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

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 cut-offs 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.