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THE ASTRONOMICAL JOURNAL VOLUME 86, NUMBER 6 JUNE 1981

APSIDAL PRECESSION OF ORBITS ABOUT AN OBLATE PLANETRICHARD GREENBERG

Planetary Science Institute, 2030 East Speedway, Suite 201, Tucson, Arizona 85719Received 19January 1981; revised Jl March 1981

ABSTRACTExpressions for the apsidal precession rates to second order in J2 appear in the literature in atleast three apparently mutually contradictory forms. These expressions are reconciled by accounting for subtle differences in the definitions of orbital elements.

I. INTRODUCTIONThe natural satellites of the outer planets can serve asprobes of the planets' gravitational fields. Satellites'orbital precession rates reflect the magnitude of variousgravitational harmonics. The rings ofUranus are composed of particles that precess in a coherent way, so eachring precesses as a unit. The resulting precession rateshave now been measured with sufficient precision todetermine, not only the gravitational harmonic coefficient J 2, but the higher-order harmonic J4 as well (Elliot

et al. 1981 ). Separation of the effects of these two harmonics is possible because their effects vary differentlywith distance r from the planet: Jz contributes a forcethat goes nearly as r 4; the J 4 force goes as r-6 .One observable orbital variation due to these oblateness terms is precession of the line of apsides, i.e., of theorientation of the major axes of the orbits. The precessionrate depends on both J2 and J4 In order to determine J4,it is necessary to account for a small par t of the J 2 contribution, the term, which has the same r dependenceas J 4 J. Elliot has privately pointed out an apparentcontradiction in the literature betweefl two evaluationsof l i erm in the expression for the apsidal precessionrate w In 1946, Brouwer published the expression

.:. 3 9 15 )w = n 2Jz Rja) 2 - F , J ~ R j a ) 4J4 Rja)4 ,(1)

wheFe R is the planet's radius, a is orbital semimajoraxis, and n is orbital mean motion. In 1959, Brouwerpublished an expression for & n which the coefficient-9 /8 f t h e J ~ term was replaced by +63/8. The properchoice of this coefficient can be crucial to the determination of J 4 . The purpose of this paper is to prevent future confusion by resolving the discrepancy betweenthese two equations and to show that in fact, for theUranian ring problem, as Elliot ):tas independentlyshown, the appropriate form of thew equation does notcontain the term at all.

11. KEPLERIAN VERSUS GEOMETRICALORBITAL ELEMENTSThe difference between Brouwer's 1946 and 1959expressions for precession rates is due to a differencebetween the definitions of semimajor axis and of meanmotion used in each expression. In 1959, a and n werethe osculating Keplerian elements; in 1946, they werethe mean distance from the planet's center and the sidereal mean motion, respectively.In order to see how different these quantities can be,

consider a satellite in equatorial orbit about an oblateplanet whose gravitational field is given by the forceF = GMm/r2 ( 1 i h R / r ) 2 , (2)

where m is the satellite's mass, r is its distance from theplanet, and M and R are the planet's mass and radius.I f the satellite is moving in a perfectly circular orbit ofradius a, the radial acceleration, n2a, must equal theforce per unit mass:n2a3 =GM t ih R/a) 2) , (3)

where n is the sidereal mean motion (i.e., angular ve-locity). Such an orbit has the shape of a type ofKeplerianorbit (i.e., circular), but it is not Keplerian; its period istoo short. The osculating Keplerian orbit is defined asthe orbit that would be followed if the perturbing forcewere instantaneously turned off (in this case, J 2 set equalto zero). Such a Keplerian orbit would be tangent to thereal trajectory, hence the term osculating. I f J 2 weresuddenly set equal to zero, the satellite would have avelocity na) too great for continued circular motion. Itslocation at that time would become the pericenter of theeccentric orbit which it must adopt. In effect, as thesatellite moves on its real circular orbit, it is always atthe pericenter of the osculating Keplerian ellipse. Theosculating orbit thus precesses at the rate n

