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State of the problem next up previous
Next: Initial conditions Up: Old neutron star distribution Previous: Introduction

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 tex2html_wrap_inline167 be a distribution function of stars in 6-dimensional phase space. Then space density is given by tex2html_wrap_inline177 . Time evolution of the distribution function is generally described by the kinetic equation

displaymath179

where tex2html_wrap_inline181 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 tex2html_wrap_inline191 (here and below we consider all stars of unit mass):

displaymath193

with tex2html_wrap_inline195 being absolute value of the initial velocity, and tex2html_wrap_inline197 - circular velocity. So the problem can be reformulated as follows: from a given distribution function tex2html_wrap_inline199 (that is, the number of stars with energy in between E and E+dE and angular momentum in between tex2html_wrap_inline191 and tex2html_wrap_inline207 ) in 2-dimensional space to find the distribution function tex2html_wrap_inline167 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

displaymath211

Introduce cylindrical space coordinates r, z, tex2html_wrap_inline217 and spherical coordinates tex2html_wrap_inline219 , tex2html_wrap_inline149 , tex2html_wrap_inline223 in the velocity space. The direction of tex2html_wrap_inline219 can be chosen arbitrary, say along orbital velocity tex2html_wrap_inline227 , tex2html_wrap_inline149 being the angle between tex2html_wrap_inline231 and tex2html_wrap_inline219 and tex2html_wrap_inline223 - azimuthal angle. Then make transformation in the velocity space from the introduced spherical coordinates to coordinates tex2html_wrap_inline237 . Using (1) calculate Jacobian tex2html_wrap_inline239 of the transform tex2html_wrap_inline241 . Finally we obtain for Q

displaymath245

Here tex2html_wrap_inline247 denotes the region in r, z-space where the motion is allowed with the given angular momentum and energy tex2html_wrap_inline251 . 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 tex2html_wrap_inline251 in the phase space tex2html_wrap_inline255 . Then the distribution function f in equation (3) can be treated as constant and is obtained directly from (3) provided tex2html_wrap_inline199 is known. The density Q can be found straightforwardly from the initial distributions p(r,z) and tex2html_wrap_inline265 using the same equation (3) with tex2html_wrap_inline267 (note that tex2html_wrap_inline265 is expressed through tex2html_wrap_inline271 , which follows from statistical calculations, and tex2html_wrap_inline273 ; see below). Finally, we obtain as a solution

displaymath275

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.


next up previous
Next: Initial conditions Up: Old neutron star distribution Previous: Introduction

Mike E. Prokhorov
Fri Jul 19 19:16:07 MSD 1996