Orbital Period Calculator

How to use this calculator

1. Choose a reference central body. For star‑centric problems, treat the Sun as 1 on the mass scale; for planets or other central bodies, you can also treat them as 1 and scale others accordingly.
2. Enter the orbital semi‑major axis in astronomical units (AU). For example, Earth is at 1 AU, Mars is at about 1.52 AU, and many hot Jupiters sit well inside 0.1 AU.
3. Enter the central mass as a relative factor. In many textbook problems you can leave this at 1 for a Sun‑like star; for a star twice as massive, you might enter 2, and so on.
4. The calculator raises the semi‑major axis to the 3/2 power to model the a³ relationship and divides by the square root of the central mass to adjust the timescale.
5. Review the resulting orbital period in years and days. Compare to known examples—like Earth’s 1‑year orbit—to sanity‑check your inputs.
6. Experiment with different semi‑major axes and central masses to see how quickly the period lengthens as you move farther out, or shortens around more massive stars.

Inputs explained

Semi-major axis (AU)

The average orbital radius measured along the semi‑major axis of the ellipse, expressed in astronomical units. One AU is approximately the mean distance between Earth and the Sun (~1.496×10¹¹ m). For circular or nearly circular orbits, this is very close to the orbit radius.

Central mass (Earth = 1)

A relative mass factor for the body being orbited. In many simple Kepler‑law problems, you can think of 1 as representing a reference central mass that makes a 1 AU orbit correspond to a 1‑year period. Increasing this value models a more massive central body, which shortens the orbital period for a fixed semi‑major axis; decreasing it models a lighter central body, which lengthens the period.

When to use it

• Estimating exoplanet orbital periods from published semi‑major axes and approximate stellar masses to understand how “years” on those worlds compare to Earth’s year.
• Checking astronomy and astrophysics homework problems that use Kepler’s third law to connect semi‑major axis and orbital period around Sun‑like or scaled stars.
• Exploring how moving a hypothetical planet inward or outward in a star system changes its year length without dealing directly with full SI constants.
• Performing quick what‑if analyses, such as how a more massive star affects orbital periods at the same distance compared with a less massive one.
• Teaching Kepler’s third law relationships in a classroom setting, where students can manipulate the semi‑major axis and central mass to see how period scales with a³ᐟ² and 1/√M.

Tips & cautions

• Keep a mental model of the basic scaling: doubling the semi‑major axis more than doubles the period (because of the 3/2 power), while doubling the central mass reduces the period by about 30% (because of the square root).
• Use known Solar System bodies as benchmarks: Earth at 1 AU and mass = 1 gives 1 year; Mars at about 1.52 AU gives a period near 1.9 years in simple models.
• For exoplanet work where stellar mass is given in units of solar masses, you can often treat the Sun as 1 and scale other stars’ masses accordingly, then plug that ratio into the central mass field.
• Remember that this calculator is based on a simplified Keplerian model. Real systems can deviate slightly due to eccentricity, additional bodies, and detailed mass distributions.
• If you need very precise orbital periods, especially for mission design or detailed ephemerides, use more complete gravitational models or published orbital data instead of this approximate scaling.
• Uses a simplified Kepler’s third law scaling with semi‑major axis and a relative central mass factor rather than the full Newtonian gravitational constant formulation.
• Assumes that the orbiting body’s mass is negligible compared to the central mass, so the simple period–axis–mass relationship holds.
• Treats the orbit as effectively Keplerian with modest eccentricity; highly eccentric orbits with strong speed variation near periapsis are not explicitly modeled in their time distribution.
• Ignores perturbations from other bodies, general relativistic corrections, and non‑spherical mass distributions—which can matter for precise work near massive or rapidly rotating objects.
• Intended for conceptual learning, homework checks, and quick order‑of‑magnitude estimates, not for spacecraft navigation or professional orbital dynamics.

Worked examples

Deep dive

FAQs

What exactly do the mass units represent?

The central mass is a relative factor chosen so that 1 AU around a central mass of 1 has a 1-year period. In many cases, you can think of 1 as a Sun-like star and scale other stars accordingly. The model does not directly use kilograms—it uses this convenient scaling for quick estimates.

Can I use this for orbits around planets instead of stars?

Yes, as long as you are consistent. You can treat your planet as the reference mass of 1 and scale other planets or central bodies relative to it. Just remember that the period returned will still be in years and days relative to that reference scaling.

How accurate is this compared to full Newtonian calculations?

For many classroom and conceptual problems, the scaling T_years ≈ a^{3/2} ÷ √M_rel provides a good approximation and matches Solar System periods reasonably well. For high-precision work or extreme systems, you’ll want to use the full Newtonian form with G and actual masses.

Does orbital eccentricity change the period?

For a given semi‑major axis and central mass, the orbital period of a Keplerian orbit is largely determined by those two parameters alone; moderate eccentricity does not significantly change the period. However, eccentricity does change how speed varies along the orbit and how long the body spends near periapsis versus apoapsis.

Can I invert this relationship to design an orbit with a specific period?

You can rearrange the scaling relationship to solve for the semi‑major axis a given a desired period and central mass, using a ≈ (T_years² × M_rel)^{1/3}. This calculator focuses on period from distance, but you can perform this inverse calculation manually or in a separate tool.

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