During a pass on April 11, 1996 a peculiarity was observed in the flash period sequence of the rocket body 95- 32 B (by Walter Nissen, Berea, Ohio, USA). Starting the measurements at 1h51m18.0s UT, the following sequence of flash periods was recorded: 15.8, 9.4, 13.1, 12.6, 14.6. Walter estimated his accuracy to be 0.8 seconds, due to the relatively round maxima.
The analysis of this pass produces results which are very similar to the results found for the 91- 29 B observations. Two very pronounced and almost identical minimal regions are found to be lying in the region (which is not very extended here, since only 5 measurements are available). The vector is parallel to the bisectrix between the vectors satellite-sun and satellite-observer. If during a pass happens to become parallel to the rotation axis direction, chances are big that the observer will see a synodic anomaly.
The minimal value of in both best-fit regions is 1.6 seconds, compared to an average of 4.85 seconds. The best-fit regions are found at (-162,9) and (18,-9), i.e. at opposite ends. Once again the best-fit regions are identical in appearance and in minimum value of , so they can not be distinguished. Interestingly, each best-fit region consists of two lobes. The lobes have a diameter of about 0.5 degrees and are centered about the the position of , i.e. the bisectrix at the second time. This is demonstrated in figure 11, which shows the contour-lines of 2, 3 and 10 seconds around the best-fit region at (-162,9). The latter contour-line connects the points, forming the typical circular regions also found in figures for other synodc anomalies. The contour-lines for 2 and 3 seconds border the best-fit regions. Figure 11 is a nice illustration of the dominant effects the quasi parallel and have on the flash periods.
Figure 11: Contour plot of as a function of and . Three contour lines are shown, for seconds. The black dots indicate the position of at the times of the flashes. The best-fit region consists of the double lobe about the dot at (-162,9).
The lobes of each best-fit region have slightly different rotation periods. The upper lobe of the (-166,18) region is centered around (-162,9.2) and has an average rotation period of 26.6 seconds, whereas the lower lobe at (-161.8,8.2) has an average period of 26.25 seconds. The upper lobe shows lower values. One lobe of the (18,-9) region is centered around (17.5,-8.8) with an average rotation period of 25.85 seconds. The other lobe is at (18.5,-9) with an average rotation period of 26.3 seconds.
Once again a very accurate determination of the direction of the rotation axis can be made, using an absolute minimum of timings, since a synodic anomaly was observed. Apparently, the sense of rotation can not be determined from synodic anomalies.
This object was also observed on April 18 at 1h38m UT (18 timings), April 20 at 20h10 UT (6 timings), April 21 at 1h19m UT (25 timings) and April 22 at 1h49m UT (10 timings). The April 20 observations was obtained by Leo Barhorst, Alkmaar, the Netherlands, but the other 3 are by the same observer as the April 11 observation. During the April 18 pass some kind of synodic anomaly was observed. The measurements were rendered quite difficult due to the appearance of secondary flashes, and probably also some diffuse reflection. Despite those difficulties, a rotation axis can be determined with relatively high precision. The depth of the minima is similar to the April 11 observations, with . One solution is at (-176,20), the other at (4,-20). Both have double lobes and have an estimated accuracy comparable to the April 11 observations (0.5 degrees).
The April 20, 21 and 22 observations do not show any sign of a synodic anomaly, but can be combined pair-wise and treated as two quasi-simultaneous observations: one for April 20/21 and one for April 21/22. On April 20/21, this gives a rotation axis at (172,48) and at (30,-18), both with an estimated accuracy of 5 to 10 degrees. The ratio is of the order 0.8. On April 21/22, a rotation axis at (163,44) and one at (26.5,-10) are found, again with estimated accuracy of 5 to 10 degrees. The ratio is of the order 0.85.
If we now combine the information of all passes of 95- 32 B between April 11 and April 22, an interesting picture emerges (see figure below). We can divide the possible directions of the rotation axis of 95- 32 B into two groups, as shown in table 1. As can be seen in the figure, solution 1 (full dots) shows a gradual change of the direction of the rotation axis at an average rate of 3.8 degrees per day. Solution 2 shows a rotation axis which moves back and forth between two positions, at an average rate of 3 degrees per day (in both directions). The direction of the normal to the orbital plane is also shown in the figure. Due to the precession of the orbital nodes, the direction of the normal changes with about 10 degrees in 11 days, hence the three positions (marked with crosses in figure).
Table 1: Direction of the rotation axis and rotation period of 95- 32 B. Solution 1 is at the left, solution 2 at the right.
The motion of the angular momentum vector (parallel to ) is governed by the torques acting on the rocket body. The torque due to the gravity gradient causes a precession of around , with a precession angular velocity :
where , and is the moment of inertia around the short body axis and is the moment of inertia around the long body axis. Also, is the angle between and . Furthermore n is the number of orbits the satellite completes per day (the 'mean motion'). Assuming the rocket body is empty, and given its dimensions are 7.4 m by 2.4 m, we find for this type of rocket. We can thus calculate the precession period of around and find it to be of the order 70 days. With , this implies the angular motion per day can be of the order 4.5 degrees.
Note though that the precession described in the above only takes place under idealized circumstances. In reality the orbital plane precesses (due to the oblateness of the Earth), so that the motion of will not be along a precession cone with constant . Furthermore, we know that the torque due to eddy currents can not be neglected. It is for example responsible for the breaking of the rotational motion of the rocket body around its own rotation axis. The eddy current torque for an end-over-end tumbling cylinder is proportional to:
where is the Earth's magnetic field. If there were no gravitational or orbital precession, the rotation axis of 95- 32 B would tend to a 'stable' solution for which . (see Technical Note 0 by Patrick Wils for details).
The motion of is in reality influenced by both torques. It is thus a mix of both tendencies, which is impossible to predict without taking recourse to numerical integration. We can conclude by saying that the motion of solution 1 looks more realistic from an intuitive viewpoint. Note that the last two points (April 20 to 22) are quite inaccurate (5 to 10 degrees error?). But solution 2 can not be excluded given the abovementioned complications.
The rotation periods can't be used to discriminate either one solution as the most probable. Theoretically speaking we expect the rotation period to slowly rise with time, due to magnetic braking. The short time (11 days) involved in our results, and the large uncertainties of the rotation periods found in our analysis (up to 0.5 or 1 seconds?), imply that no conclusions can be drawn.
Figure 12: As a function of and , the direction of the rotation axis of 95- 32 B on April 11, 18 and 21/22 (solution 1 represented by full black dots, solution 2 by stars). Also shown is the position of the normal to the orbital plane (crosses).
We feel rather confident that the rotation axis directions determined from this string of observations are reliable. The observer was fortunate enough to observe the object at a time period during which its rotation axis approximately coincided with the bisectrix regions typical for the passes he observed. The direction of the bisectrix region is partly determined by the position of the sun (and thus the time of year), and partly by the type of pass (e.g. low northbound, or high overhead, etc...). The fact that the rotation axis moved (day by day) in the same general direction as the bisectrix region, ensured that the observer was able to observe a synodic anomaly on two days: April 11 and 18. A few days after the latter, the rotation axis had 'overtaken' the bisectrix region and the observations did not show any synodic anomalies anymore, though a relatively distinct synodic effect was still observable. This combination of relatively rare conditions allowed us to determine the temporal evolution of the rotation axis with such an accuracy.