Kepler's third law
Orbital period calculator
The bigger the orbit, the longer the trip round, and the heavier the body in the middle, the faster things circle it. Give any two of orbit size, period and central mass to find the third.
Orbital period
92.82 minutes
1 h 32 min 49 s
Semi-major axis
6,791 km
from the centre
Height above surface
420 km
mean radius
Circular orbit speed
7.66 km/s
27,581 km/h
Orbits per day
15.5
Earth days of 24 h
Worked out T = 2π √(a³ ÷ GM) = 2π √((6,791,000 m)³ ÷ 3.986 × 10¹⁴ m³/s²) = 5,569 s = 92.82 minutes. Orbit around Earth.
Kepler's third law, Newton's way
In 1619 Johannes Kepler found that the square of a planet's orbital period grows with the cube of its distance from the Sun. Isaac Newton later showed why, and added the masses: T² = 4π² a³ ÷ G(M + m). Here T is the period, a the semi-major axis (half the long width of the ellipse, the average distance), M the central mass, m the orbiting mass and G the gravitational constant.
For the Sun and the planets the calculator uses their measured GM values, which are known far more precisely than G or the masses alone. Choose the surface to enter a satellite's height; the planet's mean radius is added for you.
The speed shown is for a circular orbit of that size. An elliptical orbit with the same semi-major axis has the same period, but it speeds up near the closest point and slows down far out.
Sources
- GM values and mean radii of the Sun, planets, Moon and Pluto: NASA Planetary Fact Sheets (David R. Williams, NASA Goddard Space Flight Center).
- Newtonian constant of gravitation, G = 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻²: NIST reference on constants (CODATA).
Questions and answers
Why is a geostationary orbit 35,786 km up?
A satellite that keeps pace with Earth's spin must take one sidereal day, 23 hours 56 minutes 4 seconds, per orbit. Solving for the size gives 42,164 km from Earth's centre, or 35,786 km above the equator; the calculator's 35,793 km uses Earth's mean radius rather than its equatorial one.
Why does the Moon come out at 27.3 days?
With 384,400 km and the Moon's own mass included you get 27.28 days, close to its sidereal month of 27.32 days. The Sun's pull stretches the real orbit a little, which accounts for the small difference.
How can an orbit weigh a planet?
Measure how far a moon is from its planet and how long it takes to go round, and the law gives the total mass. Choose Central mass, enter Earth's orbit of 1 au and 365.256 days, and you get the Sun's mass to within a fraction of a percent.
Does the orbiting mass matter?
Only when it is a sizeable fraction of the central mass. A spacecraft changes nothing measurable, but Jupiter shortens its own year around the Sun by about 0.05 percent.