence and performance of distributed optimization methods. One can also determine an optimal communication system design to meet the required performance for distributed optimization methods.
Conclusions
===========
The optimal coordination of growing DER penetrations requires computationally efficient models for distribution-level optimization. In this paper, we have developed a nonlinear distributed optimal power flow algorithm with convergence guarantees using network equivalence methods. Then we present sufficient conditions to guarantee the convergence of the proposed method. While our most general sufficient conditions for global convergence over time are presented for line networks, our numerical simulations demonstrate similar convergence behavior for more general, e.g., radial, networks. The numerical simulation on the IEEE 123 bus test system corroborates the theoretical analysis. The proposed distributed method is also validated by comparing the results with a centralized formulation. Developing similar sufficient conditions for global convergence of radial networks, or other general network topologies, will be of high interest.
---
1. YL is with University of Science and Technology of China, RS and AD are with EECS, Washington State University, Pullman and, BK is with Mathematics and Statistics, Washington State University, Vancouver. YL and BK acknowledge funding from NSF through grant 1819229. RS and AD acknowledge funding from DOE under contract DE-AC05-76RL01830. Corresponding author E-mail: [email protected].[↩](#fnref1)
|
arxiv_0186014
|
sustained anisotropy would change the orbital evolution of a planet within 100 au from its predicted isotropic adiabatic value.
In summary, given the mass values in Table [stelprop], the semimajor axis of an orbiting planet would likely increase by a factor of 4.4–5.6. For the lower mass stars usually considered in post-main-sequence planetary science studies, this factor is instead about 2–4.
Orbital engulfment due to tidal interactions
--------------------------------------------
As the star is losing mass and the planet’s orbit is expanding, the star is expanding as well. In fact, the stellar envelope may expand quickly enough and close enough to the planet to draw it inside. Many investigations have computed the critical engulfment distance (technically the minimum star-planet separation on the main sequence which leads to engulfment on the giant branches) for the red giant branch phase, the asymptotic giant branch phase, or both phases.
The approaches and prescriptions in the papers listed above differ. Here we seek just a rough estimate. Further, for 6 − 8*M*⊙ stars, the red giant branch phase is negligibly short. Consequently, the extremes of luminosity, temperature and radius all occur close to, but not precisely at, the tip of the asymptotic giant branch phase.
Hence, we focus on this latter phase only. carried out detailed numerical simulations which determined the critical engulfment distance of different types of planets (different masses and compositions) around asymptotic giant branch stars whose main-sequence progenitor masses went up to 5*M*⊙. They used the equilibrium tidal model of for their formalism. instead took a different approach, and derived an analytical formula for the critical engulfment distance, $d\_{\rm eng}$, as a function of several free parameters. Their prescription includes a correction for pulsations during the asymptotic giant branch phase. Neither study adopted stellar metallicity as a free parameter[10](#fn10).
Let us assume the extreme quiescent case of a planet forming on a circular orbit around stars more massive than 5*M*⊙. Extrapolating Fig. 7 of yields the following crude estimates for Jovian, Neptunian and terrestrial planets:
$$d\_{\rm eng}^{\rm (Jovian)} = 0.48 \ {\rm au} \left( \frac{M\_{\star}}{M\_{\odot}} \right) + 2.6 \ {\rm au}$$
$$\ \ \ \ \ \ \ \ \ \ \ \, = \left\lbrace 5.5, 6.0, 6.4 \right\rbrace \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Neptunian)} = 0.33 \ {\rm au} \left( \frac{M\_{\star}}{M\_{\odot}} \right) + 1.8 \ {\rm au}$$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \, = \left\lbrace 3.8, 4.1, 4.4 \right\rbrace \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Terrestrial)} = 0.28 \ {\rm au} \left( \frac{M\_{\star}}{M\_{\odot}} \right) + 1.42 \ {\rm au}$$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \, = \left\lbrace 3.1, 3.4, 3.6 \right\rbrace \ {\rm au},$$
where the evaluations above correspond to ZAMS masses of *M*⋆ = {6, 7, 8}*M*⊙.
Equations 32 and 38 of instead give
$$\frac{d\_{\rm eng}}{2 \ {\rm au} }
\approx \left[
\mathcal{P}
\left( \frac{q+1}{q + \beta - p - 1} \right)
\left( \frac{\tau}{0.1 \ {\rm Myr}} \right)
\left( \frac{M\_{\rm planet}}{M\_{\rm Jupiter}} \right)
\right]^{2/19}$$
$$\ \ \ \ \ \ \ \times
\left( \frac{M\_{\star}}{M\_{\odot}} \right)^{-1/19}
\left( \frac{R\_{\star}}{1 \ {\rm au}} \right)^{16/19}
.
\label{tidcrit}$$
In equation ([tidcrit]), *τ* represents a mass loss timescale; here we use the duration of the asymptotic giant branch phase (see Table [stelprop]). The coefficient P > 1 represents the enhancement factor due to the presence of pulsations. The indices *q*, *p* and *β* respectively represent the radial dissipation index, the mass loss index and the mass dissipation index. Although these values are free parameters, stated that the critical value of the tidal dissipation parameter, which is a component of equation ([tidcrit]), is accurate to within 50 per cent across the entire parameter space.
Our own numerical investigation reveals that the leading term with the indices varies by less than 20 per cent across all plausible values of *q*, *p* and *β*. Hence, we simply adopt *q* = 7, *β* = 2, and *p* = 0, such that the leading term is unity. We also take both *M*⋆ and *R*⋆ to represent the mass and radius of at the star at the start of the asymptotic giant branch phase. Then for ZAMS masses of *M*⋆ = {6, 7, 8}*M*⊙, we obtain
$$d\_{\rm eng}^{\rm (Jovian)} = \left\lbrace 4.0, 4.3, 4.5 \right\rbrace\mathcal{P} \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Neptunian)} = \left\lbrace 3.0, 3.2, 3.3 \right\rbrace\mathcal{P} \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Terrestrial)} = \left\lbrace 2.2, 2.4, 2.5 \right\rbrace\mathcal{P} \ {\rm au}.$$
Figure 4 of illustrates that the pulsation enhancement factor P can change the engulfment boundary by several tens of per cent. Hence, the values of $d\_{\rm eng}$ from the two different sets of assumptions and extrapolations employed here may be roughly consistent. We show the correspondence in Fig. [major].
[major]
In-spiral inside stellar envelope
---------------------------------
A planet engulfed in a stellar envelope is not necessarily immediately destroyed. The possibility exists that the planet may survive the resulting in-spiral. If so, the envelope – which is initially puffed up by the planet ingestion – would have to dissipate before the planet becomes compressed and flattened by ram pressure. Planets themselves are typically too small to unbind the envelope, and so survival would depend on a slow in-spiral.
By using orbital decay power scalings, compute, in their Figure 4, the in-spiral timescale in terms of orbits for a Jovian planet across parts of the Hertszprung-Russell diagram. In every case the number of in-spiral orbits is less than, and usually much less than, 104 orbits. At most then, the in-spiral time represents about 10 per cent of the duration of the asymptotic giant branch phase.
The in-spiral time is hence effectively negligible. This conclusion, however, is extrapolated from just the handful of papers above, which focus on the highest mass planets and stars with masses lower than those we consider here. More detailed modelling across a wider parameter space would provide a necessary complement to the tidal studies cited in Section 3.2.
Other related investigations have focused on the consequences for the parent star. illustrated that engulfment could lead to an under-massive white dwarf. Potential lithium enhancement due to planet engulfment would no longer be observable by the time the star becomes a white dwarf. However, rotational spin-up of the giant branch star due to engulfment might leave a small but lasting residue, because the spin period of the eventual white dwarf would affect tidal interactions with orbiting planets.
The residue perhaps also is manifest in the resulting change of magnetic field strength. The *Gaia*-SDSS spectroscopic white dwarf catalog in reveals that up to 50% of white dwarfs whose progenitors had masses of 6 − 8*M*⊙ have magnetic fields above the typical 1 MG detection limit. This fraction could, however, represent a signature of the large fraction of massive white dwarfs that are thought to originate from mergers.
Gravitational scattering
------------------------
Even if a planet survives tidal interactions with the star and, if applicable, in-spiral inside the envelope, the planet may then be subject to gravitational interactions with other, more distant surviving planets. The result could be ejection from the system, collision with the star or collision with the other planet.
Gravitational scattering between multiple major planets in giant branch planetary systems has been investigated in only about one dozen papers (an order of magnitude smaller than the comparable body of literature for multi-planet scattering in main-sequence planetary systems). Nevertheless, we know that in evolving single-star exo-systems, stellar mass loss can trigger instability due to changes in the Hill stability limit, Lagrange stability limit and shifting location of secular resonances. Further, the combined effect of mass loss and Galactic tides can incite instability from a distant massive planetary companion (like the theorized “Planet Nine”; see for a review) in an otherwise quiescent system.
The timescale over which the instability acts would be crucial for evolved 6 − 8*M*⊙ planetary systems. The only two of the above-cited papers which has simulated multi-planet instability in 6 − 8*M*⊙ systems are and. Figures 9-12 of reveal that instability time is unrestricted but strongly dependent on the initial separations of the planets. Further, instability is relatively rare during the giant branch phases, particularly for the shortest such phases at the highest stellar masses. This rarity suggests that gravitational instability is unlikely to feature until the star has become a white dwarf.
A plausible scenario is that gravitational instability amongst multiple planets (which survived the giant branch phases of evolution) occurred a few hundred Myr after the parent star became a white dwarf. These timescales match with instability timescales for plausible sets of initial conditions from the studies above (by extrapolating from lower stellar masses). Nevertheless, multiple planets are not actually required to pollute the white dwarf with minor planets, but can facilitate the process.
Minor planet survival
=====================
The predominant focus in the exoplanet formation literature has been on major planets. However, white dwarf metal pollution highlights the need to also model minor planet formation. In fact, almost all observations of white dwarf planetary systems are of the remnants of minor planets.
By “minor planets”, we refer to an object with a mean radius of under 103 km. This size regime encompasses moons, comets, asteroids, and large fragments of major planets. In fact, the minor planet currently orbiting the white dwarf WD 1145+017 may best be classified as an “active asteroid” whereas the one orbiting SDSS J1228+1040 is perhaps best described as a ferrous chunk of a fragmented planetary core. The third, which orbits ZTF J0139+5245, still defies classification, until at least more data is obtained.
These distinctions are becoming increasingly relevant given our detailed knowledge of white dwarf planetary systems. A related question of interest is whether white dwarf metal pollution arises from primarily intact or destroyed minor planets, and the locations at which those bodies or debris were scattered. The answer depends more strongly on radiation than gravity through the Yarkovsky and YORP effects.
The Yarkovsky and YORP effects are recoil forces and torques produced due to non-uniform scattering of absorbed radiation. The Yarkovsky effect changes the orbit of a spherical or aspherical minor planet, whereas the YORP effects spins up or/and spins down an aspherical minor planet. Both effects have been modelled for solar system asteroids in exquisite detail. Contrastingly, only a handful of basic dedicated extrasolar post-main-sequence studies have been published for the Yarkovsky effect and the YORP effect ().
This contrast does not suit the added complexity introduced in giant branch systems, when the luminosity of the parent star changes on short timescales, and can be five orders of magnitude higher than the Sun’s (see Table [stelprop]). Such high luminosities easily fling exo-asteroids out to distances of tens, hundreds or even thousands of au, regardless of the presence of major planets, which only temporarily impede such migration. These minor planets could also be propelled inwards, and the direction of motion is a function of the bulk shape, spin, surface topography and internal homogeneities of the objects.
If, however, the minor planets veer too close to a giant branch star, then they could be spun up to disruption. This rotational fission occurs when the spin of a minor planet reaches the critical spin failure rate $\omega\_{\rm fail}$. For strengthless rubble piles in the solar system, this rate corresponds to a rotational period of about 2.3 hours and has robust confirmation through observations.
demonstrated that asteroid belts within about 7 au of their parent stars along the main sequence (including the solar system’s asteroid belt) would be easily fragmented by YORP-induced rotational fission along the giant branch phases. However, they considered parent stars with main sequence masses of 1 − 5*M*⊙.
Now we extend their study to higher stellar masses. The spin of a minor planet evolves according to
$$\frac{d\omega(t)}{dt}
=
\frac
{3 \mathcal{C} \Phi }
{4 \pi \rho R^2 a(t)^2\sqrt{1 - e(t)^2} }
\left(\frac{L\_{\star}(t)}{L\_{\odot}}\right)
\label{YORPsimp}$$
where *ρ* and *R* represent the minor planet’s density and initial radius (both assumed to remain constant), and *a* and *e* represent its time-dependent semimajor axis and eccentricity. C is a constant which represents the extent of the minor planet’s asphericity, and Φ = 1 × 1017 kg ⋅ m/s2 is the Solar radiation constant.
Equation ([YORPsimp]) helps illustrate that the increasing luminosity of the star competes against the increasing semimajor axis from stellar mass loss. As previously mentioned, for adiabatic mass loss, the eccentricity remains constant. We simply set it to zero here. An analytic expression of the failure rate which includes the minor planet’s tensile, uni-axial strength *S* is
$$\omega\_{\rm fail} = \sqrt{\frac{4 \pi G \rho}{3} + \frac{4S}{3 \rho R^2}}
.$$
We now compute simple estimates of $a\_{\rm break}$, the minimum initial (ZAMS) semimajor axis for which a minor planet will survive asymptotic giant branch YORP spin-up and reach the white dwarf phase. To do so, we integrate equation ([YORPsimp]) assuming *S* = 0, *e* = *e*(*t*) = 0 and an initially non-spinning minor planet. We also assume that the asteroid does not spin down at any point during its evolution, and its asphericity is encompassed entirely within the assumed constant value of C. We consider the YORP effect in isolation, decoupled from the Yarkovsky effect; such a coupling would require a significant modelling effort, and has not yet been achieved in exoplanetary science.
Figure [minorYORP] illustrates the result in $a\_{\rm break}$–*R* space, across a range of *R* (100 m – 10 km) for which the YORP effect is significant. Coincidentally, the curves for the *Z* = 0.02, 6*M*⊙ and 7*M*⊙ cases are visually indistinguishable on the plot, as are the curves for the *Z* = 0.02, 8*M*⊙ and *Z* = 0.0001, 6*M*⊙ cases. Changing the asphericity parameter by one order of magnitude noticeably shifts the curves.
[minorYORP]
In comparison to the survival capacity of major planets, we find that across almost the entire phase space, $a\_{\rm break} > d\_{\rm eng}$ and usually, $a\_{\rm break} \gg d\_{\rm eng}$. However, in individual cases, these relations do not prevent the major planet from, for example, residing exterior to intact minor planets. Nevertheless, the relations suggest that a surviving major planet which is surrounded by fragmented debris is not an unreasonable scenario.
An important related question is whether a minor planet would be sublimated from the host star before or during YORP spin-up. In order to estimate the sublimation rate of a minor planet, we use the approximation employed by ; for a more detailed description of its physical assumptions, see Section 6.1.1. of. The radius of a minor planet evolves according to
$$\frac{dR}{dt} = -\frac
{1.5 \times 10^{10} {\rm kg} \cdot {\rm m}^{-2} {\rm s}^{-1}}
{\rho}
\sqrt{\frac{T\_{\rm sub}}{T(t)}} \,
{\rm exp}\left(-\frac{T\_{\rm sub}}{T(t)} \right)
\label{subeq}$$
where, with the the sublimation temperature $T\_{\rm sub}$ and the Stefan-Boltzmann constant *σ*, the temperature of the minor planet *T* can be approximated by
$$T(t) = \left(
\frac
{L\_{\star}(t)}
{16 \pi \sigma r(t)^2}
\right)^{1/4}
.$$
The functional form of equation ([subeq]) reveals a strong dependence on the ratio $T\_{\rm sub} /T(t)$, and the value of $T\_{\rm sub}$ is composition-dependent. For olivine, $T\_{\rm sub} \approx 6.5 \times 10^4$ K. By adopting this value, along with *ρ* = 2 g/cm3, we find that the sublimation rate is negligible across almost the entire parameter space and can be ignored. For example, at the inner edge of this space, we find that for *r* = {20, 15, 10} au, the *maximum* sublimation rate, as measured by radius decrease, across all of our *Z* = 0.02 stellar evolution tracks are, respectively, {2 × 10− 11, 5 × 10− 8, 6 × 10− 4} m/yr. Instead, for the *Z* = 0.0001, *M* = 6*M*⊙ track, there is a greater danger at 10 au, with corresponding *maximum* sublimation values of {3 × 10− 8, 3 × 10− 5, 1 × 10− 1} m/yr.
Given that these minor planets largely survive sublimation, one may inquire about the size distribution of the debris which results from YORP-based break-up. Although this topic requires further dedicated study, explored this size distribution in a basic, analytical fashion, based on the results of. estimated the total number of fissioned components of an asteroid, as well as the sizes of the components, based on a number of factors including internal strength, initial spin and stellar luminosity. The result is hence dependent on many parameters and assumptions. The resulting dust produced may be blown away due to radiation pressure by comparing the interplay between radiation and gravity ; the presence of a major planet would affect this balance, and affects the regions where dust is ejected in a non-trivial fashion (Zotos & Veras, In Preparation).
Discussion
==========
Having computed the survival limits for both major and minor planets orbiting 6*M*⊙ − 8*M*⊙ stars, we now discuss topics which are related to these findings.
Ice line locations
------------------
Giant planets are commonly assumed to have formed beyond the ice line (or ``snow line“ or ``frost line”)[11](#fn11), where volatile compounds condense. For 1*M*⊙ stars, the ice line is often assumed to reside at about 2.7 au, exterior to $d\_{\rm eng}$.
However, as the host star mass increases, the relation between the ice line distance (denoted as $d\_{\rm ice}$) and $d\_{\rm eng}$ does not become immediately clear. In the ZAMS stellar mass range considered in this paper, $d\_{\rm eng}$ for giant planets is between about 5.5 and 6.5 au (Fig. [major]).
In these same systems, the location of the ice line is determined by a combination of stellar irradiation and viscous heating. However, discs around massive stars are dispersed rapidly, allowing the effect of irradiation to dominate. In this limit, we can approximate the midplane ice-line temperature $T\_{\rm ice}$ just as did by applying the following flared-disc prescription from, and :
$$T\_{\rm ice} = T\_{\star} \left(
\frac{\alpha}{2}
\right)^{1/4}
\left(
\frac{R\_{\star}}{d\_{\rm ice}}
\right)^{3/4}$$
with
$$\alpha =
0.005 \left(\frac{d\_{\rm ice}}{1 \ {\rm au}} \right)^{-1}
+
0.05 \left(\frac{d\_{\rm ice}}{1 \ {\rm au}} \right)^{2/7}
.$$
By setting $T\_{\rm ice} = 170$ K and ZAMS values for *R*⋆ and *T*⋆ from the aforementioned sse code, we find that $d\_{\rm ice} = 2.4-3.6$ au $\approx 0.5 d\_{\rm eng}$. Hence, even in this rough approximation, the engulfment distance along the giant branch phases is more restrictive than the ice line with regard to giant planet formation location.
The theoretical considerations in this subsection help motivate dedicated planet formation studies for high host-star mass systems. We now present observational motivation.
Future white dwarf host mass increases
--------------------------------------
*Gaia* is expected to discover, through astrometry, at least a dozen giant planets orbiting white dwarfs at the final data release. If realized, this exciting prediction would revolutionize our understanding of white dwarf planetary systems[12](#fn12), but is highly unlikely to increase the mass of the most massive known white dwarf host star.
Instead, the new ``record-breaker" is more likely to arise from the steep and imminent increase in known polluted white dwarfs. This increase will arise from the larger number of new white dwarfs which have already been identified from *Gaia*. *Gaia* Data Release 2 recently uncovered an all-sky sample of ≈ 260,000 white dwarf candidates that is homogeneous down to *G* < 20 mag. From this sample, on the order of 10,000 white dwarfs with atmospheric planetary remnants are suitable to low-resolution spectroscopic study in the near future with wide-area multi-object spectroscopic surveys such as WEAVE, 4MOST, DESI and SDSS-V.
The smallest white dwarf hosts
------------------------------
The most massive known white dwarf planetary system hosts would also represent the smallest known white dwarf planetary system hosts. The classic mass-radius relation for white dwarfs
$$\frac{R\_{\star}}{R\_{\odot}} \approx 0.0127 \left(\frac{M\_{\star}}{M\_{\odot}}\right)^{-1/3}
\sqrt{1-0.607\left(\frac{M\_{\star}}{M\_{\odot}}\right)^{4/3} }$$
yields, for the range of masses in Table [stelprop], host star radii between 0.0013*R*⊙ − 0.0063*R*⊙ ≈ 900 km − 4, 450 km $\approx 0.26R\_{\rm Mars}-1.31R\_{\rm Mars}$. In contrast, a typical 0.6*M*⊙ white dwarf gives a radius of about $2.6R\_{\rm Mars}$.
