Outcomes of Grazing Impacts Between Sub-Neptunes in Kepler Multis
Jason Hwang, Sourav Chatterjee, James Lombardi Jr., Jason Steffen, Frederic Rasio
aa r X i v : . [ a s t r o - ph . E P ] N ov D RAFT VERSION N OVEMBER
30, 2017Typeset using L A TEX twocolumn style in AASTeX61
OUTCOMES OF GRAZING IMPACTS BETWEEN SUB-NEPTUNES IN KEPLER MULTIS J ASON H WANG ,
1, 2 S OURAV C HATTERJEE ,
1, 2 J AMES L OMBARDI J R ., J ASON
H. S
TEFFEN , AND F REDERIC R ASIO
1, 21
Northwestern University, Department of Physics and Astronomy, Northwestern Univsersity, 2145 Sheridan Road, Evanston, IL 60208, USA Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Department of Physics, Allegheny College, Meadville, PA 16335, USA Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 S. Maryland Pkwy., Las Vegas, NV 89154-4002, USA
ABSTRACTStudies of high-multiplicity, tightly-packed planetary systems suggest that dynamical instabilities are common and affect boththe orbits and planet structures, where the compact orbits and typically low densities make physical collisions likely outcomes.Since the structure of many of these planets is such that the mass is dominated by a rocky core, but the volume is dominatedby a tenuous gas envelope, the sticky-sphere approximation, used in dynamical integrators, may be a poor model for thesecollisions. We perform five sets of collision calculations, including detailed hydrodynamics, sampling mass ratios and core massfractions typical in Kepler Multis. In our primary set of calculations, we use Kepler-36 as a nominal remnant system, as thetwo planets have a small dynamical separation and an extreme density ratio. We use an N-body code,
Mercury 6.2 , to integrateinitially unstable systems and study the resultant collisions in detail. We use these collisions, focusing on grazing collisions,in combination with realistic planet models created using gas profiles from
Modules for Experiments in Stellar Astrophysics and core profiles using equations of state from Seager et al. (2007), to perform hydrodynamic calculations, finding scatterings,mergers, and even a potential planet-planet binary. We dynamically integrate the remnant systems, examine the stability, andestimate the final densities, finding the remnant densities are sensitive to the core masses, and collisions result in generally morestable systems. We provide prescriptions for predicting the outcomes and modeling the changes in mass and orbits followingcollisions for general use in dynamical integrators.
Keywords: equation of state – hydrodynamics – methods: numerical – planets and satellites: dynamical evolu-tion and stability, gaseous planets – stars: individual (Kepler-36) INTRODUCTIONMany studies of the dynamical evolution of high-multiplicity,tightly-packed planetary systems (known as Kepler Mul-tis) have shown that systems similar to those observedin the Kepler sample may experience planet-planet insta-bilities that result in physical collisions (Pu & Wu 2015;Volk & Gladman 2015; Hwang et al. 2017, hereafter re-ferred to as
Paper 1 ). Studies of generic planetary sys-tems show that post-disk dynamical interactions may beimportant in shaping the observed structure of KeplerMultis (Chambers et al. 1996; Juri´c & Tremaine 2008;Ford & Rasio 2008; Chatterjee et al. 2008), and additionalstudies (Thommes et al. 2008; Matsumura et al. 2010) con-firm the occurence of planet-planet interactions are con-sistent with orbits after the planetesimal disk dissipates.Fang & Margot (2012) find many Kepler Multis are dynam-ically packed, in that no additional planets may be added inbetween neighboring orbits without leading to planet-planetinteractions, suggesting many observed systems are on thecusp of dynamical instability. Studies of the structure ofthe planets in these systems have shown most of these plan-ets are made up of a high-density core with a tenuous gasenvelope dominating the volume (e.g., Wolfgang & Lopez2015; Weiss & Marcy 2014; Rogers 2015). This highly dif-ferentiated structure suggests the outcomes of planet-planetcollisions in these systems is very sensitive to the details ofthe collision, and in many cases the sticky-sphere approxima-tion, ubiquitous in dynamical integrators, may not be validfor many of these collisions. The sticky-sphere approxima-tion assumes a collision occurs in the integration when theplanets have a minimum separation less than the sum of thephysical radii, and results in a single surviving planet withmass equal to the sum of the two colliding planets’ masses,conserving center of mass, linear momentum, and angularmomentum. We perform five sets of detailed calculations,varying the mass ratio and core mass fractions, to betterunderstand and predict the outcomes of planet-planet colli-sions that may be instrumental in shaping both the orbits andplanet structures of these systems (Inamdar & Schlichting2016). For our primary suite of integrations, we use Kepler-36 as a nominal remnant of a previous planet-planet collision.To examine how the mass ratio and planet structures affectthe outcomes of these collisions, we also perform calcula-tions between planets of mass ratio q = 1 and q = 1 /
3, wherethe less-massive planet has a mass of 4 M E , and core massfractions of m c / m = 0 .
85 and m c / m = 0 . . Kepler-36 is a 2-planet system discovered by the Keplermission (Carter et al. 2012), where Kepler-36b and Kepler-36c (henceforth referred to as b and c ) have an orbital sepa-ration of less than 0 .
015 AU, and a density ratio of ρ b /ρ c =0 . b has a density consistent with little or no gas enve-lope ( ρ b = 7 . − , Carter et al. 2012), while c has a den-sity consistent with a gas mass fraction of 12% and a gasradius fraction of 55% ( ρ c = 0 .
89 g cm − ). Deck et al. (2012)show Kepler-36 likely undergoes short timescale dynamicalchaos and provide best-fit densities of ρ b = 7 .
65 g cm − and ρ c = 0 .
93 g cm − . Nagy & Ágas (2013) explore the possi-bility of additional planets in the system and use long termstability to contrain the semi-major axes of potential addi-tional planets to a < . a > .
14 AU. The highdensity difference and small dynamical separation motivatesmany studies on the formation of Kepler-36.Many formation channels have been explored to explainthe observed properties of Kepler-36. Paardekooper et al.(2013) show migration in a turbulent disc can result in sys-tems very similar to Kepler-36. Lopez & Fortney (2013) andOwen & Morton (2016) explore the possibility that the den-sity difference can be explained by evaporation of the en-velopes, and constrain the cooling timescale and initial struc-tures of b and c . Without migration, the planets’ structureis unlikely to have formed in-situ due to the combination ofsmall orbital separation and extreme density ratio, and maybe a possible remnant of a previous planet-planet collisionthat depleted the gas envelope in the smaller planet. We useKepler-36 as a potential collision remnant to guide our choiceof initial conditions for our primary set of collision calcula-tions. For this set of calculations we provide a generic an-alytic calculation of the orbits and amount of conservativemass-transfer required to create an initially unstable systemthat becomes stable after a collision resulting in two surviv-ing planets (see §A.1).Many studies have been conducted to understand the de-tails of planet-planet collisions and close encounters. Muchof the early work studies potential Moon-formation colli-sions, specifically between a proto-earth and an impactor,varying the planet structures (Hartmann & Davis 1975,Benz et al. 1986; Cameron 2000; Canup & Asphaug 2001;Canup 2004; Canup et al. 2013; Canup & Salmon 2014)and using various tabulated equations of state to handlethe abrupt changes in density and phase transitions of theimpacted rock. Liu et al. (2015) present calculations of di-rect collisions between a rocky, terrestrial planet and a gasgiant, treating the rocky core with a multi-phase equationof state (Tillotson 1962) and the gas as a polytrope with γ = 5 /
3. Podsiadlowski et al. (2010) discuss the possibilityof tidal dissipation of orbital energy leading to potentiallystable gas-giant binaries. Inamdar & Schlichting (2015) andInamdar & Schlichting (2016) combine 1-D hydrodynamiccalculations and a thermal evolution model to examine the ef-fect of collisions resulting in mergers during the giant-impactphase of planet formation, and find that these collisions resultin a large range of planet densities, similar to the observeddensities found in Kepler Multis. In this work we explore theoutcomes of grazing collisions between sub-Neptunes closeto the host star, varying the mass ratio and gas mass fractions.We present several suites of collision calculations from initialconditions sourced from dynamical integrations and find sev-eral distinct outcomes, including scatterings, mergers, and apotential planet-planet binary (formed from energy dissipa-tion due to the physical collision). We aggregate the resultsand develop a fit predicting the outcomes and changes inmass from such collisions for use in dynamical integrators.The rest of the paper is organized as follows: In §2 wediscuss the initial conditions and numerical methods usedfor our dynamical integrations of planetary systems, and ourmethod for creating 3-D planet models and numerical meth-ods used in the subsequent hydrodynamic calculations. In§3 we present the results of our hydrodynamic calculations,classifying the collision outcomes as bound planet-pairs, twosurviving planets in stable or unstable orbits, or mergers, andexplore the stability and potential observable indicators ofcollisions in both the orbits and planet structures. In §4 wepresent a prescription, aggregating the results from all sets ofcollision calculations, for predicting the outcomes and mod-eling the mass loss of planet-planet collisions for use in dy-namical integrators. Finally, we summarize our conclusionsin §5. NUMERICAL METHODSWe perform five sets of collision calculations, varying themass ratio and core mass fractions of the two-planet system.We choose our combinations of mass ratios and core massfractions to broadly cover the properties of neighboring plan-ets seen in Kepler Multis. Table 1 shows the planet masses,radii, and core properties, and host star properties used ineach set, where §A.1.1 describes how we assign the massesand core mass fractions for the Kepler-36 progenitor calcu-lations. Sub-Neptune structures are more dependent on thecore mass fraction than the total mass; despite having threetimes more gas, the 12 M E models have a physical size sim-ilar to the 4 M E models of identical gas mass fraction. Foreach set we generate a suite of dynamical integrations anduse a subset of the resultant collisions as initial conditionsfor later detailed hydrodynamic calculations.2.1. Dynamical Integrations
For each set of systems, we use a Monte Carlo methodto generate 1000 realizations in the point-mass limit, wherewe set the planets’ density to some arbitrary large value, us-ing the Burlish-Stoer integrator within the N-body dynamicspackage,
Mercury P / P = 1 . a = 0 . a (1 + e ) = a (1 − e ) to ensure fast or-bit crossing. The inclinations are randomly drawn from adistribution uniform in cos( i ), where we choose a small, butnon-zero inclination, − θ ≤ i ≤ θ ; θ = 0 . a sin( θ ) > R , where R is the radii of the planets, to simulate anearly coplanar system. Finally, we randomly sample the ar-gument of periapsis, ascending node, and true anomaly froma uniform distribution from 0 to 2 π . A collision occurs in the integration when the planets havea minimum separation less than the sum of the physical radii,calculated using MESA assuming an Earth-like core compo-sition and a core-density calculated by interpolating core-mass core-density tables in Howe et al. (2014). For each cal-culation, the first planet-planet collision is characterized bythe distance of closest approach, d min , in units of the sumof the planets’ physical radii, R and R , and the collisionenergy, E c = E k + E g in units of the binding energy of bothplanets up to a coefficient, E b = X k =1 , Gm k R k , (1)where E k and E g are the kinetic and gravitational potentialenergy of the planets in the center of mass frame, ignoringthe host star. The degree of contact, η = d min R + R , (2)characterizes the depth of the collision, where η = 1 describesan extremely grazing collision and η = 0 describes a head-oncollision.Table 3 shows the distribution of collisions for each setof calculations, categorizing the number of collisions thatresult in a direct impact of the two cores, grazing colli-sions, and near misses, defined as close encounters where1 < d min / ( R + R ) < . . We note that due to tidal bulging,many of the “near misses” would also result in physical con-tact between the planets. We find that the number of colli-sions does not decrease significantly for systems with planetsof the same mass but smaller physical size, and the fractionof collisions that result in a direct impact between the plan-ets’ cores increases with larger core mass fractions. Figure 1shows the distribution of the first planet-planet collisions ineach set of integrations. In contrast to
Paper 1 , we find thatthe number of initial collisions decreases at higher distancesof closest approach, likely due to the lower multiplicity of oursystems; since higher-multiplicity systems can go unstablewith wider orbital spacings (Gladman 1993, Chambers et al.1996), crossing orbits will extend to higher eccentricities—and hence, higher relative velocities. In §2.2, we describehow we perform detailed hydrodynamic calculations explor-ing how the distance of closest approach and collision energyaffect the outcomes, specifically if the cores merge, if theplanets survive, and how much gas is retained by the rem-nant planet(s) (Chatterjee et al. 2008).2.2.
