The Darkening Limb

Solar Dynamics Observatory

In principle, discovering new exoplanets is pretty easy. Simply measure the brightness of a star over time, and when a planet passes in front of the star, the brightness will dim slightly. The more the brightness dips, the larger the planet in relation to the star. This transit method is so effective it is how we have found the majority of exoplanets. But astronomers want to do much more than simply discover planets, and for that you need to dive into the details.

The amount of drop in a star’s brightness during a planetary transit is called the transit depth. You can see this on light curve graphs, where there is an average brightness of the star versus the maximum dip. The problem is that transit depth is pretty small. A star usually only dims by a fraction of a percent during a transit. And the brightness of a star is never constant. Flares and sunspots can cause a star to vary in brightness. Sometimes those variations are nearly as large as the transit depth itself, so things can get quite fuzzy.

Mercier, et al

This is where the statistics come in. To determine the transit depth, you take the average measured brightness of the star and compare it to the average brightness of the transit to get the depth. Obviously, that result has its own uncertainty, and that means our knowledge of the size of an exoplanet is equally uncertain. But a new study shows how you can lower that uncertainty significantly, and it involves an interesting effect known as limb darkening.¹

Wikipedia user PAR

Limb darkening is where the edge of a star appears slightly dimmer than the center of a star. The effect is due to our line of sight. Light coming from the center of a star originates deeper within the star, where things are hotter. Light coming from the edge of the star has to go through more of a star’s atmosphere to reach us, so it originates from a cooler region closer to the surface. All stars have this effect, depending on the thickness of the star’s upper atmosphere.

As this recent study notes, limb darkening affects transit depth statistically. As an exoplanet starts to transit a star, it first passes through the darkened edge region, which means the amount of dimming is less than we’d expect. Only when the planet passes through the brighter central region is the transit depth accurate. As a result, the statistics of the transit depth have a larger uncertainty.

The authors note that this uncertainty can be reduced by taking limb darkening into account. If we know the amount and scale of the limb darkening, then the data of a transit can be modified to account for it. They find that with an accurate measure of the transit depth, accuracy can be increased by a factor of five.

Unfortunately, we have only observed limb darkening for the Sun and a few other stars, such as Betelgeuse. You need to be able to see a star as more than just a point of light, and for most stars that isn’t possible. But we can simulate the limb-darkening effect. By knowing the type of star and where it is on the main sequence of stars, we can model a star’s upper atmosphere. We can determine these things by looking at the spectra of a star, so it is quite feasible to use the approach the authors suggest.