THEORETICAL BACKGROUND AND PRESENT STATUS OF THE STANFORD RELATTVTTY-GYROSCOPE EXPERIMENT
C. W. F. Everitt, W. M. Fairbank and L. I. Setoff Institute of Theoretical Physics and Department of Physics Stanford University, Stanford, California 94305.
ABSTRACT
An earlier proposal that a measurement of the precession of the spin axis of a gyroscope moving through the Earth''s gravitational field would provide a test of Einsteiifs general theory of relativity is discussed in terms of the Brans-Dicke theory and other recent developments. It is shown that an accuracy of 0.001 arc-second per year in the orientations of two or more gyroscopes is required which would also provide a rough measure of the quadrupole effect. The work of the last seven years at Stanford University on this project, including the large-scale low-temperature physics techniques and advanced techniques in electronic instrumentation and control theory, is discussed in detail. Parts of the laboratory prototype are being tested and it is hoped that a preliminary test flight will take place in 1973 or soon afterwards.
1. THEORETICAL BACKGROUND
Einstein's general theory of relativity has been of deep interest to scientists of several specialties for more than half a century. From the point of view of the physicist, a central problem consists in devising and then performing experiments that measure the rather subtle differences between the pre dictions of the relativistic and Newtonian theories of gravitation. This problem is far more difficult than the corresponding problem raised by the special theory of relativity. The differences between the predictions of the relativistic and Newtonian theories of mechanics are of order v2/c2, where v is the magnitude of the difference in velocity of two parts of the system and c is the speed of light; in recent times, many experiments have been performed in which this quantity is of order unity. In the case of the general theory of relativity, however, the corresponding parameter is
general relativity theory has recently been enhanced by an explicit formulation of the scalar-tensor theory* as a generalisation of the pure tensor theory of Einstein. This theory of Brans and Dicke expresses the amount of admixture of scalar field in terms of a positive parameter w which is infinite for the Einstein theory. On the basis of a measurement of the oblateness of the Sun and its interpretation in terms of a solar quadrupolar gravitational field, Dicke and Gol- denberg 2 have concluded that co is in the neighbourhood of 6, although this interpretation is by no means established. The value of 33 GYRO 2 A0=7 sec t= I YEAR AÔ-Q.05 sec Figure 1. Gyroscopes with spin axes parallel to the Earth's axis or perpendicular to the plane of the orbit on board a polar-orbiting satellite. of the Einstein theory by 6 to 8 per cent. There is therefore considerable interest in performing expe riments of this kind to an accuracy of 0.1 to 1 per cent. It was suggested nearly a decade ago that a measurement of the precession of the spin axis of a gyroscope moving through the Earth's gravitational field would provide a test of Einstein's theory 4. —^. -> -> In that paper it was shown that the precession angular velocity vector is given by £2T + QG + ÛM, where -> 2 -*• •> QT = (l/2mc )(F X v) (Thomas precession) ->• 2 3 •> •> QG = (3 GM/2c r )(/- X v) (geodetic precession) -*. 2 3 •>-»--»•-2 > QM = (GI/c r )[(3/-/r )(û>r) — co] (mass-current precession). •»• •> -> Here, m, r and v are the mass, coordinate, and velocity of the gyroscope, respectively, and F is any ->• non-gravitational force exerted on it; M, I, and w are the mass, moment of inertia, and rotational angular velocity vector of the Earth respectively. The Thomas precession QT is a special relativistic rather than a general relativistic effect since it does not involve G, and was discovered in connection with atomic physics 8; it is zero for a gyroscope in free fall about the Earth since then no non-gravitational support force is exerted. The geodetic precession flG is a true gravitational effect; its contribution when 6 integrated over an orbital period was first found by de Sitter . The mass-current precession S5M makes the gyroscope experiment uniquely interesting since it is the only experiment thus far proposed that is likely to distinguish between the gravitational fields produced by matter at rest and by matter in motion; £2M is related to the effect of the mass current of the rotating Sun on the orbital motion of a planet, which was first calculated by Lense and Thirring 7 but is too small to observe. When the Brans-Dicke parameter w is not infinite, the geodetic precession given above is to be multipled 1.8,9,10,11 by (4 + 3o>)/(6 + 3co). In similar fashion, the mass-current precession is to be multiplied10 •u by (3 -f- 2o>)/(4 + 2 34 effect of the mass quadrupole moment of the Earth may not be negligible. For the