Sounding of the interior structure of Galilean satellite Io using the parameters of the theory of

figure and gravitational field in the second approximation

V.N.Zharkov and T.V.Gudkova

Schmidt Institute of Physics of the Earth

B.Gruzinskaya, 10 123995 Moscow - RUSSIA

(e-mail: zharkov@ifz.ru)

All Galilean satellites are in synchronous rotation; their orbits are nearly circular and lie in the equatorial plane of Jupiter. Io is the large satellite closest to Jupiter. Therefore, the influence of Jupiter’s tidal potential on the equilibrium figure and gravitational field of Io is appreciably stronger than it is on the remaining large satellites. For the theory of Io’s figure to be consistent with currently available observational data, it must include effects of the second order in smallness. In the first approximation, the ratio of the moments J2 and C22: J2=10/3C22.

The parameters of the gravitational field for Io determined in the Galileo space mission have shown that relation holds with a high accuracy. For Io, J2 and C22 are given to the fourth decimal

place. This main relation that is used to judge whether Io has an equilibrium figure was derived in the first approximation. Therefore, the following question arises: With what accuracy is this theoretical ratio valid? To answer this question, we must construct a theory in the next (second) approximation by including the terms of order 2 ( is the small parameter of theory of figure, defined as =3π/(Gρ0τ2),

where ρ0 and τ are the average density and rotation period of Io, respectively. This is the main goal of

our analysis.

1. Introduction

The equilibrium figure of the satellite is an equipotential surface of the sum of three potentials: U = Wt + Q + V = const

the tidal potential

2

)(cosn

n

n

t ZPR

r

R

GMW ,

the centrifugal potential )(cos13 2

22

P

rQ = )(cos1

3 2

2

PR

r

R

GM

,

(the angular rotation velocity of the satellite ω is equal to the angular velocity of a synchronous satellite around the planet, and according to Kepler’s third law,

2 = 3/ RGM for synchronous satellites) the potential from mass distribution inside the satellite V (r, θ, ).

In the figure theory we pass from the actual radius r to the effective radius s defined as the radius of a sphere of equivalent volume (r, θ, ) ? (s, θ, ).

022 )(1

nnn tPssr )(1( 220 tPsss 2cos)(2

222 tPs cos)(1331 tPs

3cos)(3333 tPs )(44 tPs 2cos)(2

442 tPs )4cos)(4444 tPs .

Figure theory

To zeroth approximation the equilibrium figure of a body has the form of a sphere with the mean radius s1. In the first approximation the sphere transforms to a triaxial ellipsoid (normal figure) with the equatorial semiaxes a, b and the polar semiaxis c. In the second approximation equilibrium figure is deviated from a triaxial ellipsoid.

The figure theory is constructed by expanding

the expressions for the potentials in

powers of a small parameter 2

0

3

G ,

where 0 , are the average density and

rotation period of Io, respectively.

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The second basic problem

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The first and the second approximations

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Results

ALGEBRAIC RELATIONS – THE DUALISM IN THE EQUILIBRIUM FIGURE THEORY

The figure parameters specify the shape of the figure of an equilibrium satellite.

The gravitational moments specify the external gravitational field of an equilibrium satellite.

(A)

or inversly:

(B)

If the satellite is in hydrostatic equilibrium, then the coefficients in the expansion of its external gravitational field in terms of spherical functions can be determined by measuring its figure parameters. The reverse is also true. Relations (А) and (B) express the dualism in the figure theory.