Next I compute the elements of the osculating orbit,which will be denoted by subscript o The angular mo-912 Astron. J. 86(6), June 1981 0004-6256/81/060912-03$00.90 1981 Am. Astron. Soc. 912

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914 R. GREENBERG: APSIDAL PRECESSION

the solution of which isx = Csin n0 t , (18)

where C is a constant of integration and na is the apsidalfrequency,Ita= GMja3)i/2 ( ~ J z R j a 2 - : 2 J ~ R j a )

+ J 4 Rja )4 + O R/a)6). (19)The apsidal precession rate is the difference between thesidereal mean motion and the apsidal frequency:

w=n-na .In order to obtain 1 ), we factor out n, yielding

t =n l - nafn),(20)

(21)which is the same as (1) when expressions for na and nare inserted and terms through order R a )4 are retained.Now, in order to put (1) in terms of osculating elements, I use relations ( 6) and (7):ti = no[1 + 3Jz(Rja)2] ~ J z R j a

9 15 )X [ 1 + 3Jz Rfao)2] - 8 ~ R f a o ) - 4 J 4 Rfao)4= no Jz Rfao)2+ 6: J ~ R f a o )

- 1} J4 Rfao)4), (22)in agreement with Brouwer's (1959) result.In the problem of interpreting observation of Uranus'srings, neither (1) nor (22) is directly applicable; we haveno measurement of mean motion. Instead, we can simplyinsert expressions for na [Eq. (19)] and n [Eq. (16)] into(20) to obtain

= (GM/a 3) 1/ 2 GJz(Rja)2- 1} J4Rja)4). (23)

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Here a is the observed (not osculating Keplerian) semimajor axis of the ring. Note that the terms inn andna exactly cancel one another.Expression (23) is correctly used by Elliot et al.(1981). t is the appropriate form to use when one canmeasure only mean distance from a planet and not angular velocity, as in the case of Uranus's rings. Brouwer's1946 formula (1) would be appropriate for satellitemotion where both sidereal mean motion and semimajoraxis are observables. In principal, sufficiently precisemeasurement of it, a, and n for a single satelli te wouldpermit solution of (1) and (23) for both Jz and J4 Inpractice, data from several satellites, including nodalprecession rates, are used. Nodal precession rates aregiven by the following expressions in forms analogousto (1) and (23), respectively:. 3 15 27 )Q = -n 2Jz(Rja)2 - 4J4(Rja)4 - S J ~ R j a ) 4 ,

(24). 3 15 .Q = -(GMja3)if2 lJz(Rja)2- 4J4(Rja)4- ~ J ~ R j a (25)

Brouwer's (1959) formula (22) gives apsidal precession in terms of osculating Keplerian orbital elements.In contrast to sidereal mean motion and apparent semimajor axis, such osculating elements are not directlyobtainable from observation.I thank Alan Harris for helping me to understand howdifferent definitions of orbital elements have led toconfusion and for long encouraging me to publish anexplanatory note. James Elliot raised the issue with meagain recently. He pointed out the particular contradictions addressed in this paper.This is Planetary Science Institute Contribution No.159; PSI is a division of Science Applications, Inc. Thisresearch was supported by NASA's Planetary Astronomy Program as part of Grant No. NSG-7045.

REFERENCESBrouwer, D. (1946). The motion of a particle with negligible mass

under the gravitational attraction of a spheroid , Astron. J. 5122.Brouwer, D. (1959). Solution of the problem of artificial satellitetheory without drag, Astron. J. 64, 378.

Chandrasekhar, S. ( 1960). Principles of Stellar Dynamics (Dover,New York).Elliot, J. L., French, R. G., Frogel, J. A., Elias, J. H., Mink, D., andLiller, W. (1981). Orbit s of nine Uranian rings, Astron. J. 86,444.

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