# Gravitational Escape Velocity

by Ron Kurtus (revised 23 January 2018)

If an object is moving at a high enough velocity away from an astronomical object—such as a sun, planet, or moon—it can overcome the gravitational force and proceed into space. This is called the gravitational escape velocity and is the release velocity of a freely moving object that is sufficient to prevent the object from being overcome by gravitational force and falling back to the surface.

Typically, it is assumed that the object is moving in a vertical direction away from the massive body. However, when it is moving at an angle to the body, the escape velocity is changed.

Although the escape velocity can be considered with respect to the center of mass (CM) between the objects, it is usually measured with respect to the larger of the two objects. Also, the mass of the escaping object is considered as much less than the mass of the attracting object.

A simple equation provides the escape velocity as a function of the initial separation of the objects and the mass of the larger body. There are several conditions for the equation to be valid.

The equation allows you to calculate the escape velocity from any celestial body, provided you know the body's mass and radius, as well as the altitude of the object. Applications of the equation include the calculations of the escape velocity from the Earth, Moon and Sun.

Questions you may have include:

• What are the escape velocity equation assumptions?
• What is the equation for calculating the escape velocity?
• What are the escape velocities for the Earth, Moon and Sun?

This lesson will answer those questions. Useful tool: Units Conversion

## Escape velocity equation

When an object is projected at a sufficient velocity in a direction away from a much larger object at some altitude, it can escape the gravitational attraction between the two and fly off into space. This initial velocity is called the escape velocity.

The direction of the velocity can be in the radial or vertical direction, in the tangential direction or directions in between, as long as it results in a direction away from the larger object. Considering our convention that velocity vectors moving in the opposite direction of gravitation are negative, the standard gravitational escape velocity equation is:

ve = −√(2GM/R)

where

• ve is the vertical escape velocity in kilometers/second (km/s)
• G is the Universal Gravitational Constant = 6.674*10−20 km3/kg-s2
• M is the mass of the planet or sun in kilograms (kg)
• R is the separation in km between the centers of the objects at the point of release

### Considering altitude

When considering the separation of the objects, it is often more convenient to use the radius of the larger object plus the altitude or height instead of separation. For example, let:

R = r + h

where

• r is the radius of the planet or sun and the center of the object in km
• h is the separation from the center of the escaping object to the surface of the planet or sun in km

Thus, the equation becomes:

ve = − √[2GM/(r + h)]

This is shown in the illustration below: Rocket reaches escape velocity

### Mass of object not a factor

Surprisingly, the mass of the object projected upward is not a factor in the escape velocity. But the escape velocity does depend on the mass of the body from which it is escaping.

## Conditions and assumptions

There are several other conditions or assumptions for the gravitational escape velocity equation:

• Freely moving
• High altitude necessary for escape
• Effect of planetary rotation ignored
• Effect of other objects not included

### Freely moving

Although the object or rocket may be accelerated up to the escape velocity, any means of propulsion is turned off and the object is moving freely.

### High altitude required to reach velocity

It is usually necessary for an object to reach a high altitude before it achieves the gravitational escape velocity of the celestial body.

#### Acceleration from surface

If an object is accelerated from the surface of the planet or sun, the object will often need to travel a great distance to some high altitude in order to reach a sufficient velocity to escape.

Note: Some textbooks refer to surface escape velocity. Unfortunately, that concept is incorrect, because objects of any size do not instantaneously accelerate to such a high velocity. Also, many calculations use the escape velocity from gravity, which also are measured from the surface.

#### Higher altitudes reduces required velocity

The further the object is from the center of the celestial body, the lower the required escape velocity.

In the situation of a rocket blasting off from the Earth, Moon or some planet, the point where the engines shut off and the rocket starts coasting is where the escape velocity is determined. The higher the rocket goes before the engines shut off, the lower the required initial velocity to escape.

#### Overcoming air resistance

Also, when escaping the Earth's gravitation, air resistance must be overcome. Often a rocket will first go into orbit at some high altitude—typically, around 190 km or 120 mi, where air resistance is no longer a factor. Then it will blast off into space to reach the escape velocity.

Ions and subatomic particles escaping from Sun's gravitation are usually sent upward in turbulent solar storms until they reach the escape velocity for the altitudes reached.

### Rotation not included

The effect of planet rotation and orbital motion are not figured into the escape velocity equation we are using. Those factors can decrease or increase the escape velocity but also complicates calculations.

For example, the escape velocity of an object in the direction of the rotation of the Earth is less than when not calculating the rotation. It also varies with the latitude on Earth from which a rocket is fired. That is why it is preferred to launch rockets near the Earth's equator.

### Effect of other objects not included

The effect of gravitation forces from other objects is not considered. Gravitation from the Sun on an object leaving the Earth influences the escape velocity but is not included in our simple equation.

(See Effect of Sun on Escape Velocity from Earth for more information.)

## Summary

Gravitational escape velocity is the velocity of an object that is sufficient to escape the gravitation of a much larger body, so that it flies off into space.

The escape velocity equation is:

ve = − √(2GM/R)

or

ve = − √[2GM/(r + h)]

The escape velocity equation assumes that the object is not being propelled, it is released at a high altitude and rotation of the large body is not considered. Also, gravitation from other objects is not considered.

The equation allows you to calculate the escape velocity from any planet, moon or sun. Applications of the equation include the calculations of the escape velocity from the Earth, Moon and Sun.

## Resources and references

The following resources can be used for further study on the subject.

### Web sites

What is escape velocity? - From PhysLink

Escape Velocity - From Wikipedia

Gravitation Resources

### Books

Top-rated books on Escape Velocity and Space Travel Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

## Students and researchers

www.school-for-champions.com/science/
gravitation_escape_velocity.htm

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