4.2 Mode Analysis4 Quasi-Normal Modes of 4 Quasi-Normal Modes of

4.1 Stellar Pulsations: The Theoretical Minimum 

For the study of stellar oscillations we shall consider a spherically symmetric and static spacetime which can be described by the Schwarzschild solution outside the star, see equation (17Popup Equation). Inside the star, assuming that the stellar material is behaving like an ideal fluid, we define the energy momentum tensor

equation494

where p (r) is the pressure, tex2html_wrap_inline3184 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,

equation498

and the ``mass inside radius r '' is represented by

equation502

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

equation504

where

equation509

These equations should of course, be supplemented with an equation of state tex2html_wrap_inline3192 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 tex2html_wrap_inline3194 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

equation515

and the variation of the fluid equations of motion

equation523

while the perturbed metric will be given by equation (18Popup Equation).

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 :

  equation531

  equation537

  equation543

and the constraint

  equation549

The linear functions tex2html_wrap_inline3202, (i =1, 2, 3, 4) depend on the background model and their explicit form can be found in [118Jump To The Next Citation Point In The Article, 5Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline3212 we define the speed of sound and with a prime we denote differentiation with respect to tex2html_wrap_inline2874 :

equation554

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 (21Popup Equation) and (24Popup Equation). 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 (54Popup Equation) for H and puts the corresponding expression into tex2html_wrap_inline3220 . (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 [5Jump To The Next Citation Point In The Article].

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 (51Popup Equation) and (52Popup Equation) can be omitted (actually tex2html_wrap_inline3230 in the weak field limit [200Jump To The Next Citation Point In The Article, 5Jump To The Next Citation Point In The Article]) 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 [64Jump To The Next Citation Point In The Article]. Inversely, we can assume that the coupling between the two equations (51Popup Equation) and (52Popup Equation) describing the spacetime perturbations with the equation (53Popup Equation) 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) [22Jump To The Next Citation Point In The Article].

For the axial case the perturbations reduce to a single wave equation for the spacetime perturbations which describes toroidal deformations

  equation575

where tex2html_wrap_inline3232 . Outside the star, pressure and density are zero and this equation is reduced to the Regge-Wheeler equation, see equations (21Popup Equation) and (24Popup Equation). 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 [202Jump To The Next Citation Point In The Article] nevertheless; in this case there are still QNMs, the ones that we will describe later as `` spacetime or w -modes '' [125Jump To The Next Citation Point In The Article].

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 [53Jump To The Next Citation Point In The Article, 95Jump To The Next Citation Point In The Article] and furthermore it has been shown [23Jump To The Next Citation Point In The Article, 142Jump To The Next Citation Point In The Article] 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 [161Jump To The Next Citation Point In The Article]. The equations describing the perturbations of slowly rotating relativistic stars have been derived by Kojima [120, 121], and Chandrasekhar and Ferrari [61].



4.2 Mode Analysis4 Quasi-Normal Modes of 4 Quasi-Normal Modes of

image 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
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