MEASUREMENT PARAMETERS

OF GRAVITATIONAL

INTERACTION

ON AN EARTH-ORBITING

SATELLITE

K. A. Bronnikov, N. I. Kolosnitsyn, M. Yu. Konstantinov, V. N. Mel'nikov, and A. G. Radynov

UDC 528.27

An analysis is presented of a space experiment whose purpose is to enhance the accuracy of measurement of the gravitational constant G, improve the estimates of the fiflh force parameters a and k, and finally to verify the equivalence principle. The experiment involves exact measurements of the trajectories of a light body relative to a heavy body on a drag-free Earth satellite. Estimations of the possible effect of various factors on the relative motion trajectory are made. The equations of relative motion are derived for various types of sateUite orbits. Some preliminary estimates of the accuracy of the measurements are presented which show that the experiment proposed seems to hold promise.

This article is the first of a number of papers devoted to the Satellite Energy Exchange (SEE) project -- an experiment on an artificial Earth satellite, the essence of which has been discussed in [ 1]. It is hoped that during the experiment the gravitational constant will be measured with higher precision than has been achieved in laboratory experiments, and the inverse square law, equivalence principle, and time-stability of the gravitational constant will be verified. The current importance of these problems is due to a number of factors. First, of all fundamental physical constants the gravitational constant has been measured with the least accuracy [2], this being a result of entanglement of various experimental as well as theoretical problems. On the one hand, there is the smallness of the gravitational forces and their universality, as a result of which it is impossible to shield the gravitational interaction; on the other hand, the theoretical status of the constant is not clear since no unified theory of the fundamental physical interactions exists at present [3]. Moreover, some of the propositions of the general theory of relativity are increasingly being subjected to critical analysis. Second, in recent years doubt has been expressed regarding the classical basis of the theory of gravitation, viz., Newton's law. Essentially, this is due to the possible existence of a new macroscopic interaction with a range of the order of hundreds of meters. It is assumed that the interaction may be of a gravitational nature and depend only on the mass of the interacting bodies. In this case Newton's inverse square law would be violated, and this could be interpreted as the gravitational constant being dependent on the distance between the interacting bodies. Another possibility has also been suggested - - a dependence of the new interaction on the composition of the bodies. From this viewpoint an elucidation of the theoretical model of gravitational interaction, even at the classical level, as well as any experimental data pertaining to the problem, would be of paramount importance [41. There are also a number of questions that arise in connection with the problem of time variation of the gravitational constant. Usually implied are the cosmological variations related to the expansion of the universe [3, 5, 6]. The magnitude of the variations depends on the Hubble constant and is quite small. Nevertheless, other time variations of the gravitational constant may be possible and in particular those due to solar and other rhythms. Finally, there is the problem of the equivalence princip!e, i.e., whether the free fall of all bodies is the same. On the Earth the principle has been verified with a sufficiently high accuracy. However, the problem still persists. The cause of this is that there may be a composition-dependent physical interaction. Let us now consider the SEE project [1], according to which an artificial binary system consisting of a more massive body, the "shepherd," and a lighter body, the "particle," would orbit the Earth. Thus, a three-body system consisting of the Earth, "shepherd," and "particle" is being considered. In celestial mechanics many aspects of the three-body problem have been discussed. We shall consider those that relate to the SEE project.

Translated from Izmeritel'naya Tekhnika, No. 8, pp. 6-10, August, 1993. 0543-1972/93/360840845512.50 9 1994 Plenum Publishing Corporation

