Kepler's third law calculator
For planets round the Sun, the period in years squared equals the semi-major axis in astronomical units cubed. The general form needs the mass.
AU
years
solar masses
Period for that axis
1.881 38years
- Axis for that period
- 1.523 793 AU
- Period in days
- 687.18
- Axis in kilometres
- 227 956 130
- Mean orbital speed
- 24 127.3
- Given axis in metres
- 227 987 150 000
Show the working
- Semi-major axis = 1.524
- Or a period = 1.881
- Total system mass = 1
- Period for that axis = sqrt(1.524^3 / 1) = 1.881 38
- Axis for that period = (1.881^2 * 1)^(1/3) = 1.523 79
- Period in days = 1.881 38 * 365.25 = 687.176
- Axis in kilometres = 1.523 79 * 149597870.7 = 227 956 000
- Mean orbital speed = 1.881 38 > 0 ? 2 * pi * 1.524 * 149597870700 / (1.881 38 * 31557600) : 0 = 24 127.3
- Given axis in metres = 1.524 * 149597870700 = 227 987 000 000
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- T^2 = a^3 / M, with T in years, a in AU and M in solar masses
Mars at 1.524 AU gives 1.881 years, which is exactly what it takes.
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