Gravity research at Cottrell observatory

Acta Astronautica Vol 9, No. 2, pp. 63-,84. 1982 009~5765/82/020~63-22503.00/0 Printed in Great Britain. Pergamon Press Ltd.

GRAVITY RESEARCH AT COTTRELL OBSERVATORYt

V. S. TUMAN California State College, Stanislaus, Turlock, CA 95380, U.S.A.

(Received 26 May 1981; in revised form 6 October 1981)

Abstract--The main goal of the research at Cottrell observatory is to study the minute changes of gravity as a function of time. A 19-MHz SHE magnetometer has been modified and utilized in the cryogenic gravity meter as a detection module. The theoretical sensitivity is Ag/g = 1.6x 10 -t3. Some of the pertinent approaches to the proposed research are discussed.

I. INTRODUCTION The Frederick Gardner Cottrell observatory was officially opened for operation on 13 January 1977. The observatory is located at latitude 37°27'50 ' and longitude 121°19'10 " in Del Puerto Canyon some 40 miles east of Lick Observatory and 35 miles from the campus of California State College, Stanislaus. The observatory is housed inside a massive quartz vein, surrounded by Franciscan ser- pentine formations. The laboratory has an area of 200 sq. ft and a roof 12 ft high.

There are three concrete pads--one for the vertical seismometer, a second for the cryogenic gravity meter, and the third pad is available for colleagues and friends who might want to conduct special experiments in a low-noise environment. Two generators, a 7-kW and a 2.5-kW, are located some 100ft away from the observatory. They provide a.c. power when needed, and they are individually used to charge the batteries. All the electronic units, chart recorders, pressure and temperature control units at the observatory are operated by car battery accumulators. The batteries, however, will eventually be charged by solar cells.

The analog data from the vertical seismometer and the cryogenic gravity meter will be passed through low-pass and high-pass filters. The resulting output will then be digitized at appropriate time intervals. The digital data signal will be simultaneously recorded on a specialized magnetic tape recorder and transmitted by an FM transmitter to Cal State Stanislaus. The data will be recorded into a format acceptable to the UNIVAC computer at Jet Propulsion Laboratory (JPL) in Pasadena, California. The analysis and final interpretation of the data will be carried out in collaboration with John D. Anderson.

The main goal of the research at Cottrell observatory is to study the minute changes of gravity as a function of time. These would include the Earth eigenvibrations, tides, changes in G, and preferred frame effects, if any, in post-Newtonian theories of gravitation, and possibly the convection currents in the mantle. A 19-MHz SHE

tPaper presented at the 4th International Space Relativity Symposium of the International Astronautical Academy, XXVIIIth Congress of the International Astronautical Federa- tion, Prague, Czechoslovakia, 26 September-10 October 1977.

magnetometer has been modified and utilized in the cryogenic gravity meter as a detection module; it has a dynamic range of 2 x l0 6. It is planned to tune the gravity meter so that the maximum tidal signal coincides with the maximum dynamic range of the detection module. Under these conditions, the theoretical sensitivity of the instrument is Ag/g= 1.6x 10 -13. Although it will be difficult to attain such ultimate sensitivity, we are confident that 10-nanogal sensitivity will be achieved in the near future.

In this paper we shall describe some of the pertinent approaches to the proposed research. In Sections 2 and 3 of this article, we shall discuss briefly the equipment installed and equipment that will be installed at the Cottrell observatory. We shall also discuss the present status of the cryogenic gravity meter and the necessary improvements needed to make it a reliable unattended observatory instrument. In Section 4 we shall discuss the application of data from Cottrell observatory to gravitational physics. Finally, we shall attempt to present a fictitious highly eccentric binary black hole system which is capable of emitting a wide frequency band of gravitational radiation which could excite a special group of Earth even eigenvibrations with multipole moments. Such a band of gravitational radiation will have more energy in upper harmonics. The cross-section for higher modes of Earth eigenvibrations is smaller. The two effects may balance out such that a wide band of Earth eigenvibrations could be excited by such a fictitious black hole binary system.

2. THE OBSERVATORY, INSTRUMENTATION AND DATA ACQUISITION SYSTEM

Construction of the physical structure of the Cottrell observatory is already completed. The site is located on 25 acres belonging to the U.S. Bureau of Land Management. After two years of negotiation, Cal State Stanislaus received permission to lease the land from the government with an option to buy it in the future. A preliminary plan of the observatory is shown in Fig. 1. The background seismic and environmental noise was measured both at Cal State Stanislaus and at the Cottrell observatory. The Cottrell observatory has 36db less noise than the campus of Cal State Stanislaus.

The observatory inside the mountain has a floor area

63

64 V. S, TUMAN

j i i

c o n c r e t e pad f o r

C r y o g e n i c G r a v i t y Mete1

pad Itor Vertical Seism~meter

l ' J

e x t r a c o n c r e t e pad

roll-up steel door

Fig, 1. Plan view of Cottrell observatory.

of about 200 sq. ft and a roof 12 ft high. It will accom- modate two instruments on two permanent platforms, plus a third experimental platform for testing other instruments in a low-noise environment. The third pad is available for colleagues and friends.

Power is generated initially by a gasoline generator which charges a band of 4-, 6- and 12-V batteries. The generators are located about 100 ft from the observatory. (With the present vertical seismometer, the noise due to the generators is not detected.)

Data which is acquired at the Cottrell observatory will be sent to Cal State Stanislaus via digital radio telemetry, Information will be transmitted from two different instruments: (1) a Sprengnether long-period vertical seismometer, and (2) the cryogenic gravity meter. Primarily, we are interested in the long-period portion of the spectrum which includes information on the normal modes (eigenvibrations) and tides for the Earth and other low- frequency components of gravity.

