Exoplanet Population Inference (v2)

An updated version of my talk on exoplanet population inference given at Queen's University (Canada).

Transcript

1. EXOPLANET POPULATIONS Inferring from noisy, incomplete catalogs Dan Foreman-Mackey CCPP@NYU // github.com/dfm // @exoplaneteer // dfm.io
2. Engineering
3. Data Science
4. Photo credit: James Silvester silvesterphoto.tumblr.com not data science.
5. cb Flickr user Marcin Wichary data science.
6. data science.
7. Data Science
8. Exoplanet population inference and the abundance of Earth analogs from noisy, incomplete catalogs Foreman-Mackey, Hogg & Morton (arXiv:1406.3020)
9. The Punchline Existing methods for population inference are not robust Current data don't support claims of high Earth analog abundance There should be some transiting Earth analogs in the Kepler data
10. The Question What can we say about the population of exoplanets given all of the pixels ever downloaded by Kepler?
11. Photo credit: NASA Kepler
12. Photo credit: NASA
13. Kepler-32
14. Kepler-32
16. ... ...
17. Data from: NASA Exoplanet Archive
18. Earth
19. ... ...
20. Jupiter
21. Data from: NASA Exoplanet Archive
22. 5 10 20 30 40 50 100 200 300 400 Orbital period (days) 0.5 1 2 3 4 5 10 20 Planet size (Earth-radii) 0 10 20 30 40 50 60 70 80 90 100 Survey Completeness (C) % F o d li c in p c m r o g o f Figure credit: Petigura, Howard & Marcy (2013)
23. The Question What can we say about the population of exoplanets given all of the pixels ever downloaded by Kepler?
24. 5 10 20 30 40 50 100 200 300 400 Orbital period (days) 0.5 1 2 3 4 5 10 20 Planet size (Earth-radii) 0 10 20 30 40 50 60 70 80 90 100 Survey Completeness (C) % F o d li c in p c m r o g o f Figure credit: Petigura, Howard & Marcy (2013)
25. 6.25 12.5 25 50 100 200 400 Orbital period (days) 0.5 1 2 4 8 16 Planet size (Earth-radii) 4.9 0.6% 3.5 0.4% 0.3 0.1% 0.2 0.1% 6.6 0.9% 6.1 0.7% 0.8 0.2% 0.2 0.2% 7.7 1.3% 7.0 0.9% 0.4 0.2% 0.6 0.3% 5.8 1.6% 7.5 1.3% 1.3 0.6% 0.6 0.3% 3.2 1.6% 6.2 1.5% 2.0 0.8% 1.1 0.6% 5.0 2.1% 1.6 1.0% 1.3 0.6% 0% 1% 2% 3% 4% 5% 6% 7% 8% Planet Occurrence Fi o d a in in w e p o co B p o w th re e p R Figure credit: Petigura, Howard & Marcy (2013)
26. inverse-detection-efficiency maximum likelihood the method
27. inverse-detection-efficiency maximum likelihood the method “non-parametric” Howard et al. (2011), Dressing & Charbonneau (2013), Petigura et al. (2013), and more… ✓j = 1 j K X k=1 1[wk 2 j] Q(wk)
28. inverse-detection-efficiency maximum likelihood the method “non-parametric” Howard et al. (2011), Dressing & Charbonneau (2013), Petigura et al. (2013), and more… ✓j = 1 j K X k=1 1[wk 2 j] Q(wk) “parametric” Tabachnik & Tremaine (2002), Youdin (2011), and more… – 6 – e 2002; Youdin 2011 for some of the examples from the exoplanet litera p({ wk } | ✓ ) = exp ✓ Z ˆ ✓ ( w ) d w ◆ K Y k=1 ˆ ✓ ( wk ) .
