___- ---- THE EFFECTS OF THE NODAL REGRESSION OF THE ON THE GRAVITY OF A GYROSCOPIC James 1. Myers, Jr.

REPORT R-309 JULY,1966

!

I i This work was supported in part by the Joint Services Electronics

Pi-ograiil {E. S. Aruy, .-u. a.n ..navy, . and 'ii. S. Air Forcej under

Contract No. DA 28 043 AMC 00073(B); and in part by the National

Aeronautics and Space Administration under NASA Research Grant

NsG -443.

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NOTAT ION

A Moment of inertia about transverse axes

a Semi-major axis of ellipse

aAA a, Unit vectors i’ L C Polar moment of inertia

e Eccentricity of orbit

G Universal gravitational constant

-H Angular momentum vector

i ; angle measured from ’s polar

axis to orbital angular vector

Unit vectors in inertial coordinate system

M Mass of the earth

Components of gravity gradient moment

Radius from center of earth to center of satellite

Radius of earth Re T

I t Time

XY YY z Coordinate axes

cf Argument of perigee in orbital plane

6 Spin axis error angle for equatorial orbit (small) e 6 Spin axis error angle for polar orbit P E Gyro spin axis inclination

Coordinate transformation angle

Angle between gyro spin axis and orbital angular velocity

vector Gravity gradient precession coefficient

Right ascension of gyro; angle measured west-to-east in

earth's equatorial plane from the vernal equinox to the

line of nodes between gyro and earth equatorial planes

Y Argument of satellite in orbital plane

R Right ascension of orbit

-W Angular velocity vector of gyro precession Orbital angular velocity vector --oW w Gyro spin axis vector -S Den0 tes unit vector i

1

A gyroscopic satellite has been proposed for a test of the

general theory of relativitylY2in which the gyro "drift" rate to be

measured is less than seven seconds of arc per year. The Coordinated

Science Laboratory has proposed a simple passive gyroscope which uses

sunlight reflected from mirrors to provide optical data to determine

the spin axis orientation. 3Y4 The gravity gradient torque acting on

the satellite is one of several extraneous disturbances which can cause

spurious precession of the gyro spin axis. In this paper, general equa-

tions for the precession of a gyro satellite in a regressing orbit are

derived. These equations may be used to specify the tolerances for

initial spin axis and orbit alignments which enable an accurate measure-

ment of the relativity effect.

1. Gravity Gradient Moment

The gravity gradient moment is given by ref. 5 for the fixed

orbit configuration shown in Fig. 1. Two coordinate systems are shown,

system [2] fixed to the body spin axis, us, and system 131 fixed to the

orbital plane, or to the orbital angular velocity vector, w . The x2 -0 axis is the line of nodes between the orbital (x3 - y3) plane and the

equatorial y ) plane of the orbiting gyro. The gravity gradient (x2 - 2 moment components in the body axis system given in ref. 5 are

3GM 2 M =-- (C-A) sin0 cos0 sin w t x2 R3

M =--- GM (c-A) sine sin 2u t y2 R3 0

M =O 2 2 J 2

\

/ Orbit

Fig. 1 Planes of gyro and orbit; coordinate systems [Z] and [ 31. 3

where M is the mass of the earth and R is the distance between the centers

of the satellite and the earth. 8 is the angle between the vectors w and -S w and varies only because of precession of w . In the general case -0 , -S the satellite's orbital plane will regress about the earth's polar axis

at the rate of h degrees per year6 and will produce an additional change

in 0.

The geometry involved in the general case of interest for

regressing is illustrated in Fig. 2. An earth-based coordinate

system is fixed with z along the earth's north pole and x along the 0 0 line of vernal equinox (i.e., the line of the nodes between the ecliptic

and the earth's equatorial plane). The yo axis completes an orthogonal

right-handed system and therefore, lies in the earth's equatorial plane.

