## Mapping Extrasolar Planets III. Visible Light Surface Features from Kepler

Kepler-7b

In our previous looks at mapping extrasolar planets, we have focused on spatial variations in infrared brightness, and thus temperature, on the visible surface of extrasolar gas giant planets. The reason for this bias toward longer wavelengths is two-fold: 1) At the time, Spitzer phase curve photometry has been the dominant means of deriving crude longitudinal maps of extrasolar planets. 2) The ratio between the flux from the star and the planet is an order of magnitude less in infrared than it is in visible light – also the reason that the successes from direct imaging of extrasolar planets have been almost exclusively in infrared (the nature of the object imaged at Fomalhaut by HST in visible is unknown). Short-period extrasolar planets are thus prime candidates for secondary eclipse observations in the infrared. The following table lists several prominent planets and the eclipse depths as measured in four infrared channels by Spitzer, known as the IRAC channels (Infrared Array Camera) which have been a cornerstone of the building up of the foundation of our understanding about extrasolar planets in the past decade.

 Spitzer IRAC Eclipse Depths Planet 3.6µm 4.5µm 5.8µm 8.0µm HAT-P-7 Ab 0.098% 0.159% 0.245% 0.225% Kepler-5 b 0.103% 0.107% – – Kepler-6 b 0.069% 0.151% – – HD 189733 b 0.256% 0.214% 0.310% 0.391% HD 209458 b 0.094% 0.213% 0.301% 0.240%

Of course the eclipse depth of a planet depends on its intrinsic brightness at the observed wavelength, the brightness of the star itself, and the radius ratio between the two, and given the comparitive brightness and size of a star, it is not hard to see why the drop in flux from the system during the planet’s eclipse behind the star would be so miniscule, < 1%.

The difficulty involved is exaggerated in visible wavelengths, where the flux is dominated less by thermal emission of the two bodies and more by intrinsic processes within the star and how reflective the planet is of the star’s light. Here are some secondary eclipse depth measurements from Kepler, which observes in a filter that is approximately visible light (0.400 – 0.865µm).

 Kepler Eclipse Depths Planet ΔF HAT-P-7 Ab 0.0069% Kepler-5 b 0.0021% Kepler-6 b 0.0022% Kepler-7 b 0.0042% Kepler-12 b 0.0031%

Notice that the eclipse depths in visible light are much lower than the eclipse measurements in infrared. For planets with both Spitzer (infrared) and Kepler (visible light) eclipse depth measurements, the contrast is clear: the star-planet brightness ratio is less in infrared than it is in visible light and therefore hot Jupiters are essentially relatively brighter in infrared than in visible.

This does not exclude extrasolar planets from visible light mapping using the phase curve analysis and eclipse scanning techniques described before, it only makes it harder. Optical phase curves are weaker than infrared phase curves, so either a bigger telescope is needed to observe with enough photometric precision to resolve the phase curve in one orbit, or several orbits must be observed with existing instrumentation and allow the data to build up until a phase curve can be resolved.

Fortunately, Kepler has offered quasi-continuous coverage of the transits of many hot Jupiters over the course of four years, and in some cases, it is enough to confidently detect a phase curve. One such planet is Kepler-7 b.

Kepler-7b Phase_Curve

In this image, the green curve corresponds to the expected phase curve if the planet reflected light in a geometrically symmetric way. The red and blue curves are fitted to the data and incorporate a longitudinal offset (see Demory et al for details). The primary transit is on the left side, and the planet passes behind the star on the left side (the secondary eclipse). The depth of the primary transit is so deep on this scale that it is off the image. Notice that the phase curve is not perfectly phased to the orbit of the planet – the secondary eclipse does not occur at the peak apparent brightness. This would be comparable to the Moon being brightest not when it is full, but rather at a gibbous phase. Keep in mind that this is an optical phase curve, the brightness variations on the planet’s dayside hemisphere suggested by the phase curve corresponds to actual features you would be able to see with the human eye (and maybe a welding helmet).

