OAS NOVEMBER 20013 ASTRONOMY EZINE VOL 2 | Page 19

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to escape the Earth’s gravitational field we have to give it a velocity.
When something is moving it has kinetic energy, and that kinetic energy is given
by

where

is the object’s velocity. The object also has gravitational potential ener-

gy, as it is in a gravitational field. The gravitational potential energy is related to the gravitational
force, it is given by

which is

Notice that the gravitational potential energy is negative, whereas the kinetic energy
is always positive. The sum of the two, the total energy
by

is given

. In order for an object to escape the gravitational pull of another object, it

needs to be able to escape to infinity. If it does not escape to infinity but to a smaller distance then,
technically, it has not escaped the gravitational field of the object.
At infinity the PE is

as we are dividing by infinity. As the object travels further and further

away from its parent body it will slow down (as it is having to do work against the gravitational
force), and so the KE will get less and less. At infinity it will be zero. So we can say that, at infinity,

. But

is constant, so it is also going to be zero any distance from the

gravitational object, including at the surface of the planet.

Let us suppose the planet has a radius of
sides can cancel giving us

, we can then write

. The

and so, rearranging, we can write

velocity is then found by writing

so finally

on both

. The escape

.

The escape velocity from Earth, a White Dwarf and a Neutron Star
The equations we have just derived allows us to calculate the escape velocity from any object. We are
going to calculate the value for the Earth, a white dwarf and a neutron star.
The escape velocity from the Earth
For the Earth, the mass is

and its radius

(note, the Earth is

not spherical, it bulges at the equator, so this is an a ??????????? ???????????????????)??()??????????????????????????????????????????$????????????????????e?()????????????????????]??????????????????????????????????????????????????????????????((????????????????????????????????????????????((??]???()???????????????????????????)Q????????????????????????????)??????????????????????????????????????????M???Q????????????????????????????)?????????????????????????????M???M??????()???????????????????????M???????()=L?i%9((0