## Transits Across g-Darkened Stars

Titan transiting its oblate fluid planet (Credit: HST)

We have previously looked at modelling the transit of an extrasolar planet incorporating limb darkening. How can we build upon this? For starters, it would be nice to actually take into consideration more physical phenomena – nature is not always simple. Take for example the case of a non-spherical star. Consider the case of Saturn, whose rapid rotation distorts the planet away from a spherical shape and more into an oblate shape. Stars, being fluid bodies like gas giant planets, are subject to similar physics, and have measurable effects on the transit light curve.

How does this affect a system? A rotating fluid body like a gas giant planet or a star is subject to centrifugal force which causes the equatorial radius to be larger than the polar radius. This effect, called oblateness, can be measured with

$\displaystyle \text{oblateness} = \frac{a-b}{a}$

Where a is the equatorial radius and b is the polar radius. In the case of Saturn, the oblateness is about 0.1. Despite a higher rotational velocity, Jupiter’s higher density and surface gravity are able to keep it’s oblateness lower, but still noticeable, at about 0.06. All of the other planets have very different, non-gas-dominated compositions, and have much a lower oblateness (in all cases, less than 0.025). The sun, however, with it’s high surface gravity (28 times that of Earth) and long rotation period (about a month), is almost perfectly spherical, with an oblateness of 0.000009.

In the case of a self-luminous body like a star, oblateness will cause the surface gravity and therefore surface temperature of the star to be a function of latitude. Parts of the surface closer to the poles will be hotter than those near the equator which are pushed further from the stellar centre due to the centrifugal force imposed by the stellar rotation. The temperature of the surface at a given latitude, θ, is given by

$\displaystyle T_{eff}(\theta)=T_p\left(\frac{g(\theta)}{g_p}\right)^\beta$

Where Tg and Tp are the surface gravity and temperature at the pole, respectively, and β is the gravity darkening coefficient, which much like limb-darkening coefficients, are dependent on the properties of the star and will have to be tinkered with to get a good fit.

An example of this comes from the nearby and well-known rapidly rotating, A-type star Altair, which has been spatially resolved using interferometry, allowing for the construction of a low-resolution temperature map.

Altair (Source)

Notice the oblateness of Altair is greater than even that of Saturn. This has a strong effect on the surface temperature of the star, with the equator being over 1,000 K cooler than the poles!

Much like in the case of limb-darkening, gravity-darkening will cause the luminosity profile of the stellar disc to be concentrated in one particular area. Unlike the case of limb-darkening, however, this area need not be the centre of the disc, but could instead be anywhere on the disc. This allows for transit light curves to actually be asymmetric in the case of a misaligned planetary orbit. A transiting planet in a polar orbit of a gravity-darkened star with, for example, a 45 degree rotation axis inclination, will have its planet occulting the bright polar region before moving over the more equatorial regions, causing the minima of the transit light curve to be displaced from the transit centre over toward the ingress.

Consider for example the case of KOI-13, a rapdily rotating A-type star, much like Altair, which was discovered to have a transiting hot Jupiter by the Kepler mission.

 KOI-13 Light Curve (Source) KOI-13 Light Curve Residuals

You may need to click on the light curve to expand it to full size to see the slight difference between the gravity-darkened and non-gravity-darkened model. If nothing else, the residuals to the non-gravity-darkened model are clear that the star is not uniformly luminous.

The stars where gravity darkening will be most pronounced will of course be the stars which have high absolute rotation rates. While a high v sin Istar implies a rapid rotation, the converse is not necessarily true since the stellar inclination angle is not known a priori. However, because a gravity-darkened star appears rotationally symmetric when viewed pole-on, the effects it has on the transit light curve are greatly diminished for low values for v sin Istar. This conspires to mean that the stars for which gravity-darkening effects on the light curves of transiting planets will be those stars where Doppler spectroscopy is less effective. Thus, while the Rossiter-McLaughlin effect may not be measurable due to the stellar spectral line broadening and corresponding lack of radial velocity precision, one can still in some cases use gravity-darkening to determine the projected stellar obliquity (or misalignment between the planetary orbit axis and stellar spin axis, however one chooses to interpret the system). Your typical system for which this is applicable is a bloated hot Jupiter transiting a rapidly rotating A-type star, and at present, only a couple such such systems are known.

From the perspective of modelling, gravity darkening is essentially asymmetric limb-darkening. This post was designed to give you the idea behind gravity darkening of stars and how it can affect the light curve for a transiting planet. In a future post, we will look at modelling transiting planets orbiting gravity-darkened stars.

