Question
How to calculate the distance of a satellite in a circular orbit from the center of a planet and the escape velocity from that orbit
Answers
GPT-4o mini
To calculate the distance of a satellite in a circular orbit from the center of a planet and the escape velocity from that orbit, you can follow these steps:
### 1. Distance of the Satellite from the Center of the Planet
The distance \( r \) of a satellite in a circular orbit from the center of a planet is given by:
\[
r = R + h
\]
Where:
- \( R \) is the radius of the planet.
- \( h \) is the altitude of the satellite above the surface of the planet.
### 2. Circular Orbit Velocity
For a satellite in a circular orbit, the orbital velocity \( v \) can be calculated using the formula:
\[
v = \sqrt{\frac{GM}{r}}
\]
Where:
- \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2 \).
- \( M \) is the mass of the planet.
- \( r \) is the distance from the center of the planet, as defined above.
### 3. Escape Velocity
The escape velocity \( v_e \) from the circular orbit can be calculated using the formula:
\[
v_e = \sqrt{2 \cdot G \frac{M}{r}} = \sqrt{2} \cdot v
\]
Where \( v \) is the orbital velocity found earlier.
### Summary of Steps
1. **Calculate the Distance**:
- Determine \( R \) (the radius of the planet) and \( h \) (the altitude).
- Compute \( r = R + h \).
2. **Find Orbital Velocity**:
- Plug \( r \) into the orbital velocity formula \( v = \sqrt{\frac{GM}{r}} \).
3. **Calculate Escape Velocity**:
- Use \( v_e = \sqrt{2} \cdot v \) to find the escape velocity from that orbit.
### Example Calculation
Let's say:
- Radius of Earth \( R \approx 6.371 \times 10^6 \) m
- Altitude of the satellite \( h = 500 \times 10^3 \) m (500 km)
- Mass of Earth \( M \approx 5.972 \times 10^{24} \) kg
#### Step 1: Calculate Distance
\[
r = R + h = 6.371 \times 10^6 \, \text{m} + 500 \times 10^3 \, \text{m} = 6.871 \times 10^6 \, \text{m}
\]
#### Step 2: Calculate Orbital Velocity
\[
v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24})}{6.871 \times 10^6}} \approx 7.9 \times 10^3 \, \text{m/s}
\]
#### Step 3: Calculate Escape Velocity
\[
v_e = \sqrt{2} \cdot v \approx 1.414 \cdot (7.9 \times 10^3) \approx 1.12 \times 10^4 \, \text{m/s}
\]
This gives you the distance from the center of the Earth and the escape velocity from the orbit at the given altitude.
### 1. Distance of the Satellite from the Center of the Planet
The distance \( r \) of a satellite in a circular orbit from the center of a planet is given by:
\[
r = R + h
\]
Where:
- \( R \) is the radius of the planet.
- \( h \) is the altitude of the satellite above the surface of the planet.
### 2. Circular Orbit Velocity
For a satellite in a circular orbit, the orbital velocity \( v \) can be calculated using the formula:
\[
v = \sqrt{\frac{GM}{r}}
\]
Where:
- \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2 \).
- \( M \) is the mass of the planet.
- \( r \) is the distance from the center of the planet, as defined above.
### 3. Escape Velocity
The escape velocity \( v_e \) from the circular orbit can be calculated using the formula:
\[
v_e = \sqrt{2 \cdot G \frac{M}{r}} = \sqrt{2} \cdot v
\]
Where \( v \) is the orbital velocity found earlier.
### Summary of Steps
1. **Calculate the Distance**:
- Determine \( R \) (the radius of the planet) and \( h \) (the altitude).
- Compute \( r = R + h \).
2. **Find Orbital Velocity**:
- Plug \( r \) into the orbital velocity formula \( v = \sqrt{\frac{GM}{r}} \).
3. **Calculate Escape Velocity**:
- Use \( v_e = \sqrt{2} \cdot v \) to find the escape velocity from that orbit.
### Example Calculation
Let's say:
- Radius of Earth \( R \approx 6.371 \times 10^6 \) m
- Altitude of the satellite \( h = 500 \times 10^3 \) m (500 km)
- Mass of Earth \( M \approx 5.972 \times 10^{24} \) kg
#### Step 1: Calculate Distance
\[
r = R + h = 6.371 \times 10^6 \, \text{m} + 500 \times 10^3 \, \text{m} = 6.871 \times 10^6 \, \text{m}
\]
#### Step 2: Calculate Orbital Velocity
\[
v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24})}{6.871 \times 10^6}} \approx 7.9 \times 10^3 \, \text{m/s}
\]
#### Step 3: Calculate Escape Velocity
\[
v_e = \sqrt{2} \cdot v \approx 1.414 \cdot (7.9 \times 10^3) \approx 1.12 \times 10^4 \, \text{m/s}
\]
This gives you the distance from the center of the Earth and the escape velocity from the orbit at the given altitude.