Source: Julia Fang,1*, J. L. Margot,1,2*

*University of California, Los Angeles.

1 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA

2 Department of Earth and Space Sciences, University of California, Los Angeles, CA 90095, USA

Received 2012 July 22; accepted 2012 October 30; published 2012 November 28

Figure 1. One of two best-ﬁt models for multiplicity (left column) and inclination (right column). The multiplicity distribution is a bounded uniform distribution: for each planetary system (1) we draw a value Nmax from a modiﬁed Poisson distribution with λ = 2.25, and (2) we choose the number of planets by uniformly picking a value between 1 and Nmax. The resulting distributions of multiplicities are shown in the bottom plots. The inclination distribution is a Rayleigh of Rayleigh

distribution: (1) for each planetary system we draw a value for σ from a Rayleigh distribution with parameter σσ = 1◦, and (2) for each of its planets we draw a value for inclination from a Rayleigh distribution with parameter σ. The resulting distributions of inclinations are shown in the bottom plots.





Abstract:We investigated the underlying architecture of planetary systems by deriving the distribution of planet multiplicity (number of planets) and the distribution of orbital inclinations based on the sample of planet candidates discovered by the Kepler mission. The scope of our study included solar-like stars and planets with orbital periods less than 200 days and with radii between 1.5 and 30 Earth radii, and was based on Kepler planet candidates detected during Quarters 1 through 6. Our analysis improves on previous work by including all available quarters, extending to 200-day periods, and fitting models to observables such as normalized transit duration ratios that contain information on mutual orbital inclinations; these improvements lend to a deeper investigation of theintrinsic distributions of planetary systems. We created models of planetary systems with different distributions of planet multiplicity and orbital inclinations, simulated observations of these systems by Kepler, and compared the number and properties of the transits of detectable objects to actual Kepler planet detections. Based on the underlying distributions of our best-fit models, 75-80% of planetary systems have 1 or 2 planets with orbital periods less than 200 days. In addition, over 85% of planets have orbital inclinations less than 3 degrees. This high degree of coplanarity is comparable to that seen in our Solar System, with the exception of Mercury. These results provide important constraints and insights into theories of planet formation and evolution.CONCLUSIONSWe have investigated the underlying distributions of multiplicity and inclination of planetary systems by using the sample of planet candidates discovered by the Kepler mission during Quarters 1–6. Our study included solar-like stars and planets with orbital periods less than 200 days and with radii of 1.5–30R⊕. We created model populations represented by a total of two tunable parameters, and we ﬁtted these models to observed numbers of transiting systems and to normalized transit duration ratios. We did not include any constraints from radial velocity surveys. Below we list the main conclusions of our study.1. From our best-ﬁt models, 75%–80% of planetary systems have 1 or 2 planets with orbital periods less than 200 days. This represents the unbiased, underlying number of planets per system.2. From our best-ﬁt models, over 85% of planets have orbital inclinations less than 3◦ (relative to a common reference plane), implying a high degree of coplanarity.3. Compared to previous work, our results do not suffer from degeneracies between multiplicity and inclination. We break the degeneracy by jointly considering two types of observables that contain information on both number of planets and inclinations.4. If we extrapolate down to planet radii less than 1.5 Earth radii, the underlying multiplicity distribution is consistent with the number of planets in the solar system with orbital periods less than 200 days. If we also extrapolate to beyond 200 days, we ﬁnd that the underlying distributionof inclinations derived here is compatible with inclinations in the solar system.5. Our results are also consistent with the standard model of planet formation in a disk, followed by an evolution that did not have a signiﬁcant and lasting impact on orbital inclinations.Continued observations by the Kepler mission will improve the detectability of new candidate planets covering a larger swath of parameter space, especially to longer orbital periodsand smaller planetary radii. We anticipate that future statistical work will further boost our understanding of the underlying architecture of planetary systems.We thank Dan Fabrycky for useful discussions, as well as the entire Keplerteam for procuring such an excellent data set of planetary systems. We also thank the reviewer for helpfulcomments that improved the paper.REFERENCESBaluev, R. V. 2011,Celest. Mech. Dyn. Astron., 111, 235Batalha, N. M., Borucki, W. J., Bryson, S. T., et al. 2011, ApJ, 729, 27Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2012, arXiv:1202.5852Chambers, J. E., Wetherill, G. W., & Boss, A. P. 1996,Icarus, 119, 261Christiansen, J. L., Jenkins, J. M., Barclay, T. S., et al. 2012, arXiv:1208.0595Colon, K. D., Ford, E. B., & Morehead, R. C. 2012, arXiv: ´ 1207.2481Correia, A. C. M., Couetdic, J., Laskar, J., et al. 2010, A&A, 511, A21Sanchis-Ojeda, R., Fabrycky, D. C., Winn, J. N., et al. 2012, Nature, 487, 449Santerne, A., D´ıaz, R. F., Moutou, C., et al. 2012, A&A, 545, A76Stuart, A., & Ord, J. 1991, Kendall’s Advanced Theory of Statistics, Vol. 2, Classical Inference and Relationship, Kendall’s Advanced Theory of Statistics (5th ed.; Oxford: Oxford Univ. Press)Tremaine, S., & Dong, S. 2012, AJ, 143, 94Weissbein, A., Steinberg, E., & Sari, R. 2012, arXiv:1203.6072Winn, J. N. 2010, in Exoplanets, ed. S. Seager (Tucson, AZ: Univ. Arizona Press)Youdin, A. N. 2011, ApJ, 742, 38