
Presented at Paris conference GLOBAL HOU in
2002
FIST
LAW
or the simplest method for obtaining distances to
satellites
Ludwik Lehman
II Liceum Ogolnokształcace in
Glogow,
Polska
[Poland]
Abstract: How to obtain (using only your fist)
approximate distances to satellites passing the sky above your
head.

When you see
satellites moving slowly across the sky, you may think how far these shining
points are. It seems exceedingly difficult to measure their distances, but as
long as you do not need to be very accurate, it is not.
What are you able to
do looking at a satellite crossing the sky? First, you can estimate the time it
needs to draw some angle. Which angle? For example, you can use the fist of your
stretched hand which covers approximately 10 degrees. If you measured the time
when the satellite was close to its highest altitude, you can now easily obtain
its distance from you. How? You need only multiply the crossing time (in
seconds) by fifty and the result will be approximately equal to the satellite
distance expressed in kilometers. If you prefer to have the distance in miles,
multiply the time in seconds by thirty. Let us resume:
Satellite distance (in kilometers) = 50 x
crossing time (in seconds)
Satellite distance (in miles) = 30 x crossing
time (in seconds) (1)
Why does it work? The distance made
by a satellite during a given time can be expressed by a well-known
formula:
Distance made by the satellite = velocity x
time (2)
We know that most of
satellites are moving around the Earth in almost circular orbits. There is only
one velocity that a satellite can have in order to remain in a circular orbit
with a fixed radius. The velocity depends on the radius of the orbit, so the
satellites moving on different orbits have different velocities. Since the orbit
radii for the low-orbit satellites are almost the same, this dependence is not
very strong.
Let us
assume that without binoculars we are able to observe satellites that are not
farther than 1000 km (600 miles). The velocity which has a satellite moving 1000
km above the surface of the Earth equals 7.36 km/s. On the other hand, such the
velocity for a satellite, which passes only 100 km over the Earth surface,
equals 7.86 km/s. We see that despite the tenfold change of the distance from a
satellite to the Earth, the velocity has changed only by about 7%. Thus, we can
put the mean value of the satellite velocity into formula (2) without a
considerable loss of the accuracy. Moreover, the velocity of the observer due to
the rotation of the Earth can be neglected, because it can reach only a few
percent of the satellite velocity. Now, if we denote the angle covered by a fist
of a stretched hand as "a", we
can write:
Tan(a) =
distance made by the satellite/distance from the satellite to observer
(3)
Then from (2) and (3) we easily
get
Distance from the satellite
to the observer = velocity x time / tan(a)
(4)
If we put the average
velocity of the satellites in low orbits and tangens of 10 degrees in formula
(4), we get approximately the formula (1). This ''fist law'' is not, of course,
very precise. But you need only loudly count the seconds when a satellite passes
your fist in a stretched hand, and will soon know whether the satellite is 100,
300 or 600 miles from you. For most of the skywatchers it is really enough. If
you want to be more precise, you can try to measure the time by stop-watch and
better determine the angle covered by your own fist. In this way you will find
your personal formula for obtaining the distances to the satellites. There are
some other ways to do this even more precisely, but this is another
story.