Communication satellites are placed in a geosynchronous orbit, i. e., in a circular orbit such that they complete one full revolution about the earth in one sidereal day (23.934 h), and thus appear stationary with respect to the ground. Determine the altitude of these satellites above the surface of the earth in both SI and U. S. customary units.

Explanation:

Let the radius of orbit of geostationary satellite be R .

Time period of satellite = 2πR / V₀ where V₀ is orbital velocity

T = 2πR / √gR

T= 2πR / √(GM / R )

T = 2πR¹°⁵ / √GM  

R¹°⁵ = T x √GM  / 2π

T = 23.934 h = 23.934 x 60 x 60 s = 86126.4 s

R¹°⁵ = 86126.4 x √ ( 6.67 x 10⁻¹¹ x 5.972 x 10²⁴ )  / 2π

= 86126.4 x √ ( 398.33 x 10¹²  )  / 2π

= 86126.4 x 19.95 x 10⁶  / 2π

= 273.428 x 10⁹

R = 42.92 x 10⁶ m

= 42920 km

Radius of orbit = 42920 km

radius of earth = 6370 km

Altitude of satellite = 42920 - 6370 = 36550 km .

In US customary unit = 36550 x 10³ /.9144 yards

= 36550 x 10³ /(.9144 x 1760 ) miles

= 22771 miles .

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