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EXONESTThe Exoplanetary ExplorerKevin H. Knuth and Ben PlacekDepartment of PhysicsUniversity at Albany (SUNY)Albany NYKepler MissionThe Kepler mission, launched in 2009, aims to explore the structure and diversity of extra-solar planetary systems.

Knuth and PlacekERCIM 2014

Knuth and PlacekERCIM 2014

Knuth and PlacekERCIM 2014M = model = parametersD = dataI = prior informationBayesian InferenceBayes theorem acts as a learning rule that allows one to update their state of knowledge about the model parameter values given acquired data.PosteriorPriorLikelihoodEvidence (Z)Knuth and PlacekERCIM 2014We have developed the EXONEST Exoplanetary Explorer which is a Bayesian Inference Engine equipped with plug-and-play models of exoplanetary photometric effects.

The system will be made available to the public as open-source code so that third-party development of new photometric models of exoplanetary effects can be readily incorporated.EXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014EXONEST [1] takes Data from the Kepler Space Telescope, CoRoT, etc. and given a specified model, produces model-based parameter estimates as well as the Bayesian evidence for that modelEXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference Engine[1] Placek, Knuth, Angerhausen, 2014arxiv:1310.6764EXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014DataParameter EstimatesModel EvidenceBayesian Inference Engine

Central to EXONEST is the Bayesian Inference Engine, which is based on the Nested Sampling algorithm [2].

The specific current MATLAB implementation is the MultiNest algorithm [3,4,5], but we expect to change this in the near future to enable the handling of many more model parameters.[2] Sivia & Skilling, 2006.[3] Feroz et al., 2008. arXiv:0809.3437[4] Feroz & Hobson, 2007. arXiv:0704.3704[5] Feroz et al., 2013. arXiv:1306.2144EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Basic Stellar and Planetary Model

The basic models describing the recorded flux as originating from a star with orbiting planets are built in. This includes code for modeling the orbital dynamics, as well as describing the common parameters and their prior probabilities.

EXONESTExoplanetary ExplorerInstrument LikelihoodOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineBasic Stellar and Planetary ModelEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Instrument Likelihood

The likelihood function describes the degree to which the model is expected to describe the data. We have experimented with several different likelihoods and have found that a Gaussian likelihood with the free noise parameter sigma to be most reliable.

EXONESTExoplanetary ExplorerOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineBasic Stellar and Planetary ModelInstrument LikelihoodEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Plug-and-Play

The recorded flux can be described by user-defined plug-and-play forward models of both stellar and planetary configurations and photometric effects.

EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsBayesian Inference EnginePlug-and-PlayEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Orbital Models

For a given Keplerian planet, one of two orbital models are considered:

Circular OrbitEccentric Orbit

The Eccentric Orbit model requires an additional eccentricity parameter, which must be defined with appropriate prior values.

To be added in the future are three-body orbital models as in Trojan resonance dynamics and the Kozai mechanism.EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineOrbital ModelsEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Transit Models

Transit models accommodate both primary and secondary transits in the case of transiting planets.

EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineTransit ModelsEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsTransit ModelsAdditional ModelsPlug-and-PlayBayesian Inference EnginePhotometric Models

We have developed photometric models for several distinct physical effects. Two are related to the planet itself:

Reflected LightThermal Emissions

Two others are related to the effect that the planet has on its host star

Doppler Boosting (Beaming)Ellipsoidal Variations (Tidal Warping)

We are currently investigating additional effects and their relative magnitudes.Photometric ModelsEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Additional Models

Additional user-defined models affecting the photometric flux can be accommodated.

Presently, we are considering super-rotation, multiple star systems with star-star reflections, star spot models, optically thin coronal regions, etc.

EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Parameter Estimation

The Nested Sampling-based Inference Engine produces estimates and uncertainties of all of the model parameters employed in the forward problem.

The number of parameters employed has varied in our studies from 7 to 13 or so parameters. This is approaching the limit that MultiNest can handle, which will force us to migrate to a Nested Sampling variant that we are currently developing in-house.

EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineTransit ModelsEXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014Model Testing

The main focus of the Nested Sampling Inference Engine is the computation of the Bayesian evidence of the model.

The evidence is critical in Model Testing [6] which we are finding to be useful as a planet validation aid [1]. This has enabled us to verify circular or eccentric orbits, the relative importance of photometric effects, and even the probability that a system hosts multiple planets.EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineTransit Models[1] Placek, Knuth, Angerhausen, 2014. arxiv:1310.6764[6] Knuth et al. arxiv.org:1411.3013EXONEST Exoplanetary ExplorerKnuth and PlacekERCIM 2014ResultsKnuth and PlacekERCIM 2014KOI-13bKnuth and PlacekERCIM 2014KOI-13

This Image of the KOI-13 system was taken from the 1m RCC telescope at the Konkoly Observatory in Hungary (Szab et al. 2011)The KOI-13 system is a pair of A-type White Dwarf stars located 1630 LY from Earth. Each star is about 2 times as massive as the Sun with a temperature of about 8500K (compared to the Suns temperature of 5778K).Knuth and PlacekERCIM 2014KOI-13A and KOI-13b

