Kepler's Third Law Calculator
Calculate the orbital period from orbital radius using Kepler's third law T² = (4π²/GM)r³. Relate the orbital period and semi-major axis for planets, moons, and satellites in gravitational systems.
This free online kepler's third law calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Kepler's Third Law Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Kepler's Third Law Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Kepler's Third Law Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Kepler's Third Law Calculator when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
- •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
- •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
- •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.
About This Calculator
The Kepler's Third Law Calculator is a free, browser-based calculation tool for engineers, students, and technical professionals. Calculate the orbital period from orbital radius using Kepler's third law T² = (4π²/GM)r³. Relate the orbital period and semi-major axis for planets, moons, and satellites in gravitational systems. It implements standard formulas and supports both metric (SI) and imperial unit systems with automatic unit conversion. All calculations are performed instantly in your browser with no data sent to a server. Use this calculator as a quick reference and sanity-check tool during design, analysis, and learning. Always verify results against primary engineering references and applicable standards for any safety-critical application.
About Kepler's Third Law Calculator
The Kepler's Third Law Calculator determines the orbital period from the orbital radius and central body mass. Johannes Kepler discovered in 1619 that the square of a planet's orbital period is proportional to the cube of its semi-major axis: T² ∝ r³. Newton later showed this proportionality constant involves the central body's mass. This law enables calculating orbital periods for any satellite around any body — from the ISS around Earth to exoplanets around distant stars. It is the foundation of celestial mechanics and was historically used to weigh the Sun and planets.
The Math Behind It
Formula Reference
Kepler's Third Law
T = 2π√(r³/GM)
Variables: r = orbital radius (m), G = 6.674×10⁻¹¹, M = central body mass (kg)
Worked Examples
Example 1: Earth's Orbit
r = 1.496×10¹¹ m, M = 1.989×10³⁰ kg (Sun)
31.56 million seconds ≈ 365.3 days — one year.
Example 2: ISS Orbit
r = 6,771,000 m, M = 5.972×10²⁴ kg (Earth)
5543 seconds ≈ 92.4 minutes per orbit.
Common Mistakes & Tips
- !Using orbital altitude instead of orbital radius — add the central body's radius to the altitude.
- !Forgetting to use the correct mass — the central body, not the orbiting object.
- !Applying the simple form T² ∝ r³ when comparing orbits around different central bodies — the constant depends on M.
- !Using the wrong units — SI units (meters, kilograms, seconds) are required for the formula with G.
Related Concepts
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Frequently Asked Questions
How was Kepler's third law discovered?
Kepler spent years analyzing Tycho Brahe's precise observations of Mars. In 1619 he published the 'harmonic law' showing T² ∝ a³ for all planets. He considered this his greatest achievement, as it revealed a deep mathematical harmony in planetary motion.
Can we use this to weigh the Sun?
Yes! Rearranging: M = 4π²r³/(GT²). Using Earth's orbital data: M = 4π² × (1.496×10¹¹)³ / (6.674×10⁻¹¹ × (3.156×10⁷)²) = 1.989×10³⁰ kg. This is how the Sun's mass was first accurately determined.
Does this work for elliptical orbits?
Yes, with r replaced by the semi-major axis a (the average of closest and farthest distances from the central body). The law applies to any eccentricity from circular (e = 0) to highly elongated (e → 1).
Why are higher orbits slower?
From T = 2π√(r³/GM), period grows as r^(3/2). Both longer circumference and slower orbital speed contribute: v = √(GM/r) decreases with r, and the path 2πr increases. Together, T grows super-linearly with r.