Transit Light Curves - PDF Free Download

Transit Light Curves Szilárd Csizmadia Deutsches Zentrum für Luft- und Raumfahrt /Berlin-Adlershof, Deutschland/ Folie 1

Outline 1. Introduction: why transits? 2. Transits in the Solar System 3. Transits of Extrasolar Objects 4. Classification of transits 5. Information Extraction from Transits 5.1 Uniform stellar discs 5.2 Limb darkened discs 5.3 Stellar spots 5.4. Gravity darkened discs 5.5 Models in the past and present 6. Optimization: methods & problems 7. Exomoons & exorings 8. Summary Folie 2

Early transit observations Venus transit in 1761, 1769 Jeremiah Horrocks (1639, Venus) Folie 3

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The Astronomical Unit via the transits of Venus Folie 5

The Astronomical Unit via the transits of Venus From geogr. meas. ~0.3 AU ~0.7 AU (Kepler's third law + period measurement) Folie 6

Measuring the Atmospheric Properties of Venus utilizing its Transits (It can be extended to extra-solar planets, too) Hedelt et al. 2011, A&A Folie 7

Other usage of transits (just a few example): - measuring the speed of the light (Römer c. 1670) - testing and developing the theory of motion of satellites and other celestial objects - occultation - pair of the transit - was used to measure the speed of the gravity (Kopeikin & Fomalont 2002) - occultations also used to refine the orbits of asteroids/kuiper-belt objects as well as to measure the diameter and shape of them - popularizing astronomy Transit of the moon Sun eclipsed by the moon. Transit = kind of eclipse? Folie 8

Transit of the Earth from the L2 point of the Sun-Earth system: is it an annular eclipse? Folie 9

The benefits of exoplanet transits - it gives the inclination, radius ratio of the star/planet - we can establish that the RV-object is a planet at all (i) - inclination is necessary to determine the mass - mas and radius yield the average density: strong constrains for the internal structure - transit and occultation together give better measurement of eccentricity and argument of periastron - we learn about stellar photosphers and atmospheres via transit photometry (stellar spots, plages, faculae; limb darkening; oblateness etc.) - possibility of transit spectroscopy (atmospheric studies, search for biomarkers) - oblateness of the planet, rotational rate, albedo measurements, surfaces with different albedo/temperature; nightside radiation/nightly lights of the cities; exomoons, exorings - all of these are in principle, not in practice - Transit Timing Variations: measuring k2; other objects (moon, planet, (sub)stellar companion); mass loss via evaporation; magnetic interaction; etc. - photometric Rossiter-McLaughlin-effect (in principle; phot. prec. is not yet) Folie 10

NOTE: ALL of our knowledge about exoplanetary transits are originated from the binary star astronomy: it is our Royal Road and mine of information! Folie 11

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Orientation of the orbit i=90 t o i<>90 (few arcminutes): Plane of the sky (East) t t Gimenez and Pelayo, 1983 t p Folie 14

The definition of contacts (Winn 2010) Folie 15

(Winn 2010) Folie 16

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t t t o Folie 18

Some useful relationships Blue line: impact parameter, br s Red line: first (fourth) contact: Green line: second (third contact): Not proven here (see Milone & Kallrath 2010): Folie 19

The impact parameter b Angular momentum vector i r 90 -i br s to the observer (line of sight) Folie 20

Types of eclipses/transits Some definitions: Transit (k<<1) Annular eclipse (k<1 and k 1) R 1 : the bigger object's radius R 2 : the smaller object's radius Of course, 2nd object can be a planet, too. k = R 2 /R 1, the radius ratio (or it is the planet-to-stellar radius ratio) Total eclipse (k<1) Partial eclipse (1-k<b<1+k) Occultation (k << 1) r 1 = R 1 /A r 2 = R 2 /A, the fractional radius (A is the semi-major axis) Folie 21

The simplest model of transits/eclipses Objects are spherical, their projections are a simple disc The surface brightness distribution is uniform Time is denoted by t, the origo of the coordinate system is in the primary. Folie 22

Occurence time of the eclipses (i=90) Primary eclipse (transit): Secondary eclipse (occultation): From complicated series-calculations: Folie 25

Some very useful formulae Folie 26

Some very useful formulae Folie 27

Some very useful formulae Folie 28

By simple time-measurements you can determine eccentricity and argument of periastron: Folie 29

The shape of the transit in the case of uniform surface brightness distribution (g(v) is the phase-function) Annular eclipse/transit: Occultation: Out-of-eclipse: (See Kane & Gelino for full, correct expression) For known exoplanets (Kane & Gelino 2010): Folie 30

The partial eclipse phase is more complicated: Folie 31

The partial eclipse phase is more complicated: x β D-x α R 2 R 1 Similar for the other zone. Folie 32

The partial eclipse phase is more complicated: Folie 33

The partial eclipse phase is more complicated: Folie 34

The partial eclipse phase is more complicated: Folie 35

The partial eclipse phase is more complicated: Folie 36

The partial eclipse phase is more complicated: The partial phase is already quite complicated in the case of even a uniform disc. And: it is described by a transcendent equation so it is not invertable analytically! Folie 37

