Equations for Orbital Parameters and Derived Quantities

• $v_{1,2}$: Radial velocity of the primary (1) or secondary (2) component [$\text{km s}^{-1}$].
• $V_0$: Systemic radial velocity of the center of mass [$\text{km s}^{-1}$].
• $K_{1,2}$: Radial velocity semi-amplitude [$\text{km s}^{-1}$].
• $P$: Orbital period [days, converted to seconds for mass function calculations].
• $e$: Orbital eccentricity (dimensionless, $0 \le e < 1$).
• $\omega_{1,2}$: Argument of periastron [degrees or radians].
• $\nu$: True anomaly, the angular parameter of position at a given time [degrees or radians].
• $a_{1,2} \sin i$: Projected semi-major axis [$\text{km}$ or $\text{au}$].
• $i$: Orbital inclination angle [degrees].
• $f(m)$: Mass function, describing the minimum mass constraint of the companion [$M_{\odot}$].
• $q$: Mass ratio ($m_2 / m_1$).
• $G$: Gravitational constant.
• $\sigma_X$: Denotes the uncertainty (standard error) of parameter $X$.

Primary Component ($v_1$):

$$ v_1(\nu) = V_0 + K_1 [\cos(\omega_1 + \nu) + e \cos \omega_1] $$

Secondary Component ($v_2$):

$$ v_2(\nu) = V_0 + K_2 [\cos(\omega_2 + \nu) + e \cos \omega_2] $$

Where the argument of periastron for the secondary is shifted by $\pi$ radians ($180^\circ$):

$$ \omega_2 = \omega_1 + \pi $$

Value:

$$ a_{1,2} \sin i = \frac{K_{1,2} P}{2\pi} \sqrt{1 - e^2} $$

Uncertainty Propagation:

$$ \frac{\sigma_{a \sin i}}{a \sin i} = \sqrt{ \left( \frac{\sigma_K}{K} \right)^2 + \left( \frac{\sigma_P}{P} \right)^2 + \left( \frac{e \sigma_e}{1 - e^2} \right)^2 } $$

Value:

$$ f(m_{1,2}) = \frac{P K_{1,2}^3}{2\pi G} (1 - e^2)^{3/2} $$

Uncertainty Propagation:

$$ \frac{\sigma_{f(m)}}{f(m)} = \sqrt{ \left( \frac{\sigma_P}{P} \right)^2 + \left( \frac{3 \sigma_K}{K} \right)^2 + \left( \frac{3 e \sigma_e}{1 - e^2} \right)^2 } $$

Value:

$$ q = \frac{K_1}{K_2} $$

Uncertainty Propagation:

$$ \frac{\sigma_q}{q} = \sqrt{ \left( \frac{\sigma_{K_1}}{K_1} \right)^2 + \left( \frac{\sigma_{K_2}}{K_2} \right)^2 } $$