Satellites in Orbit

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_p² / r_s)^1/2                                (1)

where

v_s = orbital velocity (m/s)

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

r_p = radius planet (m) (earth: 6.37×10⁶ 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_s³ / (g r_p²))^1/2                         (3)

where

t_s = orbit time (s)

Height of orbit can be calculated as

h_s = r_p ((g t_s² / (4 π² 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/s²) (6.37×10⁶ m))^1/2  

          = 7905 m/s

          = 7.9 km/h

Escape velocity at sea level:

v_{s_escape} = (2 (9.81 m/s²) (6.37×10⁶ 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 = (6.37×10⁶ m) (((9.81 m/s²) (86400 s)² / (4 π²(6.37×10⁶ m)))^1/3 - 1)

    = 35968 km