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 the location of the innermost stable circular orbit (ISCO) for binary neutron stars (BNSs) in several levels of post Newtonian (PN) approximations. We reach the following conclusion at present: (1) even in the Newtonian case, there exists the ISCO for binary of sufficiently stiff equation of state (EOS). If the mass and the radius of each star are fixed, the angular velocity at the ISCO ΩISCO is larger for softer EOS: (2) when we include the first PN correction, there appear roughly two kinds of effects. One is the effect to the self-gravity of each star of binary and the other is to the gravity acting between two stars. Due to the former one, each star of binary becomes compact and the tidal effect is less effective. As a result, ΩISCO tends to be increased. On the other hand, the latter one has the property to destabilize the binary orbit, and ΩISCO tends to be decreased. If we take into account both effects, however, the former effect is stronger than the latter one, and ΩISCO becomes large with increase of the 1PN correction: (3) the feature mentioned above is more remarkable for softer EOS if the mass and radius are fixed. This is because for softer EOS, each star has the larger central density and is susceptible to the GR correction: (4) there has been no self consistent calculation including all the 2PN effects and only exist studies in which one merely includes the effect of the 2PN gravity acting between two stars. In this case, the effect has the property to destabilize the binary orbit, so that ΩISCO is always smaller than that for the Newtonian case. If we include the PN effect of the self-gravity to each star, ΩISCO will increase. §