Game Development Reference
In-Depth Information
mv
2
F
=
(11.16)
c
r
For a circle, the instantaneous radius of curvature,
r
, is simply the radius of the circle. For
a more complicated shape such as an ellipse, the radius of curvature will vary depending on
where on the ellipse the radius of curvature is being evaluated.
Circular Orbits
The easiest type of orbit to analyze is a circular orbit. The instantaneous radius of curvature of
the flight path is constant and is equal to the radius of the orbit. The velocity required to main-
tain a circular orbit,
v
c
, can be found from Equations (11.15) and (11.16).
m
3.985
e
+
14
e
v
=
6.67
e
−
11
=
(11.17)
c
r
r
As an example, the velocity required to maintain a 300
km
circular orbit around the earth
is 7737
m/s
, where
r
= (6356.8 + 300)
km
. Notice that the mass of the satellite does not affect the
velocity necessary to maintain a circular orbit.
Other Types of Orbits
A circle is not the only possible orbital shape. Orbits can also be in the shape of an ellipse, a
parabola, or a hyperbola. Most satellite orbits are circular or elliptical. There are some other
general ways to classify orbits. A
geosynchronous
orbit is one that is synchronized with the
rotation of the earth. A satellite in this type of orbit will pass over the same point on the surface
at the same time (or several times) per day.
A satellite placed in a
polar
orbit passes within 20
o
of Earth's poles. This type of orbit allows
observation of every point on the earth's surface and is commonly used for spy and weather
satellites. A
geostationary
orbit is a circular orbit that allows the satellite to travel at the same
rate as the rotation of the earth. TV broadcast satellites typically use geostationary orbits.
Escape Velocity
A satellite orbits a planet when the centripetal force generated by its motion around the planet
balances the gravitational pull of the planet. A planetary probe or lunar landing mission has to
break free of Earth's gravitational pull altogether. For a satellite to break free from the gravita-
tional pull of a planet, the kinetic energy of the satellite must be equal to the work performed
on the satellite by the gravitational pull of the planet.
1
mm
mv
2
=
6.67
e
−
11
1
(11.18)
1
2
r
The velocity at which the kinetic energy equals the gravitational work is called the
escape
velocity
,
v
esc
, and is a function of the mass of the planet and the distance the satellite is from
the center of the planet. For Earth, the escape velocity is the following:.
2
m
7.97
e
r
+
14
v
=
6.67
e
−
11
=
(11.19)
e
esc
r
