arXiv:1502.07585v1 [astro-ph.EP] 26 Feb 2015 o.Nt .Ato.Soc. Astron. R. Not. Mon. elde l 03 a rudt rdc eaierather 1998; negative a Boss produce to (e.g., argued gi- was formation 2013) positive al. a et for Helled thus forming framework, model towards CA Instability Gravitationa step in ensues. al. correlation crucial planet et planet–metallicity ant Mordasini giant the 2008; gas are 2004, a cores Lin & These Ida 2009). (e.g., much readily cores massive more 19 al. assemble environments et Ac- metal-rich Pollack Core (e.g., since to formation support planet direct for a theory provide correla- cretion to This metal-poor argued 2005). been Valenti around has & tion than Fischer 1999; (Gonzalez fre- ones metal-rich more much around detected are quently Giant correlation. such trends. such form reproduce planet must that successful theory A robust tion stochasticity. so the as above processes significance rise planet-forming they ob- special to the highly have testament in a they populations found is formation planetary trends planet statistical served of Any outcome process. 2014), the Fabrycky stochastic that & clear Winn 2013; archi- is al. it system et exoplanet Batalha of (e.g., diversity tectures bewildering the Given INTRODUCTION 1 Received Nayakshin Sergei correlati Planet-metallicity II. model. Downsizing Tidal c eateto hsc srnm,Uiest fLeicester of University Astronomy, E-mail: & Physics of Department 08RAS 2008 ealct orltoso h bevdpaesi one is planets observed the of correlations Metallicity [email protected] 000 –3(08 rne 6Mrh22 M L (MN 2021 March 26 Printed (2008) 1–13 , in,wihmyepanwyoeo h etkondrcl mgdgas imaged words: directly Key known poor. best metal the is prefere of 8799, metallicity one HR why host systems, the explain strong short. may which a window which show within time tions, not by fragments do growth damped planets core’s the gas is the giant metallicities growth cutting core’s quickly, high solid too at the contract sho r whereas metallicities This planets metallicity. low metals, Solar at smaller of the that these around fact metallicit at of the inner efficiency star by population formation the host simulated their in the in The simulations with peak not. correlation the do positive in linear planets strong non formed the a is planets planets follow between giant dominated region connection gas solid-core the smaller While that the monotonic. and find giant and gas TD, recen popu the perform on a We based of (TD). Downsizing framework calculations Tidal propos the called We in hypothesis metallicities. observations formation of range these wide of a s with interpretation host hosts metal-rich gia orbit around gas Neptune preferentially of than found co assembly are the planets in for puzzling giant prerequisite are gas fo a exoplanets planet of as correlations of cores metallicity observed theory solid accepted massive de-facto of mation the (CA), Accretion Core ABSTRACT disc l 96) ecse,L17H UK 7RH, LE1 Leicester, , a- t as hw(uhaee l 02 htpaessmaller planets that 2012) al. 4 et than (Buchhave show mass, observ FGK its method for transit metallicity Similarly, with 2008). star’s tions al. host not et the (Sousa does with stars occurrence correlate observa planet to velocity Neptune-mass seem Radial that further. show issue tions the study to reasons opacities dust chances low planet’s at increasing survival. fastest of 2011), the Bodenheimer is & (Helled cooling radiative since ets’ correlation planet–metallicity giant positive than ebeaceinwssoe opoueasrn posi- strong a produce to showed and migration was proposed. planet recently accretion includes that been pebble model have GI particular, formation ( In Instability planet Gravitational for the of model variant classical at the cores to observed. more is be what not should is there which metallicities, then CA’s higher correct If w planets. correlation is planet giant giant metallicity gas gas smaller positive the these the of for since explanation precursors CA, the of This are context metallicity. planets the Solar the in to surprising close clearly average the with ities, oee,teeaebt bevtoa n theoretical and observational both are there However, ntetertclsd,svrlipratextensions important several side, theoretical the On R ⊕ Kepler omaon ot ihawd ag fmetallic- of range wide a with hosts around form A T E tl l v2.2) file style X estv otepae’ aisrte than rather radius planet’s the to sensitive , mto,rqie for- requires rmation, l eeoe planet developed tly as lnt smaller planets tars, c tlresepara- large at nce tx fC.While CA. of ntext ial,simulated Finally, ainsynthesis lation The planets. nt nalternative an e ouain of populations sl sdriven is esult ,tesmaller the y, sashallow a ws n o even not and in planet giant h scarcity the oe grow cores e AU few ons plan- GI) ith a- is - 2 S. Nayakshin tive and not negative correlation with the star’s metallicity more rapidly (e.g., see Nayakshin 2015a). Also, radiative col- (Nayakshin 2015a,b) for coreless planets. The goal lapse of H2-dominated fragments produces a negative planet of this paper is to extend this recent work on giant planets frequency – metallicity correlation (Helled & Bodenheimer with cores and also on core-dominated (smaller) planets and 2011) because radiative cooling is more rapid at low metal- to compare the results with the observations. licities. That is clearly at odds with observations (Gonzalez The key reason why GI theory may require a major 1999; Fischer & Valenti 2005). overhaul is the realisation that GI gas fragments can also We (Nayakshin 2015a), however, considered accre- migrate from ∼ 100 AU all the way into the inner disc tion of ∼< 1 cm sized grains (usually called ”pebbles”; (Boley et al. 2010). While planet migration was a standard Johansen & Lacerda 2010; Ormel & Klahr 2010) onto the feature for the CA model since 1996 (Lin et al. 1996), GI pre-collapse fragments. While