The seismometer has two transducers for velocity and

65

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displacement. The velocity output will be amplified and passed through a low-pass filter (see Fig. 2) to simulate a worldwide standard seismographic station. Long-period displacement outputs are amplified and passed through a low-pass filter to observe the solid Earth tides and then amplified further and passed through a high-pass filter to observe the Earth's normal modes, when sufficiently excited. The temperature of the instrument is presently controlled well within 10-2°C. The instrument is sealed underground to minimize the atmospheric affect. Im- provement on pressure and temperature control of this instrument is continued. Some of the earthquake and tide data from this instrument are shown in Figs. 3-6.

The cryogenic gravity meter will have the same logic system (Fig. 7) as the seismometer in order to compare the records between the two instruments. Figure 8 shows the tide data of the cryogenic gravity meter.

Because the observatory is in a canyon, the transmitter and antenna will be located about one-half mile away on the top of a ridge.

The telemetry receiving system is shown in Fig. 7. Basically, all of the signals will be recorded on analog recorders at the observatory, but it will be possible to select one or more channels for simultaneous digital recording.

The approximate response of the overall displacement

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60

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1 I I

Gravity research at Cottrell observatory

~ I mgOI, volt

- I 0 r e g a l / v o l t

I , IOOO

PERIOD IN SECONDS

I 10,000

Fig. 2. System response (assuming recorders are -+ 10 V full scale).

66 V.S. TUMAN

":''~'i," ~ ~ '~;'-" DEC. 19,1976 Mog, 65

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California State College, Stanislaus Cottrell Observator,/

i i - ~i '~ i

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Fig. 4.

Gravity research at Cottrell observatory 67

1 8

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l - ~ TiDE DATA - - . . . .

i - r i ° f r o m C O T T R E L L O B S E R V A T O R Y * - , - - - -

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Fig. 5.

system is shown in Fig. 3. The tidal channel has a sensitivity of ~ 10 mgal/volt, which is sufficient to record the maximum peak-to-peak Earth tides at about two- thirds full scale, while the normal modes channel has a sensitivity which is about 40 db greater.

If the analog system proves to have insufficient dynamic range, the system can be converted to a low- frequency digital transmission system with the addition of universal asynchronous receiver-transmitters and analog-to-digital and digital-to-analog converters.

An FM system to receive the coded digital data from the Cottrell observatory will be installed on campus at Cal State Stanislaus. The transmitter and receivers usu- ally have a capability of handling a group of seven different signals. Most probably, we will transmit one set of earthquake data, two sets of eigenvibrations, and two sets of tide data from the cryogenic gravity meter and the long-period vertical seismometer. These data will be augmented with two other signals of pressure and temperature to monitor the observatory environment. The decoded seismic data will be converted into an analog form and will be recorded on a chart recorder for earth-

tan excellent working cryogenic gravity meter has already been developed by John Goodkind and Prothero. The instrument discussed here has a SHE magnetometer 19-MHz detection module and the ball oscillates freely rather than being locked by the feedback system.

quake displays. The eigenvibrations and the tide data will be recorded on a specialized digital magnetic recorder. Initially, the eigenvibrations will be recorded at 30-see intervals and the tide data at one-minute intervals. Cassette tapes will be utilized either on campus at Cal State Stanislaus or at JPL, to report the data in a format compatible with the JPL UNIVAC. Some data analysis and interpretation will be conducted at Cal State Stanislaus, but the major effort will be carried out at JPL. The campus observatory which will receive the data from the Cottrell observatory will be constructed this summer of 1977. The receiving unit is schematically presented in Fig. 9.

3. P R E S E N T S T A T U S O F T H E C R Y O G E N I C G R A V I T Y M E T E R

The cryogenic gravity meter is comprised of a hollow niobium ball 1 in. in diameter and a mass of 2.45 g, freely suspended in space by superconducting magnets.I" A magnetic force pushes the ball up, and the gravity force pulls the ball down. When the two opposing forces are equal, the ball is freely suspended in space. The magnetic force is extremely stable and is protected from the local environmental magnetic fluctuations by two magnetic shields. Consequently, when the Earth's gravity field decreases the ball goes up, and when it increases the ball is pulled down. The minute motions of the ball are detected by a detection module. A 19-MHz SHE mag-

68 V.S . TUMAN

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LONG PERIOD SEISMOMETER

1 DISPLACEMENT TRANSDUCER

AMPLIFIER

LOW PASS FILTERS

AMPLIFIER

HIGH PASS F ILTEI~

I VELOCITY TRANSDUCER

I AMPLIFIER L LOW PASS J FILTERS

ii J 2 3 1 15

MULTIPLEXER AND VOLTAGE CONTROLLI~D OSCILLATORS

I TRANSMITTER

1 ANTENNA

Fig. 7. Block diagram of telemetry transmitting system.

CRYOGENIC 1 GRAVITY METER

I SUPERCONOUCTING I MAGNETOMETER

LOW PASS FILTERS

I AMPLIFIER

I I HIGH PASS

FILTERS

netometer was designed and adopted as a detection module for the instrument. The magnetic environment of a pick-up coil is zeroed at a particular datum. The magnetic flux change produced by the motion of the superconducting ball is sensed by the pick-up coil. A feedback unit and a phase-sensitive detector monitors the actual motions of the hollow ball, and they are reproduced as a variable d.c. voltage which can be recorded on a chart recorder or converted to digital data and recorded on a magnetic cassette for the purpose of the analysis. The detection module has a dynamic range of 2 × 106. The instrument will be tuned such that the maximum tide data will coincide closely with maximum

amplitude of the detection module, namely -+ 5 V. This modification will be implemented once the new superin- sulated dewar of 120-1. capacity is delivered.

The output signal will be filtered by a low-pass and high-pass filter. The tide and eigenvibration signals will be analyzed in detail. The tide signal will be digitized once every minute, while eigenvibrations will be digitized once every 30 sec for the high-frequency normal modes. In order to acquaint the reader with the insturment, some of the pertinent results will be discussed below.

