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Ren pc telescope space ipn tal. et Kipping ; oqer & Mosqueira aaie al. et Sasaki Heller e ´ ; olc & Pollack Hasegawa rf eso a ,2015 6, May version Draft 2 dis- O ,no ), ABSTRACT 1,2 n ap Pudritz Ralph and , [email protected] aso in lntsmo ytmi estv othe to sensitive total is The system H moon the moons. funda- planet’s of giant location giant a of a play formation of could the lines in ice role water mental why reasons several est fsld (Σ solids of density ( beti rprinlt Σ to proportional is object otmsiemosfr to eodteieln.In line. 6 Galilean lightest ice (at two the the that beyond interesting satellites, or is it at regard, form this moons massive most eedn ailpsto fteH the of position radial ( dependent (CPD) 1999 disk al. et circumplanetary supplied tively u vra uha 5 uie ai ( radii Jupiter 150 the as in much spread gap as initially a over was up material out ’s opened extended Jupiter disk. an after circumsolar in disk formed thin supposedly optically Callisto picture, this uoa(t9 (at hywr e rmasraigcrupaeayrn in ring circumplanetary environment( spreading gas-free a mostly from a fed were as outwards formation they drifted possible proto-satellites that another suggests Yet scenario location. orbital current 10 a on gregated sities rto iktesz faottecneprr ri of ( orbit zones instanta- contemporary feeding their ac- outside the neous well thick about material accreted of optically and size Callisto an the within disk Europa, cretion migrated growingIo, the that and suggests satellites Galilean a 15 (at eo cm g 2 below aah 1981 Hayashi ehr ou nte“a-tre”mdlo nac- an of model “gas-starved” the on focus here We > . 5 R cm g 3 Jup ; uoenEteeyLreTelescope Large Extremely European au ad2002 Ward & Canup − n also(t27 (at Callisto and ) s uigtegpoeigwithin opening gap the during isk . e h opeephotoevaporation complete the ter 3 Jva lnt n n htgiant that find and planets -Jovian 7 uie a tl edn rmthe from feeding still was Jupiter ,bcuetems ftefsetgrowing fastest the of mass the because ), maidb eetbe possibly detectable, by ompanied nets. − oinds a ucetycool. sufficiently was disk jovian 1 u-iesasaehairthan heavier are stars -like o n oss yaot5 fH of % 50 about by consist and R 6 2 3 drtnigpae formation, planet nderstanding rJva xpaesatda a as acted exoplanets er-Jovian Jup c iei h P,weetesurface the where CPD, the in line ice O hl h asv on Ganymede moons massive the while , rtmsae n irtdt Callisto’s to migrated and timescale, yr notepae u otp I type to due planet the into ersz mle detectability implies size heir . h aienmosformed moons Galilean the t 1 Canada M1, 1 s ,aemsl ok ihbl den- bulk with rocky mostly are ), nrae yaotfco fthree of factor about by increases ) letkoneolntcan exoplanet known allest R oqer srd 2003a Estrada & Mosqueira Jup na ieec between difference ental lie:gsosplanets gaseous ellites: rmtepaeaycr)and core) planetary the from 3 s / 2 hssget htthe that suggests This . rd hro 2012 Charnoz & Crida n eemn h time- the determine and ) JVA PLANETS -JOVIAN . 2 2 c ie hr are There line. ice O R Jup R aedensities have ) Jup ,te ag- then ), 2 ( O Makalkin . , Show- b .In ). ). 2 Rene´ Heller & Ralph Pudritz man & Malhotra 1999).3 It has long been hypothesized (depending on their distance to the , amongst oth- that Jupiter’s CPD dissipated when the ice line was be- ers), they eventually open up a gap in the circumstel- tween the orbits of Europa and Ganymede, at about 10 lar disk and their accretion rates drop rapidly. Hence, to 15 RJup (Pollack & Reynolds 1974). Moreover, simula- the formation of moons, which grow from the accumula- tions of the orbital evolution of accreting proto-satellites tion of solids in the CPD, effectively stops at this point in viscously dominated disks around Jupiter, , and or soon thereafter. A critical link between planet and indicate a universal scaling law for the total mass moon formation is the combined effect of various energy of satellite systems (MT) around the giant planets in the sources (see the four heating terms described above) on (Canup & Ward 2006; Sasaki et al. 2010), the distribution in the CPD and the radial where M 10−4 times the (M ). position of the H O ice line. T ≈ p 2 2. In our disk model, the radial extent of the inner, op- METHODS tically thick part of the CPD, where moon formation is In the Canup & Ward (2002) model, the accretion suspected to