The deformation of each body by the tidal forces of its companion implies that the Kepler’s Third Law is no longer satisfied. It is then necessary to carry out numerical calculations to solve the problem. In the case of two stars, the system is called a tight binary system. The theoretical astrophysicists have been interested for a few years in somewhat peculiar tight binary systems, namely those made up of two black holes. These systems indeed constitute one of the principal sources of gravitational waves expected by the interferometric detectors VIRGO and LIGO, currently in construction. Gravitational waves are deformations of space-time which propagate at the speed of light. They were predicted by Einstein in 1918, as direct consequences of his theory of General Relativity. A system of two very relativistic objects, such as black holes, emits gravitational waves and thus loses energy and angular momentum. Consequently, the orbits described by each body are not exactly circular but form a slow spiral which leads to the coalescence of the two black holes. It is this cataclysmic event which is hoped to be observed using VIRGO and LIGO. A first step in the study of this phenomenon is the calculation of the orbits of the two black holes, neglecting the reaction to the gravitational radiation, i.e. by approximating the orbits with exact circles. Not only it is a tight binary system, not obeying the Kepler’s Third Law, but also an additional complication comes from the need to consider General Relativity. This results from the very nature of black holes, that the Newtonian theory of the gravitation cannot describe. Researchers from the Department of Relativistic Astrophysics and Cosmology of the Observatory of Paris have just carried out such a computation. The structure of space-time containing two black holes had already been computed by American groups, but for arbitrary motions, not expected in nature. Realistic motions must be circular orbits, the emission of gravitational waves transforming initially elliptic orbits into circular ones. The researchers of Paris Observatory obtained configurations which describe a system of two identical black holes on circular orbits. They calculated a sequence of such configurations by gradually decreasing the distance between the two black holes, in order to simulate the evolution of the system under the effect of energy loss through gravitational radiation. The orbital frequency thus obtained strongly deviates from that given by Kepler’s Third Law (by more than 100% when the black holes are close to each other). These configurations will be used as initial conditionsto compute the coalescence of the two black holes, by the researchers of the European Network of Gravitational Wave Astrophysics.
FIRST COMPUTATIONS OF THE ORBITS IN A BINARY BLACK HOLE SYSTEM
In Newtonian theory of gravitation, the determination of the orbits in a system of two point-like bodies is an elementary problem. The classical result, known as the Kepler’s Third Law, stipulates that for two bodies in circular orbit around their common centre of mass, the square of the orbital frequency is proportional to the sum of their masses and inversely proportional to the cube of their distance. When the sizes of the two bodies are not negligible compared to the distance between them, they cannot any more be considered as point-like.