Hydrodynamics
We perform 102 hydrodynamic calculations sampled fromeach set of dynamical integrations described in §2.1, using anSPH code,
StarSmasher (previously StarCrash , originallydeveloped by Rasio 1991 and later updated as described in StarSmasher is available at https://jalombar.github.io/starsmasher/.
Table 1.
System Properties
Set m m m , c / m m , c / m R R R , c / R R , c / R M ∗ T ∗ Kepler − progenitor .
67 7 .
87 0 .
95 0 .
91 2 .
65 3 .
37 0 .
56 0 .
49 1 .
071 5911 q = 1; m c = 0 .
85 4 .
00 4 .
00 0 .
85 0 .
85 4 .
21 3 .
96 0 .
33 0 .
35 1 . q = 1; m c = 0 .
95 4 .
00 4 .
00 0 .
95 0 .
95 2 .
84 2 .
72 0 .
50 0 .
52 1 . q = 1 / m c = 0 .
85 4 .
00 12 .
00 0 .
85 0 .
85 4 .
21 4 .
01 0 .
33 0 .
45 1 . q = 1 / m c = 0 .
95 4 .
00 12 .
00 0 .
95 0 .
95 2 .
84 2 .
94 0 .
50 0 .
63 1 . Set designates the set name, m and m are the masses of the inner and outer planet in Earth masses, m , c / m and m , c / m are the core mass fractions of the inner and outer planet, R and R are the radii of the inner and outer planet in Earthradii, R , c / R and R , c / R are the core radius fractions of the inner and outer planet, and M ∗ and T ∗ are the mass andtemperature of the host star in solar masses and K. Table 2.
Initial Orbits
Set a a e , min e , max e , min e , max Kepler − progenitor .
111 0 .
132 0 .
11 0 .
11 0 .
07 0 . q = 1 0 .
100 0 .
119 0 .
00 0 .
18 0 .
01 0 . q = 1 / .
100 0 .
119 0 .
00 0 .
19 0 .
00 0 . Set designates the set name, a and a are the semi-major axes of the inner andouter planet in AU, and e , min , e , max , e , min , and e , max are the minimum and max-imum eccentricities of the inner and outer planets. The initial orbits are identicalfor the sets of runs with a given mass ratio. Table 3.
Statistics of N-body Collisions
Set N total N direct impact N grazing N near miss Kepler − progenitor
502 290 148 64 q = 1; m c = 0 .
85 578 245 266 67 q = 1; m c = 0 .
85 572 303 181 81 q = 1 / m c = 0 .
95 708 360 256 92 q = 1 / m c = 0 .
95 701 447 171 83
Set designates the set name, N total is the number of collisions that occurwithin 1000 years, N direct impact is the number of collisions resulting in adirect impact between the planets’ cores, N grazing is the number of colli-sions that do not result in a direct impact between the planets’ cores, and N near miss is the number of near misses, defined as close encounters where1 < d min / ( R + R ) < . . Lombardi et al. 1999 and Faber & Rasio 2000), to treat thehydrodynamics.
StarSmasher implements variational equa-tions of motion and libraries to calculate the gravitationalforces between particles using direct summation on NVIDIAgraphics cards as described in Gaburov et al. (2010b). Usingdirect summation instead of a tree-based algorithm for grav-ity increases the accuracy of the gravity calculations at thecost of speed (Gaburov et al. 2010a). The code uses a cu- bic spline (Monaghan & Lattanzio 1985) for the smoothingkernel and an artificial viscosity prescription coupled with aBalsara Switch (Balsara 1995) to prevent unphysical inter-particle penetration, specifically described in Hwang et al.(2015). We sample collisions at varying degrees of contactand collision energies, using the position and velocity coor-dinates of the planets many dynamical times defined as thefree-fall time of a test-particle around the planet) prior to the
Figure 1.
Scatter plots showing the distance of closest approach, d min , in units of the sum of the planets’ physical radii, and collision energy, E c = E k + E g in units of the binding energy of both planets up to a coefficient, E b = Gm / R + Gm / R , where E k and E g are the kinetic andgravitational potential energy of the planets in the center of mass frame, ignoring the host star, for the first collision or near miss in each N-bodyintegration. The histograms above the scatter plots show the distribution of distances of closest approach and the histograms to the right ofthe scatter plots show the distribution of collision energies for the first collision or near miss in each integration. We show the distribution ofcollisions with a small enough distance of closest approach to result in a direct impact between the two cores (red), using the nominal coreradii given by Lissauer et al. (2013) for the Kepler-36 progenitor integrations (top left), and using the nominal core radii calculated using theassigned core mass fraction and a core density from Howe et al. (2014) for the q = 1 (bottom left) and q = 1 / < d min / ( R + R ) < .
2. Forthe more generic calculations, collisions are calculated assuming a core mass fraction of 85%. Collisions in between planets with a higher coremass fraction require a smaller distance of closest approach, and we show the minimum distance for a core mass fraction of 95% (dotted blackline). close encounter, where we treat the host star as a point-massparticle that interacts only gravitationally. We preferentiallysample collisions to explore the boundary between mergerand scattering results.2.3.
Planet Models and Equations of State
Following the method described in § A . Paper 1 , we firstuse
MESA to generate gas envelopes with a constant densitycore of mass, m c , where the core density is determined usingtables from Howe et al. (2014) assuming an Earth-like corecomposition (67 .
5% silicate mantle and 32 .
5% iron core).We irradiate the planet at the chosen semi-major axis andhost-star temperature (shown in Tables 1 and 2) for 4 . × years. We then replace the isothermal and constant-densitycores MESA generates with a differentiated core, with equa-tions of state from Seager et al. (2007), using the core massand radius as boundary conditions to solve for the iron coreand silicate-mantle mass fractions, recovering solutions veryclose to the Earth-like input values.
Planet structures are very time-sensitive. Chen & Rogers(2016) and Lopez & Fortney (2013) show the time-dependent structure of Kepler-36b and Kepler-36c (Fig-ure 1 in both papers), where Kepler-36b contracts bya factor of 7 and Kepler-36c contracts by a factor of3 between 100 Myr and 4.5 Gyr. The age of planetswhen orbital instability first develops is not well un- derstood, in part because we do not know what mighttrigger the instability in the system, just the internalsecular evolution of the system (e.g. Chatterjee et al.2008; Juri´c & Tremaine 2008; Deck et al. 2012), or anexternal trigger (e.g., Zakamska & Tremaine 2004). Ingeneral, orbital instability timescales in chaotic systemshave very broad distributions (e.g., Chatterjee et al. 2008;Zhou et al. 2007; Funk et al. 2010), so here we take a con-servative assumption, assuming the planets are old andtherefore their radii have shrunk to near equilibrium.With younger, more inflated planets, collisions happeneven faster, but this would make our initial conditionsvery arbitrary (very sensitive to exact assumptions aboutthe age of the planets at the time they collide). Collisionsoccurring between younger planets would be more dra-matic, due to lower density and hotter envelopes, and ourcalculations should be treated as a lower limit.
