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How to find a transiting exoplanets

How to find a transiting exoplanets

A colloquium about noise.

Dan Foreman-Mackey

May 16, 2017
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  1. Dan Foreman-Mackey
    Sagan Fellow / University of Washington
    @exoplaneteer / dfm.io / github.com/dfm
    How to find a transiting exoplanet
    data-driven discovery in the astronomical time domain

    View Slide

  2. Dan Foreman-Mackey
    Sagan Fellow / University of Washington
    @exoplaneteer / dfm.io / github.com/dfm
    Noise models
    and some more noise models

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  3. Let me introduce myself…

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  4. I build tools.
    and when I say "tools" I actually mean "software"…

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  5. View Slide

  6. View Slide

  7. Exoplanets

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  8. How
    We
    Find
    Exoplanets

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  9. transit

    radial velocity

    direct imaging

    microlensing

    timing
    2712
    692
    52
    40
    25
    Data Source: The Open Exoplanet Catalogue

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  10. Data Source: The Open Exoplanet Catalogue
    2000 2005 2010 2015
    year
    0
    500
    1000
    confirmed exoplanets
    transit
    RV
    microlensing
    direct imaging
    timing

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  11. Kepler
    Credit: NASA

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  12. Data Source: The NASA Exoplanet Archive; Kepler DR25; 5/13/2017
    1 10 100 1000
    orbital period [days]
    1
    10
    planet radius [R ]

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  13. So what?

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  14. The
    Population of
    Exoplanets

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  15. The population of exoplanets
    1 occurrence rates
    2 physics

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  16. Burke et al. (2015)
    the data, rises toward small planets with a = -1.8
    2
    and has a
    break near the edge of the parameter space. Given the low
    numbers of observed planet candidates in the smallest planet
    bins, the full posterior allowed behavior (1σ orange region ; 3σ
    Figure 6) the occurrence rates in the smallest Rp
    bins. (b) The
    more complicated model ensures the ability to adapt to
    variations in the PLDF in the sensitivity analysis of Section 6.2.
    (c) Previous work on Kepler planet occurrence rates indicated a
    break in the planet population for 1
    2.0 Rp
     2.8 Å
    R (Fressin
    et al. 2013; Petigura et al. 2013a, 2013b; Silburt et al. 2015).
    (d) Finally, extending this work to a larger parameter space and
    for alternative target selection samples, such as the Kepler M
    dwarf sample where a sharp break at Rp
    ∼ 2.5 Å
    R is observed
    (Dressing & Charbonneau 2013; Burke et al. 2015), the double
    power law in Rp
    is strongly (BIC >10) warranted.
    Symptomatic of the weak evidence for a broken power law
    model over the ⩽
    0.75 Rp
    ⩽ 2.5 Å
    R range, Rbrk
    is not
    constrained within the prior Rp
    limits of the parameter space.
    When Rbrk
    is near the lower and upper Rp
    limits, a1
    and a2
    also
    become poorly constrained, respectively. To provide a more
    meaningful constraint on the average power law behavior for
    Rp
    in the double power law PLDF model, we introduce aavg
    ,
    which we set to a a
    =
    avg 1
    if ⩾
    R R
    brk mid
    and a a
    =
    avg 2
    otherwise, where Rmid
    is the midpoint between the upper and
    lower limits of Rp
    . We find a = -1.54 0.5
    avg
    and
    b = -0.68 0.17 for our baseline result. We use aavg
    as a
    summary statistic for the model parameters only to enable a
    simpler comparison of our results to independent analyses of
    planet occurrence rates and to approximate the behavior for the
    power law Rp
    dependence if we had used the simpler single
    power law model. The results for a single power law model in
    both Rp
    and P
    orb
    are equivalent to the results for the double
    Figure 7. Same as Figure 6, but marginalized over 0.75 < Rp
    < 2.5 Å
    R and bins
    of dP
    orb
    = 31.25 days.
    Figure 8. Shows the underlying planet occurrence rate model. Marginalized
    over 50 < P
    orb
    < 300 days and bins of dRp
    =0.25 Å
    R planet occurrence rates
    for the model parameters that maximize the likelihood (white dash line).
    Posterior distribution for the underlying planet occurrence rate for the median
    (blue solid line), 1σ region (orange region), and 3σ region (blue region). An
    approximate PLDF based upon results from Petigura et al. (2013a) for
    comparison (dash dot line).
    Figure 9. Same as Figure 8, but marginalized over 0.75 < Rp
    < 2.5 Å
    R and bins
    of dP
    orb
    =31.25 days.
    Figure 6) the occurrence rates in the smallest Rp
    bins. (b) The
    more complicated model ensures the ability to adapt to
    variations in the PLDF in the sensitivity analysis of Section 6.2.
    (c) Previous work on Kepler planet occurrence rates indicated a
    break in the planet population for 1
    2.0 Rp
     2.8 Å
    R (Fressin
    et al. 2013; Petigura et al. 2013a, 2013b; Silburt et al. 2015).
    (d) Finally, extending this work to a larger parameter space and
    for alternative target selection samples, such as the Kepler M
    gure 7. Same as Figure 6, but marginalized over 0.75 < Rp
    < 2.5 Å
    R and bins
    dP
    orb
    = 31.25 days.
    Figure 9. Same as Figure 8, but marginalized over 0.75 < Rp
    < 2.5 Å
    R and bins
    of dP
    orb
    =31.25 days.
    he Astrophysical Journal, 809:8 (19pp), 2015 August 10 Burke et al.

