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

Early transit observations

Venus transit in 1761, 1769

Jeremiah Horrocks (1639, Venus)

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The Astronomical Unit via the transits of Venus

The Astronomical Unit via the transits of Venus

From geogr. meas.

0.3 AU 0.7 AU (Kepler's

third law period measurement)

Measuring the Atmospheric Properties of Venus

utilizing its Transits(It can be extended to

extra-solar planets, too)

Hedelt et al. 2011, AA

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?

Transit of the Earth from the L2 point of

the Sun-Earth system is it an annular eclipse?

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)

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

i90

to

iltgt90 (few arcminutes)

Plane of the sky (East)

tt

Gimenez and Pelayo, 1983

tp

The definition of contacts

(Winn 2010)

(Winn 2010)

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tt to

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)

The impact parameter b

Angular momentum vector

i

to the observer (line of sight)

90-i

bRs

r

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)

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.

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

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

Occurence time of the eclipses (i90)

- Primary eclipse (transit)
- Secondary eclipse (occultation)

From complicated series-calculations

Some very useful formulae

Some very useful formulae

Some very useful formulae

By simple time-measurements you can determine

eccentricity and argument of periastron

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)

The partial eclipse phase is more complicated

The partial eclipse phase is more complicated

D-x

x

? ?

R1

R2

Similar for the other zone.

The partial eclipse phase is more complicated

The partial eclipse phase is more complicated

The partial eclipse phase is more complicated

The partial eclipse phase is more complicated

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!

What does limb-darkening cause?

Mandel Agol 2002

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

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

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).

Example equations of the MA02 model

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)

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

The concept of effective limb darkening

The observed star the modelled star

THIS IS NOT TRUE

The concept of effective limb darkening

The observed star the unmaculated star

stellar spots

The concept of effective limb darkening

The observed star the unmaculated star

stellar spots

THIS IS TRUE

The concept of effective limb darkening

The observed star the unmaculated star

stellar spots

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.

The concept of effective limb darkening

The observed star the unmaculated star

stellar spots

The concept of effective limb darkening

The observed star the unmaculated star

stellar spots

Spots at the edge can cause effectively

limb-brightening...

See Csizmadia et al. (2012) or Barros et al.

(2011)

Gravity darkeningvon Zeipel 1924Lucy

1967Barnes 2009Claret 2011

Exomoons and exorings in the light curve

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?

Our problem is a highly nonlinear, not

invertible, multidimensional optimization

problem with many local minima. Observational

noise makes the things even more complicated.

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

Difference between local and global minima

Function value

Steepest descent

Variable

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...

The old and fast method to find the nearest

minimum(either local or global) differential

correction andLevenberg-Marquardt

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!

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.

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.

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

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.

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)

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)

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)

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.

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

Genetic Algorithms who will survive and produce

new off-springs?

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From Canto et al.

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

School Bus Routing Problem

Min C1 ( of Buses) C2 (Travel Time) s.t. Time

Window Bus Capacity

GA 409,597, HS 399,870

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

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????

Michalewicz's bivariate function

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

a/Rs

i

k

u2

u1

The final result

Csizmadia et al. 2011

Csizmadia et al. 2011

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.

Thank you for your attention!