Sharon Morsink, University of Wisconsin, Milwaukee
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 0.1 ms. Consider motion occurring at r=6M outside of a
neutron star:
test particles orbit with a frequency of ~ 1 kHz 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 more than 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. (Note, however, that there have been some evidence [4] that
the separation between the twin peaks is not constant, which casts
some doubt on the beat frequency 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 [5] 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 [6],
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 [7],
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 [8] in the situation. Bardeen and
Petterson showed
[9] 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, the standard wisdom is that the
inner part of the disk,
up to 100M must be co-planar.
Recent work [10,11] has shown that, in fact, warped disk solutions can exist
in the inner region. Moreover, they seem to suggest that it would be
quite natural for the oscillations to be produced at second harmonic of
the precession frequency, exactly what is needed!
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] D. Psaltis et. al.
astro-ph/9805084
[5] L. Stella and M. Vietri,
astro-ph/9709085
[6] S.M. Morsink and L. Stella, ApJ, submitted.
[7] W. Cui, S.N. Zhang and W. Chen,
astro-ph/9710352
[8] J. Heller, Catch 22, 1961.
[9] J.M. Bardeen and J.A. Petterson, ApJ 195, L65 (1975).
[10] D. Markovic and F.K. Lamb,
astro-ph/9801075
[11] M. Vietri and L. Stella,
astro-ph/9803089
updated July 17, 1998