How do the much smaller radii of the systems highlighted here affect the dynamical origin of metal pollution? The Roche radii of white dwarfs scale as *M*⋆1/3 regardless of the composition or spin of orbiting companions. Therefore, doubling the white dwarf mass increases the Roche radii by about 25 per cent. This level of increase is significant, and can for example, trigger thermal self-disruption of passing gaseous planets which otherwise would be safe. The larger Roche radius also provides a larger target for minor planets, which, when combined with the location requirements for major planets in these systems, places restrictions on where and when scattering occurs.
Another consequence of decreasing the size of a white dwarf and increasing its Roche radius is the resulting change in structure of a debris disc formed from the disruption of a minor planet. The outer edge would be larger than those of the currently known discs, increasing the probability of capture of solid objects ; the consequences for the inner edge is less clear because of our lack of observations between the inner disc rim and the white dwarf photosphere. The resulting disc evolution and accretion rates onto the white dwarf might also differ from what has been previously modelled.
Other features of 6*M*⊙ − 8*M*⊙ planetary systems
-------------------------------------------------
Major planets which form and reach distances of several to many au in under 70 Myr around 6*M*⊙ − 8*M*⊙ stars may help identify the dominant planet formation mechanisms in these systems. These planets formed in discs which are subject to time-varying radiative forcing at a more extreme level than their lower-host mass counterparts. When combined with photoevaporative effects from other stars in their birth cluster, outward migration of planets within these discs may be enhanced.
If the 6*M*⊙ − 8*M*⊙ progenitor host stars for polluted white dwarfs were slightly more massive, then they would have instead become neutron stars; planets are known to exist around single pulsars. These pulsar planets most likely represent products of second-generation planet formation from mergers or fallback although questions still remain [13](#fn13). Exo-asteroids may orbit pulsars below the detection threshold, although that threshold already lies at about 1 per cent of the mass of the Moon.
In contrast, both major and minor planets orbiting single white dwarfs are first-generation, because the mass in the white dwarf debris discs required to form new planets is at the level of an Io mass or higher. In contrast, the masses of white dwarfs with observed debris discs remain uncertain but are assumed to be generated instead by asteroids (with a typical mass two orders of magnitude less than Io). Further, the chemistry of any second-generation planets would be recycled from broken-up first generation bodies; Veras, Karakas & Gänsicke (submitted) demonstrate definitively that white dwarf pollution does not arise from stellar fallback.
Summary
=======
The most massive currently known progenitor of a white dwarf planetary system host ( ≈ 4*M*⊙) exceeds the maximum mass of a main-sequence major planet host ( ≈ 3*M*⊙). Because this mass difference is likely to increase with white dwarf population growth (Section 5.2), we have investigated major and minor planet survival limits for the highest possible stellar masses ( ≈ 6*M*⊙ − 8*M*⊙) which yield white dwarfs. This mass regime represents almost completely unexplored territory for planet formation theorists, and largely unexplored territory for post-main-sequence planetary evolution investigators.
We found that a major planet needs to reside beyond 3-6 au (depending on the type of planet) at the end of the short (40-70 Myr) main-sequence phase in order to survive its host star transition into a white dwarf. Minor planets could have remained intact or broken up due to YORP-induced giant branch radiation, having been subject to host star luminosities reaching up to 105*L*⊙. The boundary separating these two possibilities ( ∼ 101 − 103 au) is very sensitive to minor planet size, but is high enough to suggest that metal pollution in these systems would originate from already fragmented debris rather than intact minor planets which fragment upon reaching the white dwarf Roche radius. Our study motivates dedicated planet formation studies around the highest mass stars, and predicts that metal pollution can occur in the highest mass white dwarfs.
Acknowledgements
================
We thank the expert reviewer Fred C. Adams for helpful comments which have improved the manuscript. DV gratefully acknowledges the support of the STFC via an Ernest Rutherford Fellowship (grant ST/P003850/1). The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme no. 677706 (WD3D). Support for this work was also provided by NASA through grant number HST-GO-15073.005-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. FM acknowledges support from the Royal Society Dorothy Hodgkin Fellowship. GMK is supported by the Royal Society as a Royal Society University Research Fellow. BTG is supported by the UK Science and Technology Facilities Council grant ST/P000495.
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[lastpage]
---
1. E-mail: [email protected][↩](#fnref1)
2. STFC Ernest Rutherford Fellow[↩](#fnref2)
3. Royal Society University Research Fellow[↩](#fnref3)
4. Royal Society Dorothy Hodgkin Fellow[↩](#fnref4)
5. reported a so far unconfirmed candidate planet orbiting an asymptotic giant branch star.[↩](#fnref5)
6. considered planet formation around supermassive black holes which are just a few parsecs away from active galactic nuclei. However, the prospects of detecting planetary systems outside of the Milky Way Galaxy are currently remote, except perhaps for the Large Magellanic Cloud (which probably does not contain a central black hole; ).[↩](#fnref6)
7. have recently suggested that 20 ± 6 per cent of high-mass white dwarfs (1.08*M*⊙ − 1.23*M*⊙) originate from double white dwarf mergers. In addition, another significant fraction are likely merger byproducts merged when at least one of the stellar components was not a white dwarf.[↩](#fnref7)
8. However, see for robust evidence of a major ice giant planet orbiting a white dwarf whose progenitor mass was 1.0*M*⊙ − 1.6*M*⊙; has suggested that this giant planet is a partially destroyed and highly inflated ``Super-Puff".[↩](#fnref8)
9. The lower end of the range corresponds to warm hydrogen-atmosphere (DA class) white dwarfs.[↩](#fnref9)
10. More massive progenitor stars should have higher metallicities because they must be younger if they are evolved from single stars.[↩](#fnref10)
11. See for a challenge to the canonical theory.[↩](#fnref11)
12. Some current observational limits have been placed on the possible locations of major planets orbiting white dwarfs. Null results in a deep *Spitzer* survey can rule out at least > 10Jupiter-mass planets within the 5-50 au range around more than 40 white dwarfs. Monitoring the stable pulsation arrival times of at least 6 white dwarfs can exclude > 3Jupiter-mass planets in a range of roughly 2 − 5au. Unfortunately none of the most massive pulsating white dwarfs have stable, coherent modes that can be used to monitor for substellar companions.[↩](#fnref12)
13. The sudden stellar mass loss experienced in a supernova is typically insufficient to eject a planet on a compact orbit. The more pertinent but outstanding question of first-generation planet survival is whether the planet can physically withstand a supernova’s stellar ejecta.[↩](#fnref13)
Constraining planet formation around 6*M*⊙-8*M*⊙ stars
======================================================
[firstpage]
Identifying planets around O-type and B-type stars is inherently difficult; the most massive known planet host has a mass of only about 3*M*⊙. However, planetary systems which survive the transformation of their host stars into white dwarfs can be detected via photospheric trace metals, circumstellar dusty and gaseous discs, and transits of planetary debris crossing our line-of-sight. These signatures offer the potential to explore the efficiency of planet formation for host stars with masses up to the core-collapse boundary at ≈ 8*M*⊙, a mass regime rarely investigated in planet formation theory. Here, we establish limits on where both major and minor planets must reside around ≈ 6*M*⊙ − 8*M*⊙ stars in order to survive into the white dwarf phase. For this mass range, we find that intact terrestrial or giant planets need to leave the main sequence beyond approximate minimum star-planet separations of respectively about 3 and 6 au. In these systems, rubble pile minor planets of radii 10, 1.0, and 0.1 km would have been shorn apart by giant branch radiative YORP spin-up if they formed and remained within, respectively, tens, hundreds and thousands of au. These boundary values would help distinguish the nature of the progenitor of metal-pollution in white dwarf atmospheres. We find that planet formation around the highest mass white dwarf progenitors may be feasible, and hence encourage both dedicated planet formation investigations for these systems and spectroscopic analyses of the highest mass white dwarfs.
planets and satellites: formation – protoplanetary discs – planets and satellites: dynamical evolution and stability – planet-star interactions – stars: white dwarfs – stars: AGB and post-AGB
Introduction
============
From the first detection of exoplanetary material one century ago ( vanmaanen1917, vanmaanen1919; but recognized as such only much later) to the pioneering discoveries of exoplanets over the last three decades, we are starting to piece together an understanding of planetary formation, evolution and fate. Major and minor planets, including asteroids, comets and planetary debris, have now been observed orbiting brown dwarfs, M-type to B-type main-sequence stars, subgiant and red giant branch stars, white dwarfs, and pulsars. However, we still lack definitive detections in systems with O-type stars, asymptotic giant branch stars, and black holes[5](#fn5).
On 21 Nov 2019, the NASA Exoplanet Archive listed as ``confirmed" 4,099 major planets which have been discovered orbiting main-sequence, subgiant and giant branch stars. From that population, the most massive known planet host with a well-constrained (within 10 per cent) mass above 3.0*M*⊙ is *o* UMa b, whose host star mass is 3.09 ± 0.07*M*⊙. This detection cutoff at about 3.0*M*⊙ is not sharp, but rather represents a tail reflecting a decreasing number of discoveries as a function of stellar mass (see Fig. 5 of, Fig. 9 of and Fig. 3 of ).
This upper bound results from some combination of observational limitations and restrictions on where and how planets can form and evolve around stars more massive than the Sun, and does not change depending on stellar classification nor whether asteroseismological constraints are taken into account.
The formation of planets around (low-mass) ≤ 3*M*⊙ stars represents one of the most extensively studied aspects of exoplanetary science. The two oldest formation theories are gravitational instability (see,, and for reviews), and core accretion (see and for reviews). More recently, important physical processes such as the streaming instability and pebble accretion have provided alternatives or enhancements to the traditional formulations. Each theory, when combined with post-formation dynamical evolutionary pathways, succeeds in reproducing observed properties of some but not all of the known exoplanets.
All these formation processes require the presence of a circumstellar, protoplanetary disc, and operate in different regions of the disc on different timescales. A disc must survive long enough and be massive enough for a planet to form.
Unfortunately, like observations of main-sequence exoplanet host stars themselves, observations of protoplanetary discs are restricted by the rarity of massive host stars and the short lifetimes of their discs. Early on in the life of a stellar birth cluster, circumstellar discs have been observed to contain a wide variety of masses, up to host masses of many tens of solar masses. Young, intermediate stars such as Herbig Ae/Be stars are commonly seen to host circumstellar discs, which are almost certainly sites of planet formation (much as T Tauri stars are precursors to planet-hosts like the Sun). Several of these stars have masses greater than 3*M*⊙, suggesting that a fertile circumstellar environment could exist for high host-star mass planet formation.
Here, we present another source of motivation to study planet formation around high-mass (6*M*⊙ − 8*M*⊙) stars: the end state of these systems. As outlined in Section 2, current observations of white dwarfs with exoplanetary material expand the traditional main-sequence host-star mass range, and future observations could expand this range even further. In fact, the vast majority of $\gtrsim 3M\_{\odot}$ stars ever formed and which now reside within a local volume of hundreds of parsecs are cooling white dwarfs.
Based on these prospects, we compute survival limits for the bounding case of the highest main-sequence host star masses which would not trigger a core-collapse supernova[6](#fn6) ( ≈ 6*M*⊙ − 8*M*⊙) but instead become white dwarfs (which could eventually harbour observed metal pollution). In Sections 3 and 4 respectively we compute survival limits for both major and minor planets around these massive progenitors. We discuss our results in Section 5 and conclude in Section 6, hoping to motivate future dedicated investigations of planet formation around high-mass stars.
White dwarf planet hosts
========================
After both major and minor planets are formed and dynamically settle, the remainder of main-sequence evolution is thought to remain relatively quiescent. For example, in our solar system, over the last 4 Gyr or so, the orbits of the eight major planets have not varied enough to incite mutual scattering events. This quiet situation is very likely to continue until the end of the Sun’s main-sequence lifetime about 6 Gyr from now.
Subsequently, when the Sun ascends the giant branch phases, major changes (described in detail in Sections 3 and 4) will ensue. The surrounding bodies which survive these changes will eventually orbit a white dwarf, a stellar ember that is dense enough to stratify any accreted material into its constituent chemical elements. This property enables the detection of exoplanetary metals, because predominantly white dwarf atmospheres would otherwise be composed of only hydrogen and/or helium.
Origin of exoplanetary material
-------------------------------
These metallic debris seen in the atmospheres of single white dwarfs do not necessarily arise from major planets. In fact, signatures of individual minor planets have now been found around three different white dwarfs: WD 1145+017, SDSS J1228+1040 and ZTF J0139+5245.
These discoveries have corroborated observations of metal-polluted white dwarf photospheres. On both chemical and dynamical grounds, most of the atmospheric debris likely originates from exo-asteroids, exo-moons or their fragments from destructive giant branch radiation. Minor planets, or their fragments, can avoid being engulfed into the star during the giant branch phases only at distances beyond several or many au. Therefore, at least one (surviving) major planet must exist in these systems to perturb the minor bodies into the disruption region around the white dwarf from au-scale distances, as argued extensively in.
What if the white dwarf is currently observed to have a binary stellar companion? Then, debris in the white dwarf atmosphere could originate from the winds of its partner rather than a planetary system. However, if the companion is separated beyond a critical distance, then the accreted mass from the wind would be too small to explain the observations (and hence must arise from a planetary system). This critical distance is on the order of few au, well within the binary separation for the majority of known polluted white dwarfs in binaries.
What if, instead, a polluted white dwarf *used to* have a binary stellar companion which underwent a merger[7](#fn7)? Could the atmospheric debris then result from the merger event? The answer is no, because any metal mixing resulting from the merger would sink to the core very quickly, on a timescale which is 3-11 orders of magnitude shorter than the cooling age. Hence, no chemical signatures from the merger would be currently observable.
In summary, regardless if a white dwarf has or had binary companions, *both major and minor planets should exist in metal-polluted systems without stellar binary companions and whose main-sequence progenitor masses were 6*M*⊙ − 8*M*⊙*. So far major planets in such high mass systems have not been observed[8](#fn8). However, the statistics of high-mass metal rich white dwarfs are still poorly understood, owing to the steepness of the initial mass function and their faintness, and therefore their rarity.
170mm [stelprop]
@cccccccc@ ZAMS mass & *Z* & white dwarf mass & main sequence duration & AGB duration & Max *R* & Max *L* & Max *T*
(*M*⊙) & & (*M*⊙) & (Myr) & (Myr) & (au) & (*L*⊙) & (K)
6.00 & 0.02 & 1.14 & 66.0 & 0.56 & 5.94 & 6.16 × 104 & 4.12 × 105
6.00 & 0.0001 & 1.37 & 60.0 & 0.75 & 4.29 & 1.54 × 105 & 6.93 × 105
7.00 & 0.02 & 1.29 & 48.9 & 0.51 & 6.73 & 7.67 × 104 & 5.07 × 105
8.00 & 0.02 & 1.44 & 37.2 & 0.49 & 7.39 & 9.16 × 104 & 1.11 × 106
The most massive white dwarf host
---------------------------------
Unlike for main-sequence exoplanetary systems, white dwarf planetary systems are not yet summarized with a publicly-available database. Rather, this data is contained within different published studies.
Nevertheless, we can identify the most massive polluted white dwarf host based from individual observing campaigns. The homogeneous sample of bright, hydrogen-atmosphere white dwarfs studied in reveals that the metal polluted WD1038+633 has a robust mass determination both from Balmer line spectroscopy ($M\_{\rm WD}$ = 0.90 ± 0.01 *M*⊙; ) and from *Gaia* photometry and astrometry ($M\_{\rm WD}$ = 0.91 ± 0.01 *M*⊙; ). The progenitor mass is more uncertain, but estimate a value of ≈ 4 *M*⊙.
In a different investigation, analysed a sample of helium-atmosphere white dwarfs with metal pollution mostly drawn from the Sloan Digital Sky Survey (SDSS). Their figure 10 demonstrates that for a generous upper limit of 10% precision on *Gaia* parallax, the most massive currently known progenitor of a metal polluted white dwarf is similarly ≈ 4 *M*⊙, albeit with larger error bars on the white dwarf mass.
In either case, the most massive currently observed metal-polluted white dwarf contains a progenitor mass of about 4*M*⊙. In Section 5, we will elaborate on the exciting and imminent increase in the number of polluted white dwarfs for which spectroscopy will be obtained, and the prospects for finding even higher mass hosts.
Main-sequence progenitor masses
-------------------------------
Extrapolating measured white dwarf masses to main-sequence values requires the application of an initial-to-final-mass relation. This relation, however, is poorly constrained in the 6*M*⊙ − 8*M*⊙ regime. Observationally, Figure 5 of reveals that white dwarf masses higher than about 1.05*M*⊙ plausibly correspond to main-sequence masses of about 6*M*⊙ − 8*M*⊙, with large uncertainties; the lower mass boundary for a core-collapse supernova to occur is 8*M*⊙ ± 1*M*⊙.
Plots for the maximum envelope radius, phase durations, main sequence age, and mass loss for 6 − 8*M*⊙ stars based on the SSE stellar evolution code have already all been presented in and we do not repeat them here. Instead, we reiterate the most important of those features in Table [stelprop]. Variations in these numbers which would arise from the application of other stellar codes are small enough to not qualitatively affect our results, especially given the uncertainty in the initial-to-final-mass relation.
Short main-sequence lifetimes
-----------------------------
One set of values from Table [stelprop] which deserves special attention is the duration of the main sequence phase (35-70 Myr). Hence, subsequent to protoplanetary disc dissipation, for 6*M*⊙ − 8*M*⊙ systems, only a few to several tens of Myr would remain before the host star leaves the main sequence. This period of time might have been too short for the giant planets in our own solar system to dynamically settle. Nevertheless, significant gravitational scattering events may occur earlier in other systems, before the end of the main sequence phase.
Even during the main-sequence phase of these systems, planets are subject to other forces which are often neglected in other exoplanetary studies. In the most extreme case of a 8*M*⊙ host star, the star will actually lose 3.4 per cent of its mass by the end of the main sequence. In contrast, the Sun will lose no more than 0.06 per cent of its mass before the end of the main sequence. Further, how planet formation within protoplanetary discs is affected by such a rapidly changing massive star remains uncertain.
White dwarf disc lifetimes
--------------------------
Another important aspect of white dwarf planetary system observations is their age, because that parameter helps constrain dynamical history. The “cooling age” in particular refers to the time since the white dwarf was born. Metal polluted white dwarfs with cooling ages longer than about 100 Myr have metal sinking timescales in the atmosphere that range from just a few weeks to a few Myr [9](#fn9).
Such a short sinking timescale relative to the cooling age implies that metal pollution arises from a steady accretion stream originating in a ring- or disc-like planetary debris structure; these structures have now been observed in over 40 systems and many others are likely hidden from view. Supporting the accretion stream theory is that white dwarfs, being Earth-sized, represent targets which are too small for direct hits by minor planets scattered from distances of many au.
The lifetime of those discs provides a hint as to when the surviving major planet (or planets) perturbed a minor planet (or its fragments) towards the white dwarf. Without considering any disc replenishment mechanisms, the lifetime of these discs is estimated to be ∼ 104 − 106 yr. Because these timescales are still much smaller than a typical cooling age of at least 100 Myr, in this case the major planet did not trigger the pollution of the star immediately after it became a white dwarf, and the planet must have survived for at least 100 Myr of white dwarf cooling.
Major planet survival
=====================
Theoretical efforts to link main-sequence and white dwarf planetary systems in the highest-mass regimes have been sparse. and did previously consider the evolution of major planets around 6*M*⊙ − 8*M*⊙ host stars. Since 2013, however, the focus has shifted towards much lower stellar masses (typically 3*M*⊙ and below) because those were the only ones which matched the mounting number of observations over this period (see in particular the match between the results of and ). In this section, we provide a significantly updated assessment of the basic survival considerations from and.
As argued earlier, in order for the eventual white dwarf to be polluted at a cooling age of hundreds of Myr, at least one major planet needs to survive until the white dwarf phase, and then for at least another approximately 100 Myr. Subsequently the major planet would kick a minor planet or debris towards the white dwarf (with the travel time being less than 1 Myr). A disc is then formed, and that disc would accrete onto the star on a timescale less than about 1 Myr.
Now we detail how the major planet can survive until the white dwarf phase.
Orbital expansion due to mass loss
----------------------------------
During the giant branch phases, orbits will expand due to stellar mass loss (; ; see Section 4 of for a historical summary). For isotropic mass loss, an “adiabatic” regime may be defined where the orbital eccentricity remains nearly constant as the semimajor axis is increased. In this regime, the semimajor axis increases inversely to the mass which remains, such that, for example, a 75 per cent mass loss quadruples the semimajor axis. The boundary of the adiabatic regime scales with stellar mass. However, Figs. 14-15 of illustrate that despite the high mass loss rate from 6 − 8*M*⊙ stars, the resulting change in eccentricity for orbiting bodies within 100 au remains negligible (and in the adiabatic regime).
If the mass loss is instead anisotropic, then the character of the orbital expansion changes. However, the potential anisotropy of the mass loss from 6 − 8*M*⊙ stars is not well-constrained, and evidence from planetary nebulae remain in the realm of binary stars. Only significant sustained anisotropy would change the orbital evolution of a planet within 100 au from its predicted isotropic adiabatic value.
In summary, given the mass values in Table [stelprop], the semimajor axis of an orbiting planet would likely increase by a factor of 4.4–5.6. For the lower mass stars usually considered in post-main-sequence planetary science studies, this factor is instead about 2–4.