222222

2222

7

30

21

11

7

48

7

4

6

5ssasssJ ,

2222222222 7

13

14

5

7

8

4

1sssssC ,

222222

2244

7

9

7

15

35

72

35

36sassssJ ,

2222224242 14

5

28

3

35

12sssssC ,

. 56

3

35

322

2224444 sssC

222222

22222 7

6

7

48

7

3

63

38

7

4

6

5CCJJJs

,

222

22222222 21

19

21

11

14

1

7

8

4

1CaJCJCs ,

22222

222244 140

99

7

3

140

177

35

72

35

36CJCJJs ,

2222

2224242 14

9

140

3

14

1

35

12CaJCJCs ,

.280

3

1120

9

35

322

22224444 aCCCs

Models

The Galilean satellites of Jupiter are in a state close to hydrostatic equilibrium, then data on their

figures and gravitational fields allow us to impose a constraint on the density distribution in the

interiors of these bodies and, thereby, to make progress in modeling their internal structure. To

show the effects of the second approximation, two three-layer trial models of Io are used [2]. The

considered models of the Io’s interiors differ by the size and density of the core Io1 (the core

density c=5150 kg/m3: Fe-FeS eutectic) and Io3 (c=4600 kg/m3: FeS with an admixture of nickel),

while having the same thickness and density of the crust, and the mantle density difference is only

20 kg/m3. The models have a thin 50-km crust with the density crust=2700 kg/m3.

An important parameter of the silicate reservoir is the magnesium number Mg#, the ratio of the

number of magnesium atoms to the sum of the numbers of magnesium and iron atoms

(occasionally, this ratio multiplied by 100 is used). Theoretically, such an important parameter as the

mantle silicate iron content (Fe#=1-Mg#) in the model Io3 is equal to 0.265. This value is a factor of

two and half higher than the value for mantle silicates in the Earth. The high iron content in the Io3

model leads to high mantle density. Therefore, one can assume, that if the accuracy of the gravity

field determination for Io is significantly improved, the effects of the second approximation will put

restrictions on the value of average mantle density of Io.

Parameters of three-layer models for Io (s 1= 1821.6 km, ρ0 = 3527.5 kg m-3 ) Parameter Io 1 Io 2 Io 3

Core density ρ1 , kg m-3 Mantle density ρ1 , kg m-3 Crust density ρ1, kg m-3 Core radius sc ,km Mantle radius sm, km

2k

Core mass mc, wt %

5150 3350 2700

903.16 1781.6 1.3032

17.8

5150 3330 2700

924.73 1781.6 1.2929

19.1

4600 3320 2700

1053.29 1781.6 1.3053

25.2

.

n

The method for solving the equations of the figure theory is described in detail in [1]. A computer code was developed to calculate figures functions s4(s), s42(s) and s44(s) for the Io models. Tables 1 and 2 give results of numerical solutions of figure equations for the Io1 and Io3

models [2] at s=s1 (for visual perception tables list the normalized figure parameters and gravitational moments

. As seen from Table 1, including the second order terms in J2 and C22 decreases the values by two units in the third decimal digit. To

make clear insight into the problem, we plot normalized figure functions s2(s), s22(s) (Fig. 1a) and s4(s),s42(s) (Fig. 1b). Figure functions

s2(s),s22(s) and s31(s),s33(s) are proportional to the Love functions h2(s) and h3(s)(Fig.2) , respectively. The outer regions of Io’s interiors

influence more the Love number h3(s), then h2(s) . The same fact is seen, when comparing the figure functions s2(s), s22(s) (Fig. 1a) and

figure functions s4(s), s42(s), s44(s) (Fig. 1b). In that way, functions of the second approximations4(s), s42(s), s44(s) sound the density

distribution of the external zones to a greater extent than functions of the first approximation s2(s), s22(s).

444243331222 ,,,,,, sssssss

444243331222 ,,,,,, CCJCCCJ

Parameters Model values

- 2s , 10-3

22s , 10-3

- 2J , 10-6

22C , 10-6

- 2s , 10-3

22s , 10-3

- 2J ,10-6

22C , 10-6

1.4709

1.5287

832.25

864.90

0.0017

0.0022

1.6993

2.2378

Table 1: Normalized figure parameters sn and gravitational coefficients Jn and Cnm (the first order) and the second order corrections for the model Io1.