29. The Inverse-Detection-Efficiency Procedure ✓j = 1 j K X k=1 1[wk 2 j] Q(wk) (a weighted histogram)
30. The Inverse-Detection-Efficiency Procedure ✓j = 1 j K X k=1 1[wk 2 j] Q(wk) (a weighted histogram) BAD IDEA
31. The Inverse-Detection-Efficiency Procedure
32. The Inverse-Detection-Efficiency Procedure truth: 50 inverse-detection-efficiency gives: 28.5 ± 5.5 maximum-likelihood gives: 54.0 ± 10.4
33. A Better Inverse-Detection-Efficiency Procedure (see: dfm.io/posts/histogram1) ✓j = Nj Z j Q(w) dw
34. WHOLE STORY but that's not the yet
35. Exoplanet population inference and the abundance of Earth analogs from noisy, incomplete catalogs Foreman-Mackey, Hogg & Morton (arXiv:1406.3020)
36. 5 10 20 30 40 50 100 200 300 400 Orbital period (days) 0.5 1 2 3 4 5 10 20 Planet size (Earth-radii) 0 10 20 30 40 50 60 70 80 90 100 Survey Completeness (C) % F o d li c in p c m r o g o f Figure credit: Petigura, Howard & Marcy (2013) typical error bar
37. How do you make a histogram of noisy measurements?
38. How do you infer the True distribution of noisy measurements?
39. truth w p(w)
40. ignoring uncertainties truth w p(w)
41. ignoring uncertainties truth intuitive resampling w p(w)
42. ignoring uncertainties truth intuitive resampling w p(w) BAD IDEA
43. Hierarchical Inference
44. The Question What can we say about the population of exoplanets given all of the pixels ever downloaded by Kepler?
45. k = 1, · · · , K ✓ wk xk per-object parameters (period, radius, etc.) per-object observations global population p({ xk } | ✓ ) = Z p({ xk }, { wk } | ✓ ) d{ wk }
46. k = 1, · · · , K ✓ wk xk per-object parameters (period, radius, etc.) per-object observations global population p({ xk } | ✓ ) = Z p({ xk }, { wk } | ✓ ) d{ wk }
47. p({ xk } | ✓ ) = Z p({ xk }, { wk } | ✓ ) d{ wk } = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk }
48. p({ xk } | ✓ ) = Z p({ xk }, { wk } | ✓ ) d{ wk } = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk } HARD this is ™ (generally impossible)
49. Hogg, Myers, & Bovy (2010) Inferring the eccentricity distribution [1008.4146]
50. What is a catalog? posterior samples interim prior w (n) k ⇠ p( wk | xk, ↵ ) – 8 – d, we will reuse the hard work that went into building the c each entry in a catalog is a representation of the posterior p( wk | xk , ↵ ) = p( xk | wk ) p( wk | ↵ ) p( xk | ↵ ) ameters wk conditioned on the observations of that objec minder that the catalog was produced under a specific c tive”— interim prior p( wk | ↵ ). This prior was chosen by th s di↵erent from the likelihood p( wk | ✓ ) from Equation (2). we can use these posterior measurements to simplify Equ n many common cases, be evaluated e ciently. To find thi
51. p ( { xk } | ✓) p ( { xk } | ↵) ⇡ exp ✓ Z ˆ✓(w) dw ◆ K Y k=1 1 Nk Nk X n=1 ˆ✓(w (n) k ) p (w (n) k | ↵) p({ xk } | ✓ ) = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk } maths w (n) k ⇠ p( wk | xk, ↵ ) posterior samples includes completeness & uncertainties!