Orbital engulfment due to tidal interactions
--------------------------------------------
As the star is losing mass and the planet’s orbit is expanding, the star is expanding as well. In fact, the stellar envelope may expand quickly enough and close enough to the planet to draw it inside. Many investigations have computed the critical engulfment distance (technically the minimum star-planet separation on the main sequence which leads to engulfment on the giant branches) for the red giant branch phase, the asymptotic giant branch phase, or both phases.
The approaches and prescriptions in the papers listed above differ. Here we seek just a rough estimate. Further, for 6 − 8*M*⊙ stars, the red giant branch phase is negligibly short. Consequently, the extremes of luminosity, temperature and radius all occur close to, but not precisely at, the tip of the asymptotic giant branch phase.
Hence, we focus on this latter phase only. carried out detailed numerical simulations which determined the critical engulfment distance of different types of planets (different masses and compositions) around asymptotic giant branch stars whose main-sequence progenitor masses went up to 5*M*⊙. They used the equilibrium tidal model of for their formalism. instead took a different approach, and derived an analytical formula for the critical engulfment distance, $d\_{\rm eng}$, as a function of several free parameters. Their prescription includes a correction for pulsations during the asymptotic giant branch phase. Neither study adopted stellar metallicity as a free parameter[10](#fn10).
Let us assume the extreme quiescent case of a planet forming on a circular orbit around stars more massive than 5*M*⊙. Extrapolating Fig. 7 of yields the following crude estimates for Jovian, Neptunian and terrestrial planets:
$$d\_{\rm eng}^{\rm (Jovian)} = 0.48 \ {\rm au} \left( \frac{M\_{\star}}{M\_{\odot}} \right) + 2.6 \ {\rm au}$$
$$\ \ \ \ \ \ \ \ \ \ \ \, = \left\lbrace 5.5, 6.0, 6.4 \right\rbrace \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Neptunian)} = 0.33 \ {\rm au} \left( \frac{M\_{\star}}{M\_{\odot}} \right) + 1.8 \ {\rm au}$$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \, = \left\lbrace 3.8, 4.1, 4.4 \right\rbrace \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Terrestrial)} = 0.28 \ {\rm au} \left( \frac{M\_{\star}}{M\_{\odot}} \right) + 1.42 \ {\rm au}$$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \, = \left\lbrace 3.1, 3.4, 3.6 \right\rbrace \ {\rm au},$$
where the evaluations above correspond to ZAMS masses of *M*⋆ = {6, 7, 8}*M*⊙.
Equations 32 and 38 of instead give
$$\frac{d\_{\rm eng}}{2 \ {\rm au} }
\approx \left[
\mathcal{P}
\left( \frac{q+1}{q + \beta - p - 1} \right)
\left( \frac{\tau}{0.1 \ {\rm Myr}} \right)
\left( \frac{M\_{\rm planet}}{M\_{\rm Jupiter}} \right)
\right]^{2/19}$$
$$\ \ \ \ \ \ \ \times
\left( \frac{M\_{\star}}{M\_{\odot}} \right)^{-1/19}
\left( \frac{R\_{\star}}{1 \ {\rm au}} \right)^{16/19}
.
\label{tidcrit}$$
In equation ([tidcrit]), *τ* represents a mass loss timescale; here we use the duration of the asymptotic giant branch phase (see Table [stelprop]). The coefficient P > 1 represents the enhancement factor due to the presence of pulsations. The indices *q*, *p* and *β* respectively represent the radial dissipation index, the mass loss index and the mass dissipation index. Although these values are free parameters, stated that the critical value of the tidal dissipation parameter, which is a component of equation ([tidcrit]), is accurate to within 50 per cent across the entire parameter space.
Our own numerical investigation reveals that the leading term with the indices varies by less than 20 per cent across all plausible values of *q*, *p* and *β*. Hence, we simply adopt *q* = 7, *β* = 2, and *p* = 0, such that the leading term is unity. We also take both *M*⋆ and *R*⋆ to represent the mass and radius of at the star at the start of the asymptotic giant branch phase. Then for ZAMS masses of *M*⋆ = {6, 7, 8}*M*⊙, we obtain
$$d\_{\rm eng}^{\rm (Jovian)} = \left\lbrace 4.0, 4.3, 4.5 \right\rbrace\mathcal{P} \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Neptunian)} = \left\lbrace 3.0, 3.2, 3.3 \right\rbrace\mathcal{P} \ {\rm au},$$
$$d\_{\rm eng}^{\rm (Terrestrial)} = \left\lbrace 2.2, 2.4, 2.5 \right\rbrace\mathcal{P} \ {\rm au}.$$
Figure 4 of illustrates that the pulsation enhancement factor P can change the engulfment boundary by several tens of per cent. Hence, the values of $d\_{\rm eng}$ from the two different sets of assumptions and extrapolations employed here may be roughly consistent. We show the correspondence in Fig. [major].
[major]
In-spiral inside stellar envelope
---------------------------------
A planet engulfed in a stellar envelope is not necessarily immediately destroyed. The possibility exists that the planet may survive the resulting in-spiral. If so, the envelope – which is initially puffed up by the planet ingestion – would have to dissipate before the planet becomes compressed and flattened by ram pressure. Planets themselves are typically too small to unbind the envelope, and so survival would depend on a slow in-spiral.
By using orbital decay power scalings, compute, in their Figure 4, the in-spiral timescale in terms of orbits for a Jovian planet across parts of the Hertszprung-Russell diagram. In every case the number of in-spiral orbits is less than, and usually much less than, 104 orbits. At most then, the in-spiral time represents about 10 per cent of the duration of the asymptotic giant branch phase.
The in-spiral time is hence effectively negligible. This conclusion, however, is extrapolated from just the handful of papers above, which focus on the highest mass planets and stars with masses lower than those we consider here. More detailed modelling across a wider parameter space would provide a necessary complement to the tidal studies cited in Section 3.2.
Other related investigations have focused on the consequences for the parent star. illustrated that engulfment could lead to an under-massive white dwarf. Potential lithium enhancement due to planet engulfment would no longer be observable by the time the star becomes a white dwarf. However, rotational spin-up of the giant branch star due to engulfment might leave a small but lasting residue, because the spin period of the eventual white dwarf would affect tidal interactions with orbiting planets.
The residue perhaps also is manifest in the resulting change of magnetic field strength. The *Gaia*-SDSS spectroscopic white dwarf catalog in reveals that up to 50% of white dwarfs whose progenitors had masses of 6 − 8*M*⊙ have magnetic fields above the typical 1 MG detection limit. This fraction could, however, represent a signature of the large fraction of massive white dwarfs that are thought to originate from mergers.
Gravitational scattering
------------------------
Even if a planet survives tidal interactions with the star and, if applicable, in-spiral inside the envelope, the planet may then be subject to gravitational interactions with other, more distant surviving planets. The result could be ejection from the system, collision with the star or collision with the other planet.
Gravitational scattering between multiple major planets in giant branch planetary systems has been investigated in only about one dozen papers (an order of magnitude smaller than the comparable body of literature for multi-planet scattering in main-sequence planetary systems). Nevertheless, we know that in evolving single-star exo-systems, stellar mass loss can trigger instability due to changes in the Hill stability limit, Lagrange stability limit and shifting location of secular resonances. Further, the combined effect of mass loss and Galactic tides can incite instability from a distant massive planetary companion (like the theorized “Planet Nine”; see for a review) in an otherwise quiescent system.
The timescale over which the instability acts would be crucial for evolved 6 − 8*M*⊙ planetary systems. The only two of the above-cited papers which has simulated multi-planet instability in 6 − 8*M*⊙ systems are and. Figures 9-12 of reveal that instability time is unrestricted but strongly dependent on the initial separations of the planets. Further, instability is relatively rare during the giant branch phases, particularly for the shortest such phases at the highest stellar masses. This rarity suggests that gravitational instability is unlikely to feature until the star has become a white dwarf.
A plausible scenario is that gravitational instability amongst multiple planets (which survived the giant branch phases of evolution) occurred a few hundred Myr after the parent star became a white dwarf. These timescales match with instability timescales for plausible sets of initial conditions from the studies above (by extrapolating from lower stellar masses). Nevertheless, multiple planets are not actually required to pollute the white dwarf with minor planets, but can facilitate the process.
Minor planet survival
=====================
The predominant focus in the exoplanet formation literature has been on major planets. However, white dwarf metal pollution highlights the need to also model minor planet formation. In fact, almost all observations of white dwarf planetary systems are of the remnants of minor planets.
By “minor planets”, we refer to an object with a mean radius of under 103 km. This size regime encompasses moons, comets, asteroids, and large fragments of major planets. In fact, the minor planet currently orbiting the white dwarf WD 1145+017 may best be classified as an “active asteroid” whereas the one orbiting SDSS J1228+1040 is perhaps best described as a ferrous chunk of a fragmented planetary core. The third, which orbits ZTF J0139+5245, still defies classification, until at least more data is obtained.
These distinctions are becoming increasingly relevant given our detailed knowledge of white dwarf planetary systems. A related question of interest is whether white dwarf metal pollution arises from primarily intact or destroyed minor planets, and the locations at which those bodies or debris were scattered. The answer depends more strongly on radiation than gravity through the Yarkovsky and YORP effects.
The Yarkovsky and YORP effects are recoil forces and torques produced due to non-uniform scattering of absorbed radiation. The Yarkovsky effect changes the orbit of a spherical or aspherical minor planet, whereas the YORP effects spins up or/and spins down an aspherical minor planet. Both effects have been modelled for solar system asteroids in exquisite detail. Contrastingly, only a handful of basic dedicated extrasolar post-main-sequence studies have been published for the Yarkovsky effect and the YORP effect ().
This contrast does not suit the added complexity introduced in giant branch systems, when the luminosity of the parent star changes on short timescales, and can be five orders of magnitude higher than the Sun’s (see Table [stelprop]). Such high luminosities easily fling exo-asteroids out to distances of tens, hundreds or even thousands of au, regardless of the presence of major planets, which only temporarily impede such migration. These minor planets could also be propelled inwards, and the direction of motion is a function of the bulk shape, spin, surface topography and internal homogeneities of the objects.
If, however, the minor planets veer too close to a giant branch star, then they could be spun up to disruption. This rotational fission occurs when the spin of a minor planet reaches the critical spin failure rate $\omega\_{\rm fail}$. For strengthless rubble piles in the solar system, this rate corresponds to a rotational period of about 2.3 hours and has robust confirmation through observations.
demonstrated that asteroid belts within about 7 au of their parent stars along the main sequence (including the solar system’s asteroid belt) would be easily fragmented by YORP-induced rotational fission along the giant branch phases. However, they considered parent stars with main sequence masses of 1 − 5*M*⊙.
Now we extend their study to higher stellar masses. The spin of a minor planet evolves according to
$$\frac{d\omega(t)}{dt}
=
\frac
{3 \mathcal{C} \Phi }
{4 \pi \rho R^2 a(t)^2\sqrt{1 - e(t)^2} }
\left(\frac{L\_{\star}(t)}{L\_{\odot}}\right)
\label{YORPsimp}$$
where *ρ* and *R* represent the minor planet’s density and initial radius (both assumed to remain constant), and *a* and *e* represent its time-dependent semimajor axis and eccentricity. C is a constant which represents the extent of the minor planet’s asphericity, and Φ = 1 × 1017 kg ⋅ m/s2 is the Solar radiation constant.
Equation ([YORPsimp]) helps illustrate that the increasing luminosity of the star competes against the increasing semimajor axis from stellar mass loss. As previously mentioned, for adiabatic mass loss, the eccentricity remains constant. We simply set it to zero here. An analytic expression of the failure rate which includes the minor planet’s tensile, uni-axial strength *S* is
$$\omega\_{\rm fail} = \sqrt{\frac{4 \pi G \rho}{3} + \frac{4S}{3 \rho R^2}}
.$$
We now compute simple estimates of $a\_{\rm break}$, the minimum initial (ZAMS) semimajor axis for which a minor planet will survive asymptotic giant branch YORP spin-up and reach the white dwarf phase. To do so, we integrate equation ([YORPsimp]) assuming *S* = 0, *e* = *e*(*t*) = 0 and an initially non-spinning minor planet. We also assume that the asteroid does not spin down at any point during its evolution, and its asphericity is encompassed entirely within the assumed constant value of C. We consider the YORP effect in isolation, decoupled from the Yarkovsky effect; such a coupling would require a significant modelling effort, and has not yet been achieved in exoplanetary science.
Figure [minorYORP] illustrates the result in $a\_{\rm break}$–*R* space, across a range of *R* (100 m – 10 km) for which the YORP effect is significant. Coincidentally, the curves for the *Z* = 0.02, 6*M*⊙ and 7*M*⊙ cases are visually indistinguishable on the plot, as are the curves for the *Z* = 0.02, 8*M*⊙ and *Z* = 0.0001, 6*M*⊙ cases. Changing the asphericity parameter by one order of magnitude noticeably shifts the curves.
[minorYORP]
In comparison to the survival capacity of major planets, we find that across almost the entire phase space, $a\_{\rm break} > d\_{\rm eng}$ and usually, $a\_{\rm break} \gg d\_{\rm eng}$. However, in individual cases, these relations do not prevent the major planet from, for example, residing exterior to intact minor planets. Nevertheless, the relations suggest that a surviving major planet which is surrounded by fragmented debris is not an unreasonable scenario.
An important related question is whether a minor planet would be sublimated from the host star before or during YORP spin-up. In order to estimate the sublimation rate of a minor planet, we use the approximation employed by ; for a more detailed description of its physical assumptions, see Section 6.1.1. of. The radius of a minor planet evolves according to
$$\frac{dR}{dt} = -\frac
{1.5 \times 10^{10} {\rm kg} \cdot {\rm m}^{-2} {\rm s}^{-1}}
{\rho}
\sqrt{\frac{T\_{\rm sub}}{T(t)}} \,
{\rm exp}\left(-\frac{T\_{\rm sub}}{T(t)} \right)
\label{subeq}$$
where, with the the sublimation temperature $T\_{\rm sub}$ and the Stefan-Boltzmann constant *σ*, the temperature of the minor planet *T* can be approximated by
$$T(t) = \left(
\frac
{L\_{\star}(t)}
{16 \pi \sigma r(t)^2}
\right)^{1/4}
.$$
The functional form of equation ([subeq]) reveals a strong dependence on the ratio $T\_{\rm sub} /T(t)$, and the value of $T\_{\rm sub}$ is composition-dependent. For olivine, $T\_{\rm sub} \approx 6.5 \times 10^4$ K. By adopting this value, along with *ρ* = 2 g/cm3, we find that the sublimation rate is negligible across almost the entire parameter space and can be ignored. For example, at the inner edge of this space, we find that for *r* = {20, 15, 10} au, the *maximum* sublimation rate, as measured by radius decrease, across all of our *Z* = 0.02 stellar evolution tracks are, respectively, {2 × 10− 11, 5 × 10− 8, 6 × 10− 4} m/yr. Instead, for the *Z* = 0.0001, *M* = 6*M*⊙ track, there is a greater danger at 10 au, with corresponding *maximum* sublimation values of {3 × 10− 8, 3 × 10− 5, 1 × 10− 1} m/yr.
Given that these minor planets largely survive sublimation, one may inquire about the size distribution of the debris which results from YORP-based break-up. Although this topic requires further dedicated study, explored this size distribution in a basic, analytical fashion, based on the results of. estimated the total number of fissioned components of an asteroid, as well as the sizes of the components, based on a number of factors including internal strength, initial spin and stellar luminosity. The result is hence dependent on many parameters and assumptions. The resulting dust produced may be blown away due to radiation pressure by comparing the interplay between radiation and gravity ; the presence of a major planet would affect this balance, and affects the regions where dust is ejected in a non-trivial fashion (Zotos & Veras, In Preparation).
Discussion
==========
Having computed the survival limits for both major and minor planets orbiting 6*M*⊙ − 8*M*⊙ stars, we now discuss topics which are related to these findings.
Ice line locations
------------------
Giant planets are commonly assumed to have formed beyond the ice line (or ``snow line“ or ``frost line”)[11](#fn11), where volatile compounds condense. For 1*M*⊙ stars, the ice line is often assumed to reside at about 2.7 au, exterior to $d\_{\rm eng}$.
However, as the host star mass increases, the relation between the ice line distance (denoted as $d\_{\rm ice}$) and $d\_{\rm eng}$ does not become immediately clear. In the ZAMS stellar mass range considered in this paper, $d\_{\rm eng}$ for giant planets is between about 5.5 and 6.5 au (Fig. [major]).
In these same systems, the location of the ice line is determined by a combination of stellar irradiation and viscous heating. However, discs around massive stars are dispersed rapidly, allowing the effect of irradiation to dominate. In this limit, we can approximate the midplane ice-line temperature $T\_{\rm ice}$ just as did by applying the following flared-disc prescription from, and :
$$T\_{\rm ice} = T\_{\star} \left(
\frac{\alpha}{2}
\right)^{1/4}
\left(
\frac{R\_{\star}}{d\_{\rm ice}}
\right)^{3/4}$$
with
$$\alpha =
0.005 \left(\frac{d\_{\rm ice}}{1 \ {\rm au}} \right)^{-1}
+
0.05 \left(\frac{d\_{\rm ice}}{1 \ {\rm au}} \right)^{2/7}
.$$
By setting $T\_{\rm ice} = 170$ K and ZAMS values for *R*⋆ and *T*⋆ from the aforementioned sse code, we find that $d\_{\rm ice} = 2.4-3.6$ au $\approx 0.5 d\_{\rm eng}$. Hence, even in this rough approximation, the engulfment distance along the giant branch phases is more restrictive than the ice line with regard to giant planet formation location.
The theoretical considerations in this subsection help motivate dedicated planet formation studies for high host-star mass systems. We now present observational motivation.
Future white dwarf host mass increases
--------------------------------------
*Gaia* is expected to discover, through astrometry, at least a dozen giant planets orbiting white dwarfs at the final data release. If realized, this exciting prediction would revolutionize our understanding of white dwarf planetary systems[12](#fn12), but is highly unlikely to increase the mass of the most massive known white dwarf host star.
Instead, the new ``record-breaker" is more likely to arise from the steep and imminent increase in known polluted white dwarfs. This increase will arise from the larger number of new white dwarfs which have already been identified from *Gaia*. *Gaia* Data Release 2 recently uncovered an all-sky sample of ≈ 260,000 white dwarf candidates that is homogeneous down to *G* < 20 mag. From this sample, on the order of 10,000 white dwarfs with atmospheric planetary remnants are suitable to low-resolution spectroscopic study in the near future with wide-area multi-object spectroscopic surveys such as WEAVE, 4MOST, DESI and SDSS-V.
The smallest white dwarf hosts
------------------------------
The most massive known white dwarf planetary system hosts would also represent the smallest known white dwarf planetary system hosts. The classic mass-radius relation for white dwarfs
$$\frac{R\_{\star}}{R\_{\odot}} \approx 0.0127 \left(\frac{M\_{\star}}{M\_{\odot}}\right)^{-1/3}
\sqrt{1-0.607\left(\frac{M\_{\star}}{M\_{\odot}}\right)^{4/3} }$$
yields, for the range of masses in Table [stelprop], host star radii between 0.0013*R*⊙ − 0.0063*R*⊙ ≈ 900 km − 4, 450 km $\approx 0.26R\_{\rm Mars}-1.31R\_{\rm Mars}$. In contrast, a typical 0.6*M*⊙ white dwarf gives a radius of about $2.6R\_{\rm Mars}$.
How do the much smaller radii of the systems highlighted here affect the dynamical origin of metal pollution? The Roche radii of white dwarfs scale as *M*⋆1/3 regardless of the composition or spin of orbiting companions. Therefore, doubling the white dwarf mass increases the Roche radii by about 25 per cent. This level of increase is significant, and can for example, trigger thermal self-disruption of passing gaseous planets which otherwise would be safe. The larger Roche radius also provides a larger target for minor planets, which, when combined with the location requirements for major planets in these systems, places restrictions on where and when scattering occurs.
Another consequence of decreasing the size of a white dwarf and increasing its Roche radius is the resulting change in structure of a debris disc formed from the disruption of a minor planet. The outer edge would be larger than those of the currently known discs, increasing the probability of capture of solid objects ; the consequences for the inner edge is less clear because of our lack of observations between the inner disc rim and the white dwarf photosphere. The resulting disc evolution and accretion rates onto the white dwarf might also differ from what has been previously modelled.
Other features of 6*M*⊙ − 8*M*⊙ planetary systems
-------------------------------------------------
Major planets which form and reach distances of several to many au in under 70 Myr around 6*M*⊙ − 8*M*⊙ stars may help identify the dominant planet formation mechanisms in these systems. These planets formed in discs which are subject to time-varying radiative forcing at a more extreme level than their lower-host mass counterparts. When combined with photoevaporative effects from other stars in their birth cluster, outward migration of planets within these discs may be enhanced.
If the 6*M*⊙ − 8*M*⊙ progenitor host stars for polluted white dwarfs were slightly more massive, then they would have instead become neutron stars; planets are known to exist around single pulsars. These pulsar planets most likely represent products of second-generation planet formation from mergers or fallback although questions still remain [13](#fn13). Exo-asteroids may orbit pulsars below the detection threshold, although that threshold already lies at about 1 per cent of the mass of the Moon.
