Relativistic stellar perturbations

We have integrated the equations describing a binary system revolving in an eccentric or circular orbit [2]. These equations have been derived by a perturbative approach in the frequency domain, assuming that one of the two objects is a point-like mass which induces a perturbation on the gravitational field and on the internal structure of the other. This approach constitutes a progress with respect to the commonly used assumption that both stars are point-like masses, since we treat at least one of the two stars, whose internal structure and whose gravitational field are solutions of the fully non-linear equations of gravity, in an exact manner.

We have applied this formalism to study the perturbations of a solar-type star excited by a close orbiting planet [1]. The integration has been performed in the frequency domain. We have computed the energy spectra of the radiation emitted in gravitational waves. The purpose of this study was to understand whether the extra-solar planetary systems that have recently been discovered in large number in our galaxy could be interesting sources of gravitational waves. These systems are of particular interest because they are very close (a few parsecs from Earth), and in some of them the planet is on such a narrow orbit that the frequency of emission could be in the bandwidth of the space-based interferometer LISA. In particular we have investigated the possibility that a planet is on an orbit so close that, without being disrupted by tidal forces, it may significantly excite the stellar g-modes. We find that gravitational emission is significantly enhanced if the planet is a brown dwarf, and in this case it could move on an orbit resonant with the mode of the star emitting radiation strong enough, and for a sufficiently long period of time, to be detectable by LISA.

We now plan to study the excitation of the g-modes of neutron stars and white dwarfs due to the interaction with an orbiting companion and evaluate the consequent gravitational emission. The excitation of neutron star g-modes may be interesting for VIRGO-LIGO, because the typical frequencies range within Hz, where the ground-based interferometers are most sensitive; the excitation of white dwarf g-modes may occur in close white dwarf binaries that are quite common in our Galaxy, and may be interesting for LISA.

In [2] we considered a neutron star perturbed by a point mass moving around it on a closed orbit; we found a beating effect between the and the gravitational-wave frequencies, which is clearly visible in the waveforms (figure 1) and is not predicted by the standard Post-Newtonian approach. We also found that the gravitational radiation the system emits in the multipole is smaller than that predicted by the quadrupole formalisms, which takes into account only the radiation emitted because of the orbital motion; this effects is most significant when the two stars are close (orbital radius less than ). The star we have used as a model has a polytropic equation of state with and n=2.

  
Figure 1: The component of the gravitational wave emitted when a test mass moves on a closed or quasi-periodic orbit is plotted versus the retarded time in units of the orbital period. Since we assume that the observer is on the equatorial plane, the component vanishes. The waveform on the left refers to a circular binary with e=0 and , the one on the right to an eccentric binary with e=0.4 and periastron . The solid line corresponds to the relativistic waveform and the dashed line to the waveform computed by the quadrupole approach. Only the component of the relativistic signal is shown.

This result affects the orbital evolution of the system, especially during the last phases of the coalescence of neutron star binaries, whose signal is considered as a target for the detection of gravitational waves by ground based interferometers (VIRGO-LIGO-GEO-TAMA). Therefore, we have included in our perturbation scheme the effects of radiation reaction, and we have studied the orbital evolution of the coalescing system and computed the waveform of the emitted gravitational signals. We plan to extend our investigation to understand to what extent this result depends on the equation of state and on the finite size of the star. We will study the role the EOS of neutron stars plays on the gravitational emission, by studying the possibility of characterizing the different EOS's through integral quantities (e.g. the speed of sound in matter integrated over the whole stellar volume), to circumvent the ambiguity arising from the density dependence of matter compressibility.

In order to be extracted from the unavoidable noise of a gravitational detector, the signals emitted by astrophysical sources have to be known with an extremely high accuracy. This is due to the fact that the matched filter which is used in the data analysis is very sensitive to a mismatch of the parameters, to such an extent that even a mismatch of one cycle over in the signal produced during the coalescence of a binary system degrades the signal-to-noise ratio by a factor . At present, templates for coalescing binaries can be generated either by our perturbative approach, or by the Post-Newtonian approach, therefore we will compare the waveforms we find with the waveforms predicted by the Post-Newtonian formalism and we shall evaluate the SNR of the matched-filter used to extract the signal from the detectors' noise in the two cases. As a preliminary result, in figure 2 we show the power emitted in gravitational waves (normalized to the quadrupole power) by a neutron star, a Schwarzschild black hole with the same mass and the Post-Newtonian approximation at several orders, in the test particle limit.

In collaboration with the group at the University of Thessaloniki, we are presently investigating the possibility of simulating the evolution of a binary system, as far as the emission of gravitational waves is concerned, as a process of scattering of the quadrupole wave emitted by one star, seen as an extended body in orbital motion around the other, on the potential barrier generated by the second star and viceversa. If this approach happens to be successful, we will be able to study the latest phases of the coalescence taking into account also the tidal deformations reciprocally induced by the interacting stars and the effects they produce on the emitted radiation.

 
Figure 2: The power emitted in gravitational waves by a neutron star and a Schwarzschild black hole with the same mass perturbed by an orbiting massive test particle (both obtained summing multipoles from to ) is plotted as a function of the orbital velocity , where is the gravitational wave frequency. The power is normalized to the quadrupole emission. The results are compared with the predictions of the Post-Newtonian approximation at several orders in the test particle limit. The v-range is chosen so that the orbital frequency sweeps from 10 Hz, where the signal first enters the LIGO-VIRGO sensitivity window, to the ISCO. The post-newtonian expansion slowly converges to the Schwarzschild result, and the effects of the structure of the star on the energy output is quite clear.  

 
Figure 3: If we refine the calculations for the perturbed neutron star shown in figure 2 in the region v > 0.35, we find sharp peaks which correspond to the excitations of the f-mode of the star for and 3. In this figure we plot the emitted power in that region. The frequencies of the modes are and respectively. The peaks for are extremely sharp and can be seen only with an extremely high resolution.