AN

ORBITING THE

CLOCK

EXPERIMENT

GRAVITATIONAL

RED

TO

DETERMINE

SHIFT*

DANIEL KLEPPNER, Dept. of Physics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.

ROBERT F. C. VESSOT, Hewlett Packard Corporation, Beverly, Mass., U.S.A.

and NORMAN F. RAMSEY, Harvard University, Cambridge, Mass., U.S.A.

(Received 5 May, 1969)

Abstract. Plans are presented for an experiment to measure the gravitational red shift of the Earth by comparing a ground-based and satellite-borne hydrogen maser dock. The limiting accuracy is estimated to yield a determination of the red shift to 1 part in 105, corresponding to a clock stability of 3 parts in 10IL 1. Introduction The principle of equivalence, the assertion that there is no way of distinguishing locally between a gravitational field and an oppositely directed acceleration, plays a fundamental role in the theory of gravitation. The principle was first enunciated by Einstein (1907) as a generalization of the observed proportionality between gravitational and inertial mass. Like all the laws of physics, the principle of equivalence has its ultimate justification in experiment. Newton himself verified the proportionality of gravitational and inertial mass to an accuracy of approximately 1 part in 10 3 in a famous series of pendulum experiments (Newton, 1934). In recent years a check of this proportionality to a few parts in 1011 was carried out by Roll et al. (1964), using a refined version of the E/Stv/Ss experiment. A direct consequence of the principle of equivalence is the so-called gravitational red shift (Landau and Lifshitz, 1962). A source radiating in a gravitational field appears to an observer to be fractionally shifted in frequency by an amount ~lv/v = & o / c 2 ,

(1)

where Aq) is the gravitational potential difference between source and observer, q)s - (Po. (We assume that IAq,/c21 ~ 1.) This effect is approximately 2 parts in 10 6 for the sun and somewhat larger for other stars. Unfortunately, it is difficult to make a precise astronomical measurement of the solar red shift due to turbulence and other extraneous sources of frequency shift. A number of proposals have been made for determining the effect of gravitational * This work was supported by the National Aeronautics and Space Administration. Astrophysics and Space Science 6 (1970) 13-32; 9 D. Reidel Publishing Company, Dordrecht- Holland

14

DANIEL KLEPPNER ET AL.

potential on the rate of a clock (Ramsey, 1956; Winterberg, 1956; Singer, 1956; Hoffman, 1957; Refsdal, 1961). A direct determination of the effect of gravity on radiation has been carried out by Pound and Rebka (1960) in an experiment which was later refined by Pound and Snider (1965). These experiments used the Mossbauer effect to provide highly selective resonant absorption of gamma radiation in the Earth's gravitational field. Although the observed effect with the 75 ft. height used was only 5 parts in 1015, the results of this elegant experiment confirmed the prediction of the equivalence principle to 1%. Up to the present there has been no direct observation of the effect of gravity on the local time scale. Such a determination requires the use of separated coherent radiation sources, that is, of separated clocks. The determination would test a slightly different aspect of the principle of equivalence from that of the Pound and Rebka experiment, though, like the Pound and Rebka experiment, it would be incapable of distinguishing between different gravitational theories for which the principle of equivalence is a basic postulate. We describe here an experiment to measure the effect of gravity on the local time scale. It offers the possibility of checking the principle of equivalence to much higher precision than so far obtained. The experiment utilizes spacecraft techniques which enable clocks to operate far from the Earth, so that the full magnitude of the Earth's gravitational red shift, approximately 5 parts in 10 l~ can be observed. The clocks are hydrogen masers, which have been developed as a practical atomic clock to the point where they can achieve a stability of better than 1 part in 1013. In contrast to the present accuracy of 1%, it may be possible to measure the red shift to 10 parts per million. Such a marked increase in sensitivity would be highly desirable, but the cost of the experiment should be weighed against the high probability that it will only give the anticipated confirmation of the equivalence principle. In the following sections we describe plans for an experiment that could achieve this high accuracy. 2. Magnitude of the Relativistic Frequency Shift for a Satellite-Borne Clock In addition to the effect of gravitational potential described in Section 1, well established relativistic time dilatation effects are also significant in this experiment. The result is that a satellite-borne oscillator, when observed from the Earth, is seen to be shifted in frequency by an amount given fractionally by -=

v~ -- v~ 1 1 v2 v-~7~= ~c (cpS- %) - 2~c2 (S - v2)'

(2)

where v~ and vo are the velocities of the satellite and the observer relative to the center of mass of the Earth respectively, and ~o~ and ~0e are their respective gravitational potentials. In our case the term 'red shift' is something of a misnomer; the observer on the Earth is at a lower potential than the satellite, and the frequency is consequently increased. Strictly speaking, it is a "blue shift'. It should be emphasized that the fundamental quantity of interest is the difference of

