The figure illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and
plane polar coordinates (r,φ).
F = -(G*M*m)/r^2
The parametric equations for the orbit:
r(ψ) = a ( 1 − e cos ψ)
tan(φ/2) = [(1+e)/(1-e)]^(1/2)* tan(ψ/2)
t(ψ) = (T/2π) ( ψ − e sin ψ)
where ψ is the independent, parametric variable; a, e, and T are constants.
Prove, directly from the parametric equations, that the angular momentum L is
constant. Express L in terms of the constants of the orbit (and any other relevant parameters).
L = r x p
τnet = r x F
The Attempt at a Solution
I know that in order for L to be constant, its derivative (net torque) must be 0, but this isn't really getting me anywhere. I know torque = rxF and L = rmv, but quite frankly I can't seem to figure out how to do this using the parametric equations.
What I have done using the parametric equations is this:
L = m*r(ψ)*v(t) = m*(a*(1-e*cos(ψ)))*v(t)
From here I'm having a difficult time finding an expression for v(t). I tried using the relations
ω=((2*π)/T) and v = ω*r for the following work:
t(ψ) = (1/ω)*(ψ-esin(ψ))
==> ω = (ψ-esin(ψ))/t(ψ)
==> ω = (ψ-esin(ψ))/((T/(2π))*(ψ-esin(ψ))
==> ω = (2π/T) (this obviously backtracked)
==> v = r*(2π/T)
∴L = m*(a*(1-e*cos(ψ)))*(a*(1-e*cos(ψ)))*(2π/T)
or L = m*a2*(1-e*cos(ψ))2*(2π/T)
I'm pretty sure this isn't right because it isn't going to be constant for different r values. Any help would be appreciated.
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