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Photometric Time Series Analysis

Photometric Time Series Analysis

I gave 3-hours of lectures at the Asteroseismology and Exoplanets: Listening to the Stars and Searching for New Worlds, at the IVth Azores International Advanced School in Space Sciences. Here are the sldies

Tom Barclay

July 22, 2016
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  1. Photometric Time
    Series Analysis
    Tom Barclay
    NASA Ames Research Center
    Azores Summer School 2016
    for exoplanets

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  2. Outline
    • Transit photometry
    - history and introduction
    - Kepler data
    - pixel level data
    • Creating a light curve
    - selecting an aperture
    - detrending
    • stellar variability
    • noise sources
    • Detecting planets
    - BLS technique
    - other detection methods
    2

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  3. Outline
    3
    • Modeling a transit
    - planet parameters
    - limb darkening
    - stellar density
    • Modeling phase curves
    - reflected light
    - doppler beaming
    - ellipsoidal variations
    • Fitting data and advanced modeling
    - likelihood functions
    - MCMC modeling
    - Gaussian Processes
    • Future of transit photometry surveys

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  4. A Brief History of Cosmic Pluralism
    1592: Giordano Bruno:
    “There are countless suns and countless earths all rotating round their suns in
    exactly the same way as the seven planets of our system. We see only the suns
    because they are the largest bodies and are luminous, but their planets remain
    invisible to us because they are smaller and non-luminous. The countless worlds in
    the universe are no worse and no less inhabited than our earth”
    1726: Isaac Newton: Exoplanets!
    “If the fixed stars are the centers of similar systems, they will all be constructed
    according to a similar design and subject to the dominion of One”
    1855, 1890, 1950’s: Various Astronomers: Exoplanets!
    Other Astronomers: No.
    1952: Otto Struve: What about really short-period giant exoplanets? Maybe we
    could detect those?
    Astronomers: Now you’re just being ridiculous.
    1992: Wolszczan & Frail: No seriously guys, exoplanets.
    Astronomers: Okay. But pulsars? That doesn’t count.
    1995: Mayor & Queloz: Remember those short-period giant exoplanets…?

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  5. Why Transits
    Our most fundamental and precise knowledge of
    stars comes from eclipsing systems, even after
    more than a century since eclipses were first
    observed. The same is likely to be true for
    exoplanets.
    Josh Winn
    5

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  6. Jupiter Earth

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  7. To monitor such a large groups of stars simultaneously while
    maintaining the required photometric precision, a detector
    array coupled by a fiber-optic bundle to the focal plane of a
    moderate aperture (≈ 1 m), wide field of view (≈50°)
    telescope is required. Based on the stated assumptions, a
    detection rate of one planet per year of observation
    appears possible.
    9
    April 1984

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  8. Transits of exoplanets
    10
    Orbital Period
    Size of
    Planet
    The light curves tell us the size and orbital period of the planet.
    The orbital period can be used to estimate the planet’s surface
    temperature
    Brightness of Star
    Time (days)

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  9. The First Exo-transits
    HD 209458 b
    observed from HST
    and the ground
    11

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  10. The Transiting Pioneers
    • Many teams but Super-WASP and HATnet were the
    most successful
    • Without their efforts, no Kepler, no TESS, no PLATO
    12

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  11. Exoplanet Detections, 2015
    Earth
    >2300 Confirmed >4600 Candidates

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  12. The Kepler Field of View
    May 2009 – May 2013

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  13. Pixels!
    • Goal is to go from 2-D
    image time series to a 1-D
    time series
    • Simplest method is simply
    to sum pixels where star is
    • More complex methods
    exist
    - e.g. PSF photometry
    18

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  14. Pixels!
    • A more complex image
    • Multiple stars, more image
    motion
    19

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  15. More Pixels!
    20

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  16. Pixel Time Series
    21

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  17. Transits in Pixels
    22