845

The gist of the SEE method is to make use of a very special class of orbits in the restricted circular three-body problem, viz., orbits of the horseshoe shape [7]. In this case the particle, moving along a somewhat lower orbit than the shepherd, overtakes it and then, as a result of the interaction, goes over to a higher orbit and begins to lag. Near the turning point the relative velocities of the particle and shepherd are small and thus such a binary system located in a drag-free capsule canbe observed over a prolonged period of time. In contrast to the laboratory method, in the space method all external forces acting on the particle, which exceed considerably the force of interaction of the particle with the shepherd, can be eliminated. The tidal forces in the field of the Earth are of the same order as the particle--shepherd interaction. In the present and subsequent papers we analyze the new possibilities that the SEE concept opens up for determination of the gravitational interaction parameters. Another possibility of space determination of G is discussed in [8]. Lagrangian and Equations of Motion. In papers devoted to the three-body problem in celestial mechanics, the aim is usually to determine the motion of a body of infinitesimal mass m (satellite, asteroid) in the field of two massive bodies (e.g., the Sun and Jupiter) with comparable masses M i and Mz; this is the so-called restricted three-body problem. In the SEE project M 1 = M E (M E is the mass of the Earth), M z = M = 500 kg (heavy satellite) and m = 100 g (light satellite). Thus, M/M E = 10-23 and m/M = 10-4. The effect of M and m on M E can certainly be neglected, whereas M and m are comparable. Thus the motion of the (M, m) system in the static field of the Earth can be considered. As stated, the problem differs from the restricted three-body problem; however, a new small parameter s/R appears (notations explained below). In previous studies the separations between the three bodies were assumed to be of the same order. In the SEE approach, however, the distance of bodies M and m from the center of the Earth is - 107 m while the distance between M and m is s < 20 m. With account for the perturbing potentials of external bodies (Sun, Moon, etc.) the Lagrangianofthe "binary system" (M, m) can be written as 1

-5,

..~

...,

~

~

'

L = -~-(MR2+mrZ)+MV(R)+mV(r)+MmU(s),

(1)

where 1~, and r are the geoconcentric radius vectors of masses M and m respectively; s = r -- 1~, V(ff) is the total potential of the Earth and other bodies of the solar system; U(s) is the interaction potential for bodies M and m, and the potential does not necessarily have to be Newtonian. The quantity s = [ s I < 20 m and the characteristic length for an appreciable change of the potential V is of the order of the Earth's radius R E since the leading Newtonian term V 0 in (1) is V 0 = GME/R. The equations of motion for bodies M and m in an inertial reference system that follow from (1) are [~t=Vt(~)_

m

r'=Vd;j+M

OU as i

;

0V

Os t

Their difference is the equation for si(t), i.e., the equation of relative motion:

"~i=vi(~)-vi(~)+(M+m)

au

ds z

(2)

Evidently, the relative motion is determined by the combination of tidal forces [V(~) -- V(l~)] and interaction forces related to the potential U. Some Estimates. We make some preliminary estimations of the various contributions to the right-hand side and the possible effect of the contributions on the trajectory of relative motion of the shepherd (mass M) and particle (mass m). Estimations of this type are required for an optical choice of the method of computer modeling of the trajectories and, in particular, for neglecting undoubtedly small contributions. Separating out the dominant (spherical Newtonian) component in U and V, we obtain v(~)=aMe/R+

P"(~);

u(~*)=~Zs+~', where ~r includes the nonspherical part of the Earth's potential and the potential of external bodies such as the Sun, Moon, planets, components of the satellite, and also the hypothetic non-Newtonian component of the geopotential; 13 includes contributions due to the shepherd and particle not being ideally spherical after their preparation and also possibly a non-Newtonian component. The explicit form of the total Newtonian potential of the Earth in the axial symmetry approximation is GMe [1 1

846

TABLE 1. Estimates of Various Contributions to the Dynamics of the Relative Motion of the Particle and Shepherd Acceleration produced, cm/sec 2

Factor

Tidal forces from ~<2.10 -s potential GME/R ~3.10-s Newtonian forces between masses M and m f o r s ~ lm Quadrupole tidal I0 -8 forces Effect of higher nonspherical harmonics of the geopotential ~7.10-u Effect of the Sun ~ 3.10-~o Effect of the Moon Effect of Jupiter Effect of particle ~<5-10-" on motion of the

Resultant displacement, cm

Factor category

10:' I0 a -<,30 < 10-2 ~5.10-4 ~ 2.10-3 ~3.10 9

III Ili III IV

~ 2..10-7

IV

<0,05

III

~0,3

II

shepherd Effect of curvature of the shepherd orbit Possible violation of the equivalence I pri c~ple for N =