Table 2: Normalized figure parameters sn and gravitational coefficients Jn and Cnm (the second order) for the models Io1 and Io3

Parameters Io1 Io2

- 31s , 10-6

33s , 10-6

4s , 10-6

- 42s , 10-6

44s , 10-6

- 31C , 10-6

33C , 10-6

- 4J , 10-6

- 42C , 10-6

44C , 10-6

2.8404

3.6670

3.4666

3.9415

2.2472

1.1266

1.4545

3.0926

3.5163

1.9096

2.8334

3.6579

3.4498

3.9223

2.2190

1.1196

1.4454

3.0952

3.5014

1.9008

0 4 0 0 8 0 0 1 2 0 0 1 6 0 0

R ad iu s (k m )

0

1

2

3

4sn m

- s 2 , x10 3

s 2 2, x1 0 3

a

0 4 00 8 00 12 00 1 600

R a d iu s (k m )

0

0 .4

0 .8

1 .2s nm

s 4 , x1 0 5

- s 4 2, x1 0 6

s 4 4, x1 0 7

b

Fig. 1: The distribution of the parameters of the equilibrium figure of Io s2(s), s22 (s) (a) ands4(s), s42(s), s44(s) (b) along the planetary radius. Thfunctions s2(s), s22 (s) and s4(s), s42(s), s44(s) are not normalized.e

.

Figure 2 shows that the outer regions of Io’s interiors influence more the Love number h3(s), then h2(s). It is seen more clearly in Fig. 3. If we write the formulas for the gravitational moments in the form

1

0

1

0

3 )()( dxxfxdxJ nnn

n

1

0

1

0

3 )()( dxxfxdxC nmnmn

nm

The zeroth gravitational moment J0 is the mass of the planet, which is unity in dimensionless variables. The functions n(x)/Jn (n= 0, 2, 4) for a trial model of Io are plotted in Fig. 3. These functions have a simple physical meaning: they are the relative densities of the gravitational moment Jn, and the quantity (n(x)/Jn)dx gives the relative contribution from the region of the planetary body in the interval [x,x+Δx] to Jn. The graphs of the functions n(x)/Jn show the contributions of various zones in the planetary interior. Thus, it is evident from Fig.3, that the mantle of Io contributes significantly to the values of the gravitational moments J2 and J4, whereas the region of the core is of less importance. That is why the gravitational moments for Io 1 and Io3 models, calculated in the second approximation, differ in the third decimal digit. The Io1 and Io3 models have the same crust thickness and density, while the mantle density for the Io1 model is being only 20 kg/m3 higher than for the Io3 model.

0 4 0 0 8 0 0 1 2 0 0 1 6 0 0

R a d iu s (k m )

0

2

4

6n/J n

2

4

6

8

, x1 0 3

k g /m 3

n= 0n= 2 n= 4

Fig. 3: Functions of the relative density of the gravitational moments n(s)/Jn (n=0, 2,4) and density distribution (s) for a trial model of Io.

0 40 0 800 1200 1 600

R adius (km )

0

0 .5

1

1 .5

2

2 .5h n

h 2

h 3

Fig. 2. Love numbers h2 and h3 along the planetary radius.

The effects of the second approximation on the figure parameters and gravitational moments of the satellite Io have been considered. It turns out that the account of the second order values decrease gravitational moments J2 and

C22 by 2 units in the third decimal digit. To calculate the figure parameters s4, s42

and s44 and consequently gravitational moments J4, C42 and C44 , three integro-

differential equations [3] for trial model density distributions were first solved numerically and, as a result, all second order corrections were obtained.

We have estimated the contribution from the effects of the second approximation to the lengths of the semiaxes a, b, and c for the equilibrium figure of Io. Including the corrections of second order changes the a, b, and c semiaxes by 55, 9, and -4 m, respectively.

Conclusion

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Thank you!

10, Bolshaya Gruzinskaya str.

Moscow 123995, Russia

zharkov@ifz.ru