The importance of the gravitomagnetic field is discussed. A never-measured field of nature, the foundations of inertia in Einstein General Relativity, and a key role in theories of quasars and active galactic nuclei are important aspects of this field and are discussed. In high energy astrophysics, some theories of energy storage, power generation, jet formation and jet alignment of quasars and active galactic nuclei are based on the existence of the gravitomagnetic field of a supermassive black hole (Thorne et al. 1986). LAGEOS 3 is discussed in terms of laser ranged satellites to detect the gravitomagnetic field and supplementary inclination satellites to avoid gravity field uncertainties. Many experiments have been proposed to measure the gravitomagnetic field. The GPB experiment intends to measure the Lense-Thirring-Schiff precession of gyroscopes orbiting the earth. Polar satellites have been proposed to measure the Lense-Thirring precession of the orbital plane (an enormous gyroscope and two guided, drag-free, counter-rotating, polar satellites have been suggested to avoid orbital inclination errors.) The new idea to measure the gravitomagnetic drag of the nodes of two nonpolar, supplementary inclination, satellites is summarized

Ciufolini, Ignazio

English

NASA Technical Reports Server

N90-19961
LAGEOS IIIAND THE GRAVITOMAGNETIC FIELD
IGNAZIO CIUFOLINI !_:
Center for Space Research _:
The University of Texas at Austin J)
Austin, Texas 78712 _/ _
I. IMPORTANCE OF THE GRAVITOMAGNETIC FIELD
a) A Never-measured Field of Nature
In electrodynamics, in the frame at rest with an electrically charged sphere
we have an electric field. If we then rotate the sphere we observe a magnetic field,
which strength is proportional to the angular velocity.
Similarly, in Einstein geometrodynamics (Misner et al. 1973, Wheeler 1964)
(general relativity), a non-rotating, massive sphere produces the standard
Schwarzschild field. If we then rotate the sphere we have the occurrence of the
gravitomagnetic field, whose strength, in the weak field limit, is proportional to the
sphere angular velocity. In the weak field approximation of the Kerr metric, the
gravitomagnetic field is (Thorne et al. 1986):
--._ ---ff ^ ^
-_ ---) _ J-3(J • r)r
H=Vxl]--2 r 3 (1)
---) 2J
where 13 _=_(0,0,-_ is the gravitomagnetic or Lense-Thirring potential and J is ther 3 )
angular momentum of the central body.
We observe that gravitomagnetism and gravitational waves are the two main
aspects, still to be measured, of Einstein geometrodynamics analogous to magnetic
field and electromagnetic waves of electrodynamics,
b) dg the Foundations of Inertia in Einstein General Relativity
In Einstein geometrodynamics (Misner et al. 1973, Wheeler 1964), to solve the
initial value problem (York 1979, Choquet-Bruhat and York 1980) as a part of the
initial conditions, we need to specify on a Cauchy hypersurface Y. the conformal
mass-energy current density: J base. Through the initial conditions, with the field
equations, we then solve for the geometry g_tl_ of the universe and eventually, we
determine the local inertial frames (where gal_ _rla13) all over the spacetime. The axes
of the local inertial frames (gyroscopes) are therefore influenced, and partially
determined, by the mass-energy currents in the universe (dragging of inertial
frames).
This agrees with some kind of general relativistic formulation of the Mach
principle, according to the ideas of Einstein and Wheeler (Wheeler 1988, Carifolini
and Wheeler 1987). "Nothing would do more to demonstrate the inertia-influencing
effect of mass in motion than to detect and measure the Einstein-Thirring-Lense-
predicated gravitomagnetism of the earth" (Wheeler 1988).
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https://ntrs.nasa.gov/search.jsp?R=19900010645 2020-03-19T22:59:47+00:00Z
c) A Key Role in Theories of Quasars and Active Galactic Nuclei
In high-energy astrophysics, some theories of energy storage, power
generation, jet formation and jet alignment of quasars and active galactic nuclei are
based on the existence of the gravitomagnetic field of a supermassive black hole
(Thorne et al. 1986).
In particular, this field may explain the constant direction of the jets over
millions of light years and therefore over a time of several millions of years.
Through the standard Navier-Stokes equations, in which one includes the
gravitomagnetic field of the central object, one can show (Bardeen and Petterson
1975) that the accretion disk tends to be oriented into the equatorial plane of the
central body -- Bardeen-Petterson effect. The jets are then ejected normally to the
accretion disk, that is, normally to the equatorial plane of the central body. The
angular momentum vector of the central black hole acts, therefore, as a gyroscope
and this may explain the constant direction of the jets.
II. LAGEOS III
LASER RANGED SATELLITES TO DETECT THE GRAVITOMAGNETIC FIELD
AND SUPPLEMENTARY INCLINATION SATELLITES
TO AVOID GRAVITY FIELD UNCERTAINTIES
Many experiments have been proposed to measure the gravitomagnetic field.