gas has pressure gradient planets were somehow thought to not migrate until recently. forces that may prevent its accretion from the proto- If this were true than GI could at best account for important planetary disc onto the fragments (e.g., Nayakshin & Cha but rare giant planet systems like HR 8799 imaged directly 2013), pebbles weakly coupled to the gas by aerodynam- (e.g., Marois et al. 2008), and could never explain the Solar ical forces can still accrete onto the fragment in an anal- System: massive self-gravitating proto-planetary discs can ogy to the pebble accretion process onto massive solid cores hatch clumps only at R ∼> many tens of AU (e.g., Rice et al. (Lambrechts & Johansen 2012) in the context of the CA 2005; Rafikov 2005). model. However, simulations, starting with Vorobyov & Basu Surprisingly, pebble accretion on GI fragments was (2005, 2006), showed that massive gas fragments do migrate found to accelerate their contraction despite increasing their inward. Furthermore, Boley et al. (2010) suggested a new metallicity and hence opacity. Physically, addition of peb- way of forming terrestrial-like planets inside ∼ a few bles to the fragment increases its weight without increasing masses gas clumps by grain growth and sedimentation (see its thermal energy1. Since pre-collapse molecular gas frag- also Williams & Crampin 1971; Boss 1997), and then releas- ments are polytropes with an effective polytropic index γ ing them back into the disc by destroying the gas clumps edging ever closer to the unstable value of 4/3 as the cen- via tidal forces from the host star. Nayakshin (2010a) used tral temperature of the fragment increases towards 2000 K analytical estimates of the relevant processes and arrived at (see Nayakshin 2015a), addition of a relatively small, ∼ 10%, similar ideas, proposing the Tidal Downsizing (TD) hypoth- amount of mass in metals to the fragment usually tips it over esis for formation of all types of planets observed so far. into collapse. Higher metallicity environments provide larger The key question for a planet forming in the framework pebble accretion rates, hence Nayakshin (2015b) found, via of the TD hypothesis is how does the clump’s inward mi- a population synthesis like study of coreless gas fragments, gration time scale, which may be as short as ∼ 104 years that the frequency of gas giants survived the initial tidal (Baruteau et al. 2011; Michael et al. 2011), compare with disruption increases as metallicity of the host increases. the time scale for its internal evolution? If the fragment con- An important issue not considered by Nayakshin traction time is shorter than the inward migration time, then (2015a,b) is formation of a core within the fragment. If peb- the fragment collapses, becoming a bona fide ”hot start” bles entering the fragment accrete onto the core, the core’s protoplanet, which may mature into a gas giant planet. If accretion luminosity may be significant and this may lead the fragment contraction time is long, then the host’s tidal to expansion of the fragment instead of its contraction. A forces catch up with the fragment when the latter migrates similar effect – accretion of planetesimals onto the grow- too close to the star, and the fragment is disrupted. The ing core – is well known in the context of the CA model fragment’s short existence may however not be all in vain as and has to be taken into account when determining the fate there could be a remnant. If grains within the fragment had of the planet (e.g., Pollack et al. 1996; Alibert et al. 2005). a sufficient time to grow and sediment to the centre, getting Nayakshin (2015c, paper I hereafter) has recently presented locked into a self-gravitating massive core (see also Kuiper numerical algorithms that extend the work of Nayakshin 1951; McCrea & Williams 1965; Williams & Crampin 1971; (2015b) by adding the core formation and growth inside the Boss 1997; Helled & Schubert 2008), then the remnant of pre-collapse fragments. the disruption is the core – a practically ready rocky planet. Here we use that framework to study the planet fre- No planetesimal accretion is required for the planet to sur- quency of survival versus metallicity dependence for all types vive at that stage, although ”veneer accretion” of them or of planets rather than just the coreless giants. We also go pebbles (Johansen & Lacerda 2010) is possible. beyond of our previous work by studying not just the out- Making detailed predictions in the TD hypothesis set- come of the migration vs disruption competition, but also ting worthy of comparison to modern observations of ex- how the population of planets formed in TD model would oplanets is however not trivial. Until recently, it was as- look like in the planet mass – separation plane. sumed that collapse of the self-gravitating GI gas fragments The structure of our paper is as following. In §2 we must occur due to radiative cooling (e.g., Bodenheimer 1974; present initial conditions, assumptions and numerical algo- Bodenheimer et al. 1980; Cameron et al. 1982; Helled et al. rithms that are employ to calculate an outcome of a gas-dust 2008; Nayakshin 2011; Nayakshin & Cha 2013). The for- mation of giant gas planets in this picture is similar to 1 This is in difference to accretion of planetesimals in the CA that of low mass stars (e.g., Larson 1969), except that ac- framework, in which planetesimals impact the growing planet at cretion of gas onto the pre-collapse planets is terminated a high (up to a few km s−1) velocity and hence may actually heat very early on (e.g., Nayakshin 2010b). This formation chan- it. In contrast, pebbles sediment onto the fragment at a moderate > nel favors survival of massive (Mp ∼ a few MJ) gas gi- ∼ a few m/sec velocity, or else they fragment. Their kinetic energy ants (Forgan & Rice 2013), as more massive planets cool input into the planet is hence negligible.

