1. Timing of Transiting Planets Eric Agol, University of Washington

2. Three exoplanet problems: 1.Migration: Do Jupiters capture earths into (near) mean-motion resonance? 2.Inclination: RV measures M * sin(I) I for non-transiting planets? relative inclination of multi-planet systems? scattering vs. disk interaction 3.Planet structure: high precision radius/mass measurements (cf. Guillot, LeConte, Valencia)

3. A solution: transit timing A single planet follows Keplerian orbit - transit evenly spaced A second planet will perturb the orbit of the first, causing a change in the times between transit (Miralda-Escudé 2002, Agol, Steffen, Sari & Clarkson 2005, Holman & Murray 2005) On short timescales, months-years, can be significant, possibly measurable with CoRoT (if lucky)

4. Relative flux Time tMtM tItI Slide credit: Josh Winn tItI

5. P F = (R p /R * ) 2 Time Relative flux Slide credit: Josh Winn t1t1 t2t2

6. Transit Timing Variations (TTV) Transit Number Time minus Transit Number Time Best-Fit Periodic Orbit Transit Number Time Timing Residuals Transit Time

9. 1) Short period resonant earths? Librating resonant planets: Need observations spaced over months with seconds precision - in principle sensitive to Mars mass (Agol et al. 2005)

10. Timing precision Proportional to the photometric precision, exp Fisher matrix analysis (Carter 2008, Ford & Gaudi 2007) : This is not always achieved in practice due to correlated noise, stellar variability, spots, etc... Best precisions obtained are ~5 seconds from the ground (VLT & Magellan: Gillon et al. 2008; Winn et al. 2009) ; ~3-5 seconds with Spitzer/HST (Knutson et al. 2007; Pont et al. 2008; Agol et al. 2008) Best CoRoT timing precision is 5 seconds for CoRoT- Exo-1b; Poster P-VIII-83, Csizmadia

11. HST TTV & RV for HD 209458 TTV Analysis TTV Theory (1) TTV + RV (2) RV Theory (3) Maximum allowed mass for companion on initially circular orbit Also: HD 189733 (Pont et al. 2007)

12. Transit Timing Observation Summary Transit Light Curve (TLC) Project (Winn & Holman, PIs, 12+ papers) - achieve 5-50 sec precision CoRoT Exo-1-b, Exo-5-b: Posters P-V-51, Rauer; P-V-50, Fridlund; P-VIII-83, Csizmadia Jena telescope: Poster P-XIII-121, Raetz - XO- 1b, Tres-1, Tres-2 MOST: Miller-Ricci et al. (2008) HD 189733, 209458 GJ 436c? Alonso et al. (2009) EPOXI: Christensen et al. (2008) Spitzer: Agol et al. (2008) VLT: OGLE-111b - Diaz et al. (2008) Rien Probablement rien

13. 2) Constraining absolute mass/radius: case study of GJ 876 Super-Earth (2 day) + 2 Jupiters (30, 60 days) orbiting M- dwarf (0.3M ) Rivera et al. 2005 Pretend orbits align & transit (p~0.7%)

14. GJ 876 TTV: Libration ~600 days

15. GJ 876b,c,d: measuring mass/radius Light curve constructed for 600 days with 10 -3 day exposures (84 sec) from 26 parameters: M/R of star & planets (8), orbital elements of planets (18). Fisher matrix analysis gives an estimate of uncertainties vs. photometric errors (Carter 2008) Find fractional uncertainties on GJ 876d (7.5 M ) mass of 5 10 -4 and radius of 6 10 -4 (without radial velocity) for 1 mmag (assume stellar variability can be filtered) Try putting that on a Toblerone diagram!

16. 3) Constraining inclination: case study of HIP 14810b,c Hot Jupiter, 6.6742 days, plus longer period eccentric Jupiter, 95 days, e 0.4 Both are detected with RV; if inner planet were to transit, we would know its inclination If outer planet does not transit, it would have an uncertain inclination (I) & orientation on sky ( ) Non-transiting planet has mass M sin(I), so I affects the perturbations of inner planet due to changing mass & relative inclination

17. HIP 14810c: constraining inclination with TTV

18. Additional effects Exo-moons (Poster P-VIII-89: David Kipping) Trojans - offset in radial velocity & transit time (Poster P-III-33: Moldovan; P-III-34: Dvorak; Ford & Gaudi 2006; Ford & Holman 2007) Relativistic effects - precession of orbit (e.g. Miralda- Escudé 2002, Loeb 2005, Heyl & Gladman 2007, Pál & Koscis 2008, Jordán & Bakos 2008 ) Light-travel time - requires large planet mass & large semi-major axis (Poster P-VIII-86, Cabrera) Transit parallax (Scharf 2007) - very small Tidal evolution (Ragazzine & Wolf 2009)

19. Conclusions No convincing evidence yet for planet-induced TTV - maybe resonant terrestrial planets arent there or need longer time baseline & higher precision. In principle can constrain inclination of non-transiting planets, and mass/radius if both transit. CoRoT drawbacks: 1) duration may not be long enough; 2) low-Earth orbit limits photometric precision, which limits transit timing precision; 3) stars are fainter & variable. CoRoT advantages: 1) measuring a large number of transits for each; 3) best possibility of finding multi- planet transiting systems; 4) longer period planets (Exo-4b,6b): larger TTV

20. Extra slides

21. Four observables, four derivables: Flux decrement F Transit duration t M Ingress duration t I Period P=t 2 -t 1 b/R * =[1-t M F 1/2 /t I ] 1/2 R p /R * = F 1/2 a/R * = P/t M * = 3P/( Gt M 3 ) Seager& Mallen- Ornelas 2003 With velocity semi-amplitude, K, can also derive surface gravity of planet Can look for variations in each of these parameters - transit times are best Winn et al. (2007); Southworth et al. (2007); Beatty et al. (2007); Sozzetti et al. (2007) b=0

22. (Wright et al. 2009). Ill discuss two case studies which have a reasonable probability of transit. 28 Multi- planet RV systems

23. Earth/Super-Earth mean-motion TTV TTV for an Earth-mass planet in 2:1 mean-motion resonance - amplitude is 200 seconds - mass sensitivity scales as M trans 1/2

24. Changing proximity to inner binary alters the tidal force on inner binary. TTV induced by exterior eccentric planet (non-resonant) Agol et al. (2005)

25. Zhou et al. 2005 During migration of giant planets, terrestrial-mass cores can caught in or near low order mean- motion resonances (Narayan et al. 2004; Mandell & Sigurdsson 2004; Thommes 2005, Mandell & Raymond 2007) Can these survive for Gyr timescales? (10 11 dynamical times) Terquem & Papaloizou (2006) 3:2 2:1 1. Do (near-)resonant short period systems exist?

26. Probability of multi-planet transits Fabrycky (2008) I 5 o P transit a in /a out N = P transit = 1.7 (<100 d)

27. HIP 14810c: constraining Omega with TTV