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 calculator uses AU and years; for km / seconds workflows you’ll usually want a dedicated orbital-mechanics tool.
References
How to use this calculator effectively
This guide helps you use Kepler's Third Law Calculator (orbital period and distance) in a repeatable way: define a baseline, change one variable at a time, and interpret outputs with explicit assumptions before you share or act on results.
How it works
The page applies deterministic logic to your inputs and shows rounded output for readability. Treat it as a comparison workflow: run one baseline case, adjust a single parameter, and measure both absolute and percentage deltas. If a result seems off, verify units, time basis, and sign conventions before drawing conclusions. This approach keeps your analysis reproducible across teammates and sessions.
When to use
Use this page when you need a fast estimate, a classroom check, or a practical what-if comparison. It works best for planning and prioritization steps where you need direction and magnitude quickly before investing in deeper modeling, manual spreadsheets, or formal external review.
Common mistakes to avoid
- Changing multiple parameters at once, which hides the true cause of output movement.
- Mixing units (percent vs decimal, monthly vs yearly, gross vs net) across scenarios.
- Comparing with another tool without aligning defaults, constants, and rounding rules.
- Using rounded display values as exact downstream inputs without re-checking precision.
Interpretation and worked example
Run a baseline scenario and keep that result visible. Next, modify one assumption to reflect your realistic alternative and compare direction plus size of change. If the direction matches your domain expectation and the size is plausible, your setup is usually coherent. If not, check hidden defaults, boundary conditions, and interpretation notes before deciding which scenario to adopt.
See also
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.
What should I do first on this page?
Start with the minimum required inputs or the first action shown near the primary button. Keep optional settings at defaults for a baseline run, then change one setting at a time so you can explain what caused each output change.
Observing planning tools
If you apply this formula to observations, also check solar position, moon/tide, and timing conditions.
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