simplest case, in which the plane of the orbit is perpendicular to the quadrupole axis, this precession12>13 is D -3Q O where Q is the Earth's quadrupole moment. There is some advantage in using a polar orbit for the satellite that carries the gyroscope, since then the two principal precession vectors, Q.G and DM are perpendicular to each other. QR is perpendi cular to the plane of the orbit, and iiM is parallel to the Earth's axis. Thus, as shown in Fig. 1, gyro 1 with spin axis parallel to the Earth's axis measures only QG, while gyro 2 with spin axis perpendicular to the plane of the orbit measures only DM. The magnitude of the first precession is about 7 arc- second per year, and the time average of the second is about 0.05 arc-second per year. With this orbit, the time average of ÛQ is perpendicular to the plane of the orbit and has a magnitude of about 0.005 arc-second per year. In order to determine whether the Brans-Dicke parameter w is infinite, and its value if it is finite, it would be desirable to measure QG to an accuracy of 0.1 to 1 per cent. Then in order to establish the existence of the mass-current precession, it would be desirable to measure QM to a few per cent accuracy. Thus in both cases, an uncertainty of 0.001 arc-second per year is an ambi tious but appropriate objective, which would also provide a rough measure of the quadrupole effect. 2. EXPERIMENTAL STATUS Analysis and experimental work on the gyroscope programme has been proceeding during the past seven years in the Stanford University Department of Physics in conjunction with the Department of Aeronautics and Astronautics. A complete laboratory prototype has been designed, and parts oj it have been delivered to Stanford and are under test. We expect the evaluation of the laboratory hardware to be completed during the next two years. Meanwhile, plans for a preliminary test flight in 1973 and final flight as soon as possible thereafter, are under negotiation with the National Aero nautics and Space Administration. As discussed above, our aim in designing this experiment.has been to measure the orientations of two or more gyroscopes like those shown in Fig. 1 with respect to a star, with an accuracy of about 0.001 arc-second per year. While fully recognising the many practical obstacles that have to be over come to achieve this extreme accuracy, we believe that, at least in principle, it is within the possibilities of existing technology. In order to give some idea of the particular improvements called for, however, we may mention that the best gyroscopes currently available on Earth have residual drift rates about eight orders of magnitude greater than the design goal, and the available star-tracking telescopes have a long-term null stability about three orders of magnitude greater than this figure. Improvements of this kind in equipment can only be achieved by very precise matching of the design to the special requirements of the experiment. After prolonged thinking we have concluded that almost the only possible way of reaching these extremes of accuracy at present is by an extensive combination of three areas of technology that have opened up during the past ten or fifteen years: (1) space research, (2) large-scale low-temperature physics techniques, and (3) certain advanced techniques in electronic instrumentation and control theory. Suggestions for various methods of doing the experiment to somewhat lower accuracy have been made by a number of workers elsewhere. Without wishing to prejudge the issue, we nevertheless feel that to perform a full accuracy experiment, and probably any experiment, the arguments in favour of the special combination of techniques which we have adopted are exceedingly powerful. The gyroscope we have designed is illustrated in Fig. 2. It consists of a ball of quartz 4 cm in diameter, coated with a thin film of superconducting niobium, electrically supported by three mutually perpendicular sets of condenser plates, and spinning at about 300 revolutions per second in a vacuum. The basic ideas of the electrically suspended gyroscope were developed by Arnold Nordsieck and his colleagues at the University of Illinois in 1953. The arrangement we have adopted is a modification 35 READOUT RING INLET AND ÊLXHAUST FOR GAS SPIN-UP LEADS TO SUPPORT SUPERCONDUCTING ELECTRODES MAGNETIC SHIELD Figure 2. Gyroscope for laboratory test of relativity experiment. of the gyroscope developed by Honeywell Inc. When such a gyroscope is operated on Earth, by far the largest extraneous torques acting on it are those arising from the electrical suspension and from mass unbalance of the rotor. In space the suspension voltage may be turned down almost to zero. In fact, with appropriate attention to design details, the residual drift-rate scales as the time average of the residual accelerations on the satellite (10-9 g or less). Thus operation of the experiment in space