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Exoplanet population inference: a tutorial

Exoplanet population inference: a tutorial

My talk at the KITP exostar conference

Dan Foreman-Mackey

May 22, 2019
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  1. Today I'll mostly talk about transiting exoplanets*. The methods can

    apply more broadly . * this is what I know about and work on!
  2. 1 10 100 orbital period [days] 1 10 planet radius

    [R ] data: NASA Exoplanet Archive
  3. 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)
  4. 1 10 100 orbital period [days] 1 10 planet radius

    [R ] data: NASA Exoplanet Archive
  5. 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 100days as a function of planet size. A deep
  6. Them: * "The occurrence rate is 10%." Y'all: "what does

    it all mean?!?1?" * including me and others in the room
  7. Them: * "The occurrence rate is 10%." Y'all: "what does

    it all mean?!?1?" * including me and others in the room
  8. 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 100days as a function of planet size. A deep
  9. 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 100days as a function of planet size. A deep what do these numbers mean?
  10. 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 100days 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 .
  11. 1 Inverse detection efficiency Nexpect = 1 Ntot N X

    j=1 1 Pdet(xj) Note: don't do this!
  12. P(qj ) true number of planets nj, xj observed number

    of planets the properties of the planets and the star want have
  13. P(nj | xj , qj ) observed number of planets

    true number of planets the properties of the planets and the star
  14. value of P(nj | xj , qj ) 1 1–Pdet

    (xj ) 0 Pdet (xj ) qj = 0 1 true number of planets nj =0 1 observed number of planets
  15. 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)
  16. 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)
  17. 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!
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. Instead, take the fraction of detections and divide by the

    average detection efficiency*. * averaged over the correct distribution for all planet and star properties
  24. 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)
  25. You end up needing to do an integral over all

    the properties of all the planets and false positives that you didn't observe .
  26. If you can simulate it then you can do inference.

    a realistic catalog The promise of "likelihood-free inference".
  27. 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)
  28. 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
  29. 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
  30. Don't sum the inverse detection probabilities for your planets! *

    a more reliable estimator is just as easy to compute!
  31. p({nj }, {xj } | Q) = [1 Q P0]N0

    2 4 N1 Y j=1 Q p(xj) P(nj = 1 | xj, qj= 1) 3 5
  32. log p({nj }, {xj } | Q) = N0 log

    (1 Q P0) + N1 log Q + constant
  33. log p({nj }, {xj } | Q) = N0 log

    (1 Q P0) + N1 log Q + constant Q = N1 N0 + N1 1 P0 6= 1 N0 + N1 N1 X j=1 1 Pj
  34. 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
  35. 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