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State of the problem

The problem to be solved is to find stationary space distribution of stars which have had certain initial space and velocity distributions and move in a given potential. The potential is supposed to be axially symmetric, constant in time and not affected by this stellar population. In principle, the most straightforward way to obtain the solution is to integrate directly the equations of motion for a sufficiently large number of stars with different initial conditions. This method was indeed used in the Paczynsky (1990) and Hartmann et al. (1990) papers. Here we propose to use another method based on finding the solution of collisionless Boltzmann equation. Define be a distribution function of stars in 6-dimensional phase space. Then space density is given by . Time evolution of the distribution function is generally described by the kinetic equation

where is the potential. Axial symmetry implies that only two cylindrical coordinates r and z are required. In the case considered the problem can be significantly simplified because at least two integrals of motion exist - the full energy E and angular momentum along the z-axis (here and below we consider all stars of unit mass):

with being absolute value of the initial velocity, and - circular velocity. So the problem can be reformulated as follows: from a given distribution function (that is, the number of stars with energy in between E and E+dE and angular momentum in between and ) in 2-dimensional space to find the distribution function in 6-dimensional phase space. As we are interested in investigation of old neutron star distribution, we can safely get rid off time dependence and search for equilibrium solution of the collisionless equation.

Conservation of full number of stars gives rise to the relation

Introduce cylindrical space coordinates r, z, and spherical coordinates , , in the velocity space. The direction of can be chosen arbitrary, say along orbital velocity , being the angle between and and - azimuthal angle. Then make transformation in the velocity space from the introduced spherical coordinates to coordinates . Using (1) calculate Jacobian of the transform . Finally we obtain for Q

Here denotes the region in r, z-space where the motion is allowed with the given angular momentum and energy . Now assume that the motion in the chosen potential is ergodic, or equivalently, that a test particle moving long enough in this potential will sweep uniformly the hypersurface of constant in the phase space . Then the distribution function f in equation (3) can be treated as constant and is obtained directly from (3) provided is known. The density Q can be found straightforwardly from the initial distributions p(r,z) and using the same equation (3) with (note that is expressed through , which follows from statistical calculations, and ; see below). Finally, we obtain as a solution

Basically this result is an equivalent formulation of the ergodic hypothesis. The latter should be proved for each potential separately, because no general theorem exists for axisymmetrical potentials. We will briefly discuss this problem below.