Kepler’s Third Law Calculator

About the Kepler’s Third Law Calculator

The Kepler’s Third Law Calculator computes the orbital period (\( T \)) or semi-major axis (\( a \)) of an object in an elliptical orbit around a central body, such as a planet orbiting the Sun or a satellite orbiting Earth. This tool is designed for physics students, astronomers, and space enthusiasts studying orbital mechanics.

Formula used (Kepler’s Third Law):

• \( T^2 = \frac{4 \pi^2}{G M} a^3 \)
• Orbital Period: \( T = \sqrt{\frac{4 \pi^2 a^3}{G M}} \)
• Semi-Major Axis: \( a = \left( \frac{G M T^2}{4 \pi^2} \right)^{1/3} \)

Where:

• \( T \): Orbital period (s)
• \( a \): Semi-major axis (average distance from the orbiting object to the central body, m)
• \( G \): Gravitational constant (m³ kg⁻¹ s⁻²)
• \( M \): Mass of the central body (kg)
• \( \pi \): Mathematical constant (~3.14159)

This calculator is ideal for studying planetary orbits, satellite trajectories, and astrophysical phenomena.

• Features:
  • Calculates either orbital period (\( T \)) or semi-major axis (\( a \)).
  • Supports unit conversions: mass (kg, g), semi-major axis (m, km, AU), period (s, days, years), gravitational constant (m³ kg⁻¹ s⁻²).
  • Validates inputs: positive values for mass, semi-major axis, and period; numeric gravitational constant.
  • Keypad includes digits, decimal point, scientific notation (E), and negative sign (-).
  • Clear and backspace functionality, with a "Copy" button for results.
• Practical Applications: Useful in astronomy (e.g., calculating Earth’s orbital period around the Sun), space mission planning (e.g., satellite orbit design), and educational demonstrations of Kepler’s laws.
• How to Use:
  • Select whether to calculate Orbital Period (\( T \)) or Semi-Major Axis (\( a \)).
  • Enter the mass of the central body (\( M \)) and select the unit (kg, g).
  • Enter the semi-major axis (\( a \)) (m, km, AU) or orbital period (\( T \)) (s, days, years), depending on the calculation type.
  • Enter the gravitational constant (\( G \)) in m³ kg⁻¹ s⁻² (default: 6.6743E-11).
  • Use the keypad to insert digits, decimal point, scientific notation (E), or negative sign (-).
  • Click "Calculate" to compute the result, then use "Copy" to copy it.
  • Use "Clear" to reset, or "⌫" to delete the last character.
  • Share or embed the calculator using the action buttons.
• Helpful Tips:
  • Ensure mass, semi-major axis, and period are positive.
  • The gravitational constant can be positive (standard value) or negative (for theoretical scenarios).
  • Use scientific notation (E) for large/small values (e.g., 1.496E11 for 1 AU).
  • Final results are in SI units: period in seconds (s), semi-major axis in meters (m).
  • For circular orbits, the semi-major axis equals the orbital radius.
  • This calculator assumes the orbiting object’s mass is negligible compared to the central body.
• Examples:
  • Example 1: Earth’s Orbital Period around the Sun:
    • Calculate: Orbital Period (\( T \))
    • Input: \( M = 1.989E30 \, \text{kg} \), \( a = 1 \, \text{AU} \), \( G = 6.6743E-11 \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
    • Steps:
      • Convert: \( a = 1 \, \text{AU} = 1.496E11 \, \text{m} \)
      • Formula: \( T = \sqrt{\frac{4 \pi^2 a^3}{G M}} \)
      • Calculate: \( a^3 = (1.496E11)^3 \approx 3.347E33 \)
      • Calculate: \( G M = 6.6743E-11 \cdot 1.989E30 \approx 1.3271E20 \)
      • Calculate: \( \frac{4 \pi^2 a^3}{G M} = \frac{4 \cdot (3.14159)^2 \cdot 3.347E33}{1.3271E20} \approx 9.9518E12 \)
      • Period: \( T = \sqrt{9.9518E12} \approx 3.1568E7 \, \text{s} \)
    • Result: \( T \approx 3.1568E7 \, \text{s} \) (approximately 365.25 days or 1 year)
  • Example 2: Semi-Major Axis of a Geostationary Satellite:
    • Calculate: Semi-Major Axis (\( a \))
    • Input: \( M = 5.972E24 \, \text{kg} \), \( T = 1 \, \text{day} \), \( G = 6.6743E-11 \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
    • Steps:
      • Convert: \( T = 1 \, \text{day} = 86400 \, \text{s} \)
      • Formula: \( a = \left( \frac{G M T^2}{4 \pi^2} \right)^{1/3} \)
      • Calculate: \( T^2 = (86400)^2 \approx 7.4656E9 \)
      • Calculate: \( G M = 6.6743E-11 \cdot 5.972E24 \approx 3.9857E14 \)
      • Calculate: \( G M T^2 = 3.9857E14 \cdot 7.4656E9 \approx 2.9755E24 \)
      • Calculate: \( \frac{G M T^2}{4 \pi^2} = \frac{2.9755E24}{4 \cdot (3.14159)^2} \approx 7.5424E22 \)
      • Semi-Major Axis: \( a = (7.5424E22)^{1/3} \approx 4.2168E7 \, \text{m} \)
    • Result: \( a \approx 4.2168E7 \, \text{m} \) (approximately 42,168 km)
  • Example 3: Custom Orbit around a Planet:
    • Calculate: Orbital Period (\( T \))
    • Input: \( M = 1E23 \, \text{kg} \), \( a = 1E7 \, \text{m} \), \( G = 6.6743E-11 \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
    • Steps:
      • Calculate: \( a^3 = (1E7)^3 = 1E21 \)
      • Calculate: \( G M = 6.6743E-11 \cdot 1E23 \approx 6.6743E12 \)
      • Calculate: \( \frac{4 \pi^2 a^3}{G M} = \frac{4 \cdot (3.14159)^2 \cdot 1E21}{6.6743E12} \approx 5.9088E8 \)
      • Period: \( T = \sqrt{5.9088E8} \approx 2.4308E4 \, \text{s} \)
    • Result: \( T \approx 2.4308E4 \, \text{s} \) (approximately 6.75 hours)

Calculate orbital periods or semi-major axes for planets and satellites with detailed steps using this calculator. Share or embed it on your site!