occur, is set by the disk’s centrifugal ra- rate onto Jupiter was assumed to be time-independent. dius (rcf ). At that distance to the planet, centrifugal Canup & Ward (2006) focussed on the migration and forces on an object with specific angular momentum j growth of proto-moons, but they did not describe their are balanced by the planet’s gravitational force. Using assumptions for the temperature profile in the plane- 3D hydrodynamical simulations, Machida et al. (2008) tary accretion disk. Others used analytical descriptions calculated the circumplanetary distribution of the angu- for the temporal evolution of the accretion rates or for lar orbital momentum in the disk and demonstrated the the movement of the H2O ice lines (Makalkin & Doro- formation of an optically thick disk within about 30 RJup feeva 1995; Mousis & Gautier 2004; Canup & Ward 2006; around the planet. An analytical fit to their simulations Sasaki et al. 2010; Ogihara & Ida 2012) or they did not yields (Machida et al. 2008) consider all the energy inputs described above (Alibert et al. 2005). M (t) a 7/4 We here construct, for the first time, a semi-analytical 7.8 1011 p ⋆p m2 s−1 × MJup 1 AU model for the CPDs of Jovian and super-Jovian planets     for M

3 4 , although being very close to Jupiter (at 2.5 RJup), In particular, their isothermal model M1I, used to fit our has a very low density of about 0.86 g cm−3 (Anderson et al. 2005), Eq. (1), has a radial specific momentum distribution that is about which seems to be at odds with the compositional gradient in the 1.1 times smaller at Callisto’s orbital radius than their adiabatic . But Amalthea likely did not form at its current model M1A2. This offset means an (1.1)2 = 1.21-fold increase of orbital position as is suggested by the presence of hydrous minerals the centrifugal radius, which nicely fits to our correction factor of on its surface (Takato et al. 2004). 27/22 ≈ 1.23. Water ice lines and the formation of giant moons around super-Jovian planets 3

Recent 3D global magnetohydrodynamical (MHD) simulations by Gressel et al. (2013, see their Sect. 6.3) produce circumplanetary surface gas densities that agree 4 r 1 r 2 much better with the “gas-starved” model of Canup & Λ(r)=1 cf − 5 r − 5 r Ward (2006), which our model is derived from, than with r d  c  the “minimum mass” model of Mosqueira & Estrada (2003a).5 The latter authors argue that Callisto formed R in the low-density regions of an extended CPD with high l =1 p (4) r specific angular momentum after Jupiter opened up a − r d gap in the circumstellar disk. In their picture, the young is derived from a continuity equation for the infalling Callisto accreted material from orbital radii as wide as material and based on the angular momentum delivered 150 RJup. In our model, however, Callisto forms in the to the disk (Canup & Ward 2006; Makalkin & Dorofeeva dense, optically thick disk, where we suspect most of the 2014), and M˙ is the mass accretion rate through the solid material to pile up. Simulations by Canup & Ward CPD, assumed to be equal to the mass dictated by the (2006) and Sasaki et al. (2010) show that our assumption pre-computed planet evolution models (Mordasini 2013). can well reproduce the masses and orbits of the Galilean We set rd = RH/5, which yields rd 154 RJup for Jupiter moons. (Sasaki et al. 2010; Makalkin & Dorofeeva≈ 2014). The disk is assumed to be mostly gaseous with an ini- The effective half-thickness of the homogeneous flared tial dust-to-mass fraction X (set to 0.006 in our simula- disk, or its scale height, is derived from the solution of tions, Hasegawa & Pudritz 2013). Although we do not the vertical hydrostatic balance equation as simulate moon formation in detail, we assume that the c (T (r))r3/2 dust would gradually build , either through h(r)= s m . (5) streaming instabilities in the turbulent disk (Johansen GMp et al. 2014) or through accumulation within vortices (Klahr & Bodenheimer 2003), to name just two possible We adopt the standard assumptionp of vertical hydro- formation mechanisms. The disk is parameterized with a static balance in the disk and assume that the gas den- fixed Planck opacity (κP) in any of our simulations, but sity (ρg) in the disk decreases exponentially with distance we tested various values. The fraction of the planetary from the midplane as per light that contributes to the heating of the disk surface 2 −zs is parameterized