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吴东红 2014/05/06

On detecting terrestrial planets with timing of giant planet transits —— Eric Agol ,Jason Steffen,Re’em Sari and Will Clarkson. 吴东红 2014/05/06.

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吴东红 2014/05/06

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  1. On detecting terrestrial planets with timing of giant planet transits——Eric Agol ,Jason Steffen,Re’em Sari and Will Clarkson 吴东红 2014/05/06

  2. Eric Agol is a UW astronomer credited with discovering an earthlike planet(Kepler 62f) that’s 1,200 light years away. He’s also a man of faith — faith in science and faith in Christianity as he deals with questions such as how the universe was created. research direction Extrasolar planets-black holes-gravitational lensing

  3. outline • Introduction • Equations of motion • Several limiting cases of TTV • Applications • Conclusion

  4. 1.introduction • TTV-transit time variation Science-2012-Nesvorn-1133-6

  5. introduction • About 10 per cent of stars with known planetary companions have more than one planet.If one or both of the planets is transiting ,dynamical ineractions between the planets will alter the timing of the transits.(967/1491) • In the case that a planet is terrestrial,RV measurements may not be precise enough to detect it , and the transit probability is quite small , while transit timing of a giant companion is sensitive to the terrestrial planet’s mass.

  6. Simulation of transit timing variation curves for four different systems over the course of the nominal Kepler mission lifetime Veras 2011 fig.1

  7. 2.Equations of motion The exact Newtonian equations of motion are given by in Jacobian coordinates the transit problem is unaffected by the total velocity or position of the center of mass, so we set

  8. 3.Non-interacting planets:perturbations due to interior planet on a small orbit Approximations : the orbits of both planets are aligned in the same plane; the system is exactly edge-on; the planet and star are spherical. The equations of motion Take the limit as Timing variations that arise due to the reflex motion of the star

  9. Eccentricorbits Circular orbits The inner planet displaces the star From the barycenter of the inner Binary by an amount The velocity perpendicular to the line of sight is Where the inner binary undergoes a transit at time t0 and P1 is the Orbital period of the inner binary The timing deviation of the mth transit of the outer planet is The standard deviation of timing variations

  10. 4.Perturbations due to exterior planet on a large eccentric orbit TTV due to a perturbing planet on an eccentric orbit with a2>>a1 and e1~0 Keplerian orbit Perturbing acceleration due to the outer planet Theequationsabove Now,the time of the (N+1)th transit occurs at is the true anomaly of the inner binary at the time of the 1st transit

  11. is the true anomaly of the outer binary at the timing of the (N-1) transit The timing deviation of the (N+1)thtransit of the outer planet is Axis ratio >5.0 The standard deviation of timing variations Numerical calculations of the 3-body problem show that this approximation works extremely well in the limit r1<<r2 at the -2 contour an orbit lasting 2 years with a perturbing planet of mass M⊙ would have a transit timing standard deviation of 5 minutes

  12. 5.Perturbations for two non-resonant planets on initial circular orbits Two planets on circular orbits near a j:j+1 resonance Conjunctions occur every is the period of the outer planet The fractional distance from resonance =0 indicates the resonance The period of TTV is = Number of conjunctions per cycle Over half a cycle ,the eccentricities grows to about the variation in the instantaneous orbital frequency Fluctuations in the mean motion over a cycle:

  13. the perturbed eccentricity dominates t If the perturbed eccentricity dominates If the perturbed mean motion dominates 2:1MMR Smaller,the planets will be trapped in MMR For Halfway between resonances For lager period ratio, according to the perturbation theory , the eccentricity of the inner planet grows to For an outer transiting planet,the motion of the star dominates over the perturbation of the inner planet for Transit timing standard deviation for two planets of Mass on Initially circular orbits

