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1 cu Q).c, G r e. 8 C. CD S s : v. C.c DSCLAMER This report was prepared as an account of work sponsored by an agency of the United States Governent. Neither the United States Governent nor any agency thereof, nor any of their eployees, akes any warranty, express or iplied, or assues any legal liability or responsibility for the accuracy, copleteness, or usefulness of any inforation, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Referen- herein to any specific coercial product, process, or service by trade nae, tradeark, anufacturer, or otherwise does not necessarily constitute or iply its endorseent, recoendation, or favoring by the United States Governent or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Governent or any agency thereof.

2 Spacetie Geodesy and the LAGEOS-3 Satellite Experient Warner A. Miller*, Kaiyou Chen, Salan Habib, Arkady Kheyfets (North Carolina State University), and Daniel E. Holz (University of Chicago) Abstract This is the final report of a three-year, Laboratory-Directed Research and Developent (LDRD) project at the Los Alaos National Laboratory (LANL). LAGEOS-1 is a dense spherical satellite whose tracking accuracy is such as to yield a ediu-ter inertial reference frae and that is used as an adjunct to ore difficult and ore data-intensive absolute frae easureents. LAGEOS-3, an identical satellite to be launched into an orbit copleentary to that of LAGEOS- 1, would experience an equal and opposite classical precession to that of LAGEOS-1. Besides providing a ore accurate real-tie easureent of the earth's length of day and polar wobble, this paired-satellite syste would provide the first direct easureent of the general relativistic frae-dragging effect. Of the five doinant error sources in this experient, the largest one involves surface forces on the satellite and their consequent ipact on the orbital nodal precession. The surface forces are a function of the spin dynaics of the satellite. We have odeled the spin dynaics of a LAGEOS-type satellite and used this spin odel to estiate the ipact of the theral rocketing effect on the LAGEOS-3 experient. We have also perfored an analytic tensor expansion of Synge's world function to better reveal the nature of the predicted frae-dragging effect. We showed that this effect is not due to the Rieann curvature tensor, but rather is a "potential effect" arising fro the acceleration of the world lines in the Kerr spacetie geoetry. 1. Background and Research Objectives The proposed Laser Geodynaic Satellite Experient-3 (LAGEOS-3) would easure, for the ft tie, a new quasi-stationary property of the earth -- its gravitational agnetic *Principal investigator, e-ail: wa@lanl.gov 1

3 dipole oent (gravitoagnetis) predicted by Einstein's theory of general relativity. This gravitoagnetic field causes the local inertial fraes to be dragged around with the earth at a rate proportional to the angular oentu of the earth and inversely proportional to the cube of the distance fro the center of the earth. This will cause the line of nodes of the orbital plane of LAGEOS-3 to precess eastward at 32 dyr. Although, in this exaple, the frae-dragging effect is sall copared to the torque on the orbital plane due to the oblateness of the earth, it is an essential ingredient in the dynaics of accretion disks, binary systes, and other astrophysical phenoena. The focus of our research was twofold: (1) to develop a spin odel of LAGEOS to deterine the expected theral rocketing induced error in the experient, thereby reducing the largest source of error, and (2) to explore further the nature of the frae-dragging effect and deterine if its effect can be observed in an astrophysics setting. 2. portance to LANL's Science and Technology Base and National R&D Needs Today, alost eighty years after Einstein introduced his geoetric theory of gravity, we have just begun to easure - to verify - his gravitation theory. Of no less stature than the "tide producing" -M/r2 "electric coponent" of gravity is the inertial-frae-defining "agnetic coponent'' of gravitation -J/r3. To see this force in action, first inject a satellite into a polar orbit about an earth-like ass idealized as not spinning with respect to the distant quasars. The satellite will reain in orbit in a continuous acceleration toward the center-of-ass of the attracting body under the influence of the Newtonian l/r2 force, and its orbital plane will reain fixed in orientation with respect to distant quasars. Second, spin this central body, give it angular oentu and follow the trajectory of the satellite. ts orbital plane will experience a torque along the body's rotation axis. The orbital plane will undergo a precessional otion in the direction of the central body's rotation. The ass in otion of the central body or "ass current" produces a dipole gravitational field - the gravitoagnetic field. n the case of a satellite orbiting at two earth radii, the orbital plane will precess about the body axis of the earth at approxiately 32 dyr. This is the Lense-Thirring effect. This force has never been directly easured. A easureent of this gravitoagnetic force is arguably coparable to the pioneering work of Michael Faraday on the easureent of a agnetic force between two current-carrying wires. However, the laboratory setting for this gravity easureent will be the 4diensional curved spacetie (approxiately Kerr) geoetry enveloping the earth. The idea behind this gravity easureent is siple. Whereas the Everitt- Fairbanks experient (Standford Gravity Probe-B) proposes a polar orbiting gyroscope, the 2

