The radial variations
found at high southern latitudes, 
and
, suggest
a polytropic relation
between 
and 
in this region, given by
,
with 
; this is plotted as a solid line
in Figure 6. It must, however, be noted that, due to the spacecraft
trajectory, the data
are acquired roughly across local tubes of force, so that the validity of the
polytrope relation is based on the absence of latitudinal and temporal variation
inferred in the previous sections. It is also noteworthy that this result
refers to a well-defined wind state, i.e., stationary high-speed wind
(as shown by the
histograms in Figure 7 and in Figure 8)
near solar activity
minimum, but
concerns a limited radial range (1.52 to 2.31 AU).
For comparison, we have plotted in
Figure 6 a log-log scatterplot of the electron density as a function of
the core
temperature and the
associated best fit line (dashed-dotted line)
from the whole pole-to-pole passage of Ulysses.
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 (nonlinear) 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 
-1 the slope
to be determined, in log-log coordinates. This yields the polytrope relation
with =1.370 
0.002.
Figure 6: Scatterplot of the electron density versus the core temperature
from pole-to-pole (170,000 data points).
The high concentration of the data at low values of
the parameters corresponds to
high heliolatitudes where both the
electron density and core temperature are low.
The dashed-dotted line is the best fit polytropic model
to the data, with the deduced index of
1.37, while the solid line is the relation between and
as deduced from their radial profile southward of 
.
The thin gaps in the data are an artefact of our fitting procedure which does
not affect significantly the uncertainties of the parameters.
The correlation coefficient between and is 0.78.
Given the number of
data points (170,000), the level of significance of that correlation is very
high.
It is noteworthy that
in contrast with the high-latitude data sets considered above and
in the other sections,
these latter data, obtained in a larger dynamic range,
refer to several different types of flow coming
from both polar coronal holes, from the vicinity of the heliospheric
current sheet and from the associated stream-stream
interaction regions. As it can be seen in Figure 1 near solar equator, the
most important variation in the electron parameters corresponds to
compression due to fast-slow wind stream interactions and coronal transient
events.
The value of 
is rather close
to the value of 1.47 found by [Feldman et al., 1978a]
using and 
measured on board IMP 6, 7, and 8 in
selected stream-stream interaction regions at 1 AU,
two
years before the previous solar minimum.
[Scudder & Olbert, 1979] argued that the [Feldman et al., 1978a]
result might be flawed by a nonuniformity across streamlines. The same may be
true of our determination using the whole pole to pole data set, which has very
different coronal origins, but not of our determination of
restricted to the southern high-speed wind; in this latter case,
both the density and
temperature histograms are roughly Gaussian (as shown in Figures 7
and 8).
This latter value of the polytropic index, which is
roughly midway between isothermal and adiabatic is still rather different from
the polytropic index 
determined by
[Sittler & Scudder, 1980],
using Voyager 2 and Mariner 10 data acquired in the radial
range 0.45 to 4.76 AU at different phases of the solar cycle. However, all
these values of fall in the wide range 1.1 to 1.7 found recently by
[Phillips et al., 1995b]
using in-ecliptic observations made by ICE and Ulysses aligned in
solar longitude and 
AU apart during the declining phase of a
high-speed stream near solar activity maximum.
The large scatter of all those determinations might be due to the fact that they suffer from limited statistics and/or that flux tubes of different origins are mixed (see, e.g., [Schwenn & Marsch, 1991]). This may also mean that the solar wind cannot be described by a single polytropic index independent on heliocentric distance and flow conditions.