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  18. Transits in Pixels
    23

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  19. Transits in Pixels
    24

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  20. Pulsating M-dwarfs?
    25

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  21. A Cautionary Tale
    26

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  22. A Cautionary Tale
    27

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  23. Rotating M-dwarfs
    28

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  24. Aperture Photometry
    29

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  25. The Light Curve
    30

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  26. Light Curves
    31
    Large, close-in planets are easy to detect (cherry picking)
    Stellar variability and noise typically dominate signal …

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  27. 32
    Stellar Variability
    and Noise

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  28. Take the Kepler mission for example – the primary
    science goal was measuring the occurrence rate of
    Earth-like planets around sunlike stars
    That’s hard! Earth has an 85ppm transit
    Design needed to account for all known sources of
    noise – budget of 20ppm in 6h at Kp=12
    Stars (10ppm), Poisson (14ppm), Detector (10ppm)
    How big a telescope do we need?
    How faint a star can we look at?
    How much do we need to spend on a detector?
    Christiansen+2013
    Why do we care about noise?

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  29. Credit: Dan Foreman-Mackay, Davos
    Stars do
    stuff!
    Spacecraft do stuff!

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  30. Credit: Dan Foreman-Mackay, Davos
    Stars do
    stuff!
    Telescopes do stuff!
    Detectors do stuff!
    (Planets do stuff, too!)
    atmosphere
    Atmospheres do stuff!

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  31. Stars do
    stuff…
    They emit photons!
    Poisson noise: σ = sqrt(λ)
    You can beat it down
    with longer integrations,
    but you still need to well
    sample the shape of the
    transit…
    … basically the noise
    floor
    So why didn’t Kepler get
    there?

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  32. Stars do other stuff…
    Credit: Xavier Dumusque

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  33. Oscillations (<15 minutes, <1%)
    Granulation (15min-2days,<0.1%)
    Magnetic activity
    - spots (2days to weeks, <10%))
    - flares (stochastic,Pulsations (mins to yrs, <10s of %)
    Eclipses (hrs to yrs, <50%)
    Yikes! Luckily, we are mostly saved by timescales
    and careful target selection
    Stars do stuff…
    Credit: Arcetri Solar
    Physics Group/NSO
    Paz-Chinchon+2015
    Credit: NASA/SDO
    Davenport et al. 2014
    Molner+2014
    Christiansen, PhD thesis, 2007

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  34. Different stars do different stuff…
    Christiansen+2012
    FGK dwarf stars;
    Oscillations,
    Granulation,
    Spots
    Subgiant, giant stars;
    Oscillations, pulsations
    M dwarf stars;
    Spots, flares

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  35. Different stars do different stuff…
    Christiansen+2012

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  36. So how did we do overall?
    Kepler stellar noise budget – 10ppm
    Turns out stars are noisier than the Sun!
    Gilliland+2011

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  37. Telescopes do stuff…
    They typically have pointing jitter
    • Intra-pixel variations, e.g. Spitzer, K2 (not Kepler!)
    Vanderburg+2016

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  38. They typically have pointing jitter
    • Intra-pixel variations, e.g. Spitzer, K2 (not Kepler!)
    • Inter-pixel variations, e.g. EPOCh
    They experience thermal variations, e.g. Kepler
    Telescopes do stuff…
    Smith+2012
    Christiansen+2013

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  39. Detectors do stuff…
    Kepler Instrument Handbook, Caldwell & Van Cleve 2009

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  40. Detectors do stuff…
    Kepler Instrument Handbook, Caldwell & Van Cleve 2009

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  41. Detectors do stuff…

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  42. Noise Budget Measured
    Stellar 10.0 19.5
    Shot 14.1 16.8
    Detector 10.0 10.8
    Quarter dependent … 7.8
    Total 20.0 29.0
    Final noise contribution
    Christiansen+2013
    Gilliland+2011

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  43. Correcting the Light Curve
    48

