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Radial Velocity Curve

Understanding Apsidal Precession in Binary Star Radial Velocities

Apsidal precession, the gradual rotation of the orbit's ellipse within its plane, can occur in binary systems for various reasons. For instance, the gravitational pull from a third orbiting body, general relativistic effects, or tidal forces between stars can cause the argument of pericenter ($\omega$) to change over time. This precession leaves a distinct signature in the observed radial velocity (RV) of the stars.

This webpage provides a simple demonstration of how apsidal precession affects radial velocity measurements. We will derive the equation describing the RV curve when the argument of pericenter advances at a constant rate ($\dot{\omega}$), and examine a common approximation used in modeling.

Deriving the Radial Velocity with Constant Apsidal Precession

To find the radial velocity for a binary experiencing constant apsidal precession, we start with the time derivative of the line-of-sight position ($Z$) component of the star's Keplerian motion, adding the systemic velocity ($\gamma$). Following Equation 55 in Murray & Correia (2010, arXiv:1009.1738):

$$ V_{r} = \frac{\mathrm{d} Z}{\mathrm{d}t} + \gamma = \frac{\mathrm{d}}{\mathrm{d}t} [ r \sin (\omega +f)\sin i ] + \gamma $$

Here, $V_{r}$ is the instantaneous radial velocity, $r$ is the distance of the star from the system's barycenter, $f$ is the true anomaly, $\omega$ is the argument of pericenter, $i$ is the orbital inclination (angle between the angular momentum of the binary and the line of sight), and $\gamma$ is the systemic velocity (the velocity of the binary system's center of mass).

Now, we account for apsidal precession by letting the argument of pericenter change linearly with time: $\omega(t) = \omega_{0} + \dot{\omega} t$, where $\omega_{0}$ is the argument of pericenter at $t=0$ and $\dot{\omega}$ is the constant rate of precession. Substituting this into the equation and performing the differentiation:

$$ \begin{align} V_{r} &= \frac{\mathrm{d}}{\mathrm{d}t} [ r \sin (\omega_{0} + \dot{\omega} t +f)\sin i ] + \gamma \\ &= \sin i \left[ \frac{\mathrm{d}r}{\mathrm{d}t} \sin(\omega_{0}+\dot{\omega} t + f) + r \frac{\mathrm{d}(\omega_0 + \dot{\omega} t + f)}{\mathrm{d}t} \cos(\omega_{0}+\dot{\omega} t+ f) \right] + \gamma \\ &= \sin i [ \dot{r} \sin(\omega_{0}+\dot{\omega} t + f) + r (\dot{\omega} + \dot{f}) \cos(\omega_{0}+\dot{\omega} t+ f) ] + \gamma \\ &= \sin i [ r \dot{f} \cos(\omega_0 + \dot{\omega} t + f) + \dot{r} \sin(\omega_0 + \dot{\omega} t + f) ] + r \dot{\omega} \sin i \cos(\omega_0 + \dot{\omega} t + f) + \gamma \end{align} $$

(Note the rearrangement in the last step to group terms similar to the standard Keplerian RV derivation, plus the new term involving $\dot{\omega}$).

We use the standard expressions for $\dot{r}$ and $r\dot{f}$ (Eqs. 31 and 32 in Murray & Correia 2010):

$$ \begin{align} \dot{r} &= \frac{na}{\sqrt{1-e^2}} e \sin f \\ r\dot{f} &= \frac{na}{\sqrt{1-e^2}} (1 + e \cos f) \end{align} $$

where $n = 2\pi/P$ is the mean motion, $P$ is the orbital period, $a$ is the semi-major axis, and $e$ is the eccentricity. Substituting these into the $V_r$ expression yields:

$$ \begin{align} V_{r} &= \sin i \left[ \frac{na(1+e \cos f)}{\sqrt{1-e^2}} \cos(\omega_0+\dot{\omega}t+f) + \frac{nae \sin f}{\sqrt{1-e^2}} \sin(\omega_0+\dot{\omega}t+f) \right] \\ & \qquad + r \dot{\omega} \sin i \cos(\omega_0 + \dot{\omega} t + f) + \gamma \\ &= \frac{na \sin i}{\sqrt{1-e^2}} [ (1+e \cos f)\cos(\omega_0+\dot{\omega}t+f) + e \sin f \sin(\omega_0+\dot{\omega}t+f) ] \\ & \qquad + r \dot{\omega} \sin i \cos(\omega_0 + \dot{\omega} t + f) + \gamma \end{align} $$

