4. The gravitational strain
4.1. The equations
As 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)
where i is the angle between the spin axis and the wave propagation vector, assumed to coincide with the line of sight,
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 i of the spin axis as well for the orientation of the polarization angle . Since the signal is modulated at twice the rotation frequency, in general much shorter than the sidereal period, we have assumed also a random phase for the relative orientation of the detector with respect to the equatorial coordinate system. In the other case, the procedure adopted to calculate the total square of the strain was the following: first, we have squared equ.(19) and then performed an average in the time interval , satisfying the condition, 0.4 s one day. In this case, the cross terms give a null contribution and we obtain
This equation was used to compute the contribution from all simulated objects satisfying P 0.4 s, assuming again a random orientation for i and .
4.2. The results
The 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 h . For each experiment we have computed the distribution of values of h and then averaged the results of 500 experiments. Error bars indicate the rmsd for each bin of thickness = 0.20. The resulting distribution can be represented quite well by a Laplace law, namely
where x = log. This function corresponds to calculations performed with an ellipticity = 10-6. For other values of , it is enough to replace the constant in Eq. (21) by 23.7 log(). The probability per pulsar to have a signal above x0 is
For a continuous source, VIRGO (or LIGO) will be able to detect amplitudes of the order of h , with integration times of about 2-3 yr. From the equation above and the population number estimated previously, one should expect to detect about 2 objects if or about 12-18 pulsars if . If the average ellipticity is smaller than 10-6, most of the objects will be below the detection threshold, at least for the present antenna sensibility.
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 10-4.
© European Southern Observatory (ESO) 2000
Online publication: June 30, 2000