[Paper Review] On the self-consistent time-dependent linearized response of stellar discs to external perturbations

Dominic Dootson, John Magorrian|arXiv (Cornell University)|May 31, 2022

Stellar, planetary, and galactic studies4 citations

TL;DR

This paper presents a direct time-domain approach to solving the linearized collisionless Boltzmann equation for stellar discs, using Kalnajs' matrix method with angle-action coordinates to compute self-consistent, time-dependent responses to external perturbations. It demonstrates excellent agreement with N-body simulations for dynamical friction torque on a rotating bar, provided resonant orbit trapping is minimal.

ABSTRACT

We study the explicitly time-dependent response of a razor-thin axisymmetric disc to externally imposed perturbations by recasting the linearized Collisionless Boltzmann equation as an integral equation and applying Kalnajs' matrix method. As an application we consider the idealized problem of calculating the dynamical friction torque on a steadily rotating, two-dimensional bar. We consider two choices of basis functions in the matrix method, showing that both lead to comparable results. The torques from our linearised calculation are in excellent agreement with those measured from $N$-body simulation, as long as the bar perturbation does not resonate with a significant fraction of the disc's stars.

Motivation & Objective

• To develop a practical, time-domain method for computing the self-consistent, linearized response of stellar discs to external perturbations.
• To overcome the limitations of traditional frequency-domain approaches in scenarios involving transient or non-periodic perturbations.
• To validate the linear response framework against N-body simulations for the dynamical friction torque on a rotating bar.
• To assess the impact of resonant orbit trapping on the accuracy of linear response predictions.
• To demonstrate the computational efficiency and resolution advantages of the linear method over N-body simulations.

Proposed method

• Reformulates the linearized collisionless Boltzmann equation as a Volterra integral equation of the first kind in the time domain.
• Applies Kalnajs' matrix method using basis functions derived from potential-density pairs to represent the response in angle-action coordinates.
• Solves the resulting system of equations directly in time, avoiding Fourier transforms and their associated singularities in the response matrix.
• Computes the Green's function of the system to enable initial value problem solutions.
• Uses the response to calculate the dynamical friction torque exerted by the disc on an externally imposed rotating bar.
• Compares results with N-body simulations using a tapered Mestel disc model and tracks orbit trapping via particle counting.

Experimental results

Research questions

• RQ1 Can a direct time-domain solution of the linearized CBE accurately reproduce the dynamical friction torque on a rotating bar in a stellar disc?
• RQ2 How does the performance of the time-domain method compare to frequency-based approaches in terms of numerical stability and computational efficiency?
• RQ3 To what extent do resonantly trapped orbits cause discrepancies between linear response predictions and N-body simulations?
• RQ4 How does the self-gravity of the disc's response affect the magnitude of the dynamical friction torque?
• RQ5 Can the linear response method be efficiently extended to study the orbital decay of external perturbers in a self-consistent manner?

Key findings

• The linear response calculation reproduces N-body simulation results for dynamical friction torque with high accuracy when the bar does not significantly trap stars on resonant orbits.
• The torque is reduced by approximately a factor of two when disc self-gravity is turned off, highlighting its critical role in the response.
• The time-domain method avoids the mathematical singularities associated with the denominator in frequency-space response matrices, improving numerical stability.
• The linear response method produces smoother, less noisy torque curves than N-body simulations, demonstrating superior resolution for studying response features.
• The method is computationally efficient: kernel precomputation time is comparable to a single N-body run, and solving the response equation takes only seconds.
• The discrepancy between linear theory and N-body simulations for fast dumbbell bars is attributed to a ~10% fraction of orbits being trapped by resonance, which enhances the effective bar mass and torque.