Using the trigonometric identity $\cos(A-B) = \cos A \cos B + \sin A \sin B$ on the terms inside the square brackets (with $A = \omega_0+\dot{\omega}t+f$ and $B=f$), and recalling the relation $r = a(1-e^2)/(1+e \cos f)$:

$$ \begin{align} V_{r} &= \frac{na \sin i}{\sqrt{1-e^2}} [\cos(\omega_{0}+\dot{\omega}t+f)+e \cos(\omega_{0}+\dot{\omega}t)] \\ & \qquad + \frac{a(1-e^2)}{1+e \cos f} \dot{\omega} \sin i \cos(\omega_{0} + \dot{\omega} t + f) + \gamma \\ &= K [\cos(\omega_{0}+\dot{\omega}t+f)+e \cos(\omega_{0}+\dot{\omega}t)] \\ & \qquad + K \frac{P (1-e^{2})^{3 /2}}{2\pi (1 + e \cos f )} \dot{\omega} \cos(\omega_0 + \dot{\omega} t + f) + \gamma \end{align} $$

where $K = \frac{na \sin i}{\sqrt{1-e^2}} = \frac{2 \pi a \sin i}{P \sqrt{ 1- e^{2} }}$ is the radial velocity semi-amplitude. This is the full equation for the radial velocity under constant apsidal precession. This full equation is used to generate the plot.

A Common Approximation

You might notice that the first main term, $K [\cos(\omega_{0}+\dot{\omega}t+f)+e \cos(\omega_{0}+\dot{\omega}t)]$, looks exactly like the standard Keplerian radial velocity equation, just with $\omega$ replaced by $\omega(t) = \omega_{0} + \dot{\omega} t$

$$ V_{r, \text{Keplerian}} = K [\cos(\omega + f) + e \cos \omega] + \gamma $$

Using this form, simply substituting $\omega(t)$ for $\omega$, is a common approximation:

$$ V_{r, \text{approx}} = K [\cos(\omega_{0}+\dot{\omega}t + f) + e \cos (\omega_{0} + \dot{\omega} t)] + \gamma $$

How good is this approximation? It essentially neglects the final term from our full derivation:

$$ \text{Neglected Term} = K \frac{P (1-e^{2})^{3 /2}}{2\pi (1 + e \cos f )} \dot{\omega} \cos(\omega_0 + \dot{\omega} t + f) $$

Let's analyze the behavior using a first-order Taylor expansion in the small quantity $\dot{\omega} t$ (assuming the total precession over the time $t$ is not excessively large). Let $\omega(t) = \omega_0 + \delta \omega$, where $\delta \omega = \dot{\omega}t$.

Expanding the terms in the approximate equation:

$$ \begin{align} \cos(\omega_{0}+\dot{\omega}t + f) &\approx \cos(\omega_0 + f) - (\dot{\omega}t) \sin(\omega_0 + f) + O((\dot{\omega}t)^2) \\ e \cos(\omega_{0}+\dot{\omega} t) &\approx e \cos(\omega_0) - e(\dot{\omega}t) \sin(\omega_0) + O((\dot{\omega}t)^2) \end{align} $$

So the approximate RV contains terms that behave like $\dot{\omega}t \times (\text{sinusoid})$.

Now let's look at the leading order behavior of the neglected term. It's already proportional to $\dot{\omega}$:

$$ \text{Neglected Term} \approx K \frac{P (1-e^{2})^{3 /2}}{2\pi (1 + e \cos f )} \dot{\omega} \cos(\omega_0 + f) + O(\dot{\omega}^2) $$

Notice that the first-order correction terms derived from the approximation (highlighted in red below) grow linearly with time $t$:

$$ V_{r, \text{approx}} \approx K[\cos(\omega_0+f) + e \cos \omega_0] + \gamma \quad \textcolor{red}{- K(\dot{\omega}t)[\sin(\omega_0+f) + e\sin \omega_0]} $$

The neglected term, however, contributes a first-order correction in $\dot{\omega}$ whose amplitude does not grow with time:

$$ \text{Neglected Term} \approx \textcolor{blue}{K \frac{P (1-e^{2})^{3 /2}}{2\pi (1 + e \cos f )} \dot{\omega} \cos(\omega_0 + f)} $$

What does this mean? Over a long observational baseline (large $t$), the deviation of the true precessing RV from a simple Keplerian model (with fixed $\omega_0$) is dominated by the terms proportional to $\dot{\omega}t$ which arise from simply letting $\omega$ vary within the standard formula. The additional term we derived (Neglected Term) is also a first-order effect of $\dot{\omega}$, but its amplitude doesn't increase over time.

Therefore, for many practical purposes, especially when analyzing datasets spanning long durations where the precession effect becomes more apparent, the approximation:

$$ V_r \approx K [\cos(\omega_{0}+\dot{\omega}t + f) + e \cos (\omega_{0} + \dot{\omega} t)] + \gamma $$

captures the dominant time-varying signature of apsidal precession reasonably well, although it is not the exact expression. The script accompanying this post uses the full equation.