Systems [2] and [3] bear the same relationship to each other as shown in

Fig. 1. A new coordinate system, [l], is shown with z 1 also along the gyro spin axis, but with x along the line of nodes between the earth's 1 and gyro's equatorial planes. This is the logical system to observe

gyro motion with respect to the earth. The moment given for system [2]

can be transformed through angle 7 to system [ 13 by the transformation

~ -sin'pcx'" I,

The moment components in system [l] are now given by 4

Gyro Eauator

YO

X1 Fig. 2 Orbit and gyro frames related to inertial system; coordinate systems [O] , 111, [21, [SI. 5

M = COST M - sinT M x1 x2 y2

M = sin7 M + cosy\ M y1 x2 y2

M =M =O. =1 22

Substituting the values for M andM gives x2 y2

9 =--3~ (c-A) cosy sin0 sinL M case wOt x1 R3

+-- (c-A) sin7 sine sin 2wot R3

2 M =--3m (c-A) sin7 sine cos0 sin w t 0 Y1 R3 (3) 3GM -- (C-A) cosy\ sin0 sin 2w t. 0 R3

Cos0 , COST sin0 , and sin1 sin0 will now be found as functions of i, n,

E, and cp. From Fig. 2, the gyro spin axis unit vector is given as

G~ = sine sincpi - sine coscp'j + cos~Ct, (4) and the orbital angular velocity unit vector is

= sini si& - sini cos~j+ cosili. wO

Therefore, cos0 = ws * k0

= sini sin€ cos(Cl-cp) + cos i COS E.

From Fig. 2 a unit vector is defined along x 1 1:

B = coscpi + sinrp'j 1 . 6

From Fig. 2 it is seen that the axis is the line of nodes between x 2 the gyro equatorial plane and the orbital plane. Therefore, the x2 axis is normal to both kS and Go, and a new vector along x2 may be defined: - a z bo x = sine a -2 is 2'

Now 8 - a2 = a2 COST = sin0 COST 1 or cos1 sin0 = a1 - Go x CIS

= - sini cos€ cos(Q-cp) + cosi sine . (7 1

Also it can be seen that

Il x g2 = a2 sinn b S

= sin0 sinn Gs or sin7 sin0 G, s = hl x (io x is)

But, a1 is = 0, therefore, sin1 sine = - (a1 - Go).

Substitution of the vector components yields

sinn sin0 = - sini sin(0-cp). (8)

Equations (7) and (8) can now be substituted into the moment equations

(2) and (3): 7

2 M=-3GM (C-A)[sini cos€ cos(Q-cp) - cosi sin€] cos0 sin wot x1 R3

- --GM (c-A) sini sin(n-cp) sin 2wot * R3

GM 2 M=3 - (C-A) sini sin(n-cp) cos0 sin "Ot Y1 R3

+ e (C-A)[ sini cos€ cos(ha-cp) - cosi sin€] sin 2wot R3

M =O. z1

These equations give the gravity gradient moment for a given set of orbital parameters, i and R, and gyro spin direction E and cp.

2. Precession

The precession rate g of coordinate system [l] can be found from Euler's dynamical equation

By inspection of Fig. 2, the components of u, in system [l] are written

w = cp sin E Y1

Since the coordinate axes of system [l] lie along the principal axes

of the body, the angular momentum vector g and its derivative are given as 8

H =Ai x1 H = A @ sine Y1

H = C(0, + (I, COS E) z1

Upon substitution of these components into Eq. (12) we have *2 & + Gus (i, sin E + (C-A) tp sin E cos E

(2~- C) (I, E cos E + + sin E - CCO~i (13)

c(WS + $ cos E - (I, E sin E) =M

The angular rate of the gyro, ws, typically is more than ten orders of magnitude larger than $ or 6. As will be seen later, 4 and are of the -2 -2 order of 'p or E . Therefore, Eqs. (13) may be simplified by neglecting all terms on the left-hand side which do not contain the factor w . Now S it can be seen that M 9

Substituting Eqs. (9) and (10) into (14) and (15) gives

-3GM 2 'p = 3(9) {[sini cot E cos(~-cp) - cos ilcose sin w0 t *s +--1 sin i sin(Q-cp) sin 2w0t] 2 sin f

* 3GM CA 2 E = 3(+) Csini sin(h2-q) cos 0 sin w0t %

1 [sini cos E cos(n-cp) cosi sin sin t] (18) - -2 - €1 2u,0 where cos0 is given by Eq. (6).

