## 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)

## Mapping the Galaxy

The Milky Way Galaxy (Source)

Measuring the distance to a star makes use of astrometry – the careful monitoring of a position of a star over time. As Earth orbits the sun, it has a maximum displacement from any given position along its orbit of about 2 AU (i.e., being on the other side of the orbit). By observing the angular change in the apparent position of a star 2 AU apart, simple trigonometry can allow you to calculate the distance to the star.

In the middle of the last century (not terribly long ago from a historical perspective), we knew the distances to very few stars and knew their positions with much poorer accuracy. The FK4 catalogue catalogued the position of stars in the year 1950 with a precision of position of about 0.04 arcsec in the northern hemisphere, and a dismal 0.08 arcsec precision in the southern hemisphere. It was suggested that using a network of astrolabes over ten years could reduce the errors to about 0.03 arcsec, only marginally better. Major obstacles to the advance of stellar cartography was the typical issues that plague amateur astronomers now — atmospheric distortion of stellar images, instrumental instability, and inability for a ground-based observatory to view the entire sky.

In 1966, Pierre Lacroute came up with an idea (that he himself called “weird”) of performing the necessary measurements from a spacecraft, orbiting Earth outside the atmosphere. The idea was presented in 1967 to the IAU where it received a great deal of interest, but the technological capacity at the time (and available rocketry in France) was not accommodating to the idea. The satellite, a 140 kg spacecraft designed to observe 700 stars all over the sky with a precision of 0.01 arcsec, had stability requirements that could not be met by the Diamant rocket used by France at the time.

The idea of a spacecraft to catalogue the distances and positions of a large number of stars evolved over time and was revised and improved for the next decade, while the rest of astrophysics advanced and continued running into the problem of distance scales being poorly known.

“The determination of the extragalactic distance scale, like so many problems that occupy astronomers attention, is essentially an impossible task. The methods, the data, and the understanding are all too fragmentary at this time to allow a reliable result to be obtained. It would probably be a wise thing to stop trying for the time being and to concentrate on better establishing such things as the distance scale in our Galaxy.” — Hodge (1981)

Support for a space-based astrometry mission continued to grow and recognising that France alone did not have the resources necessary to complete the task, the European Space Agency planned and devised a new spacecraft, Hipparcos, to catalogue the positions of 100,000 stars and to determine their positions with an accuracy of 0.001 arcsec (1 milliarcsec).

Hipparcos

Hipparcos was launched on August 8, 1989 on a 3.5 year mission. It determined the positions of stars, monitored the position over the course of a half year to determine the parallax and thus distance to the star, monitored the position over the course of the entire mission to determine the proper motion of the star in space, measured the spectrum of stars to determine their composition, and performed radial velocity measurements on these stars to determine their motion toward or away from Earth. In total, 118,200 stars were observed with high precision observations (published in 1997), with another 2.5 million stars observed with lower precision (published in 2000).

Hipparcos data has practically revolutionised astronomy. With the knowledge of the positions and motions of over a hundred thousand stars in hand, we’ve been able to understand the structure and dynamics of nearby clusters, understand the local structure of the Galaxy, understand the orbits and true orientations of binary star systems, and more. Even an extrasolar planet transit was observed (though it was not known until the planet was discovered later).

This brings us to today. This Hipparcos catalogue remains as the best available source of uniform parallaxes and positions. It is time, however, to take another step forward, with greater precision, a larger sample, and newer science. The successor to Hipparcos is called GAIA – Global Astrometric Interferometer for Astrophysics – however it will not use interferometry due to a design change.

Gaia will essentially do exactly what Hipparcos did, but better. Whereas Hipparcos only measured a hundred thousand stars down to brightnesses of V = 9, Gaia will observe over a billion stars with brightnesses down to V = 20. Gaia will measure the angular position of all stars of magnitude 5.7 – 20. For stars brighter than V = 10, it will determine the position with a precision of 7 µas (microarcseconds), a precision of 12 – 25 µas down to V = 15, and 100 – 300 µas down to V = 20. It will acquire their spectrum (from 320 – 1000 nm) to determine their temperature, age, mass, and composition. It will also measure the radial velocity of stars with a precision of 1 km s-1 for V = 11.5, and 30 km s-1 for V = 17.5. Tangential velocities for 40 million stars will be measured with a precision better than 0.5 km s-1.