by a coefficient ks, typically between ρ (r)= ρ e 2h(r)2 (6) 0.1 and 0.5 (Makalkin & Dorofeeva 2014). This quan- g 0 tity must not be confused with the disk , which where z is the vertical coordinate and ρ0 the gas density can take values between almost 0 and 0.9, depending in the disk midplane. The gas surface density is given by on the wavelength and the grain properties (D’Alessio vertical integration over ρ(r, z), that is, et al. 2001). The sound velocity in the hydrogen (H) and helium (He) disk gas usually depends on the mean +inf molecular weight (µ) and the temperature of the gas, but Σg(r)= dz ρ(r, z) . (7) −inf in the disk midplane it can be approximated (Keith & Z −1 Wardle 2014) as cs = 1.9kms Tm(r)/1000K for Inserting Equation (6) into Equation (7), the latter can midplane Tm . 1000K. At these tempera- be solved for ρ0 and we obtain tures, ionization can be neglected andpµ =2.34 kg mol−1. 2 Further, the disk viscosity is given by ν = αcs /ΩK(r), 3 2 Σg(r) with ΩK(r) = GMp/r as the Keplerian orbital fre- ρ0(r)= , (8) quency. rπ 2h(r) The steady-statep gas surface density (Σg) in the opti- cally thick part of the disk can be obtained by solving which only depends on the distance r to the planet. At the continuity equation for the infalling gas at the disk’s the radiative surface level of the disk, or photospheric centrifugal radius (Canup & Ward 2006), which yields height (zs), the gas density equals 2 −zs h r 2 ρs(r)= ρ0 e 2 ( ) (9)

M˙ Λ(r) where we calculate zs as Σg(r)= (3) 3πν × l −1 2 2 zs(r) = erf 1 √2h(r) . (10) − 3 Σg(r)κP   where The latter formula is derived using the definition of the disk’s optical depth

+ inf + inf 5 More advanced 3D MHD simulations would need to take into τ = dz κPρ(r, z)= κP dz ρ(r, z) (11) account the actual formation of the planet (assumed to be a sink Zzs Zzs particle by Gressel et al. 2013) and would require resolving the inner parts of the compact disk to test whether this argument holds and our knowledge of τ = 2/3 at the radiating surface in favor of the gas-starved model. level of the disk, which gives 4 Rene´ Heller & Ralph Pudritz

is the vertical mass coordinate at zs. 2 + inf 2.2. Planet Evolution Tracks = κ dz ρ (r, z) 3 P g Zzs We use a pre-computed set of planet formation mod- els by Mordasini (2013) to feed our planet disk model −1 2 2 1 zs(r) = erf 1 √2h(r) . with the fundamental planetary properties such as the ⇔ − 3 π h(r)ρ0(r)κP r planet’s evolving radius (Rp), its mass, mass accretion   (12) rate (M˙ p), and luminosity (Lp). Figure 1 shows the evo- lution of these quantities with black solid lines indicat- Using Equation (8) for ρ0(r) in Equation (12), we obtain Equation (10). ing an accreting that ends up with one Jupiter Following the semi-analytical disk model of Makalkin mass or about 318 Earth masses (M⊕). In total, we have & Dorofeeva (2014), the disk surface temperature is given seven models at our disposal, where the planets have fi- by the energy inputs of various processes as per nal masses of 1, 2, 3, 5, 7, 10, and 12 MJup. These tracks are sensitive to the planet’s core mass, which we assume to be 33 M⊕ for all planets. Jupiter’s core mass is actu- −1 ally much lower, probably around 10 M⊕ (Guillot et al. 1+(2κPΣg(r)) Ts(r)= Fvis(r)+ Facc(r) 1997). Lower final core masses in these models translate σSB into lower planetary luminosities at any given accretion  1/4 rate. In other words, our results for the H2O ice lines 4 around the Jupiter-mass test planet are actually upper, + ksFp(r) + Tneb , (13) or outer limits, and a more realistic evaluation of the !  conditions around Jupiter would shift the ice lines closer where to the planet. Over the whole range of available planet tracks with core masses between 22 and 130 M⊕, we note that the planetary luminosities at shutdown are 10−4.11 − 3 Λ(r) 2 and 10 3.82 solar luminosities, respectively. As the dis- Fvis(r)= M˙ ΩK(r) 8π l tance of the H2O ice line in a radiation-dominated disk 1/2 ˙ 2 scales with Lp , different planetary core masses would XdχGMpM −(r/rcf ) Facc(r)= e thus affect our results by less than ten percent. 