G. Hensler et al., ON THE ISOTROPY ASSUMPTION AND THE APPLICABILITY OF THE GASDYNAMICAL EQUATIONS FOR COLLAPSING SYSTEMS, Astronomy and astrophysics, 303(1), 1995, pp. 299-306
Rotating, isothermal collapsing protostellar and protogalactic gas sys
tems are always being treated by use of the isotropic gasdynamical (Eu
ler) equations; while it is standard textbook knowledge that diffusive
and viscous effects can be neglected in most astrophysical plasmas wi
th large Reynolds numbers, or in other words non-diagonal elements of
the stress-energy tensor (second-order moments of the velocity distrib
ution function) and higher order moments vanish, it is not that clear
that the isotropy assumption is correct, i.e. the equality of the diag
onal second-order moments. Therefore we address in this paper the ques
tion whether a collapsing gas cloud remains always strictly isotropic.
Analytically the velocity field of an isothermal collapsing rotating
gas cloud is computed and it is shown that strict isotropy can be real
ized in the static case only. This means that although collisional tim
escales in the gas might be very short, the collapsing system has to d
evelop an inherent small anisotropy; the trajectory of the system in p
hase space is initially leading away from the isotropic path, staying
continuously apart from it for a small distance. We discuss this pheno
menon quantitatively by using moment equations of the Boltzmann equati
on and examine its consequences. It turns out that the deviation from
anisotropy in most cases keeps small enough to allow for a global isot
ropic treatment which is consistent with expectation; applying our res
ults to systems like cloud fluids or stellar systems, however, yields
complementary results. Additionally we prove in this paper the perform
ance of the conditions for the applicability of the gasdynamical descr
iption in a handy form and show that these conditions are usually fulf
illed. We find that for a slightly Jeans unstable system in pressure e
quilibrium the ratio of the microscopic collision time to free-fall ti
me has to be of order unity or smaller as has to be the Knudsen number
in order to ensure the validity of gasdynamic equations.