reduces the drift-rate of the gyroscope due to these causes from about 105 arc-second per year on Earth 3 to less than 10~ arc-second per year. Operation in space also has the advantage of increasing Q.T + QG by a factor of approximately 15 in comparison with a gyroscope fixed with respect to the rotating Earth. Furthermore, it eliminates errors in the telescope references due to " seeing " through the Earth's atmosphere and due to sag and creep of the telescope structure in the Earth's gravitational field. There are, however, other torques acting on the gyroscope which are not reduced in space, and it is in eliminating these that low-temperature techniques find their first important application to the experiment. One of the most significant is due to magnetic fields. In ordinary electrically- supported gyroscopes the interaction of eddy currents in the spinning metal rotor with the magnetic fields obtained by conventional shielding techniques (10~4 to 10-5 gauss) leads to residual drift-rates several orders of magnitude larger than the desired value. The present gyroscope comprises a non magnetic quartz rotor coated with a thin film of superconductor surrounded by the spherical super conducting magnetic shield also shown in Fig. 2. The only magnetic torque is one arising from the small magnetic moment generated in a spinning superconductor known as the London moment14-15. We shall show shortly that this London moment is crucially important to obtaining a satisfactory gyro readout. Its magnitude is |-R3w^3x 10"8R3co gauss cm3 where R is the radius of the gyro rotor and 36 Of other extraneous torques the most important is the one due to the interaction of the quadra- pole mass moment of the gyro rotor with the gradient of the Earth's gravitational field. The secular term in the drift rate caused by this effect is 4n2 = 3 AI GM si.n 20 a 2 I «r3 where — is the ratio of the difference between principal moments of inertia to the average value, G the Newtonian gravitational constant, M the mass of the Earth, r the orbit radius, co the spin angular velocity of the gyroscope, and a the angle between the gyro spin axis and the satellite orbit plane. In perfect polar orbit with the gyros exactly aligned as in Fig. I, this term vanishes, but in practice there are limitations to the precision of the alignment. From this equation it may be shown that — must be less than 10~6 to reach the desired performance. Gyro rotors of this degree of homogeneity have been obtained by optical selection of Schlieren quality quartz: we are also investigating the possi bility of reaching a homogeneity of one part in 108 by the use of crystalline silicon. Calculations have also been made of all other known torques acting on the gyro rotor, including random motions due to the impact of cosmic rays, gas molecules and photons from the walls of the cavity. The following are the main requirements to obtaining the desired drift performance: 1) Sphericity of rotor ~ 10~6 cm 2) Homogeneity of rotor — ~ 10-6 3) Residual magnetic fields < 10-6 gauss 4) Average residual accelerations on satellite < 10-9 g 5) Residual gas pressure < 10~8 mm Hg all of which are within the capacity of existing technology. The requirements on the sphericity and homogeneity of the gyro rotor pose a readout problem. Conventional readouts require a knowledge of the position of the axis of rotation with respect to the ball. If the moments of inertia of all axes are practically the same, it is not possible to anticipate about which axis the gyroscope will spin. The London moment provides a method of locating the spin axis without the requirement for reference marks on the surface of the ball. Figure 3 shows the proposed - readout which makes use of a superconducting loop. Shown in the figure is a spinning superconductor with a magnetic field as indicated along the axis of spin. Around the spinning sphere is a supercon ducting loop. Since the resistance of the superconducting loop is zero, any change in the flux through Figure 3. Principle of London moment readout (London moment field H = 10_Tco gauss). 37 MODULATOR DETECTOR NULLING FIELD looo J Hh< iSMJ TO AMPLIFIER Figure 4. Circuit for the measurement of the changes of flux through the superconducting loop due to changes in the orientation of the spin axis of a gyro sphere the loop caused by a change in the orientation of the spin axis of the gyro sphere will cause a current to flow in the loop which exactly cancels this change of flux. If one could read out this current, one could determine the change in orientation of the ball. Figure 4 shows the circuit we have developed to read out this current. In series with the first loop (detector) is placed a second superconducting loop (modulator). The current that flows in the two superconducting loops produces a cancelling flux which is distributed in the two loops instead of being confined to the one loop. The ratio of the