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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.
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