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

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…?

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

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

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)

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

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?

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Reﬂection/emission from the planet • Click to edit Master text styles – Second level • Third level – Fourth level • Fifth level 99 Madhusudhan & Burrows 2011

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

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

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

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

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

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

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

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

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

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)