Yesterday I asked a question on calculating the eccentricity of an exoplanet only knowing the radial velocity vs. phase graph an the mass of the star (question). The answer I got helped me a lot, but there is still a problem I can't really solve. Rob Jeffries gave a step-by-step solution on calculating the eccentricity. I can calculate the eccentric anomaly using this formula:

$$M(t) = E(t) - \epsilon\sin{E(t)}$$

And this is solvable using the Newton-Raphson method which I think I know how to use. However, this requires me to know the eccentricity which I don't know yet. Later in his answer he explains how find the eccentricity by fitting a function to the graph, but this requires me to know the eccentric anomaly at any given time, which seems paradoxical to me. How is this done?


Obviously it is not as obvious as I thought!

The radial velocity curve is defined through 6 free parameters $$V_r(t) = K\left(\cos(\omega + \nu) +e \cos \omega \right) + \gamma,$$ where $K$ is the semi-amplitude, $\gamma$ is the centre of mass radial velocity, $\omega$ is the usual angle defining the argument of the pericentre measured from the ascending node and $\nu$ is the true anomlay, which is a function of time, the fiducial time of pericentre passage $\tau$, the orbital period $p$ and the eccentricity $e$.

To proceed you estimate what all these parameters are - i.e. an initial guess.

Then, for each time $t_i$ of a data point in your RV curve you:

  1. Calculate the mean anomaly $$M(t) = \frac{2\pi}{p}(t - \tau),$$

  2. Solve "Kepler's equation" $$M(t) = E(t) - e \sin E(t)$$ numerically to give $E(t_i)$, the eccentric anomaly.

  3. Use $$\tan \frac{E(t)}{2} = \left(\frac{1+e}{1-e}\right)^{-1/2} \tan \frac{\nu(t)}{2}$$ to calculate the true anomaly $\nu(t_i)$.

  4. Calculate $V_r(t_i)$

You then calculate some figure of merit (e.g. chi-squared) and go through an iterative process to adjust the parameters and optimise the fit of model to data.

A more sophisticated discussion can be found in this paper by Beauge et al.


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