Timing Transits to Find Extrasolar Earths Eric Agol, Jason Steffen (UW) Re’em Sari (Caltech) Will Clarkson (Southampton) (MNRAS, in press)

Planetary Transits HD Seven (and counting) other transiting planets have been found

Overview: The Physics Known transiting planet Transit times are equally spaced.

Overview: The Physics Unknown perturbing planet Known transiting planet Transit times are NOT equally spaced.

Overview: The Signal Eclipse Number Time _ = Eclipse Number Time Eclipse Number Time Transit Times Best-Fit Orbit Timing Residuals

early late on time Limit: Non-interacting Planets

Limit: Widely Separated Planets Perturbing planet remains nearly constant during one orbit of transiting planet.

Changing distance to inner binary alters the effective mass of the host star. Limit: Widely Separated Planets

Circular j:j+1 Resonant Systems Conjunction changes eccentricity and period New period alters conjunction longitude When conjunction longitude changes by ~1/2 orbit, the process reverses Libration drives timing deviations: For Earth mass planet in 2:1 resonance around HD , ~3 minute signal over 150 days!

Resonant libration

Initially Circular Orbits P trans /P pert -1

Eccentricity Dependence Perturber Eccentricity Period Ratio HD perturbed by an Earth-mass planet:

Known Multi-Planet Systems

Upsilon Andromedae M 1 >0.7 M J, M 2 >1.2 M J, e 1 ≈ 0.012, e 2 ≈ 0.28, P 1 ≈ 4.6d, P 2 ≈ 241 d Not in resonance - large axis ratio -> small timing variations of inner planet (but 12% chance of transiting); outer planet has larger variations (but small transit probability):

55 Cancri M 1 >2 M J, M 2 >0.5 M J, e 1 ≈ 0.02, e 2 ≈ 0.34, P 1 ≈ 14.7d, P 2 ≈ 44 d Nearly in 3:1 resonance - large perturbations! If it were transiting (4% chance),  t of inner planet would be hours

Gliese 876 e 1 ≈0.15, e 2 ≈ 0.04, P 1 ≈ 30.1d, P 2 ≈ 61 d, transit probability ≈ 1.5% Nearly in 2:1 resonance - P lib ≈ 600 d. Long timescale variations due to precession:  t ≈ eP/  ≈ 1.4 d

Sensitivity Comparison HD Sensitivity - 10 sec rms, 10-  Astrometry 1  as Radial Velocity 0.5m/s Mass (Solar Masses) Period Ratio

Fitting Simulated Data Transit times are dirtied with Poisson noise Parameter space is sampled with simulated annealing algorithm Best fit  2 is found with downhill simplex method Confidence limits for eccentricity & semi-major axis are found by marginalizing over other parameters

Confidence Limits transit number red dots: noisy data black dots: best-fit planet For HD , noise corresponds to 10 second RMS

Confidence Limits eccentricity of perturbing planet Semi- major axis ratio of planets Parameters m m a (2.861) e e (0.015)   (0.928)   (0.84)

Applications Detection of terrestrial mass planets (although best sensitivity for resonant planets - these may be captured via migration - Narayan et al. 2004, Mandell & Sigurdsson 2004, Thommes 2005) Measurement of the mass of terrestrial planets (confirms they are not blends of Jupiter mass planets + brighter stars) Measurement of the mass-radius relation of the host star - cross-check of photospheric mass/radius measurement Measurement of inclination of non-transiting planet (if transiting planet shows timing variations)

Conclusions: Multiple planets can produce variations in the timing of eclipse due to star wobble and orbital frequency perturbations - grows with mass, period, eccentricity, and proximity to resonance This effect can be used to detect very small planets Multiple applications make this a technique well worth pursuing - terrestrial mass planets can be detected with current technology!