flux in the two loops is equal to the ratio of the inductances of the two loops. Thus the change in flux through the first loop caused by the reorientation of the ball produces a cancelling flux distribution in the two loops. If the inductance of the modulator loop is changed the current, flowing in the two loops changes and the distribution of the cancelling flux changes. If the inductance is changed 105 times per second then a 10s cycle/second alternating signal is produced which can be detected by a readout coil. The modulator consists of a long superconducting lead evaporated on a flat surface. Adjacent to it is a superconducting ground plane evaporated onto a quartz crystal. The crystal and surface are placed about 2000 A apart and the crystal is driven such that the ground plane periodically approaches and recedes from the superconducting circuit. This modulates the inductance of the circuit and causes the flux to be pumped back and forth between the two loops. The oscillating current in the two loops flows through the coil as indicated and is read out through a transformer by an amplifier. It is possible to increase the sensitivity of this circuit by placing a condenser in the circuit as indicated on the diagram. The modulating current flows in and out of the condenser plates in such a way as to provide additional parametric amplification. John Pierce17 at Stanford has worked out in detail the sensitivity of such a circuit compared with the theoretical Johnson noise in an amplifier, and finds that the mean square error in the measured flux is given by 2 The L Av 38 to below helium temperature by feeding the signal to a Josephson junction amplifier, the sensitivity can be increased to 0.001 arc-second as required by the experiment. The development of the above magnetometer is due chiefly to Dr J. Opfer. When we review the problem in a slightly wider setting, we note that while it is in a certain sense relatively easy to design a gyroscope which has a residual drift rate less than the required value, two problems in the practical operation of the gyro are supremely difficult. The first is to devise a method of readout with sufficient sensitivity and null stability which does not itself introduce drift errors. The circuit described above is unique in having an absolute null stability ensured by the special properties of quantized flux in a superconductor, and, provided the range of motion of the gyroscope does not exceed about 10 arc-minutes, the drift rate caused by the currents in the readout circuit remains below the required value of 10~16 radians per second. The second practical problem is to devise a suitable method for spinning up the gyro initially. The difficulty here is that in order to make a suc cessful gyro it is essential to mimmize all extraneous torques, whereas in order to spin it up it is necessary to introduce temporarily a very large external torque. A rough quantitative statement of the problem may be arrived at as follows. Let the spin-up torque be TS, the residual torque from the same source after spin-up be x„ the required drift rate be Q and the time required for spin-up to 300 revolutions/ second be t. Then neglecting drag torques, a very simple calculation shows that -^- ~ Qt, or taking O as about 10~16 radian per second and t as 1000 seconds, -^- < 10~13. Very few methods of achieving Xs such a level of torque switching are available for either room-temperature or low-temperature gyros copes. The method chosen for our experiment is a gas spin-up system designed by T. Dan Bracken18. The foregoing arguments indicate the significance of low-temperature techniques in the design of the gyroscope. But low temperatures play an equally significant part in the design of a satisfactory reference telescope. Near the absolute zero the coefficients of expansion of all materials decrease by several orders of magnitude in comparison with their room-temperature values, while at the same time the problem of maintaining a uniform temperature becomes increasingly easy. To give some idea of the importance of this point the following comparison is of interest. In order to obtain a telescope with a stability of 0.001 arc-second at ambient satellite temperature (250 °K), the transverse heat flux falling on the side of the satellite from the Sun's radiation would have to be cancelled to within 1 part in 108 for a quartz telescope, or 1 part in 105 for a metal one. At helium temperatures, on the other hand, the maximum allowable transverse heat flux through the same telescope is actually several orders of magnitude greater than the total heat input into the dewar. We have therefore designed the experiment in such a way that the entire core of the apparatus, telescope, gyroscopes, mounting rings, etc., are fabricated of quartz, optically contacted together and all contained at uniform temperature within the cryogenic environment. Figure 5 shows the arrangement of the telescope with a single gyroscope for l'ÇVT ***>& sr/tx-r*»**"** «—i » Figure 5. General view of apparatus for laboratory test of relativity experiment. 