### Correct Option: C. Equal to the period of the earth ### Detailed Explanation: 1. **Understanding Satellite Orbits**: - A satellite is an object that orbits a planet. The "parking orbit" typically refers to a low Earth orbit (LEO) where satellites are placed temporarily before they are moved to their final operational orbit. - The period of a satellite is the time it takes to complete one full orbit around the Earth. 2. **Kepler's Third Law of Planetary Motion**: - Kepler's Third Law states that the square of the orbital period (T) of a planet (or satellite) is directly proportional to the cube of the semi-major axis (r) of its orbit: \[ T^2 \propto r^3 \] - For satellites orbiting Earth, this law can be expressed in a more specific form: \[ T = 2\pi \sqrt{\frac{r^3}{GM}} \] where: - \( T \) is the orbital period, - \( r \) is the distance from the center of the Earth to the satellite, - \( G \) is the gravitational constant, - \( M \) is the mass of the Earth. 3. **Geostationary Orbit**: - A satellite is in a geostationary orbit when it has an orbital period equal to the rotational period of the Earth (approximately 24 hours). This means it appears to be stationary relative to a point on the Earth's surface. - For a satellite to maintain a stable orbit, its period must match the rotation of the Earth. If the satellite's period is less than 24 hours, it will appear to move westward across the sky. If it is more than 24 hours, it will appear to move eastward. 4. **Parking Orbit**: - A parking orbit is often a temporary orbit where a satellite is placed before it is moved to its final orbit. In many cases, this orbit is chosen to be equal to the Earth's rotational period to facilitate easier transfer to the final orbit. - Therefore, the correct answer is that a satellite in a parking orbit has a period equal to the period of the Earth. ### Why Other Options Are Incorrect: - **Option A: Less than the period of the earth**: - If the satellite's period is less than that of the Earth, it would not be in a stable orbit relative to the Earth's surface. It would appear to move across the sky, which is not characteristic of a parking orbit. - **Option B: More than the period of the earth**: - Similarly, if the satellite's period is greater than that of the Earth, it would also not maintain a fixed position relative to the Earth's surface. It would drift eastward, which is not the behavior expected from a satellite in a parking orbit. - **Option D: The square of the period of the earth**: - This option is incorrect because the period of a satellite cannot be expressed as the square of the period of the Earth. The relationship defined by Kepler's Third Law does not support this claim, as it relates the period to the radius of the orbit, not to the square of the period. ### Summary of Key Points: - A satellite in a parking orbit has a period equal to the Earth's rotational period. - Kepler's Third Law governs the relationship between orbital period and distance from the Earth. - A geostationary orbit is a specific case where the satellite's period matches the Earth's rotation. - Incorrect options involve periods that do not maintain a stable position relative to the Earth's surface. This understanding is crucial for grasping the dynamics of satellite motion and the principles governing orbital mechanics.