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Project Report Mon. Not. R. Astron. Soc. 000, 000{000 (0000) Printed 11 September 2014 (MN LATEX style file v2.2) In situe irregular satellite formation Xiao-Chen Zheng1;2, Hagai B. Perets3, Alice Quillen4 1 Kavili institute for astronomy and Astrophysics, Peking University, Beijing, China 2 Department of Astronomy, Peking University, Beijing, China 3Technion - Israel Institute of Technology, Haifa 32000, Israel 4 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 11 September 2014 ABSTRACT We investigate formation of satellites through collisions in a planetesimal disk that is surrounding an oblique oblate planet. We find that Kozai resonance lifted particles cause large eccentricities and inclinations in the growing satellite population and this can lead to formation of a single large moon near the Laplace radius. It could be the formation mechanism for a single large moon. Besides, retrograde moon forms in some cases motivted us a possible fomation mechanism for Triton-like irregulars. 1 INTRODUCTION 1.1 Basic physics A satellite in orbit about an oblate planet precesses about Moons around the planets are grouped into two categories, the planet's spin axis due to the planet's gravitational poten- regular and irregular. Regular moons have low inclination tial quadrupole moment. If the planet has a non-zero obliq- and eccentricity, with respect to the planet's spin axis. uity, then this axis is tilted with respect to the star's tidal In contrast, irregular satellites can have high inclinations field. The obliquity of the planet is the angle between the and eccentricities and even orbit the planet retrograde to planet's spin axis and its orbital angular momentum axis. In- the planet's orbit (for a review see Jewitt & Haghighipour side the the Laplace radius, rL, precession is predominantly 2007). Formation of regular moons is thought to take place due to planet's gravitational quadrupole moment and out- in a gaseous circum-planetary disk (e.g., Canup & Ward side this radius, due to the stellar tide. 2006) whereas the a successful model accounting for irregu- ( ) 1 2 3 5 lar moons involves capture from heliocentric orbit (see dis- 2MpJ2Rpap rL ∼ (1) cussion by Jewitt & Haghighipour 2007). M∗ Models for terrestrial planet formation often start af- where M∗ is the stellar mass, Mp is the planet's mass, Rp ter the formation of a planetesimal disk (e.g., for a recent is the planet's equatorial radius, ap the planet's semi-major review see Raymond et al. 2014). At later stages a set of axis and J2 the planet's quadrupole gravitational moment. similar sized planets (oligarchs) can be formed, when run- The quadrupole moment can be modified to take into ac- away growth is limited by the depletion of low mass objects count the contribution from moons. The planet's orbital ec- and the dispersion in the disk. Recently Perets & Payne centricity ep can be taken into account by multiplying the 2 3 2014 showed that massive moons originally formed on co- right hand side of equation 1 by (1 − ep) 10 (see equation 24 planar orbits exterior to the region of regular moons in the by Tremaine et al. 2009). solar system can scatter each other. Scattered moons, some The Laplace surface has normal (as a function of radius of which have undergone Kozai-Lidov oscillations can give a from the planet) about which the satellite's orbit precesses population of high inclination and eccentricity irregular-like due to the combined effects of stellar tide and the planet's (but prograde) moons. Scattering can also eject moons from quadrupole moment. The Laplace surface traces the pre- the planet's Hill sphere, and these if later recaptured, can dicted shape of a thin gas disk or dissipative particulate temporarily have retrograde orbits. ring surrounding the planet (e.g., Tremaine et al. 2009). Most regular planetary satellites orbit within laplace radii In this paper we investigate formation of moons in near and probably formed from a circumplanetary gas disk (e.g., a planet with non-zero obliquity. A similar setting might Canup & Ward 2006). be planet formation in a triple star system (Pelupessy & Consider a particle in a circular orbit around the planet Portegies Zwart 2013). Studies for the inclinations of plan- with semi-major axis a. We define an angle ϕ that is the etary systems affected by binary companion stars (Batygin angle between the angular momentum vector of the parti- et al. 2011; Batygin 2012; Terquem 2013; Xiang-Gruess & cle and the planet's orbital angular momentum vector. The Papaloizou 2014) could be mentioned here. Laplace surface ϕ could be writen as (using equation 7 by ⃝c 0000 RAS 2 Zheng et al. Tremaine & Davis 2014) would result in the formation of a massive moon in retro- grade