< 10-"

~7.10 -II

Note. Category of factors: I - - must be taken into account even in the simplified statement of the problem (basic problem); II - - must be taken into account in the case of "realistic" modeling; III - - may not be taken into account in modeling, but must be taken into account in real experiments; IV - - can be neglected (the boundary between categories II and III is conditional). where Jn are the numerical coefficients of the multipole moments of the Earth and J2 = 1.0826.10 - 3 , all the other Jn < 1 0 - 6 ; Pn are Legendre polynomials of the n-th degree; ~r is the latitude.

In the quadrupole approximation GM e

The results of the estimations are shown in Table 1. To be definite we assume the height of the satellite orbit h = 1500 kin.

Tidal Effect on Satellite Orbit. The modulus of the tidal acceleration, at, for a pair of particles located at a distance of R from the field source is approximately (4)

a t =.CAR; V = 2 ~ M / R 3,

where M is the mass of the field source and AR = I R1 - - R2 I is the difference of the two distances. External bodies not only affect the particle trajectory directly (as reflected in Table 1) but also as a result of tidal distortion of the satellite orbit itself. To assess this effect we note that it is related to a change in the "tidal" coefficient 3' = 3"E for the Earth in formula (4) due to a change in the distance Ro between the satellite and the center of the Earth, Aa E =AyE AR, IA?E l =

3ARo R0

'

(5)

where, as above, AR is the difference of the distances between the center of the Earth and the shepherd and the particle; AR o is the change in Ro due to the "external" tide. It is natural to assume that A R o / R o ~ a s / a N ; a s : y R o , a N : G M E /R~ ,

(6)

where a s is the tidal acceleration of the satellite relative to the Earth [formula (4) with AR = Ro]; a N is the acceleration of the satellite due to Newtonian attraction of the Earth; it determines its motion on the orbit. By comparing (4) and (6), we see that ARo/R o = 23"/3"E and hence, according to (5), A3'~ = 67; Aa B --- 67AR, i.e., the distortion of the orbit leads to additional

847

TABLE 2. Tidal Effects of External Bodies on the Satellite Orbit and, Due to Its Distortion, on the Particle Trajectory

Source

Moon Sun Jupiter

Orbit distortion ~R0, cm

1,0s ~2.10 2

Particle Resultant acceleration action of a, cm/sec 2 the displacement, cm ~2.10-9 . ~ 4-10 - l ~

,N.IJO-2 !,-, 3.10 - s

I0-:

accelerations of the particle relative to the shepherd that are about six times larger than those caused directly by the tidal effects and, it should be mentioned, possess the same periodicity. The relevant numerical estimates are presented in Table 2. Evidently, the influence of the lunar and solar tides must be taken into account. This particularly applies to: the experiment when, along with the measurements of the particle trajectory inside the capsule, a calculation of the trajectory will also be carried out based on concrete data on the satellite motion in which all possible influences will be taken into account with the highest possible accuracy. On the other hand, during the preliminary modeling there apparently will be no need to take into account those factors that produce particle displacements of no greater than 10--2 cm since this cannot influence the experimental setup. Possible Verification of the Equivalence Principle. A weak violation of the equivalence principle for bodies M and m of different chemical composition is conceivable. This would be manifest in different acceleration of the b_~odiesin gravitational fields. In the general case the violation can be described as a difference in the potentials: in (1) the term mV(r) would be replaced by m[V(~) + AV(r)]. As a result, the term AVi(~) would be added to the right-hand side of (2). In practice, for orbits that are almost circular a violation of the equivalence principle can be described satisfactorily by including the factor (1 + ~7) in the term mV(~) in the Lagrangian. Here ~7is the Eftv6s coefficient which depends on the accelerations a 1 and a 2 of the two bodies of different chemical composition located in the same gravitational field: ~/= 2(a 1 -- a2)/(a I + a2). According to the available experimental data obtained on the Earth ~/ < 10-12. With such values it is sufficient to introduce the factor (1 + rl) only in the spherical component of the potential GME/R. As a result a term --~/niGME/r2 appears in the right-hand side of (2). The estimated effect of this term for ~/ = 10-13 is given in Table 1. An estimate of the sensitivity of the SEE method to violations of the equivalence principle at distances of the order of the Earth's radius is given in [1] and is about 10--13. Kepler's third law was used in the estimation, it being assumed that the radii of the circular orbits of identical period should be different for a particle and shepherd of different chemical composition. The estimate thus obtained is of a preliminary nature and is based on the assumption of a static regime of the system. If, however, the ~/term in (2) is regarded as a dynamic acceleration that is constant in a certain (radial) direction, then it may be feasible to observe values of ~7 --- 10-16-10-17 during prolonged studies of the system. Subsequently, the possibility should be verified by the modeling discussed above. Choice of Orbit and Equation of Motion of the Shepherd. Since the effect of the particle on the shepherd is small, in order to determine the shepherd orbit at all modeling stages the familiar formulas for calculating the trajectories in the two-body problem should be applicable. In other words one may consider the shepherd orbit to be prescribed. For a gradual approach to real experimental conditions the following stages connected with a choice of the shepherd orbit and of the approximation to a realistic potential can be discerned: a) spherical potential V 0, circular orbit; b) spherical potential V 0, elliptic orbit; c) circular equatorial orbit with allowance for a quadrupole component V2; d) inclined orbit, almost circular and allowance for a quadrupole component V 2 (and possibly some higher components). Stage a) is necessary for a direct verification of the main idea of the experiment ("basic problem"). Stages c) and d) should result in a choice of an optimal scheme of the carrying out of the experiment, i.e., in a final choice of the orbit and orientation of the satellite and of the initial data on the motion of the particle in the capsule that ensure the highest accuracy and reliability for determination of the parameters required. Intermediate stages b) and c) are necessary in order to verify the effect of certain factors on the particle trajectory and for an optimal choice of the initial data. The form of the orbits can easily be written down for stages a), b), and c): a) circular orbit with V = GME/R:

848

R'=a(cosot; R=a=const

sino)t; 0);

GM E ; o)z~ aS

(7)

,

b) circular equatorial orbit in presence of a quadrupole potential [cf. (3)]: R'=a(cosot; sinc0t; 0);

GME [l+3J~ ~ R2g ] ,

R=a=const; os= ~

(8)

c) elliptic orbit with V = GME/R: R'=a(cos~--e; o)/=;--esinr

e~sin;; 0);

GM e o)2= - - -a' i - - ; e X = l / - l _ e S '

(9)

where ~" is a running parameter, e is the eccentricity, and a the major semiaxis. Equation of Relative Motion. We now consider the equation of motion of the particle relative to the shepherd (2); the orbit of the shepherd R(t) is assumed to be known. The tidal acceleration, i.e., the difference of ~te V i in the right-hand side of (2), can be expressed as a Taylor series in si/R with the center of the expansion in I~(t): 1 "S'k=Vtk(-R)sk-~'~Vikeslzse~t-...+(M+m) OU

(lO)

The first term in the right side corresponds to tidal acceleration that is linear with respect to s. The second term takes into account the curvature of the satellite orbit (cf. Table 1); the neglected terms are by at least six orders of magnitude smaller, i.e., infinitesimal. For the potential V 0 = GME/R we write down explicitly the tidal acceleration as

0

-~"

R3

3 GME

+ 2

t~* [ni(sLSs~) +2sts~)]

(si+anisn} "{"

(11) ,

where n i - Ri/R, sn = sn is the radial projection o f s . In order to analyze the trajectory of the particle in the capsule we must go over to a coordinate system coupled to the capsule and choose coordinates that are related to the orientation of the capsule. Mathematically, this consists in choosing the coordinate basis of the associated coordinate system,

7o= r~.;,., where e i is the initial coordinate basis in the inertial coordinate system with the components eik = •ik' Here and below, the indices i, j, and k refer to quantities in the inertial coordinate system and the indices a, b, and c to quantities in the coupled coordinate system. Since fa = ~a(t) is an orthonormal basis, the (fai) matrix is orthogonal: fatfbr

faffak=6t~,'~t=fat~a.