3.1 A brief description o[ the instrument The complete system is comprised of a large number

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V. S. TUMAN

TIDE DATA Sat. Ouly 28 ond Sundoy ,July 29

1979 STANFORD

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Fig. 8.

of modules, each having a different function. These modules are grouped into several units:

(1) Suspension unit (four or five superconducting magnets) (2) Accessories to the suspension unit; (a) vacuum cavity (see Figs. 10-12), (b) levitation module (photosensitive device), (c) levitation monitor (a d.c. signal indicates the position of the ball in space), (d) power supplies. (3) Detection unit: (a) detection module ( S H E 19-MHz magnetometer with dynamic range 2 × 106), (b) phase-sensitive detector and feedback module, (c) electronics (oscilloscope, oscillator and d.c. power supply) for the detection module. (4) Output unit: (a) active electronic filters, (h) chart recorders, (c) digital data acquisition system. For more details see AFCRL 72-0580.

3.2 Magnetic field calculations and conditions of stability

A brief summary of the magnetic calculations and conditions of stability will be presented below. The calculations of the magnetic field in the presence of a superconducting surface have been done by several in- vestigators. We follow here a method developed by Harding.

The applied magnetic field BA can be expressed as the gradient of a scalar potential

BA = -- V~A. (1)

The magnetostatic potential ( ~ A c a n be expressed in Legendre polynomials, with the origin at the center of the superconducting hollow ball in equilibrium position:

4~a(r, 0) = 2 b.r"Pn(cos 0). (2) n ~ O

This relation is true if the field is axially symmetric and there are no internal sources interior to r. In the presence of a superconducting ball with radius R, a field is induced in the region r > R. This field likewise can be expressed as the gradient of a scalor potential:

B~ -- - V~I, (3)

where

4', = ~o a~., ,°.(cos 0). (4)

Since under experimental conditions the ball will be displaced from the equilibrium position, it is more con- venient to express the field equations in a coordinate system with the origin at a displaced center of the ball. Thus we may write

~ anm ,~ = ~, ~ ¢,+, P, '(cos ~) cos m0. (5) n = O m = O

,,

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DISCRIMINATORI AND AMPLIFIERS

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SPAR[

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Fig. 9, Block diagram of telemetry receiving system.

,13

DIGITIZER

I MAGNETIC

TAPIE RECORDER

71

The applied field Ba can also be expressed in the bar coordinates, using the following transformation:

x =~+Xo,

y = )7, (6)

Z = 7. + Zo,

where Xo and Zo are the coordinates of the displaced center of the ball. We may write

d~l Xo de I ¢(x, y, z) = d~(-~, Y, z) +--d2 [.o=o +d-2 -o=o zo. (7) z0=O z0=O

Recognizing that the total field is Bt = BA + B x , the total field Be at r = R is given by

B -~ 2 n + l .

~ = - 2.,,=o-h-+~Rn , - { [ b " ~ o P ' l ( c ° s O ) + z ° ( n + l ) b " ÷ '

d , d , if) cos if] × ~ P. (cos O) - xob.+, ~ P. (cos

+ 1 _ [xob.+~p~(cos ~) sin ¢] 4~'~ (8) sin O J

The magnetic force exerted on the superconducting sphere is determined by integrating the Maxwell stress tensor over its surface.

72 V.S. TUMAN

Su~

flection Module Assembly

fielded Detecting Circuit

Solenoid

__Magnetic Gradient Coils

Liquid Helium Dewar

Fig. 10. Schematic of cryogenic gravitymeter.

and

Fraaial = - ~-~of f B2 sin Ocos ~R2 d(cos O) d~ p (9)

Fa,,ial = --~#o ~ J BZ cos ORZ d(cos O) d(o. (10)

Calculating B 2 from relation (9) and neglecting the second order terms in xo 2 and Zo 2 and XoZo etc., we find

Fradi,, = - 2~rR3 [b22 _ 3b,b3lxo, (11) p-o

Faxia~----41rR~ [bib2+(2b22 + 3b~b~)zo]. (12) ~o

For a gravity meter which acts like a vertical seismometer, it is necessary to have a weak K~ and a stiff horizontal spring K~.

K~ ~ K~. (20)

The period of the gravity meter is given by

where m is the mass of the niobium hollow ball.

3.3 Equation of motion [or the instrument For proper analysis of the data, it is important to relate

the output of the instrument to the ground motion. The earth motions will generate two groups of ac- celerations-namely, static and dynamic acceleration. The static acceleration is due to displacement of the cement block from the center of the Earth. If we denote the gravitational field by g, then

M G . g = - ~ - e r (22)

where M and R are the mass and the radius of the Earth, and G is the Cavendish coefficient.

M = 5.97 × 1027 gm

G = 6.67 x 10 -s cgs units

R = 6.38 x 10 ~cm

The change of gravitational acceleration due to a displacement AR is given by

2MG :.Xg = - ~ ARo, (23)

2AR Ag = - ~ g. (24)


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u P P ~ toll

;rdJ,*nt_~ Pair

Lower Coll

Co~ A

[,~dlu,,, S,

Fla~qe

Cod

Fig. 11. Lower section of the suspension unit.

This is called the static variation of the gravitational field due to a displacement AR. If the ground, however, has a periodic motion, then the displacement AR may be expressed as

A R ( t ) = ARo cos 02t, (25)

where ARo is the amplitude and to the circular frequency of the motion. The acceleration AR which will be termed dynamic change in the gravitation field is given by

A R ( t ) = - 02ZAR, (26)

o r

Agld . . . . ic = -- 022AR. (27)

Consequently, the total observed change in the earth field is given by

Agl .... + Ag/dyn = - [2~g + to2] AR. (28)

Fig. 12. Suspension unit.

Rearranging these, we get

mgltot = - ~ - 1 + A n . (29)

Note that 02 is the circular frequency of ground motion which can be of global nature--namely, a particular mode of Earth oscillation; R and g are the Earth con- stants.

