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Orbital and escape velocity

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Satellite in orbit

Orbital velocity of a satellite is at maximum at sea level and decreases with height.

Orbital velocity can be calculated as

vs = (g rp2 / rs)1/2                                (1)


vs = orbital velocity (m/s)

g = acceleration due to gravity (m/s2) (9.81 m/s2)

rp = radius planet (m) (earth: 6.37 106 m)

rs = radius satellite orbit (m)

Maximum velocity at sea level where radius planet equals radius orbit can be expressed as

vs_max = (g rp)1/2                                (1b)

Escape velocity where the satellite will leave its orbit and escape the planet gravity can be calculated as

vs_escape = (2 g rp)1/2                                (2)

Orbit periodic time can be expressed as

ts = 2 π (rs3 / (g rp2))1/2                         (3)


ts = orbit time (s)

Height of orbit can be calculated as

hs = rp ((g ts2 / (4 π2 rp))1/3 - 1)                     (4)


hs = height of orbit (m)

Example - Earth bound Satellites

Maximum velocity at sea level:

vs_max = ((9.81 m/s2) (6.37 106 m))1/2  

          = 7905 m/s

          = 7.9 km/h

Escape velocity at sea level:

vs_escape = (2 (9.81 m/s2) (6.37 106 m))1/2    

           = 11179 m/s

           = 11.2 km/h

Height of synchronous orbit can be calculated by rearranging (3) for orbit period 24 hour (86400 s) to 35700 km.

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