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Folie 1Folie 1

Transit Light Curves

Szilárd Csizmadia

Deutsches Zentrum für Luft- und Raumfahrt

/Berlin-Adlershof, Deutschland/


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Outline

1. Introduction: why transits?2. Transits in the Solar System3. Transits of Extrasolar Objects4. Classification of transits5. 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 present6. Optimization: methods & problems7. Exomoons & exorings8. Summary


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Early transit observations

Jeremiah Horrocks (1639, Venus)

Venus transit in 1761, 1769


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The Astronomical Unit via the transits of Venus


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The Astronomical Unit via the transits of Venus

~0.3 AU ~0.7 AU

(Kepler's third law + period measurement)

Fromgeogr.meas.


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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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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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Transit of theEarth from the L2 point of theSun-Earthsystem:

is it an annular eclipse?


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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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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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Orientation of the orbit

Plane of the sky (East)

v t=90 °−ω

vo=270 °−ω

i=90°

i<>90° (few arcminutes):

Gimenez and Pelayo, 1983

tp

tt

to


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The definition of contacts

(Winn 2010)


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(Winn 2010)


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tt t

o


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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):

D1,4=R1+ R2

D 2,3=R1−R2

D=( a (1−e2)1+ e cosv)

2

(1−sin2 i sin2(v+ ω))


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The impact parameter b

to the observer(line of sight)

Angular momentum vector

i

90°-i bRs

r

sin (90 °−i )=b Rs

r, r= a (1−e2)

1+ e cosv

bcircular=a cos i

R s

, belliptic=a cos i

R s

(1−e2)1+ e sinω


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Types of eclipses/transits

Transit (k<<1) Annular eclipse (k<1 and k 1)

Total eclipse (k<1)Partial eclipse (1-k<b<1+k) Occultation (k << 1)

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/R

1, the radius ratio

(or it is the planet-to-stellar radius ratio)

r1 = R

1/A r

2 = R

2/A,

the fractional radius(A is the semi-major axis)


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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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primary.• From two-body problem:

M=2πP(t−t p)

E−e sin E=M

tanv2=√1+ e

1−etan

E2


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primary.• From two-body problem:

M=2πP(t−t p)

E−e sin E=M

tanv2=√1+ e

1−etan

E2

y=r sin v

x=r cos v

r= a (1−e2)1+ e cosv

V= 2π AP √2A

r−1


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Occurence time of the eclipses (i=90)

• Primary eclipse (transit):

• Secondary eclipse (occultation):

v t=90 ˚−ω

v o=270 ˚−ω

tanv2=√1+ e

1−etan

E2

E−e sin E=M

M t , o=2πP(t−t p)=

2πn(t−t t , o)+

2πn(t t , o−t p)=

2πn(t−t t , o)+ M t , o

From complicated series-calculations:

M=v+ 2Σk=1∞ (1k + √1−e2)(− e

1+ √1−e2)k

sin kv


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Some very useful formulae

M=v+ 2Σk=1∞ (1k + √1−e 2)(− e

1+ √1−e2)k

sin kv≈v−2 e sin v+ ...

M t=2πP(t t−t p)+ M 0≈

π2−ω−2 e sin ( π

2−ω)=π

2−ω−2 e cosω

M o=2πP(t o−t p)+ M 0≈

3π2−ω−2 e sin (3π

2−ω)= 3π

2−ω+ 2 e cosω

2πP(t o−t t)≈π+ 4e cosω

2πP(t o−t t−P /2)≈4 e cosω e cosω≈

π(t o−t t−P /2)2P


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V t=2π A

P √2ar−1= 2π A

P √2(1+ e sinω)a (1−e2)

−1= 2π AP √2(1+ esinω)−1+ e2

1−e2

V t=2π A

P √1+ e 2+ 2e sinω1−e2

= 2π AP √1+ e2

1−e2(1+ 2 e sinω1+ e2 )

V t≈2π A

P √1+ e2

1−e2(1+ e sinω(1+ e2)),√1+ x≈1+ 1

2x

V o≈2π A

P √1+ e2

1−e2(1− e sinω(1+ e 2))


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V t≈2π A

P √1+ e2

1−e2(1+ e sinω(1+ e2)) V o≈

2π AP √1+ e2

1−e2(1− esinω(1+ e 2))

¿

V t−V o

V t+ V o

≈

2π AP √1+ e2

1−e2 2 e sinω/(1+ e2)

2π AP √1+ e2

1−e22

=e sinω , 1+ e2≈1+ ....

e sinω≈

2R s

Dt

−2R s

Do

2R s

Dt

+2R s

Do

=

1D t

− 1Do

1D t

+ 1Do

=Do−Dt

Do+ Dt


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

e cosω≈π(t o−t t−P /2)

2P

e sinω≈Do−Dt

Do+ Dt

Do

D t

≈ 1+ e sinω1−e sinω


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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: Lobs= L star+ g (v ) Lday , planet+ (1− g (v )) Lnight , planet

Occultation:

Lobs= L star+ (1−A) L star+ Lnight , planet

Lobs= L star

A=k2=R planet

2

R star2

g (v )=sin v sin i+ ...

