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I have two series of velocities (HJD, $v_{1}$ and $v_{2}$ ), how to fit in detail?

There is any formula on the web? $sine$ curve is easy, but I need to consider an eccentric orbit.

Suppose we can estimate $m_{1}$, $m_{2}$, and inclination angle $\approx80^{\circ}$, if we just know velocities of both of the two companions, which parameters can me derive?

What is the difference between binary velocity fitting and star+planet fitting?

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  • $\begingroup$ Are you looking for a ready made solution or do you want to understand and code something yourself? $\endgroup$ – ProfRob Jun 6 '15 at 8:38
  • $\begingroup$ @rob a formula to help me code and two questions. If there are too many details, a ready made solution is also good reference. $\endgroup$ – questionhang Jun 6 '15 at 11:59
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The radial velocity curve of a star in a binary system (with another star or a planet) 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 (its a transcendental equation, you could use Newton-Raphson or similar) to give $M(t_i)$, 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) for how closely the model and data agree 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.

If you have the RV curves of both stars, then you can fit them both simultaneously. Obviously, they have $p$, $e$, $\gamma$ and $\omega$ in common, but their RV amplitudes $K_1$ and $K_2$ will be different. The ratio of $K_1/K_2$ gives you the ratio of the two stellar masses.

If you only have one RV curve you are limited to estimating the mass function of the binary system. $$ \frac{M_2^{3} \sin^{3} i}{(M_1 + M_2)^2} = \frac{p K_1^{3}}{2\pi G},$$ where $i$ is the inclination of the orbit with respect to the line of sight. This can only give you a lower limit to $M_2$ unless $i$ is known.

Taking your specific case study. If you know $M_1$ and $i$ (this could be the case for a transiting exoplanet, or maybe a binary featuring an eclipsed black hole candidate), then the primary radial velocity curve gives you $K_1$ and hence $M_2$. If the masses and $p$ are known then Kepler's laws give the orbital separation.

There are a number of options if you want an off-the-shelf solution to fitting RV curves. Perhaps the best free one is Systemic Console.

There is no fundamental difference between analysing the RV curves of stars with exoplanets and stars with unseen (stellar) companions.

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  • $\begingroup$ So I can fit using your first formula directly? How to handle true anomlay? $\endgroup$ – questionhang Jun 11 '15 at 4:54
  • $\begingroup$ @questionhang Yes. See step 3. $\endgroup$ – ProfRob Jun 11 '15 at 6:34

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