I used the software that I described in Flash 90 to analyze Mike's observations as described in the article above. Though I suggest you read the 9-page article in Flash to know more about my method to determine the direction of the rotation axis of flashing cylinders, here's a short synopsis :

- -
- Each of Mike's timings is taken to be perfect, i.e. residuals are assumed
to be 0.
- -
- For a certain direction of the rotation axis and for each couple of flash
timings, I can calculate the rotation period (this is not equal to the flash
period), since I know the geometry at the time of the flash and the flash
period. In Mike's case, I will end up with 12 values of the rotation period
(since there are 12 couples) for *each* direction that I look at.
I calculate an average value and a standard deviation around that average
from these 12 values.
- -
- I look at all possible directions with a one degree step in right
ascension and declination. I now have average rotation periods and
standard deviations on them for 360 x 180 possible directions.

If Mike's observations were infinitely accurate, there would be one of those
360 x 180 directions for which all 12 rotation periods are equal. That is
the direction of the rotation axis, since we know that the rocket only has
**one** rotation period. In this case the standard deviation around the
average rotation period would be 0.

Naturally, the observations have some residuals on them, since the observer has a certain reaction time, etc... So we can't expect the standard deviation to be equal to 0 for one certain direction, we can only expect that deviation to be minimal for a certain direction.

The big question of course is : "how significant should the minimum be to be believable ?". If the rotation periods are only slightly more tightly grouped around the average, there is some doubt whether that *one* direction is really the rotation axis, since the improvement over other directions is only marginal.

In certain special cases, where the bisectrix between the vectors satellite-sun and satellite-observer is almost parallel to the rotation axis, the synodic effect can become extremely large. The synodic effect is just a way of saying that the time between two flashes isn't equal to the rotation period (or half of it). In these extreme geometries, the flash period can be very different from the rotation period. Walter Nissen observed this last year with 91- 29 B (see Flash 89 of Jan. 1995 and Flash 90 of Feb 1995), and the question now is whether Mike saw something similar with 92- 38 B.

In Flash 90 I plotted the reciprocal of the square of the standard deviation as a function of right ascension and declination of the rotation axis for 91- 29 B (in short ). For one certain region this value peaked quite distinctly from the surrounding region in the sky, i.e. the was maximal there. This means of course that the deviation was substantially lower than in the rest of the sky, i.e. this is the region where the rotation axis is most probably to be found. The peak was much more clearly discernible than in cases where nothing too special was seen during the pass.

The figure below shows what I found for Mike's observation, doing the same thing as for Walter's 91- 29 B observations. Two very distinct peaks can be seen at :

A : , with rot. period = and

B : , with rot. period = and

First of all, these two solutions are exactly in opposite directions. Since we only had 13 timings it wasn't possible to distinguish the *sense* of rotation. The average sigma over the whole sky is 0.82 (pretty uniformly). This means that for most of the sky the average 'residual' is 0.82 seconds. For our two limited regions (only a few degrees in radius) this drops down to 0.55 seconds. A better estimate for the 'residual' is probably the average of the *absolute* deviations. This gives less weight to bad observations (since the standard deviation is quadratic). For our regions this 'residual' is around 0.36 seconds, while the average for the whole sky is around 0.60 seconds. Similar ratios of minimum/mean were found for 91-29 B. The significance of the two directions of the rotation axis of 92- 38 B is very similar to what we found for 91- 29 B.

The determined direction critically depends on the absolute time of the flashes. Since this time was only known to within 20 seconds or so, the determined direction of the rotation axis is not very accurate. However, as test runs with slightly different times prove, the narrowness of the peaks remains the same for absolute times that do not differ by more than 1 minute from the time we took for the graph. This makes sense but I won't go into that now, since Flash is already so full.

Can we discriminate between solution A and B using the observations of February 20? The data for Feb 20 is scarce. We only have an average period (9.20), not an exact starting time and Mike's claim that the period was pretty much constant over the whole pass. Add to that the fact that solutions A and B are not very accurately known (due to the inexact starting time) and the fact that the rotation periods for A and B have error bars on them that are rather big, and it looks pretty desparate...

If we enter a series of constant 9.20 timings on Feb 20 and determine the rotation period for the directions indicated by the Feb 24 observations (i.e. solutions A and B) we get the following :

A :

B :

Given that we expect the rotation period to slowly go up as a function of time, it would seem that solution B may be preferrable. Solution B gives a period of on Feb 20 and on Feb 24. Solution A gives a period of on Feb 20 and one of on Feb 24. Given the relatively large uncertainties on the determined rotation periods, I don't think a firm choice between A and B can be made.

So, here are my tentative answers to Mike's questions :

*Of course it is possible (likely?) that A) the rotation axis is not
90 degrees from the cylinder axis *

Information on the secondary flashes would be very welcome to decide on that. I should say that the above described method assumes this angle to be 90 degrees. That is usually a good assumption since the torque produced by the Earth's magnetic field (and the eddy currents in the rocket's hull induced by the field) will cause this angle to go to 90 degrees for slender rockets. If other torques are dominating this angle may deviate from 90 degrees.

*and B) the rotation axis had a
"special" orientation during this pass that caused the appearance
to change during the observation. (See articles in Flash 89,
Jan. 95, and Flash 90, Feb. 95)*

This most certainly seems to be the case.

*So, the questions are: 1) Is the current half period 9.067, 9.20, or
9.24?*

An answer to this question is not easy, as was shown in the above. A rotation period of 9.067 seems out of the question though.

*And B) did I observe an unusual series because the axis of rotation had
a special orientation for that pass?*

I am inclined to think so. The case is almost as strong as for Walter Nissen's observation of 91- 29 B. That it isn't just as strong is, in my opinion, due to the fact that Mike did not see the same degree of 'weirdness'.

Thanks to numerours e-mail discussions with Mike McCants, the usefulness and the limitations of my method become more and more clear. It seems the method:

1. is most powerful for weird observations like Walter's and Mike's.

2. has a hard time giving us accurate rotation periods. Those have to be determined with Sat using the rot. axis. direction that my method suggests.

3. can probably indicate the rot. axis of long passes during which nothing weird happens (except for some synodic effect) only to within an accuracy of 'which quarter of the hemisphere'.

4. could be (given some slight reprogramming) adapted to analyze simultaneous observations, i.e. one single pass observed by two observers at the same time. The accuracy of the rot. axis should be a lot better than in case 3.

5. might give more accurate rot. axis (than in case 3) for quasi-simul- taneous observations, like the 82-40 J obs in Flash 90. The accuracy obtained will be worse than in case 1.

**Bart De Pontieu**

bdp@mpe.mpe-garching.mpg.de