# Escape velocity

In Astronautics, when a spacecraft (or any celestial body) is in the region gravitationally dominated by a celestial body (such as a star), it will either orbit the celestial body in an elliptical orbit or, if it is moving fast enough, away from the celestial body (possibly after first passing by it) and never return unless a third body exerts some force on it. (Ignoring the trivial case where the spacecraft's movement is aimed sufficiently close to the celestial body to crash into it.) Escape velocity, also known as parabolic velocity, refers to the precise speed at which a spacecraft must be moving so that it will escape rather than enter an elliptical orbit irrespective of which direction it is moving in. The further the spacecraft is from the celestial body, the lower the escape velocity.
An object moving at exactly escape velocity moves in a parabolic trajectory and an object moving faster than escape velocity moves in a hyperbolic trajectory. The escape velocity at any distance ${\displaystyle R}$ from the centre of a spherically symmetric celestial body with gravitational parameter ${\displaystyle \mu }$ is given by[1]. R is often taken at the surface of the celestial body.

${\displaystyle V_{e}={\sqrt {{2\mu } \over {R}}}}$ ${\displaystyle V_{e}={\sqrt {{2GM} \over {R}}}}$

## Relation to hyperbolic velocity

A spacecraft's hyperbolic velocity is exactly zero when it is moving at escape velocity, positive when it is moving faster than escape velocity and undefined when it is moving at less than escape velocity.

## Relation to specific energy

Specific energy is defined so as to be zero when a body moves at escape velocity, negative when moving at less than escape velocity and positive when moving at more than escape velocity.

## Relation to orbital velocity

The velocity of a body Vo at a distance R of the center of gravity for a circular orbit is given by

${\displaystyle V_{o}={\sqrt {{\mu } \over {R}}}}$ ${\displaystyle V_{o}=V_{e}{\sqrt {2}}}$ ${\displaystyle V_{o}={\sqrt {{GM} \over {R}}}}$

## Values of interest for a Mars mission

Note: While most of these values are known to high precision, measurements still vary between observations and the less significant digits can change as the science advances. The table below gives the gravitational parameters to six significant digits or their full available accuracy (if less) when the source was published (2011). Anyone planning an actual or paper mission should search the literature for the most accurate and recent values.
Note also that the escape velocity does not take aerodynamic drag into account, so a body at the surface of Earth or Mars would not escape if launched at exactly escape velocity.

Central body Escape velocity at surface[2] m/s Gravitational parameter

m3/s2

Sol 617 540 1,32e20
Earth 11 180 3,98e14 1 738 100
Luna 2 370 4,90e12 6 378 100
Mars 5 020 4,28e13 3 396 200
Phobos 10,3 7,16e5
Deimos 5,3 1,05e5

paper mission should search the literature for the most accurate and recent values.

## References

1. J.R. Wertz - Orbits and astrodynamics in J.R. Wertz, D.F. Everett & J.J. Pushcell eds. Space mission engineering: The new SMAD ISBN 978-1-881883-15-9 p. 201
2. N. Sarzi-Amade - Physical and orbit properties of the sun, earth, moon, and planets in J.R. Wertz, D.F. Everett & J.J. Puschell eds. Space mission engineering: The new SMAD 2011. ISBN 978-1-881883-15-9 p. 955