AN ORBITING CLOCK EXPERIMENT

15

the proper time intervals of the satellite and Earth systems. This would be apparent in an experiment in which two clocks were synchronized on Earth, and then compared after an interval during which one of the clocks was maintained at a different gravitational potential in space. Although such an experiment is appealing pedagogically, it is impractical because of the technical difficulty of operating high precision clocks at uniform rates through launching and retrieval and because of the relatively high risk of failure - a single lapse in either clock would invalidate the experiment. For this reason the clocks can better be compared continuously once the satellite is in orbit, so that in effect the proper time intervals are measured repeatedly at a high repetition rate. Such a procedure is more naturally described as a frequency comparison, rather than as a comparison of proper time intervals and for this reason the language of this paper will usually be in terms of a frequency comparison. However, it should be kept in mind that the quantity ultimately measured is the difference in proper time intervals between the satellite and Earth. As a first approximation we will treat the Earth as a spherical mass at rest and neglect all perturbations due to the quadrupole and higher moments of the Earth, as well as solar and lunar perturbations. If we designate this uncorrected red shift bySPo, we have GMe ~o = -~-

1

1 vs 2 c z'

-

(3)

where G is the gravitational constant, Me and Re are the Earth's mass and radius, respectively, and r is the radial distance to the satellite. The corrections to J o change it by less than l~o and can safely be neglected for the present. The kinetic energy, K, potential energy, V, and total energy of the satellite, E, are related by

Morn K + V =E=-G

2a

'

(4)

where a is the semi-major axis of the orbit and rn is the satellite's mass, and V is taken to be 0 at infinity. According to the virial theorem ( g ) = - 89( V ) ,

(5)

where the bracket indicates a time average. Equation (3) then yields

=

\

- ?a/"

(6)

The quantity G M e / R e c 2 = 6.94 x 10-lo is the total red shift due to the Earth, as would be observed if the satellite were at rest at a large distance from the Earth. It is known to three parts per million from current geodetic data (Kaula, 1963). The second term in the bracket accounts for the decrease in Sao due to binding of the satellite by the Earth's field. The experiment should be performed in a synchronous orbit for which a=6.61 R~, and <5P0>=5.37 x 10 -1~ As will be discussed in Section IV, the

16

DANIEL KLEPPNER ET AL.

experiment is best performed in an elliptic synchronous orbit which produces a diurnal modulation of the red shift. The amplitude of the modulated component of is approximately 3 x 10 -l~ so that if 5 P is to be measured to 10 ppm, the mean fractional frequency fluctuation between the clocks when they are repeatedly compared over a period of several hours should be 3 x 10-15. This appears to be a realistic goal in view of the present clock stability and the improvements to be described below. 3. The Hydrogen Maser Clock

The hydrogen maser has been chosen for the time standard in this experiment because it has the highest stability of any currently available time standard. Its theory and operation have been summarized in the literature (Vanier and Vessot, 1965; Kleppner et al., 1965). A brief outline of its operation is presented here, with some details on the construction of the maser for space application. A. OPERATIONOF THE MASER The hydrogen maser is an atomic beam maser which operates on the well known 21 cm hyperfine transition in the ground state of atomic hydrogen. The energy level diagram for hydrogen in its ground state is shown in Figure 1. Figure 2 schematically illustrates operation of the maser. Molecular hydrogen is dissociated in an r.f. discharge, and the emerging atoms are collimated into a beam which passes through a hexapole magnet. In the magnet atoms in the states ( F = 1, m = - 1) and ( F = 0 , m = 0 ) are deflected outwards, while those in the states ( F = l, m = 1, 0) are focused into a

rn=[

i

w AW

. . . . . . . . . .

i =0

F=[

Fig. 1. Energy level diagram for hydrogen in the ground state, zS1/z,x is a parameter proportional to the magnetic field.

AN ORBITINGCLOCKEXPERIMENT

17

Teflon-coated quartz storage bulb inside a microwave cavity which is tuned to the 1420 MHz hyperfine transition frequency. The atoms remain in the storage bulb for approximately 0.5 sec. This is sufficiently long for self-sustained oscillation to occur; the power radiated by the beam is adequate to sustain the microwave field needed to PICKUP LOOP

ATOMIC HYDROGEN SOURCE - JIZJ-~,~--/_. ! .__ . __. ................... ................. HEXAPOLE MAGNETIC

STATE SELECTOR

Fig. 2.

" I /

~[_. ~

~: : . . . .

QUARTZ ntH n . .

.

.

COATED WITH TEFLON

.

.