39 the laboratory model of the experiment. The telescope is a folded Cassegrain system of 5 |-inch aperture and 200-inch focal length. The light is divided by a beam splitter (not shown) to obtain two separate star-images, one for each readout axis. Each image is allowed to fall on the sharp edge of a roof prism, where it is again subdivided and passed to a light-chopper and photodetector at room temperature. In this way the centre of the diffraction pattern may be located with considerable preci sion. It should be emphasised that in star-tracking telescopes of this kind, the limit to performance is not the Rayleigh criterion for the separation of two star images, but the photon noise in the diffraction pattern. The present experiment will utilise as its reference a bright star near the Equator, probably Procyon. We have calculated the photon noise and shown that it is in good agreement with results obtained experimentally on a telescope simulator employing an artificial star of brightness equivalent to Procyon. With the particular arrangement adopted the resolution was about 0.01 arc-second in one second of integration time. A resolution of 0.001 arc-second is therefore obtainable by integrating for 100 seconds. The manufacturing tolerances required in the telescope are extremely severe. The image displacement corresponding to 0.001 arc-second at 200-inch focal length is about 10-6 inches. Since even the thinnest film of epoxy or other cement could readily undergo changes of the same order of magnitude due to aging, this figure establishes the vital importance of making the entire mechanical structure essentially in one piece by the optical contacting technique. Considerable thought has also been given to the problems of creep and the delayed elastic effect in the telescope structure. On Earth the telescope sags under its own weight by about 0.1 arc-second, and may be expected to display about the same amount of long-term creep during the course of a year. The weightless environment of space reduces these effects but there is still the possibility of error due to the relaxation of frozen-in strains in the material. Errors of this kind are eliminated by heat treating the mechanical structure at 1100 °C Altitude He Gos —- 4 Control Electronics Thrusters T 4 Quortz Block Cryogenic Inner Servo AGC Voltage Actuator Electronics Gyro Telescope Electronics Electronics D/A Converter Electronic Reference Accumulator Switch Generotor Digital Subtraction Dither Generotor Buffer Sampling V to F Up-Down _. Digital •*• Demod 8 Filter Converter \ Counter Data "* Output D/A \ Converter Control Logic \ \ \ Status Inputs Figure 6. Relativity instrumentation system. 40 in the vertical position for about a week at an appropiate point in the manufacturing process. One more critical tolerance is in the sharpness of the knife-edge for the roof prism. An analysis due to Dr. R. A. Nidey shows that for a 200-inch telescope the prism edge should be straight and free from nicks to within 20 millionths of an inch. In the present state of development we have obtained prisms sharp to about 50 millionths. Very great care is required in processing the signals from the gyro and telescope readouts without introducing null offsets or scaling errors in excess of 0.001 arc-second. An instrumentation system with the required properties has been devised by Mr R. A. Van Patten of the Department of Aeronautics and Astronautics at Stanford, and is illustrated in Fig. 6. Without going into details, three functions of the system deserve brief mention. The heavy lines on the diagram are the main readout loop which subtracts and sums the gyro and telescope signals with the signal from an output register. The up-down counter provides a method by which noise in the signals is progressively and continuously weighted and filtered with a loop time constant of about 20 seconds. Relativity inform ation is sampled from the output register and stored in a memory from which it is transmitted to Earth about once a day. An important feature of the arrangement selected is that even if the system is temporarily disturbed — say by the impact of a meteorite on the satellite — it can recover automatically without the introduction of errors or loss of information other than the immediate loss of data during the period of interruption. Two other control loops provide a signal for use in attitude control of the satellite and another signal for calibrating the gain of the telescope readout by direct comparison with the gyro readout. The calibration loop ensures that relativity data accurate to 0.001 arc-second are obtained even though limitations in the pointing servos may allow the telescope to swing back and forth over the line of sight to