In contrast, both major and minor planets orbiting single white dwarfs are first-generation, because the mass in the white dwarf debris discs required to form new planets is at the level of an Io mass or higher. In contrast, the masses of white dwarfs with observed debris discs remain uncertain but are assumed to be generated instead by asteroids (with a typical mass two orders of magnitude less than Io). Further, the chemistry of any second-generation planets would be recycled from broken-up first generation bodies; Veras, Karakas & Gänsicke (submitted) demonstrate definitively that white dwarf pollution does not arise from stellar fallback.
Summary
=======
The most massive currently known progenitor of a white dwarf planetary system host ( ≈ 4*M*⊙) exceeds the maximum mass of a main-sequence major planet host ( ≈ 3*M*⊙). Because this mass difference is likely to increase with white dwarf population growth (Section 5.2), we have investigated major and minor planet survival limits for the highest possible stellar masses ( ≈ 6*M*⊙ − 8*M*⊙) which yield white dwarfs. This mass regime represents almost completely unexplored territory for planet formation theorists, and largely unexplored territory for post-main-sequence planetary evolution investigators.
We found that a major planet needs to reside beyond 3-6 au (depending on the type of planet) at the end of the short (40-70 Myr) main-sequence phase in order to survive its host star transition into a white dwarf. Minor planets could have remained intact or broken up due to YORP-induced giant branch radiation, having been subject to host star luminosities reaching up to 105*L*⊙. The boundary separating these two possibilities ( ∼ 101 − 103 au) is very sensitive to minor planet size, but is high enough to suggest that metal pollution in these systems would originate from already fragmented debris rather than intact minor planets which fragment upon reaching the white dwarf Roche radius. Our study motivates dedicated planet formation studies around the highest mass stars, and predicts that metal pollution can occur in the highest mass white dwarfs.
Acknowledgements
================
We thank the expert reviewer Fred C. Adams for helpful comments which have improved the manuscript. DV gratefully acknowledges the support of the STFC via an Ernest Rutherford Fellowship (grant ST/P003850/1). The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme no. 677706 (WD3D). Support for this work was also provided by NASA through grant number HST-GO-15073.005-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. FM acknowledges support from the Royal Society Dorothy Hodgkin Fellowship. GMK is supported by the Royal Society as a Royal Society University Research Fellow. BTG is supported by the UK Science and Technology Facilities Council grant ST/P000495.
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[lastpage]
---
1. E-mail: [email protected][↩](#fnref1)
2. STFC Ernest Rutherford Fellow[↩](#fnref2)
3. Royal Society University Research Fellow[↩](#fnref3)
4. Royal Society Dorothy Hodgkin Fellow[↩](#fnref4)
5. reported a so far unconfirmed candidate planet orbiting an asymptotic giant branch star.[↩](#fnref5)
6. considered planet formation around supermassive black holes which are just a few parsecs away from active galactic nuclei. However, the prospects of detecting planetary systems outside of the Milky Way Galaxy are currently remote, except perhaps for the Large Magellanic Cloud (which probably does not contain a central black hole; ).[↩](#fnref6)
7. have recently suggested that 20 ± 6 per cent of high-mass white dwarfs (1.08*M*⊙ − 1.23*M*⊙) originate from double white dwarf mergers. In addition, another significant fraction are likely merger byproducts merged when at least one of the stellar components was not a white dwarf.[↩](#fnref7)
8. However, see for robust evidence of a major ice giant planet orbiting a white dwarf whose progenitor mass was 1.0*M*⊙ − 1.6*M*⊙; has suggested that this giant planet is a partially destroyed and highly inflated ``Super-Puff".[↩](#fnref8)
9. The lower end of the range corresponds to warm hydrogen-atmosphere (DA class) white dwarfs.[↩](#fnref9)
10. More massive progenitor stars should have higher metallicities because they must be younger if they are evolved from single stars.[↩](#fnref10)
11. See for a challenge to the canonical theory.[↩](#fnref11)
12. Some current observational limits have been placed on the possible locations of major planets orbiting white dwarfs. Null results in a deep *Spitzer* survey can rule out at least > 10Jupiter-mass planets within the 5-50 au range around more than 40 white dwarfs. Monitoring the stable pulsation arrival times of at least 6 white dwarfs can exclude > 3Jupiter-mass planets in a range of roughly 2 − 5au. Unfortunately none of the most massive pulsating white dwarfs have stable, coherent modes that can be used to monitor for substellar companions.[↩](#fnref12)
13. The sudden stellar mass loss experienced in a supernova is typically insufficient to eject a planet on a compact orbit. The more pertinent but outstanding question of first-generation planet survival is whether the planet can physically withstand a supernova’s stellar ejecta.[↩](#fnref13)
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in 12CO. By assuming the LTE conditions, optically thin emission, a rotational temperature of 80 K, and an abundance ratio of [SiO]/[H2] = 10− 7, Zapata et al. (2006) estimated the total mass, momentum, and energy in those SiO outflows to be 1.155 M⊙, 57.85 M⊙kms− 1, and 3.03 ⋅ 1046 ergs. However, as suggested by the authors, at least a factor of 50 enhancement of the SiO abundance with respect to the adopted value 10− 7 is required to explain the non–detection of 12CO, implying that the real outflow mass, momentum, and energy can be a factor of 50 lower than the estimated values. If such kind of SiO outflow systems are rare either because of the short life time in the specific phase of weak 12CO emission, or because of their formation requires unusual physical conditions, their feedback may not be significant as compared with the total outflow feedback estimated from the 12CO emission. With small size scale, the induced protostellar turbulence also dissipates faster.
The Diagnostics of Interaction Signatures
-----------------------------------------
While the interaction and shock signatures typically have the velocity of a few kms− 1 from the systematic velocity, in massive cluster forming regions, the CO isotopologues emission is extremely complicated in such velocity range, and is hard to be robustly imaged in interferometric observations. The diagnostics of the interaction signatures are therefore relying on the observations of various tracers. In the following sections, we introduce the observed features in CH3OH, HCN, and SiO, as the diagnostics in G10.6-0.4. The interaction signatures identified from the CH3OH emissions are cross compared with the velocity dispersion map of the 12CO (2-1) emission in the intermediate velocity range, to demonstrate the robustness of the diagnostics. In addition, we compare the identified interaction signatures with the 1.3 mm continuum image, which is regraded as a reliable tracer of dense molecular cores.
### The Broad CH3OH Emission
#### G10.6-0.4
By comparing the velocity integrated 12CO emission with the velocity integrated CH3OH emission, we see the association of the molecular outflows with the dense envelope. With the terminal velocity of a few tens of kilometer per second, these molecular outflows can strongly shock the ambient dense gas in the envelope. A variety of molecules are then released from the dust grain, and their abundances in gas phase can be dramatically enhanced. We diagnose the interactions between the molecular outflows and the ambient gas from the broad CH3OH 5(0,5)-4(0,4) A+ emissions. The CH3OH molecule is considered to be one of the early–type molecules, and is abundant in the dense protostellar core. In addition, its abundance can be further shock enhanced by one or two orders of magnitude. The selected transition has the lowest upper–level–energy (34.65 K) among all observed CH3OH lines, so that the excitation is least sensitive to the stellar heating.
Figure [figch3ohpv] shows a sample position-velocity (pv) diagram of the CH3OH 5(0,5)-4(0,4) A+ transition. From this figure we see two dominant signatures with distinguishable characteristics: the dense rotating envelope and the broad interaction signature. The dense envelope covers the angular scale of ∼ 20ʺ, and is characterized by high brightness and steep increase of brightness. The interaction signature, as indicated by the red arrow, is faint and is spatially localized, but covers a much larger velocity range. This broad and faint emission can be understood as the outflow wing component in the spectrum. More pv diagrams of the interaction signatures are discussed in Appendix [chapoutflowpv]. Qualitatively, one can observe that, in the same range of angular offset, the envelope and the interaction signature roughly occupy the same area (in unit of arcsecond × kms− 1) in the pv diagram, which is naturally explained by their difference in linewidth. However, the envelope is sketched by significantly more contours, which means that the area between two contour levels is much smaller.
The difference between the dense envelope and the interaction signature leads to a large contrast if we compare the velocity integrated flux between two significant contour levels, for example, the 3rd *σ* and the 6th *σ* significance levels (*f*36 hereafter). Between these two contour levels, the interaction signatures contribute to significantly more *f*36 than the dense envelope, although having lower total integrated flux. This property can be utilized to indicate the spatial distribution of the interaction signatures. Figure [figch3ohoutflow0] shows the *f*36 map of the CH3OH 5(0,5)-4(0,4) A+ transition together with the locations of the water maser sources and the locations of the 1.3 cm free-free emission peaks.
From Figure [figch3ohoutflow0] we see that this analysis is quite successful. The 0.1 pc scale hot toroid (Liu et al. 2010 a,b) which has the highest brightness temperature, cannot be seen in this map. Instead, we see significant local components (indicated by boxes) closely associated with the compact 12CO emission (for example, B2, R1, R3 outflows, and B6 outflow), which are the most likely sources to induce the interaction signatures. We note that the CH3OH emission is only significantly detected in the velocity range of -13 kms− 1 < v*l**s**r* < 10 kms− 1 while the blueshifted and the redshifted 12CO emissions are defined outside this velocity range. The consistency in their spatial distributions also indicates that the broad CH3OH emission is indeed related to the high velocity molecular outflows. We also evaluate the velocity dispersion of the emission between the 3rd *σ* and the 6th *σ* significance levels (Figure [figch3ohoutflow2]). The result consistently shows large velocity dispersion closely associated with the interaction signatures. Both Figure [figch3ohoutflow0] and Figure [figch3ohoutflow2] show that the interaction signature around the B6 outflow has a elongated distribution parallel to the global rotational axis of the dense envelope (Liu et al. 2010 a,b), and may be associated with an UC Hii region, indicated by a star symbol (Liu et al. 2010 b). Whether that signature is related to the bipolar molecular outflow or is pushed by the bipolar expansion of the ionized gas can be studied in future high resolution observations. The interaction signature indicated by the orange box corresponds to the interaction signature shown in the sample pv diagram (Figure [figch3ohpv]). The corresponding 12CO emission may be in the intermediate velocity range, and as shown in Figure [figch3ohoutflow0], is elongated in the southeast–northwest direction. It can also be visually identified in the channel map (Figure [figcochan2]) as compact components in the velocity range of 3.5–8.0 kms− 1.
The f36 analysis, especially, the velocity dispersion map (Figure 6) unveils a population of interaction signatures, which lie close to the plane of the densest flattened molecular gas (Liu et al. 2010 a, b). Those interaction signatures show locality and relatively low outflow velocities. We suggest that they are less likely due to the interaction of the wind emanated from the OB cluster embedded in the hot toroid, but rather associated with local sites of star formation. This result provides important indication that local star formation plays important roles in the feedback of kinetic energy to the dense molecular gas.
#### Cross Comparison with 12CO Emission in the Intermediate Velocity Range
In this section, we cross compare the broad CH3OH 5(0,5)-4(0,4) A+ emission (Figure [figch3ohoutflow0], [figch3ohoutflow2]) with the broad 12CO emission in the intermediate velocity range (Figure [figcomnt012]). From the velocity dispersion map of the intermediate velocity 12CO emission (Figure [figcomnt012], right), one can see two extended broad line regions, which are associated with two UC Hii regions (Hii–M: RA=18*h*10*m*28.683*s*, Decl= -19*o*55ʹ49ʺ.07; Hii–NW: RA=18*h*10*m*28.215*s*, Decl=-19*o*55ʹ44ʺ.07), respectively. The physical size scales of these two broad emission regions are both a fraction of a parsec.
The distribution of the broad 12CO emission around Hii–NW agrees well with the interaction signature traced by the *f*36 of the CH3OH 5(0,5)-4(0,4) A+ transition (Figure [figch3ohoutflow0]). The morphology can be explained by the expansional motion of the molecular/ionized gas confined by the dense gas, which has a flattened distribution in the global plane of rotation (Liu et al. 2010 a b). The broad 12CO emission around Hii–M does not have CH3OH 5(0,5)-4(0,4) A+ emission counter part, which can be explained by the lower optical depth around this specific region.
In addition to those two extended broad 12CO emission regions, we visually identify five very compact (1ʺ–2ʺ) broad 12CO emission signatures, for which the mean velocities also have significant contrasts with the ambient gas. Those regions are marked by circles in Figure [figcomnt012], and are also marked in the left most panel of Figure [figch3ohoutflow0]. The central coordinates of these five circles are: (#1) RA=18*h*10*m*29.258*s*, Decl= -19*o*55ʹ42ʺ.18; (#2) RA=18*h*10*m*29.088*s*, Decl= -19*o*55ʹ46ʺ.28; (#3) RA=18*h*10*m*29.059*s*, Decl= -19*o*55ʹ52ʺ.78; (#4) RA=18*h*10*m*29.349*s*, Decl= -19*o*55ʹ57ʺ.18; (#5) RA=18*h*10*m*28.868*s*, Decl= -19*o*55ʹ56ʺ.58, respectively (see also the channel maps in Figure [figcochan2]). Two of them (#1, #2) are closely associated with water maser sources. The east most one (#4) is associated with a *f*36 peak of the CH3OH 5(0,5)-4(0,4) A+ transition, which is marked by the yellow box in Figure [figch3ohoutflow0]. A faint *f*36 peak of the CH3OH 5(0,5)-4(0,4) A+ transition is detected around #3; and similarly with #5. These five signatures marked by circles look compact even from the 12CO (2-1) transition, which is very easy to be excited. It provides the indication that they are associated with active local dynamics. Since their mean velocities are offset from the systemic velocity of their ambient gas by a few kms− 1, we hypothesize that those signatures are the molecular gas in the local dense cores pushed by the protostellar outflows. Without the comparison with results from the outflow tracers, it is extremely difficult to recognize these interaction signatures from the 12CO (2-1) maps, and it is even more difficult to robustly interpret them when they are recognized.
We note that the missing flux and the high optical depth of the foreground gas can artificially enhance the measured velocity dispersion. However, the broad 12CO emission regions mentioned in this section have much smaller angular size scales ( < 20ʺ) than the maximum detectable angular scale of our SMA subcompact–array data ( ∼ 40ʺ), and should not be severely affected by missing flux. If the missing flux artificially enhances the measured velocity dispersion at large scale, the contrast between the measured velocity dispersion of the broad 12CO emission regions and the measured velocity dispersion of the envelope is reduced. The contrast in velocity dispersion should be higher in reality, which means the identified broad emission signatures are robust. Therefore we think the missing flux does not have a significant impact on our discussions in this section. The high optical depth of the foreground gas trims structures for all angular size scales, however, only for the emissions redder than the systemic velocity of -3 kms− 1. The mean velocities in those extended and compact broad 12CO emission regions are apparently redder than the mean velocity of the ambient gas. If the high velocity dispersion is an artifact caused by the high optical depth of the foreground gas, we expect to detect a bluer mean velocity.
#### Compare With 1.3 mm Continuum Emissions
In massive cluster forming regions, flux in 1.3 mm continuum emission is dominantly contributed by the thermal dust emission, and the free–free continuum emission from UC Hii regions (Keto, Zhang, & Kurtz 2008). To see the relation between those broad 12CO interaction signatures with the dense molecular gas, we mark the locations of those broad 12CO interaction signatures (as well as the locations of the 22 GHz water masers) on the 1.3 mm continuum image (Figure [fig1p3mm]).
From the 1.3 mm continuum image, we first see an abrupt increase of the 1.3 mm continuum flux in the middle, which suggests the distinct origins of the flux. We identify two UC Hii regions from high resolution centimeter continuum images (Keto, Ho & Haschick 1988; Guilloteau et al. 1988; Sollins et al. 2005; Liu et al. 2010abc), and mark their locations on the 1.3 mm continuum image. Those centimeter continuum emissions indicate that in the middle of the 1.3 mm continuum map, the free–free continuum emission from the UC Hii region contribute significantly. In the extended region, the 1.3 mm continuum emission is dominantly contributed by the thermal dust emission, except for a fainter UC Hii region in the northwest.
The thermal dust emission unveils abundant dense cores over a 0.5 pc region. Three dusty dense cores ( RA:18*h*10*m*29*s*.064, Decl: -19*o*55ʹ46ʺ.2; RA:18*h*10*m*29*s*.035, Decl:-19*o*55ʹ49ʺ.7; RA:18*h*10*m*28*s*.975, Decl: -19*o*55ʹ52ʺ.5) are closely associated with the identified broad interaction signatures of 12CO outflows. The core at the northeast (RA:18*h*10*m*29*s*.064, Decl: -19*o*55ʹ46ʺ.2) is further associated with a water maser source (Hofner & Churchwell 1996). The associations with water maser sources and outflow signatures indicates that those dense cores are actively forming stars. This core shows a complicated geometry, which may be explained by hierarchical fragmentations. The dense core at RA= 18*h*10*m*28*s*.975 and Decl=-19*o*55ʹ52ʺ.5 is apparently elongated, and has internal structures, which may be explained by fragmentation, or can be observationally caused by blending.
#### Compare With Low–Mass Protostellar Outflows
From the observations of the CH3OH J=2 transitions with upper–level–energy E*u* of 4.64–12.2 K, Takakuwa, Ohashi & Hirano (2003) suggests that in the Class 0 low mass protostar IRAM 04191+1522, the enhanced broad CH3OH emission trace the interactions between the protostellar outflow and its parent molecular core. Similar to what is observed in G10.6-0.4, the pv diagram of CH3OH in IRAM 04191+1522 shows the distinguishable interaction signature with the envelope component. The interaction signatures in IRAM 04191+1522 have the size scale of ∼ 0.03 pc, closely associated with the protostellar envelope. Observations in other nearby protostellar outflows (Bachiller et al. 1995, 1998, and 2001; Garay et al. 1998) show the localized enhancement of the CH3OH abundance in the strongly shocked regions around 0.1 pc from the ejecting sources. This 0.03–0.1 pc size scale correspond to the angular scale of 1ʺ–3ʺ at the distance of G10.6-0.4 (6 kpc), which is marginally resolved in our observations (Table [tableparameters]); and this scale is small as compared with the size scale of the massive envelope traced by CH3OH (20ʺ–30ʺ).
Supposedly these observed phenomena in nearby clouds and their interpretations can be extrapolated to the distant massive dense envelopes with higher temperature and high density. Then each of the detected broad CH3OH interaction signature may be associated with an independent ejecting protostar. The distribution of the observed broad CH3OH emission features in G10.6-0.4 suggests the abundant star formation activities over the entire ∼ 0.5 pc scale dense envelope. In G10.6-0.4, we already detected three UC Hii regions, indicating that multiple massive stars have already been formed. It is not surprising if there are many massive or low–mass (proto–)stars, which are still in accretion phase and are ejecting the molecular outflows. We expect higher resolution observations in lower excitation CH3OH to reveal more protostellar objects.
Note that the detection of the thermal radio jets is regarded as one of the most robust methods to identify protostellar objects in the accretion phase. However, in UC Hii regions, the free-free emission around the embedded OB cluster has the flux of a few Jy, which is of 3–4 orders of magnitude brighter than the typically flux of the thermal radio jets. The bright free-free emission associated with the OB cluster ionization can have complicated spatial distribution and geometry, which will confuse the detection of the thermal radio jets. The shock enhanced molecular emission potentially provides good complementary information in the diagnostic of the star forming activities. From Figure [figch3ohoutflow0], we see that a population of protostellar and stellar activities (four CH3OH outflow interaction signatures as indicated by three boxes and one white circle, and two UC Hii regions as indicated by stars) seem to have a coplanar distribution, suggesting a scenario of enhanced star formation in the rotationally flattened dense envelope. This scenario is supported by recent numerical hydrodynamical simulations (Peters et al. 2010).
Some protostellar and stellar activities seem to follow the filaments detected in the intermediate velocity 12CO emissions. Owing to the projection effect, we cannot robustly distinguish whether those activities are just distributed in the bulk of the envelope.
### The *HCN Outflow*
The *HCN outflow* is characterized by the significantly higher HCN (3-2) flux. Among all outflow systems detected by 12CO, the *HCN outflow* is also the unique case where we detect the SiO (8-7) emission. The deconvolved size scale of the 12CO emission is about 2ʺ, and its location is close to the central 0.1 pc scale hot toroid. Figure [fighcn] shows the velocity integrated maps of HCN (3-2), SiO (8-7), and the blueshifted 12CO, in the top and the middle panels, and shows the 3.6 cm continuum image in the bottom panel. From this figure we see the locally enhanced HCN emission, for which the location and geometry agree excellently with those of its 12CO counter part. The distribution of the SiO emission is consistent with the 12CO and HCN emission. Subjected to the non–uniform uv coverage, the SiO (8-7) data have an elongated synthesized beam, and therefore the projected geometry of the emission and the peak location are poorly constrained.