4πr2 r cf The pre-computed planetary models cover the first few 6 sin ζ(r)r + η(r) 10 yr after the onset of accretion onto the planet. We interpolate all quantities on a discrete time line with a Fp(r)= Lp 2 2 8π(r + zs )  step size of 5,000yr. At any given time, we feed Equa- (14) tions (14) with the planetary model and solve the coupled Equations (1)-(10) in an iterative framework. With Ts are the energy fluxes from viscous heating, accretion onto provided by Equation (13), we finally solve the 5th order the disk, and the planetary illumination, and Tneb de- polynomial in Equation (16) numerically. notes the background temperature of the circumstellar Once the planetary evolution models indicate that M˙ p nebula (100 K in our simulations). The geometry of the has dropped below a critical shutdown accretion rate flaring disk is expressed by the angles (M˙ shut), we assume that the formation of satellites has effectively stopped. As an example, no Ganymede-sized −1 moon can form once M˙ shut < MGan Myr (MGan being 4 Rp the mass of Ganymede) and if the disk’s remaining life ζ(r) = arctan 6 ˙ 3π r2 + z2 time is < 10 yr (see Figure 1c). As Mp determines the s ! gas surface density through Equation (3), different values ˙ dzsp zs for Mshut mean different distributions of Σg(r). In par- η(r) = arctan arctan . (15) 2 1 dr − r ticular, Σg(r = 10 RJup) equals 7.4 10 , 9.7 10 , 0 −2 × ×     and 7.8 10 kg m once M˙ shut reaches 100, 10, and × −1 Taking into account the radiative transfer within the op- 1 MGan Myr , respectively, for the planet that ends up tically thick disk with Planck opacity κP, the midplane with one (see Figure 4 in Heller & Pudritz temperature can be estimated as (Makalkin & Dorofeeva 2015). 2014) In any single simulation run, κP is assumed to be con- stant throughout the disk, and simulations of the plane- 12µχκ M˙ 2 tary H2O ice lines are terminated once the planet accretes 5 4 P 3 ˙ Tm(r) Ts(r) Tm(r)= 9 2 ΩK(r) less than a given Mshut. To obtain a realistic picture of − 2 π σSBRgγ × α a broad range of hypothetical exoplanetary disk proper- 2 Λ(r) 2 ties, we ran a suite of randomized simulations, where κP qs(r) , (16) and M˙ shut were drawn from a lognormal probability den- × l 2 −1   sity distribution. For log10(κP/[m kg ]) we assumed a where σSB is the Stefan-Boltzmann constant, RGas the mean value of 2 and a standard variation of 1, and con- ideal gas constant, γ = 1.45 the adiabatic exponent (or cerning the shutdown− accretion rate we assumed a mean ˙ −1 ratio of the heat capacities), and value of 1 for log10(Mshut/[MGan Myr ]) and a standard 4 variation of 1. qs(r)=1 To get a handle on the plausible surface absorptivi- − 3κPΣg(r) Water ice lines and the formation of giant moons around super-Jovian planets 5

ties of various disk, we tested different values of ks be- tween 0.1 and 0.5. For each of the seven test planets, we performed 120 randomized simulations of the disk evo- lution and then calculated the arithmetic mean distance of the final water ice line. The resulting distributions are skewed and non-Gaussian. Hence, we compute both the downside and the upside semi standard deviation (corre- sponding to σ/2), that is, the distance ranges that com- prise +68.3 %/2 = +34.15 % and 34.15 % of the simu- lations around the mean. We also− calculate the 2σ semi- deviation, corresponding to +95.5 %/2 = +47.75 % and 47.75 % around the mean. Downside and upside semi −deviations combined deliver an impression of the asym- metric deviations from the mean, and their sum equals that of the Gaussian standard deviations. The total, instantaneous mass of solids in the disk at the time of moon formation shutdown is given as

rice rcf Ms =2πX dr rΣg(r)+3 dr rΣg(r) , (17) Zrin Zrice !

where rin is the inner truncation radius of the disk (as- sumed at Jupiter’s corotation radius of 2.25 RJup, Canup & Ward 2002), rice is the distance of the H2O ice line, and rcf is the outer, centrifugal radius of the disk (Machida et al. 2008). Depending on the density of the disk gas and the size of the solid grains, the water sublimation temperature can vary by several degrees Kelvin (Lewis 1972; Lecar et al. 2006), but we adopt 170 K as our fidu- cial value (Hasegawa & Pudritz 2013). We simulate the evolution of the H2O ice lines in the disks around young super-Jovian planets at 5.2 astro- nomical units (AU, the distance between the Sun and the Earth) from a Sun-like star, facilitating comparison of our results to the Jovian moon system. These planets belong to the observed population of super-Jovian plan- ets at 1 AU around Sun-like stars (Hasegawa & Pudritz 2011, ≈2012; Howard 2013), and it has been shown that their satellite systems may remain intact during planet migration (Namouni 2010).