StarSmasher uses the combined gas and core profiles,and because we are interested in the evolution of thelow-density gas envelope, creates planet models withequal number-density, non-equal mass, particle-distributions(Monaghan & Price 2006) using ∼ particles per planet.Gas particles follow a semi-analytic equation of state fit to agrid of MESA models, p i = ρ i u i β i + K e ρ γ e i (cid:18) − β i ( γ e − (cid:19) , (3) u i = K e ρ γ e − i γ e − + β i k B T i µ i m H , (4)where we use γ e = 3, while K e and β i are fitted parameters.Core particles follow an equation of state from Seager et al.(2007), p i = u i ( γ c − ρ i (cid:16) − ρ ′ c ρ i (cid:17) γ c F (cid:16) − γ c , − γ c , − γ c , ρ ′ c ρ i (cid:17) , (5)where the internal energy is initially set as u i = c ρ γ c − i F (cid:16) − γ c , − γ c , − γ c , ρ ′ c ρ (cid:17) γ c − , (6) c , γ c , and ρ ′ c are constants determined by the composition,and F (1 − γ c , − γ c , − γ c , ρ ′ c ρ ) is the ordinary hypergeomet-ric function. Expressions for the mantle are like those forthe core but with coefficients applicable to a silicate com-position employed. Particles near the interfaces of the gasenvelope, silicate mantle, and iron core are treated as mixed-composition in order to resolve the high density-gradient be-tween components. For more details of how we generateour planet models, including how we fit the equation of stateused for the gas envelope, and the algorithms used to handlemixed-composition particles, see § A . Paper 1 .Figures 2 and 3 show two comparisons of the initial pro-files from
MESA and the semi-analytic core equations of state, to the planet profiles after 2000 code timescales in
Star-Smasher , relaxed in isolation to both test the stability of themodels and minimize the initial spurious noise, where a codetimescale is of order unity to the free-fall time of the planets, t free − fall ∼ (cid:16) ρ ⊙ ρ planet (cid:17) / t code . The planet models are very stableat the end of the relaxation; the radial acceleration on the par-ticles is near zero throughout the profile. The models for b and c for the Kepler-36 progenitor calculations have 12114and 8190 core particles and 87840 and 91764 gas particles,respectively.We note that much of the previous literature exploring themass-radius relationship of sub-Neptunes perform calcula-tions with both a homogeneous and a differentiated equationof state (Howe et al. 2014 and Seager et al. 2007). We em-phasize that here we are focused on differentiated cores. Ho-mogeneous cores would be denser and smaller, which wouldimpact the results of collisions simulations.2.4. Analysis of Hydro Calculations
For each output snapshot from the hydrodynamic calcula-tions, we build the planets starting from the dense iron core.For each particle, in ascending order of distance from thecenter of mass of the planet, we calculate a Jacobi Constant, C J = x + y a + a (cid:18) k r + k r (cid:19) − v x + v y + v z n a , (7)where n is the orbital frequency, k = M ∗ / ( m p + M ∗ ) and µ = m p / ( m p + M ∗ ) are the mass fractions of the star andplanet, where k + k = 1, r and r are the distances from theparticle to the host star and planet, and the position and veloc-ity vectors are defined in the corotating frame of the planetand the star, with x + y + z ≡
1. We calculate the JacobiConstant at L , assuming a circular orbit (Murray & Dermott1999), C L ≃ + / µ / − µ , (8)and assign the particle to the planet if C J > C L , updatingthe planet’s mass, position, and velocity. For each particleinitially assigned to both planets, we assign them to the planetwith lower relative orbital energy with respect to the particle, E = m (cid:18) v − Gm p r (cid:19) , (9)where m p is the mass of the planet, m is the mass of the par-ticle, and r and v are the scalar distance and relative velocitybetween the particle and the planet.We categorize the collision outcomes by examining the rel-ative orbits of the two planets, ignoring the host star, wherethe semi-major axis, a = − G ( m + m )2 E orb , (10)and the eccentricity, e = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) v G ( m + m ) − | r | (cid:19) r − r · v G ( m + m ) v (cid:12)(cid:12)(cid:12)(cid:12) , (11) Figure 2.
Radial profiles at the end of 2000 code timescales for an isolated model of a Kepler-36b progenitor, where we enable a relaxationforce (see Hwang et al. 2015 for details) for the first 100 code timescales. In each plot we mark the transitions from core to mantle and frommantle to gas envelope (vertical dashed lines). Density (top left) and internal energy (top right) profiles for the input profile (blue), generatedusing
MESA and the equations of state from Seager et al. (2007), the initial values assigned to each particle (black), and the values after 2000code timescales (red). Particle composition, x , profile (bottom left) after 2000 code timescales, where x represents the mass fraction of theparticle belonging to the heavier composition near the interface. Equilibrium profile (bottom right) after 2000 code timescales, showing theradial acceleration, in code units , from the hydrostatic force (red), the gravitational force (green), and the total force (black). These figuresshow how we assigned the particle composition and redistributed internal energy based on the smoothed densities. The model relaxes into avery stable hydrostatic equilibrium, stable for at least the timescales used for the dynamical calculations. where E orb = 12 µ v − Gm m | r | (12)is the relative orbital energy, µ is the reduced mass, v = v − v , and r = r − r . We categorize the outcome as a mergerif the relative periapsis of the two planets is less than twicethe sum of the core radii (to account for the observed tidaldeformation of the cores), as a bound planet-pair if the rela-tive orbital energy results in an apoapsis less than the mutualHill radius of the planets, and a scattering if the two planetsleave their mutual Hill spheres, where the mutual Hill radius, R H = a (cid:18) µ M ∗ (cid:19) / . (13) Figure 4 shows the orbital evolution of two collisions afterthe initial contact, one resulting in a scattering and the othera merger. COLLISIONAL OUTCOMESTables 4-6 summarize the results of our collision calcula-tions, showing the changes in mass and energy lost in thecollision as a function of the N-body collision parameters.We find that the distances of closest approach and relativecollision energies are, in general, larger than the values fromthe N-body calculations, due to resolving the colliding at-mospheres. The calculations have adequate energy conser-vation, with the maximum fractional change in total energy ∆ E tot / E tot , i < × − . Figure 5 shows the collision out-comes as a function of the distance of closest approach andcollision energy, as calculated in Mercury 6.2 , and in gen-
Figure 3.
Like Fig. 2 but for an isolated model of a Kepler-36c progenitor. eral, scatterings occur more often at larger distances of clos-est approach and collision energies, and planet-planet cap-tures (bound planet-pairs or mergers) occur at smaller dis-tances of closest approach and energies. The same initial or-bits are used for the two sets of calculations at a given massratio but different core mass fractions, and the results are verysimilar, with the only categorically different outcome occur-ing in the q = 1 / m c / m = 0 .
85 calculations results instead in a scattering in the m c / m = 0 .
95 calculations due to less energy dissipated in thelatter collision. In total we find 62 scatterings, 39 mergers,and 1 potential bound-planet-pair.Tables 7-9 summarize the orbits and planet properties ofthe remnant planets, where the angular momentum deficitis used to quantify the stability of the systems, discussed inmore detail in §3.1, and the densities are estimated using thesame method presented in
Paper 1 , where we match the to-tal mass and core mass fraction of the remnant planets to agrid of
MESA models. The initial gas mass fractions and den-sities are listed on the first row for each set of calculations.We examine each set of outcomes in more detail and discussthe orbits and planet structures of the remnants. In §4 weuse these calculations to develop prescriptions for predicting the outcomes and modeling the changes in mass during thesecollisions. 3.1.
Scatterings
To quantify the effect the collisions have on the orbits, wecalculate the angular momentum deficit, following Laskar(1997), C = n X k m k p GM ∗ a k (cid:18) − q − e k cos( i k ) (cid:19) . (14)Enforcing conservation of energy and angular momentum,we calculate the angular momentum deficit as a function ofthe period ratio, and for each set of initial and post-collisionorbits, the minimum angular momentum deficit, C min , where C min ≤ Table 4.
Results of Kepler-36 Progenitor SPH Calculations
N-body SPHRun d min E c d min E c ∆ m ∆ m ∆ E orb ǫ acc outcome1 0 . − .
019 0 .
641 0 . − .
067 0 .
009 0 . . × − bound .
628 0 .
023 0 .
606 0 . − . − .
082 0 . . × − merged . − .
054 0 . − . − . − .
030 0 . . × − merged . − .
023 0 .
668 0 . − .
067 0 .
003 0 . . × − merged . − .
071 0 . − . − . − .
293 0 . . × − merged .
522 0 .
014 0 .
548 0 . − . − .
704 0 . . × − merged . − .
010 0 .
553 0 . − . − .
629 0 . . × − merged . − .
010 0 .
610 0 . − . − .
503 0 . . × − merged . − .
041 0 . − . − . − .
077 0 . . × − merged
10 0 .
503 0 .
004 0 .
542 0 . − . − .
047 0 . . × − merged
11 0 . − .
043 0 . − . − . − .
216 0 . . × − merged
12 0 . − .
025 0 . − . − . − .
041 0 . . × − merged
13 0 .
543 0 .
001 0 .
539 0 . − . − .
742 0 . . × − merged
14 0 . − .
036 0 . − . − . − .
006 0 . . × − merged
15 0 . − .
039 0 . − . − . − .
043 0 . . × − merged
16 0 .
563 0 .
035 0 .
596 0 . − .
104 0 .
002 0 . . × − unstable
17 0 .
623 0 .
012 0 .
671 0 . − . − .
001 0 . . × − unstable
18 0 .
664 0 .
019 0 .
682 0 . − . − .
004 0 . . × − unstable
19 0 .
587 0 .
013 0 .
691 0 . − . − .
004 0 . . × − unstable
20 0 .
547 0 .
013 0 .
664 0 . − . − .
007 0 . . × − unstable
21 0 .
652 0 .
039 0 .
737 0 . − . − .
002 0 . . × − unstable
22 0 .
666 0 .
033 0 .
771 0 . − . − .
002 0 . . × − unstable
23 0 .
807 0 .
021 0 .
771 0 . − . − .
002 0 . . × − unstable
24 0 .
570 0 .
050 0 .
667 0 . − .
095 0 .
002 0 . . × − unstable
25 0 .
589 0 .
030 0 .
736 0 . − . − .
004 0 . . × − unstable
26 0 .
671 0 .
006 0 .
737 0 . − . − .
003 0 . . × − unstable
27 0 . − .
005 0 .
751 0 . − . − .
003 0 . . × − unstable
28 0 .
656 0 .
011 0 .
564 0 . − . − .
013 0 . . × − unstable
29 0 . − .
005 0 .
693 0 . − .
043 0 .
000 0 . . × − unstable
30 0 .
765 0 .
023 0 .
754 0 . − . − .
002 0 . . × − unstable
31 0 .
763 0 .
028 0 .
776 0 . − . − .
002 0 . . × − unstable
32 0 .
646 0 .
020 0 .
828 0 . − . − .
001 0 . . × − unstable
33 0 .
704 0 .
053 0 .
740 0 . − . − .
003 0 . . × − unstableRun designates the run number, d min and E c are the distance of closest approach and energy of eachcollision from the N-body and SPH calculations in units of the combined physical radii, R + R , andbinding energy, E b , described in (1), ∆ m and ∆ m are the changes in mass for the inner and outer planetin Earth masses, ∆ E orb is the change in relative planet-planet orbital energy during the collision in unitsof the binding energy, ǫ acc is the fractional change in the total energy, and outcome designates the resultof the collision. Figure 4.