In Figure [fighcnpv], we present the pv diagrams of HCN (3-2), SiO (8-7) and 12CO (2-1) around *HCN outflow*, which are all cut in the RA direction. From both panels in this figure, we can clearly see the broad outflow signatures around the angular offset of ∼ -5ʺ. The HCN (3-2) emission and the 12CO emission consistently trace the outflow to the velocity of ∼ -25 kms− 1. Owing to the detection limit, the SiO pv diagram only traces the outflow in a limited velocity range. From the pv diagram, we see that SiO is also excited in the hot toroid, which is located 2ʺ–3ʺ east of the *HCN outflow*. Assuming the LTE conditions, optically thin emission, and the excitation temperature of 100 K, we estimate the total number of the SiO molecule in *HCN outflow* to be ∼ 1.5 ⋅ 1048.
We apply the same assumptions to derive the total number of the HCN and the 12CO molecules in the *HCN outflow*. Adopting the galactic 12CO abundance (i.e. [12CO]/[H2] = 1 ⋅ 10− 4), we list the derived number and the implied abundance ratios in Table 4. In the galactic molecular clouds, the [HCN]/[12CO] ratio ranges from 10− 4 in the quiescent zone to 2.5 ⋅ 10− 3 in the hot core regions (Blake et al. 1987). The enhanced [HCN]/[H2] ratio has also been reported in other (proto–)stellar outflows (e.g. IRAS 20126: 0.1–0.2 ⋅ 10− 7, Su et al. 2007; L1157: 5 ⋅ 10− 7, Jørgensen et al. 2004). In the *HCN outflow*, the value of [HCN]/[12CO] in the blueshifted velocity range is consistent with an enhanced HCN abundance. If our assumption of [12CO]/[H2] ratio is valid, the value of [HCN]/[H2] in the *HCN outflow* is in between of the two referenced cases. Even if the assumed [12CO]/[H2] ratio is overestimated by a factor of 10 (see the discussions in Section 3.2), the derived [HCN]/[H2] ratio in the *HCN outflow* is still higher than the reported typical value of ≤ 7 ⋅ 10− 9 (Jørgensen et al. 2004). The HCN molecule is usually regarded as a dense gas tracer. Our results suggest, however, that the HCN emission can also be locally enhanced in outflows and shocks around the dense core, and therefore might not be a good tracer of the overall dynamics in massive cluster forming regions.
Among all detected high velocity outflows, the *HCN outflow* does not have specifically higher linear momentum or energy. The uniqueness in the excitation of SiO (8-7) may be explained by the heating of the central OB cluster, or be explained by the molecular gas erupted from the hot toroid with shock enhanced SiO abundance. Some hints can be seen from the 3.6 cm continuum image (Figure [fighcn], bottom panel), which shows an elongated emission feature around the location of the *HCN outflow*. Whether the elongated 3.6 cm emission feature is physically associated with *HCN outflow* or is just a projection effect can be examined by future ALMA observations of molecular lines and the hydrogen recombination lines.
We note that a few more elongated/filamentary features[6](#fn6) are resolved in the previous 1.3 cm free–free continuum observation (Sollins et al. 2005). Those elongated/filamentary features have the lengths of ∼ 1ʺ (0.03 pc), and width of ≤ 0.1ʺ (0.003 pc). The physical properties and the formation mechanism of those elongated features are still uncertain. Suppose those elongated/filamentary features of 1.3 cm free–free continuum emission are ionized gas which are undergoing thermal diffusion with velocity of 10 kms− 1, the ≤ 0.003 pc widths imply that their age is no more than 300 years. If all those elongated/filamentary features can also be explained by ionized outflows, the velocities of the outflows can be estimated by
$$\frac{\mbox{length}}{\mbox{age}} = \frac{0.03\mbox{ pc}}{300\mbox{ years}} \sim 100 \mbox{ kms$^{-1}$}.$$
This outflow velocity is consistent with the measurement in hydrogen recombination line (line–of–sight velocity 60 kms− 1, Keto & Wood 2006), although the ionized outflow is not explicitly resolved in the image. The higher brightness of those elongated/filamentary features than the ambient regions may suggest that those outflows are originally ejected in molecular form with a much higher density, and then ionized by the stellar radiation. The molecular counterparts of those ionized outflows are not necessarily detected, especially in the bipolar direction, if stellar ionization is important. We suggest that the central OB cluster may accrete the molecular gas of very compact/clumpy structures (see the NH3 opacity results, Sollins & Ho 2005), which can survive the stellar ionization. The molecular gas can then approach individual O–type stars to the distance of ∼ 10 AU, and be centrifugally accelerated to the outflow velocity. Small scale accretion disks may exist around the O–type stars. However, they are not detectable with the current instruments owing to the beam dilution.
Discussions
===========
In G10.6-0.4, previously we estimated the scalar momentum feedback from the stellar wind and from the ionized gas pressure to be of 102–103 M⊙kms− 1 (Liu et al. 2010 b). From the 12CO data presented in this paper, we conclude that the total momentum feedback from the protostellar outflows has the same order of magnitude. However, these feedback mechanisms can still play very different roles in the massive cluster forming regions, owing to their differences in physical properties.
The stellar wind is an isotropic feedback mechanism. By comparing the resulting star formation efficiency in numerical hydrodynamical simulations, Nakamura & Li (2007) suggested that the *spherical wind* cannot propagate efficiently in dense gas. The feedback from such mechanisms is trapped in small size scale, where the induced turbulence is quickly dissipated. Therefore the energy feedback from the *spherical wind* supports the cloud less efficiently. The protostellar outflow is ejected from objects embedded in the local overdensities. Its momentum, however, is typically collimated in a small solid angle, which makes it penetrate the dense gas easily and can produce large scale disturbance. The induced protostellar turbulence potentially replenishes the dissipated initial turbulence in large scale, and plays the role of regulating the cloud contraction and the star formation efficiency.
From the numerical simulation, Mac Low (1999) suggested the characteristic timescale of turbulence dissipation *t**d* and the free–fall timescale *t**f**f* have the relation:
$$t\_{d} = \left(\frac{3.9\lambda\_{d}}{M\_{rms}\lambda\_{J}}\right)t\_{ff},$$
where *λ**d* is the driving scale of the turbulence, *λ**J* is the Jeans length, and *M**r**m**s* is the rms Mach number of the turbulence. Assuming a total mass of 1000–2000 M⊙ is enclosed in the 0.5 pc scale envelope, the global free–fall timescale is about 105 years. If the averaged gas temperature is 30–50 K, the Jeans length is about 0.1 pc. The typical value of *M**r**m**s* can be 5–10[7](#fn7). Assuming the main driving source of the turbulence is the compact (proto–)stellar outflows, the value of *λ**d* can be estimated by the physical size scale of the observed outflow systems, which is about 0.06 pc (2ʺ). Therefore, the characteristic timescale *t**d* in G10.6-0.4 is about 0.23–0.47 times of t*f**f*, which is about a few times of 104 years. Among the outflow systems detected in G10.6-0.4, the maximum outflow dynamical timescale is about 104 years, which is smaller than *t**d*. The estimations of outflow energy in Section [chapestimation] suggests that the energy injection from the protostellar turbulence is capable of balancing the turbulence energy dissipation in the relevant timescale.
We provide rough estimations for the total protostellar mass from a fiducial momentum ejection efficiency I*t**o**t**a**l* = P\*M\*, where I*t**o**t**a**l* is the total ejected scalar momentum, M\* is the total stellar mass, and P\* is a proportional factor. Following Nakamura & Li (2007), we adopt P\*= 50 kms− 1. In G10.6-0.4, the projected scalar momentum in the blueshifted and the redshifted velocity ranges is 273 M⊙kms− 1, which leads to *f**l**o**s* ⋅ M\* = 5.45 M⊙, where *f**l**o**s* is the ratio of the scalar momentum contributed by the line–of–sight velocity component to the total scalar momentum. Statistically the value of *f**l**o**s* can be estimated by
$$\frac{\int\_{0}^{\frac{\pi}{2}} \sin\theta d\theta d\phi}{\int\_{0}^{\frac{\pi}{2}} 1 d\theta d\phi} \simeq 0.637,$$
assuming the inclination angle of the outflows are uniformly randomly distributed. Given the 1000–2000 M⊙ total molecular mass in the envelope, the star formation efficiency (SFE) in the past 104 years is about 0.42%–0.86%. Assuming a uniform star formation rate in the time domain, the star formation efficiency in one free–fall timescale is of the order of a few percent. This estimated SFE is comparable to the SFE in the simulations of Nakamura & Li (2007), consistently suggesting the cloud contraction is self–regulated by the local star formation. High resolution observations of the thermal dust emission to statistically study the prestellar and protostellar cores, will further improve the constraint on the efficiency of the outflow feedback. The number of the YSOs can be statistically estimated by dividing M\* with the averaged protostellar mass *m̄*. The fiducial value of *m̄* based on the Scalo (1986) initial mass function (IMF) is 0.5 M⊙. In G10.6-0.4, the value of *m̄* can be biased by the evolutionary stage, and potentially the observational selection bias that we only pick up the systems with powerful molecular outflows. Assuming the detected outflows are associated with YSOs, which on average have ∼ 1 M⊙ stellar mass, the number of the YSOs can be estimated by M\*/1.0 = 8.56. The surface density of these YSOs in the ∼ 0.5 pc region is therefore ∼ 8.56/(0.52) ∼ 34 pc− 2. The derived YSO surface density should be treated as a lower limit since our observations do not trace protostellar objects which have weak or no outflows.
Alternatively, we can assume each of the compact high velocity outflows (R1⋯4, B1⋯6, *H**C**N*) is associated with one protostellar objects. Considering also the non–identified outflow systems in the intermediate velocity range, there can be ∼ 10 protostellar objects in total in the observed region. If estimating the averaged mass of each protostellar object by 1 M⊙, our observational results suggest that an order of ∼ 10 M⊙ molecular gas are converted into (proto–)stellar objects in ∼ 104 years. The system potentially evolves into a stellar cluster with on order of 102 stars in a few t*f**f*.
The ionized gas only propagates in low density regions, which have low recombination rates. However, our previous studies (Liu et al. 2010 b) suggest that the thermal pressure of the ionized gas is capable of driving the large scale (a few times of 0.1 pc) coherent motions, such as the expansional motions of the ionized cavities or the bubble walls. The absolute velocity of such kind of motion is about a few kms− 1, and which is smaller than the thermal sound speed of the ionized gas ( ∼ 10 kms− 1). The efficiency of converting the kinetic energy in such expansional motion to the large scale turbulence energy has to be investigated in theoretical studies.
Conclusion
==========
We present the arcsecond resolution interferometry observations of the 12CO (2-1) transition. From the distribution and the mass/momentum/energy of the detected high velocity outflows, we discuss the role of the outflow feedback in the massive cluster formation region. Our main results are:
* We detect multiple high velocity outflow components in the blueshifted and the redshifted velocity ranges, respectively. The total molecular mass in these outflow systems are about 10 M⊙, and the total momentum and energy budgets are of the order of 102 M⊙kms− 1 and 1047 erg, respectively.
* No high velocity bipolar molecular outflow parallel to the global rotational axis is directly detected around the central 0.1 pc hot toroid. However, we cannot rule out the possibility that it is photo–ionized.
* The 12CO outflows are interacting with the dense molecular envelope, which can be diagnosed by the CH3OH, HCN, and the SiO shock signatures.
* From the distribution of the UC Hii regions, we know that multiple massive stars have been formed in the 0.5 pc scale massive envelope, and not only formed in the central 0.1 pc hot toroid. From the association of high velocity outflows with water maser sources, and with the interaction signatures detected in thermal molecular emission, we suggest that multiple protostellar objects with ∼ 10 M⊙ in total may also have been formed within 104 years. The star formation efficiency over one global free–fall timescale is of the order of a few percent.
* We detect strongly enhanced HCN (3-2) emission in a outflow system near the hot toroid, and therefore suggest that HCN is not a good tracer for the global dynamics.
We will follow up the polarization observation to provide thorough discussions of the energetic relation in this source. We emphasize that the discussions and conclusions in this paper are relevant to the cluster forming regions with size scale of a parsec or a fraction of a parsec, and with mass of thousands of M⊙. In less massive systems (with lower opacity), the radiative hydrodynamical simulations suggest that the heating by the stellar radiation efficiently suppress the star formation (Offner et al. 2009).
Baobab Liu thanks the SMA staff to support the observations. *Facilities:*
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A. The Channel Maps of the 12CO (2-1) and the HCN (3-2) Lines
=============================================================
We present the channel maps of the 12CO (2-1) line and the HCN (3-2) lines in this section. Figure [figcochan1], [figcochan2], [figcochan3], and [figcochan4] show the 12CO (2-1) line in the velocity channels of -131.5 kms− 1– 77 kms− 1. Contour levels are adjusted according to the rms noise. We think the channel maps in the redshifted velocity channels (9.5 kms− 1–45.5 kms− 1) are contaminated by the contribution of the foreground molecular gas, and therefore show subparsec scale diffuse emission, which are not seen in the channel maps in the blueshifted velocity range ( -124.75– -15.25 kms− 1). The absorption line of 12CO (2-1) is also detected at the location of the free-free continuum peak (ra:18:10:28.683 dec:-19:55:49.07), up to *v**l**s**r* ∼ 45 kms− 1. The HI absorption experiment (Caswell et al. 1975) consistently shows the foreground absorption up to *v**l**s**r* ∼ 45 kms− 1.
Figure [fighcnchan1] show the channel maps of the HCN (3-2) line. We note the high consistency between the HCN and the 12CO channel maps. In the HCN (3-2) channel maps, the low velocity counterpart of outflow R3 (see Section [chapname]) may be visually identified in the velocity range of 7.5 kms− 1–12 kms− 1; and the low velocity counterparts of outflow B3 and B5 may be visually identified in the velocity range of -10.5 kms− 1– -15 kms− 1; the *HCN outflow* can be seen in the same velocity range as covered by its 12CO emission
B. The Position-Velocity Diagrams of the Interaction Signatures
===============================================================
Figure [figch3ohoutflowpv] shows the pv diagrams of the CH3OH 5(0,5)-4(0,4) A+ transition, at the locations of three broad CH3OH interaction signatures, and at the location of two water maser sources. Those CH3OH interaction signatures are visually identified from the *f*36 maps (Section [chapbroadch3oh]). The position angles of the pv cuts are chosen purposely to detect the broad line emissions and to avoid the confusion with nearby interaction signatures. These figures provides complementary information in the velocity space, which cannot be explicitly seen in the *f*36 maps.
The first three panels in Figure [figch3ohoutflowpv] present the pv diagrams at three broad CH3OH interaction signatures. From these panels, we see broad emissions features of 1ʺ–2ʺ angular sizescale around the zero angular offset, The last two panels in Figure [figch3ohoutflowpv] present the pv diagrams at two water maser sources, which are associated with high velocity 12CO outflows (Section [chapname]). In panel (4), the broad emission signature can be seen around the angular offset of 0ʺ–2ʺ; in panel (5), the broad emission signature can be seen around the angular offset of 1ʺ–3ʺ.
The angular offset of 3ʺ corresponds to the physical size scale of 0.09 pc. Suppose the water maser sources mark the exact location of the stellar object, the 0.09 pc separation of the interaction signature from the water mater source is consistent with the typical size scale of the molecular outflow.
lrccccc Transition & Frequency (GHz) & E*u*/k (K) & Einstein A–Coefficient (s− 1)
CO (2-1) & 230.538 & 16.68 & 6.91 ⋅ 10− 7
HCN (3-2) & 265.886 & 25.33 & 8.36 ⋅ 10− 4
CH3OH 5(0,5)-4(0,4) E & 241.700 & 47.68 & 6.04 ⋅ 10− 5
CH3OH 5(0,5)-4(0,4) A+ & 241.791 & 34.65 & 6.05 ⋅ 10− 5
CH3OH 5(-2,4)-4(-2,3) E & 241.904 & 60.38 & 5.09 ⋅ 10− 5
CH3OH 5(2,3)-4(2,2) E & 241.905 & 57.27 & 5.03 ⋅ 10− 5
CH3OH 5(-3,3)-4(-3,2) E & 241.852 & 96.93 & 3.89 ⋅ 10− 5
SiO (8-7) & 347.331 & 74.62 & 2.20 ⋅ 10− 3
[tablemoleculelist]
[tableparameters]
[tableoutflow]
[h]
| Velocity Range | | n*H**C**N* | n12*C**O* | [HCN]/[12CO] | [HCN]/[H2] |
| --- | --- | --- | --- | --- | --- |
| Blue Shifted | (-124.75– -15.25 kms− 1) | 2.4 ⋅ 1049 | 3.0 ⋅ 1052 | 0.8 ⋅ 10− 3 | 0.8 ⋅ 10− 7 |
| All | (-124.75–58.25 kms− 1) | 1.3 ⋅ 1050 | 2.9 ⋅ 1053 | 0.4 ⋅ 10− 3 | 0.4 ⋅ 10− 7 |
[tablehcn]
![ The 1.3 mm continuum image. Contours show the levels of [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12,13,14,15, 16, 32, 64, 96]% of the peak brightness temperature 49 K. Red triangles mark the locations of the 22 GHz water maser (Hofner & Churchwell 1996). Two green crosses mark the locations of the peaks of 1.3 cm free-free continuum emission. Blue diamonds mark the locations of 5 broad ^{12}CO outflow interaction signatures (see Section [chapdiagnose]). ](1p3mm.jpg "fig:") [fig1p3mm]
[figco]
[figch3ohpv]
[figcomnt012]
[figch3ohoutflow0]
[figch3ohoutflow2]
[fighcn]
[fighcnpv]
[figcochan1]
[figcochan2]
[figcochan3]
[figcochan4]
[fighcnchan1]
[figch3ohoutflowpv]
---
1. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. More details can be referenced in Ho, Moran, & Lo (2004).[↩](#fnref1)
2. The field–of–view of our 12CO observations in G10.6-0.4 is about 1ʹ. The physical scales of the observed area and the resolution of this G10.6-0.4 data are comparable with those of the 12CO survey in OMC–2 and OMC–3, reported by Williams, Plambeck, & Heyer (2003). See Takahashi et al. (2008) for a more recent study of molecular outflows in OMC–2 and OMC–3.[↩](#fnref2)
3. We define the *massive envelope* as the region with molecular hydrogen density greater than 105 cm− 3, and with temperature greater than 30 K. Such a region can be traced by the NH3 main hyperfine inversion line, and the CH3OH 5(0,5)-4(0,4) A+ transition (Liu et al. 2010 b).[↩](#fnref3)
4. The averaged line–of–sight velocities is derived by *I*/*M*, where *M* and *I* are the mass and the line–of–sight momentum integrated in the blueshifted or the redshifted velocity range (see Table [tableoutflow].).[↩](#fnref4)
5. This value provides the rough sense of the limit of our data. Physically the outflow velocity is uncertain, which leads to the uncertainties in the estimation of the outflow dynamical timescale.[↩](#fnref5)
6. Not the arc shaped structures isolated from the free–free emission peak.[↩](#fnref6)
7. The scale of the initial turbulence in massive cluster forming regions can be referenced to Wang et al. 2008. Our present paper together with the previous observations on infrared dark cloud (Wang et al. 2008; Zhang et al. 2009) are consistent with a scenario of self regulated structure formation via turbulence dissipation and the outflow feedback.[↩](#fnref7)
High Velocity Molecular Outflows In Massive Cluster Forming Region G10.6-0.4
============================================================================
We report the arcsecond resolution SMA observations of the 12CO (2-1) transition in the massive cluster forming region G10.6-0.4. In these observations, the high velocity 12CO emission is resolved into individual outflow systems, which have a typical size scale of a few arcseconds. These molecular outflows are energetic, and are interacting with the ambient molecular gas. By inspecting the shock signatures traced by CH3OH, SiO, and HCN emissions, we suggest that abundant star formation activities are distributed over the entire 0.5 pc scale dense molecular envelope. The star formation efficiency over one global free–fall timescale (of the 0.5 pc molecular envelope, ∼ 105 years) is about a few percent. The total energy feedback of these high velocity outflows is higher than 1047 erg, which is comparable to the total kinetic energy in the rotational motion of the dense molecular envelope. From order–of–magnitude estimations, we suggest that the energy injected from the protostellar outflows is capable of balancing the turbulent energy dissipation. No high velocity bipolar molecular outflow associated with the central OB cluster is directly detected, which can be due to the photo–ionization.
Introduction
============
Theoretical studies suggest that high velocity molecular outflows play important roles in massive cluster forming regions. Massive molecular outflows associated with the accretion of the OB stars can create cavities with low molecular densities. The efficient photon leakage through the cavities potentially reduces the radiation pressure exerted on the molecular gas by a factor of 10 (Krumholz 2005), and therefore enhances the stellar accretion rate. However, if those massive molecular outflows emanated from OB stars are oriented in the bipolar direction which has lower molecular gas density, it is unclear how efficient the outflow momentum and energy can propagate and feedback into the majority of dense molecular gas. From the numerical hydrodynamical simulations, Li & Nakamura (2006) and Nakamura & Li (2007) suggested that in the parsec scale molecular cloud, the initial turbulence dissipates quickly, which leads to local contraction and star formations (see also Carroll et al. 2009). The turbulence induced by the outflows from the intermediate mass or solar mass (proto–)stars over the entire parsec scale cloud, however, quickly replenishes the initial turbulence to support the cloud. The contraction of the cloud is therefore self–regulated with a low star formation efficiency. In such scenarios, the molecular cloud can be maintained in a quasi–equilibrium state, which may be conducive to the formation of a dense molecular core in its center. Massive stars then are formed in the dense molecular core, and create bright ultracompact (UC) Hii regions via the stellar ionization.
