Clifford Will, Washington University, Saint Louis

cmw@howdy.wustl.edu

Gravitomagnetism has a history that is at least as long as that of general relativity itself. The idea that mass currents might generate the gravitational analogue of magnetic fields, and crude experiments to look for such effects predated Einstein. Soon after the publication of general relativity (GR), Lense and Thirring calculated the advance of the pericenter and line of nodes of a particle orbiting a rotating mass.

The gravitomagnetic ``dragging of inertial frames'' by rotating matter has played a part in discussions about the meaning and usefulness of Mach's principle, in astrophysical models of jets near accreting, rotating black holes, and in proposals for testing alternative theories of gravity.

It is no surprise then, that substantial effort during the past 30 or so years has gone into trying to measure gravitomagnetism. A recent preprint by Ignazio Ciufolini and colleagues [1] claims to have succeeded.

There are three main effects of gravitomagnetism in the solar-system:

1. * Precession of a gyroscope*. In the field of a body with angular
momentum , a gyroscope at a distance **r**
precesses with an angular velocity given
by
(**G=c=1**) where denotes the coefficient of frame dragging (1
in GR, in the PPN framework).
For a gyroscope in a polar
Earth orbit at 600 km altitude, the rate is 43 milliarcseconds
(mas) per year.

2. * Precession of orbital planes*.
The orbit of a particle
is a ``gyroscope'', whose axis or ``node'' (intersection of the orbit
with a reference plane) will also precess.
The rate is given by
where **a** and **e** are the semi-major axis and eccentricity of the
orbit.
For a satellite at 5000 km altitude, it amounts to about 31
mas per year.

3. * Precession of the pericenter*. In the field of a rotating body
there is an advance of
where **I** is the orbital inclination.

Since the early 1960's, measurement of the first effect has been the goal of the Stanford Gyroscope experiment (Gravity Probe B). The goal is to measure the precession of an array of gyroscopes in low Earth orbit to better than one percent. Following years of financial uncertainty, the project was endorsed in 1995 by a panel convened by the National Academy of Sciences [2], and NASA Administrator Daniel Goldin made a firm commitment to the mission. The spacecraft and payload are under construction at Stanford and Lockheed-Martin, and the project is actually slightly ahead of schedule for launch in December 1999 [3].

The paper by Ciufolini * et al.* is based on measuring
the second effect, the nodal precession. The original idea was
proposed in the late 1950s by Husein Yilmaz, and later embellished
by Richard Van Patten and Francis Everitt: measure the
precession of the plane of a satellite in polar orbit.
The multipole moments of the
Earth's gravitational field also induce orbital precession via standard
Newtonian gravity, but for polar orbits, the effects
vanish. It's crucial to suppress the Newtonian effects,
because they amount to about degrees per year.
(At 12 degrees inclination, the precession is 360 degrees
per year, permitting sun-synchronous orbits.)

Ciufolini proposed a generalization of the Yilmaz-Van Patten-Everitt idea. Since the effect of the even-order Newtonian multipoles is proportional to functions of , one can cancel the Newtonian effects using two satellites in orbits whose inclinations are supplementary (). (The Earth's odd-order multipoles, are not important). He then noted that there already existed one satellite for this purpose: the Laser Geodynamic Satellite (LAGEOS), a massive, 60 cm diameter sphere, studded with laser retro-reflectors, which was launched into a nearly circular orbit with in 1976, and soon became a central tool in geophysics and geodynamics. Low atmospheric drag, and the centimeter accuracy of laser ranging were key to its usefulness.

All that was needed for a frame-dragging test at around a 10 percent level was a LAGEOS in an orbit of inclination. Alas, this was not to be, and when LAGEOS II was launched in 1992, geophysical and political criteria dictated . Although Ciufolini and others lobbied hard for a LAGEOS III with a suitable inclination, it has not yet materialized.

Nevertheless, Ciufolini and co-workers have argued that the situation
is not hopeless. The Earth's multipole moments are known very
accurately, from decades of accurate measurements of satellite orbits
(including LAGEOS). Moments and higher are small enough and are
known well enough that their effects can be subtracted off.
Unfortunately, and are not known quite well enough. Thus
the effective * measured* nodal precession can be viewed as a
linear combination where denote the errors in and
and is the frame-dragging coefficient to be measured.
Thus there are two measurables, but three unknowns -- it's only in the
supplementary inclination case that the linear
combinations are degenerate, and can be determined uniquely with
only two observables. Given the two non-supplementary LAGEOS
satellites, one needs a third measurable. By happy chance, LAGEOS II
turned out to have a decent eccentricity -- 0.014, as compared to
0.004 for LAGEOS I. This makes its perigee advance measurable. But
the predicted advance has a different dependence on the Earth's
moments and on frame-dragging: . According to Ciufolini * et al.*, this gives the third
measurable needed.

But this quantity is the weak link in the chain for several reasons.
First, the measured orbital displacements are proportional to and **e** is still pretty small, so while the nodal precessions
could be measured to 1 mas per year, the pericenter advance was
limited to about 10 mas per year accuracy. Second, the effects of the
odd-order moments are significant for the pericenter advance. Third,
non-gravitational perturbations of the satellite, such as those
related to radiation pressure and thermal heating, affect the
pericenter advance more strongly than they do the nodal advance.
Also, tidal, secular, and seasonal variations in all the moments must
be carefully taken into account in both nodal and pericenter
precessions. The reported result for was 1.1, with a realistic
error of about 25 percent (). By contrast, researchers at
the University of Texas argue that, in view of the many error sources,
an error of 200 percent is probably more realistic [4].

As in all such satellite experiments, with many corrections to be made and subtle systematic effects to be dealt with, more data and an independent data analysis are called for to see if a LAGEOS I & II experiment can really detect gravitomagnetism. In any case, the NASA relativity mission should be much higher precision (by a factor at least 25), thought admittedly at a much higher price tag.

** References**

[1] I. Ciufolini, D. Lucchesi, F. Vespe and F. Chieppa,
gr-qc/9704065,
submitted to * Nature*
[2] Truth in advertising: panel which included the present
correspondent.
[3] See
http://stugyro.stanford.edu/RELATIVITY
[4] J. Ries, private communication.

Wed Sep 10 15:05:58 EDT 1997