Assuming that i, 0, f, and cp change much less rapidly than

N Y wot, average rates 'p and E may be found by integrating over one orbital

period T 71 cp = Gdt -T 0

For an elliptical orbit, the radius R from the center of the earth is

2 a(1-e ) R= 1 + e cos(Y-a)

where a = semi major axis of the ellipse e = eccentricity Y = wot = argument of the satellite

Cy = argument of perigee.

Also, Kepler's law of areas provides the relation 10 therefore,

--dt - dt .--dY - dY R3 R R2 dt a(l-e2) R . adt

Now, integration of (17) over one orbital period becomes an integration from 0 to 2rr in Y:

sin2 Y[l + e(cos Y cos CY + sin Y sin U)]dY a(1-e2)T 0

+- sin sinO sin Y cos Y[l + e(cos Y cos a + sin Y sin a)]dY]. sin f a(l-e2)T

The first integral yields n, and the second integral vanishes. Equation

(18) is integrated similarly, and the resulting time averages are

N + = ACsini cot E cos(n-cp) - cos i]cose (19)

where the gravity gradient precession coefficient is defined as

AS-3 GM -C-A a3(1-e2 ) 312 hS'

The orbital period T, has been eliminated by the equation 11

Equations (19) and (20) may be integrated with respect to time for any

7 Y i(t) and Q(t) to give cp and f as functions of time. cp and E are both of

the order of A, and therefore, this quantity must be small (specifically, A - <<1) for the foregoing derivation to be valid. Furthermore, the time OS ww derivatives of (19) and (20) show thatcp, f, and therefore is are of the -2 -2 order cp or E , as assumed previously.

3. Special Orbits

The relativity drift rate of the gyro spin axis will be

largest when the spin axis lies in the orbital plane.' Therefore, two

cases of special interest are an equatorial orbit and a polar orbit,

because either of these orbits will allow the gyro spin axis to lie in

the orbital plane for an extended period of time. For each of these

special cases, Eqs. (19) and (20) may be simplified and integrated

directly, as will be seen.

A. Equatorial Orbit

For an equatorial orbit, the inclination i will be assumed

small so that sin i M i,

cos i M 1.

Also, it is assumed that

TI E =:+ be'

where 6 is a small angle between the spin axis and the x - y plane, e 0 0 as shown in the sketch. 12

z w 0 -0 i

Now,

TT sin = sin(? + 6 ) = cos 6 M 1 E e e

Tr cos E = cos(- + 6 ) = -sin €ie -6 2e e .

From Eq. (6) cos @ = sin i cos(0-cp) -6 cos i e .

Since i is also a small angle,

Cos 8 M i cos(n-cp) -6 e .

f 0 1lowing v 2 2 cp = - 4i26e cos (R-cp) + i 6 e cos(n-cp)

+ i cos(fi-cp) - 13

N Since i and tie are both small, tp is larger by at least one order of

cy magnitude than e, and may now be simplified by dropping the higher order

terms in i and 6 : e N ci) -A[be - i cos(n-cp)]. I For near equatorial orbits, R changes at the rate of 6 to 9 revolutions

per year. Therefore, Eq. (24) indicates that the average rate of change

of (p is proportional to 6 the angle between the gyro spin axis and the e' earth's equatorial plane. Setting R = R + nt, Eq. (24) can be inte- 0 grated with respect to time to give Acp:

AT = A {6e- i [cos(R - (p) sin + sin(Qo- cp) ( 0 fit rat

ll By arbitrarily setting - cp = - this simplifies to 0 2'

Acp=APe+i(l t.

