A simple, heuristic derivation of the Balescu-Lenard kinetic equation for stellar systems. (arXiv:2007.07291v1 [astro-ph.GA])
<a href="http://arxiv.org/find/astro-ph/1/au:+Hamilton_C/0/1/0/all/0/1">Chris Hamilton</a> (DAMTP, Cambridge)
The unshielded nature of gravity means that stellar systems are inherently
inhomogeneous. As a result, stars do not move in straight lines. This obvious
fact severely complicates the kinetic theory of stellar systems because
position and velocity turn out to be poor coordinates with which to describe
stellar orbits – instead, one must use angle-action variables. Moreover, the
slow relaxation of star clusters and galaxies can be enhanced or suppressed by
collective interactions (‘polarisation’ effects) involving many stars
simultaneously. These collective effects are also present in plasmas; in that
case, they are accounted for by the Balescu-Lenard (BL) equation, which is a
kinetic equation in velocity space. Recently several authors have shown how to
account for both inhomogeneity and collective effects in the kinetic theory of
stellar systems by deriving an angle-action generalisation of the BL equation.
Unfortunately their derivations are long and complicated, involving multiple
coordinate transforms, contour integrals in the complex plane, and so on. On
the other hand, Rostoker’s superposition principle allows one to pretend that a
long-range interacting $N$-body system, such as a plasma or star cluster,
consists merely of uncorrelated particles that are ‘dressed’ by polarisation
clouds. In this paper we use Rostoker’s principle to provide a simple,
intuitive derivation of the BL equation for stellar systems which is much
shorter than others in the literature. It also allows us to straightforwardly
connect the BL picture of self-gravitating kinetics to the classical ‘two-body
relaxation’ theory of uncorrelated flybys pioneered by Chandrasekhar.
The unshielded nature of gravity means that stellar systems are inherently
inhomogeneous. As a result, stars do not move in straight lines. This obvious
fact severely complicates the kinetic theory of stellar systems because
position and velocity turn out to be poor coordinates with which to describe
stellar orbits – instead, one must use angle-action variables. Moreover, the
slow relaxation of star clusters and galaxies can be enhanced or suppressed by
collective interactions (‘polarisation’ effects) involving many stars
simultaneously. These collective effects are also present in plasmas; in that
case, they are accounted for by the Balescu-Lenard (BL) equation, which is a
kinetic equation in velocity space. Recently several authors have shown how to
account for both inhomogeneity and collective effects in the kinetic theory of
stellar systems by deriving an angle-action generalisation of the BL equation.
Unfortunately their derivations are long and complicated, involving multiple
coordinate transforms, contour integrals in the complex plane, and so on. On
the other hand, Rostoker’s superposition principle allows one to pretend that a
long-range interacting $N$-body system, such as a plasma or star cluster,
consists merely of uncorrelated particles that are ‘dressed’ by polarisation
clouds. In this paper we use Rostoker’s principle to provide a simple,
intuitive derivation of the BL equation for stellar systems which is much
shorter than others in the literature. It also allows us to straightforwardly
connect the BL picture of self-gravitating kinetics to the classical ‘two-body
relaxation’ theory of uncorrelated flybys pioneered by Chandrasekhar.
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