c 2008 RAS, MNRAS 000, 1–13 Planet-metallicity correlations 3

fragment being born in an outer region of a massive proto- Initial disc properties, Md=0.15 Msun planetary disc. In §3, two example planet formation tracks 10000 Σ -11 -1 are shown, one for a gas giant and another for a hot Super- Wind loss rate [10 MO • yr ] Earth planet. In §4 we present assumptions and methods 1000 for calculating a grid of models in a population synthesis like study. §5 presents the population synthesis study re-

] and dM/dt 100 sult in the planet mass versus planet-host separation plane. -2 In §6 we marginalise over the results to derive the planet- metallicity correlations for our simulated planets. Discussion [g cm 10 of main ideas and results of the paper is given in §7, whereas Σ conclusions are presented in §8. 1 0.1 1.0 10.0 100.0 r, AU

2 NUMERICAL METHODS

2.1 Disc treatment Q * t cool We follow the procedures described in paper I with several 100 cool

changes detailed now. Our goal is to simulate formation of * planets from their birth to the end of the disc migration phase. The initial disc surface density profile is given by 10 Q and t

Am Rin R Σ0(R)= 1 − exp − , (1) 1 R R ! R0 r h i where R0 is the disc length-scale, set to R0 = 100 AU, 0.1 1.0 10.0 100.0 Rin = 0.08 AU is the inner boundary radius. The constant r, AU Am is calculated so that the disc contains a given initial Rout mass Md = 2π RdRΣ0(R). The disc is ”live”, that Figure 1. Initial disc properties for Md = 0.15 M⊙. Top: Σ(R) Rin ˙ 2 is evolved by solving the time-dependent 1D viscous disc and the wind mass loss rate profile, defined as Σ(R)πR , as la- Q evolution equationsR that include the planet-disc interactions belled. Bottom: Toomre parameter, , and dimensionless cooling time of the disc, t Ω(R), versus radius. The disc is gravitation- (section 3 in paper I), and now also include the disc photo- cool ally unstable and may fragment at R ∼ 70 − 80 AU. evaporation as a sum of the UV and the X-ray driven terms, using the fits to the photo-evaporation rates Σ˙ ev(R) from Alexander & Armitage (2007) and Owen et al. (2012), re- where Q0 = 2 and Q is the local Toomre’s parameter. This spectively. The ionising photon luminosity of the star is set form is inspired by Lin & Pringle (1987) suggestion that α ∝ 42 −1 to Φion = 10 photons s , while the X-ray flux of the star Q−2 in the self-gravitating regime, on the one hand, and 30 −1 is LX = 2 × 10 erg s . more recent simulations showing that αsg saturates at about Gas fragments in the disc are born by self-gravitational 0.1 (Rice et al. 2005), on the other. instabilities, hence the disc must be initially gravitation- ally unstable in its outer region. Figure 1 shows the ini- tial disc properties for disc mass Md = 0.15 M⊙ and stel- 2.2 Planet’s birth and migration lar mass M∗ = 1 M⊙. The top panel shows the disc sur- face density, Σ(R), as well as the disc photo-evaporation The following considerations are important for both the ini- rate profile (which does not change during the simulation). tial and the final phases of our simulations. Simulations of 2 The quantity plotted with the red dotted curve is Σ˙ evπR , self-gravitating protoplanetary discs show that fragments −11 −1 in units of 10 M⊙ year . The bottom panel shows the rarely form in isolation, hence we believe that every star Toomre Q-parameter (black solid curve) and the dimension- is likely to have hatched a number of fragments early on. ∗ Presumably, most of these fragments migrate rapidly and less cooling time, tcool = tcoolΩ(R). For the disc to be self- < ∗ < perish by being disrupted (see, in particular, Vorobyov 2013) gravitating and actually fragmenting, Q ∼ 1.5 and tcool ∼ 3 (Gammie 2001). Both of these conditions are satisfied at or by being driven all the way into the star. This picture is R ∼ 70 − 80 AU. consistent with the suggestion that disruption and swallow- The disc viscosity is described with the ing of planets by their host stars power FU Ori outbursts Shakura & Sunyaev (1973) viscosity parameter, but of young protostars (Vorobyov & Basu 2006; Boley et al. we now add a self-gravity contribution to it: 2010; Nayakshin & Lodato 2012), statistics of which sug- gests that every star may have ∼ 10 − 20 of such episodes α = α0 + αsg , (2) (Hartmann & Kenyon 1996). Given that the observed giant gas planets’ frequency of where α is radius and time-independent (free) parameter 0 occurrence is < 10% (Winn & Fabrycky 2014) per star, the of the model, whereas α describes gravito-turbulence ∼ sg survival of a giant gas planet is a very rare event indeed. It Q2 is reasonable to assume that giant gas planets that survive α = 0.2 0 . (3) sg 2 2 to be observed are those that were either hatched by the Q0 + Q