The GPB experiment (Everitt 1974, Lipa et al. 1974) intends to measure the
Lense-Thirring-Schiff (Lense and Thirring 1918) precession of gyroscopes orbiting
the earth.
Polar satellites have been proposed to measure the Lense-Thirring precession
of the orbital plane -- an enormous gyroscope (Schiff 1960, Yilmaz 1959) and two
guided, drag-free, counter-rotating, polar satellites have been suggested to avoid
orbital inclination errors (Van Patten and Everitt 1976). We recall that a polar
satellite: Ix 90 °, has a null classical nodal precession.
Here, we briefly summarize the new idea (Ciufolini 1985, 1984, 1987, 1988) to
measure the gravitomagnetic drag of the nodes of two nonpolar, supplementary
inclination, laser ranged satellites. This idea can be decomposed into two parts:
(1)
(2)
Position measurements of laser ranged satellites, of LAGEOS (Smith and Dunn
1980, Yoder et al. 1983, Cohen et al. 1985) type, are accurate enough to detect
the tiny effect due to the gravitomagnetic field: the Lense-Thirring
precession.
To cancel out the uncertainties in the enormous perturbations due to the
nonsphericity of the earth gravity field, we need (Ciufolini 1985, 1984, 1987) a
new satellite, LAGEOS III, with semimajor axis and eccentricity, a and e, equal
to those of LAGEOS, but with supplementary inclination: 1111-=70 °.
The LAGEOS (Smith and Dunn 1980, Yoder et al. 1985) semimajor axis is
a = 12270 km, the period P = 3.758 h, the eccentricity e = 0.004 and the inclination
I = 109.94 ° . The period of the node P (f2)= 1046 days.
127
The Lense-Thirringnodal precessionis for LAGEOS (Ciufolini 1984).
o Lense 2GJ_
f_ ThirringLAGEOS= c2a3(l-e2)3/2 = 31 milliarcsec/year (2)
where J_ _=5.9 x 1040 g ° cm2/sec ---1.5 x 102 cm 2 is the earth angular momentum.
Unfortunately, the Lense-Thirring precession cannot be extracted from the
measured value of the LAGEOS nodal precession, because of the uncertainty in the
theoretical classical precession due to the quadrupole and higher mass moments of
the earth:
Classical
f2 _---126°/year
LAGEOS
However, a new satellite, of LAGEOS type: LAGEOS-III, with supplementary
inclination: ILAGEOS III = 70°, would have a classical precession equal in magnitude and
opposite in sign to that of LAGEOS. By contrast, since independent of the inclination,
the Lense-Thirring precession (2) would be the same, both in magnitude and sign for
the two satellites. Therefore, from the sum of the measured nodal precessions, we
should be able (Ciufolini 1985, 1984, 1987, 1988) to measure the Lense-Thirring effect.
HI. ERROR SOURCES
ORBITAL INJECTION ERRORS, GRAVITATIONAL AND
NONGRAVITATIONAL PERTURBATIONS AND MEASUREMENT ERRORS
(1) Errors from gravitational perturbations arise from uncertainties in modeling
the LAGEOS gravitational nodal perturbations. These errors may be subdivided
into"
(a) Errors from orbital injection errors (Ciufolini 1988) due to the
uncertainties in the knowledge of the static part of the even zonal
harmonic coefficients, J2n, of the earth gravity field. These errors are,
a priori, zero for the two supplementary inclination satellites. However,
any deviation of the orbital parameters of LAGEOS III from the optimal
values will introduce uncertainties which must be evaluated.
(b) Errors from other gravitational perturbations (Ciufolini 1988). Static
odd zonal harmonic perturbations; static nonzonal harmonic
perturbations; nonlinear harmonic perturbations; solid and ocean earth
tides. De Sitter (or geodetic) precession (Benotti et al. 1987); sun, moon
and planetary tidal accelerations; nonlinear, n-body, general
relativistic effects; other very tiny relativistic deviations from geodesic
motion of LAGEOS. The main error source is, however, due (Ciufolini
1987, 1988) to the uncertainties in modeling the dynamical part of the
earth gravity field, that is, to the uncertainties in modeling solid and
ocean earth tides.
(2) Errors from nongravitational perturbations (Ciufolini 1987, 1988). Direct solar
radiation pressure; earth albedo; satellite eclipses; anisotropic thermal
128
radiation; Poynting-Robertson effect; infrared radiation; atmospheric drag;
solar wind; interplanetary dust; earth magnetic field. The main error sources
are, however, due to uncertainties in modeling the earth albedo, anisotropic
thermal radiation, and atmospheric drag (Ciufolini 1987, 1988).