From Eq. (25) it is seen that for large value$ of fit, Acp is proportional

to tje.

B. Polar Orbit lT A true polar orbit (i.e., i = 7) is required for a nonre-

gressing orbit plane. The nodal regression rate of the orbit line of

nodes is given in ref. 6 by

-3 where J = 1.082 x 10 is the coefficient of the second harmonic term in 2 the earth's gravitational potential. (A more exact equation for nodal

, 14

regression, also given in ref. 6, contains terms three orders of magni-

tude smaller than Eq. (26) and will not be required in this analysis.)

The right ascension of the orbit line of nodes will now be written

hz = hz + fit, where n is the value of R at the time of injection into 0 0 orbit. Figure 3 shows a typical configuration for a near polar orbit

right after injection. Here, 6 is an error angle between the initial P orbit line of nodes and the projection of the gyro spin axis on the

earth's equatorial plane. It will be seen that the gravity gradient

precession depends on this angle and on the regression rate, h.

From Fig. 3 it can be seen that

and , therefore ,

Now, cos(n-cp) = sin(6 - bt) P

sin(hz-cp) = cos(6 - bt). P

Tr Also, assuming that i = + i', where i' is a small error in orbital in- -2 clination, we have

cos 0 = sin E sin(6 - bt) - i' COS E. P Equations (19) and (20) now become

2 , cos 2E cos E sin (6 - fit) - i sin(6 - it) P sin€ P - if2cos €1

N E = A[* sin E sin(26 - 2ht) - i' cos E cos(& - fit)] P P 15 i i c

0r bTt

Fig. 3 Initial conditions for polar orbit. I

I 16

These equations may be integrated with respect to time to give Acp and i AE as functions of time:

P

1-cos hit - cos 26 + 2i' esin f [cos ( )

sin bt] - i3 2 - sin 6 (29 1 p it

sin Gt I

I - cos 26 P c-c;;tmtj]

- 2i' cos E[ sin 6 ('-c;; "3 + cos 6

If E is allowed to vanish, these equations may give misleading

results. In particular, Eq. (29) implies that Acp increases without

limit as f --t 0. However, it must be remembered that cp and, therefore, Acp,

are undefined if f = 0 because the gyro spin axis becomes coincident with

the z - axis, as can be seen in Fig, 3. 0 It will be seen later, that for practical purposes, the nodal

regression rate, n, should be less than 45 degrees per year and, therefore,

Eq. (26) indicates that for 400-700 mi orbits, i' must be no larger than

1' or .017 radian. Consequently, for such slow regression rates, Eqs. (29)

and (30) may be simplified by dropping the terms containing i'. The

simplified expressions are ). i i

17

Some typical curves are plotted in Figs. 4 and 5 to show the

variations of Acp and Af as functions of the initial misalignment angle,

6 and the nodal regression angle, Ato In these curves, the nondimen- P' sional parameters &/At cosf and &/At sine have been plotted. These

curves illustrate the need to keep the nodal regression rate and 6 as P small as possible to avoid large values of gravity gradient precession. 18

.8 c I I I I I I I I I I

IW

Nodal regression angle, fit ( Degrees 1

I Fig. 4 Gravity gradient precession for regressing polar orbit; normalized equatorial plane component vs. regression angle.

2

-. 2

-.4

-.St I I I I I I I I I I 20 40 60 80 100 120 140 160 180 200 Nodal regression angle, ht ( Degrees) Fig. 5 Gravity gradient precession for regressing polar orbit; normalized component in plane of earth's pole and gyro spin axis vs. regression angle. i

19

REFERENCES

1. Schiff, L. I., "Possible New Experimental Test of General Relativity Theory," Phys. Rev. Letters 4, 215 (March 1, 1960).

2. Schiff , L. I. , "Motion of a Gyroscope According to Einstein's Theory of Gravitation," Proc. Nat. Acad. Sci., 46, 871 (1960).

