Let us study the relation between and . We only consider the region from +15 to centrifugal latitude, where the and measurements are independent and the uncertainties rather small. Fig. 2 shows the data plotted as temperature versus density in a log-log format, and the associated best-fit line. Since it would be inadequate to determine the slope from a classical linear least-squares fitting, because of the finite errors bars on both and , we used a standard (non-linear) approach taking into account the uncertainties on both variables ([Press et al. 1992]); to calculate the merit function, the square deviations between measurements and model are weighted by , where and are the measurement uncertainties and the slope to be determined, in log-log coordinates. This yields the polytrope relation
with (3 ).
The correlation coefficient between and is r = -0.87. Given the number of data points (46), the level of significance of that anticorrelation is very high.
Figure 2: Electron parameters measured in situ aboard Ulysses, plotted as temperature versus density, and the associated best-fit line. The data span in latitudinal distance. (The longitude varies by , centered near , and the Jovicentric distance only varies from 7.1 to 8.4 . We have used the data plotted with solid error bars in Fig.1, which are independent measurements of and .)