Let

2g 2 g - = too ,

then

2g [ 1 to2 Ag),o,---- k- +~]AR. (30)

If we write this in terms of period, we find

Agho, = - ~ (31)

where To=3580sec, a constant for the Earth; T, the period of the ground motion; i.e. the tide or a particular eigen period of the Earth. For a 12-hr tide, the dynamic effect is 1/144, which is insignificant, while for eigenvibrations with a period of 3 min, the dynamic effect is

74 V, g. TUMAN

about 400. Consequently, (To/T) 2, which we shall call the magnification factor, becomes quite significant for higher frequency eigenvibrations of the Earth.

Next we shall consider the response of the instrument to the acceleration of the cement block. From the above discussion, we find that the force-per-units mass driving the instrument is given by

Aglr=-'-" ~ 1 hR(t). (32)

The differential equation of motion for the instrument is given by

Z + 2yZ+ w~ 2 = 0, (33)

where 7 is coefficient of the friction, ~o; is the angular frequency of the undamped instrument and ~o is the angular frequency of the cement block and the particular eigenvibration of the earth, In general, for a forced vibration we may write:

g, R, Zo and 7, we can calculate the corresponding ground displacement for a specific spheroidal eigenvibration of the Earth. By utilizing different field configurations, the instrument can be tuned to have different resonant frequency. As an example let us consider when the resonant frequency of the instrument is 1 Hz and the quality factor Q is 10, we have o~; = 628, To = 3580 sec, g = 980 cm/sec 2, R = 6.35 x l0 s cm, ~ :- angular frequency of the ground and Q = 10.

In example I below, we shall consider the response of the instrument to the 12-hr tides. From eqn (40), we have

m No =

-fi !

27r 1.45 x 10 4 radians/sec, o)),ia~ 12 x 3600

o~; = 6.28 rad/sec,

2 + 27Z +w,2= - ~ ( l + ~o2) AR(t), (34) T,~ = 1 T 12'

but

hR(t) : hRo e ~ ' .

The obvious solution is given by

Z = Zoe ~' .

Substituting Z in the equation of motion, we get:

(35) AR = [1.28 X 10 y] AZ

Example 2: For 0& (36)

2zr ~O)os2 = 3 - ~ = 1.94X 10

2g , aft "~ Zo( - w 2 + 2i7w + ~o 2) = - ~- (1 1- ~o2] ARo. (37) tO_ l.1, T

Thus, the displacement of the ball:

Zo = [(~o?- ,oh 2 + 4¢o71 "~ aRo. (38)

Experimentally we measure Zo and we should calculate hRo, the motion of the ground and cement block. This is given by

[ wi2R ] AZ, ARo - L2.22 x 2g]

hRo ~- 5.77 x 10 ~

Example 3. For 0S9.

2~r ~o)os9 = ~_~ = 9.9 x 10 2,

{ [ ( °ji2 - °fl)2 + 43* %fl] 'a / heo = R-2g[l+( ~ iZo. (39)

In terms of Q, the quality factor of the instrument, we may write this as

2 2'11/2 (0)12__ 0)2) 2 O) i tO /

- Zo. (40) a R o =

In this manner, by knowing all the parameters, ~¢, to, To,

~9_= 5.66,

O)j 2 R

ARo = [3.88 x 105] h Z

Example 4. For microseisms with frequency of 1 Hz,

F oeR r ,o:R 1.-

ARo = ~ AZ,

ARo = 10 - l AZ.


In other words, the ground motion is actually amplified by a factor of Q. For this reason, it is quite evident that the quality factor affects the results at very high frequency where the environmental noise is high. Con- sequently, one should operate the instrument at very low Q in order to decrease the response of the unit to the background noise. This would allow us to filter them much easier.

In our first groups of data, we did not have any low-pass nor high-pass filters; consequently, the free oscillations of the ball were recorded on a chart recorder and then results were digitized and a Fourier analysis was performed on these data. Unadulterated Fourier analysis of the results indicated that tide data had an amplitude 10-20 times larger than eigenvibrations like 0S9. This was the reason to make a statement that we do not see the tides very well. Today, however, we know that response of the instrument to tides is I00-1000 times attenuated when compared to the eigenvibrations, as we have seen from aforementioned calculations. Thus, to see the clean tides we must perform a low-pass filtering and apply some amplification.

3.4 Calibration of the instrument We must recall that in this instrument a superconduc-

ting hollow niobium sphere is suspended over a magnetic field where the upward magnetic push cancels the downward pull of the gravitational force, Effectively, the hollow niobium sphere is suspended by a magnetic spring, with a definite spring constant. The spring constant is linearly related to the magnetic field gradient and the second derivative of the field according to equation (12). When an external force Fe acts on the suspended niobium ball, the ball is displaced from its equilibrium position until the restoring force cancels the external force.

Fe = - F z , (41)

where Fz is the restoring force:

(42)

Fz = - K A Z

Kz = spring constant,

AZ = displacement.

We note that for a unit applied force, if the spring constant is small, then the corresponding displacement AZ is large.

The detection module is sensitive to the magnetic flux change at the pickup coil. Since the superconducting niobium ball can effectively be replaced by a magnetic dipole, the magnetic flux change at the pickup coil is linearly related to the displacement of the ball. We may write:

A, h = ~:AZ, (43)

75

where the coefficient ~: is determined from the geometry of the unit. The output of the instrument is related to A~ according to the expression below:

Vo = AfbA~b = Arb(AZ, (44)

where Atb is the closed loop gain of the detection unit. Experimentally, one can determine the gain by applying a known magnetic flux at the pick-up coil and measuring the output voltage. The factor Atb for the present detection module is 1.1 x 104. Once the gain is determined for a given experimental setting, one can inversely use the gain and the output voltage to determine the input flux to the unit. When the input magnetic flux is produced by the motion of the ball, equation (43) then gives the actual displacement of the ball. From the displacement and the resonant frequency (spring constant) one can determine magnitude of the driving force acting on the ball.

In order to calibrate the unit, we must determine the coefficient ~ in equation (43), the spring constant K, and the gain of the detection unit. Let the ratio of the input flux to the output voltage for a given experimental condition be a; i.e. a = 1lAth. We begin by determination of O'.

