🌍 Kepler's Third Law Calculator
Calculate orbital period or semi-major axis using Kepler's Third Law
Example: 6,871,000 m = 500 km altitude
How to Use This Calculator
Choose Calculation Type
Select whether you want to calculate orbital period from semi-major axis, or semi-major axis from orbital period.
Enter Known Value
Input either the semi-major axis (in meters) or the orbital period (in seconds) depending on your selection. The calculator assumes Earth orbit by default.
Calculate and Interpret
Click "Calculate" to get the result. The calculator uses Kepler's Third Law, which relates the square of the orbital period to the cube of the semi-major axis.
Formula
T² = (4π² / GM) × a³
Kepler's Third Law
Where:
- T = Orbital period (seconds)
- a = Semi-major axis (meters)
- G = Gravitational constant = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻²
- M = Mass of central body (kg) - Earth: 5.972 × 10²⁴ kg
- π = Pi (3.14159...)
Rearranged Forms:
- Orbital Period: T = 2π√(a³/GM)
- Semi-Major Axis: a = (GMT²/4π²)^(1/3)
Example Calculation: LEO Period
For a satellite at 500 km altitude:
- Earth radius: 6,371,000 m
- Semi-major axis: a = 6,371,000 + 500,000 = 6,871,000 m
Calculation:
T = 2π√(a³/GM)
T = 2π√((6,871,000)³ / (6.67430×10⁻¹¹ × 5.972×10²⁴))
T ≈ 5,675 seconds
T ≈ 94.6 minutes
Kepler's Third Law in Simple Terms:
The square of the orbital period is proportional to the cube of the semi-major axis. This means:
- If you double the semi-major axis, the period increases by 2√2 ≈ 2.83 times
- If you triple the semi-major axis, the period increases by 3√3 ≈ 5.20 times
- Geostationary orbit (35,786 km) has exactly 24 hours because it follows this law
About Kepler's Third Law Calculator
Kepler's Third Law Calculator uses one of the most important laws in astronomy, discovered by Johannes Kepler in 1619. This law states that the square of the orbital period is proportional to the cube of the semi-major axis. It applies to all orbits: planets around the Sun, moons around planets, satellites around Earth, and even binary stars.
When to Use This Calculator
- Satellite Design: Calculate orbital periods for different altitudes
- Mission Planning: Determine required semi-major axis for target period
- Educational Purposes: Learn about Kepler's laws and orbital mechanics
- Astronomy: Understand planetary motion and orbital relationships
- Space Station Operations: Calculate ISS orbital characteristics
Why Use Our Calculator?
- ✅ Kepler's Third Law: Accurate implementation of this fundamental law
- ✅ Bidirectional: Calculate period from axis or axis from period
- ✅ Educational Tool: Understand the relationship between period and distance
- ✅ Real-World Values: Works for actual Earth orbits
- ✅ Free to Use: No registration required
- ✅ Mobile Friendly: Works on all devices
Understanding Kepler's Third Law
Kepler's Third Law shows a specific mathematical relationship:
- Period-Semi-Major Axis Relationship: T² ∝ a³ means period squared is proportional to semi-major axis cubed
- Applies Universally: Works for all elliptical orbits, not just circular ones
- Mass Dependency: The constant of proportionality depends on the central body's mass
- Derived from Gravity: This law follows from Newton's law of universal gravitation
- Historical Significance: Kepler discovered this empirically before Newton explained it with gravity
Kepler's Three Laws
Johannes Kepler discovered three laws of planetary motion:
- First Law: Planets move in elliptical orbits with the Sun at one focus
- Second Law: A line connecting a planet to the Sun sweeps equal areas in equal times (planets move faster when closer to the Sun)
- Third Law: The square of the orbital period is proportional to the cube of the semi-major axis (T² ∝ a³)
Real-World Examples
- Low Earth Orbit: 500 km altitude → ~94.6 minute period (ISS orbit)
- Medium Earth Orbit: 20,000 km altitude → ~12 hour period (GPS satellites)
- Geostationary Orbit: 35,786 km altitude → exactly 24 hour period
- Moon: 384,400 km from Earth → 27.3 day period
- Earth around Sun: 1 AU → 1 year period
Tips for Using This Calculator
- For circular orbits, the semi-major axis equals the orbital radius
- Remember to add Earth's radius when calculating from altitude (altitude + 6,371 km)
- The law applies to any central body - for other planets, change the mass (M) value
- Geostationary orbit is special because its period matches Earth's rotation (24 hours)
- For elliptical orbits, use the semi-major axis (average of perigee and apogee distances)
Frequently Asked Questions
What is Kepler's Third Law?
Kepler's Third Law states that the square of the orbital period (T²) is proportional to the cube of the semi-major axis (a³). Mathematically: T² = (4π²/GM) × a³, where G is the gravitational constant and M is the mass of the central body.
Why is the period squared and axis cubed?
This relationship comes from the physics of gravity. The gravitational force follows an inverse square law (F ∝ 1/r²), and when combined with Newton's laws of motion, it results in the period being proportional to the 3/2 power of the distance, which gives T² ∝ a³.
Does this work for elliptical orbits?
Yes! Kepler's Third Law works for all elliptical orbits. The semi-major axis (a) is half the longest diameter of the ellipse. For circular orbits, the semi-major axis equals the radius.
Can I use this for other planets?
Yes, but you need to change the mass (M) in the formula. For orbits around the Sun, use the Sun's mass. For orbits around Mars, use Mars' mass. The calculator currently uses Earth's mass, but you can adapt the formula for other central bodies.
Why is geostationary orbit at exactly 35,786 km?
Geostationary orbit requires a 24-hour period to match Earth's rotation. Using Kepler's Third Law with T = 24 hours (86,400 seconds), we solve for a and get 42,164 km from Earth's center, which is 35,786 km above the surface (42,164 - 6,378 km Earth radius).
How did Kepler discover this law?
Johannes Kepler discovered this law empirically in 1619 by analyzing Tycho Brahe's precise observations of planetary positions. He noticed that T² ∝ a³ for all planets. Later, Isaac Newton showed that this law follows from his law of universal gravitation.