## 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.

## Bulk Properties of the Planets

Even without going outside our solar system, we can take a look at the eight that are well known to even our children and get a preliminary grasp on the types of the planets that we can expect to find throughout the Universe. Looking at the graphic above, which has the size of our planetary system’s members to scale (but not their separations from each other), we readily note that the planets available to use for close study assume a wide variety of appearances and sizes. Furthermore, they visually seem to fall into two or three different classifications. Let’s take a more quantitative approach to comparing these planets.

The planets that we are most intuitively familiar with are the smallest members of the planetary zoo. They are the terrestrial planets, and are composed of rock and metal, silicates and iron. They have high densities and sudden phase transition boundaries between states of matter separating their solid and gaseous components (which we might call a “surface”). For our solar system, this corresponds to the planets Mercury, Venus, Earth and Mars.

Because Earth is the most massive representative of this type of planet that we have available to us for study, terrestrial planets of ~Earth-mass are generally called “Earth-like,” but this similarity is in bulk composition alone, and is not to imply habitability. The term is loose enough that even the infernally hellish Venus would fit the description. It’s important to remember that the differences between Earth and Venus are “skin-deep” so to speak. That which makes the two planets so different is confined to a thin layer of gases clinging onto their surfaces, and is a negligible fraction of the planet’s total mass. Planets that are more massive than Earth, typically about twice the mass of Earth and above, are called “super-Earths,” but the term is a loose one.

Being the prototypical terrestrial planet, the mass of Earth defines one of the more commonly used mass measurements. One “earth-mass,” $M_\oplus$, is a mass unit of mass equivalent to the mass of Earth.

Moving further out in the Solar System, we find the gas giant planets Jupiter and Saturn.

The Bulk Composition of the Gas Giants by Mass

These planets are those who have the majority of their mass in the form of “gases,” or at least those elements labelled so on the Periodic Table. As the depth into the planet increases, the atmospheric pressure increases considerably. Deep within the planet, the pressure is great enough that the gas slowly transitions into a liquid, with exotic properties that generate enormously powerful magnetospheres. At their centre may exist a core of solid material of several Earth-masses.

As a planet-forming disk evolves, it’s composition will change such that the gas-to-dust ratio drops until all of the gas has either fallen into the star, been accreted by planets, or pushed out of the planetary system through the radiation pressure of sunlight from the star. Therefore, planets that form first will more closely resemble the composition of the disk where they were formed. We can tell therefore that Jupiter formed first as its composition most closely resembles not only the sun, but that of the protoplanetary disk out of which it formed as well. In the case of gas giants, large solid cores of several Earth-mass grow in size until they begin runaway accretion of hydrogen from the disk (this typically occurs at a mass of ~10$M_\oplus$). Furthermore, Saturn formed later, as is evidenced by the lower gas-to-rock fraction.

Jupiter is the most massive example of this class of planets in our Solar system and therefore sets another unit of mass, the “Jupiter-mass,”$M_J$, roughly equal to 318$M_\oplus$,

Further out in the Solar system lie the planets that formed later, and had access to only the scraps of what was left of the hydrogen and helium in the disk. Their compositions are dominated by the ices that condensed out of the volatiles in the disk. These planets are called the “ice giants,” and are represented by Uranus and Neptune.

Bulk Composition of the Ice Giants by Mass

Like the gas giants, their atmospheres get thicker with depth but they reach a boundary that is more compositional than arising from a state of matter, where the composition goes from an H-He envelope to an icy mantle, likely dominated by water ice with other volatiles (ammonia, methane, etc). There is likely not a quick phase transition between corresponding to a surface on such planets.

Neptune is the most massive of these planets in our Solar System, and is thus the prototypical “Neptune-like planet.” A unit of mass based off Neptune can obviously be conceived, but it is not used much in practice. With Neptune having a mass of ~17$M_\oplus$, the masses for these planets are usually expressed in terms of$M_\oplus$or$M_J$.

When the masses and radii of the Solar System planets are plotted, they group fairly nicely sorted by their type.

Mass-Radius Diagram for the Solar System

In an upcoming post, we’ll look at doing this for extrasolar planets to see how our solar system fits into the range of extrasolar planet masses and radii and how well these neat cookie-cutter categories hold up in the wealth of examples of extrasolar planets with measured masses and radii to date.