Kepler's Third Law

Kepler's third law is a relation ship between the a satellite's period and the radius of its orbit. It can be described as the square of the satellite's period is proportional to the cube of the radius. To start off,

v = circumference of orbit/period of orbit
= (2(pi)r)/T,
where T is the period, r is the radius, and pi is 3.14....

We can also write,

v = ((GM/r))^(1/2),
where G is the gravitational constant and M is the larger mass or the mass of the object that the satellite is orbiting.

We can then write,

v =  (2(pi)r)/T = ((GM/r))^(1/2)

and we square both sides and solve for T, we will get

T^2 = ((4(pi)^2)/(GM))(r)^3,

which can be described as the square of the period is proportional to the cube of the radius.

Kepler's Second Law

Kepler's second law would be the law of areas. To start off we know that the hypothetical planet has an elliptical orbit. We define a particle's angular momentum to be

L = mrvsin (Beta),

where Beta is the angle between the position vector (of magnitude r) and velocity vector (of magnitude v). However the angle between the tangential velocity and position vector is always 90 degrees, which reduces the equation to

L = mrv

This implies that the only force on the satellite is the gravitational force, which acts directly toward the star or planet that is being orbited. This means that the satellite's angular momentum is conserved as it orbits.

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The satellite moves a small distance of (Delta)s = v(Delta)t during a small amount of time, (Delta)t. This motion creates a triangle of area, (Delta)A. This is the area that is swept out during (Delta)t. It can also be determined that the height of the triangle is h = (Delta)s(sin(Beta))

(Delta)A = (1/2)(base)(height)

= (1/2)(r)((Delta)s)(sin(Beta))

= (1/2)(r)(v)(sin(Beta))((Delta)t)

Which means the rate at which the are is swept out is

((Delta)A)/((Delta)t) =  (1/2)rvsin(Beta)

= (mrvsin(Beta))/(2m) = L/(2m)

From this we can see that the reason equal areas are swept out in equal amounts of time is because angular momentum is conserved. This means that Kepler's second law is a consequence of angular momentum.


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