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  44. Instrumental Systematics
    • Signals coming from the spacecraft dominate the
    signal
    - instrumental systematics also important for ground-
    based observations
    • We need to remove these instrumental signals
    49

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  45. Instrumental Systematics
    • Many ways of doing this, usually comes down to
    either:
    • decorating against an external measurement
    • using common signals amongst other stars
    50

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  46. The flux time series
    51

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  47. Fortunately stars change in the same way
    52

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  48. Principle component analysis seems to work fairly well
    53

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  49. Instrumental Systematics
    54

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  50. Instrumental Systematics
    55

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  51. Removing Stellar Signals
    • Easiest way is a high pass filter
    - not always (ever?) a good choice but often good
    enough
    - masking transits is always a good idea
    - better alternatives are using a locally conditioned
    function (i.e. polynomial, spline, etc.)
    • The right way is to build a physical model
    • Later on I’ll mention Gaussian Processes
    56

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  52. Removing Stellar Signals
    57

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  53. Removing Stellar Signals
    58

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  54. Removing Stellar Signals
    59

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  55. Detecting Planets
    • The industry standard
    is the box-least-squares
    - Kovacs 2002
    • other methods exist but
    all come down to using
    a matched filter
    60

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  56. 61
    Modeling Transits

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  57. What you observe
    • Transit duration
    • Transit depth
    • Ingress and egress
    duration
    • Time of transit mid-point
    • Orbital period
    • perhaps some phase variations and an
    occultation
    62

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  58. Modeling Transits
    63
    7/23/2012' Eric'Agol'University'of'Washington
    ' 2
    '
    Boxcar'Transit'Model:
    '
    Normalized+Flux+
    Time+
    Transits'only'give'us'quanJJes'with'dimensions'of'
    1)'Jme'';'2)'flux'';'3)'dimensionless'
    BoxNcar/pulse/topNhat'transit'shape'is'useful'for'
    transit'searches,'e.g.'BLS'or'QATS'
    From Eric Agol’s Sagan Workshop talk

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  59. Modeling Transits
    64
    7/23/2012' Eric'Agol'University'of'Washington
    ' 4
    '
    Trapezoidal'Transit'Light'Curve:
    '
    Normalized+Flux+
    Time+
    t
    I
    :'1st'contact,'start'ingress'
    t
    II
    :'2nd'contact,'end'ingress'
    t
    III
    :'3rd'contact,'start'egress'
    t
    IV
    :'4th'contact,'end'egress'
    1.  Ingress'duraJon:'t
    II
    Nt
    I
    '
    2.  Transit'duraJon:'t
    IV
    Nt
    I
    ''
    3.  Transit'‘Jme’:'(t
    II
    +t
    III
    )/2'
    4.  Orbital'period'
    t
    I
    t
    II t
    III
    t
    IV
    From Eric Agol’s Sagan Workshop talk

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  60. Modeling Transits
    65
    7/23/2012' Eric'Agol'University'of'Washington
    ' 5
    '
    Uniform'Transit'Light'Curve:
    '
    Normalized+Flux+
    Time+
    For'circular'orbit:'impact'parameter'(b/R
    *
    ),'
    velocity'of'planet'across'star'(v/R
    *
    ),'central'
    Jme'of'transit'(t
    0
    ),'and'radius'raJo'(R
    p
    /R
    *
    ).''
    With'period,'find'semiNmajor'axis'(a/R
    *
    )'
    From Eric Agol’s Sagan Workshop talk

