Let us now compare the measured density profile to the theoretical profile (23), where is deduced from the polytrope exponent found in Section 3.
Since the Ulysses trajectory was not exactly normal to equator, we first correct the variation with Jovicentric distance. This is done by normalizing the measured density to the equatorial density at the same Jovicentric distance, deduced from Voyager measurements ([Sittler and Strobel 1987]) by Bagenal (1994) (Fig.5 of that paper, without current sheet). This assumes that the variation of density with Jovicentric distance was similar for Ulysses and Voyager (apart for a constant multiplicative factor
A) in the small range of distances considered here. Such a procedure may be questionable owing to the differences in epoch and longitude of the spacecraft trajectories; it seems, however, reasonable since it allows to correct completely the small observed latitudinal asymmetry. In order to match our Ulysses results at equator, the densities of the Voyager model must be multiplied by the factor , as in Hoang et al. (1993) (i.e., the torus was denser at the time and longitude of the Ulysses traversal, compared to what Voyager measured.)
Figure 5: Theoretical density profile across the Io torus given in Eq.(23) with given in (19) - as deduced from the observed polytrope law - and , superimposed to the normalized measured density profile (corrected for the variation with Jovicentric distance). The horizontal axis is sign(z) , z being the latitudinal distance in Jovian radii. Dotted error bars identify the data for which and are measured independently. The sigma of the fit is 0.04.
Our observed profile corrected in this way is plotted in Fig.5. We have superimposed the Kappa-like profile (23), with given in ( 19), and the scale height
The distance z (in Jovian radii) is counted from the plane of symmetry of the torus as determined from our measurements, which is tilted by ( ) to the magnetic equator; this value is roughly equal to the nominal tilt of the centrifugal equator, within the precision of the magnetic field model used. All three parameters (H, A, ) are deduced from a chi-square fit involving the data points with solid error bars in Fig.5. The fit is quite good; the mean-square relative error between the measurements and the model is 0.04.
Substituting in Eq.(25) the Kappa scale height H = 0.91 found above, with the Jovian parameters m and rad/s, one finds
( being the proton mass). Assuming an effective ion mass of at ([Bagenal 1994]), this gives K, which is close to the result found by Hoang et al. (1993). It may be noted that in the same range of radial distance, the Voyager measurements imply eV and eV ([Bagenal 1994]). This gives K, which is about 2.5 times larger than our Ulysses determination. Such a difference (if it is not due to the simplifications of our model) is not surprising since (i) with a non-Maxwellian (core) distribution, the Voyager ion (core) temperature, which is determined by assuming the ion (core) distribution to be Maxwellian, is not necessarily equal to the temperature defined from the mean random energy of these particles, (ii) the spacecraft explored different longitude sectors, and (iii) the variability of the torus is known to be significant (see [Strobel 1989]): for example, the 1981 observations by Morgan (1985) also imply an ion temperature twice smaller than that derived from the Voyager analyzers. One may also note that the radio occultation experiment aboard Ulysses measured a line-of-sight electron content in the torus which similarly suggested an ion temperature twice smaller than the value inferred from Voyager measurements ([Bird et al. 1993]).
We have not included in the fitting the measurements acquired at negative latitudes beyond (dotted bars in Fig.5) because in that region the measurements of and are not independent. One may note, however, that these points also match the theoretical Kappa-like profile within the error bars, except for the three southern most points, which do not either follow the polytrope law. Noting that the Jovicentric distance of these points is nearly 9 , this might suggest a different origin or behaviour of the plasma beyond this distance, where the torus merges into the magnetodisc, and where the influence of the satellite Europa - which orbits Jupiter at 9.4 - may be significant ([Intriligator and Miller, 1982]).
It would be interesting to extend the comparison farther from the equator. This is, however, difficult, since Ulysses explored high latitudes in very different longitudes sectors and at very different Jovicentric distances. For an order of magnitude comparison, we note that Ulysses passed at at the radial distance , where the Voyager analyzers found an ion temperature roughly 0.6 times smaller than at . This yields a scale height roughly times smaller than the value above, i.e. . With this parameter, the profile (23) yields . Taking cm as determined by Voyager at that radial distance, we obtain 6 cm , which is comparable to the values measured aboard Ulysses at this location ([Desch et al. 1994]).