Exoplanet Population
Inference
A Tutorial
Dan Foreman-Mackey
CCA@Flatiron // dfm.io
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Today I'll mostly talk
about transiting
exoplanets*.
The methods can apply
more broadly .
* this is what I know about and work on!
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1
Exoplanet population
inference
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1 10 100
orbital period [days]
1
10
planet radius [R ]
data: NASA Exoplanet Archive
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leteness model
2013; Farr et al.
is shortcoming
ch pipeline and
igura et al.
.
hortcoming by
eteness of the
2014) through
s. In this study,
Kepler pipeline
rive the planet
Kepler planet
other highlight
the systematic
ce rates with
) and Dong &
ysis where we
recalculate the
input assump-
Figure 1. Fractional completeness model for the host to Kepler-22b (KIC:
10593626) in the Q1-Q16 pipeline run using the analytic model described in
Section 2.
t 10 Burke et al.
Burke, Christiansen et al. (2015)
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Take these catalogs and
get the physics of planet
formation and evolution.
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That's hard .
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1 10 100
orbital period [days]
1
10
planet radius [R ]
data: NASA Exoplanet Archive
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Fulton & Petigura (2018)
8. Planets with g
grazing transit
covariances w
darkening duri
After applying these
Where possible,
properties to the Ke
radius and temper
parameters. We cou
stellar population b
directed specificall
population. After fil
We calculated pla
efficiency methodolo
the detection sensit
recovery tests perfo
K02403.01 17.98
K00988.01 60.03
Note. This table contains
filters described in Sectio
(This table is available in
Figure 5. The distribution of close-in planet sizes. The top panel shows the
distribution from Fulton et al. (2017) and the bottom panel is the updated
distribution from this work. The solid line shows the number of planets per star
with orbital periods less than 100days as a function of planet size. A deep
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2
What is an occurrence rate?
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1
The expected number of
planets per star.
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2
The fraction of stars
with planets.
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3
The expected number of
planets per star
per unit planet property .
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4
etc.
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None of these definitions
is inherently better than
the others.
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But. They are all different .
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They have different units .
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They all depend on a
specific (often unstated)
definition of "planets" .
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So. It can be hard to
compare and understand
how they relate.
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Them: * "The occurrence
rate is 10%."
Y'all: "what does it all
mean?!?1?"
* including me and others in the room
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Them: * "The occurrence
rate is 10%."
Y'all: "what does it all
mean?!?1?"
* including me and others in the room
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Fulton & Petigura (2018)
8. Planets with g
grazing transit
covariances w
darkening duri
After applying these
Where possible,
properties to the Ke
radius and temper
parameters. We cou
stellar population b
directed specificall
population. After fil
We calculated pla
efficiency methodolo
the detection sensit
recovery tests perfo
K02403.01 17.98
K00988.01 60.03
Note. This table contains
filters described in Sectio
(This table is available in
Figure 5. The distribution of close-in planet sizes. The top panel shows the
distribution from Fulton et al. (2017) and the bottom panel is the updated
distribution from this work. The solid line shows the number of planets per star
with orbital periods less than 100days as a function of planet size. A deep
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Fulton & Petigura (2018)
8. Planets with g
grazing transit
covariances w
darkening duri
After applying these
Where possible,
properties to the Ke
radius and temper
parameters. We cou
stellar population b
directed specificall
population. After fil
We calculated pla
efficiency methodolo
the detection sensit
recovery tests perfo
K02403.01 17.98
K00988.01 60.03
Note. This table contains
filters described in Sectio
(This table is available in
Figure 5. The distribution of close-in planet sizes. The top panel shows the
distribution from Fulton et al. (2017) and the bottom panel is the updated
distribution from this work. The solid line shows the number of planets per star
with orbital periods less than 100days as a function of planet size. A deep
what do these numbers mean?
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Fulton & Petigura (2018)
8. Planets with g
grazing transit
covariances w
darkening duri
After applying these
Where possible,
properties to the Ke
radius and temper
parameters. We cou
stellar population b
directed specificall
population. After fil
We calculated pla
efficiency methodolo
the detection sensit
recovery tests perfo
K02403.01 17.98
K00988.01 60.03
Note. This table contains
filters described in Sectio
(This table is available in
Figure 5. The distribution of close-in planet sizes. The top panel shows the
distribution from Fulton et al. (2017) and the bottom panel is the updated
distribution from this work. The solid line shows the number of planets per star
with orbital periods less than 100days as a function of planet size. A deep
what do these numbers mean?
The expected number
of planets per star with
a period in the range
0–100 days and radius
in the given bin .
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Simulations
github.com/dfm/exostar19
expected number of planets per star
P(nj
| xj) =
X
qj
2{0, 1}
P(qj) P(nj
| xj, qj)
= Q P(nj
| xj, qj= 1) + (1 Q) P(nj
| xj, qj= 0)
this is the parameter
that we want to fit for!
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But. We don't know the
properties of the
unobserved planets .