c 2008 RAS, MNRAS 000, 1–13 4 S. Nayakshin disc late, or those that hang around at the outer disc for a rate. Since our discs are sampling the post self-gravitating while before migrating in, because such fragments are more phase of the disc evolution by design (see below), we may ex- likely to avoid the tidal disruption and also being driven all pect type I migration to be slower than that for Q ∼ 1 discs the way into the star. (Baruteau et al. 2011), for which fmigr ∼ 1. In the models This view is not unreasonable. Simulations show that below fmigr is randomly sampled between 1 and 10. presence of one clump in a disc may induce formation of more clumps in the same disc (e.g., Meru 2013). Mul- tiple clump formation leads to strong interactions be- 2.3 Planet’s internal evolution tween the clumps, so that some of them migrate inward faster while others are scattered on larger orbits (e.g., Planet’s internal evolution is calculated as in paper I, util- Cha & Nayakshin 2011); some of the clumps may be com- ising the ”follow the adiabats” approximation to convec- pletely ejected from the system by the clump-clump inter- tive/radiative cooling of the planet. The planet is initialised actions (Basu & Vorobyov 2012). Clumps on scattered but as a polytropic sphere of gas of homogeneous composition, still bound orbits may take a while to loose their orbital with metallicity, z0, equal to that of the parent disc, with inclinations with respect to the midplane of the disc, and a given central temperature T0 (see §4). The planet cools eccentricities, and only then start migrating in. by emission of radiation assuming interstellar dust opacity Furthermore, presence of clumps in the disc may in (Zhu et al. 2009) multiplied by the factor fop < 1 to account fact change the fragmentation properties of the disc since for grain growth and also by the metallicity of the planet in the clump induces additional perturbations (see again Meru Solar units, that is, by z/z⊙. Since the planet is surrounded 2013, , where discs were found to fragment at surprisingly by the disc, its irradiation by the disc (or by the star if the small radii in the presence of a pre-existent gas clump at planet is directly exposed to it in the final stages of calcu- larger radii). Nayakshin & Cha (2013) focused on migration lation) reduces the rate at which the planet cools, creating of a single fragment in a marginally self-gravitationally sta- a thermal bath effect (Cameron et al. 1982; Vazan & Helled ble disc, thus placing the fragment in a massive but stable 2012). disc that would not fragment on its own. They found that The planet accretes pebbles from the surrounding disc in the presence of the clump the disc became more unstable at the Hill’s accretion rate (Lambrechts & Johansen 2012) and formed additional fragments. This shows that it is pos- M˙ = 2f Σ (a)Ω R2 (5) sible to hatch new gas clumps in discs with Q> 1.5 if there z p g a H 3 1/2 are already fragments born earlier. where RH is the planet’s Hills radius, Ωa = (GM∗/a ) , Clearly, when survival of the fragments is rare, we and Σg = fg Σ(a) is the grain surface density at radius R = should not under-estimate the importance of initial condi- a. The pebble mass fraction, 0 < fp < 1, is a free parameter tions giving the clumps the best chance of surviving. There- independent of the planet’s location, fixed for a given run fore, our population synthesis model aims to account for this (see §4 below). by (a) slowing down the type I migration below its nominal Note that pebble accretion rate depends directly on the rate by some factor larger than unity (see below); (b) allow- surface density of gas around the planet. If the planet man- ing the disc mass to be below the Md = 0.15 M⊙ marginally ages to open a deep gas in the disc, Σ(a) drops to very low gravitationally unstable configuration when the fragment is values and hence M˙ z dives to negligible levels. Pebble accre- born, and (c) removing the disc instantaneously after a time tion is also turned off if and when the fragment undergoes trem randomly sampled between 0.5 and 5 Million years. the second (Hydrogen molecule dissociation) collapse. It is These choices are certainly not unique but we feel they are argued that after the collapse the planet is so compact that reasonable. An extremely rapid external photo-evaporation it should be able to accrete gas from the disc as well, so of the disc by a close-by massive star (Clarke 2007) or the that both gas and pebbles would be accreted. We currently presence of a secondary at beyond 100 AU that is later re- do not include gas accretion onto post collapse planets. moved by star-star interactions may be important for such Pebbles entering the fragment have a fixed initial size a rare population of planets as the gas giants. of apeb = 0.03 cm for this paper, and are deposited in the The disc exchanges angular momentum with the planet outer several mass grids of the planet, where they are mixed via type I and/or type II migration torques. The only dif- homogeneously with the grains already in those regions. ference from paper I is that the type I migration time scale, The grains are allowed to grow by sticking collisions and tI , is now given by sediment into the centre of the fragment into the core, as 2 2 described in paper I. Turbulence within the fragment, pa- M∗ a −1 tI = fmig Ω . (4) rameterised by a coefficient α , and convection, however, M M H2 a d p d tend to counteract grain sedimentation. We usually find The factor (1 + Mp/Md), used in paper I on the right hand that grain size must increase to about ag ∼ 1 cm before side of this equation, has been dropped, as it is argued they sediment efficiently. A further significant obstacle to that the saturation of the type I migration rate in the limit a rapid grain sedimentation into the core is the fact that Mp/Md ≫ 1 is automatically taken into account by passing grains fragment in collisions rather than stick if collision the torque of the planet to the 1D viscous disc evolution velocity is too high, as shown by laboratory experiments code. We find that when this torque is large then a gap may on dust growth (e.g., Blum & M¨unch 1993; Blum & Wurm start opening, thus reducing the torque. 2008; Beitz et al. 2011). As explained in paper I, section 4.3, In equation (4) fmigr > 1 is a free parameter that ac- grain growth turns into grain fragmentation if grain sedi- counts for a slower migration rate of the planet in a non self- mentation velocity with respect to the surrounding gas ex- −1 gravitating disc compared to the isothermal type I migration ceed umax, set in this paper to 3 m s . Grains are also

