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?

 

[Will Flannery]

The original question, 'Can time on elliptical orbit be expressed analytically?' is a little vague. I think it means can the time into an orbit, i.e. the time since the planet was at the orbit pericenter (point of closest approach) be expressed as a function of the true anomaly (the angle of the planet as measured from the foci of orbit).

Ref: 1 - https://en.wikipedia.org/wiki/Mean_anomaly
Ref: 2 - https://en.wikipedia.org/wiki/Eccentric_anomaly

From Ref. 1 the mean anomaly M is an angular expression for the time t into the orbit
M = 360/T *( t - 0), where T is the orbit period which is known.

So, if we knew the relationship between the mean anomaly and true anomaly we'd have the desired function. From Ref. 1
Mean anomaly can be calculated from the eccentricity and the true anomaly f by finding the eccentric anomaly and then using Kepler's equation. This gives, in radians:
M = atan2( ............................

The difficult question is determining the radial distance r from t. This is known as Kepler's Problem. From Ref. 2, r is related to the eccentric anomaly (see the graphical representation for a definition of eccentric anomaly) by
r = a(1 - e*cos(E) ) where a is the semi-major axis of the orbit and e is the orbit eccentricity

So we can determine t from M, and r from E.

The eccentric anomaly E is related to the mean anomaly M by Kepler's Equation
M = E( 1 -e*sin(E)), this requires a non-trivial derivation.

To determine r from t requires solving Kepler's equation for E as a function of M. Both Kepler and Newton gave numerical methods for solving the equation, with Newton's method being far superior.

There is no analytic closed form solution for Kepler's equation, Lagrange derived an infinite series solution in the 1700s, any more mathematicians have addressed the problem over the years, as covered in Solving Kepler's Equation Over Three Centuries by Colwell.