Observationally, extensive single dish surveys of the 12CO emissions found high detection rates of massive molecular outflows around the massive cluster forming regions. In addition, the ubiquitously seen bipolar profiles of the massive molecular outflows (Shepherd & Churchwell 1996 b; Beuther et al. 2002; Zhang et al. 2005; Wu et al. 2005; López-Sepulcre et al. 2009) suggest that the magnetically regulated accretion similar to that in low–mass star formation plays a role in OB star formation. Besides the massive bipolar outflows, however, owing to the great complexity of the OB cluster forming regions, identifications of outflows from individual intermediate mass or solar mass stars have been difficult, and the improvements rely on detailed analysis of interferometric data.
G10.6-0.4 is a well studied massive cluster forming region at a 6 kpc distance (Caswell et al. 1975; Ho & Haschick 1981; Ho et al. 1983; Ho et al. 1986; Keto et al. 1987; Guilloteau et al. 1988; Keto et al. 1988; Ho et al. 1994; Keto 1990; Omodaka et al. 1992; Keto 2002; Sollins et al. 2005; Sollins & Ho 2005; Keto & Wood 2006; Keto et al. 2008; Klaassen et al. 2009). In this region, a high bolometric luminosity (9.2 × 105 L⊙) and bright free-free continuum emission ( ≥ 2.4 Jy within a 0.05 pc radius, at 1.3 cm band) were detected, suggesting that a cluster of O-type stars (O6.5–B0; Ho and Haschick 1981) has formed. Our previous SMA observations in CH3OH J=5 transitions trace a massive rotating envelope ( ∼ 1000 M⊙ in the central 0.3 pc × 0.1 pc region) which shows a biconical cavity (Liu et al. 2010 b). The biconical cavity may be initially created by powerful (proto–)stellar outflows, and clues may be provided by sensitive molecular line observations to detect the high velocity emissions.
The VLA/eVLA observations of the centimeter free-free continuum emission and the CS (1-0) emission suggest that the biconical cavity is undergoing an expansional motion with the velocity of a few kms− 1, which may be powered by the ionized gas pressure. These results suggest that the pressure of the ionized gas is an important feedback mechanism in the initially low density region. In the plane of rotation, the dense gas is marginally centrifugally supported (Liu et al. 2010 a), and its global dynamics is not yet severely disturbed by the radiation pressure and the pressure of the ionized gas. In the central 0.1 pc region, a fast rotating hot toroid encircling an OB cluster is detected (Keto, Ho, & Haschick 1988; Sollins & Ho 2005; Liu et al. 2010), which is consistent with dense core formation via a global contraction of a massive envelope. The overall geometry and the dynamics qualitatively resemble the standard magnetically–regulated core collapse in the low mass star forming region.
To improve the understanding of how the envelope is shaped by the (proto–)stellar outflows, and how the global contraction is regulated, we carried out the interferometric observations of the 12CO (2-1) line in G10.6-0.4. We estimate the mass, energy, and momentum budgets in the detected outflow systems. In addition, by comparing the distribution of the high velocity 12CO (2-1) emission with the distribution of various outflow and shock tracers (CH3OH, HCN, SiO), we discuss the interactions between the outflows and the dense molecular envelope. Details about the observations are introduced in Section [chapobs]. The results are presented in Section [chapresult]. In Section [chapdiscuss], we discuss the role of the molecular outflow, and compare it with the role of the feedback from the radiation pressure, stellar wind, and the pressure of the ionized gas. A brief conclusion is provided in Section [chapconclusion].
Observations
============
We observed multiple molecular line transitions from 2005 till 2009. The properties of the observed molecular transitions are summarized in Table [tablemoleculelist]; and the observational parameters are summarized in Table [tableparameters]. We also perform high resolution observations of 3.6 cm continuum emissions. Details about calibrations and data reduction are introduced as follows.
The 1.3 mm band observation
---------------------------
We observed the CH3OH J=5 transitions and the 12CO (2-1) line toward G10.6-0.4 using the Submillimeter Array (SMA)[1](#fn1) in the compact configuration and the very extended configuration on 2009 June 10 and 2009 July 12, respectively; and observed the HCN (3-2) line with the SMA in the compact configuration on 2005 June 21 and in the extended configuration on 2005 May 9. We also observed the 12CO (2-1) lines in the SMA subcompact configuration on 2009 February 09. The frequency resolutions in these observations are 390 kHz ( ∼ 0.5 kms− 1).
The basic calibrations are carried out in `Miriad`, the self-calibration and imaging of these data are carried out in `AIPS`. Continuum emissions are averaged from the line free channels and then subtracted from the line data. We construct the continuum band visibility data at 1.3 mm from the line–free channels in the SMA subcompact, compact, and very–extended array data. We combined the data of all three array configurations, which yields a synthesized beam of 0ʺ.79 × 0ʺ.58 with position angle of 60*o*. The 1.3 mm continuum image is shown in Figure [fig1p3mm]. The discussions of this continuum image are postponed to Section [chapdiagnose].
We combine the 12CO data of all three array configurations, which yield a synthesized beam of 1ʺ.2 × 1ʺ.2 with a position angle of 87*o*; and we perform averaging per three velocity channels to enhance the signal–to–noise ratio. The observed RMS noise of the 12CO (2-1) line channel maps is 0.03 Jy/beam (0.48 K) in each 1.5 kms− 1 channels, where there is no extended strong line emissions. We combine the compact–array and the very–extended array CH3OH data, which yield 1ʺ.5 × 1ʺ.3 angular resolution and rms noise level of 0.06 Jy/beam (0.64 K) in each 0.5 km/s velocity channel. We combine the compact–array and the extended array HCN (3-2) data, yielding 1ʺ.3 × 1ʺ.1 resolution, and RMS noise of 0.24 Jy/beam (3.03 K) in each 0.5 kms− 1 channel.
The 0.85 mm band observation
----------------------------
We observed the SiO (8-7) line using the SMA in the compact configuration on 2008 October 8. The frequency resolution in this observation is 812.5 kHz ( ∼ 0.7 kms− 1).
Calibrations are carried out in `Miriad`, and imaging is carried out in AIPS. Owing to a non-uniform uv coverage in this observation, using natural weighting, we obtain a 4ʺ.9 × 1ʺ.5 synthesized beam with a position angle of 21*o*. The RMS noise in each 0.7 kms− 1 channel is 0.24 Jy/beam (0.33 K).
3.6 cm continuum observation
----------------------------
The X–band continuum emission toward G10.6-0.4 was observed in the VLA A-configuration including the VLBA Pie-Town antenna on 2005 January 2. The sources 1331+305 and 1820-254 were observed as absolute flux and gain calibrators.
The basic calibrations, self-calibration, and imaging of these data are done in `AIPS` package. The synthesized beam of the image is 0ʺ.34 × 0ʺ.19 with a position angle of -7.5*o*. The observed RMS noise of the 3.6 cm continuum image is about 0.1 mJy/beam ( ∼ 27 K in brightness temperature). We note that the 3.6 cm continuum emission is free–free emission from spectral index measurements (Keto, Zhang, & Kurtz 2008).
Results
=======
In the following sections, we inspect the spatial distribution of the high velocity 12CO outflows, with reference to the overall distribution of the dense gas in G10.6-0.4. We trace the dense molecular gas by the emission of the CH3OH J=5 transitions (Liu et al. 2010 b). The CH3OH 5(0,5)-4(0,4) A+ transition, the CH3OH 5(0,5)-4(0,4) E transition, and the blended CH3OH 5(2,3)-4(2,2) E and 5(-2,4)-4(-2,3) E transitions trace an extended 0.5 pc scale massive envelope. The highest excitation CH3OH 5(-3,3)-4(-3,2) E transition exclusively trace a < 0.1 pc scale hot toroid, which is revealed as a white compact ellipse in the middle of the CH3OH RGB image (Figure [figco]). As will be discussed in the position–velocity diagram (Figure [figch3ohpv]), the CH3OH emission shows an integrated rotational motion, and trace the warm and dense accretion flow around the OB cluster. By comparing the outflows with the CH3OH emission, we try to emphasize the formation of a cluster of stars in the self–gravitational accretion flow.
The high velocity outflows are interacting with the ambient dense gas. We diagnose the interactions between the high velocity outflows and the ambient dense gas via the broad CH3OH emission and also the distribution of the water maser sources (Hofner & Churchwell 1996). From the flux of the 12CO emission, we estimate the mass, energy, and momentum feedback. We constrain the feedback rate from the outflow dynamical timescale.
In addition, we report the detection of significantly enhanced HCN (3-2) emission, which has the distribution and the velocity range that are consistent with one of the strongest blueshifted 12CO outflow components near the central hot toroid (we call this blueshifted outflow component *HCN outflow* hereafter ). We also detect localized broad SiO (8-7) emission and geometrically elongated 3.6 cm free–free continuum emissions around *HCN outflow*. The physical implications of these observed signatures are discussed.
For the sake of clarity, the discussion in the following sections are mostly based on the velocity integrated maps. The 12CO (2-1) and the HCN (3-2) channel maps are presented in Appendix [chapchannel].
The Spatial Distribution and Morphology of the 12CO Emission
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Figure [figco] shows the velocity integrated maps of the 12CO (2-1) emission in the blueshifted, redshifted, and the intermediate velocity ranges, superposed on the dense molecular envelope as traced by the velocity integrated emission of CH3OH J=5 transitions. The intermediate velocity range is defined as -15.25–8.75 kms− 1. In this range the rms noise in channel maps is significantly worse owing to the difficulties in imaging the strong and extended emission. The blueshifted (-124.75– -15.25 kms− 1) and the redshifted (8.75– 46.25 kms− 1) velocity ranges are defined relative to the intermediate velocity range. The system velocity of G10.6-0.4 is -3 kms− 1, which is about the midpoint of the intermediate velocity range. We integrate the emission in these three limited velocity ranges such that the significant features are not diluted in the velocity space. Some highly redshifted 12CO emissions marginally detected in the velocity channels of 48.5 kms− 1– 57.5 kms− 1 (Figure [figcochan1]) will be discussed separately. Outside of these selected velocity ranges, we do not see significant detection of 12CO emission. According to the galactic rotation curve (Caswell et al. 1975; Corbel & Eikenberry 2004), the 12CO emission in the velocity range of *v**l**s**r*= -3 – 45 kms− 1 is potentially confused with the foreground molecular gas emission/absorption, which may explain why the velocity integrated map of the redshifted 12CO shows more diffuse emission than that of the blueshifted 12CO (see also the discussions in Appendix [chapchannel]).
We do not detect the collimated high velocity bipolar outflow aligned with the rotational axis of the central hot toroid. In the blueshifted 12CO velocity integrated map, which is not biased by the foreground emission/absorption, we see individual compact components. These compact components have the angular size scale of a few arcseconds. From the channel maps (Figures [figcochan1], [figcochan2], [figcochan3], [figcochan4]), one can see that these compact blueshifted outflow systems have very different coverages in the velocity space. This morphology of the high velocity 12CO emissions qualitatively look very different from the well–defined massive bipolar molecular outflow (see the case of G240.31+0.07 for an example; Qiu et al. 2009), and are difficult to be interpreted by a single wide angle wind. These result can either be explained by multiple powering sources, or can be explained by powerful non-uniform or episodic bursts. These different explanations are not mutually exclusive. Future observations with higher sensitivity and higher resolution to detect the protostellar cores and to resolve the morphology/alignment of individual outflows/jets will help to clarify the origin of the high velocity 12CO emissions. We also note that G10.6-0.4 is a quite evolved massive cluster forming region with extremely bright Hii regions. Comparing the physical size scale of the entire dense envelope ( ∼ 0.5 pc) with that of Orion [2](#fn2), it is not surprising if multiple (proto–)stellar objects have already been formed and are widely distributed in the entire massive envelope[3](#fn3). In active accretion phase, those (proto–)stellar objects will also eject molecular outflows, which will contribute to the high velocity 12CO emissions.
The size scale and the spatial distribution of the significantly detected compact redshifted 12CO emission are qualitatively consistent with the results of the blueshifted 12CO emission. However, from the channel maps (Figures [figcochan1], [figcochan2], [figcochan3], [figcochan4]), we see that the velocity coverage of the redshifted 12CO emission is about a factor of 2 smaller than the blueshifted emission. This can be either explained by the intrinsic asymmetry in some specific powering sources, or the asymmetry of the entrained ambient molecular gas. For the sake of the convenience of the following discussions, we label several compact blueshifted and redshifted components on the velocity integrated maps, which are visually identified from the channel maps. When we mention the labels like R*i*, B*i* outflow, or *HCN outflow*, we specifically refer to the 12CO emission in the corresponding velocity ranges.
By comparing the spatial distribution of the redshifted and blueshifted 12CO, we see that the compact components from these distinct velocity ranges are highly correlated. The R4 outflow and the B4 outflow are spatially close to each other, but, are isolated from the massive envelope as traced by the CH3OH emission. The averaged line–of–sight velocities[4](#fn4) of R4 outflow and the B4 outflow have comparable absolute values, which are 14.6 kms− 1 and 18.1 kms− 1, respectively. The B4 outflow is elongated in the southeast–northwest direction, which is consistent with the alignment of B4 and R4 outflows. These results suggest that R4 and B4 outflows resemble a bipolar molecular jet, and provide indirect indications that protostars form not only in the densest center of the massive envelope, but also form in other local overdensities. The B6 outflow consist a **V** shaped feature, and two other compact components north of the **V** feature. The **V** shaped feature can also be visually identified from the 12CO channel maps, in the velocity range of -19.5– -14.5 kms− 1. One appealing explanation of the morphology of the 12CO emission in B6 outflow is one collimated molecular outflow with wide angle wind. We note that spatially the B6 outflow is associated with an ultracompact HII region (see also Figures [figch3ohoutflow0], [figch3ohoutflow2]), of which we detected the arc shaped expansional signature with -5 kms− 1 < v*l**s**r* < 5 kms− 1 in the CS (1-0) position-velocity diagram (Liu et al. 2010 b). The molecular wind, the ionized gas pressure, and the ionizing photons together may have created the **V** shaped cavity.
Northeast of the hot toroid traced by the CH3OH 5(-3,3)-4(-3,2) E transition (E*u* = 96.93 K), we see abundant outflows or activities related to outflows. With the current resolution, the detailed morphology of each outflow system is poorly resolved, and from 12CO emission only it is difficult to distinguish if there is individual bipolar outflow in this region. Some hints are provided by the water maser detections. Water masers form in high density (n*H*2 = 108–109 cm− 3), and high temperature ( ∼ 2000 K) environment, and are normally found very close to protostars (Garay & Lizano 1999). In Figures [figch3ohoutflow0] and [figch3ohoutflow2], we plot the positions of water masers from Hofner & Churchwell (1996). From these figures we see that one maser source (RA:18*h*10*m*29.192*s*, Decl: -19*o*55ʹ40.93ʺ) seems to be associated with B2 or R2 outflow; and also one maser source (RA:18*h*10*m*29.144*s*, Decl: -19*o*55ʹ46.52ʺ) seems to be associated with B3 or R3 outflow. The spectrum of each of these maser sources shows a cluster of components around the cloud systemic velocity, and additionally shows some higher velocity components. These water maser results provide strong indications that the outflows detected northeast of the hot toroid are driven by local intermediate mass protostars, or even B–type massive (proto–)stars which are in active accretion phase. Other water maser sources are mostly associated with the hot toroid and the central UC Hii region. One water maser source at RA=18*h*10*m*28.328*s* and Decl= -19*o*55ʹ48.36ʺ may be associated with the *HCN outflow*.
In the velocity range of 47.75– 58.25 kms− 1, we marginally detect compact emissions around the locations of B1 and B2 outflows (Figure [figcochan1]). If the distance of G10.6-0.4 is 6 kpc, this velocity range is not likely to be affected by the emission/absorption of the foreground gas, and the highly redshifted 12CO emissions are likely to be the counter parts of blueshifted B1 and B2 outflows.
In the intermediate velocity range, the 12CO emissions show tremendous structures. We attempt to explain the results. However, it has to be borne in mind that the 12CO emissions in this velocity range are affected by the high optical depth of the foreground gas. In addition, missing the short spacing in the interferometric observations makes large scale structure invisible, and also makes the imaging extremely difficult. Therefore, the very detailed structures in individual channel maps need to be confirmed by the cross comparison with other observations. Discussions will be provided to argue that the missing short spacing problem and the foreground absorption have minor effects on our following analysis.
The velocity integrated map, the mean velocity and the velocity dispersion maps of the 12CO emission in the intermediate velocity range are provided in Figure [figcomnt012]. Based on these figures, we suggest a bimodal picture for the geometry and the dynamics. The majority of the dense gas agrees with the 0.5 pc scale envelope detected in CH3OH, and has the velocity gradient from southeast to northwest. In the envelope, a significant **V** shaped feature of outflow cavity can be seen north of the hot toroid. Northeast of the **V** shaped feature, we detect 5ʺ–10ʺ (0.15–0.3 pc) scale filamentary structures (filament *N**E*, hereafter). From the mean velocity map and the 12CO channel maps (Figure [figcochan2]), we see that filament *N**E* has brightness temperature of a few tens of Kelvin, and covers the velocity range of -16– -10 kms− 1, which is broad and is significantly bluer than the systemic velocity of the dense molecular envelope ( -3 kms− 1). A population of the redshifted gas (G10–*S**W*, hereafter) is further seen southwest of the hot toroid. The morphology of G10–*S**W* is potentially filamentary and has the northeast–southwest alignment. However, the emission from G10–*S**W* is projected against the emission from the cavity features southern to the hot toroid (Liu et al. 2010b). Some hints for the filamentary morphology of G10–*S**W* can be seen in the velocity dispersion map, which shows broad line emission regions with a narrowly elongated shape. The velocities of the filament *N**E* and the G10–*S**W* together show a gradient from the northeast to the southwest. These filamentary structures may be originated from ambient gas influenced by massive bipolar molecular outflow or the expansional motion of the ionized gas. The energetics of this dynamical feature can be approximately seen from the previous single dish measurement. By observing the 13CO (2-1) line using IRAM 30m telescope, López-Sepulcre et al. (2009) estimated the mass, momentum and energy of the molecular line wind in the velocity range of -11.6– -8.6 kms− 1 and 3.9–7.9 kms− 1 to be 200 M⊙, 1510 M⊙kms− 1, and 12 ⋅ 1046 erg, respectively. The filament *N**E* is also marginally detected in our CS (1-0) observations (Liu et al. 2010 b); however, both the filament *N**E* and the G10–*S**W* are not seen in the CH3OH images (Figure [figco]), which provides the limits on the density and the gas excitation temperature. The detailed physical conditions can be constrained by future sensitive observations in CO J=2-1 and J=3-2 isotopologues transitions. More discussions about the velocity dispersion of the intermediate velocity 12CO emission are postponed to Section [chapdiagnose].
We note that filamentary structures with the same spatial scale are also detected around the nearby massive cluster forming region, the Orion KL region (Wiseman & Ho 1996; Wiseman & Ho 1998). From the results of the NH3 observations, Wiseman & Ho (1998) suggested a bimodal pattern of velocity. In addition, they found the filaments appear to be fragmenting into chains of core, which are potentially the future or the current sites for low–mass star formation.
The Mass, Energy and Momentum of the High velocity CO Outflows
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For the blueshifted and the redshifted outflows, we assume optically thin emission with LTE conditions. We estimate the mass, momentum and energy traced by the 12CO emission by adopting the gas excitation temperature of T = 100 K and the interstellar abundance ratio [12CO]/[H2] = 10− 4. Our estimate does not explicitly consider the inclination angle, which statistically biases the total outflow momentum lower by a factor of order 1. The optically thin assumption on average may also bias the estimate lower by a factor of order 1. The angular size scale of the synthesized beam of our 12CO images is 1.2ʺ. The emissions from isolated structures with angular size scale much smaller than that of the synthesized beam are potentially beam diluted and therefore cannot be detected. In nearby clouds, the molecular outflows/jets typically have the physical size scale of a few times of 0.1 pc, which corresponds to a few times of 3ʺ at the distance of G10.6-0.4. This typical angular size scale is greater than the 1.2ʺ synthesized beam of our 12CO images. Therefore we do not consider the effect of beam dilution as a dominant bias in our estimates. The diffuse emission may not be robustly imaged in the high resolution maps. To check this effect, we image the 12CO (2-1) emission with the subcompact–array data alone, yielding 4ʺ.7 × 2ʺ.4 synthesized beam and rms noise of 0.17 Jy/beam (0.44 K) in each velocity channel of 0.5 kms− 1 width. By averaging the subcompact-array channel maps in the blueshifted and the redshifted velocity ranges, we do not find significantly more diffuse high velocity emission, thus suggesting that our estimates based on the 1ʺ.2 × 1ʺ.2 high resolution images (Section [chapobs]) are representative of the total mass, momentum, and energy feedback.