3.5 Determination of a, the ratio of the flux at the pickup coil to the output voltage

To determine the ratio of output voltage to the flux at the pickup coil, an artificial magnetic flux is generated at the pickup coil, and the corresponding output voltage is recorded on the chart recorder. The input flux is generated by applying an a.c. current to the calibration coil, which is wound directly on the pickup coil.

The magnetic flux in gauss cm 2 due to a current I in amperes in an N-turn coil with average radius R is given by:

2 ~2RIN2 A~b 10 ' (45)

where

N = 2, number of turns of the calibration coil, R = 0.755 cm, radius of the calibration coil,

for

I = 10 -6 amperes, Ath = 5.86 X 10 -6 G cm2/gA.

An a.c. current of 46 gA in the calibration coil gives rise to an output of 3.7 V. From this experiment we find that

7.3x 10 5Gcm2= 1V output,

or

a = 7.3 × 10 -5 G cm 2 per volt (old detection module).

These results are obtained under the following

76 V . S . TuMAN

experimental condition:

feedback resistance = 1 Mf~ PAR time constant = 0.3 sec PAR sensitivity = 20#V,

The determination of the coefficient a is directly related to the detection module, and it can be obtained experimentally without utilizing the rest of the equipment. The detection module is shown to be linear, and these results have already been recorded in AFCRL No. 70-0449.

3.6 Determination of the spring constant K o[ the instrument

In Section 3, we have already shown that a cryogenic gravity meter can be considered as a mass of superconducting hollow niobium sphere suspended by an electromagnetic spring. To a high degree of accuracy, ignoring the second order non-linear terms, one may write the time constant of the system as:

T ~ = 4~2mk -~, (46)

instrument, it is easier to measure the resonance period T and then calculate the spring constant from equation (46).

The hollow niobium ball is set into oscillations by an a.c. magnetic force. The force is generated by supplying the impulse coil (which is placed below the sphere) by an a.c. current. The impulse coil has the following specifications (see also Fig. 13):

N = 200 turns, inner diameter = 1.2 ram, outer diameter = 1.5 ram,

length = 3.5 ram.

The force on the ball may be expressed as the product of the magnetic dipole moment of the ball, with the vertical magnetic gradient produced by the impulse coil:

F = M \ dZ ] ' (47)

The magnetic polarization per unit volume, 13, may be expressed as:

where T = resonance period in seconds, m = mass of the niobium hollow sphere in grams, m = 2.45 gm and K-- spring constant dynes/cm.

In order to determine the spring constant K of the

~3 B / /= 81r (48)

where B is the intensity of the magnetic field supporting

4.o

3.5

3.(

2 ,

>

,-4 O >

+a

o

3'. O

O. 5

~..~_g

' ' ~-a

0 . 1 0 , 5 1 . 0 1 . 5 2 . 0

R e s o n a t o r l i ' e q u e n c y H z

Fig. 13.


the ball. The magnetic moment can be written as

M=-fff d , vol

77

where B = magnetic field strength of the suspended field and I = current in the impulse coil.

(49) Example: Consider a case where the magnetic field B is 180G, and current in the impulse coil is 2t tA, peak to peak, then

(50) AF=441 × 10±3 x 180x2x 10 6 = 1.4112 x 10_4dynes.

where Rs = radius of the hollow niobium sphere

or

M = - :~- B. (51)

The vertical component of the magnetic field, Bz, due to the impulse coil, is given as:

' t 2rrNI Z + ! r2]l/2 Z - ~

p]

With the aid of the impulse coil, a constant alternating force is applied on the ball, and the output is monitored on a chart recorder. Starting with a high-frequency, e.g. 5 Hz, the response of the ball is recorded, gradually decreasing the frequency. At a resonance, the ball yields a maximum response. As we pass over the resonance to a lower frequency, the response of the ball decreases (see Fig. 13). From such a record, we are able to determine the resonance frequency and the quality factor Q of the instrument (see Fig. 14). The spring constant K is given by

K = w2m = 4 ~ 2 f 2 m .

and magnetic gradient: From equation (46), we have:

d& ] 2rrNl d-Z-/impulse = 10

coil

, - 1 2 / i / 2 4

{(z-~) +r~/ {(/-~r~}-" 1 (52)

where N = 200--number of turns of impulse coil, r,~ = 0.135cm--average radius of the impulse coil, l = 0.35 cm--length of impulse coil, and Z = 2.05 cm--height of center of ball from impulse coil.

Thus,

dBz ~-i , ,pul . . . . , = - 0.431 I G/cm.

f = 1.1 Hz m = 2.45 gm K = 117.034 dynes/cm.

3.7 Magnetic flux change at pickup coil due to motion of ball

In this section we shall study the relationship between the amplitude of the ball and the corresponding amplitude of the signal recorded by the chart recorder or the digital unit.

The hollow niobium ball may be approximated to a magnetic dipole. The flux linked at the pickup coil is given by:

$ = f f B . r~ ds, (53)

where B is the magnetic field intensity at the pickup coil, due to the magnetic dipole of the hollow niobium sphere. Note that

B = V x A, (54)

The magnetic force on the ball is given by

. dB~ F = M -d-z-impulse coil

M=R-~-~B

for

we have

R = 1.27 cm

where A = magnetic vector potential due to the magnetic dipole of the hollow sphere.