In 2011 Kepler detected an object transiting KOI-13A at 0.0367 AU with a period of 25.4 hours (Szab et al. 2011). Initially the object was thought to be a brown dwarf.In 2012 Mislis and Hodgkin (2012) determined that the object was a Hot Jupiter with a mass of 8.3 Jupiter masses and a radius 1.4 times that of Jupiter.Image from (Szab et al. 2011)Note the spin-orbit mis-alignment.Knuth and PlacekERCIM 2014KOI-13b

Out-of-transit data for KOI-13b including fits (dark curve)Eccentric Orbit with reflected light, Doppler beaming, and ellipsoidal variations (Log Evidence: ln Z = 37748 1.1; Residual Sum of Squares: RSS = 3.45e-06)(B) Circular Orbit with reflected light, Doppler beaming, and ellipsoidal variations (Log Evidence: ln Z = 37703 0.9; Residual Sum of Squares: RSS = 3.8e-06).Bayesian model testing favors the eccentric orbit (A) by a factor of exp(45)!PhasePhase[1] Placek, Knuth, Angerhausen, 2014arxiv:1310.6764Knuth and PlacekERCIM 2014As a proof-of-concept, we examined whether the planet can be characterized using photometric effects alone (ignoring the dominant transits)Eccentric OrbitCircular OrbitKOI-13b[1] Placek, Knuth, Angerhausen, 2014arxiv:1310.6764

Ignoring transits and relying only on photometric effects we demonstrate that the planet can still be fairly well characterized.

That is, Kepler could be used to detect and characterize some non-transiting planets!Knuth and PlacekERCIM 2014(Log Evidence)HAT-P-7bKnuth and PlacekERCIM 2014

Star blocks light from planetSinusoidally-varying lightfrom planetShift in baseline flux due to thermal light emission from planet HAT-P-7b (Kepler-2b)Knuth and PlacekERCIM 2014HAT-P-7b (Kepler-2b)

[7] Placek & Knuth 2014arxiv:1409.4152Knuth and PlacekERCIM 2014HAT-P-7b (Kepler-2b)

[7] Placek & Knuth 2014arxiv:1409.4152HAT-P-7b

HAT-P-7b orbits an F8-type star in a circular orbit at 0.0377 AU with a period of 2.2 days.

We estimate HAT-P-7b to a Hot Jupiter that is 1.66 times more massive than Jupiter with 1.634 times the radius.

Its day-side temperature is 2859 33 K whereas the night-side is 1332 756K.

Knuth and PlacekERCIM 2014KIC 54*****Discovery of a Triple Star System in a 10:1 ResonanceKnuth and PlacekERCIM 2014KIC 54*****Photometric data of KIC-54***** obtained from Kepler.Quarter 13 light curve folded on the P1 = 6.45 day period, (B) Quarter 13 light curve folded on the P2 = 0.645 day period(C) is the entire Q13 light curve.

[8] Placek, Knuth, et al. 2014Knuth and PlacekERCIM 2014Digital Sky Survey (DSS)

KIC 54*****Eleven radial velocity measurements taken over the span of a week. The 6.45 day period is visible, but not the 0.645 day period.Courtesy of Geoff Marcy and Howard Issacson

[8] Placek, Knuth, et al. 2014Knuth and PlacekERCIM 2014KIC 54*****Two possible models of the system. The main star is a G-star (like our sun), at least one of the other companions (C1) is M-dwarf.

A hierarchical arrangement (C1 and C2 orbit G with 6.45 day period, and orbit one another with 0.645 day period)A planetary arrangement (C1 orbits with 6.45 day period, and C2 orbits with 0.645 day period)[8] Placek, Knuth, et al. 2014Knuth and PlacekERCIM 2014

KIC 54*****Testing the Hierarchical Model against the Planetary Model using the Radial Velocity DataThe Circular Hierarchical Model has the greatest evidence

[8] Placek, Knuth, et al. 2014Knuth and PlacekERCIM 2014KIC 54*****The KIC 54***** system is a hierarchical triple system G-star pus two co-orbiting M-dwarfs in a 1:10 resonance (P1 = 6.45 day , P2 = 0.645 day)[8] Placek, Knuth, et al. 2014Knuth and PlacekERCIM 2014

Knuth and PlacekERCIM 2014

We have developed the EXONEST Exoplanetary Explorer which is a Bayesian Inference Engine equipped with plug-and-play models of exoplanetary photometric effects.

The system will be made available to the public as open-source code so that third-party development of new photometric models of exoplanetary effects can be readily incorporated.EXONESTExoplanetary ExplorerInstrument LikelihoodBasic Stellar and Planetary ModelOrbital ModelsTransit ModelsAdditional ModelsPhotometric ModelsPlug-and-PlayBayesian Inference EngineKnuth and PlacekERCIM 2014EXONEST Exoplanetary Explorer

Thank you for your kind attentionmeh.ccSpecial thanks to:Daniel Angerhausen, Jon Jenkins, Geoff Marcy, Howard Issacson, Jeff Scargle, Michael Way, John Skilling and the Kepler Mission TeamKnuth and PlacekERCIM 2014Key to Model Parameters