4 Ciufolini LAGEOS-3LAGEOS-1 experient proposes to use the orbital plane itself as a gyroscope. The spin dynaics odeling and subsequent analysis of the nodal precession of the LAGEOS orbital plane will diinish the largest error source in the LAGEOS-3 experient. The accuracy of the knse-thirring effect can then be increased a factor of two, reducing the experiental error to a 2 percent level. 3. Scientific Approach and Results To circuvent the need for expensive satellite experients (e.g. Stanford Gravity Probe B experient and the LAGEOS-3 experient), we have undertaken an analysis of the Doppler broading of Fe lines fro an accretion disk around a spinning black hole. f we can easure the angular oentu of a black hole astronoically, then this would alleviate the need for the satellite experients. We will continue to deterine the effect on the Fe line profiles fro the angular oentu of the black hole. Recent observational results fro a Japanese x-ray satellite indicates that there is a real possibility that such a easureent can be perfored in the near future. We odeled the spin dynaics of a LAGEOS-type satellite and used this spin odel to estiate the ipact of the theral rocketing effect on the LAGEOS-3 experient (Fig. 1). We also answered the following four questions that arose fro the spin odeling: (1) Can we obtain the asyptotic solution analytically and in so doing can we understand the wobbling or slippage of the Euler angles with respect to the relatively stable total instantaneous angular velocity? (2) Can we understand why the rs fluctuations in the gravitational potential energy cause the asyptotic value of the angular oentu vector to be offset fro the orbital plane by -1 degrees? (3) Can we understand why the nutation angle () drifts initially at a rate of -2 deglyr and reaches a pseudo-stable value of deg? (4) Can we understand the fluctuations in the spin rate (v) over the first -2 yrs, which do not appear to have been detected experientally? We perfored the tensor expansion of Synge's world function and proved that the fraedragging effect arises fro the connection coefficient ters rather than fro the Rieann curvature tensor. Finally, we calculated the Doppler shift of Fe lines eitted fro an accretion disk around a spinning black hole. We have deterined the dependency of the Doppler shift on the angular oentu of the black hole using a nuerical code we developed. Figure 2 shows a contour iage generated by our code for such an accretion disk 3

5 Publications [ 13 S. Habib, D. Holz, A. Kheyfets, R. Matzner, W. Miller and B. Tolan, "Spin Dynaics of the LAGEOS Satellite in Support of a Measureent of the Earth's Gravitoagnetis," Phys. Rev., D5, 668 (1994). [2] D. Holz, K. Chen and W. Miller, "Can We Measure the Angular Moentu of a Black Hole?," subitted to Phys. Rev. Lett. (1995).

6 -. < -r Y- n Fig. 1. Dynaics of the Euler angles. The evolution of the nutation (angle of obliquity, e) of the odel satellite (upper left). n the asyptotic liit the angular velocities of nutation and precession average to zero (upper and lower right, respectively). However, the precession rate cq locks into the orbital velocity. nto the asyptotic liit (lower left): this plot deonstrates clearly the dynaics through the spin-orbit resonance phase. 5

7 Fig. 2. Contour plot of a thin accretion disk about a axially spinning black hole of ass M. The inner radius of the disk is at 3M while the outer edge is at 2M. The distortion was calculated using a general relativistic ray-tracing code developed under this project. The inclination of the photographic plate is at 6 deg (the ost probable viewing angle), while the distance fro the black hole was 1OOOM. The distortion of the disk is caused by the bending of the light rays by the spinning black hole. This is why the disk appears oblong in the "wrong" direction. The contours represent the observed Dopplershifted frequencies of the Fe lines. 6