Kepler Third Law Calculator - Period, Axis, and Mass Solver

The Kepler third law calculator is a focused tool for T^2 = 4 pi^2 a^3 / (G M). It computes the orbital period around a central body, or works in reverse to recover a missing semi-major axis or central mass. It serves students and anyone checking a textbook.

• • Homework checks: Verify a worked problem that gives a period or semi-major axis for a planet, satellite, or moon.
• • Sun-mass derivation: Recover the Sun's mass from Earth's year and one astronomical unit.
• • Satellite orbit sizing: Estimate the period or radius of a satellite around Earth, Mars, or the Moon.
• • Binary and exoplanet estimates: Use a custom central mass for comparable-mass systems.

Kepler's third law states that the square of a planet's orbital period is proportional to the cube of its semi-major axis. Newton later derived the proportionality constant, and physics problems and astronomy references use that Newton-derived form.

The calculator uses SI units internally so a single equation covers Earth satellites, the Moon, planets, and binary star systems. Displayed units are converted for readability: years and AU for solar system problems, days and hours for satellites.

For a direct T = 2 pi sqrt(a^3 / mu) solver with central body presets and altitude mode, Orbital Period Calculator offers the time-side companion to this law.

The Kepler third law calculator applies Newton's derivation. The two-body form uses G, M, m, and a to compute T. The selected mode controls which of T, a, or M is the unknown. The Kepler third law calculator supports three modes so the same form covers any textbook problem.

• T: Orbital period of the small body, in seconds.
• a: Semi-major axis, in meters. For circular orbits, this is the orbital radius.
• G: Newtonian constant, 6.6743 x 10^-11 m^3 / (kg s^2) by CODATA 2018.
• M: Mass of the central body, in kg.
• m: Mass of the orbiting body, in kg. Use 0 for planet-Sun systems.

The period mode computes T from a, M, m, and G. The axis mode solves a^3 = G (M + m) T^2 / (4 pi^2). The mass mode solves M = 4 pi^2 a^3 / (G T^2) - m. All three paths return the same T^2 / a^3 constant.

Preset masses come from NASA JPL's parameters for the Sun, Earth, Moon, Mars, and Jupiter. The custom option leaves the central mass editable for exoplanet host stars or binary systems.

According to NASA Science, the square of a planet's orbital period is proportional to the cube of the semi-major axis, the empirical form Newton derived from the law of gravitation.

The derivation of T^2 = 4 pi^2 a^3 / (G M) starts with Newton's second law, and Forces Newtons Laws Calculator works the force side of the same family of equations.

Four ideas sit underneath every calculation. They explain why the same equation fits a 90-minute Earth orbit and a 12-year Jupiter orbit.

All four concepts appear in the worked examples. Earth's year produces a T^2 / a^3 around 2.974 x 10^-19 s^2/m^3, and Mars's year at 1.523679 AU matches.

The semi-major axis is the only distance the formula needs. Eccentricity changes speed and distance during the orbit, not the total time.

Sidereal periods from Kepler's law and synodic periods from two-body alignment are different views of the same orbit timing, and Synodic Period Calculator handles the alignment side.

Pick the variable the problem gives you, then fill in the rest. The tool handles unit conversion and updates the panel as you type.

1. 1 Select the mode: Choose Period when you know a, M, m, and G. Pick Semi-major axis when you know T, M, m, and G. Pick Central body mass when you know T and a.
2. 2 Pick a central body preset: Sun for solar system planets, Earth for satellites, Moon for lunar orbiters, Mars for missions, Jupiter for its major moons, or Custom.
3. 6 Read the result and the T^2 / a^3 check: The primary result reflects the selected mode. The T^2 / a^3 constant confirms that two orbits share the same central body.
4. 4 Enter the period (for axis or mass mode): When solving for a or M, enter the known period in seconds. The display also shows days, hours, and years.
5. 5 Add the orbiting mass when relevant: Set m to 0 for planet-Sun problems. Enter the satellite's mass for low Earth orbits, or a comparable body mass for binaries.
6. 6 Read the result and the T^2 / a^3 check: The primary result reflects the selected mode. The T^2 / a^3 constant helps confirm that two orbits share the same central body.

Try a textbook exercise: choose Mode = Period, Central body = Sun, and enter a = 1.495978707 x 10^11 m. The result should be about 3.1554 x 10^7 s, or roughly 365.25 days. Switch to Mars and keep a at 1.524 x 1.495978707 x 10^11 m. The T^2 / a^3 should match Earth, while the period grows to about 687 days.

For a near-surface classroom timing example rather than a celestial orbit, Pendulum Period Calculator compares length and gravity in a small-angle setup.

The tool is short to use and explicit about inputs and units, which makes it useful for several tasks.

• • Two-way and three-way calculation: Solve for the period, semi-major axis, or central body mass without retyping a formula.
• • Universal two-body coverage: Earth satellites, lunar orbiters, solar system planets, and binary star problems use the same equation.
• • Direct unit handling: Inputs are accepted in meters, kilograms, and seconds, with the result panel showing years, days, AU, and solar masses.
• • Verification constant on screen: The displayed T^2 / a^3 lets you confirm that two orbits are around the same central body.
• • Audit trail of assumptions: Presets are visible, G is editable, and the orbiting mass field makes clear when the reduced mass correction is or is not needed.

The Kepler third law calculator is also useful for sanity checks. If a question gives a period in days and a semi-major axis in millions of km, the result panel surfaces the corresponding SI values, so it is obvious whether the inputs were entered with the right scale.

When a problem gives an unusual unit, the editable mass, period, and semi-major axis fields accept any value once the preset is set to Custom, so the same interface handles an Earth satellite at 400 km altitude or an exoplanet at 5 AU.

For timing problems near massive bodies where the orbit itself is not the variable, Gravitational Time Dilation Calculator covers a different timing concept.

Four factors change the period, axis, or mass that the Kepler third law calculator returns. Knowing them helps you decide whether a small disagreement with a published table is meaningful.

• • The two-body model assumes a point mass or spherical central body and a low-mass orbit. Real orbits around an oblate planet or in a multi-body system can drift.
• • The CODATA value of G has a relative uncertainty of about 2.2 x 10^-5. For very precise work, the uncertainty in G propagates into the result.

Preset masses use NASA JPL's published values, so switching between Sun, Earth, Moon, Mars, and Jupiter compares real bodies rather than placeholders.

Rounding causes small disagreements with mission-grade tools. The calculator returns the period to four decimal places, enough to compare against an astrodynamics reference.

According to NASA JPL Solar System Dynamics, the gravitational parameters for the Sun and planets are tracked to high precision; differences from published tables usually come from the body's reference mass rather than the formula.

According to NIST CODATA, G is 6.6743 x 10^-11 m^3/(kg s^2) in the 2018 adjustment, with a relative uncertainty of 2.2 x 10^-5.

For the conserved quantity that central-force orbits preserve, Angular Momentum Calculator works the L = m v r side of orbital motion.