Understanding Satellite Speed and Orbits

Aug 21, 2024

Calculating Satellite Speed and Orbital Period

Introduction

  • Discussion on how to calculate the speed of a satellite in a circular orbit.
  • Example: Satellite at a height of 3800 km above Earth.

Key Concepts

  • Centripetal Force: Required for circular motion, provided by gravity in this case.
  • Gravitational Force: Formula is ( F = \frac{G \cdot M \cdot m}{r^2} )
    • ( G ): Universal gravitational constant
    • ( M ): Mass of the Earth
    • ( m ): Mass of the satellite
    • ( r ): Distance between centers of Earth and satellite.

Deriving Satellite Speed Equation

  1. Set the gravitational force equal to centripetal force:
    [ \frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r} ]
  2. Cancel ( m ) and multiply both sides by ( r ):
    [ \frac{G \cdot M}{r} = v^2 ]
  3. Take the square root:
    [ v = \sqrt{\frac{G \cdot M}{r}} ]

Calculating Radius ( r )

  • Radius of Earth: ( R_e = 6.38 \times 10^6 ) m
  • Height of satellite: ( h = 3800 ) km = ( 3.8 \times 10^6 ) m
  • Total radius ( r = R_e + h = 1.018 \times 10^7 ) m.

Speed Calculation

  • Substituting values:
    • ( G = 6.67 \times 10^{-11} ) m³/kg/s²
    • ( M = 5.97 \times 10^{24} ) kg
  • Speed: ( v \approx 6254.3 ) m/s.

Calculating Period

  1. Distance traveled in one orbit: Circumference = ( 2 \pi r )
  2. Use the equation: ( D = v imes T ) to solve for period ( T ):
    [ T = \frac{2 \pi r}{v} ]
  3. Calculate ( T ): ( T \approx 10227 ) seconds.
  4. Convert to hours: ( T \approx 2.84 ) hours.

Geosynchronous Satellites

  • Definition: A satellite that remains above the same point on the Earth's equator.
  • Period of geosynchronous satellites: 24 hours.

Converting Period to Seconds

  • Period in seconds: ( 24 \times 60 \times 60 = 86400 ) seconds.

Height of Geosynchronous Satellite

  1. Relate period to radius:
    • ( r = h + R_e )
  2. Using speed equation ( v^2 = \frac{GM}{r} ):
    • Replace ( v ) with ( \frac{2 \pi r}{T} ).
  3. Derivation yields:
    [ r^{3} = \frac{G M T^{2}}{4 \pi^{2}} ]
  4. Calculate radius ( r ): ( r \approx 4.22 \times 10^7 ) m.
  5. Height: ( h = r - R_e = 3.58 \times 10^7 ) m = 35,800 km.

Speed of Geosynchronous Satellite

  • Calculate speed using the earlier derived formula:
    • ( v \approx 3072 ) m/s.

Conclusion

  • Summary of calculations:
    • Speed of satellite: 6254.3 m/s
    • Period: 2.84 hours
    • Height of geosynchronous satellite: 35,800 km
    • Speed of geosynchronous satellite: 3072 m/s.

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