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  61. Modeling Transits
    66
    7/23/2012' 6
    '
    Rela1ve+flux
    '
    Time
    +
    b/R
    *
    t
    0
    v/R
    *
    a
    R
    *
    =
    P

    v
    R
    *
    From Eric Agol’s Sagan Workshop talk

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  62. Modeling Transits
    67
    7/23/2012' 7
    '
    Rela1ve+flux
    '
    Time
    +
    t
    F
    '
    t
    T
    '
    b
    R
    *
    = 1+ΔF'2ΔF1/2
    t
    T
    2 +t
    F
    2
    t
    T
    2 't
    F
    2
    "
    #
    $
    $
    %
    &
    '
    '
    v
    R
    *
    =4 ΔF(t
    T
    2 )t
    F
    2)
    "
    #
    $
    %
    −1/2
    From Eric Agol’s Sagan Workshop talk

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  63. Modeling Transits
    68
    7/23/2012' Eric'Agol'University'of'Washington
    ' 12
    '
    LimbNdarkened'Transit'Light'Curve:
    '
    Normalized+Flux+
    Time+
    Limb'darkening'makes'life'complicated:''can'
    cause'degeneracy'between'impact'
    parameter,'limbNdarkening'parameter(s),'and
    '
    radius'raJo.'
    From Eric Agol’s Sagan Workshop talk

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  64. Why is Happening on the Star?
    69
    b, impact parameter
    • Planet-to-star
    radius ratio
    • Impact parameter
    - a function of
    inclination and
    orbital distance
    • Ingress and egress
    duration
    • Limb darkening
    • Time of transit mid-
    point
    Limb darkening
    Stellar noise

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  65. Orbital Elements
    Orbits can be
    uniquely
    described by the 6
    orbital elements
    - semimajor axis
    - eccentricity
    - inclination
    - longitude of
    ascending node
    - argument of
    periastron
    - mean anomaly
    70

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  66. What is the Planet Doing?
    71
    Winn 2010
    • The planet has an
    orbital period and
    an orbital
    distance
    • Follows Kepler’s
    Laws
    Go read Winn 2010! http://arxiv.org/pdf/1001.2010v5.pdf

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  67. What is the Planet Doing?
    72
    Winn 2010
    • The planet has an
    orbital period and
    an orbital
    distance
    • Follows Kepler’s
    Laws
    • Orbit can be
    eccentric!
    • Periastron angle
    changed duration
    Go read Winn 2010! http://arxiv.org/pdf/1001.2010v5.pdf

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  68. Transit Equations
    • A few important equations
    73
    (t)

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  69. 74
    From Eric Agol’s Sagan Workshop talk
    Limb Darkening Primer

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  70. 75
    IntegraJon'over'limb'darkening
    '
    7/23/2012' Eric'Agol'University'of'Washington
    ' 14
    '
    F(r
    1
    ,r
    2
    ,d,I(r))= rdrdφ ⋅I(r)
    visible area

    = 1
    2
    dr2 dφ ⋅
    dI(r)
    2dr
    visible area

    = π dr2
    dI(r)
    dr
    0
    r
    2
    2
    ∫ (1−δ(r
    1
    ,r,d))
    I(r)+
    r+
    AnalyJc'for'quadraJc'&'‘nonNlinear’'limbNdarkening
    '
    models'(Mandel'&'Agol'2002;'Pal'2008)'
    From Eric Agol’s Sagan Workshop talk
    Limb Darkening Primer

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  71. Standard Limb Darkening Equations
    • Uniform disk - trapezoidal model
    • Linear with stellar radius
    - Swartzshild 1906
    • Quadratic model
    - Kopal 1949
    • Non-linear limb darkening
    - Claret 2000
    76

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  72. 77
    7/23/2012' Eric'Agol'University'of'Washington
    ' 16
    '
    Choice'of'limbNdarkening'model:''
    '
    1.  If'data'quality'are'poor,'fix'to'limbNdarkening'of'
    atmosphere'models'(Claret'2000,'Sing'2011)'
    2.  If'high'quality,'may'let'parameters'float'&'fit'for'them'
    3.  Model'limbNdarkening'do'not'agree'perfectly'with'
    data,'although'3D'atmospheres'work'well'(Hayek'et'
    al.'2012)'
    4.  Unnecessary'for'secondary'eclipse'(except'for'high'S/
    N),'but'need'to'add'in'flux'from'star'
    5.  Small'planet'approximaJon:'occulted''flux'≈'(area'of'
    planetNstar'overlap)'x'(stellar'intensity'at'center'of'
    planet)'
    From Eric Agol’s Sagan Workshop talk
    Choice of Limb Darkening Model

