# Orbital mechanics solved by using substitution integrals

Show the integral below can be integrated to give Equation 1

θ(r) = ∫ {[(L/r^2) dr] / [ 2μ (E+(k/r) - (L^2/ 2μr^2))]^.5} + Constant

Using :

U= L/r and r = minimum at θ = 0

Equation 1----------cos (θ) =[ (L^2/(μkr)) - 1]/ [{1+((2EL^2)/(μk^2))}^.5]

See attached file for full problem description.

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#### Solution Preview

Referring back to Eqs. (7), (8) and (9) of the solution of the previous problem (with appropriate changes in notation), you see that:

d theta/dt = L/(m r^2) (1)

dr/dt = Plus/minus [(2/m) * (E - V(r)) ]^(1/2) (2)

where V(r) = - k/r + L^2/(2m r^2) (3)

If you divide (1) by (2) you get:

d theta/dr = L/(m r^2) * [(2/m) * (E - V(r)) ]^(-1/2) =

L/r^2 * [ 2m (E - V(r)) ]^(-1/2) =

L/r^2 * [ 2m ...

#### Solution Summary

A detailed solution is given regarding orbital mechanics.