Earth Orbit Calculator

1. What is Earth Orbit Calculator?

Definition: This calculator computes the orbital speed and orbital period of a satellite in a circular orbit around Earth, based on its height above the Earth's surface.

Purpose: It is used in aerospace engineering to determine the velocity and period of satellites, which are critical for designing orbits, such as low Earth orbit (LEO) or geostationary orbit (GEO).

2. How Does the Calculator Work?

The calculator uses the following formulas:

Formulas:

Orbital Speed: \( \text{orbital speed} = \sqrt{\frac{G \cdot M_E}{(R_E + h)}} \)
Orbital Period: \( \text{orbital period} = 2\pi \sqrt{\frac{(R_E + h)^3}{G \cdot M_E}} \)

Where:

\( G \): Gravitational constant (\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \))
\( M_E \): Earth's mass (\( M_E = 5.972 \times 10^{24} \, \text{kg} \))
\( R_E \): Earth's radius (\( R_E = 6371 \, \text{km} \))
\( h \): Height above Earth's surface (m)

Unit Conversions:

Height (\( h \)):
- 1 km = 1000 m
- 1 m = 1 m
- 1 mi = 1609.34 m
- 1 ft = 0.3048 m
Orbital Speed:
- 1 km/s = 0.001 m/s
- 1 m/s = 1 m/s
- 1 mi/s = 0.000621371 m/s
- 1 ft/s = 3.28084 m/s
Orbital Period:
- 1 s = 1 s
- 1 min = 60 s
- 1 h = 3600 s
- 1 d = 24 \times 3600 s

Steps:

Enter the height (\( h \)) above Earth's surface with its respective unit.
Convert the height to meters for calculation.
Calculate the orbital speed and orbital period using the formulas.
Convert the results to the selected units for display.
Display the results with 4 decimal places.

3. Importance of Earth Orbit Calculation

Calculating orbital speed and period is crucial for:

Satellite Deployment: Determining the required velocity to maintain a stable orbit at a given height.
Orbit Design: Planning orbits for satellites, such as geostationary satellites that need a specific period to match Earth's rotation.
Space Missions: Ensuring spacecraft can achieve the desired orbit for communication, observation, or scientific research.

4. Using the Calculator

Example: Calculate the orbital speed and period for a satellite at a height of \( h = 400 \, \text{km} \) above Earth's surface.

Enter \( h = 400 \, \text{km} \).
The calculator computes:
- Convert height to meters: \( h = 400 \times 1000 = 400000 \, \text{m} \).
- Total distance from Earth's center: \( r = R_E + h = 6371000 + 400000 = 6771000 \, \text{m} \).
- Orbital speed: \( \text{orbital speed} = \sqrt{\frac{(6.67430 \times 10^{-11}) \cdot (5.972 \times 10^{24})}{6771000}} \approx 7672.6275 \, \text{m/s} \approx 7.6726 \, \text{km/s} \).
- Orbital period: \( \text{orbital period} = 2\pi \sqrt{\frac{(6771000)^3}{(6.67430 \times 10^{-11}) \cdot (5.972 \times 10^{24})}} \approx 5545.5174 \, \text{s} \approx 92.4253 \, \text{min} \).

5. Frequently Asked Questions (FAQ)

Q: What is orbital speed?
A: Orbital speed is the velocity a satellite must maintain to stay in a circular orbit around Earth at a given height, calculated as \( \sqrt{\frac{G \cdot M_E}{(R_E + h)}} \).

Q: What is orbital period?
A: Orbital period is the time it takes for a satellite to complete one full orbit around Earth, calculated as \( 2\pi \sqrt{\frac{(R_E + h)^3}{G \cdot M_E}} \).

Q: How does the calculator handle different units?
A: The calculator allows users to input height in km, m, mi, or ft, converts them to meters for calculation, and displays orbital speed in user-selected units (km/s, m/s, mi/s, ft/s) and orbital period in user-selected units (s, min, h, d).