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Astron. Astrophys. 321, 1024-1026 (1997)

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5. Interaction with gravitational waves

The perturbation of the metric tensor associated with a weak gravitational wave propagating in the z -direction may be written as,

[EQUATION]

where [FORMULA], and [FORMULA], in a transverse-traceless gauge, is

[EQUATION]

Here [FORMULA] and [FORMULA] are the amplitudes of the two orthogonal polarizations of the wave. We shall assume [FORMULA] and the wavelength much longer than dimensions of the star (i.e. [FORMULA]. The relevant non vanishing components of the Riemann curvature tensor are then

[EQUATION]

The geodesics of a mass element at [FORMULA] will deviate from that of the center of mass at the origin upon exposure to the gravitational wave. The corresponding acceleration is

[EQUATION]

By Eq. (10), however, this can be written as

[EQUATION]

where the potential V is

[EQUATION]

Coming back to the oscillations of the star, we add the Newtonian force term associated with the acceleration of Eq. (12) to the right hand side of Eq. (1). To account for attenuation of the motion within the star we also postulate a friction force proportional to the velocity, [FORMULA], [FORMULA] =const. Thus,

[EQUATION]

This is the equation of a damped wave driven by the external force of the gravitational radiation. We solve it by expanding [FORMULA] in terms of the normal modes [FORMULA] of the [FORMULA] operator, Eq. (6). Thus

[EQUATION]

Substituting Eq. (14) in Eq. (13), multiplying the resulting expression by [FORMULA] and integrating over the volume of the star gives

[EQUATION]

5.1. Energy absorption

Time dependence of [FORMULA] and of the gravitational force is periodic. The time-averaged rate of the energy transfer to the star becomes

[EQUATION]

Substituting for [FORMULA] from Eq. (14) and simple manipulations gives

[EQUATION]

We assume that the amplitude [FORMULA] remains reasonably constant over the Lorenzian profile. One can then see that the resonances occur at [FORMULA] with half-width [FORMULA] and maxima [FORMULA]. Thus, the total energy absorption, proportional to the area [FORMULA] of the line profiles, becomes

[EQUATION]

5.2. Absorption cross section

The energy flux of the gravitational waves per unit frequency interval is

[EQUATION]

The cross section for the energy transfer from the gravitational waves to the star is the ratio of [FORMULA] to [FORMULA], the total flux across the resonant profile. Thus

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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