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Transit Light Curves Szil rd Csizmadia Deutsches Zentrum f r Luft- und Raumfahrt /Berlin-Adlershof, Deutschland/ Folie * – PowerPoint PPT presentation

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Title: 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)
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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, AA
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!
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14
Orientation of the orbit
i90
to
iltgt90 (few arcminutes)
Plane of the sky (East)
tt
Gimenez and Pelayo, 1983
tp
15
The definition of contacts
(Winn 2010)
16
(Winn 2010)
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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 (kltlt1) Annular
eclipse
(klt1 and k ? 1)

Total eclipse (klt1) Partial eclipse
(1-kltblt1k) Occultation (k ltlt 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 (i90)
  • 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
? ?
R1
R2
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 MA02 model
43
Do we know the value of limb darkening a priori?
Diamond Sing (2010) Light blue CB11,
ATLASFCM Black line CB11, ATLASL Magenta C
B11, PHOENIXL Dark blue line CB11,
PHOENIXFCM
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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,
AA 506, 1335, 2009) Recent study on 9
eclipsing binaries (A. Claret, AA 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,)
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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
51
The concept of effective limb darkening
The observed star the unmaculated star
stellar spots
THIS IS TRUE
52
The concept of effective limb darkening
The observed star the unmaculated star
stellar spots
53
The concept of effective limb darkening
The observed star the unmaculated star
stellar spots
Fstar we observe an unmaculated star ?Fplanet
we remove the light of the unmaculated surface
due to planet transit (assumption planet does
not cross the spot(s) ?Rspot2 Fstar we remove
the stellar light at the place (bspot) of the
spot ?Rspot2 Fspot we put the spot light at the
place (bspot) of the spot So, in practice, we
replaced a small part of the stellar flux with
the spot's flux.
54
The concept of effective limb darkening
The observed star the unmaculated star
stellar spots
55
The concept of effective limb darkening
The observed star the unmaculated star
stellar spots
56
Spots at the edge can cause effectively
limb-brightening...
See Csizmadia et al. (2012) or Barros et al.
(2011)
57
Gravity darkeningvon Zeipel 1924Lucy
1967Barnes 2009Claret 2011
58
Exomoons and exorings in the light curve
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?
60
Our problem is a highly nonlinear, not
invertible, multidimensional optimization
problem with many local minima. Observational
noise makes the things even more complicated.
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 Fobs the observed flux (light,
brightness etc.) Fmod the modell value for the
same ?o uncertainty of the observed data
points ?m uncertainty of the model, frequently
set to zero
62
Difference between local and global minima
Function value
Steepest descent
Variable
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...
64
The old and fast method to find the nearest
minimum(either local or global) differential
correction andLevenberg-Marquardt
65
The old and fast method to find the nearest
minimum(either local or global) differential
correction andLevenberg-Marquardt
Necessary (but not sufficient) condition for
minimum
For all parameter, so for all k!
66
The old and fast method to find the nearest
minimum(either local or global) differential
correction andLevenberg-Marquardt
1. Choose an initial p. 2. Calculate A, b and
then dp. 3. p' p dp 4. Iterate 2-3 until
convergence.
67
The old and fast method to find the nearest
minimum(either local or global) differential
correction andLevenberg-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.
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

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
    ?2min and ?2min 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.

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)

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)

72
Markov Chain Monte Carlo(with Metropolitan-Hastin
gs algorithm)
Choose x0 and s0 stepsize
The Markov-chain like in burn-in phase, but
the results are saved (the burn-in results are
forgotten!)

Burn-in phase xi1 xi r si Acceptance
?2i1 lt ?2i or if Stepsize should be
adjusted for an acceptance rate 23
The result is defined as xj MEAN(xij) ?xj
STDDEV(xij)
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.
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
75
Genetic Algorithms who will survive and produce
new off-springs?
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From Canto et al.
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

79
School Bus Routing Problem
Min C1 ( of Buses) C2 (Travel Time) s.t. Time
Window Bus Capacity
GA 409,597, HS 399,870
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)
  • ?2aim 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 ) lt 1.01

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) ? -1.801 at (2.20319...,
    1.57049...) if 0?x??????y????

82
Michalewicz's bivariate function
83
Results
Method x y d Steps Exact 2.20319 1.5704
9 - - MCMC 2.18912 0.300988 1.18959 100
000 Price (N25) 1.05775 1.57111 1.14544 250 Pric
e (N100) 2.20712 1.57936 0.00971 16 500 AGA
(N25) 2.20291 1.57080 0.00042 12 800 AGA
(N25) 2.20290 1.57080 0.00042 3225 HS
(N100) 2.20291 1.57073 0.00037 4600 HS
(N25) 2.20285 1.57072 0.00041 1300 Amoeba 2.2
0286 1.57082 0.00047 73
84
a/Rs
i
k
u2
u1
85
The final result
Csizmadia et al. 2011
86
Csizmadia et al. 2011
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.

88
Thank you for your attention!
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