• Erik Petigura

A Deep Dive into the Fulton Gap

Updated: Jun 30

A common saying in my field is “know thy star.” Often, when we discover a new planet our understanding of its properties is limited by our knowledge of its host star. One example is planet size (Rp) measured by the transit technique. The transit depth (the amount of starlight blocked) only tells you the relative size of planet and star, Rp/Rstar. Therefore, uncertainty in Rstar translates to uncertainty in Rp.

This was a major challenge during the prime Kepler­­ mission, which lasted from 2009 to 2013. At the time, most of the stars had radius uncertainties of roughly 40%. That’s a big deal because planet sizes couldn't be measured to better than 40% accuracy. (As a historical aside, Eratosthenes of Cyrene measured the size of the Earth to ~2% precision over 2000 years ago.) We know today that most Earth-size planets (Rp = 1.0 Re) are rocky and most 2.0 Re planets have gaseous envelopes. So, a planet with measured size of Rp = 1.4 Re but a 40% radius uncertainty could plausibly be 1.0 or 2.0 times the size of Earth, and thus either gaseous or rocky.

For me, one of the most personally rewarding projects of the last few years has been the California-Kepler Survey (CKS). Our team's goal was better understand the properties of some 1300 Kepler host stars (and hence the planets themselves) by analyzing stellar spectra we observed using the Keck telescope. For these stars, we managed to improve the measurements of stellar size from 40% to 10% errors. When we did this, a new feature emerged in the exoplanet population—a gap separating the population of rocky "super-Earths" and gaseous "sub-Neptunes." This feature is colloquially known as the Fulton Gap after BJ Fulton who described it in a 2017 paper he led as part of the CKS project.

Figure from Fulton et al. (2017).

One of the most remarkable aspects of this discovery was that several groups predicted this feature theoretically before it was observed [1,2,3,4]. These groups showed that X-ray and UV radiation, which stars produce copiously during their infancy, will tend to erode gaseous envelopes. This naturally herds planets into these two size groups. Since our discovery, other envelope sculpting mechanisms have been proposed [5].

To me, one of the most intriguing implications of the Fulton Gap is that Nature seems to like to producing rocky cores of a certain size and mass; if there was no preferred size, we would not see a gap. The exact mass is still debated, but it's likely around 5 times that of Earth. We don't yet know why Nature seems to prefer this mass, nor do we know why Nature abstained from making such planets in our solar system. There are still plenty of mysteries waiting to be solved.

Since 2017, I've continued to wonder whether the Fulton Gap could be wider and/or deeper than it appears above. By eye, the gap appears to span 1.5 to 2.0 Re. In a fractional sense, that's roughly 30%, only a few times larger than the ~10% uncertainties in the Rp. Even if the gap was completely devoid of planets, we would still expect to see it partially filled in due to measurement uncertainties.

The possibility that this feature could be even sharper grew more tantalizing in 2018 when Vincent Van Eylen and collaborators produced even more precise planet radii in a smaller sample of roughly 100 stars (~3% errors). The figure below shows the measured sizes of planets from the CKS sample (small points) and the asteroseismic sample (large points). The gap in the asteroseismic sample appears wider and more devoid of planets.

Figure from Van Eylen et al. (2018).

Their extra precision was enabled by two factors. First, the host stars had exquisite radius measurements using a technique called asteroseismology. Second, to perform the asteroseismic measurements Kepler observed these stars at faster cadence than most (1-min exposures instead of the typical 30-min), which enabled more precise measurements of Rp/Rstar (more on this later). Ideally, Kepler would observe all stars at short cadence, but data bandwidth limitations meant that only a small fraction of Kepler stars were observed in this way. Unfortunately, asteroseismic measurements are only available for ~5% of the Kepler planet hosts so we couldn't apply these techniques to the full CKS sample.