  14. 6.Perturbations for two planets in mean-motion resonance Two planets in a first-order j:j+1 mean-motion resonance The largest TTV of the lighter planet during the period of libration is 1/( In some cases ,the libration period of a resonant system can be much longer than the interval of observation .Thus , The TTVs cannot be easily discerned. Amplitude near the 2:1 resonance versus the difference in period from exact resonance for two systems: one with m1 = m0 (top solid) and m2 = (lower solid), and the other with m1 = m0 (top dotted) and m2 = (lower dotted). The TTV signal deeply in resonance is not as strong as that near MMR

  15. Analytical approximation of the near-resonance motion Consider a coplanar system in the vicinity of P:P+q MMR Transit timing variation of the inner planet Conical transformation Hamiltonian in keplerian orbits of the system Interaction Hamiltonian In the new variables , the Hamiltonian reads

  16. Out of MMR Inside of MMR m-e degeneracy Inside of MMR Systems deeply in MMR do Not produce strong TTVs, While those near-MMR can Generate strong sinusoidal Signals. G. Bou´e 2013

  17. 7.Non-zero eccentricities When either eccentricity is large enough , higer-order resonances and Secular perturbation become important HD 209458b Msin 0.698Mj a:0.04AU e:0 perturbed by a1M⊕ planet with various eccentricities Near the mean-motion resonances the signal is large enough that an earth-mass planet would be detectable with current technology

  18. TTV due to precession of the orbit When both planets have non-zero eccentricity ,on the secular time-scale,the precession of the orbits will lead to a significant variation in the transit timing. The period of precession: The relativistic precession rate: For P=10d ,a=0.1AU,small e Precession due to a second planet: For an earth-like planet,

  19. TTV for known multi-planet systems Gliese876 Orbits 1:2:4 resonance for Gliese 876c:b:e TTV for Gliese 876b and Gliese 876c

  20. 8.Applications Detection of terrestrial planets The possibility of detecting terrestrial planets using the transit timing technique clearly depends on strongly on: (1)The period of the transiting planet; (2)The nearness to resonance of the two planets ; (3)The eccentricities of the planets; (4)Measurement error , the intrinsic noise(stellar variability , the number of transit timing measurements) Consider the strength of TTVs produced by a terrestrial planet of mass on a Jupiter mass planet transiting a star with mass =0.1 are choosing to minimize and are the midtransit times of the N transits G. Bou´e 2013

  21. The dependence of TTVs on several parameters and transit times Consider an inner planet which transits its parent star on an initially circular ,and an external orbit at N=874 Veras 2011 fig.4 After 10 years ,there is a sharp boundary between the near-MMR regime and high-e secular regime

  22. High-resolution signal amplitudes for the region around the 3:1 PC The time scale of resonance and secular perturbation decides whether we can detect the TTV signal Veras 2011 fig.5

  23. Comparison with other terrestrial planet search techniques (1)RV (0.5m/s) (2)Astrometry (1 (3)TTV(10s) astrometry A planet can be detected at an amplitude of 10 times the noise for a given technique TTV On-resonance TTV is sensitive to much smaller Planet masses like earth. RV

  24. Confirmation of planets in near-resonance planetary pairs via TTV Transit time series of two transiting planets near first-order MMR (j-1:j) TTV , transit epoch,transit number,period of the The TTV period Kepler 49 b-c P=7.204d , A sinusoidal TTV is a strong evidence Of a pair of interacting planets Against other astrophysical false Positives. confirmation TTV anti-correlation-conservation of Energy. Xie 2012

  25. Constrain the possible additional planet in a planetary system TrEs-3-1 M. Vaˇnko 2012 Assuming that the planets are coplanar and circular

  26. 9.Conclusions • (1)Planet-planet perturbation are negligible,the main effect is the wobble of the star due to the • inner planet, • (2)the exterior planet changes the period of the interior planet by • (3)Halfway between resonances the perturbations are small ,of the order of • (4)Near-MMR systems can produce particularly strong TTVs .But if the libration period is much • longer than the interval of observation , the TTVs cannot be easily discerned. • (5)terrestrial planets can be detected when in resonance with the transiting planet.

  27. Thank you

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