EXONEST
The Exoplanetary Explorer
Kevin H. Knuth and Ben Placek
Department of Physics
University at Albany (SUNY)
Albany NY
Kepler Mission
Knuth and Placek
ERCIM 2014
The Kepler mission, launched in 2009, aims to explore the
structure and diversity of extra-solar planetary systems.
Knuth and Placek
ERCIM 2014
Knuth and Placek
ERCIM 2014
Bayesian Inference
Knuth and Placek
ERCIM 2014
Bayes theorem acts as a learning rule that allows one to
update their state of knowledge about the model
parameter values given acquired data.
Likelihood
Prior
𝑃 𝜃 𝑀, 𝐼 𝑃 𝐷 𝜃, 𝑀, 𝐼
𝑃 𝜃 𝐷, 𝑀, 𝐼 =
𝑃 𝐷 𝑀, 𝐼
Posterior
M
θ
D
I
Evidence (Z)
= model
= parameters
= data
= prior information
EXONEST Exoplanetary Explorer
Knuth and Placek
ERCIM 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.
EXONEST Exoplanetary Explorer
Knuth and Placek
ERCIM 2014
EXONEST
Exoplanetary Explorer
Data
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Parameter Estimates
Model Evidence
EXONEST [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
model
[1] Placek, Knuth, Angerhausen, 2014
arxiv:1310.6764
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Knuth and Placek
ERCIM 2014
Bayesian 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.
Orbital Models
Transit Models
Photometric Models
Additional Models
[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:1306.2144
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Knuth and Placek
ERCIM 2014
Basic 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.
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Knuth and Placek
ERCIM 2014
Instrument 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.
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Knuth and Placek
ERCIM 2014
Plug-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.
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Knuth and Placek
ERCIM 2014
Orbital Models
For a given Keplerian planet, one of two
orbital models are considered:
Circular Orbit
Eccentric 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.
EXONEST Exoplanetary Explorer
Knuth and Placek
ERCIM 2014
EXONEST
Exoplanetary Explorer
Transit Models
Bayesian Inference
Engine
Transit models accommodate both primary
and secondary transits in the case of
transiting planets.
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Knuth and Placek
ERCIM 2014
Photometric Models
We have developed photometric models
for several distinct physical effects. Two
are related to the planet itself:
Reflected Light
Thermal Emissions
Two others are related to the effect that
the planet has on its host star
Doppler Boosting (Beaming)
Ellipsoidal Variations (Tidal Warping)
Photometric Models
Additional Models
We are currently investigating additional
effects and their relative magnitudes.
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Knuth and Placek
ERCIM 2014
Additional Models
Additional user-defined models affecting
the photometric flux can be
accommodated.
Presently, we are considering superrotation, multiple star systems with starstar reflections, star spot models, optically
thin coronal regions, etc.
EXONEST Exoplanetary Explorer
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
Knuth and Placek
ERCIM 2014
Parameter 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.
EXONEST Exoplanetary Explorer
Knuth and Placek
ERCIM 2014
EXONEST
Exoplanetary Explorer
Model Testing
Bayesian Inference
Engine
The main focus of the Nested Sampling
Inference Engine is the computation of the
Bayesian evidence of the model.
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
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.
Transit Models
Photometric Models
Additional Models
[1] Placek, Knuth, Angerhausen, 2014. arxiv:1310.6764
[6] Knuth et al. arxiv.org:1411.3013
Knuth and Placek
ERCIM 2014
Results
Knuth and Placek
ERCIM 2014
KOI-13b
KOI-13
Knuth and Placek
ERCIM 2014
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 Sun’s temperature of 5778K).
This Image of the KOI-13 system was taken from the 1m RCC telescope at the
Konkoly Observatory in Hungary (Szabó et al. 2011)
KOI-13A and KOI-13b
Knuth and Placek
ERCIM 2014
In 2011 Kepler detected an object transiting KOI-13A at 0.0367 AU with a period
of 25.4 hours (Szabó et al. 2011). Initially the object was thought to be a brown dwarf.