3. RESULTS AND PREDICTIONS Figure 1(a) shows, on the largest radial scales, the Hill radius (black crosses) of the accreting . The planetary radius (black solid line) is well within the Hill , but it is quite extensive for 0.9 Myr, so much that the CPD (gray solid line) has not yet formed by that time. It only appears after 0.9 Myr of evolution of the system. Within that disk, we follow the time evolution of two features – the heat transition (orange open circles) and the H2O ice line (blue dots). The heat transition denotes the transition from the viscous to the irradiation heating regime in the disk (see Hasegawa & Pudritz 2011), and it appearsat the outer disk edge about 0.95 106 yr after the onset of accretion. It moves rapidly inwards× and within 2 104 yr it reaches the Figure 1. Evolution of a Jupiter-like model planet and its cir- inner disk edge, which sits roughly≈ × at the radius of the cumplanetary disk. Values taken from Mordasini (2013) are la- planet. At the same time ( 0.99 106 yr after the on- beled “M13”. (a) Circumplanetary disk properties. (b) Growth ≈ × of the solid core, gaseous envelope, and total mass. (c) Total mass set of accretion), the H2O ice line appears at the outer accretion rate. The dashed horizontal line indicates our fiducial disk radius and then moves slowly inward as the planet −1 shutdown rate for moon formation of 10 MGan Myr . The dashed cools. The ice line reverses its direction of movement at vertical line marks the corresponding shutdown for moon formation 6 at about 1.08 × 106 yr. (d) Planetary luminosity evolution. 1.1 10 yr due to the decreasing gas surface densities, while≈ × the opacities are assumed to be constant through- 6 Rene´ Heller & Ralph Pudritz

Figure 2. Disk temperatures around a forming Jupiter-like planet 6 10 yr after the onset of accretion. The two disk levels represent the Figure 3. Evolution of the H2O ice lines in the disks around midplane and the (at a hight zs above the midplane). super-Jovian gas planets. Black solid lines, labeled (1) - (3), indi- At a given radial distance (r) to the planet, measured in Jupiter cate the locations of the H2O ice lines assuming different mass radii, the midplane is usually warmer than the surface (see color accretion rates for the shutdown of moon formation (M˙ shut ∈ −1 bar). The orbits of Io, Europa, Ganymede, and Callisto (labelled {100, 10, 1}×MGan Myr ). The shaded area embraces the orbits I, E, G, and C, respectively) and the location of the instantaneous of Europa and Ganymede around Jupiter, where Jupiter’s H2O ice H2O ice line at ≈ 22.5 RJup are indicated in the disk midplane. line must have been at the time when the Galilean satellites com- The arrow attached to the ice line indicates that it is still moving pleted formation. The most plausible shutdown rate for the Jovian inward before it reaches its final location at roughly the orbit of system (black line with label 2) predicts ice lines between roughly Ganymede. This simulation assumes fiducial disk values (ks = 10 and 15 RJup over the whole range of super-Jovian planetary −2 2 −1 5 −2 2 −2 0.2,κP = 10 m kg ), some 10 yr before the shutdown of moon masses. Simulations assume ks = 0.2 and κP = 10 m kg . formation. for a given disk surface absorptivity (ks) and disk Planck out the disk, see Equation (13). opacity (κP). More massive planets have larger disks Figure 1(b) displays the mass evolution of the plane- and are also hotter at a given time after the onset of tary core (gray dashed line) and atmosphere (gray solid accretion, which explains the larger distance and later line). Note that the rapid accumulation of the envelope occurrence of water ice around the more massive . and the total mass at around 0.93 106 yr corresponds Solid black lines connect epochs of equal accretion rates × to the runaway accretion phase. Panel (c) presents the (1, 10, and 100 MGan per Myr). Along any given ice total mass accretion rate onto the planet (black solid line track, higher accretion rates correspond to earlier line). The dashed horizontal line shows an example for phases. The gray shaded region embraces the orbital −1 M˙ shut (here 10 MGan Myr ), which corresponds to a radii of Europa and Ganymede, between which we ex- time 1.08 Myr after the onset of accretion in that partic- pect the H2O ice line to settle. The H2O ice line around ular model. Note that shutdown accretion rates within the 1 Jupiter mass model occurs after 0.99 106 yr one order of magnitude around this fiducial value occur at the outer edge of the disk, passes through≈ the× current 0.1 - 0.3 Myr after the runaway accretion phase, that is, orbit of Ganymede, and then begins to move outwards after the planet has opened up a gap in the circumstellar around 1.1 106 yr due to the decreasing gas surface disk. In panel (d), the planetary luminosity peaks dur- densities (note,× the opacities are assumed constant). In −1 ing the runaway accretion phase and then dies off as the