Astronomy Principles and Practice Fourth Edition.pdf

The geostationary satellite 195
in an elliptic orbit is greatest when the satellite is nearest its primary, and because drag is proportional
to the square of the velocity, most orbital change in a revolution is caused at or near perigee. This,
together with the near-constant perigee height in a satellite orbit, is an opportunity to measure and
monitor the air-density at a particular height over many months. The many hundreds of satellites in
orbits of different perigee heights, therefore, provide an excellent system for building up knowledge
of the rate of change of air-density with height above the Earth’s surface. Daily changes, seasonal
changes and changes that occur due to complicated solar–terrestrial relationships can also be studied
in this way.
Fortunately, orbital changes due to the air-drag are different in character to those caused by the
Earth’s departure from a sphere.
The inclination and right ascension of the ascending node are unaffected; the argument of perigee
and the time of perigee passage suffer periodic changes. As we have seen before, not only do the
semi-major axis and eccentricity vary periodically but they decrease secularly. The rate of decrease
of semi-major axis, a, is best found by measuring the rate of decrease of the satellite’s period of
revolution T .
We have, from equation (13.31),
( )1
a
3 2
T = 2π
µ
where, in this case, µ = GM, M being the mass of the Earth.
Then a knowledge of T gives a value for a.
Formulas exist relating the drag force to the rate of change in the semi-major axis. Knowing the
satellite’s velocity, size and mass, a calculation provides the air-density.
14.7 The geostationary satellite
An orbit of particular interest for an artificial satellite is a circular one at a height above the Earth’s
surface of some 35 000 km.
Every orbit has its own period of revolution, T , given by equation (13.31). For a circular orbit of
an approximate height of 35 000 km, the period is 23 h 56 m , in other words 1 sidereal day. A satellite
placed in a circular equatorial orbit at such a height will, therefore, remain above a particular point
on the Earth’s equator. Such geostationary satellites are used for communications purposes, relaying
radio and television signals over most of the Earth. These now numerous satellites are said to occupy
the Clarke belt, so named after the science fiction writer, Arthur C Clarke, who first suggested the
use of stationary satellites for communication purposes. It may be noted that because of the relative
proximity of stationary satellites, their apparent position in the sky suffers from a substantial parallax.
14.8 Interplanetary transfer orbits
14.8.1 Introduction
The choice of orbits for most interplanetary probes may be readily understood using the concepts
discussed in this and the previous chapter. We assume in what follows that the planetary orbits are
circular and coplanar. As in many scientific problems, we begin with the simplest cases and gradually
progress to more complicated ones.
With chemical rockets, burning times are short, being at most a few minutes. During the ‘burn’,
therefore, it may be assumed that the change in the rocket’s situation is one of velocity. The ‘burn’, in
fact, is designed to change the orbit and does so by altering the velocity.