How to use (3 steps)
- Choose whether to solve for orbital period T or semi-major axis a.
- Pick the central mass μ (in solar masses) with a preset or type it, then enter the known value (a or T).
- Press Compute to get the other quantity, see steps, and compare with the Solar System. Copy URL shares the same setup.
Inputs
Quick: defaults auto-calc Earth’s orbit so results show immediately. Calculations stay in your browser.
Typical ranges: μ ≈ 0.1–10 for many stars, a ≈ 0.01–100 AU for planetary orbits, and T from hours up to thousands of years. Extreme values may be less realistic.
Results
| Quantity | Value |
|---|
Solar System comparison (a–T log plot)
Dots show log10(a) vs log10(T) in years; Kepler’s law makes them fall close to a straight line. Your orbit is highlighted.
Top-down orbit scale
Orbits are shown as circles from above. Radii use log scaling so inner and outer planets fit in one view.
| Orbit | a (AU) | T (year) |
|---|
Calculation steps
Interpretation & worked examples
What the inputs mean
- μ is the central mass in solar masses (
μ = M / M☉). - a is the semi-major axis (in AU). For an ellipse, it is not the closest or farthest distance.
- T is the orbital period (in years).
In this tool’s units, the scaling form is T² = a³ / μ, so T = √(a³/μ) and a = (μT²)^{1/3}.
Worked examples
- Earth (default): μ = 1, a = 1 AU → T ≈ 1 year.
- Mars around the Sun: μ = 1, a ≈ 1.524 AU → T ≈ 1.88 years.
- Same orbit, smaller star: μ = 0.5, a = 1 AU → T ≈ 1.41 years (longer period around a lighter star).
Common pitfalls
- Binary systems: if the orbiting body is not negligible (binary stars), use the total mass of the system.
- Real orbits: perturbations, resonance, relativity, and strong eccentricity are not modelled here.
- Units: this page uses AU and years; for km / seconds workflows you’ll usually want a dedicated orbital-mechanics tool.
References
FAQ
What is Kepler's third law?
Kepler's third law states that for objects orbiting the same central body, the square of the period T² is proportional to the cube of the semi-major axis a³. Using the gravitational constant G and central mass M, it can be written as T² = 4π² a³ / (G M).
Why can we write T² = a³/μ?
By taking Earth’s orbit (a = 1 AU, T = 1 year) as a reference and defining the mass ratio μ = M/M☉, the constants combine so that T² = a³/μ. This calculator uses that ratio form for quick scaling.
Are the orbits really circles here?
Real planetary orbits are elliptical, but many have modest eccentricity. For learning the scaling between a and T, a circular approximation using the semi-major axis is sufficient, and this tool draws the orbits as circles.
How accurate is this model?
This tool uses an idealised Keplerian model: it assumes a single massive central body, point-mass planets, and T² = a³/μ with no relativistic effects, resonances, or strong perturbations. For Solar System–like orbits it gives good approximate periods and scales, but it is not a precise orbit integrator.
Observing planning tools
If you apply this formula to observations, also check solar position, moon/tide, and timing conditions.
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