Relative planet-planet orbits (top) and final snapshots (bottom) of collisions resulting in a scattering (left) and a merger (right). Thesemi-major axes (solid blue line), and periastron and apoastron (dotted blue lines) are calculated using the relative separation (dashed blackline) and velocity vectors of the two planets. The calculation is categorized as a merger if the periapsis is less than twice the sum of the coreradii (dotted red line), as a potential planet-planet binary if the apoapsis does not exceed the mutual Hill radius between the two planets, and asa scattering if the planets leave their mutual Hill sphere. The snapshots show the logarithm of column density plots in the orbital plane, centeredon the host star (left) and the planet-planet merger (right). The axes are scaled to R ⊙ , and the column density has units of M ⊙ R − ⊙ Table 5.
Results of q = 1 SPH Calculations N-body SPHRun d min E c d min E c ∆ m ∆ m ∆ E orb ǫ acc outcome m c / m = 0 .
851 0 .
394 0 .
086 0 .
408 0 . − . − .
167 0 . . × − merged . − .
031 0 . − . − . − .
100 0 . . × − merged . − .
047 0 . − . − . − .
011 0 . . × − merged . − .
010 0 .
451 0 . − . − .
021 0 . . × − unstable .
737 0 .
050 0 .
680 0 . − . − .
001 0 . . × − unstable .
649 0 .
024 0 .
608 0 . − . − .
001 0 . . × − unstable . − .
023 0 . − . − . − .
000 0 . . × − unstable .
451 0 .
078 0 .
470 0 . − . − .
024 0 . . × − unstable .
396 0 .
065 0 .
384 0 . − . − .
063 0 . . × − unstable
10 0 .
534 0 .
106 0 .
561 0 . − . − .
003 0 . . × − unstable
11 0 .
654 0 .
105 0 .
742 0 . − . − .
000 0 . . × − unstable
12 0 .
558 0 .
067 0 .
554 0 . − . − .
003 0 . . × − unstable
13 0 .
804 0 .
042 0 .
795 0 . − . − .
000 0 . . × − unstable
14 0 .
481 0 .
027 0 .
538 0 . − . − .
004 0 . . × − unstable
15 0 .
502 0 .
063 0 .
531 0 . − . − .
003 0 . < − unstable
16 0 . − .
039 0 . − . − . − .
000 0 . . × − unstable
17 0 .
586 0 .
107 0 .
571 0 . − . − .
036 0 . . × − unstable
18 0 .
523 0 .
050 0 .
564 0 . − . − .
002 0 . . × − unstable
19 0 .
493 0 .
050 0 .
474 0 . − . − .
016 0 . . × − unstablem c / m = 0 .
951 0 .
394 0 .
086 0 .
578 0 . − . − .
036 0 . . × − merged . − .
031 0 . − . − . − .
152 0 . . × − merged . − .
010 0 .
640 0 . − . − .
005 0 . . × − unstable .
737 0 .
050 0 .
999 0 . − . − .
000 0 . . × − unstable .
649 0 .
024 0 .
888 0 . − . − .
000 0 . . × − unstable . − .
023 1 . − . − .
000 0 .
000 0 . . × − unstable .
451 0 .
078 0 .
690 0 . − . − .
006 0 . . × − unstable .
396 0 .
065 0 .
551 0 . − . − .
031 0 . . × − unstable .
534 0 .
106 0 .
816 0 . − . − .
000 0 . . × − unstable
10 0 .
654 0 .
105 1 .
090 0 . − . − .
000 0 . . × − unstable
11 0 .
558 0 .
067 0 .
803 0 . − . − .
000 0 . . × − unstable
12 0 .
804 0 .
042 1 .
168 0 . − .
000 0 .
000 0 . . × − unstable
13 0 .
481 0 .
027 0 .
779 0 . − . − .
000 0 . . × − unstable
14 0 .
502 0 .
063 0 .
767 0 . − . − .
001 0 . . × − unstable
15 0 . − .
039 0 . − . − .
000 0 .
000 0 . . × − unstable
16 0 .
586 0 .
107 1 .
090 0 . − . − .
000 0 . . × − unstable
17 0 .
523 0 .
050 0 .
823 0 . − . − .
000 0 . . × − unstable
18 0 .
493 0 .
050 0 .
693 0 . − . − .
003 0 . . × − unstable
19 0 . − .
047 1 . − . − .
000 0 .
000 0 . . × − unstableRun designates the run number, d min and E c are the distance of closest approach and energy of each collisionfrom the N-body and SPH calculations in units of the combined physical radii, R + R , and binding energy, E b ,described in (1), ∆ m and ∆ m are the changes in mass for the inner and outer planet in Earth masses, ∆ E orb isthe change in relative planet-planet orbital energy during the collision in units of the binding energy, ǫ acc is thefractional change in the total energy, and outcome designates the result of the collision. Table 6.
Results of q = 1 / N-body SPHRun d min E c d min E c ∆ m ∆ m ∆ E orb ǫ acc outcome m c / m = 0 .
851 0 . − .
018 0 . − . − . − .
016 0 . . × − merged .
445 0 .
003 0 .
530 0 . − . − .
264 0 . . × − merged . − .
043 0 . − . − . − .
172 0 . . × − merged . − .
004 0 . − . − . − .
102 0 . . × − merged . − .
022 0 . − . − . − .
038 0 . . × − merged . − .
022 0 . − . − . − .
139 0 . . × − merged . − .
023 0 . − . − .
171 0 .
088 0 . . × − merged . − .
038 0 . − . − . − .
151 0 . . × − merged . − .
019 0 . − . − . − .
088 0 . . × − merged
10 0 .
382 0 .
023 0 .
440 0 . − . − .
286 0 . . × − merged
11 0 . − .
005 0 . − . − . − .
002 0 . . × − unstable
12 0 . − .
012 0 .
850 0 . − .
019 0 .
015 0 . . × − unstable
13 0 . − .
018 0 . − . − .
032 0 .
019 0 . . × − unstable
14 0 .
643 0 .
025 0 .
678 0 . − .
039 0 .
013 0 . . × − unstable
15 0 .
608 0 .
039 0 .
631 0 . − .
057 0 .
014 0 . . × − unstable
16 0 .
668 0 .
005 0 .
668 0 . − .
036 0 .
013 0 . . × − unstablem c / m = 0 .
951 0 . − .
018 0 . − . − .
938 0 .
006 0 . . × − merged .
445 0 .
003 0 .
741 0 . − . − .
036 0 . . × − merged . − .
043 0 . − . − . − .
074 0 . . × − merged . − .
022 0 . − . − . − .
022 0 . . × − merged . − .
018 1 . − . − . − .
015 0 . . × − merged . − .
022 0 . − . − . − .
016 0 . . × − merged . − .
023 0 . − . − . − .
002 0 . . × − merged . − .
038 0 . − . − . − .
058 0 . . × − merged . − .
019 0 . − . − . − .
019 0 . . × − merged
10 0 . − .
004 0 . − . − . − .
036 0 . . × − merged
11 0 . − .
005 1 . − . − .
004 0 .
003 0 . . × − unstable
12 0 . − .
012 1 .
204 0 . − .
000 0 .
000 0 . . × − unstable
13 0 .
643 0 .
025 0 .
962 0 . − .
006 0 .
005 0 . . × − unstable
14 0 .
608 0 .
039 0 .
893 0 . − .
012 0 .
006 0 . . × − unstable
15 0 .
668 0 .
005 0 .
951 0 . − .
005 0 .
004 0 . . × − unstableRun designates the run number, d min and E c are the distance of closest approach and energy of each collisionfrom the N-body and SPH calculations in units of the combined physical radii, R + R , and binding energy, E b ,described in (1), ∆ m and ∆ m are the changes in mass for the inner and outer planet in Earth masses, ∆ E orb isthe change in relative planet-planet orbital energy during the collision in units of the binding energy, ǫ acc is thefractional change in the total energy, and outcome designates the result of the collision. Table 7.
Post-Collision Orbits and Planet Properties of Kepler-36 Progenitor Calculations
Orbit Properties Planet PropertiesRun a e a e P / P C C min m gas , m gas , ρ ρ ρ / ρ Initial Properties 0 .
233 0 .
708 1 .
12 1 .
29 1 .
151 0 .
121 0 .
028 0 .
125 0 .
042 1 .
055 0 .
003 0 .
003 0 .
024 0 .
086 1 .
875 1 .
376 0 . .
137 0 .
115 0 .
118 0 .
061 1 .
251 0 .
010 0 .
005 0 .
003 0 .
051 2 .
263 1 .
672 0 . .
120 0 .
045 0 .
127 0 .
081 1 .
083 0 .
011 0 .
010 0 .
015 0 .
065 2 .
041 1 .
571 0 . .
127 0 .
148 0 .
123 0 .
023 1 .
046 0 .
019 0 .
019 0 .
021 0 .
086 1 .
970 1 .
372 0 . .
138 0 .
109 0 .
119 0 .
097 1 .
000 0 .
018 0 .
013 0 .
008 0 .
049 2 .
216 1 .
744 0 . .
104 0 .
199 0 .
140 0 .
114 1 .
000 0 .
036 0 .
010 0 .
008 0 .
036 2 .
484 2 .
182 0 . .
141 0 .
143 0 .
116 0 .
069 1 .
000 0 .
014 0 .
006 0 .
002 0 .
033 2 .
897 2 .
270 0 . .
116 0 .
193 0 .
132 0 .
059 1 .
000 0 .
022 0 .
018 0 .
004 0 .
031 2 .
681 2 .
355 0 . .
132 0 .
115 0 .
119 0 .
022 1 .
167 0 .
007 0 .
004 0 .
008 0 .
054 2 .
290 1 .
645 0 . .
134 0 .
133 0 .
120 0 .
057 1 .
000 0 .
014 0 .
011 0 .
004 0 .
068 2 .
585 1 .
553 0 . .
118 0 .
135 0 .
127 0 .
053 1 .
122 0 .
014 0 .
012 0 .
012 0 .
050 2 .
053 1 .
639 0 . .
136 0 .
121 0 .
118 0 .
056 1 .
230 0 .
012 0 .
007 0 .
007 0 .
056 2 .
327 1 .
