Probabilistic Mass-Radius Relationship(s)

Angie Wolfgang UCSC → Penn State

NSF Postdoctoral Fellow

Eric Ford, Leslie Rogers, SAMSI Bayesian Exoplanet Populations Group

for Sub-Neptune-sized Planets

Sub-Neptune Mass & Radius

Useful to constrain possible compositions. . . but not all can have masses measured

figure courtesy of Eric Lopez

Needed: Mass-Radius Relation

?

Needed: Mass-Radius Relation

Comparing transit and RV detections

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Burke et al. 2015Mayor et al. 2011 Dynamics of

non-TTV Kepler planets

Fabrycky et al. 2014

Schlichting 2014

Formation of Kepler planets and systems

Composition Distribution =

Intrinsic Spread in M-R relationWolfgang & Lopez, 2015

Mass-Radius RelationshipHow exactly to incorporate

dispersion?Weiss & Marcy, 2014

PDF (probability density function)

Large M, R uncertainties;

C, !, "M are population-wide parameters . . .

Hierarchical Bayesian Modeling

What if there isn’t just one “true” value of # for all the data?i.e. # has its own intrinsic distribution?(Happens often for population studies)

“Regular” Bayes:p(!|x) ∝ p(x|!) p(!)posterior likelihood prior

Observables

Parameters

Example:Compositions

Hierarchical Bayesian Modeling“Regular” Bayes:p(!|x) ∝ p(x|!) p(!)posterior likelihood prior

Observables

Parameters

Hierarchical Bayes:p(!,"|x) ∝ p(x|!,") p(!|") p(")

posterior likelihood prior posterior likelihood prior

PopulationParameters

Observables

Individual Parameters

Can still use MCMC . . .“Just” adding another layer of

probabilistic structure

physics

data

HBM for Mass-Radius Relation

HBM for Mass-Radius Relation

Observers, we need your likelihoods!!Please, publish your joint posterior samples *and* your priors!!

Mobs Robs

In practice . . .

Results

Wolfgang, Rogers, & Ford, in prep.

There is intrinsic scatter in the current set of R,M measurements, which an accurate M-R relation must reflect.

probabilistic M-R relation:deterministic M-R relation:

Changing the Data Set

Changing the Data Set

Only the smallest (likely rocky) planets don’t require intrinsic dispersion to

accurately describe the M-R relation.

Probabilistic M-R Relation:

For Individual Planets:Case Study: Kepler-452 b

No mass measurement?Use probabilistic M-R relation to

constrain composition:github.com/dawolfgang/MRrelation

P(has gas) = 0.51

P(rocky) = 0.49

Jenkins et al. 2015

For Individual Planets:Case Study: Kepler-452 b

No mass measurement?Use probabilistic M-R relation to

constrain composition:github.com/dawolfgang/MRrelation

P(has gas) = 0.51

P(rocky) = 0.49

Jenkins et al. 2015

Rogers 2014:“Most 1.6 REarth Planets are Not

Rocky”

Consistent: 452b is NOT conclusively rocky: ~50% chance it is

Caveats and Future Work

• σM is constant → need to impose M > 0 and ρ < ρiron physical constraints at small radii

• Have not accounted for selection bias in sample (could explain RV, TTV discrepancy).

Wolfgang, Rogers, & Ford, arXiv:1504.07557, in review.