Thus

or

4,=ffvxA.,ids (55) f

ch = ~ A . dl (56)

The magnetic vector potential A may be expressed in terms of the magnetic dipole and the position where A is being measured. We have

F=441 x 10 3BI dynes A = M X r ir p . (57)

78 V.S. TuMAN

Fig. 14.

r is the position vector where A is being measured with reference to the center of the ball.

tAI Msin 0 = ~ , (58)

Rp = radius of pickup coil, Z = distance between pickup coil and center of hollow sphere and r = position vector,

r = Z ~ + Ro z, (59)

4, -- ~IAI dl, (60)

~b = 2~rR. JAJ, (61)

or

M sin 0 6 = 2 ~ r R p ~ . (62)

After substituting the values of M, sin 0, and r in terms of the known parameter, we get

7rR~ ReZ B d~ = (Z 2 + Ro2)~/2. (63)

To find the flux change as the ball moves up and down, we have

[0,] A ~ = ~-~ AZ, (64)

or

f3 ~rRs3Rf BZ]

Thus the magnetic flux produced at the pickup coil by the motion of the hollow niobium sphere can be expressed as


3.8 Experimental results The vertical seismometer is presently installed at the

Cottrell observatory and is tuned to 25-see periods. A (65) d.c. bridge thermister sensing device controls the tem-

perature of the seismometer to 10-2°C. We are planning to obtain other types of thermisters, such as a silicon thermister, in order to improve the temperature regula- tion to better than 10 3°C. Another unit to control the temperature environment of the outer cavity of the

(66) seismometer will be installed to improve the overall temperature control of the unit. At this stage, to save power, we are controlling th e temperature of the spring at 297.2K, some 10°C above the surrounding environment. Presently, on the average about 100 ma current is used to maintain the unit's temperature.

So far we have recorded a large number of earth- quakes from all over the world (see Figs. 4 and 5). The tides which include the solid, atmospheric tides, as well as ocean loading on coastal ranges, are given in Figs. 6 and 7. All these effects are recorded simultaneously. Efforts will be made to remove both solid tides and atmospheric tides and estimate the ocean tides loading in coastal ranges in this part of California.

A microborograph was recently installed at the Cottrell observatory and pressure fluctuation is being recorded by that instrument. Attempts will he made to remove the atmospheric effect from the total tide components. These results will also be applied to the cryogenic gravity meter.

where

[ 31rRflRfZ J[ : [ (Z 2 + Rf),:2j B, (67)

B = magnetic strength of the suspension unit

or

~= GZ,

where G is purely a geometric constant of the instrument.

For the present setup, we have

Z = 4.09 cm,

R, = 1.27 cm,

Rp = 0.755 cm.

Then

for

then

or

G = 1 , 8 × 1 0 2,

B = 100G,

= BG = 1.8,

A(;b = 1.8AZ Gcm 2

If the ball moves 1 ]~, then

A4, = 1.8 × 10 -s G cm 2,

with

the signal

A/b = 1.1 × 10 4,

Av = 1.1 × 104x 1.8x 10 -s,

Av = 198 #v/A.

Note that A:b, magnetic field B and spring constant K are all adjustable.

4. GRAVITATIONAL PHYSICS AT COTTRELL OBSERVATORY

4.1 Detection of gravitational radiation The original interest at JPL in Tuman's data was

connected with his discussion of the possibility of Earth spheroidal osci l la t ions excited by gravitational waves[20-23]. The idea was to use the Earth as a resonance detector with periods of free oscillation from a few seconds to about I hr. It is interesting that gravitational waves in the same frequency band will also influence a Doppler tracking signal returned from inter- planetary spacecraft[2,5,6]. Therefore, there is a possibility of cross-correlating data on the Earth's free oscillations with Doppler data from spacecraft, and thereby of resolving ambiguities in the detection of gravitational radiation.

4.2 Preferred frame effects A remarkable property of the theory of general rela-

tivity is that local gravitational physics, for example in the solar system, is independent of any preferred reference frame for the universal distribution of matter. It is impossible to use gravitational experiments to detect the motion of the solar system with respect to a universal system of rest. Also, the universal gravitational constant G is precisely what its name suggests. It is isotropic and is not affected by the distribution of nearby matter. Of course, these predictions of the theory may not be true, and it is important to search for the evidence of a preferred reference frame by means of gravitational experiments.

80 V.S. TuMAN

Fortunately, a genera[ post-Newtonian theoretical framework has been developed for purposes of comparing results from various gravitational experiments[27,14,28,29]. In particular, a preferred frame parameter ~2 and an anisotropy parameter ~,. enter directly into the calculation of Earth tides in the parameterized post-Newtonian (PPN) formalism. For the preferred frame parameter a2 the harmonic analysis of Wi11129] can be used to compute the frequency and amplitude of the largest tidal components for the Cottrell observatory (geocentric latitude = 37°16'43"). The results are shown in the following table for an assumed preferred frame motion of the solar system of 200 km/sec in a direction corresponding to a circular orbital motion about the center of the galaxy.

PPN Newtonian Frequency Amplitude Amplitude

(0/day) Ag/g x l0 ~ &gig X 109

30.041067 6.3a2 0.2(R2) 30.082137 19.0 a2 6.1 (K2) 15.000000 9.6a2 0.4(S)) 15.041069 67.5a2 43.1(K0

Unfortunately, the PPN tides do not produce any unique frequency component in the tidal spectrum and hence, if present, they will be evidenced by anomalous Newtonian amplitudes. Using published tidal data, Wi11129] was able to determine a bound 1~21 < 3 × 10 ̀2 by this technique. More recently Warburton and Goodkind[25] have analyzed in considerable detail an 18-month Earth tide record from their superconduction gravimeter at Pifion Flat, California. In addition to amplitude information, they also consider the differences in phase between the PPN tide and the Newtonian tide. They find that [a21 < 2 x 10 3, with 95 percent confidence and that the upper limit on ~,o is on the order of 10-3. However, they point out that their data does allow for the existence of preferred frame effects, but they cannot prove it because of uncertainties in ocean and atmospheric loading on the Earth's crust. They also point out that additional data in the future could reduce the un- certainty of geophysical effects.

This conclusion by Warburton and Goodkind makes it very important to carry out additional observations. We propose in the next two years to develop the cryogenic gravity meter into an instrument to record Earth tides at Cottrell observatory, In the meantime, data analysis methods will be developed so that we can provide independent analysis of tidal records from Cottrell over many months when those records become available in the future. If a preferred frame effect is a possibility, as claimed, then an independent analysis of tidal data is needed to complement the results of the San Diego group. For example, if they were to discover an unam- biguous preferred frame tide, the relativity community would surely demand an independent confirmation of

such a fundamental discovery. We hope that within the next two years or so to provide an independent analysis along these lines.

