The Search for Frame-Dragging by Neutron Stars and Black Holes

Sharon Morsink, University of Wisconsin, Milwaukee
morsink@pauli.phys.uwm.edu

The dragging of inertial frames has been in the news lately with recent reports that the Rossi X-ray Timing Explorer (RXTE) satellite has observed the gravitomagnetic precession of the inner edge of accretion disks around neutron stars and black holes. If verified, this would be the first observation of a strong field general relativistic effect. However, the result is far from conclusive with the present data. In this report, I'll give a short review of the observations that have been made and describe some efforts to test the hypothesis that frame-dragging has been seen. For a review of frame-dragging and efforts to measure the effect due to the Earth's motion, see Cliff Will's article in MOG [1].

The truly exciting aspect of NASA's RXTE satellite is its ability to resolve time variations in the x-ray spectrum occurring on time scales of order . Consider motion occurring at r=6M outside of a neutron star: test particles orbit with a frequency of at this radius, corresponding to a time scale well within Rossi's resolution. Within the last two years, Rossi has discovered quasi-periodic oscillations (QPOs) occurring at repetition frequencies of kHz order, suggesting that they are seeing phenomena near neutron stars or black holes. A nice review of the kHz QPO phenomenology is given by van der Klis [2]. RXTE has seen kHz QPOs from 14 sources which are neutron stars in binaries. Their partners are difficult or impossible to observe, so the masses of these neutron stars aren't known. Typically, twin peaks in the Fourier analyzed x-ray spectrum are seen in these sources. (Take a look at figure 4 of reference [2] for an example.) The peaks' frequencies (approximately 1 kHz) drift with time, but their frequency separation stays constant. A model, the sonic-point beat frequency model [3] explains the twin peak phenomenon by identifying the higher frequency peak with Keplerian motion of the accretion disk's inner edge. The peak separation is identified with the star's spin frequency. This leads to star rotation periods near 3 ms. Some of these stars are occasional x-ray bursters and an analysis of the burst spectrum leads to a spin frequency which either agrees with the peak separation or with twice the peak separation providing an independent check of the model.

Suppose that the inner section of the accretion disk is tilted out of the star's equatorial plane. If this is the case, then the frame-dragging effect will cause the plane of the orbit to precess around the star, periodically obscuring the star. We would then expect to see a peak in the power spectrum occurring at a frequency corresponding to the precession frequency. It was pointed out by Luigi Stella and Mario Vietri [4] that a peak with around the correct frequency appears in the spectrum. Moreover, they provide a consistency check. As the inner edge of accretion disk changes location (due to radiation drag), the Keplerian frequency increases approximately as (remember that the star is rotating, so this is not exact). The Lense-Thirring precession varies as , where J is the star's angular momentum. Therefore, the peak which is to be identified with Lense-Thirring precession should vary as the square of the Keplerian frequency peak. The data does show this rough trend. However, it is not this simple, since the star is not spherical, and Newtonian gravity predicts a precession due to the star's quadrupole moment which subtracts from the frame-dragging precession frequency. Depending on the equation of state assumed for the neutron star, the quadrupole precession can range from a couple percent to half of the frame-dragging precession. Using a semi-Newtonian approximation, Stella and Vietri found that if the equation of state is very stiff, the data seemed to fit well. However, a more precise calculation, using general relativity [5], shows that the quadrupole (and higher multipole moments) become very important and greatly reduce the total precession. If the equation of state is not overly stiff then the frame-dragging effect is dominant when the star is close to its maximum allowable mass. For typical equations of state, the total predicted precession frequency (including all effects) is still only half of the peak's observed frequency. There is some possibility that the factor of two could be explained by a geometric effect. The system's geometry is essentially the same when the plane of the orbit has made a half period rotation, leading to a factor of two. However, this is still a bit speculative. In any case, astronomers are analyzing the RXTE data to find the observed variation of these peaks for a number of sources. If it should turn out that the dependence of the "precession" peak with the Keplerian peak is correct, up to the factor of two, there may be some truth in the model. It should be mentioned that a similar effect has been suggested in the sources which correspond to alleged black holes [6], but in these cases there are no twin peaks, so there is really no way to test the hypothesis.

There is a bit of a Catch 22 [7] in the situation. Bardeen and Petterson showed [8] that the combination of frame-dragging and viscosity produces a torque which tends to align the disk with the star's equatorial plane, so that Lense-Thirring precession won't occur. It is this effect which is thought to keep the jets seen in active galactic nuclei aligned. Although warped, precessing disks can occur, typically the inner part of the disk, up to 100M must be co-planar. If it is possible to find a physical mechanism which will cause a perturbation to lift the inner edge of the disk, there will now be another force acting on the inner edge of the disk. The precession frequencies computed assume geodesic motion, i.e., that all other forces besides gravity are negligible. If precession occurs, the frequencies may change. Some work in this direction indicates that this is the case [9,10], in fact reducing the possible frequencies by a large factor and/or damping them strongly [10]. This is not to say that the peaks observed can't be due to frame-dragging, but it is difficult to find a physical mechanism which may cause a tilt without changing the frequencies.

In the meantime, we will have to wait for further analysis to learn whether there is a statistically significant correlation between the proposed precession peak and the Kepler peak. If so, it may be possible that frame-dragging has been observed near neutron stars.

References:

[1] C. Will, The Search for Frame-Dragging, MOG No. 10, Fall 1997.
[2] M. Van der Klis, astro-ph/9710016.
[3] M.C. Miller, F.K. Lamb and D. Psaltis, astro-ph/9609157.
[4] L. Stella and M. Vietri, astro-ph/9709085.
[5] S.M. Morsink, L. Stella and M. Vietri, in preparation.
[6] W. Cui, S.N. Zhang and W. Chen, astro-ph/9710352.
[7] J. Heller, Catch 22, 1961.
] J.M. Bardeen and J.A. Petterson, ApJ 195, L65 (1975).
[9] J.R. Ipser, ApJ 458, 508 (1996); M.C. Miller, astro-ph/9801295.
[10] D. Markovic and F.K. Lamb, astro-ph/9801075



Jorge Pullin
Sun Feb 8 20:46:09 EST 1998