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Marginalize!
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P(nj = 1) = p(xj) P(nj = 1 | xj)
= p(xj) Q P(nj = 1 | xj, qj= 1)
P(nj = 0) =
Z
p(xj) P(nj = 0 | xj) dxj
= 1 Q
Z
p(xj) P(nj = 1 | xj, qj= 1) dxj
= 1 Q P0
systems with
no planets
systems with
detected planets
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P(nj = 1) = p(xj) P(nj = 1 | xj)
= p(xj) Q P(nj = 1 | xj, qj= 1)
P(nj = 0) =
Z
p(xj) P(nj = 0 | xj) dxj
= 1 Q
Z
p(xj) P(nj = 1 | xj, qj= 1) dxj
= 1 Q P0
detection
probability
systems with
no planets
systems with
detected planets
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Put it all together.
An exercise for the reader…
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Q =
N1
N0 + N1
1
P0
6=
1
N0 + N1
N1
X
j=1
1
Pj
the
occurrence
rate
the fraction
of stars with
observed planets
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Q =
N1
N0 + N1
1
P0
6=
1
N0 + N1
N1
X
j=1
1
Pj
P0 =
Z
p(xj) P(nj = 1 | xj, qj= 1) dxj
the detection probability
averaged over the distribution
of planet and stellar properties
the
occurrence
rate
the fraction
of stars with
observed planets
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Q =
N1
N0 + N1
1
P0
6=
1
N0 + N1
N1
X
j=1
1
Pj
P0 =
Z
p(xj) P(nj = 1 | xj, qj= 1) dxj
the detection probability
averaged over the distribution
of planet and stellar properties
the
occurrence
rate
the fraction
of stars with
observed planets
Inverse detection efficiency
is not the right estimator.
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Instead, take the fraction
of detections and divide
by the average detection
efficiency*.
* averaged over the correct distribution
for all planet and star properties
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The key ingredient is the
detection efficiency model.
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leteness model
2013; Farr et al.
is shortcoming
ch pipeline and
igura et al.
.
hortcoming by
eteness of the
2014) through
s. In this study,
Kepler pipeline
rive the planet
Kepler planet
other highlight
the systematic
ce rates with
) and Dong &
ysis where we
recalculate the
input assump-
Figure 1. Fractional completeness model for the host to Kepler-22b (KIC:
10593626) in the Q1-Q16 pipeline run using the analytic model described in
Section 2.
t 10 Burke et al.
Burke, Christiansen et al. (2015)
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Remember : an occurrence
rate depends on a lot of
decisions!
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Stellar sample
Range of planet parameters
Units
Planet multiplicity
1
2
3
4
You end up needing to do
an integral over all the
properties of all the planets
and false positives that
you didn't observe .
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1
Mathematica™ can't
do that integral.
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2
Eric Agol can't
do that integral.
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3
MCMC can't
do that integral*.
* in finite time.
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This is where you use
approximate Bayesian
computation (ABC).
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This is where you use
approximate Bayesian
computation (ABC).
likelihood-free inference.
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Likelihood-free inference
is a method for doing
rigorous inference with
stochastic models .
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If you can simulate it
then you can do inference.
a realistic catalog
The promise of "likelihood-free inference".
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PLANET OCCURRENCE RATES 11
Figure 2. Inferred occurrence rates for Kepler’s DR25 planet candidates associated with high-quality FGK target stars. These rares are based
on a combined detection and vetting efficiency model that was fit to flux-level planet injection tests. The numerical values of the occurrence
Hsu et al. (2019)
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There's still lots to do!
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EPOS; Mulders et al. (2018)
no additional
s indicate the
Figure 10. Comparison of simulated planets for the example model (blue) with
detected planets (orange). The comparison region (black box) excludes hot
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EPOS; Mulders et al. (2018)
no additional
s indicate the
Figure 10. Comparison of simulated planets for the example model (blue) with
detected planets (orange). The comparison region (black box) excludes hot
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5
Take homes
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An occurrence rate needs
to come with a lot of
metadata.
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Comparing occurrence rates:
Check the units .
Check the parameter ranges .
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Don't sum the inverse
detection probabilities
for your planets!
* a more reliable estimator is just as easy to compute!
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If you're using a method
that seems intuitive , make
sure the math checks out !
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Likelihood-free inference
seems like a promising
way forward.
* a.k.a. Approximate Bayesian Computation (ABC)
Simulations
github.com/dfm/exostar19
"truth"
fraction of stars
with planets
expected number
of planets per star
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Note: this is preliminary & really just a toy…
assuming:
no mutual inclination
only geometric transit probability
0.5 < RP
/REarth
< 8; 10 < a/Rstar
< 30
Kepler data:
github.com/dfm/exostar19
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0.5 < RP
/REarth
< 8; 10 < a/Rstar
< 30
Kepler data:
github.com/dfm/exostar19
Note: this is preliminary & really just a toy…
assuming:
no mutual inclination
only geometric transit probability