The earlier theory predicts an exponential increase of the flash period with time. The evolution of the flash period of most third stages agrees well with this general theoretical prediction.

Figure 7 shows the evolution of the flash pattern of 90- 46 B. This is the cylindrical third stage of a Zenit rocket, launched by Russia on 22 May 1990, which brought Kosmos 2082 into an orbit around the Earth. The rocket is 10.4 m long, has a diameter of 3.9 meter and weighs 8300 kg after burnout. This was one of the brightest flashers visible with the naked eye. During favorable passages it showed flashes up to magnitude 2. Each mark represents one measurement of an observer. The number of days since the first flash measurement (usually just after the launch) are shown on the horizontal axis, the measured flash period (in seconds) on the vertical axis. One can clearly see the exponential increase with time, and the double periods immediately after launch.

**Fig 7 :** Flash period (in s) of 90- 46 B as a function
of days since the first flash observation. The period grows exponentially.
At long periods the flash period is more difficult to measure.

We know from chapter 7 that theory predicts :

where :

**ct** is the characteristic time

**r** is a constant depending on the rocket's dimensions and
conductivity.

The exponential increase can be demonstrated by plotting the
logarithm of the flash period on the vertical axis (fig. 8). In this way the
exponential curve is reduced to a linear one (a simple straight line!).
The increase of the period can be characterized by the slope of this straight
line, the characteristic time *ct*. As we know the characteristic time
depends
on the semi-major axis, the rocket's dimensions and conductivity. The higher the
rocket orbit, the smaller
the effect of the Earth's magnetic field, and the flatter the straight
line is.
Calculating a linear regression of the observed points in figure 8 gives the
characteristic time as a result. Since we don't know anything about the
direction of the rotation axis it is not possible to check the theory using
only this one satellite, since V is only defined up to a factor of 2.

**Fig 8 :** Logarithm of the flash period of 90- 46 B as a
function of time (in days) after the first flash observation.

If we determine the ct's of all rockets of a *big ensemble* of similar
third
stage in orbits with the same inclination (but not necessarily the same semi-
major axis), and if we divide each ct by
of the satellite in question, we get an experimental expression for .
Since the third
stages were assumed to be of the same type, they will have the same dimensions
and conductivity, hence the same r (which is still unknown to us). From
chapter 7 we know that and we also know the
average for a certain inclination i.

Two methods are now available to us to determine r for this rocket. Firstly,
we can choose r so that most of the values effectively fall between
the
bounds and . Alternatively, we can chose so that
the average of coincides with the theoretical value for
. In both cases it is tacitly assumed that we have enough satellites
so that both ways of averaging (one over *all angles* between angular
momentum and local magnetic field, the other over *all rockets* in our
ensemble) are equivalent.

We have done this for the third stage of the Russian Cosmos rocket. This cylindrical rocket (7.4 meter long, 2.4 meter diameter) has been launched several hundred times during the last thirty years. Most launches were in orbits with 74 or 82 degrees inclination (with various orbital heights).

Reliable determination of was possible for 131 rockets of the 74 degree class, and for 70 of the 82 degree class. Figure 9 shows the distribution of for the 74 degree class. The x-axis shows the values of V (in bins of 0.10), the y-axis the number of values of in a bin with value V. As we know from chapter 7, and for i = 74. We can see on figure 7 that 75%of the values are within the theoretically predicted range. The values that exceed correspond to a ct which is lower than theoretically expected. The values that are lower than correspond to very high ct's. Both groups are exceptions to the theory of chapter 7.

**Fig 9 :** Histogram of for 131 Cosmos rockets with 74
degrees inclination. Theory expects :

If we look in detail at the satellites which have exceptionally low ct's, we notice that most of these rockets have undergone an acceleration before or during the interval in which the ct was determined. This probably means that in addition to the magneto-torque, other torques were acting on the rocket (due to the fuel- leak). This explains the lower than expected ct-values. Some of the rockets with higher ct's were already several years in space when the ct was determined. This contrasts with most other ct-determinations which usually correspond to the first year after launch. It might be that the electrical conductivity changes due to a prolonged presence in space due to interaction with the highly reactive atomic oxygen. This would influence the constant Q considerably, thus invalidating our assumption that the material constants are the same for all rockets.

