Satellite Orbits
Introduction
A geostationary satellite is one which stays in a
fixed position relative to an observer on the Earth. In other words, from
your position on the Earth, the satellite always stays in the same position
in the sky. This means that you do not have to change the position of your
satellite dish, to track the satellite, as the satellite is not moving
relative to you. There is only one orbit above the Earth that a geostationary
satellite can exist and this orbit must be circular. All other orbits,
circular or elliptical are non-geostationary. To determine the height above
the Earth where this orbit exists, consideration is given to the forces
which exist to permit the satellite to remain in a circular orbit at a
fixed height above the mean equatorial radius of the Earth. These forces
are the radial force towards the centre of the Earth due to the revolving
satellite, and the gravitational force which exists between two masses.
Both of these forces are the same. That is, the centripetal force acting
radially inward Fc, is given by:
Newtons
where ms is the mass of the satellite
(kg), w is the angular velocity in radians/s
and r is the distance of the satellite from the centre of the Earth (m).
This centripetal force is the force exerted by gravity on the satellite,
holding it in a circular orbit, that is FG , where:
Newtons
mE is the mass of the Earth (kg) and
G is the gravitational constant (m3 kg-1 s-2).
Since Fc = FG, the distance
of the satellite from the centre of the Earth, the satellite orbital radius
r, can be determined. That is:
metres
since, w = 2p
f, where f is the frequency of the satellite rotation in Hz. The mean distance
that the satellite is above the equator, that is the height of the satellite
h, is equal to the distance that the satellite is from the centre of the
Earth less the radius of the earth ae. That is:
metres
Note that the satellite radius, or the height
of the satellite above the equator is independent of the mass of the satellite
ms. Given that G = 6.670 x 10-11 m3 kg-1
s-2, mE = 5.976 x 1024 kg, or more accurately
GmE = 3.98603 x 1014 m3 s-2,
and that ae = 6.3878160 x 106 m. From these constants,
and knowing that the condition for stationarity is that the satellite must
rotate at the same speed as the Earth rotates, that is once every 24 hours
expressed in seconds, giving f = (1/(24 x 60 x 60)) Hz, the radius of the
satellite and the height of the satellite above the equator can be determined.
That is:
The Applet
The applet is designed to permit the satellite radius
to be changed and the plane of the satellite through the centre of the
Earth, with respect to the equatorial plane, to be changed. The visual
rotational speed of the satellite and the Earth can also be altered using
the first scrollbar.
Geostationary orbit
With the third scrollbar set to a satellite radius
of 42.2 x 1000 km, and the second scrollbar set for a plane of tilt of
zero degrees from the equatorial plane, a geostationary orbit can be obtained.
Setting the first scrollbar to 160, means that each frame of rotation (of
which there are 24) takes 160 ms. This frame rate gives quite a fast rotation
of both the Earth and satellite. On this setting, after the "start" radio
button has been activated there is an initial synchronization flickering.
This is not meant to be the creation of the universe. The synchronization
flickering is worst for the Netscape browser. It is recommended that to
minimize this flickering Microsoft's Internet Explorer be used. Once away,
the satellite which starts from the left-hand side of the screen, rotates
in synchronization with the Earth. As the Earth will only rotate once,
it can be noted that each time the "start" radio button is activated, the
satellite will end up back in the same position on the left-hand side of
the screen after also rotating once with the rotating Earth. That is there
has been one rotation of the satellite for one rotation of the Earth. The
satellite has been deliberately placed above Borneo and will stay there
for the entire duration of a single rotation. Increasing the value of the
first scrollbar enables the rotational motion to slow down for better inspection.
The yellow print-out on the screen indicates the height of the satellite
above the Earth. This indication should be around 35.8 x 1000 km for a
satellite radius of 42.2 x 1000 km setting of the satellite radius.
Non-Geostationary orbit of radius less than the geostationary
orbit
With the radius set for a distance less than 42.2
x 1000 km, the satellite will rotate more than once around the Earth for
a single rotation of the Earth. For example, if the radius is made 21.1
x 1000 km, that is half the radius, it is expect that there would be (1/2)-3/2
= 2.83 rotations to each Earth rotation. This follows from the equation
for the satellite radius given above, that is
giving, relative to a reference frequency fo
and radius ro, which is that of the geostationary orbit in this
case, the relationship
Similarly, to find the radius at which there would
be only two rotations of the satellite for one rotation of the Earth, use
of the following equation is used:
giving the radius as (2)-2/3 that of
the geostationary satellite radius. That is 0.63ro = 26.6 x
1000 km.
Non-Geostationary orbit of radius greater than the
geostationary orbit
With the radius set for a distance greater than 42.2
x 1000 km, the satellite will rotate less than once around the Earth for
a single rotation of the Earth. For example, if the radius is made 60 x
1000 km, that is 1.42 times the geostationary satellite radius, it is expect
that there would be (1.42)-3/2 = 0.59 of a rotation to each
Earth rotation. This can be tested using the applet and estimating where
the blue coloured satellite stops after the Earth has finished its rotation.
Non-Geostationary orbit of inclined satellite orbital
plane
By adjusting the angle, as done by sliding the second
scrollbar, the satellite's orbital plane can be rotated up to 360 degrees
from the equatorial plane of the Earth. The radius of the satellite can
also be adjusted for greater or less radius than the geostationary orbit
radius. This permits a visualization of how different orbits other than
the geostationary orbit can never be synchronized with a fixed position
on the Earth.
.
The source
code (version Rev.1 98/08/08) is available
according to the GNU
Public License.
Tony Townsend,
tonyart@ieee.org