635 0 . .
124 0 .
134 0 .
126 0 .
088 1 .
000 0 .
017 0 .
017 0 .
002 0 .
023 3 .
274 2 .
421 0 . .
167 0 .
249 0 .
109 0 .
167 1 .
000 0 .
099 0 .
026 0 .
040 0 .
088 1 .
633 1 .
340 0 . .
099 0 .
265 0 .
149 0 .
171 1 .
000 0 .
099 0 .
032 0 .
034 0 .
079 1 .
729 1 .
464 0 . .
120 0 .
008 0 .
126 0 .
004 1 .
071 0 . − .
001 0 .
013 0 .
081 2 .
451 1 .
451 0 . .
119 0 .
057 0 .
126 0 .
006 1 .
092 0 .
003 0 .
001 0 .
026 0 .
087 1 .
843 1 .
357 0 . .
121 0 .
036 0 .
125 0 .
030 1 .
047 0 .
002 0 .
002 0 .
038 0 .
088 1 .
682 1 .
332 0 . .
128 0 .
017 0 .
121 0 .
045 1 .
093 0 .
003 0 .
002 0 .
036 0 .
088 1 .
706 1 .
331 0 . .
122 0 .
034 0 .
124 0 .
004 1 .
025 0 .
001 0 .
001 0 .
010 0 .
080 2 .
957 1 .
464 0 . .
121 0 .
040 0 .
125 0 .
042 1 .
057 0 .
004 0 .
003 0 .
042 0 .
089 1 .
554 1 .
321 0 . .
121 0 .
055 0 .
125 0 .
027 1 .
045 0 .
004 0 .
003 0 .
043 0 .
089 1 .
549 1 .
320 0 . .
132 0 .
083 0 .
119 0 .
027 1 .
157 0 .
007 0 .
003 0 .
043 0 .
089 1 .
545 1 .
320 0 . .
120 0 .
003 0 .
125 0 .
006 1 .
061 0 . − .
001 0 .
015 0 .
083 2 .
325 1 .
432 0 . .
134 0 .
081 0 .
118 0 .
043 1 .
202 0 .
008 0 .
002 0 .
040 0 .
088 1 .
634 1 .
330 0 . .
128 0 .
032 0 .
121 0 .
048 1 .
091 0 .
004 0 .
003 0 .
041 0 .
088 1 .
603 1 .
327 0 . .
121 0 .
041 0 .
125 0 .
036 1 .
049 0 .
003 0 .
003 0 .
043 0 .
089 1 .
546 1 .
324 0 . .
130 0 .
039 0 .
120 0 .
024 1 .
127 0 . − .
000 0 .
008 0 .
079 3 .
680 1 .
472 0 . .
122 0 .
027 0 .
124 0 .
030 1 .
034 0 .
002 0 .
002 0 .
029 0 .
087 1 .
795 1 .
353 0 . .
135 0 .
086 0 .
117 0 .
058 1 .
236 0 .
011 0 .
003 0 .
043 0 .
089 1 .
547 1 .
320 0 . .
130 0 .
076 0 .
120 0 .
030 1 .
129 0 .
006 0 .
004 0 .
043 0 .
089 1 .
547 1 .
319 0 . .
120 0 .
044 0 .
126 0 .
048 1 .
084 0 .
005 0 .
004 0 .
043 0 .
089 1 .
544 1 .
315 0 . .
135 0 .
092 0 .
118 0 .
048 1 .
232 0 .
011 0 .
003 0 .
042 0 .
089 1 .
580 1 .
324 0 . Run designates the run number, a , e , m gas , and ρ are the semi-major axis (in AU), eccentricity, gas mass (in Earthmasses), and density (in CGS) of the inner out and outer planets, P / P is the period ratio, C and C min are the angularmomentum deficit and minimum possible angular momentum deficit, normalized as described in §3.1, and ρ / ρ is thedensity ratio. The initial gas mass fractions and densities are listed on the first row of each set of calculations. Table 8.
Post-Collision Orbits and Planet Properties of q = 1 Calculations Orbit Properties Planet PropertiesRun a e a e P / P C C min m gas , m gas , ρ ρ ρ / ρ m c / m = 0 .
85 Initial Properties 0 .
600 0 .
600 0 .
509 0 .
509 1 .
001 0 .
102 0 .
060 0 .
118 0 .
148 1 .
000 0 .
016 0 .
011 0 .
079 0 .
067 0 .
829 0 .
856 1 . .
117 0 .
145 0 .
102 0 .
048 1 .
000 0 .
006 0 .
005 0 .
006 0 .
009 1 .
705 1 .
962 1 . .
127 0 .
109 0 .
096 0 .
194 1 .
000 0 .
032 0 .
013 0 .
139 0 .
139 0 .
503 0 .
504 1 . .
112 0 .
033 0 .
105 0 .
078 1 .
107 0 .
005 0 .
004 0 .
136 0 .
135 0 .
499 0 .
498 0 . .
113 0 .
042 0 .
105 0 .
062 1 .
110 0 .
004 0 .
002 0 .
144 0 .
144 0 .
508 0 .
508 1 . .
104 0 .
095 0 .
114 0 .
022 1 .
141 0 .
006 0 .
004 0 .
144 0 .
143 0 .
508 0 .
508 1 . .
106 0 .
066 0 .
112 0 .
082 1 .
078 0 .
007 0 .
007 0 .
144 0 .
144 0 .
509 0 .
508 1 . .
113 0 .
107 0 .
105 0 .
053 1 .
117 0 .
009 0 .
008 0 .
135 0 .
134 0 .
498 0 .
496 0 . .
111 0 .
041 0 .
106 0 .
042 1 .
075 0 .
002 0 .
002 0 .
117 0 .
116 0 .
474 0 .
473 0 . .
105 0 .
067 0 .
112 0 .
092 1 .
103 0 .
009 0 .
007 0 .
143 0 .
143 0 .
508 0 .
508 1 . .
106 0 .
096 0 .
112 0 .
054 1 .
083 0 .
008 0 .
007 0 .
144 0 .
144 0 .
509 0 .
508 1 . .
101 0 .
127 0 .
118 0 .
076 1 .
271 0 .
014 0 .
008 0 .
143 0 .
143 0 .
508 0 .
508 0 . .
105 0 .
040 0 .
113 0 .
091 1 .
122 0 .
007 0 .
005 0 .
144 0 .
144 0 .
509 0 .
508 1 . .
117 0 .
103 0 .
101 0 .
059 1 .
251 0 .
009 0 .
004 0 .
143 0 .
142 0 .
507 0 .
507 0 . .
102 0 .
017 0 .
117 0 .
144 1 .
232 0 .
014 0 .
010 0 .
143 0 .
142 0 .
507 0 .
507 1 . .
111 0 .
092 0 .
106 0 .
042 1 .
065 0 .
007 0 .
006 0 .
144 0 .
144 0 .
508 0 .
508 1 . .
106 0 .
082 0 .
113 0 .
082 1 .
099 0 .
009 0 .
008 0 .
132 0 .
128 0 .
493 0 .
488 0 . .
107 0 .
054 0 .
111 0 .
104 1 .
058 0 .
009 0 .
009 0 .
143 0 .
143 0 .
508 0 .
508 1 . .
110 0 .
056 0 .
107 0 .
057 1 .
049 0 .
004 0 .
004 0 .
137 0 .
137 0 .
501 0 .
501 1 . m c / m = 0 .
95 Initial Properties 0 .
200 0 .
200 0 .
977 0 .
977 1 .
001 0 .
114 0 .
034 0 .
104 0 .
136 1 .
000 0 .
013 0 .
010 0 .
025 0 .
028 1 .
648 1 .
605 0 . .
106 0 .
064 0 .
112 0 .
105 1 .
000 0 .
004 0 .
004 0 .
003 0 .
002 2 .
119 2 .
225 1 . .
112 0 .
039 0 .
106 0 .
072 1 .
084 0 .
004 0 .
004 0 .
045 0 .
044 1 .
305 1 .
314 1 . .
112 0 .
041 0 .
105 0 .
062 1 .
107 0 .
004 0 .
003 0 .
047 0 .
047 1 .
198 1 .
208 1 . .
104 0 .
098 0 .
114 0 .
022 1 .
152 0 .
007 0 .
004 0 .
047 0 .
047 1 .
200 1 .
209 1 . .
106 0 .
069 0 .
111 0 .
079 1 .
069 0 .
007 0 .
007 0 .
047 0 .
047 1 .
198 1 .
208 1 . .
114 0 .
113 0 .
104 0 .
046 1 .
142 0 .
010 0 .
008 0 .
044 0 .
044 1 .
322 1 .
345 1 . .
111 0 .
040 0 .
106 0 .
042 1 .
071 0 .
002 0 .
002 0 .
032 0 .
031 1 .
568 1 .
579 1 . .
105 0 .
066 0 .
112 0 .
091 1 .
104 0 .
008 0 .
007 0 .
046 0 .
046 1 .
210 1 .
219 1 . .
106 0 .
097 0 .
112 0 .
053 1 .
086 0 .
008 0 .
007 0 .
047 0 .
047 1 .
198 1 .
208 1 . .
101 0 .
128 0 .
118 0 .
076 1 .
277 0 .
014 0 .
008 0 .
047 0 .
046 1 .
206 1 .
216 1 . .
105 0 .
041 0 .
113 0 .
092 1 .
123 0 .
007 0 .
005 0 .
047 0 .
047 1 .
198 1 .
208 1 . .
118 0 .
104 0 .
101 0 .
061 1 .
260 0 .
010 0 .
004 0 .
046 0 .
046 1 .
211 1 .
219 1 . .
101 0 .
015 0 .
117 0 .
147 1 .
245 0 .
015 0 .
010 0 .
046 0 .
046 1 .
213 1 .
222 1 . .
112 0 .
073 0 .
105 0 .
072 1 .
103 0 .
007 0 .
006 0 .
047 0 .
047 1 .
198 1 .
208 1 . .
106 0 .
097 0 .
112 0 .
053 1 .
086 0 .
008 0 .
007 0 .
047 0 .
047 1 .
198 1 .
208 1 . .
107 0 .
051 0 .
111 0 .
107 1 .
063 0 .
009 0 .
009 0 .
047 0 .
046 1 .
206 1 .
215 1 . .
111 0 .
064 0 .
106 0 .
