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Formation of terrestrial planets in disks with different surface density profiles

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Abstract

We present the results of an extensive study of the final stage of terrestrial planet formation in disks with different surface density profiles and for different orbital configurations of Jupiter and Saturn. We carried out simulations in the context of the classical model with disk surface densities proportional to \({r^{-0.5}}, {r^{-1}}\) and \({r^{-1.5}}\), and also using partially depleted, non-uniform disks as in the recent model of Mars formation by Izidoro et al. (Astrophys J 782:31, 2014). The purpose of our study is to determine how the final assembly of planets and their physical properties are affected by the total mass of the disk and its radial profile. Because as a result of the interactions of giant planets with the protoplanetary disk, secular resonances will also play important roles in the orbital assembly and properties of the final terrestrial planets, we will study the effect of these resonances as well. In that respect, we divide this study into two parts. When using a partially depleted disk (Part 1), we are particularly interested in examining the effect of secular resonances on the formation of Mars and orbital stability of terrestrial planets. When using the disk in the classical model (Part 2), our goal is to determine trends that may exist between the disk surface density profile and the final properties of terrestrial planets. In the context of the depleted disk model, results of our study show that in general, the \(\nu _5\) resonance does not have a significant effect on the dynamics of planetesimals and planetary embryos, and the final orbits of terrestrial planets. However, \(\nu _6\) and \(\nu _{16}\) resonances play important roles in clearing their affecting areas. While these resonances do not alter the orbits of Mars and other terrestrial planets, they strongly deplete the region of the asteroid belt ensuring that no additional mass will be scattered into the accretion zone of Mars so that it can maintain its mass and orbital stability. In the context of the classical model, the effects of these resonances are stronger in disks with less steep surface density profiles. Our results indicate that when considering the classical model (Part 2), the final planetary systems do not seem to show a trend between the disk surface density profile and the mean number of the final planets, their masses, time of formation, and distances to the central star. Some small correlations were observed where, for instance, in disks with steeper surface density profiles, the final planets were drier, or their water contents decreased when Saturn was added to the simulations. However, in general, the final orbital and physical properties of terrestrial planets seem to vary from one system to another and depend on the mass of the disk, the spatial distribution of protoplanetary bodies (i.e., disk surface density profile), and the initial orbital configuration of giant planets. We present results of our simulations and discuss their implications for the formation of Mars and other terrestrial planets, as well as the physical properties of these objects such as their masses and water contents.

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Notes

  1. Because in the Nice model, Jupiter and Saturn are initially in circular orbits, \(\nu _6\) does not manifest itself clearly. In order to make this resonance show, we slightly increased the eccentricities of the two giant planets to 0.01. This is consistent with the initial orbital eccentricities of these planets in the Nice model-II (Levison et al. 2011).

  2. In all our numerical integrations throughout this study, we determine the locations of secular resonances by monitoring the precession rates of protoplanetary bodies and planetesimals, and comparing them with those of Jupiter and Saturn. Secular resonances increase the orbital eccentricities and inclinations of their affected bodies. We also monitor these quantities and mark the time and semimajor axis at which eccentricity and inclination are maximally enhanced.

  3. According to the calculations by Nagasawa et al. (2005), the outward displacement of \(\nu _6\) in our disk models should be no more than 0.1 AU. This outward displacement would be smaller in disks with less steep surface density profiles (i.e., lower values of \(\alpha \)). While in our simulations, the \(\nu _6\) resonance did move outward, the amount of its displacement was slightly larger than that suggested by Nagasawa et al. (2005). For example, in a disk with \(\alpha =1.5\) and considering Jupiter and Saturn in their current orbits, \(\nu _6\) appeared around 2.35 AU, which corresponds to an outward displacement of 0.2–0.3 AU compared to the location of this resonance in simulations considering test particles and same configuration for giant planets. Our results indicate that the calculations by Nagasawa et al. (2005) underestimates the outward sweeping of secular resonances by not considering the disk gravitational potential.

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Acknowledgments

We are indebted to A. Izidoro for his help with making the figures and analysis of the results, and for his invaluable comments that greatly improved our manuscript. We are also thankful to the referees for their constructive suggestions and recommendations. NH acknowledges support from the NASA ADAP Program under Grant NNX13AF20G, NASA PAST Program under Grant NNX14AJ38G, HST Grant HST-GO-12548.06-A, and NASA Astrobiology Institute under Cooperative Agreement NNA09DA77A at the Institute for Astronomy, University of Hawaii. OCW acknowledges support from FAPESP (PROC. 2011/08171-3) and CNPq (Proc. 312813/2013-9). Support for program HST-GO-12548.06-A was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy Incorporated, under NASA Grant NAS5-26555. NH would also like to thank the Alexander von Humboldt Foundation, and the Kavli Institute for Theoretical Physics at the University of California-Santa Barbara (KITP) for their kind hospitality during the final stage of this project. KITP visiting program is supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.

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Authors and Affiliations

  1. Institute for Astronomy, University of Hawaii, Honolulu, HI, 96822, USA

    Nader Haghighipour

  2. Grupo de Dinámica Orbital & Planetologia, UNESP - Univ. Estadual Paulista, Guaratinguetá, SP, CEP 12516-410, Brazil

    Othon C. Winter

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  1. Nader Haghighipour
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Correspondence to Nader Haghighipour.

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Haghighipour, N., Winter, O.C. Formation of terrestrial planets in disks with different surface density profiles. Celest Mech Dyn Astr 124, 235–268 (2016). https://doi.org/10.1007/s10569-015-9663-y

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