## Dancing in the Dark

Epislon Eridani, the nearest known planet-hosting star other than the sun.

The motion of a star around the barycentre of a planetary system is not necessarily confined to detectability through indirect methods only. The barycentric motion of a star in a planetary system can, with sufficient precision, be directly observed. Logically, the larger the star’s barycentric semi-major axis, the more easy it would be to observe such a motion. However very long, multi-decade period orbits are harder to observe simply because they progress at timescales comparable to the human lifespan. As such, astrometry is biased toward intermediate-period, massive planets.

Often, the barycentric motion of a star will be tiny. Our sun being an example rarely diverges from two solar-radii from the Solar System barycenter. Across the extreme distances, these changes are difficult to resolve, leading to large error bars that can swamp out the real orbital motion. For this reason, it is often the case that a large number of measurements are required to build up confidence in the detection of an astrometric signal, much as with the Doppler spectroscopy method.

While the above plot is messy, it makes clear that the stellar orbit is far from face-on. It gives us a rough measurement of the stellar orbit’s inclination.

The advantages of astrometry justify the difficulty in performing it for planetary systems. By directly observing the barycentric motion of a star, it is possible to reconstruct the orbit of the orbiting planet fully in 3D (while distinguishing which node is the ascending node, $\it{\Omega}$, will require a combination of astrometric and radial velocity data). Because astrometry provides the full 3D orbit, the inclination may be measured and thus the true mass of the planet is found, resolving the inclination degeneracy that plagues Doppler spectroscopy.

Because the star is not the only thing in motion, some work is needed before jumping into searching for exoplanet-induced astrometric signals. Firstly, the proper motion (the natural drift of the position of the star in the sky as the result of each star’s independent galactocentric orbit) of the star must be modelled out. Additionally, the orbital motion of Earth around the Sun will cause another signal in the astrometric data that must be modelled out. After this is done, whatever remains must be the intrinsic motion of the star under the influence of other bodies.

To give some idea of comparison, from “above” the solar system, the sun’s astrometric signal would look like this.

The large yellow circle represents the diameter of the sun, and the black line is the path it takes, due to the influence of the planets in our solar system. Jupiter and Saturn dominate the astrometric signal. The other planets aren’t detectable in this timespan. Notice that the amplitude of the astrometric variation is comparable to the diameter of the star itself. More specifically, for a star with a distance$D$and a single orbiting companion, the astrometric amplitude can be estimated as

$\displaystyle \alpha = \arctan{\left( \frac{a * q}{D(q+1)} \right)}$

where q is the mass ratio between the planet and the star$M_p / M_*$, and a is given as

$\displaystyle a = (G(M_* + M_p))^{1/3} (P/2 \pi)^{2/3}$

The need for precision thus makes itself apparent. So far, very few extrasolar planets have been detected this way. On the positive side, radial velocity candidates whose true masses are significantly higher than their RV-derived minimum mass have larger than expected amplitudes. This makes astrometry quite effective at determining if exoplanet candidates are actually low mass or even main sequence stars. In most cases, astrometry can set an upper limit to the astrometric amplitude, which translates directly into an upper limit for the planet’s mass. This can permit even a non-detection to secure the planetary nature of a candidate extrasolar planet.