c 2008 RAS, MNRAS 000, 1–13 Planet-metallicity correlations 5 vaporised if the surrouding gas temperature is too high for the given grain species. Disc evolution Three grain species are included in the model: wa- ter ice, CHON, and silicates (”rocks”). The relative abun- 1000.0 t= 1 Myr 1.32 Myr dances of the species are 0.5, 0.25 and 0.25, respectively. The 1.50 Myr ] 100.0 grain growth, fragmentation, vaporisation, and sedimenta- -2 1.65 Myr tion equations are solved for each species separately since 10.0 [g cm

these three species have very different material and thermal Σ properties. 1.0 Grains reaching the centre are allowed to accrete onto (a) the solid core there, whose initial mass is set to a very 0.1 −4 ”small” value (10 M⊕). Growing core is expected to 0.1 1.0 10.0 100.0 radiate some of its gravitational potential energy away, R, AU but a self-concistent modelling of energy transfer within Gas giant planet formation the core is fraught with many physical uncertainties (e.g., Disc evaporated Stamenkovi´cet al. 2012). As in paper I (section 4.4), we 10.000 Gap openned parameterise the energy release by the core via the Kelvin- Planet collapses Helmholtz contraction time, tkh, of the solid core, which is 1.000 5 fixed for all the runs here at tmin = 3 × 10 years. The lu- minosity released by the core is an internal energy source 0.100 for the contracting fragment, analogous to that of a stel- Separation, a

Separation and radii [AU] 0.010 lar core, and may even exceed the luminosity of the whole Radius, Rp Hill’s radius, R planet, causing its expansion. (b) H 0.001 1.0 (c)

3 EXAMPLE RUNS ] Earth Before the grid of models is discussed, we present two ex- Total Core mass ample planet formation tracks that illustrate the sequence Rocks CHON of events in the TD model for planet formation. 0.1 Core mass [M 3.1 A gas giant planet

Figure 2 shows an example selected from the grid of runs 500 1000 1500 3 in which the end result is a gas giant planet of mass Mp = time [10 years] 1.17 MJ at separation of a = 2.7 AU. The initial mass of the planet is Mp = 1.0 MJ, so all the extra mass in the end of the run comes from accretion of pebbles from the disc. The Figure 2. Evolution of the disc (panel a) and the embedded fragment (panles b and c) that survives to be a gas giant planet initial mass of the disc is Md = 0.074 M⊙, planet location (see §3.1 for detail). Panel (a) shows disc surface density profiles a = 83 AU, and the initial planet’s central temperature 0 at times t = 0 (solid curve) plus several later times as labelled in Tc = 254 K. The disc metallicity is Solar, that is, z = 0.015, the legend. The position of the planet at corresponding times is the pebble mass fraction is fp = 0.086, disc viscosity pa- marked by a cross of same collor at the bottom of the panel. Panel −3 rameter α0 = 2.9 × 10 . The disc removal time is trem = 3 (b) shows the planet’s separation, radius and the Hills radius, Million years, but in this simulation the disc dissipates ear- whereas panel (c) shows the mass of the core versus time. lier due to photo-evaporation, as we shall see below. Further, the opacity reduction factor fop = 0.5, the planet’s type I migration speed is moderated by factor fmigr = 8.05, which gen becomes atomic and partially ionized. During collapse is on a high side for the grid of models, so this run presents the fragment contracts to the radius of just a few times that an example of a rather slowly migrating clump. of Jupiter, and then continues to contract further due to a Fig. 2a presents the disc surface density profile at five rapid radiative cooling. This fragment is never challenged different times. The initial condition at t = 0 is shown with by tidal forces of the star as the planet’s Hill radius (red the black solid curve, all the other curves are marked by their dotted curve) is always much larger than Rp. respective times in the legend. The crosses on the bottom The next notable transition in the planet-disc system of the panel show the position of the planet at times colour- occurs at t ≈ 1 Million years, when the planet opens a deep coded in the same way as the surface density curves. gap in the disc. The planet is located at a ≈ 27 AU at Next panel, fig. 2b, reports the radial location of the that time. From that time on, the planet migrates in type planet (black solid curve), the planet’s Hill (red dotted) and II regime. Note that at time t = 1.32 Million years (the its actual (Rp, blue dashed) radii, all ploted versus time. dashed blue curve in fig. 2a), the gap becomes deeper than Boxed text and the corresponding arrows mark the times of it was before, and the inner disc has a significant gas deficit three important transitions in this run. In particular, the at a ∼ 2 AU. This is not only due to the gap openning; the fragment is initially molecular Hydrogen dominated, col- photo-evaporation of the disc is especially strong at a ∼ a lapses at time t = 0.55 Million years, at which point Hydro- few AU, and is negligible within 1 AU, as can be seen from