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  73. 78
    Modeling Transits
    7/23/2012' Eric'Agol'University'of'Washington
    ' 15
    '
    Sky'separaJon'of'planets'versus'Jme:'
    1.  Straight'line'transit:'
    –'fine'for'a/R*
    1,'e'small'
    2.  Circular'orbit:'
    3.  Keplerian'orbit'–'requires'Kepler'solver'(m'&'e''f);'7'
    parameters'(Murray'&'Dermos):'
    4.  NNbody'integrator'(for'3+'bodies,'precession,'GR,'etc.):'
    7nN1'parameters'(Fabrycky)'
    • Integrate'over'each'exposure'unJl'converged'(Kipping'2010)
    '
    r
    sky
    /R
    *
    = v /R
    *
    ( )2
    (t −t
    0
    )2
    + b /R
    *
    ( )2
    r
    sky
    /R
    *
    = a /R
    *
    1−sin2 icos2 2π(t −t
    0
    )/P
    ( )
    r
    sky
    /R
    *
    = a /R
    *
    1−sin2 isin2
    ω + f
    ( )
    From Eric Agol’s Sagan Workshop talk

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  74. Modeling Transits
    • Almost everyone uses a parameterization of a limb
    darkened transit developed by Mandel & Agol (2002)
    • It parameterizes a transit by:
    - the projected distance between the center of the planet and
    the center of the star in stellar radii
    - the radius ratio of the star and planet
    - limb darkening parameters
    • It makes assumptions that you should be aware of!
    - the star has uniform brightness behind the planet
    - planet is dark
    - limb darkening can be parameterized by a simple function
    • Must, must, must include integration time
    79

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  75. Modeling Transits
    • I’m going to be using ktransit, many other codes exist
    - github.com/mrtommyb/ktransit
    • Here is a simple model
    M.add_star(
    rho=1.5, # mean stellar density in cgs units
    ld1=0.2, # ld1--4 are limb darkening coefficients
    ld2=0.4, # if only ld1 and ld2 are non-zero then a quadratic limb darkening law is used
    ld3=0.0, # if all four parameters are non-zero we use non-linear flavour limb darkening
    ld4=0.0,
    dil=0.0, # a dilution factor: 0.0 -> transit not diluted, 0.5 -> transit 50% diluted
    zpt=0.0 # a photometric zeropoint, in case the normalisation was wonky
    )
    M.add_planet(
    T0=1.0, # a transit mid-time
    period=1.0, # an orbital period in days
    impact=0.1, # an impact parameter
    rprs=0.1, # planet stellar radius ratio
    ecosw=0.0, # eccentricity vector
    esinw=0.0,
    occ=0.0) # a secondary eclipse depth in ppm
    80

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  76. A Wealth of Information
    ld1=0.0
    ld2=0.0
    ld3=0.0
    ld4=0.0
    period=1.0
    impact=0.1
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    81
    Uniform Disk

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  77. A Wealth of Information
    ld1=0.6
    ld2=0.0
    ld3=0.0
    ld4=0.0
    period=1.0
    impact=0.1
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    82
    Linear Law

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  78. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1.0
    impact=0.1
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    83
    Quadratic Law

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  79. A Wealth of Information
    ld1=0.46
    ld2=0.13
    ld3=0.40
    ld4=-0.25
    period=1.0
    impact=0.1
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    84
    4-parameter Law

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  80. A Wealth of Information
    ld1=0.46
    ld2=0.13
    ld3=0.40
    ld4=-0.25
    period=1.0
    impact=0.1
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    85
    4-parameter Law
    Deviations of a few 100 ppm