052 1 .
074 0 .
005 0 .
004 0 .
045 0 .
045 1 .
271 1 .
281 1 . .
111 0 .
051 0 .
106 0 .
077 1 .
075 0 .
006 0 .
005 0 .
047 0 .
047 1 .
198 1 .
208 1 . Run designates the run number, a , e , m gas , and ρ are the semi-major axis (in AU), eccentricity, gas mass (in Earth masses), anddensity (in CGS) of the inner out and outer planets, P / P is the period ratio, C and C min are the angular momentum deficit andminimum possible angular momentum deficit, normalized as described in §3.1, and ρ / ρ is the density ratio. The initial gasmass fractions and densities are listed on the first row of each set of calculations. Table 9.
Post-Collision Orbits and Planet Properties of q = 1 / Orbit Properties Planet PropertiesRun a e a e P / P C C min m gas , m gas , ρ ρ ρ / ρ m c / m = 0 .
85 Initial Properties 0 .
200 0 .
600 0 .
509 1 .
204 2 .
371 0 .
153 0 .
288 0 .
106 0 .
095 1 .
000 0 .
046 0 .
022 0 .
006 0 .
140 1 .
849 1 .
224 0 . .
117 0 .
211 0 .
114 0 .
112 1 .
000 0 .
039 0 .
039 0 .
005 0 .
108 1 .
893 1 .
252 0 . .
142 0 .
246 0 .
108 0 .
009 1 .
509 0 .
019 0 .
006 0 .
003 0 .
112 1 .
821 1 .
254 0 . .
109 0 .
152 0 .
115 0 .
052 1 .
083 0 .
015 0 .
014 0 .
012 0 .
127 1 .
667 1 .
228 0 . .
110 0 .
068 0 .
115 0 .
088 1 .
075 0 .
018 0 .
017 0 .
021 0 .
136 1 .
307 1 .
226 0 . .
148 0 .
234 0 .
107 0 .
141 1 .
000 0 .
055 0 .
038 0 .
002 0 .
119 2 .
101 1 .
237 0 . .
125 0 .
139 0 .
111 0 .
113 1 .
200 0 .
032 0 .
029 0 .
006 0 .
148 1 .
813 1 .
223 0 . .
133 0 .
196 0 .
109 0 .
074 1 .
000 0 .
023 0 .
016 0 .
008 0 .
124 2 .
050 1 .
226 0 . .
133 0 .
116 0 .
110 0 .
147 1 .
000 0 .
047 0 .
039 0 .
008 0 .
132 2 .
122 1 .
224 0 . .
165 0 .
339 0 .
104 0 .
063 2 .
010 0 .
040 0 .
009 0 .
001 0 .
109 2 .
492 1 .
246 0 . .
095 0 .
339 0 .
123 0 .
065 1 .
469 0 .
082 0 .
056 0 .
138 0 .
146 0 .
502 1 .
215 2 . .
111 0 .
103 0 .
115 0 .
113 1 .
043 0 .
033 0 .
033 0 .
136 0 .
148 0 .
500 1 .
212 2 . .
101 0 .
035 0 .
119 0 .
123 1 .
283 0 .
032 0 .
022 0 .
131 0 .
149 0 .
491 1 .
211 2 . .
127 0 .
088 0 .
110 0 .
062 1 .
237 0 .
013 0 .
006 0 .
127 0 .
148 0 .
487 1 .
213 2 . .
123 0 .
111 0 .
111 0 .
006 1 .
170 0 .
009 0 .
004 0 .
119 0 .
148 0 .
476 1 .
212 2 . .
112 0 .
127 0 .
114 0 .
084 1 .
035 0 .
025 0 .
025 0 .
129 0 .
148 0 .
489 1 .
212 2 . m c / m = 0 .
95 Initial Properties 0 .
600 1 .
800 1 .
204 2 .
314 2 .
371 0 .
143 0 .
244 0 .
107 0 .
075 1 .
536 0 .
035 0 .
018 0 .
010 0 .
046 1 .
854 2 .
729 1 . .
136 0 .
262 0 .
108 0 .
054 1 .
415 0 .
028 0 .
018 0 .
003 0 .
042 2 .
064 2 .
922 1 . .
150 0 .
288 0 .
106 0 .
004 1 .
669 0 .
027 0 .
007 0 .
002 0 .
040 2 .
598 3 .
000 1 . .
104 0 .
140 0 .
118 0 .
093 1 .
206 0 .
026 0 .
022 0 .
018 0 .
044 1 .
551 2 .
805 1 . .
083 0 .
277 0 .
132 0 .
210 1 .
000 0 .
141 0 .
053 0 .
038 0 .
045 1 .
484 2 .
769 1 . .
121 0 .
174 0 .
113 0 .
100 1 .
113 0 .
032 0 .
031 0 .
012 0 .
045 1 .
610 2 .
787 1 . .
089 0 .
352 0 .
127 0 .
129 1 .
000 0 .
111 0 .
061 0 .
038 0 .
047 1 .
488 2 .
615 1 . .
115 0 .
095 0 .
115 0 .
083 1 .
010 0 .
017 0 .
017 0 .
008 0 .
042 2 .
195 2 .
954 1 . .
127 0 .
105 0 .
112 0 .
140 1 .
000 0 .
043 0 .
040 0 .
005 0 .
041 1 .
956 2 .
987 1 . .
110 0 .
175 0 .
116 0 .
052 1 .
086 0 .
017 0 .
017 0 .
012 0 .
044 1 .
637 2 .
842 1 . .
118 0 .
142 0 .
112 0 .
104 1 .
077 0 .
036 0 .
035 0 .
045 0 .
048 1 .
289 2 .
545 1 . .
106 0 .
138 0 .
116 0 .
106 1 .
146 0 .
036 0 .
033 0 .
047 0 .
047 1 .
207 2 .
585 2 . .
128 0 .
094 0 .
110 0 .
064 1 .
253 0 .
014 0 .
006 0 .
044 0 .
048 1 .
330 2 .
524 1 . .
124 0 .
116 0 .
111 0 .
008 1 .
187 0 .
010 0 .
005 0 .
041 0 .
048 1 .
432 2 .
506 1 . .
112 0 .
131 0 .
114 0 .
083 1 .
034 0 .
026 0 .
025 0 .
044 0 .
048 1 .
319 2 .
529 1 . Run designates the run number, a , e , m gas , and ρ are the semi-major axis (in AU), eccentricity, gas mass (in Earth masses), anddensity (in CGS) of the inner out and outer planets, P / P is the period ratio, C and C min are the angular momentum deficit andminimum possible angular momentum deficit, normalized as described in §3.1, and ρ / ρ is the density ratio. The initial gasmass fractions and densities are listed on the first row of each set of calculations. Figure 5.
Results of collisions as a function of the degree of contact, η = d min / ( R + R ) , and the collision energy in units of the binding energy,as described in (1), from the N-body calculations. In the Kepler-36 progenitor calculations (top left), we find 14 mergers ( × ), 1 bound planet-pair ( ⋆ ), and 18 scatterings (circle), in the q = 1; m c = 0 .
85 (middle left) calculations we find 3 mergers and 16 scatterings, in the q = 1; m c = 0 . q = 1 / m c = 0 .
85 (bottom left) calculations we find 9 mergers and 7scatterings, and in the q = 1 / m c = 0 .
95 (bottom right) calculations we find 10 mergers and 5 scatterings. Of the 63 collisions resulting in ascattering, 28 exhibit a flipped orbit (red edge-color), where the initially outer planet becomes the inner planet after the collision. m ∼ . M E ),tend to lose some fraction of their gas envelopes, and only thelargest planets ( m = 12 . M E ) gain mass during the collision.We develop a model for predicting the change in mass duringa collision in §4. Using the same method as Paper 1 , wefind that the density ratios tend to become more extreme forsystems with mass ratios, q = 1, with the lower-mass planetslosing more mass than the higher-mass companions. Theseresults suggest that two very tightly-packed planets with alarge density ratio (e.g. Kepler-36; Carter et al. 2012) maybe evidence of a previous planet-planet collision (Liu et al.2016, Inamdar & Schlichting 2016).3.2. Mergers
Collisions that eventually result in the merger of the twocores generally undergo multiple episodes of physical colli-sions at the periapsis of each planet-planet orbit, losing somerelative orbital energy at each periapsis passage. While ourtreatment of the cores does not allow us to fully integratethrough the merger, we are able to follow the changes in massand orbital energy in each orbit until the cores physically col-lide. In previous versions of these calculations, where theequations of state of both the core and gas are approximatedsimply as polytropes, we found several distinct outcomes, in-cluding fragmentation of the less-massive core, where somemass is accreted onto the more massive core and the remain-der becomes bound, forming a planet-planet binary with amore extreme mass-ratio. Preferential shedding of the mantlemay lead to remnants where the less-massive planet surviveswith a much higher iron-core mass fraction, and the more-massive planet has an excess of mantle material. Follow-upstudies using equations of state that more accurately modeldirect core-core collisions are necessary to fully understandthe details of these outcomes.3.3.