(3) Errors from measurement uncertainties in the orbital parameters (Ciufolini
1988). These errors are due to the uncertainties in the measurement of the
LAGEOS orbital parameters and, in particular, to the errors in the
measurement of the inclination I and the nodal longitude f_, relative to an
asympotic inertial frame (Ciufolini 1988).
IV. PRELIMINARY ERROR ANALYSIS
A PRELIMINARY 10% MAXIMUM ERROR,
OVER THE PERIOD OF THE NODE
A major problem in modeling the LAGEOS orbit is the average secular decrease
(Cohen et al. 1985, Rubincam 1988) of the LAGEOS semimajor axis of about 1 mm per
day. This corresponds to an along-track acceleration of about -3 × 10-1° cm/sec 2.
This acceleration may be explained by three mechanisms (Rubincam 1988, Afonso e t
al. 1988):
(1) Neutral and charged particle drag (Rubincam 1982, Afonso et al. 1985).
(2) Thermal drag from the earth infrared radiation (Rubincam, 1987 and 1988)
and the thermal lag of the LAGEOS retroreflectors, due to their thermal inertia.
The re-emission causes an along-track acceleration opposite to the satellite
motion.
(3) Thermal drag from the sun radiation plus satellite eclipses by the earth
(Afonso et al. 1988). The sun radiation is absorbed and then re-emitted by the
satellite. This phenomenon may be important when the satellite orbits are
partially in the shadow of the earth.
The effect of these three perturbations on the node has been investigated in
relation to the supplementary inclination configuration (Ciufolini 1988, Farinella
1988). The preliminary result is that their effect should not be larger than 1%
Lense-Thirring due to particle drag (Ciufolini 1987, 1988). This is negligible over the
period of the node for two supplementary inclination satellites due to infrared
radiation (Ciufolini 1988, Farinella 1988) and not larger than
2% t2 Lense-Thirring due to thermal drag plus satellite eclipses (Ciufolini 1988).
To support these preliminary figures, we observe that to give a secular nodal
precession, a force must: 1) be perpendicular to the orbital plane, and 2) change
sign in half a period. Even in this worst case, an acceleration with amplitude
-3 x 10-1° cm/sec 2 can, at most, cause a LAGEOS nodal precession of
5% _ Lense-Thirring, as it is easily calculated (Ciufolini 1988) from the Lagrange
equation for the node. Ciufolini (1988) has carried out a comprehensive
preliminary analysis. The result is that, using 2 nonpolar, laser ranged satellites
with supplementary inclinations, the maximum error, over the period of the node of
129
-3 years, should not be larger than N10% of the gravitomagnetic effect to be
measured.
A study group, composedof University of Texas at Austin and Italian scientists,
is, at present, performing a comprehensive computer simulation and covariance
analysis of the experiment.
ACKNOWLEDGEMENTS
I thank C. Alley, E. Amaldi, J. Anderson, P. Bender, N. Cabibbo, M. Cerdonio,
D. Christodoulidis, R. Eanes, F. Everitt, P. Farinella, P. Knocke, L. Guerriero,
R. Hellings, H. Mark, R. Matzner, K. Nordtvedt, T. Regge, D. Rubincam, I. Shapiro,
D. Smith, K. Thorne, M. Watkins, S. Weinberg, J. A. Wheeler, and C. Will.
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130
DISCUSSION
SHAPIRO: LAGEOS-II will of course help to understand the partially unmodeled
secular variation of the semimajor axis.
CIUFOLINI: Certainly; however this unmodeledvariation should not be a problem for
the gravitomagnetic experiment. In fact, even in the worst case that a comparable
acceleration would be acting perpendicularly to the orbital plane and changing sign
in half a period, that is, even in the caseof a maximum contribution to the nodal rate,
the total nodal drag due to this accelerationwould not be more than 5% of the Lense-
Thirring effect.
NORDTVEDT:Why do you quote a dJ2/J 2 uncertainty limitation, when you might use
the LAGEOS orbit, itself, as a measure of J2?
CIUFOLINI: Unfortunately the nodel precession and the rates of change with time of
the other orbital parameters are due, not only to the earth quadrupole moment, but to
other harmonics too. A solution would be to orbit several high-altitude, laser-ranged
satellites, of LAGEOS type, to measure each even zonal harmonic coefficient, to the
proper order, plus one satellite to measure the Lense-Thirring effect. Another
solution is to orbit polar satellites (Yilmaz 1959, Everitt and Van Patten 1976), since
they have a null classical precession. Another solution is to use supplementary
inclination satellites to cancel out the classical precession (LAGEOS III, Ciufolini
1984).
SHAPIRO: (Comment on question by K. Nordtvedt). The even zonal harmonics (J2,
J4 .... ) affect other orbital parameters, as well as W, and these effects for the orbit of
LAGEOS I, as well as for other satellites, allow estimates to be made of the values of J2,
J4 ..... It was to the residual errors, or uncertainties, from these estimates that
Ignazio referred.
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