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Transit Timing Variations
Szilárd Csizmadia
Institut for Planetary Research,
German Aerospace Center
Berlin, Germany
[email protected]
Jena University
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' – T0) + 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:
O0 = T0
For large N: The
O1 = T0 + P0 + 
period variation is
O2 = T0 + P0 +  + P0 + 2
half of the
 arithmetical series for 
quadratic
term!
ON = T0 + NP0 + P0N(N-1)
ON  T0 + NP0 + P0N2 
Real example for O-C variations
SZ Lyn (pulsating
star 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,
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 very
important because
the amplitude of the
perturbation can became
very high.
(See your textbooks...)
The higher the libration
the higher the O-C amplitude.
If the mean motion are in
p:q resonance (p, q are small
integers) there is a resonance
and the consequence is the
libration.
The p:q smaller the libration's
amplitude 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/1112.216
5
Kepler-11 system
Nature 470, 52, 2011
CoRoT-1b (A&A 510, A94, 2010)
A&A 528, A53 (2011)
HAT-P-13b
Pal et al. MNRAS413, l42, 2011
HAT-P-13b
Nascimbeni et al. A&A 532, A24, 2011