What does limb-darkening cause? Mandel & Agol 2002 Folie 38

More precise approximation of the stellar radiation and thus the light curve shape: Limb darkening + small planet approximation Total flux of the star: Blocked flux of a small planet: Relative flux decrease: Folie 39

More precise = more complicated If we take into account, that the stellar intensity is not constant behind the planet, we can reach even higher precision, but this requires to introduce: - elliptic functions to describe the light curve shape (e.g. Mandel & Agol 2002) - Jacobi-polynomials as parts of infinite series for the same purpose (Kopal 1989; Gimenez 2006) - applying semi-analytic approximations (EBOP: Netzel & Davies 1979, 1981; JKTEBOP Southworth 2006) - using fully numerical codes (Wilson & Devinney 1971; Wilson 1979; Linnel 1989; Djurasevic 1992; Orosz & Hausschildt 2000; Prsa & Zwitter 2006; Csizmadia et al. 2009 - etc). Folie 41

Example: equations of the M&A02 model: Folie 42

Do we know the value of limb darkening a priori? Diamond: Sing (2010) Light blue: C&B11, ATLAS+FCM Black line: C&B11, ATLAS+L Magenta: C&B11, PHOENIX+L Dark blue line: C&B11, PHOENIX+FCM Folie 43

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Probing the limb darkening theories on exoplanets and eclipsing binary stars Careful analysis with quadratic LD-law of HD 209 458 : "It seems that the current atmosphere models are unable to explain the specific intensity distribution of HD 209458." (A. Claret, A&A 506, 1335, 2009) Recent study on 9 eclipsing binaries (A. Claret, A&A 482, 259, 2008): Folie 45

Effect of stellar spots Concept of effective limb darkening (??) Limb darkening is a function of temperature, surface gravity and chemical composition. Stellar spots are always present: size, darkness, lifetime etc. can be very different. u eff = f(t star, T spot, Area spot, u star, u spot, ) Folie 46

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The concept of effective limb darkening The observed star = the modelled star Folie 48

The concept of effective limb darkening The observed star = the modelled star THIS IS NOT TRUE Folie 49

The concept of effective limb darkening The observed star = the unmaculated star + stellar spots Folie 50

The concept of effective limb darkening The observed star = the unmaculated star + stellar spots THIS IS TRUE Folie 51

The concept of effective limb darkening The observed star = the unmaculated star + stellar spots Folie 52

The concept of effective limb darkening The observed star = the unmaculated star + stellar spots F star : we observe an unmaculated star ΔF planet : we remove the light of the unmaculated surface due to planet transit (assumption: planet does not cross the spot(s) πr spot 2 F star : we remove the stellar light at the place (b spot ) of the spot πr spot 2 F spot : we put the spot light at the place (b spot ) of the spot So, in practice, we replaced a small part of the stellar flux with the spot's flux. Folie 53

The concept of effective limb darkening The observed star = the unmaculated star + stellar spots Folie 54

The concept of effective limb darkening The observed star = the unmaculated star + stellar spots Folie 55

Spots at the edge can cause effectively limbbrightening... See Csizmadia et al. (2012) or Barros et al. (2011) Folie 56

Gravity darkening von Zeipel 1924 Lucy 1967 Barnes 2009 Claret 2011 Folie 57

Exomoons and exorings in the light curve Folie 58

The big question(s) How to find the best agreement??? Is the best agreement the solution itself? How big is our error? How fast is our code? Folie 59

Our problem is a highly nonlinear, not invertible, multidimensional optimization problem with many local minima. Observational noise makes the things even more complicated. Folie 60

How to find the solution if one has this more precise, but more complicated functions? To minimize: N: number of observed data points P: number of free parameters i: index of the point F obs : the observed flux (light, brightness etc.) F mod : the modell value for the same σ o : uncertainty of the observed data points σ m : uncertainty of the model, frequently set to zero Folie 61

Difference between local and global minima Function value Steepest descent Variable Folie 62

A time-consuming, but global minimum-finder method: grids How to do it: choose regurarly or randonly enough tests in the parameters space Advantage: it finds the global minimum (if the number of trials are big enough) Disadvantage: the required time tends to infinity... Folie 63

The old and fast method to find the nearest minimum (either local or global): differential correction and Levenberg-Marquardt Folie 64

The old and fast method to find the nearest minimum (either local or global): differential correction and Levenberg-Marquardt Necessary (but not sufficient) condition for minimum: For all parameter, so for all k! Folie 65

The old and fast method to find the nearest minimum (either local or global): differential correction and Levenberg-Marquardt 1. Choose an initial p. 2. Calculate A, b and then dp. 3. p' = p + dp 4. Iterate 2-3 until convergence. Folie 66