4.3 Geophysical effects The increased sensitivity of the cryogenic gravity

meter over conventional instruments demands a more careful attention than in the past to geophysical effects, such as ocean and atmospheric loading. Warburton and Goodkind and their group at UCSD have faced this problem with data from their superconducting gravimeter and have published their data analysis methods in the literature[24-26]. We can make use of their findings in our own data analysis.

We have completed computer software to evaluate the Newtonian tide for a rigid Earth as a function of time. The method of computation is described briefly in the next section.

4.4 Calculation of Earth tides Rigid body tides have been calculated for the direct

attraction of the Sun and Moon on the gravity meter. It has been necessary to consider tidal effects at a level well below a microgal. For comparison purposes, the direct attraction of the Sun and Moon on a gravity meter will cause a maximum variation of about 275 ~zgal. The required accuracy in the theoretical tide has been achieved by computing a vertical component in the acceleration of gravity which is produced by a combination of the attraction of the Moon and Sun. The classical developments of the tide generating potential have not been used, Instead, the variations in the acceleration of gravity have been computed directly by means of the expression,

~g ( R ) 3 1 - / r \ 3 r ( 1 r3cosz] , - = " [ tv ) g (68)

where m is the mass of the perturbing body (Sun or Moon), r is the distance between the center of the Earth and the perturbing body, R is the distance of the observatory from the center of the Earth, and r' and z are respectively the distance and zenith angle between the observatory, and perturbing body.

The geocentric positions of the Moon and Sun at any instant of time have been obtained from JPL ephemeris tapes. Positions of both the Moon and Sun on these tapes have been computed by an n-body numerical in- tegration and hence do not depend on any analytical theory for the lunar or solar motions. The computation of the position of the observatory in inertial space takes into account the variations in the rate of rotation of the Earth and also the wandering of the pole. Both of these effects are important at the microgal level.

The subtraction of two quantities near unity in the second term of equation (68) can be avoided by defining a small parameter q.

, ,691,

From the law of cosines,


where

r 'z = r 2 + R 2 - 2Rr cos z, (70)

and r' can be eliminated in equation (69) to yield:

.=

and hence,

(f,,)3 = (1 + 2q)3/2. (72)

The expression for Ag/g used in the computation is derived from equations (68) and (72):

(2 Z - r ) f R ( Ag _ m R3 [, q)],] g (l+ -q r[:-cosz cos

(73)

3 + 6q + 4q 2 (74) f(q) = 1 + (1 + 2q) 3/2'

computed by equation (71).

81

n4 { [ J n z(ne)- 2eJ, ,(ne) g(ne) =

+2 J,(ne) + 2eJ,+,(ne)- J, ÷2(ne)] 2 n J

+ (1 - e2)[.l,_dne) - 2J,(ne) + J,+z(ne)] 2

4 [j,(ne)]2}. +V

Peters and Mathews calculated the value of g(ne) for values of eccentricity of 0.2, 0.5 and 0.7. We have extended these calculations to e = 0.9L 0.96 and 0.98. Note that the major axis a and minor axis b of the Keplerian ellipses are related to the eccentricity e of the orbit by

For

4.5 Gravitational radiation from a highly eccentric black hole binary system and

With our cryogenic gravity meter we had obtained some data, the Fourier analysis of which can be inter- preted as very low energy excitation among the Earth eigenvibrations [20]. A very dramatic structure, however, was observed among these eigenvibrations, which we have called a parity structure. Namely, within a certain frequency band, the even eigenvibrations with quadru- pole and multipole moments had been slightly more excited than their neighboring odd eigenvibrations of the Earth. The mechanism for this parity structure among the Earth eigenvibrations has remained a mystery. We had speculated that one remote possibility could be ascribed for to the intense tensor gravitational radiation[20].

In their classical paper, Peters and Mathews[15] have calculated the gravitational radiation from point masses in a Keplerian orbit. These authors pointed out that such a binary system would radiate gravitational radiation at fundamental frequency OJo and higher harmonics moo associated with period of the orbit:

b = a ~/(I - e 2)

e = 0.95, b = 0.31225a; e = 0.96, b = 0.28a; e = 0.98, b = 0.199a;

e = 0.99, b = 0.141a.

The function g(ne) for eccentricities of 0.90, 0.95, 0.96 and 0.98 are given in Fig. 15.

The closest approach between a centrally attracting body, a black hole, for example and the orbiting object, is given by 3, where

3 = a(1 - e)

e = 0.95 6 = 0.05a.

Naturally for realistic orbital motion,

[ G(m, + m2)]'/z where O,o = L S r J '

where m, and mz are the masses, a is the major axis of the ellipse, and G is the gravitational constant. In fact, they came to an interesting conclusion that for highly eccentric orbit the power radiated in higher harmonics can be much higher than in lower fundamental harmonics. The power radiated for the nth harmonic is given by:

p ( n ) _ 1~_~_~,1"32 G 4 m,Zm22(m,a 5 + mz)} g(ne),

6 >> (r, + ro),

2MG r~- c2

is the Schwarzschild radius of the central body with mass M, and ro is the radius of the orbiting body.

A fictitious black hole binary system of central mass M = 106 solar mass and an orbiting object of one solar mass rotating in a Keplerian orbit with eccentricity of 0.95 and a period of 19,480 sec or 5.411 hr seems to fit our former data[20]. Such a giant binary black hole system will generate gravitational radiation with a power

45g.

ago

~_

350

300

/

250,

----

Z

200,

E ~5

0¢---

----

1-

I00,

~------

< h \

97

i .

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

I

< ,-]

C,4

- o •

0

Fig.

15.

