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Gravitational Escape Velocity for a Black Hole
by Ron Kurtus (revised 18 August 2010)
A Black Hole is a sun or star that has collapsed on itself, such that its gravitational field is so strong than not even light can escape its pull. Thus, 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 substitutes the speed of light for the velocity. The defining size 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. There is a mini-quiz near the end of the lesson.
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Escape velocity equation
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 equation
The escape velocity equation for a sun or star under normal conditions is:
ve = √(2GM/R)
where
- ve is the escape velocity in kilometers/second (km/s)
- G is the Universal Gravitational Constant = 6.67*10−20 km3/kg-s2
- M is the mass of the sun or star in kilograms (kg)
- R is the distance from the center of mass of the sun to the center of the object in kilometers (km)
Note: Since ve is in km/s, G is stated in km3/kg-s2 and R in km.
Black Hole equation
If the mass of the star was compressed to such a small size or high density that 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.
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 velocity equation for a Black Hole is:
ve > c (escape velocity greater than c)
or
c < √(2GM/R) (c less than escape velocity)
where c is the speed of light in a vacuum (approximately 300,000 km/s or
186,000 mi/s).
Finding the radius for a given mass
An interesting application of the escape velocity for a Black Hole is finding the radius of such an object, provided you know its mass.
Squaring both sides of the equation and rearranging the items results in the equation:
R < 2GM/c2
Substituting in values, you get:
R < 2*(6.67*10−20 km3/kg-s2)M/(3*105 km/s)2
R < (1.5*10−30)M km
Thus, given the mass, you can find the radius.
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:
Re = 2GM/c2
Black Hole and Schwarzschild radius
Black Hole with mass of our Sun
An application of the event horizon equation is if the mass of the black hole equaled the mass of our Sun (2*1030 kg). In that case, its Schwarzschild radius or event horizon would be:
Re = (1.5*10−30)*(2*1030) km = 3 km
In other words, a star with the mass of our Sun with its matter compressed to a radius of 3 km or less 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 its mass were compressed enough to make it a Black Hole.
The escape velocity equation for a Black Hole is:
c < √(2GM/R)
The Schwarzschild radius is:
Re = 2GM/c2
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Resources
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
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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