TUNED RF CAVITY

Schematic diagram of hydrogen maser.

induce the hyperfine transition. The long interaction time produces a very narrow resonance width, typically 1 Hz. Since the transition frequency is 1.4 x 10 9 H z , the intrinsic precision of the device is very high; two masers can readily be compared to 1 part in 1014 in an observation time of 100 sec. A cutaway view of a maser designed for space operation is shown in Figure 3. B. THE HYDROGENMASERFOR SATELLITEAPPLICATIONS A brief description of the features of the hydrogen maser is given below. A more detailed description has been given by Vessot et al. (1967). 1. Atomic Hydrogen Source and State Selector The maser requires a flux of approximately 1013 atoms/sec in the state ( F = 1, m=0). This is provided by a source consisting of a discharge chamber, 2.5 cm in diameter, supplied with molecular hydrogen at 0.05 mm pressure. An r.f. discharge is maintained in the chamber by a 100 mHz oscillator. A few watts of d.c. power is required. Off axis flow of hydrogen from the discharge is reduced by a multitube collimator to minimize the gas load. The hexapole magnet has a gap diameter of 3.2 ram, a length of 7.6 cm, and provides a peak field at the pole tips of approximately 8k gauss. A stopping disc at the down stream end of the magnet serves to prevent undissociated hydrogen from entering the bulb. Hydrogen for several years' operation is stored under pressure in a small tank and is metered out to the discharge at a controllable rate through a heated palladium-silver pellet. 2. Storage Bulb The storage bulb retains the radiating atoms for the relatively long period required for oscillation. Atoms remain in the bulb for about 0.5 sec, during which time they under-

18

DANIEL KLEPPNER ET AL.

Fig. 3. Cutaway view of maser for space applications. (1) Hydrogen discharge source; (2) vacuum pump; (3) hexapole magnetic state selector; (4) magnetic shields; (5) solenoid to provide small ambient field; (6) coupling loop for reactance cavity tuner; (7) loop for exciting field dependent 'Zeeman' transition; (8) rf coupling loop; (9) cavity; (10) quartz storage bulb; (11) heater windings.

go about 104 collisions with the walls. The chief limitation on tile effective storage time, at present, is loss of free atoms at the walls of the bulb. This reduces the radiation lifetime to 0.3 sec. The storage bulb is made of quartz and is 15 cm in diameter. Its surface is coated with Teflon in order to minimize perturbations during wall collisions. The effect of the wall collisions is to decrease the radiation frequency by about 2 parts in 1011. The wall shift appears to be stable in time and since the relativistic frequency shift will be modulated by the elliptic motion of the satellite, any possible long term changes in the wall shift are not expected to introduce significant error.

AN ORBITING CLOCK EXPERIMENT

19

3. Microwave Cavity The atoms radiate in a microwave cavity which is tuned to the transition frequency. The cavity is probably the most critical single component since long term stability of the clock depends directly on the cavity's dimensional stability. If the cavity is mistuned it will cause a frequency error because of the pulling effect of a maser cavity. An automatic tuning system to overcome this will be described below. The cavity operates the cylindrical TE011 mode. It is made of Cer-Vit, a material which has a temperature coefficient less than {~- that of fused quartz. Power is coupled from the cavity by a small loop in the end plate. A second loop couples the cavity to a varactor diode, providing electrical fine tuning of its resonant frequency.

4. Vacuum System Although it may seem like a luxury to put a vacuum system aboard a satellite, there are several compelling reasons for doing so. The chief problem with exhausting the maser to space is that contamination of the storage bulb by outgassing from the satellite could seriously degrade the maser, or even prevent it from operating. In any case, the maser would have to be vacuum isolated until it was in space, for the risk of contamination during preparation and launch would be prohibitive. These factors, together with the necessity of operating the maser until shortly before launch in order to check its performance and to get the best possible frequency calibration, argue strongly for a self-contained vacuum system. The vacuum is maintained by an ion pump designed to fit the maser configuration. It has a six-element array about a central chamber and provides a pumping speed of 150 liters/sec. Under normal operating conditions the pump draws 2 ma from a 1.5 kV power supply.

5. Magnetic Shields There is a small quadratic dependence of the atomic resonance frequency on the ambient magnetic field. For this reason the cavity is surrounded by a set of three concentric Moly-Permalloy shields. A solenoid within the inner shield provides a uniform ambient field of 1 milligauss. This produces a fixed fractional offset of the frequency of 2 • 10 -12. The field can be monitored on command by introducing a signal to drive the low frequency transition ( F = 1, m = 1 ) ~ ( F = 1, m=0). This line has a field dependence of 1.4 kHz/milligauss, and a width of about 5 Hz. Provision must be made for demagnetizing the shields once the satellite is in orbit. This can be accomplished by passing an alternating current of about 400 amperes along the shield axis and decreasing it to zero over a period of approximately 15 sec.

