* Moscow State Academy of Control Devices Design and Informatics, Stromynka 20, Moscow 107846, Russia
** Space Research Institute Russian Academy of Sciences, Profsojusnaja 84/32, Moscow 117810, Russia
In recent years, substantial information concerning the Io torus has been published. Much new information has been obtained by the Galileo mission (e.g. http://www.jpl.nasa.gov/galileo). Nevertheless, numerous unanswered questions remain.
Explaining the observed steady state rotation of the torus seems to be difficult. The angular velocity of the torus must be in accordance with the celestial mechanics laws, of the same orders as Io's one. But the radial plane corresponding to the eccentricity and maximum thickness of the torus rotates in the inertial medium with an angular velocity approximately 4.5 times greater then Io's orbital angular velocity, being quite stable relative to Jupiter.
Some attempts have been made to explain the picture being observed by hypothesis of continuous ejection of the mass through the outer border of the torus and replacement via Io's volcano activity. Suitable calculations give the value approximately 1000 kg/s as the estimation of the mass debit [Hill et al. 1991; Schneider et al., 1991]. Experimental results from Galileo place the debit closer to 10 kg/s, two orders of magnitude smaller [Russell et al., 1997]. This makes the conception of strictly non steady torus vulnerable to criticism.
We wish to straighten out the situation describing the main geometric and kinematic features of the Io torus using a steady state model based on some of Alfven's ideas [Alfven, 1981]. Three key aspects of our approach are:
A mathematical model of the plasma ring corresponding to these assumptions has already been presented by Rabinovich at al. [1992; 1996]. We improve the model for undisturbed motion by taking into account the eccentricity of the dipole and using some features of the classical "problem of two immovable centers" which was formulated and solved by Leonard Euler more than two centuries ago.
In our case, the geometrical center of the planet and the dipole play the role of the two centers, whereas the gravitational and the ponderomotive ones related to the midpoint of torus elements play the role of the forces. We propose that the repulsive ponderomotive force is smaller than the attractive gravitational force.
Let us use the following transformations:
It is possible to prove that the position of the last point is stable up to first order relative to the eccentricity of the dipole, and the resultant modulus is stable up to the zeroth order. Nonsteady terms could be included in the equations of disturbed motion of the torus, but this is not the point of this paper.
Let use as a mathematical model for the undisturbed motion of the torus the differential equation of the relative motion of material point projected on the torus- central radius-vector in a coordinate system rotating together with the planet. The absolute angular velocity of the torus and the Alfven waves velocity relative to the middle line of the torus are the main parameters of the problem.
The sum of the following two steady state rotations is the exact solution of the basic equation:
We may identify the last rotation as a "corotation" of the torus and the sum of two rotations mentioned above as a "partial corotation" [Alfven, 1981].
Table 1 presents the calculated values of some torus' parameters related to its middle line obtained by aforementioned mathematical model.
1 Circular velocity (km/s) 17.27 2 Circular velocity, tacking into account the magnetic field (km/s) 10.70 3 Corotation velocity (km/s) 80.65 4 Alfven's waves velocity (km/s) 7.831 5 Plasma density g/cm3 4.246 10-17
The basic data for the calculations were taken from [Russel et al., 1997] (for torus geometry) and [Kaufmann, 1979; Whiple, 1981; Burgess, 1977] (for the Jovian systems parameters).
Figure 1 presents the kinematic picture of the partial corotation of the torus corresponding to the described model for the mean values of the torus parameters related to its central line.
Summarizing the results, the steady state model considered here gives a possible explanation for the main natural features of geometry and kinematics of the Io torus without any additional physical hypotheses. Comparison with new experimental data should provide a possibility for checking the improved model. Acknowledgments The authors thank the Russian Foundation for Fundamental Research (grant 97-01- 00536) for support of this investigation.
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Fig. 1.. Five positions of the Io torus, Io and Jupiter in the inertial medium corresponding to one rotation of Jupiter around its axis (the scale is distorted for clarity).
Dated December 1997