EXTRACTING PLANET MASS AND ECCENTRICITY FROM TTV DATA

By Y. Lithwick, Ji-Wei Xie, Yanqin Wu. 2012

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Most planet pairs in the Kepler data that have measured transit time variations (TTVs) are near first-order mean-motion resonances. We derive analytical formulae for their TTV signals. We separate planet eccentricity into free and forced parts, where the forced part is purely due to the planets' proximity to resonance. This separation yields simple analytical formulae. The phase of the TTV depends sensitively on the presence of free eccentricity: if the free eccentricity vanishes, the TTV will be in phase with the longitude of conjunctions. This effect is easily detectable in current TTV data. The amplitude of the TTV depends on planet mass and free eccentricity, and it determines planet mass uniquely only when the free eccentricity is sufficiently small. We analyze the TTV signals of six short-period Kepler pairs. We find that three of these pairs (Kepler 18, 24, 25) have a TTV phase consistent with zero. The other three (Kepler 23, 28, 32) have small TTV phases, but ones that are distinctly non-zero. We deduce that the free eccentricities of the planets are small, ≲ 0.01, but not always vanishing. Furthermore, as a consequence of this, we deduce that the true masses of the planets are fairly accurately determined by the TTV amplitudes, within a factor of ≲ 2. The smallness of the free eccentricities suggests that the planets have experienced substantial dissipation. This is consistent with the hypothesis that the observed pile-up of Kepler pairs near mean-motion resonances is caused by resonant repulsion. But the fact that some of the planets have non-vanishing free eccentricity suggests that after resonant repulsion occurred there was a subsequent phase in the planets' evolution when their eccentricities were modestly excited, perhaps by interplanetary interactions.

Subfields: Astronomy and Astrophysics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/f4c81b3af5b4388ad40bfd77a488eb29fb7fbbec

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Citations in the corpus

Kepler Planet Masses and Eccentricities from TTV Analysis (2016, methodology): “The forced components are computed analytically as the eccentricities induced by the nearest three first-order MMRs (see, e.g., Equation (13) of Lithwick et al. 2012); other nonresonant contributions to the forced eccentricity are negligible.”
THE RADIUS DISTRIBUTION OF PLANETS AROUND COOL STARS (2013, background): “…MC 249-17, Pasadena, CA 91125 multi systems have been measured from the effects of mutual gravitational interactions (e.g. Cochran et al. 2011, Lithwick et al. 2012, Nesvorny et al. 2013) providing insight into the composition and atmospheric evolution of these planets (Wu & Lithwick 2013;…”
TRANSIT TIMING OBSERVATIONS FROM KEPLER. VIII. CATALOG OF TRANSIT TIMING MEASUREMENTS OF THE FIRST TWELVE QUARTERS (2013, background): “However, as pointed out by F11 (see also Agol et al. 2005; Holman & Murray 2005; Lithwick et al. 2012), the expected timescale of the dynamical interaction between planets is in most cases larger than the orbital period of the transiting planet.”
ECCENTRICITY FROM TRANSIT PHOTOMETRY: SMALL PLANETS IN KEPLER MULTI-PLANET SYSTEMS HAVE LOW ECCENTRICITIES (2015, background): “While these two parameters are often correlated, sometimes eccentricity information can nevertheless be inferred using statistical arguments (e.g. Lithwick et al. 2012; Wu & Lithwick 2013), or from the “chopping” effect (e.g. Deck & Agol 2015).”
The nature of the TRAPPIST-1 exoplanets. (2018, background): “Degeneracies commonly plague TTV inversion problems (Agol & Fabrycky 2017), in particular, between mass and eccentricity (Lithwick et al. 2012).”
One of the closest exoplanet pairs to the 3:2 mean motion resonance: K2-19b and c (2015, methodology): “The outer, smaller planet c can be expected to show even greater TTVs (scaled up by the mass ratio of the two planets), although these have not yet been observed.”
Spacing of Kepler Planets: Sculpting by Dynamical Instability (2015, background): “Recent studies using both transit timing variation (Lithwick et al. 2012; Hadden & Lithwick 2014) and radial velocity (Weiss & Marcy 2014) techniques have shown that planet masses scales approximately linearly with planet radii, or M/M⊕ ≈ 3.0R/R⊕.”
WASP-47: A HOT JUPITER SYSTEM WITH TWO ADDITIONAL PLANETS DISCOVERED BY K2 (2015, methodology): “This is consistent with the TTV super-period we expect for planets orbiting in this configuration, which we calculate to be PTTV = 52.67 days using Equation 7 of Lithwick et al. (2012).”
THE MASS OF KOI-94d AND A RELATION FOR PLANET RADIUS, MASS, AND INCIDENT FLUX (2013, methodology): “We use the prescription of Equations 8 and 9 from Lithwick et al. (2012) to predict the anti-correlated TTV signals produced during the interaction of planets c and d between 650 and 800 days.”
HUBBLE SPACE TELESCOPE NEAR-IR TRANSMISSION SPECTROSCOPY OF THE SUPER-EARTH HD 97658B (2014, methodology): “…obtained using either the radial velocity or transit timing techniques, it is possible to constrain the average densities and corresponding bulk compositions of the super-Earths in the Kepler sample (e.g., Lithwick et al. 2012; Hadden & Lithwick 2013; Weiss & Marcy 2014; Marcy et al. 2014).”
About 30% of Sun-like Stars Have Kepler-like Planetary Systems: A Study of Their Intrinsic Architecture (2018, background): “To enable TTV detections, the planet transits tend to be deeper and have longer periods (Lithwick et al. 2012; see also Equation (14)).”
CHARACTERIZING THE COOL KOIs. IV. KEPLER-32 AS A PROTOTYPE FOR THE FORMATION OF COMPACT PLANETARY SYSTEMS THROUGHOUT THE GALAXY (2012, background): “The free eccentricities of Kepler-32b and c have recently been estimated to have non-zero values from the phase of the TTV signals (Lithwick et al. 2012; Wu & Lithwick 2012).”
TRANSIT TIMING OBSERVATIONS FROM KEPLER. IX. CATALOG OF THE FULL LONG-CADENCE DATA SET (2016, methodology): “a Orbital period of the inner planet. b Orbital period of the outer planet. c Normalized distance to resonance defined as D º -- 1P P j j 12 1 (Lithwick et al. 2012). d Resonance type. e The TTV period of the inner planet (found by modeling the data). f The TTV period uncertainty of the inner…”
DENSITIES AND ECCENTRICITIES OF 139 KEPLER PLANETS FROM TRANSIT TIME VARIATIONS (2013, methodology): “If these transit time variations (TTVs) can be detected, they can be used to characterize planets (Agol et al. 2005; Holman & Murray 2005; Steffen et al. 2013, 2012; Fabrycky et al. 2012; Nesvornỳ et al. 2012; Lithwick et al. 2012; Xie 2013).”
DENSITY AND ECCENTRICITY OF KEPLER PLANETS (2012, background): “In Lithwick et al. (2012) (hereafter Paper1) we analyze the TTV (transit time variation) signals of six planet pairs and show that the TTV phases are consistent with most planets having in general small eccentricities.”