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1.2 Problem posed by compact binary systems

Inspiralling compact binaries, containing neutron stars and/or black holes, are promising sources of gravitational waves detectable by the detectors LIGO, VIRGO, GEO and TAMA. The two compact objects steadily lose their orbital binding energy by emission of gravitational radiation; as a result, the orbital separation between them decreases, and the orbital frequency increases. Thus, the frequency of the gravitational-wave signal, which equals twice the orbital frequency for the dominant harmonics, “chirps” in time (i.e. the signal becomes higher and higher pitched) until the two objects collide and merge.

The orbit of most inspiralling compact binaries can be considered to be circular, apart from the gradual inspiral, because the gravitational radiation reaction forces tend to circularize the motion rapidly. For instance, the eccentricity of the orbit of the Hulse-Taylor binary pulsar is presently e0 = 0.617. At the time when the gravitational waves emitted by the binary system will become visible by the detectors, i.e. when the signal frequency reaches about 10 Hz (in a few hundred million years from now), the eccentricity will be -6 e = 5.3 × 10 - a value calculated from the Peters [177] law, which is itself based on the quadrupole formula (2View Equation).

The main point about modelling the inspiralling compact binary is that a model made of two structureless point particles, characterized solely by two mass parameters m1 and m2 (and possibly two spins), is sufficient. Indeed, most of the non-gravitational effects usually plaguing the dynamics of binary star systems, such as the effects of a magnetic field, of an interstellar medium, and so on, are dominated by gravitational effects. However, the real justification for a model of point particles is that the effects due to the finite size of the compact bodies are small. Consider for instance the influence of the Newtonian quadrupole moments Q1 and Q2 induced by tidal interaction between two neutron stars. Let a 1 and a 2 be the radius of the stars, and L the distance between the two centers of mass. We have, for tidal moments,

5 5 Q = k m a1-, Q = k m a2-, (6) 1 1 2L3 2 2 1L3
where k 1 and k 2 are the star’s dimensionless (second) Love numbers [162], which depend on their internal structure, and are, typically, of the order unity. On the other hand, for compact objects, we can introduce their “compactness”, defined by the dimensionless ratios
K1 = Gm1--, K2 = Gm2--, (7) a1c2 a2c2
which equal ~ 0.2 for neutron stars (depending on their equation of state). The quadrupoles Q 1 and Q2 will affect both sides of Equation (5View Equation), i.e. the Newtonian binding energy E of the two bodies, and the emitted total gravitational flux L as computed using the Newtonian quadrupole formula (4View Equation). It is known that for inspiralling compact binaries the neutron stars are not co-rotating because the tidal synchronization time is much larger than the time left till the coalescence. As shown by Kochanek [147] the best models for the fluid motion inside the two neutron stars are the so-called Roche-Riemann ellipsoids, which have tidally locked figures (the quadrupole moments face each other at any instant during the inspiral), but for which the fluid motion has zero circulation in the inertial frame. In the Newtonian approximation we find that within such a model (in the case of two identical neutron stars) the orbital phase, deduced from Equation (5View Equation), reads
{ } finite size 1 (x )5 f - f0 = - ---5/2 1 + constk --- , (8) 8x K
where x = (Gmw/c3)2/3 is a standard dimensionless post-Newtonian parameter ~ 1/c2 (w is the orbital frequency), and where k is the Love number and K is the compactness of the neutron star. The first term in the right-hand side of Equation (8View Equation) corresponds to the gravitational-wave damping of two point masses; the second term is the finite-size effect, which appears as a relative correction, proportional to (x/K)5, to the latter radiation damping effect. Because the finite-size effect is purely Newtonian, its relative correction 5 ~ (x/K) should not depend on c; and indeed the factors 1/c2 cancel out in the ratio x/K. However, the compactness K of compact objects is by Equation (7View Equation) of the order unity (or, say, one half), therefore the 1/c2 it contains should not be taken into account numerically in this case, and so the real order of magnitude of the relative contribution of the finite-size effect in Equation (8View Equation) is given by 5 x alone. This means that for compact objects the finite-size effect should be comparable, numerically, to a post-Newtonian correction of magnitude x5 ~ 1/c10 namely 5PN order3. This is a much higher post-Newtonian order than the one at which we shall investigate the gravitational effects on the phasing formula. Using k'= _ const k ~ 1 and K ~ 0.2 for neutron stars (and the bandwidth of a VIRGO detector between 10 Hz and 1000 Hz), we find that the cumulative phase error due to the finite-size effect amounts to less that one orbital rotation over a total of ~ 16, 000 produced by the gravitational-wave damping of point masses. The conclusion is that the finite-size effect can in general be neglected in comparison with purely gravitational-wave damping effects. But note that for non-compact or moderately compact objects (such as white dwarfs for instance) the Newtonian tidal interaction dominates over the radiation damping.

The inspiralling compact binaries are ideally suited for application of a high-order post-Newtonian wave generation formalism. The main reason is that these systems are very relativistic, with orbital velocities as high as 0.5c in the last rotations (as compared to -3 ~ 10 c for the binary pulsar), and it is not surprising that the quadrupole-moment formalism (2View Equation, 3View Equation, 4View Equation, 5View Equation) constitutes a poor description of the emitted gravitational waves, since many post-Newtonian corrections play a substantial role. This expectation has been confirmed in recent years by several measurement-analyses [77Jump To The Next Citation Point78Jump To The Next Citation Point111Jump To The Next Citation Point79Jump To The Next Citation Point203Jump To The Next Citation Point183Jump To The Next Citation Point184Jump To The Next Citation Point152Jump To The Next Citation Point92Jump To The Next Citation Point], which have demonstrated that the post-Newtonian precision needed to implement successively the optimal filtering technique in the LIGO/VIRGO detectors corresponds grossly, in the case of neutron-star binaries, to the 3PN approximation, or 6 1/c beyond the quadrupole moment approximation. Such a high precision is necessary because of the large number of orbital rotations that will be monitored in the detector’s frequency bandwidth (~ 16,000 in the case of neutron stars), giving the possibility of measuring very accurately the orbital phase of the binary. Thus, the 3PN order is required mostly to compute the time evolution of the orbital phase, which depends, via the energy equation (5View Equation), on the center-of-mass binding energy E and the total gravitational-wave energy flux L.

In summary, the theoretical problem posed by inspiralling compact binaries is two-fold: On the one hand E, and on the other hand L, are to be deduced from general relativity with the 3PN precision or better. To obtain E we must control the 3PN equations of motion of the binary in the case of general, not necessarily circular, orbits. As for L it necessitates the application of a 3PN wave generation formalism (actually, things are more complicated because the equations of motion are also needed during the computation of the flux). It is quite interesting that such a high order approximation as the 3PN one should be needed in preparation for LIGO and VIRGO data analysis. As we shall see, the signal from compact binaries contains at the 3PN order the signature of several non-linear effects which are specific to general relativity. Therefore, we have here the possibility of probing, experimentally, some aspects of the non-linear structure of Einstein’s theory [47Jump To The Next Citation Point48Jump To The Next Citation Point].

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