the star over a range of as much as 0.1 arc-second. One very important feature of the experiment is that there is an automatic absolute check of the accuracy of the readout and instrumentation systems in space. This is obtained by observation of signals corresponding to the aberration of starlight. The motion of the Earth around the Sun leads to an annual aberration of ± 20 arc-seconds, and the motion of the satellite around the Earth leads to an orbital aberration with 90-minute period of about ± 5 arc-seconds. Both of these numbers may be calculated with very great precision. In the system we have designed the telescope is maintained pointing to the star to within 0.1 arc-second and the aberration signal appears in the gyro readout and hence ultimately in the output register of the instrumentation system, each element of which has been designed to be accurate and linear to within 0.001 arc-second over a range of ± 50 arc-seconds. Both aberration signals are therefore reproduced with full accuracy in the output register as they occur, and any unanticipated change in the calibrations of the circuits can be immediately detected and cor rected for in data analysis on the ground. An overall view of the satellite as presently conceived is given in Fig. 7. It comprises a dewar vessel containing about 300 litres of liquid helium, with one telescope and four gyroscopes grouped in two pairs, spinning in opposite directions to give a double check on each relativity effect. An interesting feature of the design is that the helium gas evaporating from the dewar acts as the supply for a gas-jet attitude-control system for the satellite. The pointing accuracy so obtained is about + 2 arc-seconds. In order to obtain the required accuracy of ± 0.1 arc-second or better for the telescope, an inner fine-pointing servo has been added driven by superconducting magnetic actuators inside the cryogenic environment. Analysis of the performance of the attitude control and inner pointing servos has been carried out in the Department of Aeronautics and Astronautics at Stanford by Dr D. B. DeBra in conjunction with Messrs. J. Witsmeer, J. Knudson and J. Carroll. In addition to the four gyroscopes the satellite also contains an internal zero-g proof mass as shown in Fig. 7. By measuring the location of the proof mass in its housing it is possible to add translational control to the gas jet system and compensate for residual accelerations due to aerodynamic drag, solar-radiation pressure, etc. The reduction in average acceleration should lead to an improvement of one order of magnitude or more in the drift rate of the gyroscope. The application of the zero-g principle to a satellite is being developed by members of the Department of Aeronautics and Astronautics independently of the present experiment. It is expected that a test flight of such a satellite will be made in October 1970. The design of the helium dewar for the flight experiment has several interesting features. In particular, the special properties of the superfluid helium have made possible a novel solution to the 41 INSTRUMENTATION AND TELEMETRY TENSION WIRE ERECTED IN SPACE INLET FOR LIOUID HELIUM TRANSFER TUBE; -INFLATABLE PILLOWS SUPER1NSULATI0N ATTITUDE CONTROL JET (I of 12) -ONE FOOT Figure 1. General view of relativity satellite. problem of containing a liquid in the weightless environment. The dewar is sealed with a fine porous dlug of high thermal conductivity, through which only the superfluid component can pass. With an appropriate choice of design parameters the flow is made self-regulating through the equilibrium between the thermo-mechanical and the ordinary pressure gradients. The helium emerges from the plug, evaporates on the outside, and then refrigerates the dewar by conduction through the material of the plug. Inside the dewar thermal equilibrium is maintained by the action of the creeping helium film. Experiments with the superfluid plug have been carried out at Stanford by Mr P. Selzer. Other special features of the design include the use of gold-coated windows to reduce the radiation to the telescope and the use of dual heat-exchangers to recover refrigeration from the sensible heat of the vapour. ACKNO WLEDGEMENTS This research was sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, and by NASA, under AFOSR Contracts F44620-68C-0075, AF 33 (615J-67C-1245, and NASA Grant No. NGR 05020-019. In addition to the names mentioned in the text above we would like to offer a special word of acknowledgement to Professor R. H. Cannon of the Department of Aeronautics and Astronautics at Stanford for his continued support and interest in the programme. 42 REFERENCES 1. Brans, C. Phys. Rev. 124, p. 925 (1961). Dicke, R. H. 2. Dicke, R. H. Phys. Rev. Lett. 18, p. 313 (1967). Goldenberg, H. M. 3. Einstein, A. Ann. Physik 35, p. 898 (1911). 4. Schiff,L. I. 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