c 2008 RAS, MNRAS 000, 1–13 6 S. Nayakshin

fig. 1. The planet-facing side of the disc inside the planet’s orbit thus loose mass rapidly due to photo-evaporation. Disc evolution At this point in time, the inner disc is drained roughly 1000 t= 0.18 Myr equally efficiently by both accretion onto the star and photo- 0.23 Myr evaporation. By time t = 1.5 Million years, there is a com- 1.20 Myr ]

-2 1.50 Myr plete gap between the planet and the star. The outer disc is 100 then evaporated too, and the planet stops migrating com- Tidal disruption

[g cm Gap closed pletely. Σ Finally, fig. 2c shows the core mass assembled within the 10 fragment, and its decomposition on the silicates (“rocks”) (a) plus water ice and CHON. Since the fragment heats up quickly when pebble accretion commences, water ice grains 0.1 1.0 10.0 100.0 vaporise before they could sediment into the core, hence R, AU Super Earth planet formation they do not contribute to the core at all. CHON grains 100.0 (green dashed curve) sediment while the centre of the frag- ment is cooler than T ≈ 400 K, and then only the rocks Gap openned are refractory enough to continue to accrete onto the core. 10.0 Tidal disruption This particular planet did not manage to assemble a mas- Gap closed sive solid core before it collapses (when all grains not yet locked into the core are vaporised); the final core mass is 1.0 only Mc = 0.55 M⊕. Separation, a The final metallicity of the planet is z = 0.158, about Radius, R Separation and radii [AU] p 0.1 (b) Hill’s radius, R ten times that of the initial (Solar) disc metallicity. This H example thus shows how a metal-rich gas giant planet with a low mass core can be assembled in the context of the TD (c) hypothesis. ] Earth 1.0 Total Core mass 3.2 A hot Super-Earth planet Rocks CHON Figure 3 demonstrates how a hot Super-Earth planet can be

assembled in the framework of TD. The initial mass of the Core mass [M planet is again Mp = 1.0 MJ. The initial mass of the disc is 0.1 Md = 0.066 M⊙, planet’s location is a0 = 108 AU, and the initial planet’s central temperature Tc = 282 K. The disc 0 100 200 300 400 500 600 metallicity is z = 1.7 times Solar, the pebble mass fraction time [103 years] −3 is fp = 0.08, disc viscosity parameter α0 = 2.3 × 10 . The disc removal time is trem ≈ 5 Million years, but again disc dissipates earlier due to photo-evaporation. The planet’s Figure 3. Same as Fig. 2, but for a simulation that forms a type I migration speed is barely reduced from the isothermal hot Super-Earth planet as the end result (§3.2). The blue box in the top panel (a) shows where and when the gas fragment is result, e.g., factor fmigr = 1.3, so the fragment migrates tidally disrupted, which leaves behind a massive Mc ∼ 6.4M⊕ in much sooner than it does in §3.1. The maximum grain core that continues to migrate, and finally arrives in the “hot sedimentation velocity for this simulation is set to vmax = region”, a = 0.23, by the time the disc is dispersed. 10 m s−1, which, combined with a higher z value, encourages a faster solid core growth. It appears that the much more rapid migration is to from the star). Now, when pebble accretion onto the frag- blame for most of the differences in the fate of the fragment ment stops, its contraction stops as well. However, since the here as compared to the much more slowly migrating one in fragment is metal-rich, the core of the fragment continues §3.1. Fig. 3 shows that the fragment finds itself in the inner to grow in mass rapidly and its luminosity becomes suffi- disc much quicker than the one in Fig. 2 did. The gap in ciently high to start puffing the fragment up. This increase the disc is openned by the fragment at around a = 7.2 AU, in the fragment’s radis after the disc gap openning is easy to at time t = 0.18 Million years. Significantly, the fragment is spot in Fig. 3b. Since the fragment continues to migrate in, still in the pre-collapsestate at that moment, and it also has now in type II regime, the planet’s Hills radius continues to a massive solid core, Mc = 4.8 M⊕ at the time of the gap shrink, whilst the planet’s radius, Rp, increases with time. openning. The more massive core is due to the higher value A tidal disruption is thus unavoidable, and indeed occurs vmax in this simulation. when the fragment reaches a = 3.16 AU. Contraction of the fragment in this simulation occurs The core’s mass is then equal to Mc = 6.4 M⊕. As the almost entirely due to pebble accretion because radiative fragment is disrupted, its mass plumets to Mc (in this paper cooling is very inefficient. This is because the fragment’s we neglect a possible tightly bound dense gas “atmosphere” dust opacity is high, especially so due to pebble accretion: by around the core, which is not likely to be massive as our t = 0.18 Million years the fragment metallicity is z = 0.144. cores are quite bright). With this reduced mass, the planet The fragment is also basked in the radiation field of the disc is no longer able to maintain the deep gap in the disc, which (which at the outer disc radii is dominated by irradiation is swiftly closed (see the blue dashed curve in Fig. 3a). The