For transformation of (10) in the coupled coordinate system both sides should be expressed in terms o r s a = fai Si. This yields I

oU

+ -~ fatfbffckVtj~(R)sbsC+...+ (M+m)osa

(12)

The tidal terms for the potential V 0 resemble the right-hand side of (11) that includes contributions that are linear and quadratic with respect to sa and in which sa must be substituted for s i. ...)

The concrete choice of fa (orientation of the capsule) for an arbitrary orbit can be made in various ways. We shall use the basis completely associated with the radius vector R or with the direction from the shepherd to the center of the Earth: 9

,

7, =----~/I-~ ; 7s--~'- ~/i~;

(13)

849

A merit of (13) is the simplicity of the calculations; only data on the orbit R(t) are required. We can now specify the form of Eqs. (12) for orbits a, b, and c. a) Circular orbit (7), the "basic problem." The matrices in (12) are

(! _ii) 0

; (

(!o0)

)=--0) g

10

0

(14)

,

00

where the sign _ in (13) is c._hosenso that the direction of f l is opposite to that of the motion of the satellite. In a right-handoriented basis we then have f3 = e 3- For the coordinates (x, y, z) of the particle we obtain 3Brat a4

x--2c@=

x 7

+(M+m)Us

(15)

;

y+2o~=3o)~y+ 2B ~i- (xa--2Y2+Zg)+(M+m)Us ~s 9. z=-~

~-

3BVz a4

+UVI+m)U~

z T

(16)

;

(17)

'

where s = (x 2 + y2 + z2)1/2, B = GM E, co2 = B / a 3, Us _ dU/ds. The terms containing the squares of s a are 106 times smaller than the linear terms since s / a - 10 - 6 . b) Circular orbit (8); inclusion of the quadrupole potential does not affect the form of Eqs. (15)-(17) except for the cases when the term 6Cy/a is added to the right-hand side of(16) or - - 6 C z / a is added to the right-hand side of (17); here C = 0.5BJ2RE 2. The expression for co should now be changed to co2 = (B/a 3) + (3C/a5). c) Elliptic orbit (9) "~1= ~--1 ( e ' s i n ~ , e--cos~, 0); ~=

~-1 ( c o s t - - e , e'sin~,O) ;

j), wherefl = 1 - - e c o s ( . Therefore

~.-

(ili) 0 0

; (fa f 0 ) = - - - 7

(i es,n0 0) -- esin~

.

--e' 0

0 0

(18)

,

and the final form of the equations of motion is x--

(1--ecos~)~

--

--e~x+e(xc~

(1--ecos~) 4

"/)+ (1----'-~'~)z -- (1--ecos~)~ 9.

z

+(M+m)Us

(3--e2)v+2e(e'xsing--Vc~ o~z

~

(x~--2YZ+z2)

3o)2yz

-~ ;

+(M-Fm)UsT;

z

(l_ecos~)3 + a ( l _ e c o s ~ ) ~ W ( M + m ) U s T ;

where, it will be recalled, w2 = GME/a 3 and the relation between the parameter ( and t is cot = ~ - - e sin (. A violation of the equivalence principle at distances of the order of the radius of the Earth would lead to the appearance of an additional term --~/GMEni/R 2 in the right-hand side of the equations of motion (14) (general case). As a result, in the concrete cases a, b, c considered here it is only the y components of the equations that would change: the right-hand sides of (13) and (18) would be supplemented by the terms --~r

and--

~lco~a

(l_ecos~)~

,

(20)

respectively, where ~7 is the E6tv6s coefficient. E r r o r in M e a s u r i n g G on the Basis of T h r e e or Four Points. The equations of motion of the particle cannot be solved in quadratures even in the simplest case of (19)-(21). The problem is not made easier by making auxiliary simplifying assumptions 850

such as neglect of terms quadratic in s a, coplanarity of the motion of the particle and shepherd (z - 0) and the Newtonian nature of the interaction (U = G/s). The equations in this case have this form

l

x--2~'y=--Ax/s~;

(21)

y-b2o3x=3o~Zy--Ayl s ~ , j

where A = G(M + m), s 2 = x 2 + y2 with a motion interval (22)

x= q-y=--3o3=y ~ - - 2 A / s = c o n s t .

