Self-consistent dynamical models with a finite extent -- V. Smooth radial truncations and phase-space consistency

TL;DR

This work addresses constructing self-consistent spherical models with finite radial extent by replacing sharp outer truncations with a smooth, infinitely differentiable taper ${\cal S}$ controlled by truncation radius $r_{\text{T}}$ and sharpness $\xi$, applicable to arbitrary density profiles and implemented in SpheCow. Applying the method to the Hernquist model, the authors show that softly truncated systems can be isotropic or support Osipkov--Merritt anisotropies, with a characteristic bump–dip feature in the distribution function near the truncation energy ${\cal E}_{\text{T}}$ and a calculable critical sharpness $\xi_{\text{crit}}(r_{\text{T}},\lambda)$ delimiting consistency. The results demonstrate that soft truncations dramatically enlarge the space of dynamically viable finite-extent models, while also highlighting that phase-space consistency does not guarantee long-term stability due to potential secular instabilities linked to DF features. The approach provides a general, physically motivated framework for constructing finite-extent spherical models with controlled outer-edge behaviour and is ready for use in analytical studies and equilibrium initial-condition generation for simulations, with a clear diagnostic via the distribution function and potential stability considerations.

Abstract

Many stellar systems exhibit a finite spatial extent, yet constructing self-consistent spherical models with a prescribed outer boundary is non-trivial because sharp density cutoffs introduce discontinuities that lead to inconsistencies in the associated distribution function. In this paper we show that these difficulties arise from the abruptness of the truncation rather than from the finite extent itself. We introduce a general and infinitely differentiable radial truncation scheme that can be applied to any density profile, and illustrate its behaviour using the Hernquist model. We find that softly truncated models are dynamically consistent provided that the truncation is sufficiently gradual, and we determine the corresponding critical truncation sharpness. Their distribution functions display a characteristic bump-dip feature near the truncation energy that signals the transition between consistent and inconsistent cases. In contrast to sharply truncated models, softly truncated systems can support an extensive family of Osipkov-Merritt orbital structures, including moderately radial ones. Soft truncations therefore offer a general and physically motivated route to constructing finite-extent dynamical models with well-controlled outer-edge behaviour.

Figures (6)

• Figure 1: Illustration of the truncation function ${\cal{S}}(y)$. The truncation radius is fixed at $r_{{\text{T}}} = 2$, whereas the truncation sharpness takes different values ranging from 0.5 to 1.
• Figure 2: Basic properties of softly truncated Hernquist models. Properties shown are the density, mass profile, circular velocity curve, gravitational potential, surface density, and projected mass profile. The thick golden line corresponds to the standard Hernquist model without truncation; the thinner lines correspond to softly truncated models with different values for the truncation radius $r_{{\text{T}}}$ and the truncation sharpness $\xi$. The latter ranges from 0.5 (lightest shade) to 1 (darkest shade) in steps of 0.1. We adopt normalised units with $G = M = b = 1$.
• Figure 3: Dynamical properties of softly truncated Hernquist models with an isotropic orbital structure. Properties shown are the velocity dispersion profile, phase-space distribution function, and differential energy distribution. The different lines have the same meaning as in Fig. \ref{['Hernquist_basic.fig']}. For the sharp truncation model ($\xi=1$), the distribution function and differential energy distribution are not shown.
• Figure 4: Zoom into the bump--dip region of the differential energy distribution of softly truncated Hernquist models with an isotropic orbital structure. The $y$-axis is shown in linear scaling. The different lines have the same meaning as in Figs. \ref{['Hernquist_basic.fig']} and \ref{['Hernquist_iso.fig']}.
• Figure 5: The critical truncation sharpness $\xi_{\text{crit}}(r_{{\text{T}}},\lambda)$ for truncated Hernquist models with different truncation radii and different Osipkov--Merritt parameters. All Osipkov--Merritt models with $\xi\leqslant\xi_{\text{crit}}$ are consistent. The dotted grey line corresponds to $\lambda=0$, i.e. isotropic models.