where
p
(r) is the pressure,
is the total energy density. Then from the conservation of the
energy-momentum and the condition for hydrostatic equilibrium we
can derive the Tolman-Oppenheimer-Volkov (TOV) equations for the
interior of a spherically symmetric star in equilibrium.
Specifically,
and the ``mass inside radius r '' is represented by
This means that the total mass of the star is M = m (R), with R being the star's radius. To determine a stellar model we must solve
where
These equations should of course, be supplemented with an
equation of state
as input. Usually is sufficient to use a one-parameter equation
of state to model neutron stars, since the typical thermal
energies are much smaller than the Fermi energy. The polytropic
equation of state
where
K
is the polytropic constant and
N
the polytropic exponent, is used in most of the studies. The
existence of a unique global solution of the Einstein equations
for a given equation of state and a given value of the central
density has been proven by Rendall and Schmidt [174].
If we assume a small variation in the fluid or/and in the spacetime we must deal with the perturbed Einstein equations
and the variation of the fluid equations of motion
while the perturbed metric will be given by equation (18).
Following the procedure of the previous section one can decompose the perturbation equations into spherical harmonics. This decomposition leads to two classes of oscillations according to the parity of the harmonics (exactly as for the black hole case). The first ones called even (or spheroidal, or polar) produce spheroidal deformations on the fluid, while the second are the odd (or toroidal, or axial) which produce toroidal deformations.
For the polar case one can use certain combinations of the metric perturbations as unknowns, and the linearized field equations inside the star will be equivalent to the following system of three wave equations for unknowns S, F, H :
and the constraint
The linear functions
, (i
=1, 2, 3, 4) depend on the background model and their explicit
form can be found in [118
,
5
]. The functions
S
and
F
correspond to the perturbations of the spacetime while the
function
H
is proportional to the density perturbation and is only defined
on the background star. With
we define the speed of sound and with a prime we denote
differentiation with respect to
:
Outside the star there are only perturbations of the
spacetime. These are described by a single wave equation, the
Zerilli equation mentioned in the previous section, see
equations (21) and (24
). In [118] it was shown that (for background stars whose boundary density
is positive) the above system - together with the geometrical
transition conditions at the boundary of the star and regularity
conditions at the center - admits a well posed Cauchy problem.
The constraint is preserved under the evolution. We see that two
variables propagate along light characteristics and the density
H
propagates with the sound velocity of the background star.
It is possible to eliminate the constraint - first done by
Moncrief [152] - if one solves the constraint (54) for
H
and puts the corresponding expression into
. (The characteristics for
F
change then to sound characteristics inside the star and light
characteristics outside.) This way one has just to solve two
coupled wave equations for
S
and
F
with unconstrained data, and to calculate
H
using the constraint from the solution of the two wave
equations. Again the explicit form of the equation can be found
in [5
].
Turning next to quasi-normal modes in the spirit of section 2, we can Laplace transform the two wave equations and obtain a system of ordinary differential equations which is of fourth order. The Green function can be constructed from solutions of the homogeneous equations (having the appropriate behavior at the center and infinity) and its analytic continuation may have poles defining the quasi-normal mode frequencies.
From the form of the above equations one can easily see two
limiting cases. Let us first assume that the gravitational field
is very weak. Then equation (51) and (52
) can be omitted (actually
in the weak field limit [200
,
5
]) and we find that one equation is enough to describe (with
acceptable accuracy) the oscillations of the fluid. This approach
is known as the Cowling approximation [64
]. Inversely, we can assume that the coupling between the two
equations (51
) and (52
) describing the spacetime perturbations with the equation (53
) is weak and consequently derive all the features of the
spacetime perturbations from only the two of them. This is what
is called the ``inverse Cowling approximation'' (ICA) [22
].
For the axial case the perturbations reduce to a single wave equation for the spacetime perturbations which describes toroidal deformations
where
. Outside the star, pressure and density are zero and this
equation is reduced to the Regge-Wheeler equation, see
equations (21
) and (24
). In Newtonian theory, if the star is non-rotating and the
static model is a perfect fluid (i.e. shear stresses are absent),
the
axial
oscillations are a trivial solution of zero frequency to the
perturbation equations and the variations of pressure and density
are zero. Nevertheless, the variation of the velocity field is
not zero and produces non-oscillatory eddy motions. This means
that there are no oscillatory velocity fields. In the
relativistic case the picture is identical [202
] nevertheless; in this case there are still QNMs, the ones that
we will describe later as ``
spacetime or
w
-modes
'' [125
].
When the star is set in slow rotation then the axial modes are
no longer degenerate, but instead a new family of modes emerges,
the so-called
r
-modes. An interesting property of these modes that has been
pointed out by Andersson [14,
94] is that these modes are generically unstable due the
Chandrasekhar-Friedman-Schutz instability [53,
95
] and furthermore it has been shown [23
,
142
] that these modes can potentially restrict the rotation period
of newly formed neutron stars and also that they can radiate away
detectable amounts of gravitational radiation [161
]. The equations describing the perturbations of slowly rotating
relativistic stars have been derived by Kojima [120,
121], and Chandrasekhar and Ferrari [61].
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Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |