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• Nonlinear least-squares optimization results
• Markov Chain Monte Carlo solver

The radial velocity technique is the cornerstone of modern astronomy. Before the advent of NASA's Kepler efforts, radial velocity measurements served as the primary method of exoplanet detection: Campbell et al. 1988, Latham et al. 1989, Mayor & Queloz 1995, Marcy & Butler 1996., Akeson et al. 2013. The mathematical solution of the two-body problem shows that when a planet orbits a star in an elliptical path, the star also moves in an ellipse about a common center of mass. The technique relies on measuring the Doppler shifts in the line spectra of stars and assessing their recessional velocities in these elliptical orbits. This is calculated using the relativistic Doppler equation.

$$ \begin{equation} z \equiv \frac{\lambda_{\mathrm{obs}} - \lambda_{\mathrm{rest}}}{\lambda_{\mathrm{rest}}} = \frac{1}{\gamma(1+v/c)} - 1 \quad \mathrm{where} \quad \gamma = \frac{1}{1-(v/c)^2} - 1 \end{equation} $$

where $\gamma$ is the Lorentz factor. Then fitting this data for the orbital parameters yields information of any bodies orbiting it. The velocity, $v$, is a function of time which takes a periodic solution since for one half of the orbit, the spectral lines will be blue shifted and for the other half, redshifted. The motion of a body around another can then be understood in the context of the two-body problem; the solution to which was first presented by Sir Isaac Newton in 1687 [1]. Newton demonstrated that the elliptical motion of bodies and Kepler's three laws (1609, 1619) could both be described by his law for gravitation. But what Newton achieved was only showing that Kepler's observations could be explained by an inverse-square law. To actually find the positions and velocities of bodies in the two-body problem is not quite as straightforward; this is termed the Kepler problem. The problem is six-dimensional, describing orbits in a binary system by six parameters. These are shown in the table below. The period is not strictly an orbital element but I have included it here for completeness.

These parameters can completely define an orbit but are not yet whole. The Keplerian problem can be reduced to the solutions of primarily three equations, [2].

$$ \begin{gather} M = E - e\sin E \tag{1.1}\\\ \nu = 2\arctan \left(\sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}\right) \tag{1.2} \\\ \dot{\mathrm{z}} = \mathrm{v} = \kappa + K\left[\cos(\nu(t) + \omega) + e\cos(\omega)\right] \tag{1.3} \end{gather} $$

where $M$ is the mean anomaly, $E$ is the eccentric anomaly, $\nu$ is the true anomaly, $K$ is the velocity semi-amplitude, i.e., a measure of the "wobble" induced on the star from an orbiting body,

$$ K = \frac{n a\sin i}{\sqrt{1-e^2}}, $$

$\kappa$ is the long-term mean velocity from the binary, and $n$ is the mean motion $n = 2\pi/P$. Equation 1.2 is known as Kepler’s equation and no closed-form solution to it exists. Physically, Kepler's equation demonstrates the relationship between the polar coordinates defining a celestial body and the time elapsed from a given moment. Equation 1.2 is derived from arguments of the true anomaly which goes beyond the mathematical scope of this exploration [5]. Equation 1.3 is the radial velocity equation. The data is fit for the parameters in that equation. All the parameters outlined in Table 1 are shown in a diagram of the Keplerian orbit below. It is worth noting that $\Omega$ does not appear in expressions that solve for the radial velocity. This is obvious since the orientation of the orbit in the plane of the sky does not affect the observations. This also extends itself to imply that $\Omega$ cannot be found from radial velocity data.

Kepler orbital parameters. Left A three-dimensional view of the Kepler problem with relevant parameters labeled. The apastron is the point of furthest distance from the center of gravity. The reference plane is considered to be equatorial, i.e., when the inclination is equal to zero. This is often a very difficult quantity to measure and therefore only the projected value of the semi-major axis asini is measurable gamma denotes the direction pointing towards the observer, i.e., Earth. The diagram is not drawn to scale. Right A two-dimensional projection of the Kepler problem with the true anomaly indicated. The inferior conjunction is the closest point at which the planet crosses the observer's line of sight.

As the star moves along the smaller elliptical orbit depicted by the dashed line in the Left figure, its atomic spectra will shift as well. The radial velocity of celestial bodies are measured by taking high-resolution spectra; this is typically done by superimposing Echelle spectograph. These are then compared with laboratory-measured spectral line wavelengths. Radial velocity measurements made in this manner are termed as absolute radial velocities. Such measurements are limited by the wavelength calibration of the spectrograph and typically have accuracies of the order of 100m/s (Chubak et al., 2012). An alternative method is to measure differential radial velocities, i.e., the change in redshift between two epochs. This paper employs data from the latter method since they provide accuracies than two orders of magnitude greater than absolute radial velocities. Another factor worth considering is that as observatory taking the spectographs is itself moving — the Earth rotates at ~300m/s and orbits the Sun at ~30km/s —the measured radial velocity of a perfectly stable star will appear to vary on diurnal and annual timescales. Since even the Earth takes an elliptic orbit, its velocity is not constant and would increase at the point of perihelion. Such measurements of radial velocity are corrected such that the measurements are as an observer would have measured in an inertial reference frame. This is termed as barycentric correction following Wright and Eastman (2014).

Orbital Anomalies

The anomalies are angular measures that define the position of a mass moving along a Keplerian orbit. These are derived from the seven primary parameters in Table 1. The mean anomaly, $M$, is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periastron. It is the angular distance from the center of gravity which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit [3, 4]. Kepler’s second law postulates that planets sweep equal areas in equal times --- that is to say that celestial bodies do not move uniformly in orbit but speed up at periastron and slow down at the apastron. To circumvent this irregularity, the mean anomaly is introduced. It is far easier to calculate angles in a circular orbit and later using it with the eccentric anomaly to arrive at the true anomaly. The eccentric anomaly, $E$, is one of the angles on the right-angled triangle where one vertex lies at the center of the ellipse, the adjacent side lies on the semi-major axis, the hypotenuse of length $a$ lies on the auxiliary circle of the ellipse, and the opposite side is perpendicular to the semi-major axis passing through the point $P$, the location of the orbiting body. It is essentially like the true anomaly but with the exception of the center being located at the center of the ellipse as opposed to the center of mass. The true anomaly is the angle made between the semi-major axis and the position of the orbiting body as seen from the periastron.

Mass function determination.

$$ f \equiv \frac{(M_2\sin i)^3}{(M_1 + M_2)^2} = \frac{PK^3}{2\pi G}(1-e^2)^{(3/2)} $$