Optimization problems in astronomy Folie 68

Goals The optimization should: be fast (in CPU time = number of steps x time required for one step) capture all the global minima (values between χ 2 min and χ2 min + 1) produce maps of the phase-space (parameter-space, hyperspace) capture the best fit(s) however, no standard method exists main problem: each hyperspace is different and that is why it requires its own methods/settings that is why no general receipt, new methods are tried and developed "no free lunch"-theorem of mathematics: whatever optimization method is used, we cannot avoid the problem that it takes time or we have a fast method, but we do not catch the best fit. Folie 69

What is Optimization in other words? Procedure to find the parameters which produce the local (or global) maximum/minimum of a function In the astronomical inverse problem we are (usually) interested in the global minimum of the χ 2 -function. Finding Best Solution Minimal Cost (Design) Minimal Error (Parameter Calibration) Maximal Profit (Management) Maximal Utility (Economics) Folie 70

Optimization algorithms used for transiting exoplanets MCMC (HAT, WASP teams, and CoRoT-4b, 5b, 12b, partially 6b, 11b) Amoeba (all CoRoT-planets, except 4b, 5b, 12b, 13b) Harmony Search (for 13b, as well as an additional independent methods for 6b-11b) I tried (based on binary star astronomy experience): MCMC Amoeba Price AGA HS (first time in astronomy) Differential corrections (probably good for high S/N, not mentioned hereafter) Daemon (not good for us, not mentioned hereafter) Folie 71

Markov Chain Monte Carlo (with Metropolitan-Hastings algorithm) Choose x 0 and s 0 stepsize Burn-in phase: x i+1 = x i + r s i Acceptance: χ 2 i+1 < χ2 i or if Stepsize should be adjusted for an acceptance rate ~23% The Markov-chain: like in burn-in phase, but the results are saved (the burn-in results are forgotten!) The result is defined as: x j = MEAN(x ij ) Δx j = STDDEV(x ij ) Folie 72

Disadvantages: - the two distributions should be nearly the same (P is the probability distribution in reality, Q is the same for the calculated models.) - the sampling of the whole parameter space is not well done, infinitely long time is required to sample the whole hyperspace - if the chain is not long enough, then it is more probable that we find a local minimum instead of the global one. Folie 73

Amoeba - very simple - depends on the starting values - you have to restart it with different starting numbers several times (~1000) - the sampling of the parameter space is questionable, uniqueness is not warranted and not checked Folie 74

Genetic Algorithms: who will survive and produce new off-springs? Folie 75

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From Canto et al. Folie 77

The big family of genetic algorithms ~ 1970 Price (1979; sometimes it is used for eclipsing binaries) GA (in astronomy; 1995, Charbonneau) HS (2001) AGA (2010)... many more Folie 78

School Bus Routing Problem 8 4 8 15 10 20 5 5 4 9 10 School 5 7 10 4 15 20 10 4 5 5 6 6 7 5 7 10 4 15 5 5 Depot 1 3 5 2 3 8 Min C1 (# of Buses) + C2 (Travel Time) s.t. Time Window & Bus Capacity GA = $409,597, HS = $399,870 Folie 79

Stopping criteria more seriously: Supervisor is unpatient or proceeding's deadline (the worst things what you can imagine) Number of iterations (e.g. in MCMC or the previous astronomer's advice) Marquardt-lambda is smaller than machine's accuracy (Milone et al. 1998) χ 2 aim is reached (sometimes it is not possible) Standard deviations of the parameters are within a prescribed values Changes are smaller than the scatter of the fit (it can be dangerous...) Convergence: changes in parameters is within a prescribed value (this value can be related to the scatter of the actual parameter values) Zola et al. (2002): max( χ 2 ) / min( χ 2 ) < 1.01 Folie 80

Comparison of methods MCMC Price AGA HS Test: where is the global minimum of Michalewicz's bivariate function: We know that f(x,y) -1.801 at (2.20319..., 1.57049...) if 0 x π, 0 y π,. Folie 81

Michalewicz's bivariate function Folie 82

Results Method x y d Steps Exact 2.20319 1.57049 - - MCMC 2.18912 0.300988 1.18959 100 000 Price (N=25) 1.05775 1.57111 1.14544 250 Price (N=100) 2.20712 1.57936 0.00971 16 500 AGA (N=25) 2.20291 1.57080 0.00042 12 800 AGA (N=25) 2.20290 1.57080 0.00042 3225 HS (N=100) 2.20291 1.57073 0.00037 4600 HS (N=25) 2.20285 1.57072 0.00041 1300 Amoeba 2.20286 1.57082 0.00047 73 Folie 83

a/rs i k u 1 u 2 Folie 84

The final result Csizmadia et al. 2011 Folie 85

Csizmadia et al. 2011 Folie 86

Summary (i) Transits (and occultation) are the mine of information of our knowledge about transits. (ii) You can learn the most on transiting exoplanets. Other kinds of exoplanets are very important, but transiting ones tell you more about themselves. (iii) Transits (and occultations) are geometric events. However, to fully understand them, you have to know more about stellar physics than the planet itself... (iv) To analyze transits in detail, experience and carefullness are needed behind the theoretical knowledge about optimization problems. Folie 87

Thank you for your attention! Folie 88