The Kepler-7 b phase curve shows us that the brightest part of the planet’s dayside atmosphere is on the westward side of the dayside hemisphere. Because this phase curve is an accumulation of 14 quarters of Kepler data, it is best thought of as an “average” phase curve over the course of 3.5 years, and therefore the longitudinally resolved visible light map of the planet is an average of the planet’s surface brightness over 3.5 years. Since the planet does not contribute to the phase curve of the system during secondary eclipse, the scatter of the data during that secondary eclipse is a good representative of the overall data scatter. It may come from instrumental noise or stellar noise, but whatever its origin, the fact that it is not obviously different from the scatter in the phase curve when the planet’s brightness is contributing implies that the surface features resolved here are both stable and long-lived.

Kepler-7b longitudinal brightness distribution

What could be the cause of this bright area on the planet? Demory et al explain:

Kepler-7b may be relatively more likely to show the effects of cloud opacity than other hot Jupiters. The planet’s incident flux level is such that model profiles cross silicate condensation curves in the upper, observable atmosphere, making these clouds a possible explanation. The same would not be true for warmer planets (where temperatures would be too hot for dayside clouds) or for cooler planets (where silicates would only be present in the deep, unobservable atmosphere). Furthermore, the planet’s very low surface gravity may play an important role in hampering sedimentation of particles out of the atmosphere.

Now how do we know that this bright spot is not simply due to thermal emission? Some hot Jupiters are sufficiently hot that the glow from their heat in visible light can affect, or even dominate their eclipse depths. The obvious answer would be to check Kepler-7 b’s eclipse depths in the infrared, and this is a job for Spitzer. Spitzer observed the secondary eclipse of Kepler-7 b in both 3.6 µm and 4.5 µm, and in both wavelengths, the eclipse of the planet behind the star was not confidently detected. This means that the planet isn’t just not hot enough to produce the brightness asymmetry in the Kepler phase curve, but it’s eclipse couldn’t even be confidently detected in infrared. This firmly rules out thermal emission as the source of the optical phase curve asymmetry.

This represents the first time that visible light “surface” features have been identified on an extrasolar planet. This is but a baby-step forward in our ability to map extrasolar planets, but it is a milestone nonetheless. Kepler data may be able to detect phase curve asymmetries in other hot Jupiter systems (or if we are lucky, smaller worlds!), and this can significantly contribute to our understanding of the atmospheres of these planets.

## Modelling the Transit Light Curve

The Sun (Credit: NASA)

We looked at a simplistic understanding of planetary transits and transit light curves (here). In the interests of accuracy, it would be worth investigating a more formal understanding of the transit light curve. Specifically, this post will be aimed at actually modelling transit light curve while taking into account some more complicated dynamics such as limb darkening.

The actual shape of the transit light curve can be mathematically described by a function of the sky-projected centre-to-centre distance between the star and the planet, z, where z = 0 is understood to be the mid-transit of a centrally-crossing transit and the impact parameter, b is understood to be the minimum value of z throughout the transit, and the transit starts at about z ~ 1. More accurately, first and fourth contact start at $z = R_* + R_p$, second and third contacts start at $z = R_* - R_p$. Furthermore, let’s define p as the ratio of the radii of the planet and the star, $p = R_p/R_*$.

Let us now define a function of the total fractional area of the star that is not obscured by the planet. Normally you might be inclined to feel that $(R_p/R_*)^2$ suffices, however it describes only the area ratio between second and third contacts, and is thus inadequate to describe the entire transit. We therefore will construct the amount of obscured stellar area as a piecewise function of p and z whose intervals describe different parts of the transit. It will be given as the ratio between the total area and the unobscured area subtracted away from the whole total area. Specifically, $A(p,z) = 1 - \lambda(p,z)$.

$\displaystyle \lambda(p,z) = \begin{cases} 0, & 1 + p < z\\ \frac{1}{\pi} \left[ p^2\kappa_0+\kappa_1 - \sqrt{\frac{4z^2-(1+z^2-p^2)^2}{4}} \right], & \lvert 1 - p\lvert < z \le 1 + p\\ p^2, & z \le 1 - p\\ 1, & z \le p - 1 \end{cases}$

Where $\kappa_1 = cos^{-1}[(1 - p^2 + z^2)/2z]$ and $\kappa_0 = cos^{-1}[(p^2 + z^2 - 1)/2pz]$. There’s a lot going on here but it isn’t too complicated to understand. The first case, $1 + p < z$, describes when the planet is completely off the stellar disc, before first contact hence there is no (zero) obscured area. The second case is the most complicated and describes the time between first and second (third and fourth) contacts. The third case is what we covered earlier, where the amount of obscured area is simply the square of the ratio of their radii, and therefore simply $p^2$. The final case is only satisfied in the (unlikely) condition that the planet-to-star radius ratio > 1, and the entire area of the stellar disc is obscured.

