6.6 Gravity and orbits

Satellite Orbits

• Orbital Distance (r): Distance from the center of the Earth (or another planet) to the satellite.

• Orbital Period (t_sat): Time taken for the satellite to complete one full orbit.

Synchronous Orbit

• Definition: A specific distance (rsynchronous) where tsat = t_earth (the rotation period of the Earth).

• Constant Position: Satellite appears stationary above the same point on Earth due to synchronized periods.

• Application: Commonly used in telecommunications.

Equation for Synchronous Orbit

• Synchronous radius:
  $r_{synchronous} = <br>oot{3}{ rac{g m t^2}{4 ext{π}^2}}$

• Derived from:
  $t^2 = rac{4 ext{π}^2 r^3}{g m}$

• Impact of Distance: Increasing r results in increased t, losing sync with Earth's rotation.

Height Calculation

• Height (h):
  $h = r - r_{Earth}$

• Calculated using:

• Mass of Earth (m) = $5.97 imes 10^{24} kg$

• Gravitational constant (g) = $6.67 imes 10^{-11} m^3 kg^{-1} s^{-2}$

• Earth rotation period (t_earth) = 86400 seconds.

• Result:

  • $r_{synchronous}$ yields approx $4.22 imes 10^{7} m$.

  • Height (h) is approx $3.59 imes 10^{7} m$ or 35,900 km.

Mars Synchronous Orbit

• Task: Calculate the rotation period (t_Mars) for a satellite in synchronous orbit.

• Equation:
  $r<em>{synchronous} = oot{3}{ rac{g m t</em>{Mars}^2}{4 ext{π}^2}}$

• Velocity Equation:
  $v_{sat} = <br>oot{g m/r}$

• Isolate r using:
  $r = rac{g m}{v_{sat}^2}$

• Solve for t_Mars using values for Mars' mass and known velocity.

Final Result for Mars

• Calculated t_Mars:

  • Approx $88,470 s$ or $~24.6 hours$.