Ellipse problem

[h4tt3n]

Hello,

 

Recently I stumbled across a math problem, that I hope you can help me with. It is really about defining a planetary orbit from very little data, but since all orbits are ellipses I might as well post it here too (it has also been posted in the astronomy subforum).

 

I'm building a simple 2D program that simulates gravitiational attraction between celestial bodies. The program is strictly Newtonian - it calculates all movement using F = G*(M+m)/d^2 - but I'd like to calculate backwards so to speak and find the details about any orbit's eccentricity, orbital period, orbit centre and foci ect ect.

 

All the data I have is:

 

-body's velocity vector length and direction

 

-body's distance to sun

 

-mass of all bodies

 

-x, y coordinates of all bodies

 

and derived from above data

 

-orbit semimajor axis

 

The data I don't have but wish I did is:

 

-Orbit's semiminor axis

 

-Orbit's eccentricity

 

-Orbit's centre coordinates

 

-Orbit's angle relative to a pre-defined frame of reference

 

I just cant figure out how to isolate these values, but except for that the program works fine and is great fun to play with. The problem is solvable though, since different orbits have unique coorelation between body distance to sun and velocity depending on eccentricity. Please help!

 

Best regards,

 

Michael

[Tom Mattson]

but since all orbits are ellipses

 

Not true. In Newtonian dynamics an orbit can be any conic section.

 

Does that change your line of questioning at all?

[h4tt3n]

Ah, you're right, but it doesn't change my line of question. Of course an orbit can be parabolic and hyperbolic too, but in those cases you can't define semimajor axis, semiminor axis and eccentricity, and for the same reason I'm not interested in those specific cases. My main interest is to define any closed orbit.

 

Michael