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Vortrag > Autor > Dokumentname > Datum Folie 1 Transit Light Curves Szilárd Csizmadia Deutsches Zentrum für Luft- und Raumfahrt /Berlin-Adlershof, Deutschland/

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Vortrag > Autor > Dokumentname > Datum Folie 2 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

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Vortrag > Autor > Dokumentname > Datum Folie 3 Early transit observations Jeremiah Horrocks (1639, Venus) Venus transit in 1761, 1769

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Vortrag > Autor > Dokumentname > Datum Folie 4

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Vortrag > Autor > Dokumentname > Datum Folie 5 The Astronomical Unit via the transits of Venus

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

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

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Vortrag > Autor > Dokumentname > Datum Folie 8 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?

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Vortrag > Autor > Dokumentname > Datum Folie 9 Transit of the Earth from the L2 point of the Sun-Earth system: is it an annular eclipse?

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Vortrag > Autor > Dokumentname > Datum Folie 10 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)

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

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Vortrag > Autor > Dokumentname > Datum Folie 12

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Vortrag > Autor > Dokumentname > Datum Folie 13

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Vortrag > Autor > Dokumentname > Datum Folie 14 Orientation of the orbit Plane of the sky (East) i=90° i<>90° (few arcminutes): Gimenez and Pelayo, 1983 tptp t toto

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Vortrag > Autor > Dokumentname > Datum Folie 15 The definition of contacts (Winn 2010)

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Vortrag > Autor > Dokumentname > Datum Folie 16 (Winn 2010)

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Vortrag > Autor > Dokumentname > Datum Folie 17

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Vortrag > Autor > Dokumentname > Datum Folie 18 t t t o

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Vortrag > Autor > Dokumentname > Datum Folie 19 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):

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Vortrag > Autor > Dokumentname > Datum Folie 20 The impact parameter b to the observer (line of sight) Angular momentum vector i 90°-i bR s r

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Vortrag > Autor > Dokumentname > Datum Folie 21 Types of eclipses/transits Transit (k<<1) Annular eclipse (k<1 and k 1) Total eclipse (k<1) Partial eclipse (1-k**
**

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Vortrag > Autor > Dokumentname > Datum Folie 22 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.

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Vortrag > Autor > Dokumentname > Datum Folie 23 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. From two-body problem:

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Vortrag > Autor > Dokumentname > Datum Folie 24 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. From two-body problem:

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

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Vortrag > Autor > Dokumentname > Datum Folie 26 Some very useful formulae

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Vortrag > Autor > Dokumentname > Datum Folie 27 Some very useful formulae

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Vortrag > Autor > Dokumentname > Datum Folie 28 Some very useful formulae

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Vortrag > Autor > Dokumentname > Datum Folie 29 By simple time-measurements you can determine eccentricity and argument of periastron:

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

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Vortrag > Autor > Dokumentname > Datum Folie 31 The partial eclipse phase is more complicated:

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Vortrag > Autor > Dokumentname > Datum Folie 32 The partial eclipse phase is more complicated: R1R1 R2R2 D-x x Similar for the other zone.

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Vortrag > Autor > Dokumentname > Datum Folie 33 The partial eclipse phase is more complicated:

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Vortrag > Autor > Dokumentname > Datum Folie 34 The partial eclipse phase is more complicated:

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Vortrag > Autor > Dokumentname > Datum Folie 35 The partial eclipse phase is more complicated:

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Vortrag > Autor > Dokumentname > Datum Folie 36 The partial eclipse phase is more complicated:

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Vortrag > Autor > Dokumentname > Datum Folie 37 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!

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Vortrag > Autor > Dokumentname > Datum Folie 38 What does limb-darkening cause? Mandel & Agol 2002

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Vortrag > Autor > Dokumentname > Datum Folie 39 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:

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Vortrag > Autor > Dokumentname > Datum Folie 40 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:

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Vortrag > Autor > Dokumentname > Datum Folie 41 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 etc).

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Vortrag > Autor > Dokumentname > Datum Folie 42 Example: equations of the M&A02 model:

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Vortrag > Autor > Dokumentname > Datum Folie 43 Do we know the value of limb darkening a priori? Diamond: Sing (2010) Light blue:C&B11, ATLAS+FCMBlack line: C&B11, ATLAS+L Magenta:C&B11, PHOENIX+LDark blue line:C&B11, PHOENIX+FCM

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Vortrag > Autor > Dokumentname > Datum Folie 44

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

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Vortrag > Autor > Dokumentname > Datum Folie 46 Effect of stellar spots u eff = f(T star, T spot, Area spot, u star, u spot, ) 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.

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Vortrag > Autor > Dokumentname > Datum Folie 47

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Vortrag > Autor > Dokumentname > Datum Folie 48 The concept of effective limb darkening The observed star = the modelled star

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Vortrag > Autor > Dokumentname > Datum Folie 49 The concept of effective limb darkening The observed star = the modelled star THIS IS NOT TRUE

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Vortrag > Autor > Dokumentname > Datum Folie 50 The concept of effective limb darkening The observed star = the unmaculated star + stellar spots

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Vortrag > Autor > Dokumentname > Datum Folie 51 The concept of effective limb darkening The observed star = the unmaculated star + stellar spots THIS IS TRUE

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Vortrag > Autor > Dokumentname > Datum Folie 52 The concept of effective limb darkening The observed star = the unmaculated star + stellar spots

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Vortrag > Autor > Dokumentname > Datum Folie 53 The concept of effective limb darkening The observed star = the unmaculated star + stellar spots F star : we observe an unmaculated star F planet : we remove the light of the unmaculated surface due to planet transit (assumption: planet does not cross the spot(s) R spot 2 F star : we remove the stellar light at the place (b spot ) of the spot R 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.