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  17. Kepler
    and the
    Transit
    Method
    (the spacecraft)

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  18. Credit: NASA/European Space Agency

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  19. Jupiter
    Credit: NASA/European Space Agency

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  20. Jupiter
    Earth
    Credit: NASA/European Space Agency

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  21. View Slide

  22. 1.0 0.5 0.0 0.5 1.0
    time since transit [days]
    100
    50
    0
    relative brightness [ppm]

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  23. …but this is the real world. A few problems:
    1 Timing
    2 Geometry
    3 Spacecraft motion
    4 Intrinsic brightness variation

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  24. …but this is the real world. A few problems:
    1 Timing
    2 Geometry
    3 Spacecraft motion
    4 Intrinsic brightness variation
    transit probability

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  25. …but this is the real world. A few problems:
    1 Timing
    2 Geometry
    3 Spacecraft motion
    4 Intrinsic brightness variation
    transit probability
    noise!

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  26. Credit: NASA

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  27. Credit: NASA
    190,000 stars
    for 4 years
    at 30 minute cadence
    with 10-3 pixel pointing precision

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  28. Data Source: The NASA Exoplanet Archive; Kepler DR25; 5/13/2017
    1 10 100 1000
    orbital period [days]
    1
    10
    planet radius [R ]

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  29. 1 Kepler (2009)
    2 K2 (2014)
    3 TESS (2018)
    4 PLATO (2025)

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  30. Population
    Inference

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  31. Ingredients
    1 Systematic target selection &
    catalog of stellar properties
    2 Systematic catalog of planets
    3 Quantified completeness &
    reliability
    4 False positive rates & other
    effects (e.g. multiplicity)

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  32. Data Source: The NASA Exoplanet Archive; Kepler DR25; 5/13/2017
    1 10 100 1000
    orbital period [days]
    1
    10
    planet radius [R ]

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  33. Burke, et al. (2015)
    model
    et al.
    ming
    e and
    al.
    g by
    f the
    ough
    tudy,
    peline
    planet
    planet
    hlight
    matic
    with
    ng &
    e we
    e the
    ump-
    icity,
    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.
    Burke et al.

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  34. We need
    1 Fully automated methods for
    planet discovery
    2 Rigorous methods for
    population inference

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  35. How to
    Find a
    Transiting
    Exoplanet

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  36. Science.
    physics
    data

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  37. Science.
    physics
    data
    a model

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  38. View Slide

  39. View Slide

  40. View Slide

  41. View Slide

  42. star

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  43. spacecraft star

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  44. detector
    spacecraft star

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  45. detector
    spacecraft star
    planet?

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  46. +
    planet star spacecraft detector observation
    + + =

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  47. +
    planet star spacecraft detector observation
    + + =
    PHYSICS

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  48. +
    planet star spacecraft detector observation
    + + =
    PHYSICS ????

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  49. The way we draw transits…

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  50. …and the way we should draw transits
    interesting
    boring boring

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  51. +
    planet star spacecraft detector observation
    + + =
    PHYSICS DATA-DRIVEN MODELS

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  52. +
    planet star spacecraft detector observation
    + + =
    PHYSICS DATA-DRIVEN MODELS
    (Gaussian Process)

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  53. How to find a transiting exoplanet
    1 Fit & remove data-driven

    noise model
    2 Matched filter grid search for
    candidate signals
    3 Vet candidates to remove

    false alarms

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  54. Scalable
    Methods
    An aside...

