## 4. The gravitational strain## 4.1. The equationsAs we have mentioned above, pulsars could emit GW by having a time-varying quadrupole moment produced either by a slight asymmetry in the equatorial plane (assumed to be orthogonal to the spin axis) or by a misalignment between the symmetry and angular momentum axes, case in which a wobble is induced in the star motion. In the former situation the GW frequency is equal to twice the rotation frequency, whereas in the latter two modes are possible: one in which the GW have the same frequency as the rotation, and another in which the GW have twice the rotation frequency. The first mode dominates by far at small wobble angles while the importance of the second increases for large values. Here we neglect the possible precessional motion and, in this case, the two polarization components of GW emitted by a rotating neutron star are (Zimmerman & Szedenits 1979; Bonazzola & Gourgoulhon 1996) and where G is the gravitation constant, c is the velocity of light, r is the distance to The source, is the angular rotation velocity of the pulsar and the ellipticity is defined as with the being the principal moments of inertia of the star. The detected signal by an interferometric antenna is where and are the beam factors of the interferometer, which are functions of the zenith distance , the azimuth as well as of the wave polarization plane orientation . Notice that the angles and are functions of time due to the Earth's rotation, introducing a modulation of the signal. The explicit functions for the beam factors, taking into account the geographic localization and the orientation of the VIRGO antenna were taken from Jaranowski et al. (1998). Concerning the detection strategy, recall that the population
derived from our simulations of potential GW emitters is in the range
5100 - 7800. In this case, according to the conclusions by GBG97, it
becomes advantageous to search for individual detections instead of
the total square amplitude. Here we have simulated both strategies. In
the former case, the strain amplitude was calculated for each pulsar
satisfying the condition P 0.4 s,
using Eq. (19) and assuming a random orientation for the inclination
This equation was used to compute the contribution from all
simulated objects satisfying P
0.4 s, assuming again a random orientation for ## 4.2. The resultsThe main differences between the present approach and previous calculations should be emphasized. Our procedure allows a more realistic estimate of the rotation period distribution, as well the number of pulsars able to contribute to the gravitational strain. Additionally the spatial distribution of those pulsars throughout the galactic disc changes for each simulation, although their average properties remaining constant. It is thus preferable to present the statistics of our numerical experiments, from which is possible to estimate the probability of having a signal above a given threshold. In Fig. 7 we give the statistics for the gravitational strain
where x = log. This function
corresponds to calculations performed with an ellipticity
= 10 For a continuous source, VIRGO (or LIGO) will be able to detect
amplitudes of the order of
In Fig. 8 we present the statistics for the square of the
amplitude, resulting from the different spatial distribution of the
pulsars in each numerical experiment. We emphasize that the amplitude
of the signal is due essentially to a few pulsars, in agreement with
the conclusions derived from the statistics of single objects and with
the analytical study by de Freitas Pacheco & Horvath (1997). As a
consequence, the sidereal modulation is not fixed by the galactic
center-anticenter asymmetry, but by those few dominant objects. No
typical modulation curve was obtained, since the relative positions of
these pulsars vary from experiment to experiment. Using the equations
by GBG97 for the signal-to-noise ratio, one should expect to detect a
signal of with the presently planned
VIRGO sensibility. Our simulations indicate that, if
a signal of such an amplitude has
weak probability to be detected (about 1/100) and signals of the
required amplitude can only be obtained if the average pulsar
ellipticity is of the order of 10
© European Southern Observatory (ESO) 2000 Online publication: June 30, 2000 |