6. Temperature Contol Due to their random thermal motion the radiating atoms undergo a time dilatation effect similar to that produced by the motion of the satellite in its orbit. The frequency shift is proportional to the temperature, and is, fractionally, - 1.4 • 10-13 ~ In order

20

DANIEL KLEPPNER ET AL.

to avoid a fractional change in frequency of, for instance, 2 x 10 -15 the temperature of the storage bulb must be held constant to 0.014~ This is accomplished in several stages of progressively finer control with conventional techniques. C. A U T O M A T I C T U N I N G SYSTEM

As mentioned above, long term stability of the maser appears to be limited by cavity stability. The effect of mistuning the cavity is to shift the oscillation frequency by an amount v -

Vo = (v~ -

Vo) Av.IA~,

where Vo is the atomic resonance frequency, vc is the resonance frequency of the cavity, v is the oscillation frequency, and A v, and A vo are the atomic and cavity resonance widths, respectively. Under typical conditions, A va/Av~=3 x 10 -5. Thus, to keep the maser tuned to 3 x 10 -15, the cavity must stay tuned to 10 -l~ Fortunately there is a simple test to determine whether or not the cavity is correctly tuned. If the atomic resonance width, A v,, is varied, the oscillation frequency will remain constant providing v~= Vo. This criterion is valid even when effects of a frequency shift due to hydrogen spin-exchange collisions are included (Crampton, 1967). This is most easily accomplished by increasing the hydrogen flux until the resonance line is broadened by spin-exchange scattering. The test requires a secondary frequency reference for comparison while the flux is changed. The most convenient such reference is a second hydrogen maser, and for this reason and the desire for a redundant frequency standard it is desirable to use a pair of masers in the satellite. The masers are tuned by an automatic system which alternately varies the flux to each maser and provides the necessary correction signal to the cavity. Tuning is done continuously, with a typical tuning cycle of once per minute. Details of such a system are given by Vessot et al. (1967). D . P E R F O R M A N C E OF T H E MASERS

Although a complete set of satellite-borne masers has yet to be constructed and tested there is a considerable amount of stability data available from existing masers. Data taken in 1965 (Vessot et al., 1966) and more recently (Vessot et al., 1968) have shown the fractional frequency stability for periods of time from 103 to 10" sec to be better than 1 part in 101.. The data shown in Figure 4 gives the standard deviation of the fractional frequency stability between pairs of adjacent samples each averaged over time z. It is seen that there are two distinct branches to the plot. Below 100 sec averaging time the data shows a z-1 behavior that is associated with the additive white noise in a bandwidth of 2 Hz. For z greater than 100 sec the data shows the effect of a I / f dependence of the spectral density of the output frequency. This type of variation is due to time dependent systematic effects such as cavity mistuning, magnetic field variations, etc., that, in principle, can be better controlled. The chief source of this kind of instability is the cavity and it is expected that the use of the low expansion material and the automatic tuning system will improve the performance substantially.

AN ORBITING CLOCK EXPERIMENT z o b-

W

123 yr..) Z W

wn: b.J z 0

21

i.,..,.-12.,m I I IlllIII t I 111,1~I E 1 I IIII_-- LEAST SQUARE FIT TO DATA CALCULATED FREQUENCY lO-13_i\ , \ " - ~ D E V I A T I O N -

I 0 -14 --

\\ \ \ \ \ \ ~ \\\

p-

nIt_

F tO -15

I I l llIlll I

Fig. 4.

\XN\\\ I

I l lIrlIl

Xr

I r frl

I0 I00 I000 AVERAGING TIME, SECONDS

Frequency comparison of two hydrogen masers. Rms deviation of the fractional frequency difference between pairs of adjacent samples each averaged over r sec.

4. The Choice of an Orbit A. T H E CASE FOR A S Y N C H R O N O U S ORBIT

The red shift increases with gravitational potential, and this inescapable fact dictates the use of a 'high' orbit, with a >>R, as contrasted to a 'low' orbit close to the Earth. The disadvantage of a 'high' orbit is the relatively large energy requirement for launching. Since the observed red shift is itself proportional to the satellite energy, however, any decrease in the launching energy necessarily results in a corresponding decrease in the experiment's sensitivity. Although 5: achieves its maximum value when the satellite is in an escape orbit, the requirement for communicating with the satellite over an extended period dictates a closed orbit. With consideration of these requirements a synchronous orbit naturally presents itself. The semi-major axis of a synchronous orbit is 6.61 Re, so that from Equation (6) ( 5 : o ) = 5 . 3 7 • -1~ 7 7 ~ of the maximum possible value. Consequently, the experiment has close to its full sensitivity. In addition, a synchronous orbit has the important practical advantage of allowing uninterrupted communication between the ground station and satellite, which substantially simplifies the problem of comparing the satellite and ground-based clocks. Furthermore it allows data to be collected continuously, in contrast to the case when the satellite is only intermittently visible.

22

DANIEL KLEPPNER ET AL.

