∫

In order to solve this integral we make the substitutions

(Eq'n 1)

and

(Eq'n 2)

We can rewrite Equation 2 as

(Eq'n 3)

so our integral becomes

(Eq'n 4)

But we have already solved the two integrals on the right side of that equation, so we have

(Eq'n 5)

Example Application:

Time Elapsed on an Orbit

Johannes Kepler's second law of planetary motion gives us

(Eq'n 6)

in which
L=v_{0}r_{0}=vr (v_{0}
representing the velocity of an orbiting body as it passes through the orbit's
peritelion and r_{0} representing the radius of the orbit at its
peritelion), a constant of the motion, and
θ
represents the angle along the orbit through which the body has moved past
peritelion. That equation gives us the angular speed of the body on the orbit as

(Eq'n 7)

Applying the Pythagorean theorem to a description of the orbiting body's velocity gives us

(Eq'n 8)

which we can solve for dt, obtaining

(Eq'n 9)

If we apply the law of conservation of energy to a very small body revolving in the gravitational field of a much larger body of mass M, then we have

(Eq'n 10)

We can solve that equation for v^{2}
and substitute the result into Equation 9 to get

(Eq'n 11)

And finally we integrate that equation to obtain

(Eq'n 12)

Thus we can calculate the time that elapses between a body's passage through its orbit's peritelion and its passage through a point a given distance r from the orbit's prime focus. If we express that radial distance through the standard description of an orbital radius,

(Eq'n 13)

then we can calculate the elapsed time as a
function of the angular displacement of the body along the orbit.

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