Folie 1

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

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

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

5. The Astronomical Unit via the transits of Venus

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

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

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?

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

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)

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!

14. Orientation of the orbit i=90° to i<>90° (few arcminutes): Plane of the sky (East) tt Gimenez and Pelayo, 1983 tp

15. The definition of contacts (Winn 2010)

16. (Winn 2010)

18. tt to

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

20. The impact parameter b Angular momentum vector i to the observer (line of sight) 90°-i bRs r

21. Types of eclipses/transits Some definitions: R1: the bigger object's radius R2: the smaller object's radius Of course, 2nd object can be a planet, too. k = R2/R1, the radius ratio (or it is the planet-to-stellar radius ratio) r1 = R1/A r2 = R2/A, the fractional radius (A is the semi-major axis) Transit (k<<1) Annular eclipse (k<1 and k  1) Total eclipse (k<1) Partial eclipse (1-k<b<1+k) Occultation (k << 1)

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.

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:

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:

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

26. Some very useful formulae

27. Some very useful formulae

28. Some very useful formulae

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

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

31. The partial eclipse phase is more complicated:

32. The partial eclipse phase is more complicated: D-x x  R2 R1 Similar for the other zone.

33. The partial eclipse phase is more complicated:

34. The partial eclipse phase is more complicated:

35. The partial eclipse phase is more complicated:

36. The partial eclipse phase is more complicated:

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!

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

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:

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:

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. 2009 - etc).

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

43. 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

45. 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):

46. 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. ueff = f(Tstar, Tspot, Areaspot, ustar, uspot,)

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

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

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