Orbital Speed Calculator: v = sqrt(GM / r)

How fast a satellite must move to stay in a circular orbit.

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

  • v = \sqrt{\frac{GM}{r}}
  • r = \frac{GM}{v^2}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the orbital speed: orbital speed = (sqrt(G) x mass of the planet^0.5) / (orbit radius, from the centre of the planet^0.5)
  2. Put the values in (every value in its SI unit): orbital speed = (sqrt(G) x 5.972e+24^0.5) / (6.771e+6^0.5)
  3. The answer: orbital speed: 7672.49 m/s
  4. The same answer in other units: 7.67249 km/s; 27621 km/h; 17162.9 mph

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? Mass of the planet; Orbital speed (leave it as it is if this is what you are finding) 7672.6 m/s; Orbit radius, from the centre of the planet (leave it as it is if this is what you are finding) 6771000 m.

  1. The formula, arranged to find the mass of the planet: mass of the planet = ((orbital speed x orbit radius, from the centre of the planet^0.5) / (sqrt(G)))^2
  2. Put the values in (every value in its SI unit): mass of the planet = ((7672.6 x 6.771e+6^0.5) / (sqrt(G)))^2
  3. The answer: mass of the planet: 5.97217e+24 kg

Common questions

What is the speed of a satellite in low Earth orbit?

About 7.7 km/s, which is about 27 600 km/h, for an orbit 400 km up.

Why does a higher orbit have a lower speed?

Gravity is weaker further away, so less speed is needed to balance it with the circular motion.

How is it related to escape velocity?

The escape velocity is the square root of 2 times bigger than the circular orbit speed at the same radius.

Related calculators

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

Further reading: Any physics text (circular orbits).

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