Black Holes

Jillian's Guide to Black Holes: Forming - Types - Outside - Inside - Finding - References - Websites

How fast do you have to go so you don't fall back down onto something?

Escape Velocity

Velescape = √(2GM/R)

G stands for the gravitational constant, 6.67 x 10-11 m3/(kg s2)
M stands for the mass of the object from which you want to escape
R stands for the radius of the object from which you want to escape

Everyday experience tells us that if we throw a ball up, it will come back down. The ball doesn't "escape" from the Earth because Earth's gravity is strong and we didn't throw the ball very quickly. The word "escape" in this case means "doesn't fall back down". What about satellites orbiting the Earth? They certainly don't fall back down (at least, hopefully they don't!). When we send satellites into orbit, we "throw" them very quickly with powerful rockets, so they're going fast enough that they orbit the Earth instead of falling back down to it.

What if you were on the Moon or an asteroid? Would you have to throw something just as fast to escape from it? The speed at which you have to move to escape from an object's gravity depends on two things: the mass and the radius of the object from which you want to escape. The mass term is in the top half of the equation, so a larger mass means a larger escape velocity; but since the radius term is in the bottom half of the equation, a smaller radius means a larger escape velocity.

Let's play with this a bit! For instance, consider two planets: one made of rock and ice and one made of iron. If the planets are the same size, then the iron planet will have more mass. You will have to throw something much faster to escape from the iron planet than from the rock and ice planet. Likewise, if the planets are the same mass, the iron planet will be smaller, so you will have to throw something faster to escape from it.

Consider something like our Sun, with 1.99 x 1030 kg (or 4.39 x 1030 pounds) of mass and a radius of 695,500 km (432,164 mi). If you were on the surface of the Sun (ouch, hot!), you would have to throw a baseball 618 km/s (about 1.4 million mph) to make it orbit the Sun. That's fast! Okay, what about a white dwarf with the mass of the Sun? A 1 solar mass white dwarf is about the size of the Earth, so the escape velocity would be 6,451 km/s (over 14 million mph)! What about a 1 solar mass neutron star --- those are only 162,931 km/s (about 364 million mph)! For the same mass, the escape velocity increases as the size decreases. When you reach the size of a black hole, the escape velocity is greater than the speed of light! This is why we call them "black": we can't see the surface because light can't escape from the surface.

Interestingly, the speed doesn't depend on the mass of the object that wants to escape. If you want to launch a rocket or a baseball into orbit around the Earth, you just have to get them moving fast enough. A rocket, however, is much heavier than a baseball, so it takes a lot of energy to get a rocket moving that fast!

As one final note, I have been giving you escape velocities from the surface of the object, keeping with the example of throwing a baseball. However, the radius term in the equation is really just the distance between the baseball and the center of the object from which you want it to escape. If you stand on the surface of the object, then you use the radius of the object. If you are 2 miles above the surface, you use the radius of the object plus 2 miles. Thus, throwing a baseball fast enough to achieve escape velocity on the surface of the Earth requires you to throw it 11.2 km/s (that's 25,053.7 mph), but from the top of the Earth's atmosphere (~480 kmor 300 mi high) )you would only have to throw it 10.8 km/s (24,158.9 mph). If we consider a rocket, at the time it shuts off its engines, it only has to be going faster than the escape velocity at its current distance above the Earth.


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