B. ADVANTAGE OF AN ELLIPTIC ORBIT

Although the average value of 5o0 depends only on the period of the orbit, the instantaneous value varies with the orbital position. The following result follows from Equations (3) and (4): 5o~

-

~+2a-

;

(7)

5~ achieves its maximum value at apogee and its minimum value at perigee. The extremes of 5oo are shown for various eccentricities in Figure 5. The advantage of an eccentric orbit is apparent: the diurnal variation of the observed red shift modulates the quantity being measured. Not only does this reduce concern about the absolute I

i

I o. I

I 0.2

I

I

l

I

I

I

/

0

o

4

x

L3 I 09

o o Fig. 5.

013 0.4 ECCENTRICITY

I Q),5

I 0,6

0.7

Extremes of red shift vs. eccentricity. The upper and lower branches give the red shift at apogee and perigee, respectively.

calibration of the satellite and ground-based clocks, it provides a much more detailed observation of the red shift than is possible for a fixed offset. For instance, since the kinetic and gravitational effects dominate at different points of the orbit, they can be examined separately. Furthermore, by repeating the measurement over many periods of observation it should be possible to improve the final results by statistically averaging

AN ORBITING CLOCKEXPERIMENT

23

60

]

. . . . . . (b)

--DIURNAL VARIATION \ IN RED SHIFT,x I0 iO \\

50-

"

-

40

6 -• m_

\\\

5

Xxxx 3o

._~ ~)

/(;

\\

W~ w (o)

"~c

I

............. -""x

/ \

iJJE3

/

\\\/

-

/

4

c~

-

z -- I--

#20

-3o

\\X\\\

E IO

J

~

\ \\ ',\

2 ~

J

<~ z

\\

0

\\

-IO --J~Jl

I

I

I

I

I

I

0

o.~

o.2

o,3

0.4

o.5

o.6

\

\\

g

0 0.7

ECCENTRICITY Fig. 6. Diurnal variation in red shift, and minimum angle of elevation, for synchronous orbit inclined at 28.5 ~ as viewed by an Earth station at latitude 20 ~ (curve a) and at 0 ~ (curve b). Perigee is at the ground station longitude in the Northern hemisphere.

TABLE I Orbital configuration Period Inclination Eccentricity Perigee Apogee Extremes of R Minimum Maximum Maximum Doppler shift (2.5 GHz carrier) Minimum angle of elevation

24 hours (sidereal day) 28.5 ~ 0.52 2.17 Re 10.04 Re 3.091 • 10-10 6.084 • 10-1~ 18.65 kHz 14.9 ~

the individual measurements, thereby decreasing the effect of random variations in the clock rates as well as other random sources of error. C. ORBITAL CONFIGURATION

F o r reasons discussed in the last section, it is clear that an orbit with the highest

possible eccentricity is desirable. The limiting requirement here is the minimum

24

D A N I E L K L E P P N E R E T AL.

tolerable altitude h~in, where h is the angle between the line of sight of the satellite and the horizon. For a given value of hmin, the highest eccentricity is achieved in an equatorial orbit in combination with an equatorial ground station. However, the advantage, compared to an orbit inclined at the launch latitude, 28.5 ~ is only marginal for hmin ~< ~<15 ~ and since an inclined orbit permits approximately 50% more payload than an equatorial orbit, the inclined orbit is obviously preferable. The latitude of the ground station also affects the satellite visibility. Increasing latitude requires progressively lower eccentricity in order to maintain a fixed value of

6L

l

I

;

I

t

I

I

I

L

I

~

1

I

-

5

~

26

-22

SHIFT

S

0

-- 16 -,~

2D4

b-kL

x

-

:1:

12 ~

cr Ld _J

PPLER SHIFT --

8

a_ o_ 0 I

V

o

F i g . 7.

I

-

l

l

2

I

4

1

I

I

6 8 TIME (HOURS)

I0

12

I

4

0

Red shift and Doppler shift as a function of time. Inclination 28.5~ Earth station latitude 20~ eccentricity, 0.52; carrier frequency 2.5 GHz.

hmln. This effect is illustrated in Figure 6, where the diurnal variation in ~9~ and the minimum altitude for various ground station latitudes are plotted vs. eccentricity. For purposes of discussion, we will assume that the ground station is at 20~ and that the minimum altitude is 15 ~ A summary of the characteristics of such a configuration is given in Table I, and a plot of the red shift, and the instantaneous Doppler shift, are shown for a 12 hour period in Figure 7. 5. C o m m u n i c a t i o n s

between

the Satellite and Earth-Station

Clocks

Comparing two hydrogen masers in the laboratory presents no special problems;

