figure parameters of ganymede

Acta Geod. Geoph. Hung., Vol. 32(1-2}, pp. 225-233 {1997}

FIGURE PARAMETERS OF GANYMEDE

M BURSAl

[Manuscript received April 28, 1997]

The second zonal and second sectorial Stokes parameters of Ganymede determined on the basis of Galileo spacecraft orbit dynamics (June and September 1996) are used to determine the triaxial level ellipsoid of Ganymede. The polar and the equatorial flattening are of the same order in magnitude, about 5 X 10-4, the secular Love and tidal numbers amount 0.80 and they clarify the equilibrium state of Ganymede. The tidal rotational and orbital dynamics and tidal evolution of the Jupiter-Ganymede system differ significantly from those of the Earth-Moon system.

Keywords: Ganymede; Jupiter; satellites; figure parameters

1. Introduction

The Galileo spacecraft orbit dynamics when passing close to Jupiter's satellite Ganymede in June and September 1996 resulted in determining the second zonal J~o) and the second sectorial J2,2 Stokes parameters of the largest satellite in the Solar System as follows (Anderson et al. 1996)

J~o) = -(127.44 ± 0.64) x 10-6 ,

J~o) = J 2,2 = (38.18 ± 0.21) x 10-6 •

(1)

(2)

The above values are conventional, not normalized. They make it possible to treat the triaxiality of Ganymede, the figure of which was believed spherical in the pre-Galileo spacecraft era (Davies et al. 1996). It is just the aim of the paper.

2. Dynamic parameters of the Jupiter-Ganymede system

The rotation axes of Jupiter (J) and Ganymede (G) are nearly parallel. The geocentric equatorial coordinates (declination, right ascension) of the north poles of J and G are (Davies et al. 1996)

OJ = 64.49° + 0.003°T , O'J = 268.05° - 0.009°T ,

oG = 64.57° + 0.003°T , O'G = 268.200T

(epoch 2000 January 1.5, T in Julian centuries).

(3)

(4)

The inclination i of the orbital plane of Ganymede is small enough (Burns 1986),

i = 0.195° , (5)

1 Astronomicallnstitute, Acad. Sci. Czech Republic, Bocni 11/1401, 141 31 Praha 4, Czech Republic, E-mail: [email protected]

1217-8977/97/$ 5.00 @1997 Akadimiai Kiad6, Budapest

226 MBUR5A

to be assumed to equal zero in the figure theory, i.e. G assumed to be orbiting in the equatorial plane of J. The eccentricity of the orbit is (Burns 1986) e < 0.002 and it can be assumed to be circular in the figure theory. The Jovicentric gravitational constant of Jupiter is (Campbell and Synnott 1985)

(6)

and that of Ganymede (Anderson et al. 1996)

GMa = (9 886.6 ± 0.5) x 109m3s-2; (7)

G stands for the Newtonian gravitational constant. The distance ~JG between the mass centers of the bodies in question (Burns 1986)

~Ja = 1 070 X 106 m. (8)

The Ganymede's angular velocity of rotation wa being equal to its mean motion nG is (Davies et al. 1996)

wa = na = 1.01644341 x 1O-5rad S-1 ,

and its mean radius (Davies et al. 1996)

Ra = (2 634 ± 10) km.

The parameter in the potential of the centrifugal forces of G

- - - 3 q = wl;Rb = nl;Rb = G(MJ + Ma) ( Ra )

GMa GMa GMa 6.Ja

comes out, if values (7), (9) and (10) are adopted, as

q = 1.909 707 x 10-4 •

(9)

(10)

(11)

(12)

Parameters (1), (2) and (7) define the external gravitational potential V of G, smoothed in retaining harmonic terms n = 2 only, parameter (12) defines its potential of centrifugal forces Q. Its tidal potential Vt due to Jupiter is as follows

Vt = ~~ L~=2 (~r [P~O)(sin6GJ)p~O)(sin<p)+ +2 L~=1 ~~+:~: P~k)(sin 6GJ)P~k)(sin <p) cos k(A + TGJ)] ;

(13)

p~k) denotes the associated Legendre function of degree n and order k, {} stands for the centric radius-vector of the potential point P on the physical surface of Ganymede, <P is the centric latitude of P, 6GJ and TGJ are equatorial coordinates (declination and hour angle respectively) of J centered at the mass center of G. Because of synchronous rotation of G orbiting in the equatorial plane of J, we may

Acta Geod. Geoph. Hung. 3£, 1997

FIGURE PARAMETERS OF GANYMEDE 227

impose SGJ = 0, as well as, TGJ = 0, the primary meridian of G be that containing the mass center of J. Then all odd zonal harmonic terms in (13) equal zero. Zero are also all the even tesseral harmonic terms if order Ie is odd, as well as, all the odd tesseral harmonic terms if Ie is even. Because e/ .:;lJG ~ 0.0025, we limit ourselves to ij = 3 in (13): .