orbit could be a prospective scenario for Triton-like cos 2ϕ − a5=2r5 cot 2ϕ = p L : (2) satelliate formation without applying any other energy dis- sin 2ϕp sipation mechanism. However, the observed Triton is much where ϕp is the azimuthal angles of the planet, the obliquity closer to Neptune compare to the simulation and the inli- 1 − ∗ − nation as well as the mass in our mimic result have non- is ϕp ϕ = ϕp 2 π ignorable devergency with observed value. Actually, the lat- ter difference could contribute to the model dependence and in the simulation we only consider the inelastic collision so 2 NUMERICAL SIMULATIONS the collison outcomes are resonable to be different to the 2.1 Initial settings real case, but how could Triton migrate into such a inner region is still a puzzle, additional migration scenario is nec- The code Mercury (Chambers 1999) was modified to take cessary to be considered, tidal dissipation could be the most into account the oblateness of the planet by adding a luminous proposal. quadrupole moment to the gravitational potential. When In Fig.2, we study how moons growth with time in the two particles collide they stick together forming a single top plot, while in the bottom pattern, their individual semi- body with total mass the sum of each impactor. major axis evolving are also presenting in detail. As same Simulations begin with a single planet in orbit about a as the inferring by Fig.1, most moon mass prefer to concen- star. The planet is initial surrounded by a disk of planetes- trate in the inner region but there are large amount of small imals in circular orbits that are in the planet/star orbital planetesimals bordely distribute in the few times of laplace plane. Parameters for the simulations are listed in Table 1. radii to few tens percent of planet Hill radius with moder- The density assumed for planetesimals is that of the largest ate excitation in orbit which could be the origin of irrgular −3 moon and is between 1-2 g cm . Initial planetesimal sepa- group. ration was 7 rH with the mutual Hill radius For comprison, planetsimals in Neptune system are also 2µ 1=3 ai + ai+1 ai+1 − ai initially at the anylatic laplace plane and the equator plane, RH = ( ) ; k = (3) 3 2 RH shown in Fig .3 and Fig.4, respectively. The final mass distri- bution in two plots show similar features, there is one mas- We ran three sets of simulations: sive object form at the reference plane within one laplace • Series A: started with planetesimals initial in the radii in both cases, while beyond the region, small plan- planet's orbital plane. etesimals are randomly dispersed around the circum-stellar • Series B: with planetesimals started in the Laplace sur- plane which is consistent to the A1 series. The absence of face. retrograde object in B1 and C1 series could contribute to • Series C: with planetesimals started in a planet perpen- the overlap of laplace surface with the primary location of dicular to the planet's spin axis (the obliquity plane). plantesimals within laplace radii, therefore the inclination of most planetesimals in this region are impossible to be The A1 simulation uses parameters for Neptune. greatly excietd by quadropole torque of host planet. We also run a couple of test simulations with the same To have a basic idea that how could the planet's obliq- parameters as these series but a zero obliquity. uity take effect on moon formation, in Fig.5, we consider the planetesimals evolving around one zero-obliquity planet. As what we expected, collisions of planetesimals in such a non- 2.2 Results and discussion J2 system are not as effeciency as A1, B1 and C1 series Figure 1 shows the result of the A1 simulations. Instead resulting in the absence of massive moon after 1 Myr evolu- of the analyatic laplace surface derived by equation 2, the tion. Therefore, we could rougly conclude that the planet's planetesimals are initially located at the reference plane, in obliquity has significant influnece on the observed configu- another words, they are primarily considering to be born ration of surrounding moons. at the circumstellar disk. After 1 Myr evolving, there is a Besides, the moon formation around Saturn system is massive object formed within one rL, caused by increase also studied in Fig.6, Fig.7 and Fig.8. The planetsimals are in collisions probability due to Kozai oscillations near rL. initially located at reference plane, laplace surface and equa- Beyond the laplace radii, most plantesimals are smaller in tor plane, respectively. In both cases, after 1 Myr evolution, mass, and even one or two dominated bodies formed, they there is no massive moon formed within one laplace radii, it are prefering to orbit around rL rather than outer region.
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