this graph, M˙ shut 10 MGan Myr can well explain the planet opens up a gap in the circumstellar disk, which mentioned properties≈ in the Galilean system. starves the CPD. In Figure 4, we present the locations of the ice lines in Figure 2 shows a snapshot of the temperature struc- a more global picture, obtained by performing 120 ran- ture of the disk surface and midplane around a Jupiter- domized disk simulations for each planet, where M˙ shut 6 mass planet, 10 yr after the onset of mass accretion. and κP were drawn from a lognormal probability distri- The location of the instantaneous water ice line is indi- bution. We also simulated several plausible surface ab- cated with a white dotted line, and the current positions sorptivities of the disk (D’Alessio et al. 2001; Makalkin of the Galilean satellites are shown with black dotted & Dorofeeva 2014) (0.1 ks 0.5), which resulted in ice lines. Over the next hundred thousand years, the heating line locations similar to≤ those≤ shown in Figure 4, where rates drop and the ice line moves inward to Ganymede’s ks = 0.2. The mean orbital radius of the ice line at the present orbit as the planet’s accretion rate decreases due time of shutdown around the 1 MJup planet is almost to its opening of a gap in the circumsolar disk. We argue precisely at Ganymede’s orbit around Jupiter, which we that the growing Ganymede moved with the ice line trap claim is no mere coincidence. Most importantly, despite and was parked in its present orbit when the circumjo- a variation of M˙ shut by two orders of magnitude and con- vian disk dissipated. Due to the rapid decrease of mass sidering more than one order of magnitude in planetary accretion onto the planet after gap opening (up to about masses, the final distances of the H2O ice lines only vary 4 an order of magnitude per 10 yr), this process can be between about 15 and 30 RJup. Hence, regardless of the reasonably approximated as an instant shutdown on the actual value of M˙ shut, the transition from rocky to icy time scales of planet formation (several 106 yr), although moons around giant planets at several AU from Sun-like it is truly a gradual process. stars should occur at planetary distances similar to the Figure 3 shows the radial positions of the ice lines one observed in the Galilean system. around super-Jovian planets as a function of time and We ascribe this result to the fact that the planetary lu- Water ice lines and the formation of giant moons around super-Jovian planets 7

Figure 4. Distance of the H2O ice lines at the shutdown of moon Figure 5. Instantaneous mass of solids in the disks around super- formation around super-Jovian planets. The solid line indicates Jovian planets at moon formation shutdown. Solid and dashed the mean, while shaded areas denote the statistical scatter (dark lines refer to disk absorptivities of ks = 0.2 and 0.4, respec- gray 1 σ, light gray 2 σ) in our simulations, based on the posterior tively. The ordinate scales in units of the total mass contained distribution of the disk Planck mean opacity (κP) and the shut- in the Galilean moons (≈ 2.65 MGan). For any given shutdown down accretion rate for moon formation (M˙ shut). The dashed line rate, we find a linear increase in the mass of solids at the time represents the size of the optically thick part of the circumplan- of moon formation shutdown as a function of planetary mass, in etary disk, or its centrifugal radius. All planets are assumed to agreement with previous simulations for the solar system giants orbit a Sun-like star at a distance of 5.2 AU and ks is set to 0.2. (Canup & Ward 2006; Sasaki et al. 2010). Super-Jovian planets Labeled circles at 1 MJup denote the orbits of the Galilean satellite of 10 MJup should thus have moon systems with total masses of −3 Io, Europa, Ganymede, and Callisto. Orange indicates rocky com- ≈ 10 ×MJup, or 3 times the mass of Mars. position, blue represents H2O-rich composition. Circle sizes scale with moon radii. Note that Ganymede sits almost exactly on the 4.1. Accretion and Migration of both Planets and Moons circumjovian ice line. While Canup & Ward (2002) stated that accretion minosity is the dominant heat source at the time of moon −7 −1 4 −1 rates of 2 10 MJup yr (about 2.6 10 MGan Myr ) formation shutdown. Planetary luminosity, in turn, is best reproduced× the disk conditions in× which the Galilean determined by accretion (and gravitational shrinking), ˙ satellites formed, our calculations predict a shutdown hence a given Mshut translates into similar luminosities accretion rate that is considerably lower, closer to and similar ice line radii for all super-Jovian planets. −1 10 MGan Myr . The difference in these results is mainly Planets above 1 MJup have substantially larger parts of owed to two facts. First, Canup & Ward (2002) only con- their disks beyond their water ice lines (note the logarith- sidered viscous heating.6 Our additional heating terms mic scale in Figure 4) and