Bound Planet-Pair
While most collisions resulting in the two planets becom-ing a bound pair have an orbit with a periapsis that resultsin a core-core collision, and a likely merger, one calcula-tion shows a potentially long-lived planet-planet binary. Fol-lowing Podsiadlowski et al. (2010), we define the inner bi-nary as the two bound planets and the outer binary as thebound planet-pair and the host star, and compare the maxi-mum apoapsis of the inner binary to the mutual Hill radius atthe periapsis of the outer binary, r , apo = a in (1 + e in ) (cid:18) m m + m (cid:19) , (15) R H , peri = a out (1 − e out ) (cid:18) µ M ∗ (cid:19) / , (16)where m > m , and a in , e in , a out , and e out are the semi-majoraxes and eccentricities of the inner and outer binary. We esti- mate the inner binary’s maximum apoapsis by enforcing themaximum distance of a planet from the center of mass of theinner binary be less than the Hill radius at the outer binary’speriapsis, r max , apo = a out (1 − e out ) (cid:18) µ M ∗ (cid:19) / (cid:18) m m + m (cid:19) . (17)Figure 7 shows the orbital evolution of the inner and outerbinaries of this calculation, and we see that the orbit of theplanet-planet binary has a periapsis large enough to preventthe cores from colliding and an apoapsis small enough thatthe planets remain inside the mutual Hill sphere. Determingthe long-term stability of this planet-planet binary is complexand requires including tides and decay of the inner binary’sorbits due to the gas envelope. RECIPES TO PREDICT COLLISIONAL OUTCOMESThe details of the collision are important in determiningboth the outcomes and the structures of the remnant planet(s).The maximum relative energy of a bound planet-planet pair(resulting in either a merger or a long-lived planet-planet bi-nary) can be calculated, as the maximum orbital separationmust be less than the mutual Hill Radius, E max = − Gm m a max . (18)Using (15) and (16), a max = a out (cid:18) − e out + e in (cid:19) (cid:18) m m + m (cid:19) (cid:18) µ M ∗ (cid:19) / , (19)where a out , e out , a in , and e in are the semi-major axes and ec-centricities of the outer (planets’ center of mass and host star)and inner (planet-planet) binaries, and m , m , and µ are theplanet masses and reduced mass after the collision, where m < m . In each planet-planet collision, some amount ofenergy, E d , is lost from the planets’ orbit, and we predict thatthe remnant will result in a scattering event, where both plan-ets survive, if E c − E d > E max , (20)and may otherwise become a capture, either a merger orbound planet-pair.4.1. Fitting Energy Dissipated and Change in Mass
To predict the outcomes of a generic collision, we examinethe energy dissipated and changes in mass during the firstperiastron passage of the two planets. Figures 8-10 showthe energy dissipated and changes in mass for each planetas a function of the distance of closest approach. We fit theenergy dissipated and change in mass of each planet witha power law as a function of distance of closest approach, f = Ad k , discarding collisions where we do not fully resolvethe first passage, or with a distance of closest approach closeenough to cause contact between the physical cores, as thisintroduces additional physics and requires a different fit. We8 Figure 6.
Solutions of period ratio, P / P , and angular momentum deficit, C , scaled by C = M E (GM ⊙ AU) / , before the collision (green)and after the collision (red), along with the coordinates of the first and last snapshots in the SPH calculation (black points). The solution for aHill-stable orbit is shown for comparison (solid black line) for the pre-collision masses of each calculation, and we see that, in general, systemsbecome more stable after the collision. Angular momentum deficits below zero are unphysical, and orbits with solutions where C <9
Solutions of period ratio, P / P , and angular momentum deficit, C , scaled by C = M E (GM ⊙ AU) / , before the collision (green)and after the collision (red), along with the coordinates of the first and last snapshots in the SPH calculation (black points). The solution for aHill-stable orbit is shown for comparison (solid black line) for the pre-collision masses of each calculation, and we see that, in general, systemsbecome more stable after the collision. Angular momentum deficits below zero are unphysical, and orbits with solutions where C <9 Figure 7.
Orbital evolution of a collison resulting in a potentialplanet-planet binary. The planet-star orbits (top), with the semi-major axes (solid line), and periastron and apoastron (dotted lines).The planet-planet orbit (bottom) after the collision shows the physi-cal separation (dashed black line), semi-major axes (blue), and peri-astron and apoastron (dotted lines) of the planet-planet binary. Thesum of the physical size of the cores (dotted red line) and the maxi-mum periapsis (dotted black line) from (17) for comparison. use only scattering collisions to fit the changes in mass, as weare interested in the masses of each planet after leaving theirmutual Hill sphere. Table 10 summarizes the best fit valuesfor each set of calculations.We find that higher mass ratios and core mass fractionsexhibit a steeper exponent in the power law for both the en-ergy dissipated and change in mass. The change in mass ofthe less-massive planet is predicted well by the power-lawfit, but we find that the change in mass of the more-massiveplanet in q = 1 calculations is not as well modeled by a sin-gle power-law and likely depends on additional factors, forexample, the angle of impact relative to the host star.4.2. Prescription
For each set of collisions, we develop models using theinitial energy, distance of closest approach, masses, and or-
Table 10.
Best Fits for Energy Dissipated andChange in Mass
Set
A kE d / E b = A E η k E Kepler − Progenitor . × − − . q = 1; m c = 0 .
85 1 . × − − . q = 1; m c = 0 .
95 2 . × − − . q = 1 / m c = 0 .
85 1 . × − − . q = 1 / m c = 0 .
95 2 . × − − . m lost , / m = A m , η k m , Kepler − Progenitor . × − − . q = 1; m c = 0 .
85 1 . × − − . q = 1; m c = 0 .
95 6 . × − − . q = 1 / m c = 0 .
85 5 . × − − . q = 1 / m c = 0 .
95 1 . × − − . m lost , / m = A m , η γ m , Kepler − Progenitor . × − − . q = 1; m c = 0 .
85 1 . × − − . q = 1; m c = 0 .
95 9 . × − − . q = 1 / m c = 0 . − . × − . q = 1 / m c = 0 . − . × − − . Set designates the set name, and A and k are dimen-sionless variables used in models predicting the energydissipated and changes in mass as a function of the dis-tance of closest approach. bits to predict the outcome and final masses and orbits of theremnant planet(s). Using (18) and (19), the critical initialcollision energy required for a scattering may be expressed, E c = E esc + E d , (21)where E esc = − Gm ( m + m )2 a out + e in − e out (cid:18) µ M ∗ (cid:19) − / , (22)and m , m , and E d may be estimated as a function of d min using the fits to discussed in the previous subsection.Figure 11 shows, for each set of calculations, the criticalinitial collision energy in excess of the energy required fora planet to leave the mutual Hill sphere (to normalize thedistance from the host star), as a function of the distance ofclosest approach, separating the predicted captures and scat-terings. We see that the potential planet-planet binary is veryclose to the critical energy separating captures and scatter-ings. The model accurately predicts 98 out of 102 outcomes,and we examine closely the outcomes that are misclassified.Of the four misclassified outcomes, two are mergers that oc-cur after initially leaving the mutual Hill sphere, and can be0 Figure 8.
Change of orbital energy of the planets as a function of the degree of contact, η = d min / ( R + R ), where the symbols designate theoutcomes as described in Figure 5. We show the fits, E d / E b = A E η k E , (dashed black lines) where the best-fit values are reported in Table 10. Figure 9.
Fractional change in mass after the first close encounter, of the inner (left) and outer (right) planet as a function of the degree ofcontact, η = d min / ( R + R ), for the Kepler-36 progenitor (top), q = 1; m c = 0 .
85 (middle), and q = 1; m c = 0 .
95 (bottom) calculations, where thesymbols designate the outcomes as described in Figure 5. We show the fits, m lost , i / m i = A m , i η k m , i , (dashed black lines) where the best-fit valuesare reported in Table 10. Figure 10.
Fractional change in mass after the first close encounter, of the inner (left) and outer (right) planet as a function of the degree ofcontact, η = d min / ( R + R ), for the q = 1 / m c = 0 .
85 (top), and q = 1 / m c = 0 .
95 (bottom) calculations. Symbols and fits are as in Fig. 9. attributed to a second, separate close encounter, and one is acollision very close to the critical collision energy, resultingin a potentially long-lived planet-planet binary. This modelmay be adapted for a generic collision between sub-Neptunesusing the fits from the calculations most similar to the targetsystem. Figure 12 shows the distribution of collisions fromour dynamical integrations discussed in §2.1 with the pre-dicted critical collision energy for separating grazing colli-sions resulting in mergers from scatterings. Table 11 sum-marizes the outcomes of grazing collisions and near misses,where 72% - 96% of grazing collisions result in a scattering,motivating an improvement on the sticky-sphere approxima-tion. Based on the predicted outcome, the final propertiesof the collision remnant may be estimated, specifically, theamount of gas retained by a merger remnant, and the energyand gas lost during a scattering.4.2.1.
Scatterings
We find that a majority of grazing collisions where thecores do not physically come into contact result in a scatter-ing with both planets leaving their mutual Hill sphere. Dur-ing the scattering, both planets lose mass and orbital energy, which is deposited into unbinding gas and also converted intointernal energy.The change in mass and orbital energy during a scatteringcollision can significantly affect the dynamical evolution ofthe system. The prescription for treating scattering collisionsis as follows, using as inputs the degree of contact, masses,mass ratio, and core mass fractions of the planets:1) Integrate until reaching the projected minimum sepa-ration.2) Use the fits presented in Table 10 to estimate the massfraction, m lost , i , and orbital energy, E d , lost from bothplanets. The best fit values may be chosen either fromthe set of collisions most similar to the two collidingplanets or using a linear interpolation.
3) Calculate the remnant masses for both planets m ′ i = m i (1 − m lost , i ( η )) . (23)For cases where the mass ratio of the two planets islarge ( q < / Figure 11.
The outcomes of each collision, designated as described in 5, as a function of the degree of contact, η = d min / ( R + R ) , and thecollision energy in units of the binding energy, as described by eq. (1), from the SPH calculations. The collision energy is normalized by theescape energy as described by eq. (22). We show the predicted critical collision energy separating scattering collisions from captures (dashedblack lines), and we see that the model accurately predicts the outcome in 98 out of 102 collisions, where 2 of the misclassified outcomes arethe result of a second, separate close encounter, and 1 results in the only bound-planet pair found in our set of calculations. Figure 12.
Distribution of collisions characterized as in Figure 1, with the predicted critical initial collision energy described by eq. (21) (dashedblack line) separating grazing collisions resulting in mergers (blue) and scatterings (green). Table 11 summarizes the predicted outcomes, where72% to 96% of the grazing collisions result in a scattering, and are poorly modeled by the sticky-sphere approximation. Table 11.
Predicted Outcomes of Grazing Col-lisions
Set N grazing N scattering Kepler − progenitor
212 167 q = 1; m c = 0 .
85 333 320 q = 1; m c = 0 .
95 269 255 q = 1 / m c = 0 .
85 348 295 q = 1 / m c = 0 .
95 254 183
Set designates the set name, N grazing is the num-ber of grazing collisions (including near misses de-scribed in Table 3) that occur within 1000 years and N scattering is the number of grazing collisions thatresult in a scattering, where the planets leave theirmutual Hill radii without merging.
