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 (17View Equation). Inside the star, assuming that the stellar material is behaving like an ideal fluid, we define the energy momentum tensor
T μν = (ρ + p)u μuν + pgμν, (44 )
where p(r) is the pressure, ρ(r) 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,
− λ 2m (r) e = 1 − ------, (45 ) r
and the “mass inside radius r” is represented by
∫ r m (r) = 4π ρr2dr. (46 ) 0
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
dp- ρ-+-p-dν- dr = − 2 dr, (47 )
dν- 2eλ(m--+-4πpr3)- dr = r2 . (48 )
These equations should of course, be supplemented with an equation of state p = p (ρ, ...) 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 p = K ρ1+1∕N 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

( μ 8πG-- μ) δ Gν − c4 Tν = 0, (49 )
and the variation of the fluid equations of motion
( μ ) δ T ν;μ = 0, (50 )
while the perturbed metric will be given by Equation (18View 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:

2 2 − -1 ∂-S-+ ∂-S-+ L (S,F, ℓ) = 0, (51 ) c2 ∂2t ∂2r∗ 1
2 2 − 1-∂-F- + ∂-F--+ L2 (S,F, H,ℓ) = 0, (52 ) c2∂2t ∂2r∗
2 2 − --1--∂-H--+ ∂-H--+ L (H, H ′,S,S ′,F, F′,ℓ) = 0, (53 ) (cs)2 ∂2t ∂2r∗ 3
and the constraint
∂2F-- ′ ′ ∂2r∗ + L4(F,F ,S,S ,H, ℓ) = 0. (54 )
The linear functions Li, (i = 1,2,3,4) depend on the background model and their explicit form can be found in [118Jump To The Next Citation Point6Jump To The Next Citation Point]. 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 cs we define the speed of sound and with a prime we denote differentiation with respect to r∗:
-∂--= e(v−λ)∕2-∂-. (55 ) ∂r∗ ∂r
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 (21View Equation) and (24View 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 (54View Equation) for H and puts the corresponding expression into L2. (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 [6Jump To The Next Citation Point].

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 (51View Equation) and (52View Equation) can be omitted (actually S → 0 in the weak field limit [200Jump To The Next Citation Point6Jump To The Next Citation Point]) 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]. Inversely, we can assume that the coupling between the two Equations (51View Equation) and (52View Equation) describing the spacetime perturbations with the Equation (53View 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].

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

2 2 v[ ] − -1 ∂-X--+ ∂-X--+ e- ℓ(ℓ + 1)r + r3(ρ − p) − 6M = 0, (56 ) c2 ∂2t ∂2r∗ r3
where X ∼ h rϕ. Outside the star, pressure and density are zero and this equation is reduced to the Regge–Wheeler equation, see Equations (21View Equation) and (24View 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] 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].

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 [1494] is that these modes are generically unstable due the Chandrasekhar–Friedman–Schutz instability [53Jump To The Next Citation Point95Jump To The Next Citation Point] and furthermore it has been shown [23Jump To The Next Citation Point142Jump To The Next Citation Point] 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]. The equations describing the perturbations of slowly rotating relativistic stars have been derived by Kojima [120121], and Chandrasekhar and Ferrari [61].

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