Abstract
Collisionless stellar systems are driven towards equilibr ium by mixing of phase-space elements. I show that the excess-mass function D( f ) = R ¯ F(x,v)> f ( ¯ F(x, v)− f ) d 3 xd 3 v (with ¯ F(x, v) the coarse-grained distribution function) always decreas es on mixing. D( f ) gives the excess mass from values of ¯ F(x, v)> f . This novel form of the mixing theorem extends the maximum phase-space density argument to all values of f . The excess-mass function can be computed from N-body simulations and is additive: the excess mass of a combination of non-overlapping systems is the sum of their individual D( f ). I propose a novel interpretation for the coarsegrained distribution function, which avoids conceptual problems with the mixing theorem. As an example application, I show that for self-gravitating cusps (ρ∝ r −γ as r→ 0) the excess mass D∝ f −2(3−γ)/(6−γ) as f →∞, i.e. steeper cusps are less mixed than shallower ones, independent of the shape of surfaces of constant density or details of the distribution function (e.g. anisotropy). This property, together with the additi vity of D( f ) and the mixing theorem, implies that a merger remnant cannot have a cusp steeper than the steepest of its progenitors. Furthermore, I argue that the remnant’s cusp should not be sh allower either, implying that the steepest cusp always survives.
Highlights
The dynamical state of a stellar system is completely described by its ‘fine-grained’ distribution function, F(x, v, t), which refers to the phase-space density at point (x, v) and time t
The main objective of galactic dynamics is to solve this system of equations
A stellar system out of equilibrium is driven towards equilibrium by way of mixing its phase-space densities in a process of violent relaxation
Summary
The dynamical state of a stellar system is completely described by its ‘fine-grained’ distribution function, F(x, v, t), which refers to the phase-space density at point (x, v) and time t. Lynden-Bell (1967) derived a distribution function for the end-state of violent relaxation assuming the conservation of phasespace volumes of given density according to equation (1). Another approach to the dynamics of violent relaxation was taken by Chavanis (1998) in deriving a time evolution equation for F. While this is a promising attempt, its practicality is limited and unlikely to surpass that of N-body simulations. The purpose of this paper is to present (see Section 2) a novel approach to mixing which is conceptually different from that of Tremaine et al and avoids their conceptual problems This leads to a novel form of the mixing theorem in terms of a new concept, the excess-mass function, which is simple to apply and easy to interpret.
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