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  55. Medium data; big questions…
    1 Kepler
    2 K2
    3 TESS
    190,000 stars
    60,000 obs. per star
    250,000 stars
    4,000 obs. per star
    500,000 stars
    20,000 obs. per star
    approximately…

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  56. Scaling of Gaussian Processes
    O(N3)
    Cholesky factorization

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  57. Scaling of Gaussian Processes
    O(N3)
    Cholesky factorization
    O
    (
    N
    log
    2 N
    )
    Approximate methods
    Ambikasaran, DFM, et al. (2016); arXiv:1403.6015

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  58. Scaling of Gaussian Processes
    O(N3)
    Cholesky factorization
    O
    (
    N
    log
    2 N
    )
    Approximate methods
    Ambikasaran, DFM, et al. (2016); arXiv:1403.6015
    O(N)
    Exploiting structure of specific 1D kernels
    DFM, et al. (submitted); arXiv:1703.09710

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  59. DFM, et al. (submitted); arXiv:1703.09710
    102 103 104 105
    number of data points [N]
    10 5
    10 4
    10 3
    10 2
    10 1
    100
    computational cost [seconds]
    1
    2
    4
    8
    16
    32
    64
    128
    256
    direct
    O(N)
    100
    numb
    github.com/dfm/celerite

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  60. Pause…
    time for some examples!

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  61. The
    Frequency
    of Jupiter
    Analogs
    1

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  62. Tim Morton (Princeton)

    David Hogg (NYU)
    Eric Agol (UW)
    Bernhard Schölkopf (MPIS)
    in collaboration with…
    DFM, et al. (2016)
    arXiv:1607.08237

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  63. 1 10 100
    orbital period [days]
    1
    10
    planet radius [R ]
    Data Source: The NASA Exoplanet Archive

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  64. Data Source: The NASA Exoplanet Archive
    1 10 100
    orbital period [days]
    1
    10
    planet radius [R ]

    View Slide

  65. Data Source: The NASA Exoplanet Archive
    1 10 100 1000 10000
    orbital period [days]
    1
    10
    planet radius [R ]

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  66. Data Source: The NASA Exoplanet Archive
    1 10 100 1000 10000
    orbital period [days]
    1
    10
    planet radius [R ]

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  67. Why Kepler?
    Radial velocity, microlensing, etc. better suited…

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  68. 1 Systematic target selection &
    catalog of stellar properties
    2 Systematic catalog of planets
    3 Quantified completeness &
    reliability
    4 False positive rates & other
    effects (e.g. multiplicity)

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  69. Data Source: The NASA Exoplanet Archive
    1 10 100 1000 10000
    orbital period [days]
    1
    10
    planet radius [R ]

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  70. DFM et al. (2016); arXiv:1607.08237
    1 10 100 1000 10000
    orbital period [days]
    1
    10
    planet radius [R ]
    Data Source: The NASA Exoplanet Archive

    View Slide

  71. How to find a transiting exoplanet
    1 Fit & remove data-driven

    noise model
    2 Matched filter grid search for
    candidate signals
    3 Vet candidates to remove

    false alarms

    View Slide

  72. +
    planet star spacecraft detector observation
    + + =
    PHYSICS
    GAUSSIAN

    PROCESS
    CAUSAL

    MODEL (PCA)
    PHOTON

    NOISE

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  73. How to find a transiting exoplanet
    1 Fit & remove data-driven

    noise model
    2 Matched filter grid search for
    candidate signals
    3 Vet candidates to remove

    false alarms

    View Slide

  74. DFM, et al. (2016)
    40 20 0 20 40
    hours since event
    (a) variability
    KIC 7220674
    40 20 0 20 40
    hours since event
    (b) step
    KIC 8631697
    40 20 0 20 40
    hours since event
    (c) box
    KIC 5521451
    40 20 0 20 40
    hours since event
    (d) transit
    KIC 8505215

