# 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 *