For known exoplanets (Kane & Gelino 2010):

Lday , planet , Lnight , planet≪ L star , (L planet / L star≈10−5..−7)

(See Kane & Gelino for full, correct expression)


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The partial eclipse phase is more complicated:

Lobs= L star (1− A( t ))+ g (v ) Lday , planet+ (1− g (v )) Lnight , planet

Lobs≃ L star (1− A( t ))


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(D− x )2+ y 2=R12

x 2+ y 2=R 22

R1

R2

D-xx

Aeclipsed=Sector−Triangle

Aeclipsed=R22−1

2R2

2 sin

Similar for the other zone.


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(D− x )2+ y 2=R12

x 2+ y 2=R 22

D 2−2Dx=R12−R 2

2

x= D2−

R12(1−k 2)

2D


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(D− x )2+ y 2=R12

x 2+ y 2=R 22

D 2−2Dx=R12−R 2

2

x= D2−

R12(1−k 2)

2D

cos=D−xR1

= D2 R1

+R1

2D(1−k 2)=1

2(DR1

+R1

D(1−k 2))

cos= xR2

= xkR1

= D2 k R1

−R1

2

2Dk R1

(1−k 2)= 12k (D

R1

−R1

D(1−k 2))


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cos= 12(D

R1

+R1

D(1−k 2))

A=(R12− 1

2R1

2 sin)+ (R22− 1

2R2

2 sin)

D= A(1−e2)1+ e cosv

√1−sin2 i sin2(v+ ω)

cos= 12k(D

R1

−R1

D(1−k 2))


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A=(R12− 1

2R1

2 sin)+ (R22− 1

2R2

2 sin)

A=R12((−1

2sin)+ 1

k(−1

2sin))


√1−sin2 i sin2(v+ ω)


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A=(R12− 1

2R1

2 sin)+ (R22− 1

2R2

2 sin)

A=R12((−1

2sin)+ 1

k(−1

2sin))


√1−sin2 i sin2(v+ ω) The partial phase is already quitecomplicated in the case of even a uniform disc. And: it is describedby a transcendent equation so it isnot invertable analytically!


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What does limb-darkening cause?

Mandel & Agol 2002


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More precise approximation of the stellar radiation and thus the light curve shape: Limb darkening + small planet approximation

I (μ)= I 0(1−u1(1−μ)−u2(1−μ)2)μ , I 0≈

2hc2

λ5

1

ehc / kT λ−1

Total flux of the star:

Blocked flux of a small planet:

F star=∫0

2π

∫0

R star

I (μ)dr d φ=π Rstar2 I 0(1− u1

3−

u2

6 )

Relative flux decrease:

F star=π R planet2 (1−u1(1−√1−d 2)−u2(1−√1−d 2)2)√1−d 2

Δ FF star

=R planet

2

R star2

1−u1(1−√1−d 2)−u2(1−√1−d 2)2√1−d 2

1−u1

3−

u2

6


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More precise approximation of the stellar radiation and thus the light curve shape: Limb darkening + small planet approximation

I (μ)= I 0(1−u1(1−μ)−u2(1−μ)2)μ , I 0≈

2hc2

λ5

1

ehc / kT λ−1

Total flux of the star:

Blocked flux of a small planet:

F star=∫0

2π

∫0

R star

I (μ)dr d φ=π Rstar2 I 0(1− u1

3−

u2

6 )

Relative flux decrease:

F star=π R planet2 (1−u1(1−√1−d 2)−u2(1−√1−d 2)2)√1−d 2

Δ FF star

=R planet

2

R star2

1−u1(1−√1−d 2)−u2(1−√1−d 2)2√1−d 2

1−u1

3−

u2

6


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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 thisrequires 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).


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Example: equations of the M&A02 model:


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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


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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):

Probing the limb darkening theories on exoplanets and eclipsing binary stars


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Effect of stellar spots

ustar

u spot

ueff

= f(Tstar

, Tspot

, Areaspot

, ustar

, uspot,

)

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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The concept of effective limb darkening

The observed star = the modelled star


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The observed star = the modelled star

THIS IS NOT TRUE


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The observed star = the unmaculated star + stellar spots


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THIS IS TRUE


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F observed=F star−Δ F planet (t )−π R spot2 Δ F star (b spot)+ π R spot

2 F spot


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F observed=F star−Δ F planet (t )−π R spot2 Δ F star (b spot)+ π R spot

2 F spot

Fstar

: we observe an unmaculated star

ΔFplanet

: we remove the light of the unmaculated surface due to

planet transit (assumption: planet does not cross the spot(s)πR

spot2 F

star: we remove the stellar light at the place (b

spot) of the spot

πRspot

2 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.