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  75. 12 Foreman-Mackey, Hogg, Morton,
    et al.
    0.50
    0.25
    0.00
    10321319
    1.2
    0.6
    0.0
    10287723
    1.6
    0.8
    0.0
    8505215
    0.8
    0.0
    6551440
    0.8
    0.0
    8738735 3
    2
    1
    0
    8800954
    4
    2
    0
    10187159
    4
    2
    0
    3218908
    3.0
    1.5
    0.0
    4754460
    5.0
    2.5
    0.0
    8410697
    4
    2
    0
    10842718
    8
    4
    0
    11709124
    16
    8
    0
    3239945
    4
    2
    0
    8426957
    50
    25
    0
    9306307
    80
    40
    0
    10602068
    Figure 3. Sections of PDC light curve centered on each candidate (black) with the posterior-median
    transit model over-plotted (orange). Candidates with two transits are folded on the posterior-median
    DFM, et al. (2016)

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  76. 1 Systematic target selection &
    catalog of stellar properties
    2 Systematic catalog of planets
    3 Quantified completeness &
    reliability
    4 False positive rates & other
    effects (e.g. multiplicity)

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  77. nuisance
    boring boring

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  78. DFM, et al. (2016)
    3 5 10 20
    period [years]
    0.2
    0.5
    1.0
    2.0
    RP
    /RJ
    0.048
    0.211
    0.499
    0.669
    0.727
    0.710
    0.635
    0.046
    0.194
    0.468
    0.616
    0.657
    0.630
    0.569
    0.043
    0.193
    0.460
    0.605
    0.623
    0.591
    0.520
    0.038
    0.174
    0.433
    0.529
    0.529
    0.492
    0.427
    0.0
    0.3
    0.6
    0.0 0.3 0.6

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  79. DFM, et al. (2016)
    2.00 ± 0.72 planets
    per G/K- dwarf
    occurrence rate in range:

    2 – 25 years, 0.1 – 1 RJ

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  80. EVEREST:
    A Noise
    Model for K2
    2

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  81. Credit: NASA
    R.I.P.
    Kepler

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  82. cbna
    Flickr user Aamir Choudhry
    K2

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  83. https://keplerscience.arc.nasa.gov/k2-fields.html

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  84. Adapted from a similar figure by Ian Crossfield
    baseline
    number of targets
    TESS

    K2

    Kepler

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  85. 3.4
    3.6
    3.8
    4.0
    log10
    Te↵
    0
    2
    4
    log10
    g
    Kepler
    3.4
    3.6
    3.8
    4.0
    log10
    Te↵
    K2
    Data Source: The NASA Exoplanet Archive; 5/13/2017

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  86. View Slide

  87. View Slide

  88. 4000
    2000
    0
    2000
    4000 raw: 301 ppm
    EPIC 201374602; Kp = 11.5 mag
    10 20 30 40 50 60 70 80
    time [BJD - 2456808]
    400
    0
    400 residuals: 35 ppm
    relative brightness [ppm]
    4000
    2000
    0
    2000
    4000 raw: 301 ppm
    EPIC 201374602; Kp = 11.5 mag
    10 20 30 40 50 60 70 80
    time [BJD - 2456808]
    400
    0
    400 residuals: 35 ppm
    relative brightness [ppm]

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  89. cbna
    Flickr user Aamir Choudhry
    Luger, et al. (2016, 2017)
    led by…

    Rodrigo Luger & Ethan Kruse

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  90. +
    planet star spacecraft detector observation
    + + =
    PHYSICS
    GAUSSIAN

    PROCESS
    CAUSAL

    MODEL +

    PIXEL-LEVEL

    DECORRELATION
    PHOTON

    NOISE
    inspired by: Vanderburg & Johnson (2014)
    Crossfield, et al. (2015)
    Aigrain, et al. (2015)
    DFM, et al. (2015)
    Deming, et al. (2015)
    + more

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  91. Figure credit: Rodrigo Luger
    Ideal Observed

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  92. Pixel-level decorrelation (PLD)
    if background is correctly subtracted,

    and astrophysical signal is multiplicative,

    then
    the fractional astrophysical contribution

    is equal in all pixels.
    Deming, et al. (2015); Luger, et al. (2016, 2017)
    ˆ
    pn(t) =
    pn(t)
    PN
    k=1
    pn(t)
    estimator for
    instrumental
    signal estimator for
    astrophysical
    signal
    pixel time series

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  93. =
    Figure credit: Rodrigo Luger; Deming, et al. (2015); Luger, et al. (2016, 2017)
    Pixel-level decorrelation (PLD)
    ÷