In 2018, the European-led Gaia spacecraft enabled a new breakthrough because it could precisely measure distances to Kepler field stars, which are typically a few thousand light-years from Earth. Knowledge of a star's distance is very useful in measuring its size. Gaia uses parallax which is the apparent shifting of a star due to Gaia's changing perspective as it orbits the Sun. (Parallax is also plays an important role in human depth perception.) More distant stars shift by smaller amounts, and a typical Kepler star at 3000 light-years shifts by a mere 0.001 arc-seconds. This shift is equivalent to a size of a dime on top of the Eiffel tower as seen from New York City!

BJ and I teamed up to refine the CKS stellar sizes by incorporating Gaia measurements to see if we could further resolve the gap. Planet radius errors are due to a combination of uncertainty on Rstar and Rp/Rstar. We estimated these errors at ~10% and ~5%, respectively, and meaning the stellar uncertainties were the "tallest tent pole." We knew we could improve them significantly with Gaia and thus improve the planet size precision.

On April 25, 2018, the date of the Gaia data release, we woke up at 3AM because we were so excited to see what this new data might reveal. In a few days, we reduced the stellar size uncertainties from 10% to 2% and recomputed the planet radii. To our slight disappointment, the sharpness of the gap did not change dramatically.

CKS planet sample before (left) and after (right) the incorporation of Gaia distances. The radius valley is not noticeably sharper in the second sample.

What accounts for the discrepancy between our view and the asteroseismic view? In Fulton and Petigura (2018), F18 hereafter, we attributed these differences to differences in the host stars. Most of the CKS stars are between 0.8 and 1.2 solar-masses, while the Van Eylen et al. (2018) asteroseismic sample, V18 hereafter, included more massive stars of 1.0 to 1.4 solar-masses. Variations in the location or width of the gap with stellar mass could explain the different views. The figure below indicates that the gap moves toward larger planet sizes with increasing stellar mass.

Figure from Fulton and Petigura (2018)

But that's not the full story.

In a recent paper, I took a deep dive into the V18 and F18 samples and found an additional factor that contributes to the different views. I found that we had underestimated the contribution of Rp/Rstar errors to Rp uncertainties in F18. We adopted the values of Rp/Rstar and their uncertainties measured by the Kepler project and tabulated on the Exoplanet Archive. I found that these are underestimated by about a factor of two. So in F18, instead of achieving ~5% errors on planet size, we were, in fact, stuck at 10%. In other words, the our improvement when incorporating Gaia parallaxes was much more modest than we thought

These errors in Rp relate to subtleties in how Rp/Rstar is measured from a transit. If the stellar disk was uniformly bright, then Rp/Rstar would be directly related to the depth of the transit. However, stars are not uniformly bright disks as this picture of the Sun (taken during a rare transit of Venus) shows


This effect is known as limb-darkening. The light emitting "surface" of the Sun (the "photosphere") is slightly farther from the center of the Sun at the limb (the edge) than at the middle of the stellar disk. The photosphere at the edge is slightly cooler and hence darker. So to accurately measure Rp/Rstar, one must determine whether the planet transits centrally or at some higher impact parameter and then account for the brightness of the star under that transit path.

Unfortunately, impact parameter b, which ranges from 0 (central) to 1 (grazing), is very challenging to measure for most Kepler planets. The uncertainty in b results in an uncertainty on Rp/Rstar because the two parameters are covariant, as the diagram below illustrates.

Transit 1 has a high impact parameter (b = 0.9). Transit 2 is central (b = 0.0), and we have selected Rp/Rstar and the velocity of the planet such that the two long-cadence light curves are in near-perfect agreement (top right). Note that sizes differ by ~20%. The differences between the two transits are more pronounced in short cadence data (lower right). Figure from Petigura (2020); sizes not to scale.

For most Kepler planets, we cannot tell the difference between Transit 1 and Transit 2, even though the planets differ in size by 20%—the same as the width of the radius gap! In principle, this degeneracy can be broken if the light curve is observed at short cadence, although this also requires data of exceptional quality. The high quality (low noise) and short cadence sampling of the asteroseismic sample allowed V18 to break this degeneracy and achieve such precise Rp/Rstar.

