Doug Eardley, ITP, UCSB
doug@itp.ucsb.edu
In decaying binary systems consisting of neutron stars or black holes, a lot can happen near the innermost stable circular orbit, or ISCO, as a number of theorists have shown over the last couple of years. A number of questions still remain open though: Where the ISCO is located, what physics determines it, and what observable effects ensue. First of all, the ISCO is not even precisely defined.
For the special case of test particle orbiting around a black hole (or sufficiently compact neutron star) the ISCO is well defined, as we all learned in our relativity courses: at r=6M for a non-rotating black hole, or r=M for a maximally rotating black hole. But the binary orbit of two massive stars evolves due to gravitational radiation reaction, and the ISCO is really the fuzzily-defined radius where slow orbital shrinkage goes over to rapid plunge. Because energy loss by gravitational waves is a bit inefficient even under the best of circumstances, the orbital shrinkage is probably slow enough that the ISCO is reasonably well defined in practice: For instance, if a distant gravity wave observer plots plots gravity wave amplitude as a function of time, a distinct feature will appear. The ``ISCO radius" is a phrase commonly used, but must be deprecated because there is no unique (or at least generally agreed upon) way to measure it invariantly.
Sharpening the issue further is its relevance to the Grand Challenge, as was reported by Sam Finn in MOG8. The first 3D codes able to evolve such a binary must begin near the ISCO, so the numerical relativists need to know where it is, and how to model the system for initial conditions near it. The gravity waveforms observable in LIGO, VIRGO and LISA will depend on this issue (Flanagan & Hughes, gr-qc/9701039), and so may gamma-ray bursts (Rees, astro-ph/9701162).
The purpose of this report is to give a quick and nontechnical introduction to these rapidly evolving issues, along with links to the literature for those who want to delve into the whole story. I have probably missed some references; if so, let me know.
The earliest approach to the ISCO is the post-Newtonian one, which has gone through a long development, most notably by Will and collaborators. Two recent papers which reference this body of work are Lai & Wiseman (gr-qc/9609014), and Will & Wiseman ( Phys. Rev. D54, p.4813, 1996; gr-qc/9608012). If one ignores radiation, the ISCO is well defined here, and if radiation reaction is included, the ISCO is indeed ``reasonably well defined" ( Phys. Rev. D47, p.3281, 1993). Sensible answers emerge, but the calculations must be carried to high order, and theoretically speaking it not clear they converge. For binaries involving neutron stars, an important physical effect is the tidal interaction between the two stars, which may disrupt the stars (as pointed out in Newtonian theory by Dai, Rasio and Shapiro, Ap.J. 420, p.811, 1994) before the ISCO is reached, and which also can affect the location of the ISCO.
More recently, Taniguchi and Nakamura
(astro-ph/9609009)
have developed an analytic method based on fluid ellipsoids and
a pseudo-relativistic form of the Newtonian gravitational potential,
of the form 
. This potential doesn't
enjoy a clear justification from GR theory (beyond a reasonably
accurate reproduction of test particle orbits) but at least it has
the virtue of simplicity. They are able to do a parameter study
of tidal effects as a function of orbital compactness, and they clearly
show in their models the dividing line between cases where tidal effects
dominate in creating the ISCO, and cases where relativistic orbital
effects dominate.
Turning to full GR theory, one way to make the ISCO well defined is to put our binary into a large cavity with walls made of a mythical material that perfectly reflects gravity waves, and seek a rigidly rotating configuration of stars and standing waves. (This idea goes back 30 years to Thorne and nonspherical modes of neutron stars.) If the cavity is too big, then the total mass-energy of the standing waves dominates the problem --- in the limit of an infinite cavity, we get a non-asymptotically flat spacetime. However, thanks to GW inefficiency, the cavity can be made rather large. The best approach of this kind seems that of Blackburn and Detweiler ( Phys. Rev. D46, p.2318, 1992). When applied to the ISCO radius, though, it gives a value which is much lower than that from any other method.
Another way to define the ISCO precisely is to go to some kind of
radiation-less approximation to full GR. (The waves can then be painted
in later.) The most ambitious version to date of such an approach is that
of Wilson & Mathews ( Phys. Rev. Lett. 75, p.4161, 1995),
and Wilson, Mathews, & Marronetti ( Phys. Rev. D54, p.1317,
1996;
gr-qc/9601017),
who numerically construct a family of curved but non-radiating spacetimes
containing fully hydrodynamic neutron stars. They find a rather large ISCO
radius --- in fact, large enough that the orbital angular momentum J still
exceeds 
, where M is the total mass-energy of the binary system, so
that the stars are forbidden to plunge directly to a Kerr black hole!
Another important result comes out of their work: The neutron stars
themselves may go radially unstable and begin collapsing to black holes,
before the ISCO is reached. The neutron star binary problem may reduce
itself to the black hole binary problem! If so, matter is removed from
the game, and cannot carry angular momentum to infinity, or help make gamma
ray bursts. Again, however, it's not clear how good the approximation
is. For further aspects see
gr-qc/9512009,
gr-qc/9601019,
gr-qc/9603043,
gr-qc/9701033.
As should be clear, a number of resourceful groups have studied this
problem by an amazing variety of methods, none rigorous to date --- and
it is sometimes hard to inter-compare results. Here is a suggestion:
Everyone at work on this problem will benefit if all groups report,
at a minimum, the invariant observables M, J, and 
(noting
that quadrupole gravity waves will show up at 
), for each
orbital configuration, and especially for the ISCO:
For a given binary, the evolutionary sequence of orbital configurations
should obey the Thorne-Zel'dovich law, 
. I say
should rather than does, because to my knowledge no-one has
proved this law for configurations which are not strictly stationary, rigidly
rotating, asymptotically flat solutions of the full Einstein equations ---
so binaries are not yet covered, and in particular, radiation-less schemes
are not yet covered. This provides a good sanity check on the numerical
calculations, assuming validity. To prove more general validity poses a
good research problem.
In summary, here is a list of important questions that remain open:
? ---
If so, do unexpected GW signals emerge after the ISCO to carry off the
excess J?