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F observed=F star (u star)−Δ F planet (t , b planet , ustar)−πR spot

2 Δ F star(ustar , bspot)+ π Rspot2 F spot(uspot ,bspot)

F modelled=F star (ueff )−Δ F planet (t , b planet , ueff )


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F observed=F star (u star)−Δ F planet (t , b planet , ustar)−πR spot

2 Δ F star(ustar , bspot)+ π Rspot2 F spot(uspot ,bspot)

F modelled=F star (ueff )−Δ F planet (t , b planet , ueff )

F star≡F modelled


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Spots at the edge can cause effectively limb-brightening...

See Csizmadia et al. (2012)

or Barros et al. (2011)


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Gravity darkening

von Zeipel 1924Lucy 1967Barnes 2009Claret 2011


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Exomoons and exorings in the light curve


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The big question(s)

F star≃F modelled

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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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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How to find the solution if one has this more precise, but more complicated functions?

To minimize:

N: number of observed data pointsP: number of free parametersi: index of the point F

obs: 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

χ2= 1N−P−1

Σi=1N (F obs , i−F mod , i

√o , i2 + m ,i

2 )2


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Difference between local and global minima

Variable

Functionvalue

Steepest descent


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A time-consuming, but global minimum-finder method: grids

How to do it: chooseregurarly or randonlyenough tests in theparameters space

Advantage: it findsthe global minimum(if the number of trialsare big enough)

Disadvantage:the required time tends to infinity...


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

F mod=F mod ,0+ Σ j=1P dF mod

dp j

Δ p j

S≡Σ(F obs−F mod )2=(F obs−F mod ,0+ Σ j=1

P dF mod

dp j

Δ p j)2

=minimum


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dp j

Δ p j

S≡Σ(F obs−F mod )2=(F obs−F mod ,0+ Σ j=1

P dF mod

dp j

Δ p j)2

=minimum

Necessary (but not sufficient) condition for minimum: dSdpk

=0

For all parameter, so for all k!


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dp j

Δ p j

Δ p=A−1 b

A jk=Σi

dF i

dp j

dF i

dpk

b k=Σi(F obs−F mod ,0)idF i

dpk

1. Choose an initial p.2. Calculate A, b and then dp.3. p' = p + dp4. Iterate 2-3 until convergence.


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dp j

Δ p j

Δ p=A−1 b

A jk=Σi

dF i

dp j

dF i

dpk

b k=ΣI (F obs−F mod ,0)idF i

dpk

1. Choose an initial p.2. Calculate A, b and then dp.3. p' = p + dp4. Iterate 2-3 until convergence.

Levenberg-Marquardt:

Lambda can be variable.

A ii' =(1+ λ) Aii


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Optimization problems in astronomy

Optimization is used in all field of astronomy (not a complete list):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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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 χ2min 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.


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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 χ2-function.

Finding Best Solution

Minimal Cost (Design)

Minimal Error (Parameter Calibration)

Maximal Profit (Management)

Maximal Utility (Economics)


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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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Markov Chain Monte Carlo(with Metropolitan-Hastings algorithm)

Choose x0 and s0 stepsize

Burn-in phase:xi+1 = xi + r si

Acceptance: χ2i+1

< χ2i or if

Stepsize should be adjusted foran acceptance rate ~23%

e−χi+ 1

2

2T > R=[0...1]

The Markov-chain:

like in burn-in phase, but theresults are saved

(the burn-in results are forgotten!)

The result is defined as:

xj = MEAN(x

ij)

Δxj = STDDEV(x

ij)


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Disadvantages:- the two distributions shouldbe nearly the same(P is the probability distribution in reality, Q is the same for the calculated models.)

- the sampling of the wholeparameter space is not well done,infinitely long time is requiredto sample the whole hyperspace

- if the chain is not long enough,then it is more probable that we finda local minimum instead of the global one.


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Amoeba

- very simple

- depends on the starting values

- you have to restartit with different startingnumbers several times(~1000)

- the sampling of theparameter space is questionable, uniqueness is not warranted and not checked


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


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Folie 77From Canto et al.


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The big family of genetic algorithms

~ 1970Price (1979; sometimes it is used for eclipsing

binaries)GA (in astronomy; 1995, Charbonneau)HS (2001)AGA (2010)... many more


Folie 79

School Bus Routing Problem

GA = $409,597, HS = $399,870

Depot

School

1 2 3

4 5 6 7

8 9 10

7

5 8

5 4 5

3

4 5 6

85 7 4

5 45

10 15 5

10 15 20 10

15 10 20

Min C1 (# of Buses) + C2 (Travel Time)

s.t. Time Window & Bus Capacity


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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)

• χ2aim 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( χ2 ) / min( χ2 ) < 1.01


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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 0xπyπ

f ( x , y)=−sin x sin20 xπ−sin y sin20 2y2

π


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Michalewicz's bivariate function


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Results

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


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a/Rs

u1

u2

k

i


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The final result

Csizmadia et al. 2011


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Csizmadia et al. 2011


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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 88

Thank you for your attention!

vortrag > autor > dokumentname > datum folie 1 transit light curves szilárd csizmadia...

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