4) Decrease the magnitude of the relative velocity be-tween the two planets to account for the decrease inorbital energy, conserving energy (accounting for theenergy lost), v ′ v = s µ ′ (cid:20) Gd min v (cid:0) m m − m ′ m ′ (cid:1) + µ − E d ( η ) v (cid:21) , (24)where µ ′ and v ′ are the remnant reduced-mass andmagnitude of relative velocity.5) Numerically solve for the remnant velocities of eachplanet, conserving specific angular momentum, v ′ = | v ′ − v ′ | , (25) m r × v + m r × v m + m = m ′ r × v ′ + m ′ r × v ′ m ′ + m ′ , (26)where v ′ i = v ′ i ˆv i is the remnant velocity vector parallelto the initial velocity vector.6) (Optional) Adjust the radii of both planets to accountfor the loss of gas. This step assumes that subsequentcollisions occur after the planets have relaxed back intoequilibrium.Using this prescription should result in a material differ-ence in dynamical integrations that exhibit scattering colli-sions, generally driving the planets into more stable orbits,and in cases with a subsequent merger, less-massive rem-nants. 4.2.2. Mergers
While we do not integrate collisions resulting in a phys-ical collision between the cores through the merger due tothe computational cost, we find that the short term change inmass is largely determined by the planets’ core masses. 4 M E planets tend to lose a majority of their envelope in addition to some mantle, while 12 M E planets do not exhibit signif-icant changes in mass. Our method for treating for mergerevents is the same as from Paper 1 , ejecting the gas fromboth planets and using the sticky-sphere approximation as-suming most of the core mass ( ∼ − CONCLUSIONSWe conduct a detailed study on collisions between twoplanets in initially unstable orbits, varying the mass ratiosand core mass fractions of the planets to sample a range oftypical neighbors in Kepler Multis, and summarize our find-ings as follows:1) 102 collision calculations result in 62 scatterings, 39mergers, and 1 bound planet-pair.2) While every scattering remnant eventually resulted inan eventual crossing orbit during post-collision dy-namical integrations, we find that collisions tend tostabilize the system. Further collisions may eventuallyresult in a stable system with two surviving planets.3) The collisions that result in mergers tend to eject a ma-jority of the gas from the less-massive planet. Futurecalculations with a better treatment of the core are re-quired to fully resolve planet-planet mergers.4) One collision results in a potentially long-lived planet-planet binary with an eccentric orbit such that the pe-riapsis avoids collision of the planets’ cores and theapoapsis is small enough that the planets remain insidethe mutual Hill sphere.5) The outcome of a collision depends very sensitively onthe distance of closest approach and the relative energybetween the two planets at the collision; specifically,6 high relative energies and large distances of closest ap-proach lead to more scatterings and low relative ener-gies and low distances of closest approach are morelikely to result in a merger or bound planet-pair.6) After a collision, the density ratio of q = 1 planets tendto become more extreme, with the higher-mass coreretaining more of, or even accreting, the disrupted gas,and the lower-mass core losing a higher fraction of gas;we found a minimum density ratio of ρ /ρ = 0 . ρ /ρ = 1 . Future Work
Given our choice of equations of state to model the core,which were chosen to provide a stable boundary condition,we focused on studying grazing collisions and were not ableto integrate core-core impacts through the entire merger. Im-proving the algorithms used to resolve interfaces betweencomponents and improving our core equations of state to bet-ter models shocks and mixing, will allow the study of thedeeper impacts, for example the long term evolution of merg-ers.In this work we focused on performing many calculationsto better understand the outcomes of collisions after the first periastron passage and we do not resolve collisions past afew orbital periods of the innermost planet. Calculations thatintegrate scattering collisions for long timescales are impor-tant in fully understanding the behavior of the disrupted gas,particularly if the gas eventually falls back onto one of theplanets. A study of the long-term evolution of the gas willrequire additional physics such as a model to handle stellarwinds.We studied in detail five sets of 2-planet systems, madeup of two sub-Neptunes, where both planets have a tenuousgas envelope that dominates the volume, varying the massratios and gas mass fractions. However, there still remainsmuch work to be done to better understand the likely com-mon collisions in high-multiplicity, tightly-packed systems.
Performing similar calculations at lower planet-ages willincrease the size of the planet envelopes, resulting in col-lisions at larger distances of closest approach, and thechange in structure will likely lead to qualitatively dif-ferent outcomes. Increasing the semi-major axes of theplanets will lower the impact of the host star, and includ-ing terrestrial planets and gas giants will also expand thescope of this type of study.
The existence of long-lived planet-planet binaries is an in-teresting problem and the absence of an observation allowsus to place an upper limit on the occurence rate. A detailedtreatment of the stability of such a system could provide in-sight to the types of collisions that are able to create thesephenomena. 5.2.
Acknowledgements
This research was supported in part through the compu-tational resources and staff contributions provided for theQuest high performance computing facility at NorthwesternUniversity, which is jointly supported by the Office of theProvost, the Office for Research, and Northwestern Univer-sity Information Technology. FAR and JAH were supportedby NASA Grant NNX12AI86G. JAH was also supported byan NSF GK-12 Fellowship funded through NSF Award DGE-0948017 to Northwestern University. JCL was supportedby NSF grant number AST-1313091. JHS was supportedby NASA grants NNX16AK08G and NNX16AK32G. Wethank Joshua Fixelle for useful discussions while develop-ing the equations of state used in the SPH calculations andFrancesca Valsecchi for help with using
MESA to generatesub-Neptune envelopes. We thank the anonymous refereefor providing very thorough and helpful feedback leading tomajor improvements in this manuscript. This work used theSPLASH visualization software (Price 2007).7APPENDIX A. APPENDIXA.1.
Kepler-36 Progenitor Properties
In this section we show that an initially unstable system that undergoes a planet-planet collision can result in two planets withthe same angular momentum and orbital energy distribution as an observed and dynamically stable system, using Kepler-36as our nominal system. We calculate the orbital elements and planet masses of potential prognenitor systems to the currentlyobserved Kepler-36 system, assuming conservative mass transfer, m = m , o − ∆ m , (A1) m = m , o + ∆ m , (A2)where m and m are the initial planet masses, m , o and m , o are the observed planet masses, and ∆ m is the amount of masstransfer, conservation of angular momentum, h = X k =1 , µ k , o q G ( M ∗ + m k , o ) a k , o (1 − e k , o ) , (A3)where M ∗ = 1 . M ⊙ is the mass of the host star, a , o = 0 . a , o = 0 . e , o = 0 .
04 and e , o = 0 .
04 are the observed upper limits to the eccentricities, and µ , o and µ , o are the reduced masses (Carter et al.2012). We assume that the collision results in the loss of some orbital energy, E = − ǫ X k =1 , GM ∗ m k , o a k , o , (A4)where ǫ is the fraction of orbital energy conserved in the collision.A.1.1. Mass Transfer
We estimate the distribution of gas in the progenitor Kepler-36 system, assuming the currently observed distribution of gas is aresult of conservative mass transfer in a prior planet-planet collision. Carter et al. (2012) find the mass and radius measurementof b is consistent with a planet with little or no gas envelope and Howe et al. (2014) predict a core mass fraction of m c / M = 0 . c . We generate a series of models for c using MESA (Paxton et al. 2011; Paxton et al. 2013; Paxton et al. 2015), varying thecore mass fraction and using the core mass and core-density relationship from Howe et al. (2014) assuming a 32 .
5% iron and67 .
5% perovskite differentiated core. Figure 13 shows the density of our models as a function of the core mass fraction, where m , c / m , o = 0 .
882 is consistent with the observed density, which is reasonably close to the value reported by Howe et al. (2014),with differences potentially arising from our model including irradiation from the host star leading to a larger radius.We use the result from Lee & Chiang (2015) to estimate the initial mass of the gas envelopes, m g = m , g + m , g , (A5) m g = ∆ m (cid:18) + (cid:18) m , c m , c (cid:19) α (cid:19) , (A6)where m g is the gas mass observed on c , m , g and m , g are the gas mass of the progenitor planets, and α = 2 .
6, and estimate thatthe mass lost by b is ∆ m = 0 . M ⊕ . A.1.2.
Orbital Elements
We estimate the orbital elements of a dynamically unstable progenitor system that, after a collision, results in a dynamicallystable system, specifically a system resembling Kepler-36, assuming conservative mass transfer and conservation of angularmomentum. Because the progenitor system must allow for a crossing orbit, a (1 + e ) = a (1 − e ) , (A7)8 Figure 13.
Density as a function of core mass, using a series of
MESA models (blue circles) to generate a fit (solid green). We find thatthe observed density of ρ = 0 .
895 g cm − (solid black) corresponds to a fit core mass fraction of m c / M = 0 .
882 (dashed green), which agreesreasonably well with the core mass fraction prediction of m c / M = 0 .
861 from Howe et al. (2014). and we assume angular momentum is conserved, we can solve for the range of inner semi-major axes, a = h G (1 + e ) (cid:16) µ p ( M ∗ + m , o − ∆ m )(1 − e ) + µ p ( M ∗ + m , o + ∆ m )(2 − (1 + e ) T − / ) (cid:17) , (A8)where e and e are the eccentricities of b and c , and T is the period ratio. For each solution we find the fractional loss in orbitalenergy required to move from our proposed progenitor system to the observed Kepler-36 system, ǫ E = − M ∗ ( m + m T − / )2 E o a . (A9)We use the constraints that the progenitor system does not initially have a crossing orbit, f = a (1 − e ) − a (1 + e ) ≥ , (A10)and ∂ f ∂ e = a (cid:18) h e m h e m − (cid:19) , (A11)to determine the boundaries on e and e , as a function of both ∆ m and T , where h and h are the angular momentum of theplanets at the crossing orbit solution. Using ∆ m = 0 . M ⊕ from §A.1.1, T = 1 .
3, and maximizing conservation of energy, ǫ E = 0 . a = 0 .
111 AU, a = 0 .
132 AU, e ≥ .
11, and e ≤ . Balsara, D. S. 1995, Journal of Computational Physics, 121, 357Benz, W., Slattery, W. L., & Cameron, A. G. W. 1986, Icarus, 66,515Cameron, A. G. W. 2000, Origin of the Earth and Moon, 133Canup, R. M., & Asphaug, E. 2001, Nature, 412, 708Canup, R. M. 2004, ARA&A, 42, 441Canup, R. M., Barr, A. C., & Crawford, D. A. 2013, Icarus, 222,200 Canup, R. M., & Salmon, J. 2014, AAS/Division for PlanetarySciences Meeting Abstracts, 46, 9