52. the original "interim" prior the observable rate density likelihood of pixels given population expected # of observable exoplanets p ( { xk } | ✓) p ( { xk } | ↵) ⇡ exp ✓ Z ˆ✓(w) dw ◆ K Y k=1 1 Nk Nk X n=1 ˆ✓(w (n) k ) p (w (n) k | ↵) sum over posterior samples product over objects The "Money Equation™"
53. github.com/dfm/exopop
54. the original "interim" prior the observable rate density likelihood of pixels given population expected # of observable exoplanets p ( { xk } | ✓) p ( { xk } | ↵) ⇡ exp ✓ Z ˆ✓(w) dw ◆ K Y k=1 1 Nk Nk X n=1 ˆ✓(w (n) k ) p (w (n) k | ↵) sum over posterior samples product over objects The "Money Equation™"
55. The Rate Density Model e sample. hing to note here is that ˆ ✓ is the rate density of exo observe taking into account the geometric transit probabil cies. In practice, we can model the observable rate density ˆ ✓ ( w ) = Qc ( w ) ✓ ( w ) he detection e ciency (including transit probability) at w ant to infer: the True occurrence rate density. We haven’t al form for ✓ ( w ) and all of this derivation is equally appli ensity as, for example, a broken power law or a histogram d rate density ˆ is a quantitative description of the rate n the Petigura et al. (2013b) catalog; it is not a description ✓(w) = dN dw The "observable" rate density where
56. The Rate Density Model method. Instead, we could use a functional form for the ameters along with the parameters of the rate density. all the results in this Article, we’ll model the rate dens nction ✓ ( w ) = 8 > > > > > < > > > > > : exp(✓1 ) w 2 1 , exp(✓2 ) w 2 2 , · · · exp(✓J ) w 2 J , 0 otherwise eters ✓j are the log step heights and the bins j are fixed (looks like a histogram)
57. The Rate Density Model p(✓) = GP(✓; ) prior on the log bin heights p( ✓ | { xk }) / p( ✓ ) p({ xk } | ✓ ) use MCMC to sample the posterior PDF for the bin heights
58. 6.25 12.5 25 50 100 200 400 Orbital period (days) 0.5 1 2 4 8 16 Planet size (Earth-radii) 4.9 0.6% 3.5 0.4% 0.3 0.1% 0.2 0.1% 6.6 0.9% 6.1 0.7% 0.8 0.2% 0.2 0.2% 7.7 1.3% 7.0 0.9% 0.4 0.2% 0.6 0.3% 5.8 1.6% 7.5 1.3% 1.3 0.6% 0.6 0.3% 3.2 1.6% 6.2 1.5% 2.0 0.8% 1.1 0.6% 5.0 2.1% 1.6 1.0% 1.3 0.6% 0% 1% 2% 3% 4% 5% 6% 7% 8% Planet Occurrence Fi o d a in in w e p o co B p o w th re e p R Figure credit: Petigura, Howard & Marcy (2013)
59. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.00 0.15 0.30 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
60. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.00 0.15 0.30 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
61. 0 1 2 3 ln R/R 10 3 10 2 10 1 100 (ln R/R ) all 6.25  P/day < 25 25  P/day < 100 100  P/day < 400 1 10 R/R Figure credit: Foreman-Mackey, Hogg & Morton (2014)
62. 2 3 4 5 6 ln P/day 10 3 10 2 10 1 (ln P/day) all 0.5  R/R < 2 2  R/R < 8 8  R/R < 32 10 100 P/day Figure credit: Foreman-Mackey, Hogg & Morton (2014)
63. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.00 0.15 0.30 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
64. Figure credit: Foreman-Mackey, Hogg & Morton (2014) 7 6 5 4 3 2 1 ln 0.0 0.1 0.2 0.3 0.4 0.5 0.6 p(ln ) 10 3 10 2 10 1
65. Figure credit: Foreman-Mackey, Hogg & Morton (2014) 7 6 5 4 3 2 1 ln 0.0 0.1 0.2 0.3 0.4 0.5 0.6 p(ln ) 10 3 10 2 10 1 our result has large fractional uncertainty—w his is shown in Figure 9 where we compare the to the published value and uncertainty. Earth analogs is = 0.019+0.019 0.010 nat 2 ates that this quantity is a rate density, per na radius. Converted to these units, Petigura e same quantity (indicated as the vertical lines the expected number of Earth-like exoplanets per Sun-like star per ln-radius per ln-period
66. 5 4 3 2 ln Petigura et al. (2013) Dong & Zhu (2013) linear extrapolation negligible uncertainties Foreman-Mackey et al. (2014) Figure credit: Foreman-Mackey, Hogg & Morton (2014)
67. Figure credit: Petigura, Howard & Marcy (2013) of stars having nearly Earth-size planets ð1 − 2 R⊕Þ with any orbital period up to a maximum period, P, on the h size ð1 − 2 R⊕Þ are included. This cumulative distribution reaches 20.2% at P = 50 d, meaning 20.4% of Sun-like Petigura et al. assumed that the period distribution of small planets is flat from 50d-400d
