Calculating eccentric anomaly using the Newton-Raphson method

Question

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?

Answer 1

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.