Planet x orbiting the the earths orbital 180 degrees off phase

[ponces]

Homework Statement

hi guys this is my first post, and it relates to classical mechanics. the problems asks that if there is a planet, planet x, with the same mass as that of the Earth and orbiting the same orbit as that of the Earth but 180 degres off from earth, so that the if the Earth is at the perihelion, planet x is at aphelion. the problems says that because of the elliptical nature of the orbit, from the Earth there'll be a time in which planet x can be observed and not hidden by the sun, it then goes on to ask to find the maximum angle between the sun and planet x as viewed from earth.

Homework Equations

I know that for an elliptical orbit I can determined the angular and radial change as a function of time for both planets. so my plan is to relate the radial distance of from sun to Earth and from sun to planet x, and then relate them by the cosine law function to find the angle that I need. then I will differentiate the equation with respect to time and set it equal to zero to find the minima and maxima of the angle.
guys I apologize if I didn't writte any equations, this is my first post and am learning to use the latex program. but I will appreciate it if you could tell me if I am in the right path. the problems is from Goldstein 3rd classical mechanics 3.17

The Attempt at a Solution

 

[BvU]

Welcome to PF. You sure that picture isn't copyrighted ?

Anyway, look up a few goodies on elliptical orbits, e.g. here ; might save you a lot of work.

 

[gneill]

You won't find a closed form solution for the position of the objects in orbit at a given time. You'll run into what is called The Kepler problem aka The Prediction Problem. The speed of an object in an elliptical orbit varies throughout the orbit (faster near periapsis, slower near apoapsis).

A numerical approach is doable though if you're handy at programming or using a package like matlab.

 

[BvU]

Beg to differ with gneill: the 180 degrees given mean that there is an obvious point in the elliptical trajectory where the angle is maximum. In the link that is nicely pictured in the figure under "useful ellipse factoid". No calculations, no differentiation necessary. But I could be wrong, so check it out!

 

[gneill]

Beg to differ with gneill: the 180 degrees given mean that there is an obvious point in the elliptical trajectory where the angle is maximum. In the link that is nicely pictured in the figure under "useful ellipse factoid". No calculations, no differentiation necessary. But I could be wrong, so check it out!

The planets are only 180° apart on the orbit when they are at opposite ends of the major axis. The one at perihelion will be moving faster than the one at aphelion and they will be out of sync by some amount that varies until they're in the similar configuration six months later (positions swapped).

It's this being out of synch with the 180° starting separation that would allow one to glimpse the other. The varying speeds on the orbit complicate pinning down the maximum elongation. I can't off hand think of a way to tie the value to the geometry of the ellipse, but it will be observed by the one nearest perihelion. For an orbit with a low eccentricity such as the Earth's orbit the variation from the 180° is probably going to be quite small, maybe a couple of degrees at a guess.

 

[BvU]

Oh boy, this is more complicated indeed. Sorry to have made such uncorroborated suggestions !
g is right in post #3 and I am too naive in post #2. Worse: dead wrong in post #4. (fortunately with a caveat in the last few words)

Nice chance for me to refresh Kepler orbits, but no help for ponce (no time at the moment...)