Similar results have been obtained for the 82 degree class (figure 10). Again about 75% fall in the predicted range for V (1.12 to 2.40). The explanation for the deviations is similar. The peaks in both figure 9 and 10 are probably not significant, but due to the low number of rockets involved. More observations are needed to obtain better statistics.

**Fig 10 :** Histogram of for 70 Cosmos rockets with 82
degrees inclination. Theory expects :

So it seems there are quite a few exceptions to the theoretical behavior. One detailed example can be seen in figure 11, which shows the evolution of the logarithm of the flash period as a function of time for the octet rocket 87- 51 J. This Russian rocket brought eight small spherical military communication satellites (0.9 m diameter, 40 kg) in an orbit around the Earth on 16 June 1987. The orbital height of this Kosmos rocket is 1600 km, considerably higher than most of the tracked objects, hence the very slow increase. However, one notices that the flash period remains almost constant during more than a year. It is not clear whether this is an example of the predicted 'wavy' character (see chapter 7) due to a changing ct, or due to electrostatic charging of the rocket.

There are several charging mechanisms for a rocket. Incoming photons can excite electrons out of the hull (the photo-electric effect) causing the rocket to be positively charged. Low-energy electrons (e.g. from the radiation belts) can collide with the rocket, leaving their negative charge on the surface. The collision can also cause secondary electrons to be torn off of the surface (thereby positively charging it). Finally, interaction with a dense thermal plasma can neutralize charges on the rocket. It has been calculated that a charge which is uniformly distributed over the surface of the rocket can cause the rotation period to remain constant.

From 1992 onwards the flash period is on the rise again. A problem with
following this effect is the lower accuracy of the *long* flash periods.
At minute-long periods, it is impossible to measure more than just a few
periods, since the rocket remains above the horizon only a few minutes.

**Fig 11 :** The logarithm of the flash period of 87- 51 J as
a function of days after the first measurement.
The period stays constant for over a year (between 900 and 1300 days.

A second example of a strange flash period evolution can be seen in figure 12. This is the payload MIDAS 6, 63- 14 A. MIDAS-satellites are American military satellites with presumably a 9 meter length and 1.5 meter diameter, which weigh 2000 kg. MIDAS 4 was launched into orbit by means of an Atlas Agena B rocket on 9 May 1963. (The Atlas Agena B was also used during the Gemini project as target rocket for practicing docking manoeuvers).

Here the flash period decreases monotonously! (notice the long time scale). This implies a continuous acceleration during the past 30 years. The last few years the curve has become flatter, but the acceleration still continues. Although still a hot topic, the discussion has focused on a fuel leak (a bit improbable, since a vast amount of fuel would be needed to sustain the effect during this long period), or a thermal or radiation effect.

**Fig 12 :** Flash period of 63- 14 A as a function of time (in
days). The flash period has been decreasing for more than 30 years.

Another MIDAS, 61- 18 A shows prominent oscillations (figure 13) which turn out to be correlated with the angle between the orbital plane and the direction of the Sun. It has been suggested cautiously that this is due to an effect of radiation pressure, but this is not certain. More observations are needed. It could also be an example of periodic fuel-freezing and evaporation.

**Fig 13 :** Flash period of MIDAS 3 (61- 18 A) as a
function of days
since the first measurement. The flash period seems to oscillate around an
ever increasing average.

Another bizarre example (figure 14) is the behavior of the octet rocket 81-116 J. Although launched at the end of 1981, it was still flashing in 1989, and underwent a sudden decrease in March 1990, next an increase again until February 1991 when a swift decrease (acceleration) put this to an end. Since then, the flash period is slowly increasing again, but we remain alert.

**Fig 14 :** Flash period of 81-116 J as a function of days
since the first measurement. Two sudden periods of acceleration have been
observed so far.

Although single period decreases can easily be explained by fuel leaks, there has been no explanation so far for a double decrease with an interval of one year. It has already happened several times, and currently the theory of a periodic sublimation of frozen fuel is gaining ground. More research is required.

A regular fuel leak has happened many times just after the launch. It is sometimes commanded in order to avoid explosion. Figure 15 shows a striking example of this. It is the case of 71- 86 J, which underwent a sharp decrease in rotation period a few weeks after launch.

**Fig 15 :** Flash period of 71- 86 J as a function of days
since the launch. A fuel leak caused a gradual decrease of the period.
After fuel exhaustion the magnetic friction starts to dominate.

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