Question

How to calculate the orbital period of a satellite in a circular orbit around the Earth given Earth's mass and radius

Answer 1

To calculate the orbital period of a satellite in a circular orbit around the Earth, you can use Kepler's third law, which is simplified for circular orbits. The formula for the orbital period \( T \) is given by:

\[
T = 2\pi \sqrt{\frac{r^3}{GM}}
\]

where:
- \( T \) is the orbital period,
- \( r \) is the distance from the center of the Earth to the satellite (which is equal to Earth's radius plus the altitude of the satellite),
- \( G \) is the universal gravitational constant (approximately \( 6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2} \)),
- \( M \) is the mass of the Earth (approximately \( 5.972 \times 10^{24} \, \text{kg} \)).

### Steps to Calculate the Orbital Period:

1. **Determine the Variables**:
- Identify the mass of the Earth \( M \).
- Identify the radius of the Earth \( R_E \) (approximately \( 6.371 \times 10^{6} \, \text{m} \)).
- Determine the altitude \( h \) of the satellite above the Earth's surface.
- Calculate the total distance from the center of the Earth to the satellite:
\[
r = R_E + h
\]

2. **Plug Values into the Formula**:
- Use \( r \) in the formula for \( T \).

3. **Perform the Calculation**:
- Compute \( r^3 \),
- Multiply by \( GM \),
- Take the square root, and multiply by \( 2\pi \) to find \( T \).

### Example Calculation

Assume a satellite is in orbit at an altitude of 300 km above the Earth's surface.

1. **Calculate \( r \)**:
- \( R_E \approx 6.371 \times 10^{6} \, \text{m} \)
- \( h = 300 \times 10^{3} \, \text{m} \)
- \( r = R_E + h = 6.371 \times 10^{6} \, \text{m} + 300 \times 10^{3} \, \text{m} = 6.671 \times 10^{6} \, \text{m} \)

2. **Calculate \( T \)**:
- \( G = 6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2} \)
- \( M = 5.972 \times 10^{24} \, \text{kg} \)
- Compute \( T \):
\[
T = 2\pi \sqrt{\frac{(6.671 \times 10^{6})^3}{(6.674 \times 10^{-11})(5.972 \times 10^{24})}}
\]
- Calculate \( r^3 \):
\[
r^3 \approx 2.973 \times 10^{20} \, \text{m}^3
\]
- Calculate \( GM \):
\[
GM \approx 3.986 \times 10^{14} \, \text{m}^3\, \text{s}^{-2}
\]
- Now plug in the values:
\[
T = 2\pi \sqrt{\frac{2.973 \times 10^{20}}{3.986 \times 10^{14}}}
\]

Finally, calculate \( T \) to find the orbital period in seconds. You can further convert this to minutes if needed (by dividing by 60).

### Conclusion:
This approach will give you the orbital period of a satellite based on its altitude and the parameters of the Earth.