thus have much more material (illumination from the planet, accretion onto the disk and available for the formation of giant, water-rich analogs of stellar illumination) contribute additional heat, which Ganymede and Callisto. imply smaller accretion rates to let the H2O ice lines Figure 5 shows the total mass of solids at the time of move close enough to the Jupiter-like planet. Second, the moon formation shutdown around super-Jovian planets. parameterization of planetary illumination in the Canup Intriguingly, for any given shutdown accretion rate the & Ward (2002) model is different from ours. While they total mass of solids scales proportionally to the planetary assume an r−3/4 dependence of the midplane tempera- mass. This result is not trivial, as the mass of solids ture from the planet (r being the planetary radial dis- depends on the location of the H2O ice line at shutdown. ˙ tance), we do not apply any pre-described r-dependence. Assuming that Mshut is similar among all super-Jovian In particular, T (r) cannot be described properly by a planets, we confirm that the M 10−4 M scaling law m T ∝ p simple polynomial due to the different slopes of the var- observed in the solar system also applies for extrasolar ious heat sources as a function of planetary distance. super- (Canup & Ward 2006; Sasaki et al. 2010). Previous models assume that type I migration of In addition to the evolution of the H2O ice lines, we the forming moons leads to a continuous rapid loss of also tracked the movements of the heat transitions, a proto-satellites into the planet (Canup & Ward 2002; specific location within the disk, where the heating from Mosqueira & Estrada 2003a,b; Alibert et al. 2005; Sasaki planetary irradiation is superseded by viscous heating. et al. 2010). (Alibert et al. 2005) considered Jupiter’s Heat transitions cross the disk within only about 104 yr 5 accretion disk as a closed system after the circumstel- (see Figure 1a), several 10 yr before the shutdown of lar accretion disk had been photo-evaporated, whereas moon formation, and thereby cannot possibly act as Sasaki et al. (2010) described accretion onto Jupiter moon traps. Their rapid movement is owed to the abrupt with an analytical model. In the Mosqueira & Estrada starving of the planetary disk due to the gap opening of (2003a,b)(ME) model, satellites migrate via type I but the circumstellar disk, whereas the much slower photoe- perturb the gas as they migrate and eventually stall and vaporation of the latter yields a much slower motion of open a gap, ensuring their survival. In opposition to the the circumstellar heat trap. The ineffectiveness of heat Canup & Ward (CW) theory, their model does not pos- traps for satellites reflects a key distinction between the tulate “generations” of satellites, which are subsequently processes of moon formation and for- mation. 6 Canup & Ward (2002) discuss the contribution of planetary luminosity to the disk’s energy budget, but for their computations 4. DISCUSSION of the gas surface densities they ignore it. 8 Rene´ Heller & Ralph Pudritz lost into the planet, because satellite formation doesn’t one might suggest that Callisto and Ganymede formed start until the accretion inflow onto the planet wanes. after the GT from newly accreted planetesimals into a There are two difficulties with the CW picture. First, still active, gaseous disk around Jupiter. But then Io and type I migration can be drastically slowed down as grow- Europa might have been substantially enriched in water, ing giant moons get trapped by the ice lines or at the too. Tanigawa et al. (2014, see their Figure 8) found inner truncation radius of the disk7. Thus it is not obvi- that accretion via gas drag is most efficient ous that a conveyor belt of moons into their host planets between 0.005 and 0.001 Hill radii (RH) or about 4 to is ever established. Second, our Figure 5 also contra- 8 RJup where gas densities are relatively high. dicts this scenario, because the instantaneous mass of To come straight to the point, our preliminary stud- solids in the disk during the end stages of moon forma- ies suggest that in the GT paradigm, the icy Galilean tion (or planetary accretion) is not sufficient to form the satellites must have formed prior to Jupiter’s excursion last generation of moons. In other words, whenever the to the inner solar system (Heller et al. 2015, in prep.). instantaneous mass of solids contained in the circumjo- This illustrates the great potential of moons to constrain vian disk was similar to the total mass of the Galilean planet formation, which is particularly interesting for the moons, the correspondingly high accretion rates caused GT scenario where the timing of migration and planetary the H2O ice line to be far beyond the orbits of Europa accretion is yet hardly constrained otherwise (Raymond and Ganymede. & Morbidelli 2014). We infer, therefore, that the final moon population around Jupiter and other Jovian