Chapter 5. Location of the Innermost Stable Circular Orbit of Binary Neutron Stars in the Post Newtonian Approximations of General Relativity

In this paper, we present results obtained from our recent studies on thelocation of the innermost stable circular orbit (ISCO) for binary neutron stars(BNSs) in several levels of post Newtonian (PN) approximations. We reach thefollowing conclusion at present: (1) even in the Newtonian case, there existsthe ISCO for binary of sufficiently stiff equation of state (EOS). If the massand the radius of each star are fixed, the angular velocity at the ISCO$\Omega_{ISCO}$ is larger for softer EOS: (2) when we include the first PNcorrection, there appear roughly two kinds of effects. One is the effect to theself-gravity of each star of binary and the other is to the gravity actingbetween two stars. Due to the former one, each star of binary becomes compactand the tidal effect is less effective. As a result, $\Omega_{ISCO}$ tends tobe increased. On the other hand, the latter one has the property to destabilizethe binary orbit, and $\Omega_{ISCO}$ tends to be decreased. If we take intoaccount both effects, however, the former effect is stronger than the latterone, and $\Omega_{ISCO}$ becomes large with increase of the 1PN correction: (3)the feature mentioned above is more remarkable for softer EOS if the mass andradius are fixed. This is because for softer EOS, each star has the largercentral density and is susceptible to the GR correction: (4) there has been noself consistent calculation including all the 2PN effects and only existstudies in which one merely includes the effect of the 2PN gravity actingbetween two stars. In this case, the effect has the property to destabilize thebinary orbit, so that $\Omega_{ISCO}$ is always smaller than that for theNewtonian case. If we include the PN effect of the self-gravity to each star,$\Omega_{ISCO}$ will increase.Comment: 33 pages ptptex file, 29 figures, to appear in Progress of Theoretical Physics Supplement No.128 (1997) `Perturbative and Numerical Approaches to Gravitational Radiation