Vt = c;:::; {(*f [-~pJo)(sin¢)+~PJ2)(sin¢)cos2A] +

+ (~r [-~pJl)(sin¢)cosA+ i4PJ3)(sin¢)COs3Al).

3. Triaxial level ellipsoid of Ganymede

(14)

The sum of potentials V, Q and Vt represent the total gravity potential W at P(e, ¢,A):

W = G~g{I+3Jl-3q+(Jl2J~O)-~Jl-3q)pJO)(sin¢)+

+ (Jl 2 J~2) + iJl-3q) PJ2)(sin ¢) cos 2A + (15)

+ *Jl-3qG [-~pp)(sin¢)cosA+ 214PJ3)(sin¢)cos3A]} ,

Jl = RG/ e· We wish to treat the equipotential surface W = constant = Wo representing the boundary surface of G. Let us select it so that

GMG 1 2 -2 GMG ( 1) Wo = RG + aWGRG = RG 1 + aq , (16)

numerically (17)

This value is considered as fundamental constant in the figure solution. Having selected the equipotential surface we can treat its figure. We wish to

represent it by the best fitting triaxial level ellipsoid E(a > b > c) centered at the mass center of G, with the largest axis pointing toward the mass center of Jupiter. Its semi-axes and flattenings a = (a - b)/a, al = (a - c)/a are defined by five adopted constants: GMG, Wo, WG, J~o), J 2,2, values (7), (17), (9), (1) and (2).

Gravity potential on the surface of E is constant, it is equal to the potential at the basic equipotential surface W = Woo After inserting {! = {!E (radius vector of surface E) and W = Wo into Eq. (15), and assuming the total gravity potential be constant on E, one gets, after some tremendous algebra,

a = GMg [1- v 2io) + 3/12i2) + ~/I-3q + L/l4(J(O»2_ WO 2 2 2 40 2

_ ~/l4 i o) J(2) 1033/14(i2»2 _ 13/1-1 J(o)q + 457/1-1 J(2)q+ 2 2 2 70 2 14 2 35 2

+ 1~~/I-6q2~~~Ff+ 2~1552/1-9q3 - ~~~~/I-6q2Fl +

+ 9003 -3 F2 301 -6 2 l:' + 2817 -3 F l:' + 81 F2 l:' ] 3920/1 q 1 - 392/1 q .1'2 1960/1 q 1.1'2 140 1.1'2 ,

(18)

