## Orbit Projections in the Sky

Projection (Source)

Whether you are determining the barycentric motion of a star with µas precision to determine the orbit and mass of an unseen planetary companion, or directly observing planetary companions orbiting dozens of AU from their stars over the course of some decent fraction of a decade to pin down their orbits, you are basically trying to determine the motion of an object in the sky. While the orbit of the object is three-dimensional, the path it takes in the sky is not (at least not for monitoring nearby objects, especially in the solar system). An integral part of binary star astrometry is therefore projecting a 3D orbit onto a 2D plane, and fitting data to that observed 2D orbit. Since a star plus planet system is physically analogous to a binary star with a high mass ratio, the mathematics ends up being the same.

With our focus thus far in this blog on Doppler spectroscopy and transit photometry, the orbital parameters we have concerned ourselves with are the semi-major axis, a, the eccentricity of the orbit e, the inclination of the orbit, i, and the longitude of periapsis ω, which define the size of the orbit, the deviation from circular, the angle between the orbit plane and the plane of the sky, and the angle of the periapsis of the orbit from the observer, respectively. For the longitude of periapsis, ω=0° is defined in such a way that the line connecting the star to the periapsis of the orbit is perpendicular to the line of sight — the Earth-star-planet angle is a right angle. ω=90° after a rotation of the orbit 90° in the orbit plane in such a way to where if the inclination of the orbit is 90°, the periapsis would be between Earth and the star, and the transit midpoint would occur at periapsis.

Changing the longitude of periapsis rotates the orbit in the orbit plane

We have virtually ignored the ascending node, Ω, which defines the rotation of the orbit plane around the line of sight. For some inclination near 90°, when Ω=0°, the planet orbits “up-and-down” and when Ω=90°, the planet orbits “left-and-right.” Note that this does not necessarily mean a polar or equatorial orbit around the star, as in this discussion, we are agnostic about the orientation of the stellar rotation axis. The ascending node of a planetary orbit has been mostly ignored in this blog because it does not actually affect either the Doppler behaviour of the star, or for the most part the shape of the transit light curve. Technically, with all else held constant, varying Ω for a transiting planet will affect the projected angle between the star spin axis and the planet orbital axis, λ, which is detectable as the Rossiter-McLaughlin effect with Doppler spectroscopy (See here). However, since we do not know the orientation of the stellar spin axis in space, we aren’t able to fit Ω as an adjustable parameter to the data.

Changing the ascending node rotates the orbit around the line of sight

Plotting 3D orbits as they appear on the (locally) two-dimensional plane of the sky to determine it’s orbit is the astrometric equivalent of calculating the radial velocity behaviour of a star. We input orbital parameters into a model and try to fit the data to that model to assess how well it approximates reality.

First, we will need to define a couple of functions that address the orbital motion of the object in the x and y dimensions in the sky, let’s call these functions $f_x(t)$ and $f_y(t)$.

$\displaystyle f_x(t)=\cos E - e \\ f_y(t)=\sqrt{1-e^2} \sin E$

where e is the eccentricity of the orbit, t is the time of calculation, and E is the eccentric anomaly as derived and calculated in the same way as in the Doppler spectroscopy method discussed here.

Now we will need to bring in transformations to describe the deprojection of the 3D ellipse into a 2D plane. These take the form of the Theile-Innes constants:

$\displaystyle A=a(\cos\omega\cos\Omega-\sin\omega\sin\Omega\cos i)\\ B=a(\cos\omega\sin\Omega+\sin\omega\cos\Omega\cos i)\\ F=a(-\sin\omega\cos\Omega-\cos\omega\sin\Omega\cos i)\\ G=a(-\sin\omega\sin\Omega+\cos\omega\cos\Omega\cos i)$

Where a is the semi-major axis of the orbit, ω is the longitude of periapsis, Ω is the ascending node and i is the inclination of the orbit. We now combine the two to get the position of the object in the x- and y- axes as a function of time:

$\displaystyle x(t)=B f_x(t)+Gf_y(t)\\y(t)=Af_x(t)+Ff_y(t)$

We are now in a position to evaluate the goodness-of-fit of a given model to data using essentially the same statistical tools and techniques described in this post. The observed-minus-calculated for an astrometric position is in this case of course determined with Pythagorean Theorem, $O-C=\sqrt{\Delta x^2 + \Delta y^2}$. As an example, here are the positions of a star in a 15 year orbit around the supermassive black hole in the centre of the Galaxy.

Star orbiting SgrA* (source)