AN ORBITING CLOCK EXPERIMENT

25

matters are far different when the clocks are separated by 40000 km and one is aboard a fast moving satellite. However, as we shall show in this section, the problem is not intractable, and, in fact the full potential accuracy of the clocks can be utilized. A related problem, determining the orbital parameters necessary for predicting the red shift, will be discussed in the next section. To put the present problem into perspective, we will assume in this section that the clocks are instantaneously stable to a few parts in 1014, and that with suitable averaging of random fluctuations, their rates may be compared to a few parts in 1015. In an ideal communications system the averaging period required to determine the relative clock rates would depend only on the clocks themselves. However, in practice the averaging time will depend chiefly on the communications system, and from this point of view the problem becomes one of comparing the frequency of two supposedly ideal clocks to 2 x 10 -15 in the shortest possible time. Time signals could take the form of pulses transmitted between the satellite and ground. If the range resolution of the satellite is Ar, then the time resolution is AT= Ar/c and the minimum observation time (given by AT/Tmln=2 x 10 -15) is Tmi~= Ar/(2 x 10-15c). Current range resolution is 30 m, which yields Train=5 x 107 sec= 1.6 years. Clearly, this is unsatisfactory. Fortunately, the situation is really much more favorable. The saving grace is that absolute range information is not required here - it is the differential range which is significant. This is apparent when one considers a hypothetical satellite at a fixed range - obviously the range plays no direct role when information on time intervals is exchanged. However, allowance must be made for any change in the range during the interval. This naturally suggests a phase coherent communications system, sometimes referred to as a Doppler tracking system. A signal governed by the satellite's clock is continuously transmitted to the ground. This signal carries information on the clock's phase, but the phase information is masked by large phase changes due to the satellite's motion and changes in the propagating medium. These unwanted phase variations are determined by continuously transmitting a second signal from the Earth to the satellite which then relays it back to Earth. The phase of the signal received at the ground station changes by 4 r~ every time the range changes by one wavelength, and this information is used to correct the timing signal. As long as the tracking system maintains phase coherence, the differential range is known to one quarter wavelength. For an S band carrier frequency, the resolution is at least 3 cm, giving a minimum required observation time of TmJn= 5 x 104 sec, or 13 hours. (An even shorter time is possible. This figure assumes that the phase uncertainty is 7r; if the system tracks with reasonably high signal to noise, the phase uncertainty can be less.) This estimate rests on two important assumptions: (a) that phase coherence can be maintained over a period greater than Train, and (b) that phase fluctuations which are internally generated by the tracking system are not a serious problem. With regard to (a), it should be pointed out that the system bandwidth is so small that a high signal-

26

DANIEL KLEPPNER ET AL.

to-noise ratio can be attained with only modest power. Also, the phase is highly predictable, so that in principle the signal can be interrupted for a short time without introducing an error due to 'skipping' cycles. As for (b), conventional phase coherent techniques should be adequate to keep jitter within tolerable limits. Some of the features needed for our purpose appear in a system discussed by Badessa et al. (1960), the chief difference being that the present system is designed for continuous tracking, rather than the transmission of intermittent time signals. A simplified schematic diagram of a communications system for comparing the satellite and ground-based clocks is presented in Figure 8. Both clocks are assumed to have the same proper frequency, %. The satellite broadcasts a signal at its clock frequency, and this signal is received at the ground at frequency v~. In addition the Earth station transmits a tracking signal at frequency vo to the satellite. The satellite relays this signal to the ground, where it is received at frequency v~. These frequencies are related as follows: (a) 'Tracking'Signal. If we refer all velocities to the geocenter, the following result is obtained from straightforward application of special relativity: ,

-

(8)

where the symbols are defined by Figure 9. This result is in approximate agreement with general relativity, and is sufficiently accurate for our purpose. (b) 'Clock' Signal. The following expression follows directly from the principle of equivalence. Alternatively, it can be obtained from any gravitational theory which

SATELLITE TRANSPONDER

SATELLITE CLOCK

I

II

uo

AIXER

=

GROUND CLOCK ~o

= MIXER

II

I '~'0- ~0

V o - Uo

MIXER

ii / ~'o -Vo 2

Fig. 8.

-

/ ,

A~O6T = ~0

Uo+Uo 2

Schematic diagram of communications system.

AN ORBITING CLOCK EXPERIMENT

27

satisfies the principle of equivalence when it is taken to first order in v2/c2 and qUc2, where ~o is the Newtonian potential (Weinberg, 1969). (Cutler (1969) has demonstrated this explicitly for the theory of general relativity.)