c 2008 RAS, MNRAS 000, 1–13 Planet-metallicity correlations 7 mass of the core is however sufficiently high for it to continue Table 1. Fixed parameters of the simulations and their values: its migration in type I now, on a longer time scale yet fast maximum velocity for sticking rather than fragmentation, vmax, −1 enough to arrive at a = 0.23 AU by the time the disc is inm s ; core’s Kelvin-Helmholtz contraction time, tkh, in years; dispersed at t = 1.5 Million years. opacity reduction factor due to dust growth, fop; pebble radius, The composition of the core is almost exclusively rock, apeb, in cm; ionising photon flux from the star, Φion, photons −1 −1 with CHON and especially water ice grains unable to sedi- s ; X-ray luminosity of the star, LX , in erg s . ment because the core heats up to temperatures above ∼ 400 K quickly (cf. paper I for more detail). The core’s compo- Parameter vmax tkh fop apeb Φion LX sition is significantly different from that predicted by the Value 3 3 × 105 0.5 0.03 1042 2 × 1032 Core Accretion paradigm, in which ices are very impor- tant (Pollack et al. 1996; Alibert et al. 2005) for assembly of cores as massive as this one. This may be a testable dif- Table 2. Randomly varied parameters of the grid of models. The ference between the models if composition of Super-Earths first row gives parameter names, the next two their minimum is constrained observationally, although the potential pres- and maximum values. The columns are: α0, the disc viscosity pa- ence of a Hydrogen/He atmosphere on top of the core may rameter; fp, the pebble mass fraction; factor; a0 [AU], the initial complicate differentiation between the models. position of the fragment; Md, the initial mass of the disc, in units of M⊙; fmigr, the reduction multiplier for type I planet migra- tion; trem, Million years, the time for disc removal; αd, turbulence parameter within the fragment. 4 THE POPULATION SYNTHESIS GRID OF Parameter α fp a0 Md fmigr trem αd MODELS Min 0.001 0.05 60 0.05 1 0.5 10−4 A population synthesis involves investigation of the model’s Max 0.01 0.2 120 0.2 10 5 0.003 parameter space, which is usually large, by randomly sam- pling it (e.g., Mordasini et al. 2009) in order to pull out sta- tistical trends of the model that may not be distinguisheable Similarly, random uniformly distributed variables are gener- from just a handful of runs. In following this approach, we ated for all the other entries in Table 2. fix some parameters of the model in order to reduce the pa- Having defined the parameters for a simulation, we rameter space to a manageable but physically meaningful model one gas fragment per disc at a time. The fragment minimum. starts with a given initial total (H/He + grains) mass, and Table 1 lists these fixed parameters, their meanings and has the same metallicity as that of the disc. The initial values. The latter are chosen to be “reasonable” based on temperature is randomly picked (as described in equation experiment (e.g., vmax) or on what is commongly used in 6 above), between T0(Mp), given by literature (e.g., L ). While these values are not unique, we X M 1/2 found by varying them that none of the parameters in Table T (M ) = 100 K p , (7) 0 p 0.5 M 1 change the metallicity trends that are the focus of this pa- J h i per significantly, although quantitatively there are of course and twice T0(Mp). Here the mass dependence of T0 reflects changes in the results. For example, the normalisation of the fact that more massive fragments cool more rapidly (e.g., the giant planet number per star surviving in the end of the Helled et al. 2008; Nayakshin 2010b), and are also hotter simulations does depend on how the disc is removed in the at formation due to the adiabatic compression. The choice end of the simulations, and hence on LX and Φion, but the made in equation 7 does not pre-determines or drives the metallicity trends of the survived planets are the same. main results of this paper. One parameter in Table 1 that is difficult to constrain Finally, we wish to investigate how our results depend from first principles and which does influences the results is on the metallicity of the host star/disc and the initial mass fop, the grain opacity reduction due to grain growth. The of the fragment. Therefore, the random models are repeated latter is fixed here on a rather large value, fop = 0.5, which for a grid of the metallicity and the initial fragment’s mass (i−3)/2 essentially precludes a significant opacity reduction. This values. The metallicity grid is given by zi/z⊙ = 3 , is done following the arguments presented in paper I: the with i = 1, 2, ... 5. The initial planet mass grid is given by observed metallicity trends are incompatible with low dust Mp/ MJ = 0.5, 0.7, 1 and 2. For each of the combinations opacity models and presumably imply that dust fragmen- of z and Mp from this grid, e.g., z = 3z⊙ and Mp = 1 MJ, tation (and not only dust growth) is occuring within the the total of 243 models are run. This gives the total of 4860 fragments to keep the opacity at relatively large values. runs. A typical run takes about 2 hours of physical time on Table 2 lists the set of variable parameters, and their a single CPU, although some runs (usually with higher disc minimum and maximum values. The parameters for a run mass and viscosity, as that leads to smaller time steps) had are then randomly picked in the logarithmic space between to be executed for up to 48 hours. these respective minimum and maximum values. For exam- ple, the disc viscosity parameter, α0, is very poorly known for protoplanetary discs (and any other astrophysical discs). 5 RESULTS We introduce a random variable ξα uniformly distributed Figure 4 presents the results of the 4860 runs in the planet between 0 and 1, draw a random value ξα, and define α0 for a given simulation by mass versus planet-star separation plane. In order to facil- itate a visual comparison to the exoplanetary data, giant ln α0 = ξα ln αmin + (1 − ξα) ln αmax . (6) planet masses are shown as Mp sin i, where i is inclination

