Which version of the Kepler Problem is correct?

[Will Flannery]

TL;DR Summary

Two versions of the solution exist, r and θ as a function of time, and 1/r as a function of θ. Which one is correct?

On one hand:
wiki - Kepler Problem - There doesn't seem to be a clear statement of what the problem is. There is a section on the solution which is given as a function u(θ) where u = 1/r.

In Classical Dynamics, Thornton, et. al., the section on Planetary motion - Kepler's Problem - "The equation for the path of a particle moving under the influence of a central force ... ", followed by a derivation of u(θ) where u = 1/r (eq 8.41)

On the other hand:
In R. Fitzpatrick's Kepler Problem we read "In a nutshell, the so-called Kepler problem consists of determining the radial and angular coordinates, r and θ, respectively, of an object in a Keplerian orbit about the Sun as a function of time." Kepler's equation is derived, and a numerical method is given for solving it.

From On Newton's Solution to Kepler's Problem - The Monthly Notes of the Royal Astronomical Society (1882) - "The equation to be solved by successive approximation is x - e sin x = z where e is the eccentricity, z is the known mean anomaly, and x is the eccentric anomaly to be determined." The mean anomaly plays the role of time, and the eccentric anomaly plays the role of position.

https://ocw.aprende.org/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/central-force-motion/central-force-motion-and-the-kepler-problem/MIT8_01SC_coursenotes28.pdf- "Since Johannes Kepler first formulated the laws that describe planetary motion, scientists endeavored to solve for the equation of motion of the planets. In his honor, this problem has been named The Kepler Problem." **However, the solution derived is u(θ) and I don't see any derivation of r and θ as a function of time in the entire module.

wiki - Equations of motion - "In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time."

I'm sure that Fitzpatrick's and the Royal Society's, r and θ as a function of time, is correct, but I'm wondering how the alternate solution, 1/r as a function of θ, came into acceptance.

 

[kuruman]

What if you did a parametric polar plot of r using the Fitzpatrick equation and a second polar plot using the Kepler equation. If the two give the same orbit, then both are correct.

 

[phyzguy]

Well, I think finding the solution of u(Θ) is the easiest part of the problem, but only solves part of the problem. It gives you the shape of the orbit, but doesn't give you a way to determine where the planet is along the orbit as a function of time. You need this in order to truly solve the Kepler problem.

 

[hutchphd]

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Both solutions are "correct". Particular needs determine which form is better.
If you wish to draw the path then $r(\theta)$ is probably most useful. If you need to know times of events then the parameterization in terms of t is required
I don't see why a global value judgement is either important or interesting.

.

 

[robphy]

Summary:: Two versions of the solution exist, r and θ as a function of time, and 1/r as a function of θ. Which one is correct?

[snip]

I'm sure that Fitzpatrick's and the Royal Society's, r and θ as a function of time, is correct, but I'm wondering how the alternate solution, 1/r as a function of θ, came into acceptance.

Note that https://en.wikipedia.org/wiki/Kepler_problem# and other $u=\displaystyle\frac{1}{r}$ approaches obtain a convenient differential equation whose solution is
$$u \equiv \frac{1}{r} = -\frac{km}{L^2} (1+e\cos(\theta-\theta_0))$$

You can see the same type of equation as (254) in http://farside.ph.utexas.edu/teaching/336k/lectures/node40.html