Student uses NASA data to reveal new details on planets in other solar systems AAS_Publishing
1. IntroductionThrough the detection of thousands of transiting planets—coupled with a well-studied completeness function—the Kepler mission has facilitated detailed investigations of planetary demographics. Using planet radius and period constraints from the Kepler data, prior studies demonstrated the prevalence of small planets , constrained the multiplicity distribution of planetary systems , identified the Kepler dichotomy , and revealed a gap in the radius distribution of planets between 1.
Thus far, masses have been constrained in >50 multiplanet systems using TTVs. These studies were restricted to systems in which the TTVs are particularly strong. Of the 145 planets in 55 systems characterized by Hadden & Lithwick , the average excess scatter ratio is >12, where the excess scatter ratio is defined as the standard deviation of the observed TTVs divided by the mean error of the mid-transit time measurements; for the remaining systems, the mean excess scatter ratio is ∼2.
Finally, systems where a pair of adjacent planets is within 1% of a first-order mean-motion resonance are rejected. These systems likely require tuned orbital elements for long-term stability, which is incompatible with the broad and independent priors on orbital elements adopted herein. For instance, Gillon et al. initially found that most samples from their fits to TRAPPIST-1 become unstable within 0.5 Myr; however, Tamayo et al.
This method is motivated by the apparent excess of scatter in KOIs' TTV variance relative to their mean measurement uncertainties; see Figure 1, the distributions of excess scatter ratio for planets in single-planet and multiplanet systems. However, as noted in Section 2, the excess scatter ratio is a qualitatively useful, but nonrigorous, summary of TTV time series, because it does not consider the number of observed transits.
In Section 3.1, we first outline the decomposition of TTVs into three independent components: noise from measurement uncertainties, stellar noise, and planet–planet perturbations. In Section 3.2, we constrain the distribution of stellar noise for the Kepler sample via hierarchical Bayesian inference of the single-planet systems. Then, in Section 3.3, we discuss our method of generating posteriors on planet mass in multiplanet systems using low-S/N TTVs.
where Φplanets,i,j is the mid-transit time variation induced by planet–planet perturbations, and Φ⋆,i,j is the mid-transit time variation due to activity on the host star's surface . Prior studies of TTV mass estimation often hinge on modeling Φplanets,i,j as a function of time and either neglect Φ⋆,i,j or statistically account for Φ⋆,i,j as white noise. However, such modeling of Φplanets,i,j relies on high-S/N data.
where σmid,i,j 3 is the uncertainty in the mid-transit time measurement. σmid,i,j is influenced by the transit S/N, the uncertainty in detrending long-term stellar activity from Kepler light curves, and the finite integration time of Kepler . For planet i around star j, σmid,i,j is set equal to the mean of the mid-transit time measurement uncertainties from Holczer et al. .
3.2. Underlying Distribution of Transit Timing Variations from Stellar NoiseUsing the above decomposition of , planet mass can be constrained by isolating the contribution from Vplanets,i,j . This relies on first inferring the distribution of V⋆,i,j for the Kepler sample. To infer the V⋆,i,j distribution, we consider the subpopulation of single-planet systems. In principle, systems with only one detected planet may harbor several nontransiting planets.
The above procedure is motivated by the expectation that stellar noise sources induce scatter into observed TTV time series. In practice, the hierarchical model constrains the distribution of excess scatter relative to the reported measurement uncertainties but is agnostic to the origin of the excess scatter.
δ is the Dirac delta function, and is the TTV variance for the ith planet calculated from forward modeling of the system for a sufficiently long baseline. To evaluate , we integrate systems for 10,000 days using TTVfaster . Compared to other methods of TTV calculation, TTVfaster is computationally efficient, is publicly accessible, and scales to higher multiplicities.Assuming V⋆,i,j is uncorrelated between planets is an incomplete treatment of the TTV signal.
For a significant fraction of planets , low-S/N TTVs place informative constraints on planetary composition. In Figure 7, we present M95% for each planet alongside the planets' radii. For a planet of radius R, the low-S/N TTV-derived mass constraint is considered informative if M95% 20 planets. Similar to the selection of informative mass constraints, planets are considered likely volatile-rich if M95% 30% of the sample).
For this analysis, we consider the synthetic Kepler catalogs of the Exoplanet System Simulator . In this model, planets' orbital periods and radii are drawn from a series of underlying distributions, while the planets' eccentricities and mutual inclinations are assigned such that the system has an angular momentum deficit at the critical value for stability.
5.2. Alternate Treatment of Stellar NoiseIn Section 3, the TTV variance of a given planet was decomposed into three sources: noise from measurement uncertainties , stellar noise , and planet–planet perturbations . The distribution of V⋆,i,j was then inferred via hierarchical modeling of the sample of systems with only one detected transiting planet. In practice, the hierarchical model inferred the distribution of excess scatter relative to the reported measurement uncertainties.
Following the Monte Carlo procedure outlined in Section 3.3, we generate posteriors for each multiplanet system in our sample using Equation as the likelihood function. In Figure 10, we present the upper bounds on planet mass M95% derived using our standard treatment of V⋆,i,j against the upper bounds derived using our alternate treatment.
For high-S/N TTVs, model posteriors often show strong planet–planet correlations, particularly between planet phases . However, since our method only considers TTV amplitude, correlations between adjacent planets' longitudes of periastron likely indicate the system is near resonant, and that the system goes unstable for certain longitude configurations.
6. ConclusionsPrior studies inferred planet mass from TTVs by modeling TTVs as a function of time using either numerical integration , first-order in eccentricity and mass analytic solutions , or a linear combination of basis functions . However, such methods are restricted to high-S/N TTVs.