However, even without a numerical investigation of the equations, a preliminary estimate of the possible accuracy of measurement of G made by studying small segments of the trajectory can be deduced from the equations. For such segmems and a fixed time interval r, we obtain in the finite differences approximation

1

"xCt)= -if- [xCt.q-~)--x(t) l ; ;c'(t) =

xr~+-r)-- x(t)

(23)

t

Xr

.tO. [XU+~.t)--SX(t+Z) +X(t)l

and analogously for any other variable. Let us assume that the masses M and m, as well as the time intervals, are measured exactly. The error in measurement of G on the basis of three points (t, t + r, t + 2r) due to the error in determining the coordinate 6 / c a n be estimated by using Eq. (21): 6G

O

6A

--

6(~s

a

+s

lx+2@t

3(3s

6x

+-2-

9

(24)

We assume in the estimation that x = s, 8s = 6x = 6l. Furthermore, according to (23), @ = 28l/r, 6X = 46l/~, and

since necessarily r << 1/co (the interval must be much smaller than the orbital period which is also the characteristic evolution time for the particle), we may put 6(R + 2c<~) = 46l/r 2. Finally, substituting the right-hand side of (21) for R + 2coy, we obtain 6G

a

46l

-

46[

s2

s ~-7"7

-"

For a numerical estimation we assume 6I = 10- 6 cm, s = 5 m, r = 200 sec, M = 500 kg. This gives ~SG/G ~ 4Hs2/r2GM = 8.10-4. A similar estimate can be arrived at by equating the left-hand sides of (22) for some two segments of the trajectory located at equal distances s 1 and s2 from the shepherd. After some transformations and leaving only the dominant contribution to the error and for the sake of definiteness assuming s 2 = 2s 1 = s, we obtain the following estimate: ~G/G ~ 16sVe6I/GMr, where ,V is a certain mean velocity along the trajectory. Setting ~l = 10- 6 cm, s = 3 m, r = 200 sec, 9 = 0.03 cm/sec, M = 500 kg we obtain 6G/G = 0.7.10 - 4 . Such accuracy is certainly insufficient, but as a preliminary estimate it seems to confirm the feasibility of the method. We will later show that by means of numerical experiments and application of the least-squares method, the real error AG/G can be reduced to 10- 6 . The sensitivity of the SEE method with respect to violation of the equivalence principle (parameter ~/) and to variations of the hypothetical fifth force will also be assessed.

REFERENCES

.

A. J. Sanders and W. E. Deeds, Phys. Rev. D, 46, 489 (1992).

2.

V. N. Mel'nikov, in: Gravitational Measurements, Fundamental Metrology, and Constants, V. de Sabbata and V. N. Mel'nikov (eds.), Kluywer Acad. Publ., Dortrecht (1988).

3.

K. P. Stanyukovich and V. N. Mel'nikov, Hydrodynamics, Fields, and Constants in the Theory of Gravitation [in Russian], t~nergoatomizdat, Moscow (1983).

4.

V. de Sabbata, V. N. Mel'nikov, and P. I. P r o m , Progr. Theor. Phys., 88, 623 (1992). V. N. Mel'nikov and A. G. Radynov, Izmer. Tekh., No. 4, 16 (1985).

5.

851

.

7. 8.

852

V. N. Mel'nikov, in: Advances in Science and Technology, Series Classical Field Theory and Theory of Gravitation, Vol. 1, Gravitation and Cosmology [in Russian] (1991), p. 49. G. H. Darwin, Acta Math., 21, 99 (1987). P. N. Antoniuk, K. A. Bronnikov, and V. N. Mel'nikov, Izmer. Tekh., No. 8, 3 (1993).