The Transit Timing method: How to detect extrasolar ... - Jean Coupon
Jun 26, 2006 - 2-year observations (170 transits). A HD209458b transit observed with the. HST. Charbonneau et al (2005). 1. Effects of a perturbing planet on ...
The Transit Timing method: How to detect extrasolar planets? Jean Coupon (Supervisor: Norman W. Murray)
Canadian Institute for Theoretical Astrophysics Institut d’Astrophysique de Paris June 26th, 2006
Context
M. Holman, N. Murray (2005) and E. Agol et al (2005): one can detect extrasolar planets with the Transit Timing variations, and in some cases, terrestrial mass planets.
Introduction Idea: Transit timings are perturbated in case of a second planet.
Motivations: •New method to discover extrasolar planets •Spaced based mission Kepler •Possibility to detect terrestrial mass planets
Problem: How to extract orbital elements and the mass of the perturbing planet?
Introduction 1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies 1.3 Numerical results
2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit
3. Terrestrial mass planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
1. Effects of a perturbing planet on the transit timings
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
1.1 Descriptions and assumptions
1.2 Parameters dependencies 1.3 Numerical results
Transit n
2. Estimation of the parameters of the perturbing planet 2.1 Method
Transit n+1
Observer
2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions
Transit interval:
3.2 Numerical results 3.3 Observable systems
•Direct effect (interactions with the perturbing planet) •Indirect effect (motion of the star)
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
1.1 Descriptions and assumptions
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance
Main assumptions: 1. Edge-on system and coplanar orbits. 2. General relativity and tidal effects neglected 3. Gaussian noise (σ = 5 sec) 4. The transiting planet is a HD209458b-type planet (T1 = 3.52 days, a1 =4.5 10-2 AU, e1 = 0.025) 5. 2-year observations (170 transits)
3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
A HD209458b transit observed with the HST. Charbonneau et al (2005)
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies
1.2 Parameters dependencies M. Holman, N. Murray:
1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
Mass, m2: amplitude of variations is a linear function of the mass. Eccentricity, e2: amplitude is a monotone increasing function. Semi-major axis, a2: amplitude is a monotone decreasing function (except for resonances).
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
1.2 Parameters dependencies
1.2 Parameters dependencies 1.3 Numerical results
Numerical computation of the maximum of the variations
2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
Analytical expression
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
1.2 Parameters dependencies Mean-anomaly (at t = 0), M2: the changes in this parameter don’t modify the amplitude but can affect the pattern of the transit timings data set. Argument of periapse, ω2: when ω1 = ω2, amplitude is minimum because the direct effect tends to oppose the indirect effect.
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
1.2 Parameters dependencies
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
1.3 Numerical results
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
Planets on circular orbits
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
1.3 Numerical results
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
Perturbing planet on an eccentric orbit
2. Estimation of the parameters of the perturbing planet
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
2.1 Method Extract the maximum of information from the transit timing set of data. Put some a priori information on the system. Invert the problem by a χ2 minimization.
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
2.2 Planets on circular orbits
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
We cannot distinguish the period of the transiting planet e2 must be small. Variations are big enough to see the pattern of the transit timings.
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
2.3 Perturbing planet on an eccentric orbit
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
Large eccentric orbit
Periapse time passage
Period of the transiting planet
3. Terrestrial planet in meanmotion resonance
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
3.1 New assumptions The previous analytic expression doesn’t work anymore We must take into account the variations on the perturbing planet, because perturbations are amplified at each conjunction Example: 2:1 resonance
t = T1
t = 2T1
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
3.1 New assumptions
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits
The resonant argument φ, will depend on: •Masses of both planets •Orbital elements of both planets
2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance
dφ/dt leads to the libration frequency
3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
We will see the libration period.
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions
3.2 Numerical results
1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
m2 = 10-3M°
Libration period
1. Effects of a perturbing planet on the transit timings 1.1 Descriptions and assumptions 1.2 Parameters dependencies 1.3 Numerical results 2. Estimation of the parameters of the perturbing planet 2.1 Method 2.2 Planets on circular orbits 2.3 Perturbing planet on an eccentric orbit 3. Terrestrial planet in mean-motion resonance 3.1 New assumptions 3.2 Numerical results 3.3 Observable systems
3.2 Observable systems Numerical computations give m2, min = 5.10-6M° Maximum timing deviation = 25 sec
detectable !!!
Conclusion A new method to detect extrasolar planets: Efficient method for most of the systems. However one must have a good accuracy for the measurements. Leads to a lot of information on the parameters of the perturbing planet.
Conclusion A way to discover small planets: The resonant perturbations are degenerated and are special cases… …but Earth-mass planets can be detected by this method!!
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