As has been indicated before, the flash period is not necessarily equal to the rotation period. This is due to changing illumination conditions as the satellite moves across the sky. To understand the so-called synodic effect we have to understand why satellites flash.
Most rockets and payloads give off specular flashes. Their metallic surfaces act as a mirror for the sunlight. The observer can see this reflected sunlight only if the reflecting surface is perpendicular to the bisectrix of the angle between the line from satellite to sun and the line from satellite to observer (see figure 5).
Fig. 5: Reflection of sunlight on a cylindrical body. Only for a certain value of will the reflected sunlight be visible to the observer.
Let's assume the satellite-sun-observer geometry as a given and constant in time, which it isn't in reality. The reflection condition would hardly ever be satisfied if it wasn't for the fact that most satellites rotate around some axis. Due to this rotation the reflecting surfaces continuously change direction during one rotation period, which gives more possibilities to fulfill the reflection condition. If the reflecting surfaces are flat (as for box shaped satellites), there is no guarantee that the reflection condition will be fulfilled during one rotation. Cylindrical objects however have reflecting surfaces in many more directions at any time than satellites with flat sides. Most rockets and payloads are basically cylindrical. It can be proven that rotating cylindrical objects will give off between zero and two flashes per period (the ends of the cylinder are assumed non-reflective).
If the rotation axis is perpendicular to the long body axis (i.e. the satellite is tumbling or alternatively , with the angle between long axis and rotation axis) there will always be two flashes per period. Moreover, the two flashes will always be symmetrical. This means that the time difference between the two flashes will be exactly half of the rotation period. If the bisectrix is parallel to the rotation axis (in the case of a tumbling cylinder) the rocket will give off one long and continuous flash, i.e. it will be steady. Most third stages are tumbling.
Spinning cylinders (i.e. the rotation axis is along the body axis or ) don't give off regular flashes. Of course, spinning satellites usually have booms, solar panels or other structures that cause asymmetries in the cylindrical body and hence regular flashes can occur.
Cylinders that have a rotational state that is between spinning and tumbling do not necessarily give off two flashes per period, nor are the flashes necessarily symmetrical.
So far we have assumed the satellite-sun-observer geometry to be constant in time. In reality the satellite moves with respect to the observer. In the above we saw that a tumbling rocket would give off a new flash after exactly one half of the rotation period . Now, however, the satellite-observer line has changed direction (due to the satellite's motion) and the flash will not necessarily occur. It will probably occur a little bit later or earlier than expected. This means the measured flash period is no longer the same as the rotation period (or half of it). This is what we call the synodic effect. Figure 6 features an example of the synodic effect.
Fig. 6: Between the two flashes the satellite rotates through only 140 degrees, not 180, because orbital motion has changed its geometry with respect to the observer. The actual rotation period exceeds the flash period by a factor of = 1.29.
The size of the synodic effect depends on the attitude of the satellite with respect to the sun, the observer and the rotation axis. In general the size of the synodic effect is hard to predict since we don't know the direction of the rotation axis. However, as a general rule one can say that the synodic effect will usually be larger for objects at low orbital heights, since they move faster across the sky and the geometry can thus change more drastically during one rotation period. Objects with long rotation periods are also more prone to a larger synodic effect, for essentially the same reason (more change of geometry per period since the period is long).
Satellites with rotation periods below 10 s are usually not very much influenced by the synodic effect, i.e. the synodic effect is smaller than our measuring accuracy. Objects with longer periods can show flash periods which are different from the rotation period by several seconds. For certain extreme geometries, the synodic effect can become as large as half a rotation period. The flashes will then (for a short time) appear at seemingly irregular intervals.
Rockets that are not tumbling (i.e. the angle between the rotation axis and the body axis is not exactly 90 degrees) usually show asymmetric flashes, i.e. the time between two flashes is not equal to half of the rotation period. If you count an odd number of periods, you could very well come up with a flash period that is very different from the rotation period. You can circumvent this problem by counting an even number of periods, since even for non-tumbling rockets the time between the first and the third flash is the rotation period (influenced by the synodic effect, of course).
If secondary structures are visible it is wise to count an even number of flashes as well, since you don't know what causes the secondary flashes. It is very common to see secondary flashes 'appear' or 'disappear' during a pass. This is due to the changing illumination of the object. Making split timings can safeguard you against the confusion caused by the secondary flashes.
Using a modern stopwatch with a memory of fifty split timings you can time every flash. It is then much easier to check how big the synodic effect is and which secondary flashes are visible. The example below is a good illustration of the influence of the synodic effect. Mike McCants observed the rocket 81-116 J on February 10, 1991. He started his stopwatch on the first flash and measured flashes at the times given in Table 4.
Table 4 : Flash timings of 81-116 J
The first column gives the number of the flash, the second column is the time in seconds and the last column is the time divided by the flash period (which was 12.71 s). As we can see from the last column there were secondary flashes at first (flash 1, 3) which then disappeared and later re-emerged (flash 8). Also, the synodic effect is not very big, maximally 5.00-4.98 = 0.02 (i.e. 2%) of the period.
The following pages cover the theoretical aspects of tumbling satellites.