For a typical CKS planet, we can't tell the difference from a high-b and a low-b transit from the light curve alone, so we must assume that they are equally likely. There is a standard statistical technique for characterizing the set of credible fits to the data called Markov Chain Monte Carlo or MCMC. Here's the set of credible Rp/Rstar and b values for one Kepler Object of Interest, KOI-85.03.

Figure from Petigura (2020)

The blue region shows the range of credible b and Rp/Rstar values for KOI-85.03 based on the long cadence light curve. Impact parameter is only weakly constrained, and the constraints on Rp/Rstar must absorb this uncertainty. If b is actually between 0 and 0.7, then the uncertainty on b doesn't make a big difference in Rp/Rstar. But if the transit is indeed high-b then we will make a large error in Rp/Rstar. In the case of this transit, if b was actually 0.9, then the true Rp/Rstar should be 1.14% instead of our inferred value of 0.96±0.02. In other words, we would report a value for the planet radius that is 18% too small. If this planet was a sub-Neptune sitting on the upper edge of the Fulton Gap, it would spill into the gap. Because planetary systems should be randomly oriented with respect to the earth, we expect ~20% of the planets to have b = 0.8–1.0; thus ~20% of Kepler planets have radii that are underestimated by 10% to 20%. These act to fill in the gap.

In an ideal world, we would precisely measure the impact parameters of all Kepler planets and correctly their light curves. Unfortunately, there just isn't enough information in the light curves to do that. It turns out we can remove the high-b planets so they don't contaminate the population. But how can we identify and remove the high-b planets if we can't measure b in the first place? There is one additional piece of information that we can bring to bear: the transit duration. Given accurate knowledge of a star's mass and radius (thanks, Gaia) and assuming a circular orbit, one can measure b from the transit duration.

The stellar mass and radius, along with a planet's orbital period will tells you its orbital speed. That's Kepler's Third Law (Kepler the astronomer, not Kepler the spacecraft). For a given speed, a shorter duration means a shorter transit path or higher b. So transit duration can serve as a proxy for b.

In reality, it's not so simple. If a planet's orbit is eccentric (non-circular) its speed will change throughout its orbit (Kepler's Second Law). However, various groups have characterized the distribution of Kepler eccentricities. I built on this work to characterize how effectively transit duration can be used to filter out high-b planets. It works pretty well because most Kepler planets are not too eccentric. I used this technique to clean the sample of Kepler planets from BJ and my 2018 paper and compare it to the V18 sample. Here's what I found:

As a point of reference, here is the F18 sample.

You can see the two planet populations, but the boundaries are far less distinct than in V18. Let's first remove the effect of stellar mass and try to approximate the V18 host star sample:

The two populations are starting to separate, due to the stellar mass dependence of the radius gap. Finally, let's use transit duration to remove as many b > 0.8 planets as possible:

The two populations become still more distinct. This brings the CKS sample into closer agreement with what V18 found in the asteroseismic sample.

So what's the significance of all this? We now have a new technique to sharpen our view of the exoplanet population. Various planet formation models make predictions for how the boundaries of the super-Earth/sub-Neptune population should change with properties like stellar mass, metallicity, and age. Our sharpened view should enhance our ability to test these predictions.

We should also investigate the planets that remain in the gap. Perhaps they are currently experiencing the final stages of envelope loss and are in the process of "hopping" the gap. If true, they may present an opportunity to study mass loss in realtime. We could observe a short-lived but critical period in a sub-Neptune's life as it transforms into a super-Earth. Observations by JWST or other space-based telescopes will be particularly valuable.

While it may seem obsessive to try to measure the size of an exoplanet thousands of light years away to a precision of a few percent, in doing so we have uncovered new features in the exoplanet population that any realistic model of planet formation must reproduce. I wonder what other features are still hidden beneath the blur of our measurement uncertainties.