The gravitational instability of an infinite homogeneous, finitely
conducting,
rotating, collisionless, anisotropic-pressure plasma in the presence of
a uniform
magnetic field with finite-ion-Larmor-radius (FLR) corrections and generalized
polytropic laws is investigated. The polytropic laws are considered for
the
pressure components in directions parallel and perpendicular to the magnetic
field. The method of normal-mode analysis is applied to derive the dispersion
relation. Wave propagation is considered for both parallel and perpendicular
axes of rotation. Longitudinal and transverse modes of propagation are
discussed separately. The effects of rotation, finite electrical resistivity,
FLR
corrections and polytropic indices on the gravitational, firehose and mirror
instabilities are discussed. The stability of the system is discussed by
applying
the Routh–Hurwitz criterion. Extensive numerical treatment of the
dispersion
relation leads to several interesting results. For the transverse mode
of
propagation with the axis of rotation parallel to the magnetic field, it
is
observed that rotation stabilizes the system by decreasing the critical
Jeans
wavenumber. It is also seen that the region of instability and the value
of the
critical Jeans wavenumber are larger for the Chew–Goldberger–Low
(CGL) set
of equations in comparison with the magnetohydrodynamic (MHD) set of
equations. It is found that the effect of FLR corrections is significant
only in the
low-wavelength range, and produces a stabilizing influence. For the transverse
mode of propagation with the axis of rotation parallel to the magnetic
field, the
finite electrical resistivity removes the polytropic index [nu] from the condition
for
instability. The inclusion of rotation alone or FLR corrections alone or
both
together does not affect the condition for mirror instability. The growth
rate of
the mirror instability is modified owing to uniform rotation or FLR corrections
or both together. We note that the condition of mirror instability depends
upon
the polytropic indices. We also note that neither the mirror instability
nor the
firehose instability can be observed for the isotropic MHD set of equations.
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