Section 3.8: Satellite Dynamics
Satellites are bodies that orbit a planet or star under the influence of gravity. Their dynamics involve orbital speed, altitude, and the forces required to maintain orbit.
Key Relationships:
- Orbital speed: \( v = \sqrt{\frac{GM}{r}} \)
- Orbital period: \( T = 2\pi \sqrt{\frac{r^3}{GM}} \)
- Centripetal acceleration: \( a_c = \frac{v^2}{r} \)
Geostationary Satellites:
- Orbit period = Earth's rotation period (24 h)
- Orbit radius \( r = \sqrt[3]{\frac{GMT^2}{4\pi^2}} \)
- Appears stationary relative to Earth's surface
Example 1 — Satellite Period
A satellite orbits Earth at 2 × 10^7 m from the center. Find its orbital period.
\( T = 2\pi \sqrt{\frac{r^3}{GM}} = 2\pi \sqrt{\frac{(2\times10^7)^3}{6.674\times10^{-11} \cdot 5.97\times10^{24}}} \approx 2.24 \times 10^4 \text{ s} \approx 6.2 \text{ h} \)
Practice Problems
- Determine the orbital speed for a satellite 500 km above Earth's surface.
- Calculate the period of a geostationary satellite.
- A satellite experiences centripetal acceleration of 0.2 m/s². Find its orbital speed if orbit radius is 1 × 10^7 m.
- Determine the altitude for a satellite with a 12-hour orbit.
- A satellite's orbit radius doubles. How does its orbital speed change?