Gaia (Source: ESA)

While the stellar astrophysics enabled by Gaia will be revolutionary in its own right, the unprecedented astrometric precision also makes the mission interesting from an extrasolar planet perspective. Hipparcos was not able to discover any planets on its own, but it was marginally helpful for extrasolar planet science. Planets detected with radial velocity have unknown true masses. The greater the true mass of the planet, the greater the astrometric amplitude of the barycentric motion of the star is (see this post where astrometry is discussed in the context of planet detection). Planets of especially high true masses would therefore have a chance of having their star’s barycentric motion detectable to Hipparcos. Otherwise, Hipparcos data could be used to set upper limits to the true mass of the planet, by knowing that it’s astrometric effect must be sufficiently low so as to not have been detected by Hipparcos (an upper limit to the astrometric amplitude and thus the planetary mass).

The astrometric precision and vast number of targets available to Gaia will allow for the detection of a large number of planets. Astrometry is, of course, less biased toward high values of the planetary orbital inclination, and will permit us to know the true mass of the planet and orientation of the orbit in 3D space. Still, several complications are expected to arise based on nearly two decades of radial velocity experience.

Just like with radial velocity (and, actually, science in general), models will need to be fitted to data points to yield high-quality fits, however as Doppler spectroscopy has shown us, planetary systems can often feature several components all contributing to the barycentric velocity profile of the star, complicating radial velocity fitting in the same way it can be expected to complicate astrometric fitting. Radial velocity surveys can often produce more than one model that fit the data nicely, where both models may disagree on certain aspects of the orbit, or even number of planets. Astrometry is likely to be prone to the same problems. In the case of astrometry, it may even be harder because of the greater number of free parameters – ascending node, inclination, etc, issues that need to be modelled for an astrometric fit that could usually be ignored for a radial velocity fit.

These challenges can be addressed and handled, and the Gaia data will be wonderfully productive to extrasolar planet science. It is hard to know how many planets we can expect Gaia to discover, because statistics for planets in intermediate-period orbits are still unconstrained, but with the accuracy and large number of stars Gaia will observe, it is likely that Gaia will discover thousands of giant planets. It will be sensitive to Jupiter analogues out to 200 parsecs.

Gaia Results (Source: Sozzetti (2010)

What about transiting planets? A transit of HD 209458 b was squeezed out of Hipparcos data, which was not at all optimised for transiting planet science. Can Gaia be expected to detect transiting planets? As far as photometric precision, Gaia is expected to achieve 1 mmag precision for most objects Gaia will observe, down to V ~ 15, and 10 mmag precision at the worst case of V ~ 20. For most hot Jupiter systems, mmag precision is indeed sufficient for transit detections. The next major issue is cadence.

Focused transit searches tend to be high-cadence, narrow field observations, whereas Gaia is an all-sky, low cadence observatory. On average, each star will be observed by Gaia 70 times, giving us 70 measurements for a light curve of any given star with a baseline of five years. While 70 measurements spread out over five years seems dismal (and let’s not sugar-coat the issue — for a transit search, it is dismal, but Gaia is not designed to be a transit search mission), but for a planet in a short period orbit, perhaps three or four measurements may occur while the planet is transiting. Obviously, the longer the orbital period, the less a fraction of the planet’s orbital period is spent in-transit, and the fewer transits will be observed by Gaia. Since only 70 measurements will be taken, Gaia is severely biased toward short-period transiting planets.

Early studies suggested wildly fantastic transiting planet yields. Høg (2002) estimated over a half million hot Jupiters and thousands of planets in longer periods would be found, based on the (unrealistic) assumption that a transit could be identified based on a single data point and other oversimplifications. Robichon (2002) suggested that Gaia will detect 4,000 – 40,000 transiting hot Jupiters under the assumption that each star would receive an average of 130 measurements, however the currently planned Gaia mission has instead 70 measurements per star.

Dzigan & Zucker suggest that Gaia could potentially detect sub-Jupiter-sized planets around smaller stars, and that a ground-based follow-up campaign can easily observe hints of transiting planets that show up in Gaia data. They also suggest that a few hundred to a few thousand hot Jupiters could be found in Gaia photometry.

While Gaia will perform km s-1 radial velocity measurements on millions of stars, this precision level is simply not sufficient to detect even hot Jupiters. It will, however, be able to tell if a transiting planet candidate is a brown dwarf instead, or an eclipsing binary star, allowing for one method of ruling out false positives. Interestingly, the astrometric fit to the orbit of a planet will have the inclination of the planetary orbit sufficiently well-characterised that a list of planets that are likely to transit can be compiled and followed-up with ground-based radial velocity and photometry. These long-period transiting planets will certainly prove valuable – they will be likely to host detectable rings and moons.

ESA will launch Gaia on a Soyuz ST-B rocket in November of this year. It will take five years after a commissioning phase for the total extrasolar planets science results to become known. It will be very exciting to see what giant planets exist in the solar neighbourhood. They will attract interest in follow-up observations to discover smaller, inner worlds that may exist. Gaia has the potential for flagging the first solar system analogues in the solar neighbourhood for dedicated study.

## 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.