3. Cooper, D. H. , "Passive Polyhedral Gyro Satellite," Coordinated Science Laboratory Report 1-128, (Feb. 15, 1965).

4. Courtney-Pratt, J. S., Hett, J. H., McLaughlin, J. W., "Optical Measurements on Telstar to Determine the Orientation of the Spin Axis and the Spin Rate," Journ SMPTE 72 (June, 1963).

5. Scarborough , J . B. , "The Gyroscope , Theory and Applications ,'I Interscience Publishers (1958).

6. Wolverton, R. W. , (Editor) , "Flight Performance Handbook for Orbital Operations," John Wiley & Sons, (1963).

7. Thomas, L. C., Cappellari, J. 0. "Attitude Determination and Prediction of Spin-Stabilized ," Bell System Tech. Journ. pp. 1679-80 (July, 1964). I I I I Distribution list as of May 1,1966 I 1 Dr. Edward M. Reiliey 1 APGC (PGBPS-12) 1 Commandant Asst. Director (Research) Eglin AFB, Florida 32542 U. S. Army Comand and General Staff College Ofc. of Defense Res. 6 Engrg. Attn: Secretary I Department of ~efense AFETR Tschnical Library Fort Leavenworth, Kansas 66270 i Washinktan, D. c. 20301 (Em,W-135) Patrick AFB, Florida 32925 1 Dr. H. Robl, Deputy Cnief Scientist 1 Office of Deputy Director U. S. Ary Research Office (Durham) (Research ad InfoImation Rm 3D1037) 1 AFETR (ETLLG-I) Box CM, Duke Station De part re" t of Defense STINFO officer (for Library) Durham, Sorth Carolina 27706 The Pentagon Patrick AFB, Florida 32925 Washington, D. C. 20301 1 Cmanding Officer 1 AFCRL (CRMXIX) U. S Army Research Office (Durham) 1 Direct or AFCRL Research Libram, Stop 29 Attn: CRD-At-IP (Richard 0. 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1. ORIGINATING 4VXIVITY (Corporate author) 21. REPORT SECURITY CLASSIFICATION University of Illinois Unclassified Coordinated Science Laboratory 2b. -P

I THE EFFECTS OF THE NODAL REGRESSION OF THE ORBIT ON THE GRAVITY GRADIENT PRECESSION OF A GYROSCOPIC SATELLITE 4. DESCRIPTIVE "ES (Type of report and inclusive dates)

5. AuTHOR(S) (Last name. first name, initial)

6. REPORT DATE 7. TOTAL NO. OF WES 7b. NO. OF REFS. Julv. 1966 19 7 ea. QmT- OR -ANT NO. I*.ORIQINATOR'S REPORT NUMEERCS) ,&,,&043 AMC 00073(E) I 2001450lB31F R-309 I' I 9b.OTHER REPORT NO(S) ( Any other numbera thmt may b. wsstgrud this -r d. I 10. AVAILABILITY/ LIMITATION NOTICES

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71. SUPPLEMENTARY NOTES 12.SPONSORlNG MILITARY ACTIVITY Joint Services Electronics Program thru U. S . Army Electronics Command PV 07703 13. ABSTRACT I A gyroscopic satellite has been proposed for a test of the general I theory of relativity in which the gyro "drift" rate to be measured is les I than seven seconds of arc per year. The Coordinated Science Laboratory I has proposed a simple passive gyroscope which uses sunlight reflected fro I mirrors to provide optical data to determine the spin axis orientation. I The gravity gradient torque acting on the satellite is one of several I extraneous disturbances which can cause spurious precession of the gyro I spin axis. In this paper, general equations for the precession of a gyro I satellite in a regressing orbit are derived. These equations may be used I to specify the tolerances for initial spin axis and orbit alignments whic enable an accurate measurement of the relativity effect. (Author)

I

DD 1473 Security Claulftratlon Security Classification

KEY WORDS

gyroscopic sate 11 ite general theory of relativity 1 passive gyroscope gravity gradient torque precession of the gyro spin axis

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