Flashing satellites are always under the influence of the synodic effect, the effect that is responsible for the difference between the rotation period and the flash period. The flash period is the time between two flashes as seen by the observer, the rotation period is the time the rocket needs to turn around its axis once. If the satellite were stationary with respect to the observer, the flash period would be equal to the rotation period. So, if we observe flashing geostationary satellites, the synodic effect is non- existent.
But we usually observe flashing satellites that make a pass lasting only a few minutes. During this time the satellite-sun-observer geometry can change considerably. I refer you to Chapter 6 of the English language brochure which explains in more detail how the synodic effect influences the flash period. What I would like to do in this article is to quantify the synodic effect (under very general conditions). Using the graphical representations below, it should be possible for every observer to estimate the 'average' size of the synodic effect for a certain pass.
Let us assume the satellite is in the zenith of the observer. It is in an approximately circular orbit and it is moving with an angular velocity v across the observer's sky. If the rotation period is P seconds, what is then the size of the synodic effect for any given direction of the rotation axis? Obviously the size of the synodic effect will strongly depend on the direction of the rotation axis. Since we want to make general plots, that are valid in any situation, we will express the direction of the rotation axis in a local coordinate system.
The center of the coordinate system is the satellite. The x-axis points from the satellite to the observer. The y-axis is parallel to the velocity vector of the satellite, i.e. it is directed along the direction the satellite is travelling to. Since we assumed the orbit to be circular, we know the x-axis is perpendicular to the y-axis. The z-axis completes the coordinate system to a right-handed system. The figure below shows the used coordinate system. The xy-plane is of course the orbital plane of the satellite.
We can now define the direction of the rotation axis as a function of this
coordinate system, as shown in the figure above. The 'right ascension'
and the 'declination' 
aren't the real right ascension and
declination, but are the angles defined in our local system. If 
is
not equal to 0, then the rotation axis does not lie in the orbital plane.
And if 
, then the rotation axis lies in the orbital
plane of the satellite.
If 
and 
then the observer is looking right along the
rotation axis. For 
and 
the rotation axis is
perpendicular to the direction the observer is looking along.
means the rotation axis is perpendicular to the orbital
plane.
What we have calculated is the following. We assume that the first flash
occurs right at zenital position. For all possible directions of the
rotation axis (in our local system), we then calculate how long it will take
until the next flash. This is done by iteration.
The time between the two flashes is the flash period as the observer would
measure it.
We can then easily calculate the percentage wise difference between
the rotation period P and the flash period FP as :
This is of course the percentage wise size of the synodic effect,
with a sign to show whether the flash period was shorter (-) or longer
(+) than the rotation period. The division by 2 of the rotation period
occurs because there are two flashes per period. We calculate the time
between the two first flashes. If there were no synodic effect, the
time between these flashes would be 
.
So, we can plot the synodic effect as a function of 
and 
.
There is however one more unknown : the Sun's position. The Sun can take
a whole range of positions, but there is one (approximate) constant in the
sun's position : it is usually about 20 degrees below the horizon of the
observer whenever we observe. The plane of the observer's horizon is
parallel to the yz-plane, since the x-axis is the local vertical (the
satellite is in the zenithal position!). So the solar position is
assumed to make an angle of 20 degrees with the yz-plane.
What I have done is to calculate the synodic effect for three different solar
positions, all with the above constraint.
In the first geometry the sun is assumed to be (as viewed from the satellite)
in the orbital plane, at a position of 
and 
.
The second geometry has the sun perpendicular to the satellite's velocity
vector (the y-axis) with 
and 
.
In this case the satellite is illuminated 'from the side'.
The third geometry is an intermediate between these two extreme geometries.
See also the figure above for details on the geometries.
Below you can find three plots. The x-axis of each plot is the 'right ascension', the y-axis is the 'declination'. For a certain direction of the rotation axis (given by x and y on the plot) the plot represents the (signed) synodic effect as defined above.
The three plots were calculated for the following parameters : v = 0.5 degrees/ s and P = 20 seconds. (i.e. flashes about every 10 seconds)
The plots are rectangular, which means they are very distorted at the poles (similar to the distortion a Mercator projection undergoes at the poles of a map).
Some general remarks :
1. Each of these plots has 'weird points' where the synodic effect is maximal in absolute value and changes sign very fast. The position of these weird points changes slightly for the various solar positions and is defined by the bisectrix between the sat-sun and sat-obs vector (x-axis). The 'weird' points are the same that were responsible for e.g. Walter Nissen's 91- 29 B and Mike McCants' 92- 38 B observations.
2. For most directions of the rot. axis the synodic effect is of the order 1 to 2 %. This makes 0.1 to 0.2 seconds on a 10 second period, certainly more than our 'measuring accuracy'.
3. The solution where the satellite is illuminated from the side (geometry 2) shows larger synodic effect values (on average). In fact, passes where the sun illuminates the satellite from the side have (on average) synodic effects twice as much as passes for which the sun is in the orbital plane. Since the sun is usually in the west or east for 'twilight' observations, this means that north-south passes will show a larger synodic effect than east-west passes. Since there are not that many low-inclination satellites, this also means that people at low latitudes will usually suffer more from the synodic effect than people at higher latitudes. Of course, in summer satellites can be seen practically all night, and the sun's direction can in those cases also be north (or south on the southern hemisphere). In those cases east-west passes will show more synodic effect than north-south passes.
4. The solution where the sun is 'in' the orbital plane shows the smallest synodic effect values on average (of all sun geometries). There are always weird regions where the synodic effect for this geometry is larger than the average se for the perpendicular direction, but those regions are limited.
5. These plots remain the identical for cases where the product of v and P remains constant, i.e. v P = constant. If you halve v and double P you get the same synodic effect, i.e. the same plots.
6. Within the bounds of the assumptions made, the scaling of the results is
straightforward. Say : 
and 
, where
and 
are the values that I used for the plots above.
Then the plots scale as 
.
E.g. we halve v (
) and triple Prot (
), then
all labels on the contour lines should be multiplied with 1.5
7. For all sun geometries the regions around declination 
have the
maximal 'normal'
synodic effect, i.e. the largest synodic effect in 'not-weird' regions of
the sky. This result makes sense intuitively as well.
Geometry 2 is equivalent to a northbound pass in the morning or a southbound pass in the evening. The results for other geometries (e.g. northbound pass in the evening) can be deduced from the Geometry 2 plot. I have not made any attempt to be politically correct in a hemisphere sense, i.e. all sentences with 'south', 'east', etc... in them are valid for the northern hemisphere only.
I hope these plots clarify the synodic effect a bit. I welcome suggestions and comments.
Bart De Pontieu