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Vortrag > Autor > Dokumentname > Datum Folie 54 The concept of effective limb darkening The observed star = the unmaculated star + stellar spots

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Vortrag > Autor > Dokumentname > Datum Folie 55 The concept of effective limb darkening The observed star = the unmaculated star + stellar spots

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Vortrag > Autor > Dokumentname > Datum Folie 56 Spots at the edge can cause effectively limb- brightening... See Csizmadia et al. (2012) or Barros et al. (2011)

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Vortrag > Autor > Dokumentname > Datum Folie 57 Gravity darkening von Zeipel 1924 Lucy 1967 Barnes 2009 Claret 2011

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Vortrag > Autor > Dokumentname > Datum Folie 58 Exomoons and exorings in the light curve

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Vortrag > Autor > Dokumentname > Datum Folie 59 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?

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

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Vortrag > Autor > Dokumentname > Datum Folie 61 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 o : uncertainty of the observed data points m : uncertainty of the model, frequently set to zero

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Vortrag > Autor > Dokumentname > Datum Folie 62 Difference between local and global minima Variable Function value Steepest descent

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Vortrag > Autor > Dokumentname > Datum Folie 63 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...

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

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Vortrag > Autor > Dokumentname > Datum Folie 65 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!

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Vortrag > Autor > Dokumentname > Datum Folie 66 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.

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Vortrag > Autor > Dokumentname > Datum Folie 67 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. Levenberg-Marquardt: Lambda can be variable.

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Vortrag > Autor > Dokumentname > Datum Folie 68 Optimization problems in astronomy Optimization is used in all field of astronomy (not a complete list): in cosmology (e.g. analyzing CBE, WMAP, Planck data) extragalactic distance scale (e.g. Ia SNae distance scale problem, fitting the light curve with templates) galactic astronomy (e.g. fitting isochromes to open/globular cluster's HRD, even in extragalctic scales (e.g. S96 open cluster in gx. NGC 2403, Vinkó,..., Csizmadia,... et al. 2009, ApJ) determining the age of a single star (e.g. host stars of exoplanets!) with isochrone-fitting fitting frequencies of an RR Lyrae type star (e.g. Dékány & Kovács 2009): age, mass, radius, internal structure and evolutionary status of a star binary star astronomy, transiting exoplanets (light curve fit) the most basic tool for an astronomer who works with data

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Vortrag > Autor > Dokumentname > Datum Folie 69 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 2 min and 2 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.

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Vortrag > Autor > Dokumentname > Datum Folie 70 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 2 -function. Finding Best Solution Minimal Cost (Design) Minimal Error (Parameter Calibration) Maximal Profit (Management) Maximal Utility (Economics)

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Vortrag > Autor > Dokumentname > Datum Folie 71 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)

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Vortrag > Autor > Dokumentname > Datum Folie 72 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: 2 i+1 < 2 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 ) x j = STDDEV(x ij )

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Vortrag > Autor > Dokumentname > Datum Folie 73 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.

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Vortrag > Autor > Dokumentname > Datum Folie 74 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

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Vortrag > Autor > Dokumentname > Datum Folie 75 Genetic Algorithms: who will survive and produce new off-springs?

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Vortrag > Autor > Dokumentname > Datum Folie 76

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Vortrag > Autor > Dokumentname > Datum Folie 77 From Canto et al.

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Vortrag > Autor > Dokumentname > Datum Folie 78 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

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Vortrag > Autor > Dokumentname > Datum Folie 79 School Bus Routing Problem GA = $409,597, HS = $399,870 Depot School Min C1 (# of Buses) + C2 (Travel Time) s.t. Time Window & Bus Capacity

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Vortrag > Autor > Dokumentname > Datum Folie 80 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) 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( 2 ) / min( 2 ) < 1.01

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Vortrag > Autor > Dokumentname > Datum Folie 81 Comparison of methods MCMC Price AGA HS Test: where is the global minimum of Michalewicz's bivariate function: We know that f(x,y) at ( , ) if 0 x y

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Vortrag > Autor > Dokumentname > Datum Folie 82 Michalewicz's bivariate function

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Vortrag > Autor > Dokumentname > Datum Folie 83 Results MethodxydSteps Exact MCMC Price (N=25) Price (N=100) AGA (N=25) AGA (N=25) HS (N=100) HS (N=25) Amoeba

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Vortrag > Autor > Dokumentname > Datum Folie 84 a/Rs u1u1 u2u2 k i

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Vortrag > Autor > Dokumentname > Datum Folie 85 The final result Csizmadia et al. 2011

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Vortrag > Autor > Dokumentname > Datum Folie 86 Csizmadia et al. 2011

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Vortrag > Autor > Dokumentname > Datum Folie 87 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.

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Vortrag > Autor > Dokumentname > Datum Folie 88 Thank you for your attention!

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