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  94. +
    planet star spacecraft detector observation
    + + =
    PHYSICS
    GAUSSIAN

    PROCESS
    CAUSAL

    MODEL +

    PIXEL-LEVEL

    DECORRELATION
    PHOTON

    NOISE

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  95. Luger, et al. (2016); see also Aigrain, et al. (2015)

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  96. EVEREST
    +
    planet star spacecraft detector observation
    + + =
    PHYSICS
    GAUSSIAN

    PROCESS
    CAUSAL

    MODEL +

    PIXEL-LEVEL

    DECORRELATION
    PHOTON

    NOISE

    View Slide

  97. EVEREST
    +
    planet star spacecraft detector observation
    + + =
    PHYSICS
    GAUSSIAN

    PROCESS
    CAUSAL

    MODEL +

    PIXEL-LEVEL

    DECORRELATION
    PHOTON

    NOISE

    View Slide

  98. g. 3.—
    Cross-validation procedure for first order PLD o
    03150 (WASP-47 e), a campaign 3 planet host. Show
    ter v in the validation set (red) and the scatter in the
    (blue) as a function of , the prior amplitude for
    Luger, Kruse, DFM, et al. (2017)

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  99. Luger, Kruse, DFM, et al. (2017)
    Kp = 15; for campaigns 3, 4, and 8,
    EVEREST
    recovers the
    Kepler
    precision dow
    of (variable) giant stars, leading to a higher average CDPP, while campaign 7
    change in the orientation of the spacecraft and excess jitter.
    Fig. 20.—
    The same as Figure 19, but comparing the CDPP of
    all
    K2 stars to that of
    Kepler
    .
    EVEREST 2.0
    recovers the original
    Kepler
    photometric precision down to at least Kp = 14, and past
    contam
    the in
    which
    inated
    valida
    fects o
    overfit
    spacec
    get ap
    of the
    apertu
    a time
    overfit
    §
    3.7, o
    this be
    In F
    ing bin
    overfit
    light c
    binary

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  100. EVEREST 2.0 7
    This pro-
    ma of the v
    n
    , and that
    e of the seg-
    3, where we
    sections for
    e minimum
    al line indi-
    se between
    and slight
    re conserva-
    ith nPLD to
    report our
    and a com-
    arisons with
    curves. We
    proxy 6 hr
    h we calcu-
    we smooth
    clip outliers
    deviation in
    Luger, Kruse, DFM, et al. (2017)

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  101. Data Source: The NASA Exoplanet Archive; Kepler DR25; 5/13/2017
    1 10 100 1000
    orbital period [days]
    1
    10
    planet radius [R ]

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  102. 1 10 100 1000
    orbital period [days]
    1
    10
    planet radius [R ]
    Data Source: The NASA Exoplanet Archive; Kepler DR25; 5/13/2017
    Kruse, et al. (in prep)

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  103. 1 10 100 1000
    orbital period [days]
    1
    10
    planet radius [R ]
    Data Source: The NASA Exoplanet Archive; Kepler DR25; 5/13/2017
    Kruse, et al. (in prep)
    800 candidates
    500 new

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  104. Population inference?
    as a function of host star properties?

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  105. 40 50 60 70 80
    0.985
    0.990
    0.995
    1.000
    1.005
    90 100 110
    0.985
    0.990
    0.995
    1.000
    1.005
    −0.05 0.00 0.05
    0.90
    0.92
    0.94
    0.96
    0.98
    1.00
    −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06
    −0.06
    −0.04
    −0.02
    0.00
    0.02
    0.04
    0.06
    .
    .
    .
    .
    a
    b
    c d
    K2 long cadence data
    Barycentric Julian Date − 2,457,700 [day]
    Relative brightness
    Relative brightness
    1b 1c 1d 1e 1f 1g 1h
    1b 1c 1d 1e 1f 1g 1h
    Time from mid−transit [day]
    Relative brightness
    transit 1
    transit 2
    transit 3
    transit 4
    folded lightcurve
    Orbital separation [AU]
    Figure 1:
    a, b
    : Long cadence K2 light curve detrended with EVEREST and with stellar variability
    removed. Data points are in black, and our highest likelihood transit model for all seven planets
    TRAPPIST-1h: Luger, Sestovic, Kruse, et al. (2017); arXiv:1703.04166
    embargoed