Image from (Szabó et al. 2011)
Note the spin-orbit mis-alignment.
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.
Knuth and Placek
ERCIM 2014
KOI-13b
As a proof-of-concept, we examined whether the planet can be characterized
using photometric effects alone (ignoring the dominant transits)
Eccentric Orbit
Phase
Circular Orbit
Phase
Out-of-transit data for KOI-13b including fits (dark curve)
(A) 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)!
[1] Placek, Knuth, Angerhausen, 2014
arxiv:1310.6764
KOI-13b
Knuth and Placek
ERCIM 2014
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 nontransiting planets!
(Log Evidence)
[1] Placek, Knuth, Angerhausen, 2014
arxiv:1310.6764
Knuth and Placek
ERCIM 2014
HAT-P-7b
HAT-P-7b (Kepler-2b)
Knuth and Placek
ERCIM 2014
Sinusoidally-varying light
from planet
Shift in baseline flux due to
thermal light emission
from planet
Star blocks light from planet
HAT-P-7b (Kepler-2b)
Knuth and Placek
ERCIM 2014
[7] Placek & Knuth 2014
arxiv:1409.4152
HAT-P-7b (Kepler-2b)
Knuth and Placek
ERCIM 2014
HAT-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.
[7] Placek & Knuth 2014
arxiv:1409.4152
Knuth and Placek
ERCIM 2014
KIC 54*****
Discovery of a Triple Star System in a 10:1 Resonance
KIC 54*****
Knuth and Placek
ERCIM 2014
Digital Sky Survey (DSS)
Photometric data of KIC-54***** obtained from Kepler.
(A) 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. 2014
KIC 54*****
Knuth and Placek
ERCIM 2014
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. 2014
KIC 54*****
Knuth and Placek
ERCIM 2014
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) A hierarchical arrangement
(C1 and C2 orbit G with 6.45 day period, and orbit one another with 0.645 day period)
(B) A planetary arrangement
(C1 orbits with 6.45 day period, and C2 orbits with 0.645 day period)
[8] Placek, Knuth, et al. 2014
KIC 54*****
Knuth and Placek
ERCIM 2014
Testing the Hierarchical Model against the Planetary Model using the Radial Velocity Data
The Circular Hierarchical Model has the greatest evidence
[8] Placek, Knuth, et al. 2014
KIC 54*****
Knuth and Placek
ERCIM 2014
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. 2014
Knuth and Placek
ERCIM 2014
EXONEST Exoplanetary Explorer
Knuth and Placek
ERCIM 2014
EXONEST
Exoplanetary Explorer
Bayesian Inference
Engine
Basic Stellar and
Planetary Model
Instrument Likelihood
Plug-and-Play
Orbital Models
Transit Models
Photometric Models
Additional Models
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.
Thank you for your kind attention
meh.cc
Special thanks to:
Daniel Angerhausen, Jon Jenkins, Geoff Marcy, Howard Issacson, Jeff Scargle, Michael
Way, John Skilling and the Kepler Mission Team
Knuth and Placek
ERCIM 2014
Key to Model Parameters
M* or Ms
R* or Rs
Mass of Star
Radius of Star
(Solar mass units)
(Solar radii)
Mp
Rp
Mass of Planet
Radius of Planet
(Jupiter mass units)
(Jupiter radii)
Ag
Td
Tn
Geometric Albedo
Day-side Temperature
Night-side Temperature
(unit less [0,1])
(Kelvin)
(Kelvin)
T or P
e
i
Orbital Period
Orbital Eccentricity
Inclination of Orbital Plane
(days)
(unit less [0,1])
(degrees)
Mo
ω
K
𝛾
Mean Anomaly
Argument of the Perisatron
Radial Velocity Semi-amplitude
Systemic Velocity (Radial Component)
(radians)
(radians)
(m/s)
(m/s)
σ
Noise level
(parts per million)