Orbital Period Calculator: T = 2 pi sqrt(r^3 / GM)

The time one orbit takes, from the orbit radius and the mass of the planet.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • T = 2\pi\sqrt{\frac{r^3}{GM}}
  • r = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the orbital period: orbital period = (2 pi / sqrt(G) x orbit radius, from the centre^1.5) / (mass of the planet or star^0.5)
  2. Put the values in (every value in its SI unit): orbital period = (2 pi / sqrt(G) x 6.771e+6^1.5) / (5.972e+24^0.5)
  3. The answer: orbital period: 5544.93 s
  4. The same answer in other units: 92.4156 minutes; 1.54026 hours; 0.0641775 days

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Orbit radius, from the centre; Orbital period (leave it as it is if this is what you are finding) 5546 s; Mass of the planet or star (leave it as it is if this is what you are finding) 5.972E+24 kg.

  1. The formula, arranged to find the orbit radius, from the centre: orbit radius, from the centre = ((orbital period x mass of the planet or star^0.5) / (2 pi / sqrt(G)))^0.66667
  2. Put the values in (every value in its SI unit): orbit radius, from the centre = ((5546 x 5.972e+24^0.5) / (2 pi / sqrt(G)))^0.66667
  3. The answer: orbit radius, from the centre: 6.77187e+6 m
  4. The same answer in other units: 6771.87 km

Common questions

What is Kepler's third law?

The square of the orbital period is proportional to the cube of the semi-major axis of the orbit.

How long does the space station take to orbit the Earth?

About 92 minutes, at a height of about 400 km.

How high is a geostationary orbit?

About 35 786 km above the surface, or 42 164 km from the centre of the Earth.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics text (Kepler's laws).

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