Section 2.3: Planetary Motion
Planetary motion describes how planets move around the Sun. The motion obeys Newton’s law of gravitation and Kepler’s laws, resulting in elliptical orbits and varying orbital speeds.
\( F = G \frac{M m}{r^2} \)
- Planets move in ellipses with the Sun at one focus.
- Line joining planet and Sun sweeps equal areas in equal time.
- Square of orbital period \( T^2 \propto r^3 \), where \( r \) is mean orbital radius.
Example 1
A planet orbits a star at a distance of 1.5×1011 m with a mass of 6×1024 kg. Find the gravitational force acting on it if the star has mass 2×1030 kg.
\( F = G \frac{M m}{r^2} = 6.674\times10^{-11} \frac{2\times10^{30} * 6\times10^{24}}{(1.5\times10^{11})^2} \approx 3.56 \times 10^{22} \text{ N} \)
Example 2
Find the orbital period of a satellite orbiting Earth at 7000 km from its center.
Orbital speed: \( v = \sqrt{\frac{GM}{r}} \)
Orbital period: \( T = \frac{2 \pi r}{v} = 2 \pi r \sqrt{\frac{r}{GM}} \)
Substitute \( r = 7\times10^6 \) m, \( M = 5.97\times10^{24} \) kg, \( G = 6.674\times10^{-11} \)
\( T \approx 5820 \text{ s} \approx 1.62 \text{ hours} \)
Practice Problems
- Calculate the gravitational force between Earth and Moon.
- A satellite orbits at 300 km above Earth. Compute its orbital period.
- Using Kepler’s 3rd law, find the period of a planet at twice Earth’s orbital radius.
- Find the velocity of a planet orbiting at 1 AU around a star of 2×1030 kg.
- Compute centripetal acceleration of Earth in its orbit around the Sun.
- A satellite has mass 500 kg at 1000 km altitude. Find orbital speed.
- Determine radius of circular orbit for a 1000 kg satellite with 8×106 N centripetal force.
- Using Kepler’s 2nd law, compute area swept by planet in 30 days if orbit radius is 1.5×1011 m.
- A planet has orbital period 687 days. Find its mean orbital radius.
- Calculate gravitational force on a 1000 kg satellite orbiting at 5000 km from planet center.