How to use the orbital mechanics calculator
- Choose the central body: Earth, the Moon, Mars, Venus, Jupiter or your own values for its gravitational parameter GM and radius.
- Choose a circular or elliptical orbit.
- For a circular orbit, enter the altitude above the surface. For an ellipse, enter the lowest altitude (periapsis) and the highest altitude (apoapsis).
- Read the orbital speed and period, the escape speed at that radius and, for an ellipse, the speeds at both ends, the semi-major axis and the eccentricity.
Formula
circular speed v = square root of (GM / r) period T = 2 pi square root of (r^3 / GM)
escape speed = square root of (2 GM / r) = square root of 2 x circular speed
ellipse: a = (rp + ra) / 2 speed at distance r: v = square root of (GM (2 / r - 1 / a))
GM is the gravitational parameter of the central body in cubic kilometers per second squared, r the distance from its center (radius plus altitude), and rp and ra the closest and farthest distances. Kepler's third law says T squared over a cubed is constant for a body.
A low orbit and a long ellipse around Earth
At 400 km above Earth the distance from the center is 6,778.137 km. The circular speed is 7.66856 km/s, the period is 92.56 minutes (about 15.56 orbits a day) and the escape speed is 10.845 km/s. At 35,786 km the speed falls to 3.07466 km/s and the period is 23.934 hours, close to one rotation of Earth, which is why satellites there seem to hover.
An ellipse from 300 km to 35,786 km has a semi-major axis of 24,421.137 km, an eccentricity of 0.72654 and a period of 10.55 hours. The speed is 10.1515 km/s at the low point and 1.6078 km/s at the high point, and speed times distance is equal at both ends. Around the Moon at 100 km, the circular speed is 1.6335 km/s.
Why higher means slower
A satellite in a higher orbit moves more slowly and takes longer to go around, because gravity is weaker there and the path is longer. It needs more energy to reach but less speed to stay. Escape speed at any radius is always about 41% more than circular speed, the factor square root of 2.
In an ellipse the body speeds up as it falls toward the center and slows as it climbs away, always keeping speed times distance constant at the two ends. This is Kepler's second law in simple form.
Limits of the model
The page uses the two-body model: a point mass, no atmosphere, no other bodies and a perfectly round body. Real orbits near a planet are changed by drag, the flattening of the planet and the pull of the Moon and Sun, and an orbit that low would decay quickly in practice. The body constants are published reference values rounded for this page. For motion near the ground, see the Projectile Motion Calculator, and for the Sun's timing on Earth the Sunrise and Sunset Calculator.
More engineering tools
For wire runs, cross sections and bends, try the Wire Voltage Drop Calculator, Section Properties Calculator and Sheet Metal Bend Allowance Calculator.
Assumptions and limits
- The central body and the orbiting object are treated as point masses, and the object has negligible mass.
- Altitude is measured from the radius shown for each body (Earth uses its equatorial radius).
- There is no atmosphere, and no pull from any other body.
- Orbits are closed ellipses, so the apoapsis must be at least the periapsis.
- The example altitudes are an illustration. Replace them with your own.
Frequently asked questions
What is the gravitational parameter GM?
It is the universal gravitational constant times the mass of the body. It is known more precisely than either one alone, which is why orbit calculations use it directly.
Why does a geostationary orbit take 23.934 hours?
That is Earth's rotation period relative to the stars (a sidereal day), about 4 minutes shorter than the 24-hour solar day. A satellite with that period stays over the same spot on the equator.
Can I use this for a planet not in the list?
Yes. Choose your own values and enter the gravitational parameter in km cubed per second squared and the radius in km. The same formulas apply to any central body.
What is the eccentricity?
It describes how stretched the ellipse is: 0 is a circle and values near 1 are very long and thin. It equals (apoapsis distance - periapsis distance) divided by their sum.