68. 2 3 4 5 6 ln P/day 10 3 10 2 10 1 (ln P/day) all 0.5  R/R < 2 2  R/R < 8 8  R/R < 32 10 100 P/day Figure credit: Foreman-Mackey, Hogg & Morton (2014)
69. 6.25 12.5 25 50 100 200 400 Orbital period (days) 0.5 1 2 4 8 16 Planet size (Earth-radii) 4.9 0.6% 3.5 0.4% 0.3 0.1% 0.2 0.1% 6.6 0.9% 6.1 0.7% 0.8 0.2% 0.2 0.2% 7.7 1.3% 7.0 0.9% 0.4 0.2% 0.6 0.3% 5.8 1.6% 7.5 1.3% 1.3 0.6% 0.6 0.3% 3.2 1.6% 6.2 1.5% 2.0 0.8% 1.1 0.6% 5.0 2.1% 1.6 1.0% 1.3 0.6% 0% 1% 2% 3% 4% 5% 6% 7% 8% Planet Occurrence Fi o d a in in w e p o co B p o w th re e p R Figure credit: Petigura, Howard & Marcy (2013)
70. 5 4 3 2 ln Petigura et al. (2013) Dong & Zhu (2013) linear extrapolation negligible uncertainties Foreman-Mackey et al. (2014) Figure credit: Foreman-Mackey, Hogg & Morton (2014)
71. 5 4 3 2 ln Petigura et al. (2013) Dong & Zhu (2013) linear extrapolation negligible uncertainties Foreman-Mackey et al. (2014) Figure credit: Foreman-Mackey, Hogg & Morton (2014)
72. 5 4 3 2 ln Petigura et al. (2013) Dong & Zhu (2013) linear extrapolation negligible uncertainties Foreman-Mackey et al. (2014) Figure credit: Foreman-Mackey, Hogg & Morton (2014)
73. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.00 0.15 0.30 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
74. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.00 0.15 0.30 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
75. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.00 0.15 0.30 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014) nsity is exactly what Petigura’s extrapolation model predicts but, for comparison, we so integrate our inferred rate density over their choice of “Earth-like” bin (200  P/d 0 and 1  R/R < 2) to find a rate of Earth analogs. The published rate is 0.057+ Petigura et al. 2013b) and our posterior constraint is Z 400 day P=200 day Z 2 R R=1 R ✓ (ln P, ln R) d[ln R] d[ln P] = 0.019+0.010 0.008 . 9. Comparison with previous work Our inferred rate density of Earth analogs (Equation 22) is not consistent with previo ublished results. In particular, our result is completely inconsistent with the earlier r ased on exactly the same dataset (Petigura et al. 2013b). This inconsistency is du e di↵erent assumptions made but it merits some investigation. The two key di↵ere tween our analysis and previous work are (a) the form of the extrapolation function ) the presence of measurement uncertainties on the planet radii. The two main assumptions that we relax in this Article are the extrapolation
76. The Number of Transiting Earths nets places a probabilistic constraint on the number of xisting Kepler dataset. If we adopt the definition of “ 013b, 200  P/day < 400 and 1  R/R < 2), and integ sity function and the geometric transit probability (Equa the expected number of Earth-like exoplanets transiting e stars chosen by Petigura et al. (2013b) is N , transiting = 10.6+5.9 4.5 ainties are only on the expectation value and don’t inc e. This is an exciting result because it means that, if we planet search pipelines to small planets orbiting on long Earth analogs in the existing data. Furthermore, because ting systems in the catalog, the True expected number of biting Sun-like stars is probably larger than the values in Let's go find them!
77. The Punchline Existing methods for population inference are not robust Current data don't support claims of high Earth analog abundance There should be some transiting Earth analogs in the Kepler data
78. Extras
79. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.0 0.2 0.4 p(ln P/day) 0.0 0.4 0.8 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
80. 7 6 5 4 3 2 1 ln 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p(ln ) 10 3 10 2 10 1 Figure credit: Foreman-Mackey, Hogg & Morton (2014)
81. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln P/day 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ln R/R 0.0 0.2 0.4 p(ln P/day) 0.0 0.5 1.0 p(ln R/R ) 10 100 P [days] 1 10 R [R ] Figure credit: Foreman-Mackey, Hogg & Morton (2014)
82. 7 6 5 4 3 2 1 ln 0.0 0.1 0.2 0.3 0.4 0.5 0.6 p(ln ) 10 3 10 2 10 1 Figure credit: Foreman-Mackey, Hogg & Morton (2014)