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# Gravitational Escape Velocity from a Black Hole

by Ron Kurtus (revised 8 March 2011)

A *Black Hole* is a very massive sun or star that has collapsed on itself, such that its gravitational field is so strong than not even light can escape its pull. It is called a "black hole" because that is how it appears to telescopes.

The equation for the gravitational escape velocity of a Black Hole uses the speed of light as the highest possible velocity for material trying to escape the star. The defining separation from the center of a Black Hole is called the *event horizon* or *Schwarzschild radius* and can be determined from the escape velocity equation.

You can also use the equation to calculate the size of the Sun if its mass were compressed enough to make it a Black Hole.

Questions you may have include:

- What is the escape velocity equation for a Black Hole?
- What is the Schwarzschild radius of a Black Hole?
- How big would a Black Hole be if it had the mass of our Sun?

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

## Black Hole escape velocity

The equation for the escape velocity of a Black hole is obtained by substituting the speed of light in the standard escape velocity equation.

### Standard escape velocity equation

The standard escape velocity equation of a small object from a celestial body is:

v_{e}= − √(2GM/R)

where

**v**is the escape velocity in kilometers/second (km/s)_{e}**G**is the Universal Gravitational Constant = 6.674*10^{−20 }km^{3}/kg-s^{2}**M**is the mass of the sun or star in kilograms (kg)**R**is the separation between the center of the sun and the center of the object in km

Note:vis negative, since the direction is away from the center of the celestial body. Also, since_{e}vis in km/s,_{e}Gis stated in km^{3}/kg-s^{2}andRin km.

### Black Hole equation

If the mass of the star was compressed to such a small size or high density that the magnitude of the escape velocity was greater than the speed of light, any particles or objects projected upward from its surface could not escape the gravitational pull:

s_{e}> c

where

**s**is escape speed or magnitude of_{e}**v**_{e}-
**c**is the speed of light in a vacuum = 2.997*10^{5}km/s

Einstein proved in his *General Theory of Relativity* that light is affected by gravitation. This means that even light or electromagnetic waves could not escape from a Black Hole.

Thus, the escape speed equation for a Black Hole is:

s_{e}= √(2GM/R) > c

This equation means that when the values of **M** and **R** for a celestial object are such that **√(2GM/R)** is greater than the speed of light, nothing can escape the body—not even light.

## Finding the radius for a given mass

An interesting application of the Black Hole escape velocity equation is finding the radius of the body, provided you know its mass. Squaring both sides of the equation and rearranging the items:

c^{2}< 2GM/R

Rc^{2}< 2GM

This results in the equation:

R < 2GM/c^{2}

Substituting in values for **G** and **c**, you get:

R <2*(6.674*10^{−20}km^{3}/kg-s^{2})*M/(2.997*10^{5}km/s)^{2}

R <(1.486*10^{−30})*Mkm

Thus, given the mass, you can find the required radius for the body to be a Black Hole.

### Relativity and Schwarzschild radius

Although, our escape velocity equation for a Black Hole given is based on the classical equations and not the relativistic, it is still valid.

In 1916, scientist Karl Schwarzschild derived what is called the *Schwarzschild radius* from Einstein's gravitational field equations in the General theory of Relativity. The Schwarzschild radius represents the *event horizon* of a Black Hole or the limiting radius where nothing can leave. Its equation is:

R_{S}= 2GM/c^{2}

In the explanation of Overview of Gravitational Escape Velocity, it was shown that at greater altitudes, the required escape velocity decreases. The Schwarzschild radius represents a limiting separation from the center for light to escape from a Black Hole. At separations less than this radius, the required escape velocity is *greater* than the speed of light.

Black Hole and Schwarzschild radius

## Black Hole with mass of our Sun

One application of the event horizon equation would be if the mass of the black hole equaled the mass of our Sun (1.988*10^{30} kg). In that case, its Schwarzschild radius or event horizon would be:

R(1.486*10_{S}=^{−30})*(1.988*10^{30}) km = 2.954 km

In other words, a star with the mass of our Sun with its matter compressed to a radius of less than 2.954 km would be a Black Hole, because the escape velocity would be greater than the speed of light.

## Summary

A Black Hole has a gravitational field is so strong than not even light can escape its pull. The gravitational escape velocity equation for a Black Hole substitutes the speed of light for the velocity.

The event horizon or Schwarzschild radius is the defining size of a Black Hole and can be determined from the escape velocity equation. You can use the equation to calculate the size of the Sun if it's mass were compressed enough to make it a Black Hole.

The escape speed equation for a Black Hole is:

s_{e}= √(2GM/R) > c

The Schwarzschild radius is:

R_{S}= 2GM/c^{2}

Be friendly to those around you

## 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

### Books

You can purchase these books in your local bookstore or through Amazon.com.

**Top-rated books on Escape Velocity and Space Travel**

## Questions and comments

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## Gravitational Escape Velocity from a Black Hole