ActA Geo4. Geoph. Hung. 32, 1997

228 MBUR5A

a = _;!v2i o) + 3v2J(2) + 2v-3q _ ;!V4(J(o»2 + ~v-lJ(O)q+ 2 2 2 8 2 28 2

+ 2. v4J(O)i2) + 1!v4(i2»2 + ~v-1J(2)q _ ..Lv - 6q2 _ 63 F3+ 2 2 2 14 2 14 2 28 80 1

+ ~~~~v-9q3 _ ~~~~1I-6q2 Fl + 1~12~7 v-3qFr + ~:~V-6q2 F2+

+3i95ill-3qFIF2 - "i Fr F2,

al = 6v2J~2) + ~1I-3q + lill-1J~2)q _ 676V4(J~2»2+

+~1I-6q2 + 3469411-6q2 F2 + 24790v-3qFIF2 - :~ F12 F2.

Fl = v2h(o) _ !1I-3q,

F2 = v2J2 (o) + ~v-3q,

- Wo v=RG GMG ·

(19)

(20)

(21)

(22)

Terms containing q3, (J~O»3 etc. have been retained, however, n = 3 tidal terms in (14) neglected.

Numerically, a = 2 635 057 m, (23)

1/0: = 1 454.5, (24)

1/0:1 = 1 939.7, (25)

and/or b = 2 633 699 m , (26)

c = 2 633 245 m . (27)

Consequently, volume T defined by equipotential surface W = Wo , value (17), comes out as

T = 7.655 x 1Q19m3

and, in view of (7), the mean density 0' of Ganymede

value identical to that by Anderson et al. (1996) .

(28)

(29)

Because of rotational/orbital resonance of Ganymede, we may assume the tidal and rotational distortions due to the tidal friction be responsible for its actual figure during the tidal evolution of Jupiter/Ganymede system. The tidal and rotational distortion oJ~o) and tidal distortion OJ2.2 in the Stokes parameters J~o) and h.2 are as

- 3

oio) = -~k _ ~k GMJ (RG) 2 3,q 2 t GMG ~JG '

(30)

(31)

Acta Geod. Geopk. Hung. 3~. J 997


k. and kt are secular Love numbers equal 1.5 if homogeneous body and zero if the mass is concentrated at the center. Let the conditions be imposed as

(32)

i.e. let us assume the tidal and rotational distortions be fully responsible for the actual figure of Ganymede. Then one gets

k, = -3 J~o) + 2J2 ,2 = 0.80, q

h,2 kt = 4- = 0.80 . q

(33)

(34)

It means, if (32), the mass concentration toward the center discovered by Anderson et al. (1996) is evident. The same conclusion can be drawn on the basis of the coefficient of inertia (C is the maximum principal moment of inertia)

This value is practically the same as derived by Anderson et al. (1996). Note that the ratious

Otlq = 3.60, Otdq = 2.70,

as well as, - J~o) lOt = 0.19

correspond well to the equilibrium figure (Bursa 1994).

(35)

0(36)

(37)

4. Tidal variation within Jupiter-Ganymede system due to tidal friction

The long-term tidal variation in ~JG can be derived from the Lagrangian planetary equation for the time variation in the semimajor axis of the orbit. The perturbing function in that case is the Jupiter-Ganymede tidal force function ~ VJG

which reads, in view of the above 8GJ = 0, TGJ = 0, as well as, 8JG = 0 and TJG = 0 (8JG stands for jovicentric declination, TJG for jovicentric hour angle of the center of mass of Ganymede), as follows

GM2 (ft )5 kJ.:!.!.:!.J......!!:L (1 +;! cos2cJ) + ~JQ ~JQ 4 4

+k GM~ (.&...)5 (1 + 3 2) G ~JQ ~JQ 4 4 cos CG , (38)

RJ = 69 911 km (39)

Acta Geod. Geoph. Hung. 32, 1997

230 M BuRSA

is the mean radius of Jupiter (Davies et al. 1996), kJ and kG stand for the Love number for Jupiter and Ganymede respectively, !J and !G for their phase lag angles. Then the Lagrangian equation gives (KopaI1978)

~ = _2tl.1/ 2 [G(M + M )]-1/2 M.r.±& [86VJQ + 86VIQ] = dl . JG J G MJMo 8£J 8£0

= 6 [G(MJ+Mo)] 1/2 [GMQ (...!.iL.)5 (kJ!J) + 6JO GMJ 6JO

+g~h (:J~)5 (kG!G)] .

(40)

The estimate for the Love number of Jupiter by Yoder (1979) is

kJ = 0.5, (41)

for its phase lag angle by Greenberg (1982)

!J = 2.6 X 10-7 , (42)

and, if so, (43)

As regards Ganymede,

kG!G = 0 (44)

at present, because of its rotational/orbital resonance. However, kG!G =1= 0 should be assumed in the past and it should be taken into account in treating the tidal evolution of Jupiter-Ganymede system. Even if a rough estimate, a dynamical model should be adopted. Let the mass, the mean radius and the principal moment of inertia be constant during the tidal evolution, values (7), (10), and (35), or

C = 3.20 X 1035 kg m2 • (45)

Let the angular velocity of rotation wG(to) of primeval Ganymede be at Epoch

WG(to) = 2.181662 x 10-4 S-1 (46)

i.e. the period of its rotation 8h. Then the primeval spin angular momentum LG (to) comes out as

(47)

and it can be compared to its present value

(48)

the decrease comes out as

(49)

Acta Geod. Geoph. Hung. 32, 1997


It equals the tidal increase ilLJG in the orbital angular momentum LJG of the Jupiter-Ganymede system

LJG = :J~ [G(MJ + MG)ilJG]1/2 , J+ G

ilLJG = -ilLG .

(50)

(51)

Then the tidal increase oilJG in the Jupiter-Ganymede system can be estimated

oil = 2ilL 6.1/ 2 [G(MJ + MG)p/2 = 82 5 k JG JG JG GMJMG . m, (52)

and the decrease in the mechanical energy oEJG of the system due to the tidal friction

