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FARSIDE EXPLORER: UNIQUE SCIENCE FROM A
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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, T.V. Gudkova, Schmidt Institute of Physics of the Earth, B.Gruzinskaya 10, Moscow 123995, Russia. Contact: zharkov@ifz.ru
Introduction: 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 the Galilean satellites determined in the Galileo space mission have shown that this relation holds with a high accuracy. if the Galilean satellites of Ju- piter are in a state close to hydrostatic equilibrium, then data on their figures and gravi- tational 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 [1].
To show the effects of the second approximation, two three-layer trial models of Io are used [2]. They differ by the size and density of the core Io1 (the core density rc=5150 kg/m3: Fe-FeS eutectic) and Io3 (rc=4600 kg/m3: fes with an admixture of nickel), while having the same thickness and density of the crust (50-km crust , rcrust=2700 kg/
m3), and the mantle density difference is only 20 kg/m3.
Figure theory: The equilibrium figure of the satellite is an equipotential sur- face of the sum of three potentials: U W Q V const= t+ + = , the tidal potencial
2
(cos )
n
t n
n
GM r
W P Z
R R
∞
=
=
∑
, the centrifugal potencial[ ]
2[ ]
2 2
2 2
1 (cos ) 1 (cos )
3 3
r GM r
Q P P
R R
ω θ θ
= - = - (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, ω2=GM/r3 for synchronous satellites), and the potencial from mass distribution inside the satellite V(r,u,φ).
In the figure theory we pass from the actual radius к to the effective radius s defined as the radius of a sphere of equivalent volume (r,u,φ)à(s,u,φ):
2 1
2 2 0 2 2 22 2 31 3
0
3 2 2
33 3 4 4 42 4 44 4
1 ( ) (1 ( ) ( )cos 2 ( )cos
( )cos3 ( ) ( )cos 2 ( )cos 4 )
n n n
r s s P t s s s P t s P t s P t s P t s P t s P t s P t
φ φ
φ φ φ
∞
=
= + = + + + + +
+ + +
∑
The figure theory is constructed by expanding the expressions for the potencials in powers of a small parameter a=3 /p Gr t0 2, where ρ0, τ are the average density and rotation period of io, respectively.
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: 55, 9, and -4 m, respectively.
Dualism in the figure theory: The values of the figure functions at x = 1 are called the figure parameters. Figure parameters specify the shape of the figure of an equilibrium satellite. the values snm(s) at the surface of the body determine the Jn and Cnm coef- ficients (gravitational moments) in the expansion of the external gravity field. If the sat- ellite 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.
Results: To calculate the figure parameters s4, s42 and s44 and consequently gravi- tational moments J4, C42 and C44 , three integro-differential equations [3] for trial model
Fig. 1. To zeroth approximation the equilibrium figure of a body has the form of a sphere with the mean radius s1. In the first ap- proximation 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.
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corrections were obtained. figure functions s s2( ),s s22( ) ands s31( ), s s33( ) are pro- portional to the love functions h2(s) and h3(s), respectively. The 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 1
3
0 0
( ) n ( )
n n n
J =
∫
d x d x + f x =∫
c dx.the zeroth gravitational moment J0 is the mass of the planet, which is unity in dimen- sionless variables. The functions χn(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 (χn(x)/Jn)dx gives the relative contribu- tion from the region of the planetary body in the interval [x,x+Δx] to Jn. the graphs of the functions χn(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 impor- tance. That is why the gravitational moments for Io 1 and Io3 models, calculated in the second approximation, differ in the third decimal digit. these 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.
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 den- sity. 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 restric- tions on the value of average mantle density of io.
note how strongly the parameters depend on small variations in the mantle density of the satellite at constant crust thickness and density. To quantitatively estimate this effect, let us form two logarithmic derivatives based on the Io models: δk2/ k2=1.324 δρm/ ρm, δmc/mc=-12.23 δρm/ρm, where ρm is the density of the mantle, and / mc is the core mass.
We see that small variations in the number k2 change greatly the core mass.
Conclusion: The analysis of the effects of the second approximation on the figure pa- rameters and gravitational moments of the satellite io has been done. the considered
models mainly differ in the core density and the core size. 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. the effects of third and forth harmonics are determined mostly by outer layers of io. to reveal the difference in density distribution, the gravitational moments J4, C42 and C44 should be determined to accuracy with three or four decimal digits. then the mantle density could be known with good precision.
Acknowledgements: This research was made possible partly by Grant No. 09-02-00128 from the russian fund for fundamental research.
References: [1] Zharkov, V.N. A theory of the equilibrium figure and gravitational field of the Galilean satellite Io: the second approximation, Astronomy lettes, 30, 496-507. [2] Zharkov, V.N., Karamursov, B.S., Models, figures and gravitational moments of Jupiter’s satellites Io and Eu- ropa. Astronomy Letters. 32. 495-505,2006. [3]Zharkov,V.N.,Gudkova,T.V.Models, figures, and gravitational moments of Jupiter’s satellite Io:effects of second approximation, PSS, 58,1381- 1390, 2010.
Fig. 2. Love numbers h2 and h3 along the
planetary radius. Fig.3. Functions of the relative density of the gravitational moments cn(s)/Jn (n=0, 2,4) and density distribution r(s) for a trial model of io.
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