or super-Jovian exo- 4.2. Parameterization of the Disk planets must, at least to a large extent, have built during Finally, we must address a technical issue, namely, our the ongoing, final accretion process of the planet, when − it was still fed from the circumstellar disk.8 In order choice of the α parameter (10 3). While this is consis- to counteract the inwards flow due to type I migration, tent with many previous studies, how would a variation we suggest a new picture in which the circumplanetary of α change our results? Magnetorotational instabili- H2O ice line and the inner cavity of Jupiter’s accretion ties might be restricted to the upper layers of CPDs, disk have acted as migration traps. This important hy- where they become sufficiently ionized (mostly by cosmic pothesis needs to be tested in future studies. The effect high-energy radiation and stellar X-rays). Magnetic tur- of an inner cavity will also need to be addressed, as it bulence and viscous heating in the disk midplane might might have been essential to prevent Io and Europa from thus be substantially lower than in our model (Fujii et al. plunging into Jupiter. 2014). On the other hand, Gressel et al. (2013) modeled In our picture, Io should have formed dry and its mi- the magnetic stresses in CPDs with a 3D magnetohydro- gration might have been stopped at the inner truncation dynamic model and inferred α values of 0.01 and larger, radius of Jupiter’s accretion disk, at a few Jupiter radii which would strongly enhance viscous heating. Obvi- (Takata & Stevenson 1996). It did not form wet and then ously, sophisticated numerical simulations of giant planet lose its water through tidal heating. Ganymede may have accretion do not yet consistently describe the magnetic formed at the water ice line in the circumjovian disk, properties of the disks and the associated α values. where it has forced Io and Europa in the 1:2:4 orbital Given that circumstellar disks are almost certainly mean motion resonance (Laplace et al. 1829). From a magnetized, CPD can be expected to have inherited mag- formation point of view, we suggest that Io and Europa netic fields from this source. This makes it likely that be regarded as moon analogs of the terrestrial planets, magnetized disk winds can be driven off the CPD (Fendt whereas Ganymede and Callisto resemble the precursors 2003; Pudritz et al. 2007) which can carry significant of giant planets. amounts of angular momentum. Even in the limit of Our combination of planet formation tracks and a CPD very low ionization, Bai & Stone (2013) demonstrated model enables new constraints on planet formation from that magnetized disk winds will transport disk angular moon observations. As just one example, the “Grand momentum at the rates needed to allow accretion onto Tack” (GT) model suggests that Jupiter migrated as the central object. However, independent of these uncer- close as about 1.5AU to the Sun before it reversed its tainties, the final positions of the H2O ice line produced migration due to a mutual with Saturn in our simulations turn out to depend mostly on plane- (Walsh et al. 2011). In the proximity of the Sun, however, tary illumination, because viscous heating becomes neg- solar illumination should have depleted the circumjovian ligible almost immediately following gap opening. Hence, accretion disk from water ices during the end stages of even substantial variations of α by a factor of ten would Jupiter’s accretion (Heller et al. 2015, in prep.). Thus, hardly change our results for the ice line locations at Ganymede and Callisto would have formed in a dry envi- moon formation shutdown since these must develop in ronment during the GT, which is at odds with their high radiatively dominated disk structure (but it would al- H2O ice contents. They can also hardly have formed over ter them substantially in the viscous-dominated regime millions of years (Mosqueira & Estrada 2003a) thereafter, before and during runaway accretion). because Jupiter’s CPD (now truncated from its environ- Our assumption of a constant Planck opacity through- ment by a gap) still would have been dry. Alternatively, out the disk is simplistic and ignores the effects of grain growth, grain distribution within the disk, as well as the 7 An inner cavity can be caused by magnetic coupling between evolution of the disk properties. In a more consistent the rotating planet and the disk (Takata & Stevenson 1996), and it model, κ depends on both the planetary distance and can be an important aspect to explain the formation of the Galilean P satellites (Sasaki et al. 2010) distance from the midplane, which might entail signifi- 8 This conclusion is similar to that proposed by the ME model, cant modifications in the temperature distribution that but for reasons that are very different. we predict. Water ice lines and the formation of giant moons around super-Jovian planets 9

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