1/~0=(~-~2~92/s Vo

(9)

+ 2~03/c2 - ~ )

112"e23/"

When this result is expanded to first order, comparison with Equation (2) reveals that it is 1 + 5a. Equations (8) and (9) refer to signals received simultaneously at the Earth station. For the moment we will neglect motion and acceleration of the Earth station

SATELLITE

2

E23

--

--

~

~

~I1~iiiii

I

~3

\

^

EARTH \ STATION \ Fig. 9. Relative velocities of Earth station and satellite Ii=V/C. during the time of transmission, in which case ~23 = - ~12, and l~l =113- If we let 6s=l~2-~a2, 5e=111.~2 the following result for the output signal of the system shown in Figure 8 is obtained after some manipulation:

,

AVo v~ - voV~ 1 - ~ \%- - 1

)

= s ~ ( 1 - (ss - 8~

+ 5~(5s - 8~ [1 - (Ss - 8 ~

(10)

In this expression, quantities smaller than O (5 3) have been dropped. The first term is the desired red shift, multiplied by a small Doppler correction. The second term is a residual uncorrected Doppler term resulting from the velocity of the ground station.

28

DANIEL KLEPPNER ET AL.

If we let Y = 6 (6s- 6), then -

Arout Vo

x

1

y.

(11)

1 = (< - <)

It can be shown that effects due to acceleration and translation of the ground station during the time of transit are comparable to 50, which means that they must be measured with care. This is facilitated by operating the communications system in a completely symmetrical fashion, which allows the role of spacecraft and Earth station to be identical. Only one additional communications channel is needed for this, an elaboration which is also desirable because of the redundancy it affords. It is interesting to note that the corrections for source acceleration are different for the Earth and satellite. Although only the relative velocity affects the observed red shift, the accelerations of the two stations are individually significant and lead to slightly different corrections. Operation of such a system depends on the ability to maintain phase coherence over extended times. The criterion for this is that random fluctuations of the phase during the round trip time for an Earth-satellite signal, approximately 0.2 sec, be less than 1 rad. Otherwise cycle 'skipping' can take place and the phase becomes ambiguous by an unknown integral number of half wavelengths. Turbulence in the troposphere and the ionosphere is the dominant source of phase fluctuations. Their effect is reviewed by Badessa et aL (1960) and, more recently, by Evans and Hagfors (1968). Briefly, it appears that short term fluctuations should be under 1 rad, and that coherence can be maintained over long periods. 5. Corrections to the Red Shift

Major contributions to the red shift were described in Section 2. In this section we discuss a number of small corrections and sources of possible systematic error. Since the fractional sensitivity of the experiment may approach 3 x 10 -15, corrections of 1 x 10 -15 are significant, and effects 10 times smaller should not be neglected. The following list indicates the relative contributions of these corrections. In most cases an upper limit to the correction will be given, rather than an exact calculation. Unless otherwise indicated, the corrections can be calculated to adequate precision. The superscripts e and s will indicate whether the correction applies to the earth station or satellite, respectively. (a) Motion of the Earth station. From Equation (2), the contribution to 5p due to motion of the Earth station is 2.38 x 10-lZ. 65~; = 89(re~c) 2 = 89(coeR e sin ~o/c) z = 1.19 x 10-12 sin2 q0,

(12)

where toe is the Earth's rotational velocity and q5 is the colatitude of the ground station. Note that this correction is properly included in Equation (10). (b) Quadrupole moment of the Earth. If the quadrupole contribution to the Earth's

29

AN ORBITING CLOCK EXPERIMENT

potential is included, we have ~o(r)-

[

MeG l - J 2 ~ r

]

89

where the quadrupole constant is (King-Hele

,

(13)

et al., 1965)

j o = (1082.64 +__.03) x 10 -6 . This correction affects both the ground station and satellite: namely, 66:~ = 3.76 x t0 -13 d~

(14)

~ 9.0 x 10-14(perigee).

(c) Higher order contributions to the Earth's potential. About 40 higher order contributions to the Earth's potential have been evaluated (Izsak, 1965). The largest of these is j 0 = - ( 2 . 5 _ 0 . 1 ) x 10 -6. Its contribution to cS:Te is less than 10 -15, and it, along with all higher moments, can safely be neglected in calculating the potential at the satellite. (d) Effect of the Sun. The gravitational potential of the Sun at the surface of the Earth is approximately 15 times larger than the Earth's potential. The variation in the Sun's potential as the satellite circles the Earth produces a change in the red shift of approximately 5 x 10 - l a . However, as pointed out by Hoffman (196t), the observed frequency variation would be considerably less due to an ahnost complete cancellation of the gravitational effect by the relativistic Doppler shift due to motion of the Earthsatellite system around the Sun. This result can also be deduced as a simple consequence of the equivalence principle by the following argument. The Earth and the satellite are both in free fall towards the Sun. I f we think of them as members of an isolated system, it is readily seen that their common acceleration in the Sun's field can have no observable effect. Thus the linear variation of the Sun's potential makes no contribution to the red shift, though higher order terms do contribute. This is a good example of the local nature of the equivalence principle: to the extent that curvature of the Sun's potential can be neglected, the Earth-satellite system can be regarded as local. A quantitative estimate of the Sun's effect is easily obtained. For simplicity, we will treat the extreme case where the Sun, Earth and satellite lie on a straight line. Let rs_e= distance from Sun to Earth, rs_~= distance from sun to satellite, and r = distance from Earth to satellite. Then rs_e = rs_e___r. The acceleration of the Earth due to the Sun is ae=MsG/rs_e2, where Ms = the Sun's mass. The acceleration of the satellite due to the Sun is as=MsG/rs.e z. The acceleration of the satellite as observed from the Earth is a's=as-ae. I f we regard this acceleration as the result of a field having an effective potential q0eff,we have

