Formula & Calculator
Kepler's Third Law
Relates a body's orbital period to the semi-major axis of its orbit.
Interpretation
T² = (4π²/GM) a³. Relates orbital period (T) and semi‑major axis (a). For objects orbiting a central mass M. Used in celestial mechanics and exoplanet detection.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| T | Orbital period | s |
| G | Gravitational constant | |
| M | Central mass | kg |
| a | Semi-major axis | m |
What it means
Kepler’s third law states that the square of the orbital period (T) of a planet is proportional to the cube of the semi‑major axis (a) of its orbit. The constant of proportionality depends on the central mass (M) and the gravitational constant (G). This law was derived empirically by Johannes Kepler and later explained by Newtonian gravity. It is fundamental to celestial mechanics, used to determine masses of planets, stars, and even black holes. In astronomy, it enables the calculation of orbital distances from periods (or vice versa), and is essential for understanding the dynamics of binary star systems, exoplanets, and satellite orbits. The simplified form T² = a³ (years and AU) applies to the solar system. Understanding this law is crucial for astrophysicists and planetary scientists.
Worked example
Kepler's Third Law – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| G (m³/kg·s²) | 6.67430e-11 |
| M (kg) | 1.989e30 |
| a (m) | 1.496e11 |
| Parameter | Value |
|---|---|
| G | 6.67430e-11 |
| M | 5.972e24 |
| a | 6.771e6 |
Common mistakes
- Units: T in seconds, a in metres, M in kg, G in m³/(kg·s²).
- Period squared: T² is proportional to a³ – not a².
- Orbiting mass: M is the mass of the central body (e.g., Sun) – not the orbiting object.
- Simplified form: For solar system (a in AU, T in years): T² = a³ (when M = 1 solar mass).
- Elliptical orbits: a is the semi‑major axis, not the average radius.
Applications
Kepler's third law, T² = (4π²/GM) a³, relates the orbital period (T) of a planet or satellite to the semi‑major axis (a) of its orbit, with the constant depending on the mass of the central body (M). This law is the foundation of orbital mechanics, enabling astronomers to determine the masses of planets, stars, and even black holes from the orbits of their companions. Space mission planners use it to calculate the required orbital parameters for satellites and interplanetary probes. By measuring the orbital period and distance, scientists can infer the mass of the central object, which is crucial for understanding planetary systems and stellar evolution. This law also underpins the discovery of exoplanets via radial velocity and transit timing variations. Understanding Kepler's third law is essential for any work involving celestial dynamics.
- Determination of planetary and stellar masses from orbital motion
- Design of satellite orbits and interplanetary trajectories
- Discovery and characterisation of exoplanets
- Calculation of asteroid and comet orbital elements
- Fundamental education in celestial mechanics
Frequently Asked Questions
T² = (4π² / GM) a³. It describes the relationship between a body's orbital period (T) and the semi‑major axis (a) of its orbit. It applies to any two bodies orbiting under mutual gravity and is fundamental to celestial mechanics.
- T – orbital period (time for one complete orbit).
- a – semi‑major axis of the orbit (average distance from the central body).
- G – gravitational constant (6.674×10⁻¹¹ N·m²/kg²).
- M – mass of the central body (e.g., the Sun, Earth, or a planet).
When using years for T and astronomical units (AU) for a, the law simplifies to T² = a³ (since the constants cancel in these units). This form is extremely useful for quick calculations in the solar system.
Kepler's Third Law holds for any elliptical orbit, with a being the semi‑major axis. The period depends only on the semi‑major axis, not on the eccentricity. This is a remarkable result: two orbits with the same a have the same period, regardless of how elongated they are.
Starting from Newton's law of gravitation and centripetal force for a circular orbit: m v²/a = GMm/a². With v = 2πa/T, we get 4π²a/T² = GM/a², which rearranges to T² = (4π²/GM) a³. The derivation extends to ellipses via calculus.
Using Earth's mass (M_E ≈ 5.972×10²⁴ kg), we can compute the orbital period for any given semi‑major axis. For a geostationary orbit, where T = 24 hours, solving gives a ≈ 42,164 km (about 35,786 km above Earth's surface).
Since T² ∝ a³, it follows that T ∝ a^(3/2). This means that orbital period increases faster than the size of the orbit. For instance, doubling the semi‑major axis increases the period by a factor of 2^(3/2) ≈ 2.83.
By measuring the orbital period and semi‑major axis of a satellite or planet, we can compute the mass of the central body: M = (4π² a³) / (G T²). This is how we determine masses of planets, stars, and even supermassive black holes (using orbiting stars or gas).
Kepler's Third Law relates the period to the semi‑major axis. The vis‑viva equation gives the speed at any point in an orbit: v² = GM (2/r − 1/a). They complement each other: one gives the time scale, the other gives the instantaneous velocity.
For two stars orbiting their common centre of mass, the law is modified: T² = (4π² / (G(M₁+M₂))) a³, where a is the separation between the stars and M₁+M₂ is the total mass. This allows astronomers to measure stellar masses from binary orbits.