Abstract
Linear perturbation is used to investigate the effect of gravitational softening on the retrieved two-armed spiral eigenmodes of razor-thin stellar discs. We explore four softening kernels with different degrees of gravity bias, and with/without compact support (compact in the sense that they yield exactly Newtonian forces outside the softening kernel). These kernels are applied to two disc galaxy models with well-known unsoftened unstable modes. We illustrate quantitatively the importance of a vanishing linear gravity bias to yield accurate frequency estimates of the unstable modes. As such, Plummer softening, while very popular amongst simulators, performs poorly in our tests. The best results, with excellent agreement between the softened and unsoftened mode properties, are obtained with softening kernels that have a reduced gravity bias, obtained by compensating for the sub-Newtonian forces at small interparticle distances with slightly super-Newtonian forces at radii near the softening length. We present examples of such kernels that, moreover, are analytically simple and computationally cheap. Finally, these results light the way to the construction of softening methods with even smaller gravity bias, although at the price of increasingly complex kernels.
Highlights
The evolution of a collisionless stellar system is determined by the collisionless Boltzmann equation (CBE)
The direct numerical integration of the CBE in six-dimensional phase space is in general impossible because under the CBE the distribution function (DF) develops ever finer structures owing to phase mixing or chaotic mixing
The two interloping modes at ω3 = 0.384 + 0.103 i and ω4 = 0.323 + 0.075 i have not been described in the literature before. We confirmed that they are robust to changes of the numerical parameters in the code and that they exert a zero total torque on the disc, as they should, and see no reason to discard them as spurious (Polyachenko & Just 2015)
Summary
The evolution of a collisionless stellar system is determined by the collisionless Boltzmann equation (CBE) ∂F(x, v, t) ∂t + v · ∂F(x, v, t) ∂x ∂V (x, t) ∂x ∂F(x, v, ∂v t) = (1). F (x, v, t) is the distribution function (DF), which gives the stellar phase-space density at location (x, v) and time t. The direct numerical integration of the CBE in six-dimensional phase space is in general impossible because under the CBE the DF develops ever finer structures owing to phase mixing or chaotic mixing. Numerical schemes that smooth out such fine structure (whereby violating the CBE) are possible but taxing (Yoshikawa, Yoshida & Umemura 2013; Schaller et al 2014; Colombi et al 2015)
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