The 12CO outflows in the intermediate velocity range may be some energetic systems with small inclination angle; and observationally, systems in this velocity range are severely confused with the global dynamics of the dense gas (Liu et al. 2010a, b). Therefore the direct analysis of the 12CO outflow momentum and energy in the intermediate velocity range is difficult. In addition, the 12CO emission is optically thick in this velocity range. Without the complementary information of the 13CO or C18O observations, the optically depth is uncertain. Statistically, we expect the outflow feedback from those systems with low inclination angle quantitatively has the same order of magnitudes as the feedback from outflows which have high inclination angles and are detected in the blueshifted and the redshifted velocity ranges. Or, there can be some less energetic outflow systems, as those detected in OMC–2 and OMC–3 (see Williams, Plambeck, & Heyer 2003, and therein). The contributions of such systems are not significant as compared with the energetic high velocity outflows.
We estimate the deconvolved size scale of each outflow system by performing 2D Gaussian fitting of the velocity integrated maps. The outflow dynamical timescale is estimated by dividing the major axis of the fitted Gaussian with the averaged velocity. The dominant uncertainties in the estimation of the dynamical timescale are caused by the inclination, which is not measured in these observations. Also, owing to the limited angular resolution, we are only able to measure the dynamical timescale greater than 1700 years[5](#fn5), assuming a typical outflow velocity of ∼ 20 kms− 1.
The dynamical timescale, mass, momentum, and energy of each significant and compact outflow system in G10.6-0.4 are summarized in Table [tableoutflow]. The individual outflow systems have a typical mass of ∼ 1 M⊙, and momentum of a few tens of M⊙kms− 1. Since we only estimate the physical properties of the outflow systems in the blueshifted and the redshifted velocity ranges while the real outflow systems continue into the intermediate velocity range, these quantities can be underestimated. To provide the sense of how much bias is made by considering only the limited velocity ranges, we estimate the physical properties of the *HCN outflow* both in the blueshifted velocity range and in the entire velocity range with 12CO (2-1) detection. By including the intermediate velocity range, the dense gas in the massive envelope significantly contributes to the 12CO flux, and the estimated mass should be treated as the upper limit of the molecular mass in the outflow system. From Table [tableoutflow], we see that the results with the inclusion of the intermediate velocity range has a factor of 10 higher mass, a factor of 4 higher momentum, and a factor of 2.7 higher energy. These results can be understood since the kinetic energy is scaled with the square of velocity, and therefore the low velocity outflow gas in the intermediate velocity range does not contribute to a lot of momentum and energy, as long as its mass is not way much higher than the high velocity components. Thus we conclude that our estimations of the total energy and momentum feedback based on the high velocity components only is accurate up to an order of magnitude. The estimated physical parameters of *HCN outflow* (Table [tableoutflow]) can be uncertain owing to the strong photo–dissociation. However, it is unlikely that the total molecular mass integrated in the line–of–sight of *HCN outflow* can be comparably high as the mass enclosed by the nearby hot toroid ( ∼ 200 M⊙). Based on this argument, we suggest a lower limit of the [12CO]/[H2] abundance ratio in *HCN outflow* to be ∼ 10− 5. We expect the [12CO]/[H2] abundance ratio in other outflow systems to be less biased from the typical value of 10− 4.
We estimate the outflow mass, momentum, and energy in the entire blueshifted and the redshifted velocity range by integrating the flux in the whole field of view. For the blueshifted outflows, the total mass is about 6.2 M⊙; the total line–of–sight momentum is 163 M⊙kms− 1; the total energy is 6.6 ⋅ 1046 erg. For the redshifted (also including the velocity range of 46.25–58.25 kms− 1) outflows, the total mass is about 4.4 M⊙; the total line–of–sight momentum is 109 M⊙kms− 1; the total energy is 3.2 ⋅ 1046 erg. The estimated mass includes all entrained and shocked gas in the outflows which can be detected in 12CO emission. The dynamical timescale of these outflow systems has the order of 103–104 years. In the blueshifted velocity range, the visually identified compact systems (B1⋯6, *H**C**N*–B outflows) contribute to 89% of the total outflow mass; 84% of the outflow momentum; and 72% of the outflow energy. However, in the redshifted velocity range, the visually identified compact systems (R1⋯4 outflows) only contribute to 31% of the total outflow mass; 20% of the outflow momentum; and 16% of the outflow energy. The small fractional contribution of the compact systems (R1⋯4 outflows) in the redshifted velocity range can be explained by the significant contribution of the diffuse emission (Figure [figcochan1]) in the velocity range of 8.75 – 46.25 kms− 1. Those diffuse emission can be contributed by both the foreground molecular emission and the diffuse molecular outflows, which are not distinguishable with the data we currently have. Additionally, the flux of the molecular outflows in the redshifted velocity range is also attenuated by the foreground absorption. Since the order of magnitudes of the total mass, momentum, and energy in the redshifted velocity range are still comparable with those in the blueshifted velocity range, we suggest that the contribution of the foreground emission are roughly compensated by the attenuation of the foreground absorption. Therefore, we accept the estimated total outflow mass, momentum, and energy in the redshifted velocity range without modeling and correcting for the foreground effect, and we argue that the relevant discussions are not seriously biased.
The total molecular mass in the region with CH3OH detection (Figure [figco]) is about 1000–2000 M⊙ (Liu et al. 2010 a). Assuming the majority of mass is moving with 2–4 kms− 1 absolute velocity (Liu et al. 2010), the momentum and kinetic energy in this system has the order of magnitudes of 2000–8000 M⊙ ⋅ kms− 1 and 4–32 ⋅ 1046 erg. The mass of the outflows is about a few percent of the total molecular mass in this region. However, energy feedback from the outflows is comparable to the kinetic energy of the filament NE and G10–SW (see Section [chapname]), and is a significant fraction of the total kinetic energy of the massive envelope. Note the system is marginally centrifugally supported (Liu et al. 2010), and the gravitational potential energy has the same order of magnitude as the rotational kinetic energy. If the initial turbulence is induced by the global gravitational contraction, its kinetic energy should have the same order of magnitude as the gravitational potential energy, or less than that.
Our estimations of the outflow feedback parameters are based on the 12CO (2-1) flux. In the Orion 1S region ( ∼ 450 pc from the sun), the high resolution observation of the SiO (5-4) line unveils a cluster of molecular outflows, which are not detected in the 12CO emission (Zapata et al. 2006). Such kind of systems may be rare, and from that observation, have the dynamical ages of few hundreds of years, and have physical size scales of 460–2700 AU. Although our 12CO data achieve a factor of 2 lower rms noise level than the 12CO data reported in Zapata et al. (2006), the greater distance and hence more severe beam dilution makes those system non–detectable in G10.6-0.4 in 12CO. By assuming the LTE conditions, optically thin emission, a rotational temperature of 80 K, and an abundance ratio of [SiO]/[H2] = 10− 7, Zapata et al. (2006) estimated the total mass, momentum, and energy in those SiO outflows to be 1.155 M⊙, 57.85 M⊙kms− 1, and 3.03 ⋅ 1046 ergs. However, as suggested by the authors, at least a factor of 50 enhancement of the SiO abundance with respect to the adopted value 10− 7 is required to explain the non–detection of 12CO, implying that the real outflow mass, momentum, and energy can be a factor of 50 lower than the estimated values. If such kind of SiO outflow systems are rare either because of the short life time in the specific phase of weak 12CO emission, or because of their formation requires unusual physical conditions, their feedback may not be significant as compared with the total outflow feedback estimated from the 12CO emission. With small size scale, the induced protostellar turbulence also dissipates faster.
The Diagnostics of Interaction Signatures
-----------------------------------------
While the interaction and shock signatures typically have the velocity of a few kms− 1 from the systematic velocity, in massive cluster forming regions, the CO isotopologues emission is extremely complicated in such velocity range, and is hard to be robustly imaged in interferometric observations. The diagnostics of the interaction signatures are therefore relying on the observations of various tracers. In the following sections, we introduce the observed features in CH3OH, HCN, and SiO, as the diagnostics in G10.6-0.4. The interaction signatures identified from the CH3OH emissions are cross compared with the velocity dispersion map of the 12CO (2-1) emission in the intermediate velocity range, to demonstrate the robustness of the diagnostics. In addition, we compare the identified interaction signatures with the 1.3 mm continuum image, which is regraded as a reliable tracer of dense molecular cores.
### The Broad CH3OH Emission
#### G10.6-0.4
By comparing the velocity integrated 12CO emission with the velocity integrated CH3OH emission, we see the association of the molecular outflows with the dense envelope. With the terminal velocity of a few tens of kilometer per second, these molecular outflows can strongly shock the ambient dense gas in the envelope. A variety of molecules are then released from the dust grain, and their abundances in gas phase can be dramatically enhanced. We diagnose the interactions between the molecular outflows and the ambient gas from the broad CH3OH 5(0,5)-4(0,4) A+ emissions. The CH3OH molecule is considered to be one of the early–type molecules, and is abundant in the dense protostellar core. In addition, its abundance can be further shock enhanced by one or two orders of magnitude. The selected transition has the lowest upper–level–energy (34.65 K) among all observed CH3OH lines, so that the excitation is least sensitive to the stellar heating.
Figure [figch3ohpv] shows a sample position-velocity (pv) diagram of the CH3OH 5(0,5)-4(0,4) A+ transition. From this figure we see two dominant signatures with distinguishable characteristics: the dense rotating envelope and the broad interaction signature. The dense envelope covers the angular scale of ∼ 20ʺ, and is characterized by high brightness and steep increase of brightness. The interaction signature, as indicated by the red arrow, is faint and is spatially localized, but covers a much larger velocity range. This broad and faint emission can be understood as the outflow wing component in the spectrum. More pv diagrams of the interaction signatures are discussed in Appendix [chapoutflowpv]. Qualitatively, one can observe that, in the same range of angular offset, the envelope and the interaction signature roughly occupy the same area (in unit of arcsecond × kms− 1) in the pv diagram, which is naturally explained by their difference in linewidth. However, the envelope is sketched by significantly more contours, which means that the area between two contour levels is much smaller.
The difference between the dense envelope and the interaction signature leads to a large contrast if we compare the velocity integrated flux between two significant contour levels, for example, the 3rd *σ* and the 6th *σ* significance levels (*f*36 hereafter). Between these two contour levels, the interaction signatures contribute to significantly more *f*36 than the dense envelope, although having lower total integrated flux. This property can be utilized to indicate the spatial distribution of the interaction signatures. Figure [figch3ohoutflow0] shows the *f*36 map of the CH3OH 5(0,5)-4(0,4) A+ transition together with the locations of the water maser sources and the locations of the 1.3 cm free-free emission peaks.
From Figure [figch3ohoutflow0] we see that this analysis is quite successful. The 0.1 pc scale hot toroid (Liu et al. 2010 a,b) which has the highest brightness temperature, cannot be seen in this map. Instead, we see significant local components (indicated by boxes) closely associated with the compact 12CO emission (for example, B2, R1, R3 outflows, and B6 outflow), which are the most likely sources to induce the interaction signatures. We note that the CH3OH emission is only significantly detected in the velocity range of -13 kms− 1 < v*l**s**r* < 10 kms− 1 while the blueshifted and the redshifted 12CO emissions are defined outside this velocity range. The consistency in their spatial distributions also indicates that the broad CH3OH emission is indeed related to the high velocity molecular outflows. We also evaluate the velocity dispersion of the emission between the 3rd *σ* and the 6th *σ* significance levels (Figure [figch3ohoutflow2]). The result consistently shows large velocity dispersion closely associated with the interaction signatures. Both Figure [figch3ohoutflow0] and Figure [figch3ohoutflow2] show that the interaction signature around the B6 outflow has a elongated distribution parallel to the global rotational axis of the dense envelope (Liu et al. 2010 a,b), and may be associated with an UC Hii region, indicated by a star symbol (Liu et al. 2010 b). Whether that signature is related to the bipolar molecular outflow or is pushed by the bipolar expansion of the ionized gas can be studied in future high resolution observations. The interaction signature indicated by the orange box corresponds to the interaction signature shown in the sample pv diagram (Figure [figch3ohpv]). The corresponding 12CO emission may be in the intermediate velocity range, and as shown in Figure [figch3ohoutflow0], is elongated in the southeast–northwest direction. It can also be visually identified in the channel map (Figure [figcochan2]) as compact components in the velocity range of 3.5–8.0 kms− 1.
The f36 analysis, especially, the velocity dispersion map (Figure 6) unveils a population of interaction signatures, which lie close to the plane of the densest flattened molecular gas (Liu et al. 2010 a, b). Those interaction signatures show locality and relatively low outflow velocities. We suggest that they are less likely due to the interaction of the wind emanated from the OB cluster embedded in the hot toroid, but rather associated with local sites of star formation. This result provides important indication that local star formation plays important roles in the feedback of kinetic energy to the dense molecular gas.
#### Cross Comparison with 12CO Emission in the Intermediate Velocity Range
In this section, we cross compare the broad CH3OH 5(0,5)-4(0,4) A+ emission (Figure [figch3ohoutflow0], [figch3ohoutflow2]) with the broad 12CO emission in the intermediate velocity range (Figure [figcomnt012]). From the velocity dispersion map of the intermediate velocity 12CO emission (Figure [figcomnt012], right), one can see two extended broad line regions, which are associated with two UC Hii regions (Hii–M: RA=18*h*10*m*28.683*s*, Decl= -19*o*55ʹ49ʺ.07; Hii–NW: RA=18*h*10*m*28.215*s*, Decl=-19*o*55ʹ44ʺ.07), respectively. The physical size scales of these two broad emission regions are both a fraction of a parsec.
The distribution of the broad 12CO emission around Hii–NW agrees well with the interaction signature traced by the *f*36 of the CH3OH 5(0,5)-4(0,4) A+ transition (Figure [figch3ohoutflow0]). The morphology can be explained by the expansional motion of the molecular/ionized gas confined by the dense gas, which has a flattened distribution in the global plane of rotation (Liu et al. 2010 a b). The broad 12CO emission around Hii–M does not have CH3OH 5(0,5)-4(0,4) A+ emission counter part, which can be explained by the lower optical depth around this specific region.
In addition to those two extended broad 12CO emission regions, we visually identify five very compact (1ʺ–2ʺ) broad 12CO emission signatures, for which the mean velocities also have significant contrasts with the ambient gas. Those regions are marked by circles in Figure [figcomnt012], and are also marked in the left most panel of Figure [figch3ohoutflow0]. The central coordinates of these five circles are: (#1) RA=18*h*10*m*29.258*s*, Decl= -19*o*55ʹ42ʺ.18; (#2) RA=18*h*10*m*29.088*s*, Decl= -19*o*55ʹ46ʺ.28; (#3) RA=18*h*10*m*29.059*s*, Decl= -19*o*55ʹ52ʺ.78; (#4) RA=18*h*10*m*29.349*s*, Decl= -19*o*55ʹ57ʺ.18; (#5) RA=18*h*10*m*28.868*s*, Decl= -19*o*55ʹ56ʺ.58, respectively (see also the channel maps in Figure [figcochan2]). Two of them (#1, #2) are closely associated with water maser sources. The east most one (#4) is associated with a *f*36 peak of the CH3OH 5(0,5)-4(0,4) A+ transition, which is marked by the yellow box in Figure [figch3ohoutflow0]. A faint *f*36 peak of the CH3OH 5(0,5)-4(0,4) A+ transition is detected around #3; and similarly with #5. These five signatures marked by circles look compact even from the 12CO (2-1) transition, which is very easy to be excited. It provides the indication that they are associated with active local dynamics. Since their mean velocities are offset from the systemic velocity of their ambient gas by a few kms− 1, we hypothesize that those signatures are the molecular gas in the local dense cores pushed by the protostellar outflows. Without the comparison with results from the outflow tracers, it is extremely difficult to recognize these interaction signatures from the 12CO (2-1) maps, and it is even more difficult to robustly interpret them when they are recognized.
We note that the missing flux and the high optical depth of the foreground gas can artificially enhance the measured velocity dispersion. However, the broad 12CO emission regions mentioned in this section have much smaller angular size scales ( < 20ʺ) than the maximum detectable angular scale of our SMA subcompact–array data ( ∼ 40ʺ), and should not be severely affected by missing flux. If the missing flux artificially enhances the measured velocity dispersion at large scale, the contrast between the measured velocity dispersion of the broad 12CO emission regions and the measured velocity dispersion of the envelope is reduced. The contrast in velocity dispersion should be higher in reality, which means the identified broad emission signatures are robust. Therefore we think the missing flux does not have a significant impact on our discussions in this section. The high optical depth of the foreground gas trims structures for all angular size scales, however, only for the emissions redder than the systemic velocity of -3 kms− 1. The mean velocities in those extended and compact broad 12CO emission regions are apparently redder than the mean velocity of the ambient gas. If the high velocity dispersion is an artifact caused by the high optical depth of the foreground gas, we expect to detect a bluer mean velocity.
#### Compare With 1.3 mm Continuum Emissions
In massive cluster forming regions, flux in 1.3 mm continuum emission is dominantly contributed by the thermal dust emission, and the free–free continuum emission from UC Hii regions (Keto, Zhang, & Kurtz 2008). To see the relation between those broad 12CO interaction signatures with the dense molecular gas, we mark the locations of those broad 12CO interaction signatures (as well as the locations of the 22 GHz water masers) on the 1.3 mm continuum image (Figure [fig1p3mm]).
From the 1.3 mm continuum image, we first see an abrupt increase of the 1.3 mm continuum flux in the middle, which suggests the distinct origins of the flux. We identify two UC Hii regions from high resolution centimeter continuum images (Keto, Ho & Haschick 1988; Guilloteau et al. 1988; Sollins et al. 2005; Liu et al. 2010abc), and mark their locations on the 1.3 mm continuum image. Those centimeter continuum emissions indicate that in the middle of the 1.3 mm continuum map, the free–free continuum emission from the UC Hii region contribute significantly. In the extended region, the 1.3 mm continuum emission is dominantly contributed by the thermal dust emission, except for a fainter UC Hii region in the northwest.
The thermal dust emission unveils abundant dense cores over a 0.5 pc region. Three dusty dense cores ( RA:18*h*10*m*29*s*.064, Decl: -19*o*55ʹ46ʺ.2; RA:18*h*10*m*29*s*.035, Decl:-19*o*55ʹ49ʺ.7; RA:18*h*10*m*28*s*.975, Decl: -19*o*55ʹ52ʺ.5) are closely associated with the identified broad interaction signatures of 12CO outflows. The core at the northeast (RA:18*h*10*m*29*s*.064, Decl: -19*o*55ʹ46ʺ.2) is further associated with a water maser source (Hofner & Churchwell 1996). The associations with water maser sources and outflow signatures indicates that those dense cores are actively forming stars. This core shows a complicated geometry, which may be explained by hierarchical fragmentations. The dense core at RA= 18*h*10*m*28*s*.975 and Decl=-19*o*55ʹ52ʺ.5 is apparently elongated, and has internal structures, which may be explained by fragmentation, or can be observationally caused by blending.
#### Compare With Low–Mass Protostellar Outflows
From the observations of the CH3OH J=2 transitions with upper–level–energy E*u* of 4.64–12.2 K, Takakuwa, Ohashi & Hirano (2003) suggests that in the Class 0 low mass protostar IRAM 04191+1522, the enhanced broad CH3OH emission trace the interactions between the protostellar outflow and its parent molecular core. Similar to what is observed in G10.6-0.4, the pv diagram of CH3OH in IRAM 04191+1522 shows the distinguishable interaction signature with the envelope component. The interaction signatures in IRAM 04191+1522 have the size scale of ∼ 0.03 pc, closely associated with the protostellar envelope. Observations in other nearby protostellar outflows (Bachiller et al. 1995, 1998, and 2001; Garay et al. 1998) show the localized enhancement of the CH3OH abundance in the strongly shocked regions around 0.1 pc from the ejecting sources. This 0.03–0.1 pc size scale correspond to the angular scale of 1ʺ–3ʺ at the distance of G10.6-0.4 (6 kpc), which is marginally resolved in our observations (Table [tableparameters]); and this scale is small as compared with the size scale of the massive envelope traced by CH3OH (20ʺ–30ʺ).
Supposedly these observed phenomena in nearby clouds and their interpretations can be extrapolated to the distant massive dense envelopes with higher temperature and high density. Then each of the detected broad CH3OH interaction signature may be associated with an independent ejecting protostar. The distribution of the observed broad CH3OH emission features in G10.6-0.4 suggests the abundant star formation activities over the entire ∼ 0.5 pc scale dense envelope. In G10.6-0.4, we already detected three UC Hii regions, indicating that multiple massive stars have already been formed. It is not surprising if there are many massive or low–mass (proto–)stars, which are still in accretion phase and are ejecting the molecular outflows. We expect higher resolution observations in lower excitation CH3OH to reveal more protostellar objects.
