Grab Your Umbrellas, ’Cause It’s Raining Grazing Planets!

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Title: Accurate Modeling of Grazing Transits Using Umbrella Sampling
Author: Gregory J. Gilbert
Author’s Institution: University of Chicago
Status: Published in AJ

Today’s author uses umbrellas to accurately model the planets that “graze” their stellar hosts.

Planets That Graze on Their Stars

Roughly 75% of all known exoplanets were discovered via transit surveys. These surveys monitor many stars at once to look for dips in brightness that could be caused by a planet passing, or “transiting,” in front of a star. Although rare, some of these planets only “graze” their host stars, meaning that they only partially transit their parent star’s disk (check out this astrobite to learn more about a specific case of a grazing planet).

In astronomical terms, “grazing” planets are defined as those that have an impact parameter that is larger than the ratio of the planet’s radius to the star’s radius. The impact parameter is defined as the distance between the center of the stellar disk and the center of the planetary disk at conjunction, where conjunction is the point in a planet’s orbit where it is most closely aligned with its star, as viewed from Earth. A perfectly centered transit has an impact parameter of 0 while a transit in which only half of the planetary disk passes in front of the stellar disk has an impact parameter of 1.

plot of flux over time for different impact parameters

Figure 1: The impact parameter (the distance between the centers of the stellar and planetary disks at conjunction) changes the shape of a transiting planet’s light curve. On this plot, the flux, or brightness, of the star normalized to 1 is on the y-axis. The time before and after the transit in hours is on the x-axis. Planets that have a high impact parameter graze the disk of their host star during their transit, making it more difficult to characterize a planet using its light curve. [Gilbert 2022]

Figure 1 demonstrates how the shape of the light curve from a transiting planet changes as a function of the impact parameter. The depth of the dip in a light curve allows astronomers to estimate the planet’s radius relative to the star, but this estimation becomes more difficult if the planet is grazing. For example, the light curve of a smaller, non-grazing planet could look the same as the light curve from a larger, grazing planet. One therefore needs to simulate grazing transits even in cases where it is unlikely that the planet grazes its host star.

However, today’s author shows that standard Monte Carlo methods, which are frequently used by exoplanet scientists to model grazing planets, can lead to unreliable results! Identical runs of the same model can return differing results, or results where it is not obvious that the model is wrong (Figure 2). When dealing with a handful of planets, one can let the simulation run for a longer period of time or add additional data, such as the spectrum of the star, to the model. However, for larger samples, a more efficient method is needed. What can astronomers do instead?

plots of posterior distributions

Figure 2: Plots of the posterior distributions from four identical Monte Carlo simulations. The parameters explored are the impact parameter, b, and the ratio of the planet’s radius to the star’s radius, r. Although the simulations are identical at the start, they devolve into four wildly different scenarios. In Panel A, the simulation is mostly consistent with a non-grazing planet (b < 1). In Panel B, the simulation fails to explore entirely whether the planet is grazing or not. In Panel C, the simulation gets caught at the boundary between a grazing and non-grazing planet. In Panel D, the simulation has a bimodal posterior distribution that barely explores whether the planet is grazing at all. [Gilbert 2022]


They can use umbrella sampling! Umbrella sampling is a technique that has been used in other scientific fields for decades, but not by astronomers until recently (specifically, Matthews et al. (2018) was the first to introduce umbrella sampling to the field of astronomy). This technique splits a distribution into sub-regions, draws samples from each of these sub-regions independently, and recombines these samples into a single posterior distribution (Figure 3). The author finds that this technique returns more reliable results than those from standard Monte Carlo methods (Figure 4).

plots of posterior distributions

Figure 3: On the top left, the target distribution is split into three sub-regions, each of which is assigned a function. On the top right, after sampling from each of these sub-regions independently, each sub-region is assigned a biased distribution. On the bottom left, the three unbiased sub-distributions are shown. On the bottom right, the three unbiased sub-distributions are combined into a single posterior distribution. [Gilbert 2022]

plots of posterior distributions

Figure 4: Posterior distributions of radius, impact parameter, and transit duration for a mini-Neptune orbiting a K-dwarf star. The vertical dashed lines represent ground-truth values for this system. These plots demonstrate how standard Monte Carlo methods fail to properly explore the parameters of grazing planets and show that umbrella sampling produces more robust results! [Gilbert 2022]

A good deal of math is needed to properly weight the sub-regions relative to one another; these calculations are described in detail in the article, and a step-by-step tutorial can be found on the author’s GitHub. Nonetheless, the math is worth it — this technique can be used to explore any complicated distribution, so it can be used in fields beyond exoplanet science. This means you should get out your umbrellas, ‘cause it’s gonna be raining grazing planets!

Original astrobite edited by Jana Steuer.

About the author, Catherine Clark:

Catherine Clark is a PhD candidate at Northern Arizona University and Lowell Observatory. Her research focuses on the smallest, coldest, faintest stars, and she uses high-resolution imaging techniques to look for them in multi-star systems. She is also working on a Graduate Certificate in Science Communication. Previously she attended the University of Michigan, where she studied astronomy and astrophysics as well as Spanish. Outside of research, she enjoys spending time outdoors hiking and photographing, and spending time indoors playing games and playing with her cats.