Calculate Cosmic Velocities

Calculator for the first and second cosmic velocity. The first cosmic velocity is the circular velocity. This is the speed, which an object (e.g. rocket, satellite) must have to fly around a celestial body in a circular orbit. The second cosmic velocity is the escape velocity. This is the lowest speed, which an object needs to escape the gravity of a celestial body. When the speed is lower than the first cosmic velocity, the object will fall down. With a speed between the first and the second, the orbit will be an ellipse. With exactly the second cosmic velocity, the object will leave in a parabolic trajectory, with a higher speed on a hyperbolic trajectory.

The formulas are:

First cosmic velocity = √ gravitational constant * mass / radius
v₁ = √ G * M / r

Second cosmic velocity = √ 2 * gravitational constant * mass / radius
v₂ = √ 2 * G * M / r

Gravitational constant G = 6.6743 * 10^-11 m³/(kg*s²) = 0.000000000066743 m³/(kg*s²)

Example: Earth radius is 6371 km. A satellite orbiting Earth at a height of 1000 km has an orbit radius of 7361 km. Its circular velocity is 7.36 kilometers per second, its escape velocity is 10.41 kilometers per second.

The term cosmic velocity came into use in the 19th century to describe very high speeds such as those found in meteorites and comets. Today they are mainly used in connection with the mass and radius of the Earth, but they can be calculated for all celestial bodies for which these values ​​are known. For black holes, the second cosmic speed is higher than the speed of light.