Note that the detection of the thermal radio jets is regarded as one of the most robust methods to identify protostellar objects in the accretion phase. However, in UC Hii regions, the free-free emission around the embedded OB cluster has the flux of a few Jy, which is of 3–4 orders of magnitude brighter than the typically flux of the thermal radio jets. The bright free-free emission associated with the OB cluster ionization can have complicated spatial distribution and geometry, which will confuse the detection of the thermal radio jets. The shock enhanced molecular emission potentially provides good complementary information in the diagnostic of the star forming activities. From Figure [figch3ohoutflow0], we see that a population of protostellar and stellar activities (four CH3OH outflow interaction signatures as indicated by three boxes and one white circle, and two UC Hii regions as indicated by stars) seem to have a coplanar distribution, suggesting a scenario of enhanced star formation in the rotationally flattened dense envelope. This scenario is supported by recent numerical hydrodynamical simulations (Peters et al. 2010).
Some protostellar and stellar activities seem to follow the filaments detected in the intermediate velocity 12CO emissions. Owing to the projection effect, we cannot robustly distinguish whether those activities are just distributed in the bulk of the envelope.
### The *HCN Outflow*
The *HCN outflow* is characterized by the significantly higher HCN (3-2) flux. Among all outflow systems detected by 12CO, the *HCN outflow* is also the unique case where we detect the SiO (8-7) emission. The deconvolved size scale of the 12CO emission is about 2ʺ, and its location is close to the central 0.1 pc scale hot toroid. Figure [fighcn] shows the velocity integrated maps of HCN (3-2), SiO (8-7), and the blueshifted 12CO, in the top and the middle panels, and shows the 3.6 cm continuum image in the bottom panel. From this figure we see the locally enhanced HCN emission, for which the location and geometry agree excellently with those of its 12CO counter part. The distribution of the SiO emission is consistent with the 12CO and HCN emission. Subjected to the non–uniform uv coverage, the SiO (8-7) data have an elongated synthesized beam, and therefore the projected geometry of the emission and the peak location are poorly constrained.
In Figure [fighcnpv], we present the pv diagrams of HCN (3-2), SiO (8-7) and 12CO (2-1) around *HCN outflow*, which are all cut in the RA direction. From both panels in this figure, we can clearly see the broad outflow signatures around the angular offset of ∼ -5ʺ. The HCN (3-2) emission and the 12CO emission consistently trace the outflow to the velocity of ∼ -25 kms− 1. Owing to the detection limit, the SiO pv diagram only traces the outflow in a limited velocity range. From the pv diagram, we see that SiO is also excited in the hot toroid, which is located 2ʺ–3ʺ east of the *HCN outflow*. Assuming the LTE conditions, optically thin emission, and the excitation temperature of 100 K, we estimate the total number of the SiO molecule in *HCN outflow* to be ∼ 1.5 ⋅ 1048.
We apply the same assumptions to derive the total number of the HCN and the 12CO molecules in the *HCN outflow*. Adopting the galactic 12CO abundance (i.e. [12CO]/[H2] = 1 ⋅ 10− 4), we list the derived number and the implied abundance ratios in Table 4. In the galactic molecular clouds, the [HCN]/[12CO] ratio ranges from 10− 4 in the quiescent zone to 2.5 ⋅ 10− 3 in the hot core regions (Blake et al. 1987). The enhanced [HCN]/[H2] ratio has also been reported in other (proto–)stellar outflows (e.g. IRAS 20126: 0.1–0.2 ⋅ 10− 7, Su et al. 2007; L1157: 5 ⋅ 10− 7, Jørgensen et al. 2004). In the *HCN outflow*, the value of [HCN]/[12CO] in the blueshifted velocity range is consistent with an enhanced HCN abundance. If our assumption of [12CO]/[H2] ratio is valid, the value of [HCN]/[H2] in the *HCN outflow* is in between of the two referenced cases. Even if the assumed [12CO]/[H2] ratio is overestimated by a factor of 10 (see the discussions in Section 3.2), the derived [HCN]/[H2] ratio in the *HCN outflow* is still higher than the reported typical value of ≤ 7 ⋅ 10− 9 (Jørgensen et al. 2004). The HCN molecule is usually regarded as a dense gas tracer. Our results suggest, however, that the HCN emission can also be locally enhanced in outflows and shocks around the dense core, and therefore might not be a good tracer of the overall dynamics in massive cluster forming regions.
Among all detected high velocity outflows, the *HCN outflow* does not have specifically higher linear momentum or energy. The uniqueness in the excitation of SiO (8-7) may be explained by the heating of the central OB cluster, or be explained by the molecular gas erupted from the hot toroid with shock enhanced SiO abundance. Some hints can be seen from the 3.6 cm continuum image (Figure [fighcn], bottom panel), which shows an elongated emission feature around the location of the *HCN outflow*. Whether the elongated 3.6 cm emission feature is physically associated with *HCN outflow* or is just a projection effect can be examined by future ALMA observations of molecular lines and the hydrogen recombination lines.
We note that a few more elongated/filamentary features[6](#fn6) are resolved in the previous 1.3 cm free–free continuum observation (Sollins et al. 2005). Those elongated/filamentary features have the lengths of ∼ 1ʺ (0.03 pc), and width of ≤ 0.1ʺ (0.003 pc). The physical properties and the formation mechanism of those elongated features are still uncertain. Suppose those elongated/filamentary features of 1.3 cm free–free continuum emission are ionized gas which are undergoing thermal diffusion with velocity of 10 kms− 1, the ≤ 0.003 pc widths imply that their age is no more than 300 years. If all those elongated/filamentary features can also be explained by ionized outflows, the velocities of the outflows can be estimated by
$$\frac{\mbox{length}}{\mbox{age}} = \frac{0.03\mbox{ pc}}{300\mbox{ years}} \sim 100 \mbox{ kms$^{-1}$}.$$
This outflow velocity is consistent with the measurement in hydrogen recombination line (line–of–sight velocity 60 kms− 1, Keto & Wood 2006), although the ionized outflow is not explicitly resolved in the image. The higher brightness of those elongated/filamentary features than the ambient regions may suggest that those outflows are originally ejected in molecular form with a much higher density, and then ionized by the stellar radiation. The molecular counterparts of those ionized outflows are not necessarily detected, especially in the bipolar direction, if stellar ionization is important. We suggest that the central OB cluster may accrete the molecular gas of very compact/clumpy structures (see the NH3 opacity results, Sollins & Ho 2005), which can survive the stellar ionization. The molecular gas can then approach individual O–type stars to the distance of ∼ 10 AU, and be centrifugally accelerated to the outflow velocity. Small scale accretion disks may exist around the O–type stars. However, they are not detectable with the current instruments owing to the beam dilution.
Discussions
===========
In G10.6-0.4, previously we estimated the scalar momentum feedback from the stellar wind and from the ionized gas pressure to be of 102–103 M⊙kms− 1 (Liu et al. 2010 b). From the 12CO data presented in this paper, we conclude that the total momentum feedback from the protostellar outflows has the same order of magnitude. However, these feedback mechanisms can still play very different roles in the massive cluster forming regions, owing to their differences in physical properties.
The stellar wind is an isotropic feedback mechanism. By comparing the resulting star formation efficiency in numerical hydrodynamical simulations, Nakamura & Li (2007) suggested that the *spherical wind* cannot propagate efficiently in dense gas. The feedback from such mechanisms is trapped in small size scale, where the induced turbulence is quickly dissipated. Therefore the energy feedback from the *spherical wind* supports the cloud less efficiently. The protostellar outflow is ejected from objects embedded in the local overdensities. Its momentum, however, is typically collimated in a small solid angle, which makes it penetrate the dense gas easily and can produce large scale disturbance. The induced protostellar turbulence potentially replenishes the dissipated initial turbulence in large scale, and plays the role of regulating the cloud contraction and the star formation efficiency.
From the numerical simulation, Mac Low (1999) suggested the characteristic timescale of turbulence dissipation *t**d* and the free–fall timescale *t**f**f* have the relation:
$$t\_{d} = \left(\frac{3.9\lambda\_{d}}{M\_{rms}\lambda\_{J}}\right)t\_{ff},$$
where *λ**d* is the driving scale of the turbulence, *λ**J* is the Jeans length, and *M**r**m**s* is the rms Mach number of the turbulence. Assuming a total mass of 1000–2000 M⊙ is enclosed in the 0.5 pc scale envelope, the global free–fall timescale is about 105 years. If the averaged gas temperature is 30–50 K, the Jeans length is about 0.1 pc. The typical value of *M**r**m**s* can be 5–10[7](#fn7). Assuming the main driving source of the turbulence is the compact (proto–)stellar outflows, the value of *λ**d* can be estimated by the physical size scale of the observed outflow systems, which is about 0.06 pc (2ʺ). Therefore, the characteristic timescale *t**d* in G10.6-0.4 is about 0.23–0.47 times of t*f**f*, which is about a few times of 104 years. Among the outflow systems detected in G10.6-0.4, the maximum outflow dynamical timescale is about 104 years, which is smaller than *t**d*. The estimations of outflow energy in Section [chapestimation] suggests that the energy injection from the protostellar turbulence is capable of balancing the turbulence energy dissipation in the relevant timescale.
We provide rough estimations for the total protostellar mass from a fiducial momentum ejection efficiency I*t**o**t**a**l* = P\*M\*, where I*t**o**t**a**l* is the total ejected scalar momentum, M\* is the total stellar mass, and P\* is a proportional factor. Following Nakamura & Li (2007), we adopt P\*= 50 kms− 1. In G10.6-0.4, the projected scalar momentum in the blueshifted and the redshifted velocity ranges is 273 M⊙kms− 1, which leads to *f**l**o**s* ⋅ M\* = 5.45 M⊙, where *f**l**o**s* is the ratio of the scalar momentum contributed by the line–of–sight velocity component to the total scalar momentum. Statistically the value of *f**l**o**s* can be estimated by
$$\frac{\int\_{0}^{\frac{\pi}{2}} \sin\theta d\theta d\phi}{\int\_{0}^{\frac{\pi}{2}} 1 d\theta d\phi} \simeq 0.637,$$
assuming the inclination angle of the outflows are uniformly randomly distributed. Given the 1000–2000 M⊙ total molecular mass in the envelope, the star formation efficiency (SFE) in the past 104 years is about 0.42%–0.86%. Assuming a uniform star formation rate in the time domain, the star formation efficiency in one free–fall timescale is of the order of a few percent. This estimated SFE is comparable to the SFE in the simulations of Nakamura & Li (2007), consistently suggesting the cloud contraction is self–regulated by the local star formation. High resolution observations of the thermal dust emission to statistically study the prestellar and protostellar cores, will further improve the constraint on the efficiency of the outflow feedback. The number of the YSOs can be statistically estimated by dividing M\* with the averaged protostellar mass *m̄*. The fiducial value of *m̄* based on the Scalo (1986) initial mass function (IMF) is 0.5 M⊙. In G10.6-0.4, the value of *m̄* can be biased by the evolutionary stage, and potentially the observational selection bias that we only pick up the systems with powerful molecular outflows. Assuming the detected outflows are associated with YSOs, which on average have ∼ 1 M⊙ stellar mass, the number of the YSOs can be estimated by M\*/1.0 = 8.56. The surface density of these YSOs in the ∼ 0.5 pc region is therefore ∼ 8.56/(0.52) ∼ 34 pc− 2. The derived YSO surface density should be treated as a lower limit since our observations do not trace protostellar objects which have weak or no outflows.
Alternatively, we can assume each of the compact high velocity outflows (R1⋯4, B1⋯6, *H**C**N*) is associated with one protostellar objects. Considering also the non–identified outflow systems in the intermediate velocity range, there can be ∼ 10 protostellar objects in total in the observed region. If estimating the averaged mass of each protostellar object by 1 M⊙, our observational results suggest that an order of ∼ 10 M⊙ molecular gas are converted into (proto–)stellar objects in ∼ 104 years. The system potentially evolves into a stellar cluster with on order of 102 stars in a few t*f**f*.
The ionized gas only propagates in low density regions, which have low recombination rates. However, our previous studies (Liu et al. 2010 b) suggest that the thermal pressure of the ionized gas is capable of driving the large scale (a few times of 0.1 pc) coherent motions, such as the expansional motions of the ionized cavities or the bubble walls. The absolute velocity of such kind of motion is about a few kms− 1, and which is smaller than the thermal sound speed of the ionized gas ( ∼ 10 kms− 1). The efficiency of converting the kinetic energy in such expansional motion to the large scale turbulence energy has to be investigated in theoretical studies.
Conclusion
==========
We present the arcsecond resolution interferometry observations of the 12CO (2-1) transition. From the distribution and the mass/momentum/energy of the detected high velocity outflows, we discuss the role of the outflow feedback in the massive cluster formation region. Our main results are:
* We detect multiple high velocity outflow components in the blueshifted and the redshifted velocity ranges, respectively. The total molecular mass in these outflow systems are about 10 M⊙, and the total momentum and energy budgets are of the order of 102 M⊙kms− 1 and 1047 erg, respectively.
* No high velocity bipolar molecular outflow parallel to the global rotational axis is directly detected around the central 0.1 pc hot toroid. However, we cannot rule out the possibility that it is photo–ionized.
* The 12CO outflows are interacting with the dense molecular envelope, which can be diagnosed by the CH3OH, HCN, and the SiO shock signatures.
* From the distribution of the UC Hii regions, we know that multiple massive stars have been formed in the 0.5 pc scale massive envelope, and not only formed in the central 0.1 pc hot toroid. From the association of high velocity outflows with water maser sources, and with the interaction signatures detected in thermal molecular emission, we suggest that multiple protostellar objects with ∼ 10 M⊙ in total may also have been formed within 104 years. The star formation efficiency over one global free–fall timescale is of the order of a few percent.
* We detect strongly enhanced HCN (3-2) emission in a outflow system near the hot toroid, and therefore suggest that HCN is not a good tracer for the global dynamics.
We will follow up the polarization observation to provide thorough discussions of the energetic relation in this source. We emphasize that the discussions and conclusions in this paper are relevant to the cluster forming regions with size scale of a parsec or a fraction of a parsec, and with mass of thousands of M⊙. In less massive systems (with lower opacity), the radiative hydrodynamical simulations suggest that the heating by the stellar radiation efficiently suppress the star formation (Offner et al. 2009).
Baobab Liu thanks the SMA staff to support the observations. *Facilities:*
natexlab#1#1
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A. The Channel Maps of the 12CO (2-1) and the HCN (3-2) Lines
=============================================================
We present the channel maps of the 12CO (2-1) line and the HCN (3-2) lines in this section. Figure [figcochan1], [figcochan2], [figcochan3], and [figcochan4] show the 12CO (2-1) line in the velocity channels of -131.5 kms− 1– 77 kms− 1. Contour levels are adjusted according to the rms noise. We think the channel maps in the redshifted velocity channels (9.5 kms− 1–45.5 kms− 1) are contaminated by the contribution of the foreground molecular gas, and therefore show subparsec scale diffuse emission, which are not seen in the channel maps in the blueshifted velocity range ( -124.75– -15.25 kms− 1). The absorption line of 12CO (2-1) is also detected at the location of the free-free continuum peak (ra:18:10:28.683 dec:-19:55:49.07), up to *v**l**s**r* ∼ 45 kms− 1. The HI absorption experiment (Caswell et al. 1975) consistently shows the foreground absorption up to *v**l**s**r* ∼ 45 kms− 1.
Figure [fighcnchan1] show the channel maps of the HCN (3-2) line. We note the high consistency between the HCN and the 12CO channel maps. In the HCN (3-2) channel maps, the low velocity counterpart of outflow R3 (see Section [chapname]) may be visually identified in the velocity range of 7.5 kms− 1–12 kms− 1; and the low velocity counterparts of outflow B3 and B5 may be visually identified in the velocity range of -10.5 kms− 1– -15 kms− 1; the *HCN outflow* can be seen in the same velocity range as covered by its 12CO emission
B. The Position-Velocity Diagrams of the Interaction Signatures
===============================================================
Figure [figch3ohoutflowpv] shows the pv diagrams of the CH3OH 5(0,5)-4(0,4) A+ transition, at the locations of three broad CH3OH interaction signatures, and at the location of two water maser sources. Those CH3OH interaction signatures are visually identified from the *f*36 maps (Section [chapbroadch3oh]). The position angles of the pv cuts are chosen purposely to detect the broad line emissions and to avoid the confusion with nearby interaction signatures. These figures provides complementary information in the velocity space, which cannot be explicitly seen in the *f*36 maps.
The first three panels in Figure [figch3ohoutflowpv] present the pv diagrams at three broad CH3OH interaction signatures. From these panels, we see broad emissions features of 1ʺ–2ʺ angular sizescale around the zero angular offset, The last two panels in Figure [figch3ohoutflowpv] present the pv diagrams at two water maser sources, which are associated with high velocity 12CO outflows (Section [chapname]). In panel (4), the broad emission signature can be seen around the angular offset of 0ʺ–2ʺ; in panel (5), the broad emission signature can be seen around the angular offset of 1ʺ–3ʺ.
The angular offset of 3ʺ corresponds to the physical size scale of 0.09 pc. Suppose the water maser sources mark the exact location of the stellar object, the 0.09 pc separation of the interaction signature from the water mater source is consistent with the typical size scale of the molecular outflow.
lrccccc Transition & Frequency (GHz) & E*u*/k (K) & Einstein A–Coefficient (s− 1)
CO (2-1) & 230.538 & 16.68 & 6.91 ⋅ 10− 7
HCN (3-2) & 265.886 & 25.33 & 8.36 ⋅ 10− 4
CH3OH 5(0,5)-4(0,4) E & 241.700 & 47.68 & 6.04 ⋅ 10− 5
CH3OH 5(0,5)-4(0,4) A+ & 241.791 & 34.65 & 6.05 ⋅ 10− 5
CH3OH 5(-2,4)-4(-2,3) E & 241.904 & 60.38 & 5.09 ⋅ 10− 5
CH3OH 5(2,3)-4(2,2) E & 241.905 & 57.27 & 5.03 ⋅ 10− 5
CH3OH 5(-3,3)-4(-3,2) E & 241.852 & 96.93 & 3.89 ⋅ 10− 5
SiO (8-7) & 347.331 & 74.62 & 2.20 ⋅ 10− 3
[tablemoleculelist]
[tableparameters]
[tableoutflow]
[h]
| Velocity Range | | n*H**C**N* | n12*C**O* | [HCN]/[12CO] | [HCN]/[H2] |
| --- | --- | --- | --- | --- | --- |
| Blue Shifted | (-124.75– -15.25 kms− 1) | 2.4 ⋅ 1049 | 3.0 ⋅ 1052 | 0.8 ⋅ 10− 3 | 0.8 ⋅ 10− 7 |
| All | (-124.75–58.25 kms− 1) | 1.3 ⋅ 1050 | 2.9 ⋅ 1053 | 0.4 ⋅ 10− 3 | 0.4 ⋅ 10− 7 |
[tablehcn]
![ The 1.3 mm continuum image. Contours show the levels of [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12,13,14,15, 16, 32, 64, 96]% of the peak brightness temperature 49 K. Red triangles mark the locations of the 22 GHz water maser (Hofner & Churchwell 1996). Two green crosses mark the locations of the peaks of 1.3 cm free-free continuum emission. Blue diamonds mark the locations of 5 broad ^{12}CO outflow interaction signatures (see Section [chapdiagnose]). ](1p3mm.jpg "fig:") [fig1p3mm]
[figco]
[figch3ohpv]
[figcomnt012]
[figch3ohoutflow0]
[figch3ohoutflow2]
[fighcn]
[fighcnpv]
[figcochan1]
[figcochan2]
[figcochan3]
[figcochan4]
[fighcnchan1]
[figch3ohoutflowpv]
---
1. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. More details can be referenced in Ho, Moran, & Lo (2004).[↩](#fnref1)
2. The field–of–view of our 12CO observations in G10.6-0.4 is about 1ʹ. The physical scales of the observed area and the resolution of this G10.6-0.4 data are comparable with those of the 12CO survey in OMC–2 and OMC–3, reported by Williams, Plambeck, & Heyer (2003). See Takahashi et al. (2008) for a more recent study of molecular outflows in OMC–2 and OMC–3.[↩](#fnref2)
3. We define the *massive envelope* as the region with molecular hydrogen density greater than 105 cm− 3, and with temperature greater than 30 K. Such a region can be traced by the NH3 main hyperfine inversion line, and the CH3OH 5(0,5)-4(0,4) A+ transition (Liu et al. 2010 b).[↩](#fnref3)
4. The averaged line–of–sight velocities is derived by *I*/*M*, where *M* and *I* are the mass and the line–of–sight momentum integrated in the blueshifted or the redshifted velocity range (see Table [tableoutflow].).[↩](#fnref4)
5. This value provides the rough sense of the limit of our data. Physically the outflow velocity is uncertain, which leads to the uncertainties in the estimation of the outflow dynamical timescale.[↩](#fnref5)
6. Not the arc shaped structures isolated from the free–free emission peak.[↩](#fnref6)
7. The scale of the initial turbulence in massive cluster forming regions can be referenced to Wang et al. 2008. Our present paper together with the previous observations on infrared dark cloud (Wang et al. 2008; Zhang et al. 2009) are consistent with a scenario of self regulated structure formation via turbulence dissipation and the outflow feedback.[↩](#fnref7)
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arxiv_0186016
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