oEJG = !C [w~ -w~(to)] -! GMJMG [A~a - A,;(fo)] =

= -6.92 X 1027 kg m2s-2 ,

(53)

(54)

Assuming tidal evolution time, e.g., as t - to = 1.5 X 109y, the integral mean value of (kGcG) can be estimated as

- 1 6.LG6.~G _ -8 kGtG = "3 GM} Rb(t _ to) - 2.30 x 10 . (55)

The long-term variation in the J-G distance (40), estimated with the use of present (kJcJ) value (43) and the integral mean hypothetical value (55), comes out as

d6.JG -1 ---"'dt = 0.003m cy . (56)

5. Discussion. Ganymede-Moon comparison

In view of values (33)-(37), the conclusion can be drawn about the hydrostatic equilibrium of Ganymede at present (Anderson et a1. 1996). Et least, there is no contradiction among its figure and orbital/rotational parameters from the equilibrium point of view.

A comparison with the Moon suggests itself, because of the orbital/rotational resonance, i.e. the similarity of the rotational/orbital dynamics, and because of the similarity in the size. The second degree lunar Stokes coefficients are (McCarthy 1992)

(J~o»{ = -202.151 X 10-6 ,

(J2,2){ = 22.302 x 10-6 ,

the selenocentric gravitational constant

(57)

(58)

(59)

Acta Geod. Geoph. Hung. 3£, 1997

232 M BuRSA

and (Davies et al. 1996)

W( = n( = 2.661 696 96 x 1O-6rad s-l ,

R( = 1 737 400 m,

the secular Love numbers of the Moon come out absurd as

(k,)( = 62.4,

(k,)( = 11.8.

(60)

(61)

(62)

(63)

(64)

It only supports the well-known fact that the present state of the Moon's body is far from the ideal hydrostatic equilibrium, e.g., the observed ratio {C - A)/B (C > B > A) are principal moments of inertia of the Moon) turns out to be 17 times as large as the hydrostatic one (Kopal 1966). If the tidal forces were assumed to be fully responsible for the second sectorial Stokes coefficient (58), then the Earth-Moon distance should be much smaller than its present value 384 400 km, about 192 700 km, and its period should amount 9.74 days. If both, tidal, as well as, rotational distortions were fully responsible for both second order Stokes coefficients, values (57) and (58), than the period comes out 5.5 days at the EarthMoon distance 131 500 km, i.e. at 20.6 Earth's radii. This relative "equilibrium distance" of the Moon is close to that of Ganymede at present, which amounts 15.3 Jupiter's radii. Also the actual rotational-orbital period of Ganymede, which amounts 7.15 days, is close to the Moon's equilibrium period above.

However, the Earth-Moon system and the Jupiter-Ganymede system differ significantly as regards their tidal friction dynamics: (kJ cJ), Eq. (43), compared to that of the Earth (ke ce) ,... 0.013 is five orders smaller in magnitude.

6. Conclusions

1. The polar and the equatorial flattenings of Ganymede are of the same order in magnitude: 5 x 10-4 •

2. The secular Love numbers of Ganymede, due to the rotational (ks) and tidal (kd distortions, come out as ks = kt = 0.80 and support the static equilibrium state of Ganymede.

3. The rotational and tidal distortions can be considered as being fully responsible for the second zonal and the second sectorial Stokes parameters in the gravitational potential, values (1) and (2).

4. The actual figure parameters of Ganymede are close to those defining the equlibrium figure.

Acta Geod. Geoph. Hung. 3!, 1997


5. There is a large difference between the Jupiter-Ganymede and the EarthMoon orbital/rotational tidal dynamics, even if similarities in some dynamical parameters.

6. The product of the Love number and the phase lag angle of Jupiter amounts only 10-5 in' relation to that of the Earth.

7. The increase of the Ganymede-Jupiter distance during the tidal evolution of the system due to the tidal friction can be estimated about 82.5 km only.

References

Anderson J D, Lau E L, Sjogren W L, Schubert G, Moore W B 1996: Nature, 384, 541-543. Burns J A 1986: In: Satellites, J A Burns, M S Mathews eds, The University of Arizona

Press, Tucson, 1-38. Bursa M 1994: Studia geoph. et geod., 38, 7-22. Campbell J K, Synnott S P 1985: Astron J., 90, 364-372. Davies M E, Abalakin V K, Bursa M, Lieske J H, Morando B, Morrison D, Seidelmann P K,

Sinclair A T, Yallop B, Tjuflin Y S 1996: Cele8tial Mechanic8 and Dynamical Adronomy, 63, 127-148.

Greenberg R 1982: In: Satellites of Jupiter, D Morrison ed., The University of Arizona Press, Tucson, Arizona, 65.

Kopal Z 1966: An introduction to the study of the Moon. D Reidel Publishing Company, Dordrecht

Kopal Z 1978: Dynamics of close binary systems. D Reidel Publishing Company, Dordrecht McCarthy D D ed. 1992: IERS Standards. Observatoire de Paris Yoder C F 1979: Nature, 279, 767.

ACt4 Geod. Geoph. Hung. 9f, 1997

figure parameters of ganymede

Documents