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.