Kinetic theory of spatially inhomogeneous stellar systems without collective effects

We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For $N\rightarrow +\infty$, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the Vlasov-Landau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the Lenard Balescu-type kinetic equation recently derived by Heyvaerts (2010) since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the diffusion coefficient and friction force

A particle method for the homogeneous Landau equation

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.Comment: 27 pages, 14 figures, debloated some figures, improved explanations in sections 2, 3, and

Dressed diffusion and friction coefficients in inhomogeneous multicomponent self-gravitating systems

General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstrates their equivalence. The corresponding stochastic Langevin equations are presented: they can be a practical alternative to numerically solving the inhomogeneous Fokker-Planck and Balescu-Lenard equations. The present formalism allows for a self-consistent description of the secular evolution of different populations covering a spectrum of masses, with a proper accounting of the induced secular mass segregation, which should be of interest to various astrophysical contexts, from galactic centers to protostellar discs.Comment: 27 pages, 1 figur

Escape of stars from gravitational clusters in the Chandrasekhar model

We study the evaporation of stars from globular clusters using the simplified Chandrasekhar model. This is an analytically tractable model giving reasonable agreement with more sophisticated models that require complicated numerical integrations. In the Chandrasekhar model: (i) the stellar system is assumed to be infinite and homogeneous (ii) the evolution of the velocity distribution of stars f(v,t) is governed by a Fokker-Planck equation, the so-called Kramers-Chandrasekhar equation (iii) the velocities |v| that are above a threshold value R>0 (escape velocity) are not counted in the statistical distribution of the system. In fact, high velocity stars leave the system, due to free evaporation or to the attraction of a neighboring galaxy (tidal effects). Accordingly, the total mass and energy of the system decrease in time. If the star dynamics is described by the Kramers-Chandrasekhar equation, the mass decreases to zero exponentially rapidly. Our goal is to obtain non-perturbative analytical results that complement the seminal studies of Chandrasekhar, Michie and King valid for large times $t\to+\infty$ and large escape velocities $R\to +\infty$. In particular, we obtain an exact semi-explicit solution of the Kramers-Chandrasekhar equation with the absorbing boundary condition f(R,t)=0. We use it to obtain an explicit expression of the mass loss at any time t when $R\to +\infty$. We also derive an exact integral equation giving the exponential evaporation rate $\lambda(R)$, and the corresponding eigenfunction $f_{\lambda}(v)$, when $t\to +\infty$ for any sufficiently large value of the escape velocity R. For $R\to +\infty$, we obtain an explicit expression of the evaporation rate that refines the Chandrasekhar results