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. 2016 Oct 11;113(41):11431-11435.
doi: 10.1073/pnas.1604692113. Epub 2016 Sep 26.

Exoplanet Orbital Eccentricities Derived From LAMOST-Kepler Analysis

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Free PMC article

Exoplanet Orbital Eccentricities Derived From LAMOST-Kepler Analysis

Ji-Wei Xie et al. Proc Natl Acad Sci U S A. .
Free PMC article

Abstract

The nearly circular (mean eccentricity [Formula: see text]) and coplanar (mean mutual inclination [Formula: see text]) orbits of the solar system planets motivated Kant and Laplace to hypothesize that planets are formed in disks, which has developed into the widely accepted theory of planet formation. The first several hundred extrasolar planets (mostly Jovian) discovered using the radial velocity (RV) technique are commonly on eccentric orbits ([Formula: see text]). This raises a fundamental question: Are the solar system and its formation special? The Kepler mission has found thousands of transiting planets dominated by sub-Neptunes, but most of their orbital eccentricities remain unknown. By using the precise spectroscopic host star parameters from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) observations, we measure the eccentricity distributions for a large (698) and homogeneous Kepler planet sample with transit duration statistics. Nearly half of the planets are in systems with single transiting planets (singles), whereas the other half are multiple transiting planets (multiples). We find an eccentricity dichotomy: on average, Kepler singles are on eccentric orbits with [Formula: see text] 0.3, whereas the multiples are on nearly circular [Formula: see text] and coplanar [Formula: see text] degree) orbits similar to those of the solar system planets. Our results are consistent with previous studies of smaller samples and individual systems. We also show that Kepler multiples and solar system objects follow a common relation [[Formula: see text](1-2)[Formula: see text]] between mean eccentricities and mutual inclinations. The prevalence of circular orbits and the common relation may imply that the solar system is not so atypical in the galaxy after all.

Keywords: exoplanets; orbital eccentricities; planetary dynamics; solar system; transit.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Transit duration ratio statistics of Kepler single transiting planets. (A) The cumulative distribution of the observed transit duration T normalized by the expected values for circular and edge-on orbit T0 for the Kepler singles (red). The circular-orbit model (dot-dashed) is clearly ruled out. The models shown (gray scale) assume Rayleigh eccentricity distributions with mean eccentricities e¯ varying between 0 and 0.6 (dashed). The best-fit model with e¯=0.32 is shown in blue. The observed distribution has relatively large deviations from the best-fit model in the range of T/T01.02.0, which may indicate either the breakdown of the assumed model Rayleigh distribution or the need for more than one underlying population (see Fig. 4 and related discussion in the main text). (B) Relative likelihood in logarithm as a function of e¯. The blue, green, and red hatched regions indicate the 68.3% (1σ), 95.4% (2σ), and 99.7% (3σ) confidence levels.
Fig. 2.
Fig. 2.
Transit duration ratio statistics of Kepler multiple transiting planets. (A) Cumulative distribution of observed transit duration ratios T/T0 for Kepler multiples is shown in red. It is fitted with models assuming Rayleigh distributions in eccentricities and mutual inclinations. A range of circular-orbit models with e¯=0 and i¯ between 0° and 10° are shown in green. The orbits of Kepler multiples are consistent with being circular and nearly coplanar. The best-fit model (blue) has e¯=0.04 and i¯=0.024(1.4°). For comparison, a range of models with mean eccentricities between 0 and 0.5 and inclinations between 0° and 10° are shown in gray scale. (B) Contours of relative likelihood in logarithm in the i¯e¯ plane. The blue, green, and red contours indicate the 68.3% (1σ), 95.4% (2σ), and 99.7% (3σ) confidence levels of e¯ and i¯. The blue star marks the best-fit values.
Fig. 3.
Fig. 3.
Mean eccentricity e¯ (blue markers) and inclination i¯ (red markers) as a function of transiting multiplicity Np. The filled circles show the best fit, and the error bars indicate the 68% confidence interval. As can be seen, Fig. 3 reveals an abrupt transition of e¯ rather than a smooth correlation with Np (see more discussion in SI Appendix, section 5.1).
Fig. 4.
Fig. 4.
Modeling transit duration ratio distribution (T/T0) of single-transiting systems with a two-population model. (A) Similar to Fig. 1A except that we model the observed distribution with a two-population model. We fix the dynamical cold population with an eccentricity distribution corresponding to the best fit of multiples as shown in Fig. 2 and set another dynamically hot population with Rayleigh eccentricity distribution with mean e¯hot. The fraction (Fhot) and mean eccentricity (e¯hot) of the hot population are two free parameters. Note that the Rayleigh distribution approaches the thermal distribution at e¯0.6. (B) Contours of relative likelihood in logarithm in the Fhote¯hot plane. The blue, green, and red hatched regions indicate the 68.3% (1σ), 95.4% (2σ), and 99.7% (3σ) confidence levels. The magenta line (A) and magenta star symbol (B) indicate the best fit of the two-population model, whereas the blue line and star are for the one-population model. The differences in Bayesian information criterion (BIC) between the best fits of the one-population and two-population models is ΔBIC=2Δln(L)ln(Nobs) = 40.2, which indicates a significant improvement in fitting.
Fig. 5.
Fig. 5.
The mean eccentricity and inclination of Kepler multiples fit into the pattern of the solar system objects. The thin filled circle and error bars show the mean values and 68% confidence intervals of orbital eccentricity and inclination distributions for solar system objects, including eight planets (blue); regular moons of Jupiter, Saturn, and Uranus (light green, yellow, and black); main belt asteroids (green); and trans-Neptune objects (TNOs; both the classical Kuiper Belt objects with orbital semimajor axes 30–55 AU shown in red and the scattered disk objects with semimajor axes >55 AU shown in cyan). They follow an approximately linear relation with e¯(1–2)i¯ (dashed and dotted lines). The thick purple filled circle and error bar (and arrow) show the eccentricity and inclination constraints of Kepler multiples: 0.006 rad (0.3°)<i¯< 0.038 rad (2.2°) and e¯<0.07 derived in this work. The Kepler multiples fall on the linear relation of the solar system objects, and they are on similarly circular and coplanar orbits to the solar system planets.

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