_v_ = (2π_r_)/(_T_),
since the distance the satellite travels to complete an orbit is 2π_r_,
and _T_ is the time of one complete revolution. Thus we have the
equality
√((_gR_²)/(_r_)) = (2π_r_)/(_T_),
and, solving for _r_, we get
_r_ = (_gR_²_T_²)/(4π²)(^⅓)
Since we are interested in a synchronous satellite, _T_ in this case
will be 24 hours. We can now find _r_ (using _g_ = 32 feet per second
per second and _R_ = 3960 miles), and then obtain the distance _r_ -
_R_, which will be 22,300 miles. By using our previous formulas, we also
can find the velocity of a satellite moving in this orbit, which will
turn out to be _v_ = 6870 miles per hour.
[Illustration: _One possible method of using synchronous satellites.
Signals from New York (N) to Paris (P) would go via satellite S₁;
signals from New York to Calcutta (C) would go via satellites S₁ and
S₂._]
_The illustration above_ gives a rough idea of how a synchronous
satellite system might be set up. Three communications satellites, _S_₁,
_S_₂, and _S_₃, are above the equator in fixed positions equal distances
apart and 22,300 miles up. Located in this manner, they would cover the
major part of the earth’s surface. From a point directly beneath it, the
distance would be 22,300 miles to a satellite; from other points the
slant range would be greater. Signals sent from, say, New York (point N)
to Paris (point P) would be reflected via satellite _S_₁. In doing this,
they would travel a total distance of about 46,000 miles. Because we
can’t send signals any faster than the speed of light (186,000 miles per
second), it would take at least a quarter of a second for a signal to go
this far. For communicating a much greater distance, say from New York
to Calcutta (point C), the signal path would use two satellites, _S_₁
and _S_₂. In this case, the total distance traveled by a signal would be
more than 90,000 miles, and the one-way time delay would be about half a
second.