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  81. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1, 10, 100
    impact=0.1
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    86
    Orbital periods of 1, 10 100 days

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  82. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1.0
    impact=0.1, 0.4, 0.85
    rprs=0.1
    ecosw=0.0
    esinw=0.0
    87
    Inclinations of 0.1, 0.4, 0.85

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  83. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1.0
    impact=0.1
    rprs=0.1, 0.05, 0.01
    ecosw=0.0
    esinw=0.0
    88
    Rp/r* of 0.1, 0.05, 0.01

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  84. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1.0
    impact=0.1
    rprs=0.1, 0.05, 0.01
    ecosw=0.0, 0.3, 0.7
    esinw=0.0
    89
    ecosw of 0.0, 0.3, 0.7

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  85. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1.0
    impact=0.1
    rprs=0.1, 0.05, 0.01
    ecosw=0.0, 0.3, 0.7
    esinw=0.0
    90
    ecosw of 0.0, 0.3, 0.7

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  86. A Wealth of Information
    ld1=0.4
    ld2=0.26
    ld3=0.0
    ld4=-0.0
    period=1.0
    impact=0.1
    rprs=0.1, 0.05, 0.01
    ecosw=0.0, 0.3, 0.7
    esinw=0.0, 0.3, 0.7
    91
    esinw of 0.0, 0.3, 0.7

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  87. Modeling Stellar Density
    • As Dan Huber
    mentioned, you can
    model stellar density
    using the transit light
    curve
    - Seager & Mallén-
    Ornelas (2002)
    92

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  88. Planets orbiting the Sun at 1AU
    93

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  89. Planets orbiting M0-type star receiving
    1x Earth insolation
    94

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  90. 95
    Modeling
    Phases

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  91. Phase variations
    96

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  92. Tidally induced ellipsoidal variations
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    – Second level
    • Third level
    – Fourth level
    • Fifth level
    97

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  93. Doppler beaming
    • From the reflex
    motion of the
    star owing to a
    planet
    • Combination of
    two effects
    – one relativistic and
    one classical
    98
    Classical
    • As the star moves towards/away us the
    light is blue/red-shifted
    • The spectrum of the star moves in/out of
    the Kepler passband the star gets
    brighter/fainter
    Relativistic
    • As star moves towards us the light
    is beamed in our direction
    • As it moves away light is beamed in
    other direction, star gets fainter

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  94. Reflection/emission from the planet
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    – Second level
    • Third level
    – Fourth level
    • Fifth level
    99
    Madhusudhan & Burrows 2011

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  95. Phase variations
    • Build a simple model (really simple!!)
    • Assume phase variations are a combination of a few sinusoidal functions
    100
    Ellipsoidal variations Doppler beaming
    Reflection/emission
    from planet
    Shown here is a
    Lambertian phase
    function

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  96. What we can learn from phase
    variations
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    – Second level
    • Third level
    – Fourth level
    • Fifth level
    101

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  97. Using the occultation and reflection
    102
    The occultation tells us the
    planet-star contrast.
    The reflection/emission amplitude
    tells us the day-night planet
    contrast

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  98. Kepler-13b
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    – Second level
    • Third level
    – Fourth level
    • Fifth level
    103
    Used to derive a mass from beaming of
    9.2±1.1 MJup
    Shporer et al. 2011

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  99. Phase variations seen from TrES-2b
    • We detect significant
    ellipsoidal, beaming and
    reflection from TrES-2b.
    • A radial velocity amplitude
    consistent with ground
    based RVs
    104
    Photometry Ground-based RVs
    Barclay et al., 2012

