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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
  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
  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
  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…?
  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
  6. None
  7. None
  8. Jupiter Earth

  9. 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
  10. 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)
  11. The First Exo-transits HD 209458 b observed from HST and

    the ground 11
  12. The Transiting Pioneers • Many teams but Super-WASP and HATnet

    were the most successful • Without their efforts, no Kepler, no TESS, no PLATO 12
  13. Today!

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

  15. None
  16. The Kepler Field of View May 2009 – May 2013

  17. None
  18. 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
  19. Pixels! • A more complex image • Multiple stars, more

    image motion 19
  20. More Pixels! 20

  21. Pixel Time Series 21

  22. Transits in Pixels 22

  23. Transits in Pixels 23

  24. Transits in Pixels 24

  25. Pulsating M-dwarfs? 25

  26. A Cautionary Tale 26

  27. A Cautionary Tale 27

  28. Rotating M-dwarfs 28

  29. Aperture Photometry 29

  30. The Light Curve 30

  31. Light Curves 31 Large, close-in planets are easy to detect

    (cherry picking) Stellar variability and noise typically dominate signal …
  32. 32 Stellar Variability and Noise

  33. 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?
  34. Credit: Dan Foreman-Mackay, Davos Stars do stuff! Spacecraft do stuff!

  35. Credit: Dan Foreman-Mackay, Davos Stars do stuff! Telescopes do stuff!

    Detectors do stuff! (Planets do stuff, too!) atmosphere Atmospheres do stuff!
  36. 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?
  37. Stars do other stuff… Credit: Xavier Dumusque

  38. Oscillations (<15 minutes, <1%) Granulation (15min-2days,<0.1%) Magnetic activity - spots

    (2days to weeks, <10%)) - flares (stochastic,<few%) 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
  39. Different stars do different stuff… Christiansen+2012 FGK dwarf stars; Oscillations,

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

  41. So how did we do overall? Kepler stellar noise budget

    – 10ppm Turns out stars are noisier than the Sun! Gilliland+2011
  42. Telescopes do stuff… They typically have pointing jitter • Intra-pixel

    variations, e.g. Spitzer, K2 (not Kepler!) Vanderburg+2016
  43. 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
  44. Detectors do stuff… Kepler Instrument Handbook, Caldwell & Van Cleve

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

    2009
  46. Detectors do stuff…

  47. 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
  48. Correcting the Light Curve 48

  49. 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
  50. Instrumental Systematics • Many ways of doing this, usually comes

    down to either: • decorating against an external measurement • using common signals amongst other stars 50
  51. The flux time series 51

  52. Fortunately stars change in the same way 52

  53. Principle component analysis seems to work fairly well 53

  54. Instrumental Systematics 54

  55. Instrumental Systematics 55

  56. 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
  57. Removing Stellar Signals 57

  58. Removing Stellar Signals 58

  59. Removing Stellar Signals 59

  60. 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
  61. 61 Modeling Transits

  62. 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
  63. 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
  64. 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
  65. 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
  66. Modeling Transits 66 7/23/2012' 6 ' Rela1ve+flux ' Time +

    b/R * t 0 v/R * a R * = P 2π v R * From Eric Agol’s Sagan Workshop talk
  67. 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
  68. 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
  69. 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
  70. 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
  71. 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
  72. 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
  73. Transit Equations • A few important equations 73 (t)

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

  75. 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
  76. 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
  77. 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
  78. 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
  79. 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
  80. 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
  81. 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
  82. 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
  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 ecosw=0.0 esinw=0.0 83 Quadratic Law
  84. 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
  85. 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
  86. 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
  87. 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
  88. 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
  89. 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
  90. 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
  91. 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
  92. Modeling Stellar Density • As Dan Huber mentioned, you can

    model stellar density using the transit light curve - Seager & Mallén- Ornelas (2002) 92
  93. Planets orbiting the Sun at 1AU 93

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

  95. 95 Modeling Phases

  96. Phase variations 96

  97. Tidally induced ellipsoidal variations • Click to edit Master text

    styles – Second level • Third level – Fourth level • Fifth level 97
  98. 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
  99. Reflection/emission from the planet • Click to edit Master text

    styles – Second level • Third level – Fourth level • Fifth level 99 Madhusudhan & Burrows 2011
  100. 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
  101. What we can learn from phase variations • Click to

    edit Master text styles – Second level • Third level – Fourth level • Fifth level 101
  102. 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
  103. Kepler-13b • Click to edit Master text styles – 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
  104. 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
  105. 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
  106. Fitting a Transit to Data 106

  107. 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
  108. 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
  109. 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
  110. 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
  111. Using MCMC analyses • Let use Markov-Chain Monte Carlo -

    I use emcee 111
  112. Gaussian Processes 112 Credit to Dan Foreman-Mackey search for his

    speaker deck page for more info
  113. Gaussian Processes 113

  114. Gaussian Processes 114

  115. Gaussian Processes 115 [Ai-yi]T

  116. Gaussian Processes 116

  117. Gaussian Processes 117

  118. Gaussian Processes 118

  119. Gaussian Processes 119

  120. Gaussian Processes 120

  121. Gaussian Processes 121

  122. Gaussian Processes 122

  123. GPs and Red Giants 123

  124. GPs and Red Giants 124

  125. GPs and Red Giants 125

  126. GPs and Red Giants 126

  127. GPs and Red Giants 127

  128. GPs and Red Giants 128

  129. The Future 129 TESS PLATO TRAPPIST NGTS Mearth

  130. References 130 • Carol Haswell - Transiting Exoplanets • Michael

    Perryman - Exoplanet Handbook • Sara Seager et al. - Exoplanets
  131. extra slides 131

  132. Including Radial Velocity Data 132 ESO

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

  134. 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)