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  106. These
    Noise
    Models are
    Models of
    Stars
    3

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  107. nuisance!
    interesting interesting

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  108. Suzanne Aigrain (Oxford)

    Vinesh Rajpaul (Oxford)

    Eric Agol (UW)
    Sivaram Ambikasaran (Indian Inst. of Sci.)
    in collaboration with…
    Angus, et al. (submitted)
    DFM, et al. (submitted)
    Ruth Angus (Columbia)
    led by…

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  109. Figure credit: Ruth Angus
    100 101
    Age (Gyr)
    101
    102
    Rotation period (days)
    Coma Berenices
    Praesepe
    Hyades
    NGC 6811
    NGC 6819
    The Sun
    Asteroseismic targets
    M67 (Esselstein, in prep)

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  110. Angus, et al. (submitted); github.com/RuthAngus/GProtation

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  111. ctive model should
    ers and be flexible
    QP behaviour. A
    irements. We thus
    a method to prob-
    ation periods. This
    e rotation period,
    rtainty.
    arning community
    iology, geophysics
    used in the stellar
    e stellar variability
    l. 2012; Haywood
    5; Haywood 2015;
    t al. 2015; Rajpaul
    eful in regression
    cifically when the
    variate Gaussian. If
    n in
    N
    dimensions,
    can describe that
    ocesses is provided
    tween data points
    demonstration, we
    ight curve of KIC
    s once every ⇠ 30.5
    FGK stars. Clearly,
    summit of the Mauna Loa volcano in Hawaii (data from Keeling
    and Whorf 2004) using a kernel which is the product of a periodic
    and a SE kernel: the QP kernel. This kernel is defined as
    ki
    ,
    j
    =
    A
    exp
    2
    6
    6
    6
    6
    4
    (
    xi xj
    )2
    2
    l
    2
    2 sin2
    ⇡(
    xi xj
    )
    P
    !3
    7
    7
    7
    7
    5
    + 2
    ij
    . (2)
    It is the product of the SE kernel function, which describes the
    overall covariance decay, and an exponentiated, squared, sinusoidal
    kernel function that describes the periodic covariance structure.
    P
    can be interpreted as the rotation period of the star, and controls
    the amplitude of the sin2 term. If is very large, only points almost
    exactly one period away are tightly correlated and points that are
    slightly more or less than one period away are very loosely cor-
    related. If is small, points separated by one period are tightly
    correlated, and points separated by slightly more or less are still
    highly correlated, although less so. In other words, large values of
    lead to periodic variations with increasingly complex harmonic con-
    tent. This kernel function allows two data points that are separated
    in time by one rotation period to be tightly correlated, while also
    allowing points separated by half a period to be weakly correlated.
    The additional parameter captures white noise by adding a term
    to the diagonal of the covariance matrix. This can be interpreted
    to represent underestimation of observational uncertainties — if
    the uncertainties reported on the data are too small, it will be non-
    zero — or it can capture any remaining “jitter,” or residuals not
    captured by the e ective GP model. We use this QP kernel function
    (Equation 2) to produce the GP model that fits the Kepler light curve
    0 20 40
    time [days]
    1.0
    0.5
    0.0
    0.5
    1.0
    relative flux [ppt]
    Kepler light curve
    10 1 100
    ! [days 1]
    10 3
    10 2
    10 1
    S(!)
    power spectrum
    0
    0.000
    0.025
    0.050
    0.075
    0.100
    0.125
    k(⌧)
    3.50 3.75 4.00 4.25
    rotation period [days]
    Angus, et al. (submitted); github.com/RuthAngus/GProtation

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  112. 0 1 2 3 4
    ln(Injected Period)
    2
    0
    2
    4
    6
    ln(Recovered Period)
    7
    6
    5
    4
    3
    ln (Amplitude)
    Angus, et al. (submitted); github.com/RuthAngus/GProtation

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  113. github.com/
    dfm/peerless
    rodluger/everest
    RuthAngus/GProtation
    dfm/celerite
    Jupiter analogs
    K2 de-trending
    GP models of rotation
    fast 1D GPs
    Open science

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  114. Summary
    1 Find exoplanets
    2 Learn about stars
    Build data-driven noise models and…
    Dan Foreman-Mackey
    Sagan Fellow / University of Washington
    @exoplaneteer / dfm.io / github.com/dfm

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