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  100. Fitting a Transit to Data
    • Simple: use a Gaussian log-likelihood
    - aka a chi-squared
    - optimize using your favorite optimization scheme
    loglike = (
    - (0.5 * num_data_points)
    * np.log(2. * np.pi)
    - 0.5 * np.sum((model_lc - flux)**2
    / ferr)
    )
    negloglike = -loglike
    • Usually does fine if you avoid including eccentricity
    105

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  101. Fitting a Transit to Data
    106

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  102. Building a Likelihood Function
    • Ok but we can do better - Bayesian, use priors
    • If we have the stellar density from seismology, use a gaussian prior on
    that
    • Let’s keep things physical
    • Planet should not enter the star so:
    ecc < (1.-(1./ar))):
    • The center of the star should be the brightest point*
    • ld1 > 0.0
    • specific intensity to remain above zero
    • ld1 + ld2 < 1.0
    • do not allow limb-brightened profiles*
    • ld1 + 2.*ld2 > 0.0
    • we should also include tophat priors to keep all parameters sensible
    • eccentricity should be < 1.0
    • We should be sampling in log-space of most physical parameters
    (Jeffreys Priors)
    107

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  103. Building a Likelihood Function
    • About that eccentricity…
    - I sample in esinw and
    ecosw, other options
    apply
    • this is biased
    - however we can assert a
    prior on e that gets mostly
    around this bias
    - logecc = -
    np.log(ecc)
    108

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  104. Uncertainties!
    • Probably wise to never trust an observer’s
    uncertainties
    • We can model away their optimism
    • sigma_obs -> (sigma_obs^2 + sigma_mod^2)^0.5
    • sigma_mod is a model parameter
    • could represent either underestimated uncertainness
    or deficiencies in the model
    109

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  105. Uncertainties!
    • Probably wise to never trust an observer’s uncertainties
    • We can model away their optimism
    • sigma_obs -> (sigma_obs^2 + sigma_mod^2)^0.5
    • sigma_mod is a model parameter
    - could represent either underestimated uncertainness or
    deficiencies in the model
    - sample in log-space
    - loglc = (
    - (npt_lc/2.)*np.log(2.*np.pi)
    - 0.5 * np.sum(np.log(sigma_obs^2 + sigma_mod^2))
    - 0.5 * np.sum((model_lc - flux)**2 / err_jit2)
    )
    110

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  106. Using MCMC analyses
    • Let use Markov-Chain Monte Carlo
    - I use emcee
    111

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  107. Gaussian Processes
    112
    Credit to Dan Foreman-Mackey
    search for his speaker deck page for more info

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  108. Gaussian Processes
    113

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  109. Gaussian Processes
    114

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  110. Gaussian Processes
    115
    [Ai-yi]T

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  111. Gaussian Processes
    116

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  112. Gaussian Processes
    117

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  113. Gaussian Processes
    118

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  114. Gaussian Processes
    119

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  115. Gaussian Processes
    120

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  116. Gaussian Processes
    121

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  117. Gaussian Processes
    122

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  118. GPs and Red Giants
    123

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  119. GPs and Red Giants
    124

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  120. GPs and Red Giants
    125

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  121. GPs and Red Giants
    126

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  122. GPs and Red Giants
    127

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  123. GPs and Red Giants
    128

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  124. The Future
    129
    TESS
    PLATO
    TRAPPIST
    NGTS
    Mearth

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  125. References
    130
    • Carol Haswell - Transiting Exoplanets
    • Michael Perryman - Exoplanet Handbook
    • Sara Seager et al. - Exoplanets

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  126. extra slides
    131

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  127. Including Radial Velocity Data
    132
    ESO

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  128. Modeling Transit Timing Variations
    133
    Transit Timing Variations (TTVs)

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  129. Stellar Blends
    134
    Planet or Blend?
    Eclipsing Binary Physically bound or
    Chance alignment
    Primary
    Star
    Secondary
    Star (MS or not)
    Tertiary
    Star or planet
    An observed periodic transit signal could be due to:
    Transiting Planet
    (or planetary size object)

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