<p>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 $e$.</p>

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

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

<ol>
<li><p>Calculate the mean anomaly
$$M(t) = \frac{2\pi}{p}(t - \tau),$$</p></li>
<li><p>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.</p></li>
<li><p>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 $\nu(t_i)$.</p></li>
<li><p>Calculate $V_r(t_i)$ </p></li>
</ol>

<p>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.</p>

<p>A more sophisticated discussion can be found in this paper by <a href="http://www.gehirn-und-geist.de/sixcms/media.php/370/Leseprobe.406372.pdf" rel="nofollow">Beauge et al.</a></p>

<p>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.</p>

<p>If you only have one RV curve you are limited to estimating the <a href="http://cmi2.yale.edu/bh/week4/pages/massfunction.html" rel="nofollow">mass function</a> 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.</p>

<p>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.</p>

<p>There are a number of options if you want an off-the-shelf solution to fitting RV curves. Perhaps the best free one is <a href="http://oklo.org/downloadable-console/" rel="nofollow">Systemic Console</a>.</p>

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

CC BY-SA 3.0
2015-06-11T04:54:07.620Z
So I can fit using your first formula directly?  How to handle true anomlay?
questionhang
CC BY-SA 3.0
2015-06-11T06:34:59.910Z
@questionhang Yes. See step 3.
ProfRob
CC BY-SA 3.0
2015-06-10T21:56:13.877Z
2015-06-10T21:56:13.877Z
ProfRob
<p>I have two series of velocities (HJD, $v_{1}$ and $v_{2}$ ), how to fit in detail?<br/></p>

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

<p>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?<br/></p>

<p>What is the difference between binary velocity fitting and star+planet fitting?<br/></p>

CC BY-SA 3.0
2015-06-06T08:38:58.453Z
Are you looking for a ready made solution or do you want to understand and code something yourself?
ProfRob
CC BY-SA 3.0
2015-06-06T11:59:56.917Z
@rob  a formula to help me code and two questions. If there are too many details, a ready made solution is also good reference.
questionhang
<velocity>
radial velocity fitting of a binary
questionhang
# radial velocity fitting of a binary

(Asked by: username_0 on 2015-06-06)

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?

**Tags:** velocity
> **Comments:**
> **username_1**: Are you looking for a ready made solution or do you want to understand and code something yourself?  
> **username_0**: @rob  a formula to help me code and two questions. If there are too many details, a ready made solution is also good reference.  

---

## 1 answers

### Answer by username_1 on 2015-06-10. Score: 3

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.](http://www.gehirn-und-geist.de/sixcms/media.php/370/Leseprobe.406372.pdf)

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](http://cmi2.yale.edu/bh/week4/pages/massfunction.html) 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](http://oklo.org/downloadable-console/).

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

> **Comments:**
> **username_0**: So I can fit using your first formula directly?  How to handle true anomlay?  
> **username_1**: @questionhang Yes. See step 3.  

---

