Can time on elliptical orbit be expressed analytically?

[snorkack]

Can time on elliptical orbit be expressed analytically? Which relations are capable of analytic expression?
The distance from focus can be expressed as a function of position angle θ:
r=a(1-e²)/(1+e cos θ)
The length linearly along the ellipse famously cannot be expressed analytically.
The total time spent on ellipse depends on a alone (Kepler 3rd).
If the angular speed at any r of θ were known then the angular speed at any other r would be because r∂θ/∂t=cost for any a and e.
But is there any analytic expression to find r∂θ/∂t given a and e?
Also, is there any way to find
₀^θ∫∂θ/∂t, or ₀^t∫∂t/∂θ? These are different questions because many analytic expressions cannot be reversed.

 

[anuttarasammyak]

Have you investigated Kepler’s second law of constant area speed for your problem already ?

 

[pasmith]

If your inverse square central force problem is governed by $$\ddot r - r\dot \theta^2 = -\frac{K}{r^2}$$ with $L = r^2\dot\theta$ constant and $\theta(0) = 0$ then we have $$L^2 = K\ell$$ where the semi-latus rectum $\ell$ of the elliptical orbit is given in terms of the semi-major axis $a$ and eccentricity $0 \leq e < 1$ by $$\ell = a(1 - e^2).$$ We then have $$\begin{split} Lt &= \int_0^t L\,dt \\ &= \int_0^{\theta(t)} r^2 \,d\theta \\ &= \int_0^\theta \frac{\ell^2}{(1 + e\cos \theta)^2}\,d\theta.\end{split}$$

 

[snorkack]

Found the answer to two questions (out of the three).
Kepler equation:
M=E-e*sin E
where M is mean anomaly and E is an intermediate expression called "elliptic anomaly".
E can be expressed through true anomaly and vice versa:
sin E=√(1-e²)*sin θ/(1+e*cos θ)
tan θ/2=√((1+e)/(1-e))*tan E/2
Given E, or θ, finding M is straightforward.
Finding E (and thus θ) given M is an insoluble problem after over four centuries.

 

[snorkack]

Found the answer to two questions (out of the three).
Given E, or θ, finding M is straightforward.
Finding E (and thus θ) given M is an insoluble problem after over four centuries.

The answer to third question is also easy on some reflection.
While θ as a function of M is insoluble, M as a function of θ is soluble. But since M can be expressed as a function of θ, you can express ∂M/∂θ as function of θ - you can take a derivative of every function but you cannot take an integral of every function. But since ∂θ/∂M is a simple reciprocal of ∂M/∂θ, you can calculate it, as long as you are calculating it as a function of θ rather than M (which is insoluble).

 

[hutchphd]

So do you have a result?