Dep

ende

nce

on N

. G

(N, E

) of

the

rel

ativ

e po

wer

in

diffe

rent

har

mon

ics

for

£

o


Table I. Calculations of energy dumped in Earth eigenvibrations from a fictitious black hole binary system

Coefficient of Earth elgen- Corresponding Cross g(nc) Energy coupled to vibrations models harmonic section for c=0.95 eigenvibrations

0S2 6 108 .06 6 x 106

054 12-13 25 x I06 .25 6.25 x 106

056 20 11 x I06 1.14 12.54 x 106

oS8 27 6.2 x 106 3.06 18.97 x I06

oSIo 34 4 x 106 6.10 24.40 x 106

0S12 38 2.8 x 106 8.31 23.27 x 106

0514 44 2.0 x 106 12.19 24.38 x 106

0516 48 1.5 x 106 15.08 22.62 x 106

0518 52 1.2 x 106 18.16 21.79 x 106

oS20 56 1.0 x 106 20.58 20.58 x 106

0522 60 0.8 x 106 24.72 19.78 x 106

oSz4 64 .69 x 106 28.10 19.39 x 106

0S26 68 .59 x 106 31.50 18.59 x 106

0528 71 .51 x 106 34.03 17.36 x 106

0528 74 .44 x 106 36.52 16.67 x 106

0S30 78 .39 x 106 39.78 15.51 x 106

83

of 2.2 × 1045 ergs/sec in the region of the Earth eigenvibrations oS8 to oS28.

In Table 1, we have calculated the corresponding harmonics of the model which would excite the even eigenvibrations of the Earth. The cross section of the oS2 mode was calculated in detail, using the physical charac- teristic of the Earth's internal structure. This was found to be about l0 s cm 2. It is believed that the cross section for polarized gravitational radiation falls as inverse square of the harmonics order[24], the cross section for upper harmonics all the way to oS3o are estimated. These are tabulated in column 3. In column 4, we record the power coefficient g(ne) which was expressed above and is calculated for e = 0.95 (see Fig. 15). In column 5, we give the product of columns 3 and 4, which yields effective power coefficient for different Earth eigenvibrations. Note that a band of eigenvibrations with proper intensity can be excited.

The gravitational flux density required to maintain a parity structure between even and odd eigenvibrations is about 103ergs/cm2sec[20]. Thus, a giant binary black hole source of this nature at 440 light years away is capable of exciting the Earth and gives rise to the observed parity structure.

Acknowledgements--I am very grateful to the Research Cor- poration for their generous grant to establish and operate tem- porarily the Cottrell observatory. I am also grateful to the Cali- fornia State College, Stanislaus administrators for their help and cooperation and for some discussions with Dr. Robert Uhrham- met and Professor B. Belt, of U. C. Berkeley.

Some of the research carried out for this paper was also supported by a grant from the Caltech President's Fund.

I acknowledge numerous helpful discussions and encourage- ment from J. D. Anderson of JPL and Professor C. W. F. Everitt of the Physics Department at Standord University. Special thanks are also due to Professor William M. Fairbank of Stan- ford University for his help, guidance and cooperation.

This paper was completely revised and edited for publication during my tenure (1980-81) at California Institute of Technology, Jet Propulsion Laboratory, with a grant from the National Research Council of U.S.A.

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(1%1). 4. B. A. Bolt and J. Dorman, .L. Geophys. Res. 66, 2%5 (1961). 5, R. W. Davies, In Colloques Int. CNRS No. 220, (Edited by

Y. Choquet-Bruhat), Paris (1974). 6. F. B. Estabrook and H. D. Wahlquist, G. R. S. 6, 439-447

(1975). 7. A. F. Hebard, Ph.D. Dissertation, Stanford University (1972). 8. T. J. Herron, I. Tolstoy and D. W. Kraft, J. Geophys Res 74,

1321-1329 (1%9). 9. B. V. Jackson and L. B. Slichter, J. Geophys Res 79, 1711-

1715 (1974). 10. I. M. Longman, J. Geophys Res 67, 845--850 (1962). 11. I. M. Longman, J. Geophys Res 68, 485--4% (1%3). 12. P. Melchior, The Earth Tides, Pergamon Press, New York

(1966). 13. Moore, Ph.D. Dissertation, Princeton University (1%9). 14. K. Nordtvedt, Jr. and C. M. Will, Ap. J. 177, 775-786 (1972). 15. P. C. Peters and J. Mathews, Phys, Rev. 131,435--440 (1%3). 16. E. S. Robinson, J. Geophys Res 79, 4418--4424 (1974). 17. K. S. Thorne, Orange Aid Preprint No. 462, Caltech (1976). 18. K. S. Thorne and V. B. Braginsky, Ap. J. (Lett) 204, LI

(1976).

84 V,S. TuMAN

19. K. S. Thorne, Personal communication (1973). 20. V. S. Tuman, Nature 229, 618 (1971). 21. V. S. Tuman, In Proceedings of the Experimental Tests of

Gravitation Theories, R. Davies, Ed., JPL TM 33-499 (1971). 22. V. S, Tuman, In Experimental Gravitation (Edited by B.

Bertotti). Academic Press, New York (1974). 23, V. S. Tuman, In Observation of Earth eigenvibrations pos-

sibly excited by gravity waves, in Nature Physical Sci. 23, 104 (1971).

24. R. J. Warburton, C. Beaumont and J. M. Goodkind, Geophys. J. R. Astr. Soc. 43, 707-720 (1975).

25. R. J. Warburton and J. M. Goodkind, Ap. J. 208, 881 (1976). 26. R. J. Warburton and J. M. Goodkind, Geophys. J. R. Astr.

Soc. 48, 281-292 (1977). 27. C. M. Will, Ap. J. 169, 141 (1971). 28. C. M. Will, Ap. J. 185, 31 (1973). 29. C. M. Will, In Experimental Gravitation (Edited by B. Ber-

totti). Academic Press, New York (1974).

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Gravity research at Cottrell observatory