Question
What is the formula to calculate the period of a satellite in a circular orbit around the Earth?
Answers
Answered by
GPT-4o mini
The formula to calculate the period \( T \) of a satellite in a circular orbit around the Earth is derived from Kepler's third law of planetary motion and can be expressed as:
\[
T = 2\pi \sqrt{\frac{r^3}{GM}}
\]
where:
- \( T \) is the orbital period (time it takes to complete one orbit),
- \( r \) is the distance from the center of the Earth to the satellite (which is the sum of the Earth's radius and the altitude of the satellite),
- \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \),
- \( M \) is the mass of the Earth, approximately \( 5.972 \times 10^{24} \, \text{kg} \).
In practical terms, if you know the altitude of the satellite above the Earth's surface, you can calculate \( r \) as:
\[
r = R_e + h
\]
where:
- \( R_e \) is the Earth's average radius (approximately \( 6.371 \times 10^6 \, \text{m} \)),
- \( h \) is the altitude of the satellite above the Earth's surface in meters.
Once you have \( r \), you can plug it into the formula to get the orbital period \( T \).
\[
T = 2\pi \sqrt{\frac{r^3}{GM}}
\]
where:
- \( T \) is the orbital period (time it takes to complete one orbit),
- \( r \) is the distance from the center of the Earth to the satellite (which is the sum of the Earth's radius and the altitude of the satellite),
- \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \),
- \( M \) is the mass of the Earth, approximately \( 5.972 \times 10^{24} \, \text{kg} \).
In practical terms, if you know the altitude of the satellite above the Earth's surface, you can calculate \( r \) as:
\[
r = R_e + h
\]
where:
- \( R_e \) is the Earth's average radius (approximately \( 6.371 \times 10^6 \, \text{m} \)),
- \( h \) is the altitude of the satellite above the Earth's surface in meters.
Once you have \( r \), you can plug it into the formula to get the orbital period \( T \).