Satellites in Orbit
Orbital and escape velocity for geostationary satellites.
Orbital velocity of a satellite is at maximum at sea level and decreases with height.
Orbital velocity can be calculated as
v_{s} = (g r_{p}^{2}/ r_{s})^{1/2} (1)
where
v_{s} = orbital velocity (m/s)
g = acceleration due to gravity (m/s^{2}) (9.81 m/s^{2})
r_{p} = radius planet (m) (earth: 6.37 10^{6} m)
r_{s} = radius satellite orbit (m)
Maximum velocity at sea level where radius planet equals radius orbit can be expressed as
v_{s_max} = (g r_{p})^{1/2} (1b)
Escape velocity where the satellite will leave its orbit and escape the planet gravity can be calculated as
v_{s_escape} = (2 g r_{p})^{1/2} (2)
Orbit periodic time can be expressed as
t_{s} = 2 π (r_{s}^{3} / (g r_{p}^{2}))^{1/2} (3)
where
t_{s} = orbit time (s)
Height of orbit can be calculated as
h_{s} = r_{p }((g t_{s}^{2} / (4 π^{2} r_{p}))^{1/3}  1) (4)
where
h_{s} = height of orbit (m)
Example  Earth bound Satellites
Maximum velocity at sea level:
v_{s_max} = ((9.81 m/s^{2}) (6.37 10^{6} m))^{1/2}
= 7905 m/s
= 7.9 km/h
Escape velocity at sea level:
v_{s_escape} = (2 (9.81 m/s^{2}) (6.37 10^{6} m))^{1/2}
= 11179 m/s
= 11.2 km/h
Height of the synchronous orbit for a geostationary satellite can be calculated by using (4) for an orbit period of 24 hours or 86400 s:
h_{s} = (6.37 10^{6} m) (((9.81 m/s^{2}) (86400 s)^{2} / (4 π^{2}(6.37 10^{6} m)))^{1/3}  1)
= 35968 km
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