- VOo

= OM,

r

2

(rs.,

,.)2

,

(15)

30

DANIEL KLEPPNER ET AL.

f r o m which it follows that rs_ e q- r

_

aMs rS. e

(

1+

-

('2;)

+...

(16)

This expression yields q%ff at noon and midnight. At sunrise and sunset, qOe~is found by evaluating the above expression at r = 0 . The m a x i m u m variation in ~oef~(r) is and the fractional contribution it makes to the red shift of the satellite at apogee is

GMsrZ/rs_~3,

sun-

= 1.83 X 10 -15

(17)

Ys.eC2

Similarly, the effect of the Sun at the Earth's surface yields

GMs g(~.,) = rS_~C2

&9~

1.81 x 10-17

(18)

The same argument can be applied to the Moon, and gives 65~

= 3.28 X 10 - l s

65P~aoo~ = 3.91 x 10-17.

(19) (20)

(e) Diurnal variations in the Earth's potential. The gravitational field at the Earth's surface varies during the course of a day, the chief variation occurring with the 12 hour tidal period. This is a serious source of possible error, for though the red shift of the ground station does not directly enter the final result, any temporal variations in it would be indistinguishable from a variation in the satellite's red shift. This effect is expected to be small, though possibly not negligible, by the following argument: The quadrupole contribution to the Earth's potential is approximately 1 x 10-3. It produces a correction to See of 3.8 x 10 -13, as given by Equation (14), and a fractional correction to the Earth's field of 1 x 10- 3. Tidal variation in g are typically 3 x 10- 6g, and since the driving field (which is, in fact, given by Equation 15) has the same symmetry as the Earth's centrifugal field, the tidal potential is expected to change 5ee by about 1 x 10-15. Although the tidal variations occur dominantly with a 12 hour period, the 24 hour component can be significant and should not be neglected. 6. Tracking Requirements

The satellite's range must be predicted to high accuracy in order to predict 5P0 to the required accuracy of 10 -15 . F r o m Equation (7) we find that the required range resolution for this is given by

Ar/r g

6

x 10 .6 .

Taking a mean value of r = 6.6 R e ~ 4 x 107m, we obtain Ar ~ 240 m .

AN ORBITING CLOCK EXPERIMENT

31

This is well within the capabilities of present radar ranging techniques. Note that once the range has been established, the Doppler tracking system will supply continuous differential range information, so that continuous absolute ranging is not required. The semi-major axis must also be determined to comparable accuracy. This can be inferred from range measurements at apogee and perigee, or can be independently determined from an optical determination of the period. A complication in applying the above is that the range is measured with respect to the ground station, whereas the required quantity is the distance between the satellite and geocenter. However, since the geoid is known to approximately 7 m (Rapp, 1968), the transformation can be made without difficulty.

Acknowledgements The authors wish to thank Robert J. Rorden for pointing out the usefulness of coherent phase tracking in the present experiment, and to acknowledge helpful discussions with Nat Edmonson, Jr., John V. Evans, John G. Gregory, Russel D. Skelton and Fred Wills. They are indebted to Martin W. Levine and Arthur O. McCoubrey for aid in the design of the experiment, and to Steven Weinberg for a helpful discussion on relativity. They particularly wish to thank Leonard S. Cutler for making numerous contributions to the design and understanding of this experiment, and Ernest J. Ott for encouraging and supporting this work.

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32

DANIELKLEPPNERET AL.

Vanier, J. and Vessot, R. F. C.: 1965, Appl. Phys. Lett. 4, 122-123. Vessot, R. F. C. et al.: 1966, IEEE Trans. on Inst. andMeas., Vol. IM-15, No. 4, 165. Vessot, R. F. C., Levine, M., Mueller, L., and Baker, M. : 1967, Proceedings of the 21st Annual Symposium on Frequency Control, U.S. Army Electronics Command, Ft. Monmouth, N.J., pp. 512-542. Vessot, R. F. C., Levine, M., Cutler, L., and Baker, M. : 1968, Proceedings of the 22rid Annual Symposium on Frequency Control, U.S. Army Electronics Command, Ft. Monmouth, N.J., pp. 605-620. Weinberg, Steven: 1969, private communication. Winterberg, F. : 1956, Astronautiea Aeta 2, 25.