I'm working on this project and I've kind of come to a roadblock.(adsbygoogle = window.adsbygoogle || []).push({});

Here's a little background: we have a 1 solar mass star and a planet orbiting it. The planet is defined by 5 orbital elements (period, mass, eccentricity, longitude of periastron, and mean anomaly). I wrote a code that will plot the radial velocity of the planet vs. time. I'm using the systemic console (www.oklo.org). I don't know how familiar you guys are with it; this is my first post here.

Now the radial velocity of the planet tells us the radial velocity of the star. The only difference is the semi-major axis is reduced by 1/(1 + Ms/Mp) and the path is flipped by 180°.

Here's the problem: I plot the radial velocity vs. time and I get an ok looking graph. Then I go to do a fitting test. The amplitude is dead on. So that's correct. But every time, the phase is off! Sometimes by ~30°, sometimes by ~180°. What's going on?

Here's my math (skip to the bottom to see the phase): I'm using the bookSolar System Dynamicsby Murray and Dermott. There's a neat matrix and I'm taking the radial velocity to be along the X axis. The equation for that is...

X = r(cosΩ*cos(ω + f) - sinΩ*sin(ω + f)*cosI)

where r is the radius, Ω is the longitude of ascending node, ω is the longitude of periastron, I is the inclination angle, and f is the true anomaly. Ω = I = 0, so our equation is reduced to...

X = r*cos(ω + f)

Take a derivative to get radial velocity. r and f change with time so you have to product rule.

There's useful formulas in the book for r dot and r*(f dot), but my final solution for the phase of the planet is...

-[e*sinω + sin(ω +f)]

e is eccentricity.

Just switch the sign for the radial velocity of the star. Now f can't be explained simply in terms of the mean anomaly, but it can with the eccentric anomaly, E.

tan(f/2) = sqrt[(1+e)/(1-e)]*tan(M/2)

It's messy but my new phase for the star looks like...

e*sinω + sin(ω + [2 * arctan(sqrt[(1+e)/(1-e)]*tan(E/2))])

If we consider a perfectly circular orbit (e = 0), then arctan-tan eliminate, 2s cancel and we have ω + M on the inside, as I'd expect.

I related E to M using Newton-Raphson iterations, and checked for convergence using Kepler's Equation. And it works. I think I'm overlooking something really obvious. I've been staring at this for a few days.

Anyone brave enough to conquer my wall of text and lend me some insight? It would be greatly appreciated! Thank you!

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# Orbit in space (extrasolar planets)

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