Calculating z as a function of time may be done with knowledge of the impact parameter and use of Pythagorean theorem.

For a transiting planet with an impact parameter of zero and a radius ratio of $R_p/R_* = 0.1$ (producing a 1% transit depth), the above model produces the following light curve plot.

Simulated Light Curve

Real light curves are actually curved, however, whereas this model is clearly much more “rigid” in shape. The reason for this is an effect known as limb darkening, where the outer layers of the star are not able to isometrically scatter light from underneath, causing less light from the limb of the star to reach the observer than from the centre of the stellar disc. For this reason, stars generally appear dimmer at their limbs and brighter in their centres. The magnitude of this difference will depend on a number of factors intrinsic to each star.

Proper handling of limb darkening will be required to obtain high-fidelity fits to high-accuracy data. A particularly popular and effective model for limb darkening was presented by Claret (2000). We can consider a four-parameter limb-darkening law describing the relative brightness of a point on the star expressed as a function of the angle between the observer, the centre of the star, and a line between the stellar centre and the given point on the star.

$\displaystyle I(r) = 1 - \sum^4_{n=1} c_n(1 - \mu^{n/2})$

Where $\mu$ is given as $\cos \theta = \sqrt{1 - r^2}$, and each value of c is called a “limb darkening coefficient.” In the case of the centre of the stellar disc, $I(0) = 1$. Each star will have a unique set of limb darkening coefficients, and they will change depending on what wavelength the transit is being observed in (due to how the properties of the stellar photosphere’s light scattering varies with wavelength). One could choose to have as many limb darkening parameters (and thus coefficients) as desired, but a high number is rarely required to adequately fit transit light curve data.

Limb darkening must be taken into account to get accurate measurements of radius ratios of transiting objects, and thus planetary radii. Because limb darkening will concentrate the brightness of the star into the centre of the disk, a planet that would cause a 1% transit depth in a uniformly bright star would cause a deeper transit depth in a central transit around limb-darkened star, and a shallower transit depth in a grazing transit around a limb-darkened star. Proper care in modelling the limb-darkening behaviour of the star is therefore required to ensure good measurement of the planetary radius.

We see therefore that the brightness difference is not a pure function of the ratio of the stellar and planetary radii. $\Delta F = (R_p/R_*)^2$ is only an approximation, and only correct for the unrealistic case of a uniformly bright stellar disc.

So what can we do to get a more comprehensive representation of a light curve? Let’s modify the equation for $\Delta F$ we’re familiar with and consider brightnesses instead of just purely area ratios. Let’s take our limb-darkening equation as the radius of the star and determine the area, since it describes the brightness of the star from centre to limb. With this equation as the stellar “radius,” we use some simple calculus to expand it to the “brightness area” of the circle that is the stellar disc. This calculates the total brightness of the star while taking into consideration the changing brightness behaviour from centre to limb. The actual transit depth of a planet at any given distance to the centre of the star (represented as z again) is simply the ratio of the total stellar brightness in the area of the star under the planet to the total brightness of the stellar disc. Therefore, we draft a new function based off our original understanding of $\Delta F$, while taking into effect the above model for how much of the star’s area is covered at any given time in the transit, as well as bringing in our limb-darkening model to describe the brightness distribution of the stellar disc:

$\displaystyle F(p,z) = \left[ \int^1_0 dr 2rI(r) \right]^{-1} \int^1_0 dr I(r)\frac{d[A(p/r,z/r)r^2]}{dr}$

And it makes quite a difference, too. By incorporating limb darkening coefficients, we’re able to much more accurately simulate the transit of a planet across a limb-darkened star. The diagram below shows the same planet (with p = 0.1) transiting stars of different limb-darkening parameters. The shape and modification to the transit depth is readily apparent.

Effects of Limb-Darkening coefficients