Transit Timing Variations

Szilárd Csizmadia Jena UniversityInstitut for Planetary Research,German Aerospace CenterBerlin, Germanyszilard.csizmadia@dlr.de 2011 Jan 11

Transits & Eclipses

Some real examples of light curves (1)

STARE from ground

HST from space

Some real examples of light curves (2)

From ground

From space

What is O-C?

O: observed midtime of a transit,

of an eclipse, of a light maxima of

a pulsating star, of any kind of a

signal...

C: calculated time of this signal

(linear, quadratic, periodic, etc.

ephemeris)

Pronounce: “O minus C”

An example

C0 = T0

C1 = T1 + P

C2 = T2 + P + P = T2 + 2P

CN = T0 + NP

T0: epoch, P: period, N: cycle number

An other example (P=2 days)

What is the big advantage of O-C? It is accumulating that is why we can study very small effects (smaller than that of the precision of the individual measurements). Example 1:

O = T0' + NP'

- C = T0 + NP

O – C = (T0' – T

0) + N(P' – P) Wrong period: linear O-C

Wrong epoch: zero-point shift

For instance: P' – P = 1 second (10-5 days), and our

precision is about 20 seconds, then you have to wait

for 20 minima to find the period is wrong (but 60 better)

Example2 : the period increases with a small part in every cycle:

P' = P0 (1 + N)

Then:O

0 = T

0

O1 = T

0 + P

0 +

O2 = T

0 + P

0 + + P

0 + 2

arithmetical series for O

N = T

0 + NP

0 + P

0N(N-1)

ON T

0 + NP

0 + P

0N2

For large N: The

period variation is

half of the

quadratic term!

Real example for O-C variations

SZ Lyn (pulsatingstar in a non-eclipsing binary)

Derekas et al.A&A 402, 733 (2003)

The companion star was discovered from the O-C diagram!

Can we discover other objects around a star, like a planet,

using the O-C diagram?

The answer is definitely YES!

The uncertain case of CM Dra

A&A 460, 583 (2008)

How it looks like...

(G. Perez, SMM/IAC)

Another very "certain" case: V391 Peg

Nature 449, 189 (2007)

At this moment we have only theoretical calculations:

star + transiting planet + another planet

For perturbation calculations, see:

general case: Borkovits et al. (A&A 398, 1091, 2003)

coplanar case in circular orbits: Agol et al. (MNRAS 359, 567, 2005)

A general configuration

Perturbation is stronger in case of conjuctions

(because the mutual distance is smaller, forces are stronger!)

The effect is not symmetric: before conjuction the planet is accelerated,

after that it is decelerated.

The O-C diagram amplitude (and its shape!)

can be calculated

applying the equations

of celestial mechanics –

generally it means numerical integrations of the equations of motion.

Third order analytic theory and code for analysis:

Borkovits et al. (A&A 398, 1091, 2003) Simplified equations:

Agol et al. (MNRAS 359, 567, 2005)

Kozai - mechanism

It is an important mechanism, if the mutual inclination is greater than 40° between the two planets: eccentricity will grow up to the vicinity of 1 (!) periodically. (Inclination also changes.)

Perhaps this is the explanation of some of the observed very high eccentricities (up to 0.92) in some exoplanetary system?

Resonances are veryimportant becausethe amplitude of theperturbation can becamevery high.

(See your textbooks...)

The higher the libration the higher the O-C amplitude.

If the mean motion are inp:q resonance (p, q are smallintegers) there is a resonanceand the consequence is the libration.

The p:q smaller the libration'samplitude higher.

Trojans (p:q = 1:1): the librational amplitude can be as high as 350°!

A very important table

Kirste, S. Bachelor thesis, 2008

Kepler-19b

Kepler-20 system

Kepler-20 system

http://arxiv.org/abs/1112.2165

Kepler-11 system

Nature 470, 52, 2011

CoRoT-1b (A&A 510, A94, 2010)

A&A 528, A53 (2011)

Pal et al. MNRAS413, l42, 2011

HAT-P-13b

HAT-P-13b

Nascimbeni et al. A&A 532, A24, 2011