To plot the data (e.g. 
) as a function of all possible directions of the rotation axis, we have
used the Hammer-Aitoff projection of the whole celestial globe. The Hammer-Aitoff projection
produces an equal area map of the entire celestial globe, making it particularly useful for our
purpose, which is to accurately depict the probability the satellite's rotation axis points to a certain
region of the sky.
Figure 2 is typical for the plots used in this appendix. It shows the celestial grid in
the background (dotted-black), with the poles at the upper and lower extreme of the plot. Overimposed
are a contour plot of 
(full-white) and a color coding of 
to improve visibility of
the information contained in the contours. In the latter color coding, black regions always indicate
regions with higher 
, whereas white regions have a low 
.
If we now study Figure 2 in detail, we find 4 regions which are conspicuous: two black
regions and two white regions. The two black regions coincide with the direction of the bisectrix
at various times. Note that
there are two regions because of the ambiguity connected with the sense of rotation (one
is for right-handed rotation, the other for left-handed rotation). The two regions are at
opposite ends of one direction. In these regions the standard deviation 
is maximal, i.e. the
rotation periods determined for these directions do not coincide at all. They represent the worst fit,
and are the least likely directions for the rotation axis. The maximum standard deviation is of the
order 10 s, which is extremely large with respect to a rotation period of 40 s. Note however that the
values here are very spiky (not smooth at all), as evidenced by the tiny white contour lines
at the center of the black regions. This lack of smoothness is due to the fact that at these locations
a slight change of direction of the rotation axis can change the observed flash periods strongly. In
mathematical terms from (3): 
is almost parallel with 
.
The other two conspicuous regions are white, and represent regions with minimal 
, i.e. regions
of best fit. One is located at (180,50), the other at (-10,-30). The depth of the minimum is
comparable in both regions, with an absolute minimum of 
s. Both regions measure
about 3 degrees in diameter, though the exact boundaries are necessarily vague. The average
over the entire celestial globe is 1.83 s, so that in the minimal regions a reduction of the
standard deviation of 18 % is observed. The reduction may not seem very significant, but we should note
that the data of the first pass were randomized by adding a time difference of between -1.5 and 1.5
seconds to each time. The minimum standard deviation of 1.52 is thus easily explained by the
introduced randomization. Nevertheless, our method has so far given us two regions, of which we know
one to be correct (180,50). Given just the data of the first pass, it is impossible to decide which
of the two regions contains the real direction of the rotation axis.
Figure 2: Contour plot of the standard deviation 
(seconds) as a function of the direction
of the rotation axis, for the first pass of 72- 57 J. The x-axis shows the right ascension
(-180 to 180) with respect to the vernal equinox point, the y-axis shows the declination
(-90 to 90) with respect to the
celestial equator. Black dotted lines are grid lines, full white lines are contour lines. The color
coding changes from black indicating high 
('bad' directions) to white indicating
low 
('best' directions).
Fortunately, data of a second pass are available. Figure 3 shows 
for the second
pass. The satellite-observer geometry has changed considerably with respect to the first pass, which
is reflected in the different location of the two black ('worst-fit') regions. Note that 
is the bisectrix of 
, which can change considerably from pass to pass, and 
which for
observations at the same time of year is almost constant. If the observations are obtained by the
same observer, then the change of 
is also limited. We come back to this below.
The data in Figure 3 further show that the location of one of the white ('best-fit')
regions has changed considerably, whereas the white region containing the real solution has stayed in
place at (180,50). The minimum of 
is 0.31 s, whereas the average 
over the entire
celestial globe is 0.74 s. Note that the average is calculated by taking into account the solid angle
each bin (of 2 x 2 degrees) covers. The lower 
values are, of course, due to the fact that
the randomization of this pass was limited to 
s. The reduction of 
at the minima
is now almost 60 %, which is much higher than for the results of the first pass.
Figure: Contour plot of the standard deviation 
(seconds) as a function of the direction
of the rotation axis, for the second pass of 72- 57 J. The x-axis shows 
and the y-axis
. See the legend to Figure 2 for details.
Using the results of both passes, we find that there is one region which is the 'best-fit' for both cases.
This region is centered around (180,50) (with a radius of 2 degrees), which coincides with the real
solution. Since it is not necessarily true that there is always
a common region which is easy to identify, we decided to plot the quantity 
to develop a
more objective tool for determining the 'best-fit' region. We define:
where 
and 
, the resp. total number of timings determined during pass 1 and 2.
is the average standard deviation for the first pass, 
is the same for the second pass. Note that the average is over all possible directions of 
.
If we plot 
, we find Figure 4. Now it is clear that only one region sticks
out, the one centered around (180,50). Note though that the definition of 
normalizes both
passes so that they are weighted inversely proportional to 
. For cases where
real data are used, one might want to consider
giving one of the passes even less weight than the other, if the data are considered to be less reliable.
Figure 4 shows four regions of 
(two for each pass). We can see that at the center of
those regions, some directions are much better fits than the surrounding directions. However, none of
those directions minimizes 
to the extent the (180,50) region does.
Figure: Contour plot of the standard deviation 
(seconds) as a function of the direction
of the rotation axis, for both passes of 72- 57 J. The x-axis shows 
and the y-axis
. See caption of Figure 2 for details.