c 2008 RAS, MNRAS 000, 1–13 8 S. Nayakshin of the system to the observer, generated randomly assuming we essentially simulate the end phase of the disc evolution. a uniform distribution for cos i. This is done because many There is no reason why fragments could not be born be- of the known giant planets were detected by the radial veloc- fore the phase we are simulating. These would migrate even ity method and so also have the unknown sin i factor. The faster, and most likely would all end up either being tidally gas giants are defined here as fragments that did not have disrupted or being driven all the way to the inner bound- a tidal disruption and hence are all more massive than our ary. Hence the number of giant planets migrating all the minimum starting fragment mass, e.g., 0.5 MJ. All of these way into the star is still an under-estimate. This is consis- planets are above the horizontal dashed line. tent with the view presented earlier – that destroyed gas Cores, the remnants of tidal disruption of the initial giant planets cause FU Ori outbursts and that statistically gas fragments, are the planets below the dashed line. In Fig. speaking, there may be many such events per protostar. 4, all of the survived giant planets are shown, but, for the sake of clarity, only a third of the cores, randomly selected from the full sample, is plotted. The mass of the cores, the scale for which is given in units of M⊕ on the right vertical 6 METALLICITY CORRELATIONS axis, is not multiplied by the sin i factor since many of the 6.1 Small separations observed planets of this mass are detected by the transit method which is not affected by the inclination uncertainty. Figure 5 shows the number of planets formed in our runs at We emphasise that in this paper no consideration for separations 0.1 AU

c 2008 RAS, MNRAS 000, 1–13 Planet-metallicity correlations 9

TD + Pebble Accretion 1000.0

1.0000

100.0

0.1000 Gas Giants ] ] Jup

10.0 Earth

sin i [M 0.0100 p Mass [M M 1.0

0.0010 z = 1/3 Solid core planets z = 0.57 z = 1 z= 1.7 0.1 z = 3 0.0001 0.1 1.0 10.0 100.0 a [AU]

Figure 4. Planet mass versus planet-host separation for the pebble accretion grid of models for different metallicities, as shown in the legend in Solar metallicity units. For the gas giant planets (historically found by RV methods) Mp sin i is used as proxy for Mp, where sin i is randomly generated orbital inclination with respect to the line of sight. For core-dominated planets, located below the dash-dotted line, Mp in units of Earth mass (on the right vertical axis) is plotted. the time window for grains to sediment down into the core the fragment does not migrate it is not tidally disrupted in is shorter because the fragment heats up too quickly and the these isolated runs. grains are eventually vaporised which clearly stops accretion Figure 6 shows the results of this series of runs for three of the core. initial planet masses, Mp = 0.5, 1 and 2 MJ. The top panel This physics is internal to the fragment and has noth- shows the mass of the core assembled inside the fragment by ing to do with the fragment-disc interaction. Therefore, to the time it collapses versus the metal loading time on the investigate it in greater detail, we perform a series of runs in horizontal axis. For Mp = 1 MJ, the core mass is essentially which we turn off fragment’s migration and keep the frag- independent of tz. The bottom panel of Fig. 6 explains the ment’s pebble accretion rate constant, parameterised, as in reason for this finding to some degree. The panel shows the Nayakshin (2015a), in terms of the ”metal loading time”, tz, time it took the fragment to collapse. Unsurprisingly, the defined by faster the metals are added to the planet, the faster it col- lapses, so that the core indeed has a shorter time window to dMz z⊙Mp grow. For M = 1 M the two trends – the faster core growth = , (8) p J dt tz and the shorter time available for that – nearly cancel each other out, yielding a nearly constant result. For Mp = 2 MJ, where z⊙ = 0.015 is the Solar metallicity. Thus tz is the time the dependence is mostly flat, but at the longest tz the core scale on which the planet’s metal content increases by the mass actually decreases. This is correlated with the time it amount contained in that planet at Solar metallicity. Since takes for the fragment to collapse (the bottom panel of Fig.

c 2008 RAS, MNRAS 000, 1–13 10 S. Nayakshin

6). Physically, for Mp = 2 MJ, the longest tz planets collapse faster because their contraction is dominated by radiative cooling rather than pebble accretion. Radiative contraction Gas giants 100 Super Earths is faster at lower metallicities (Helled & Bodenheimer 2011; Earths Nayakshin 2015a), and hence the result. Finally, for the low- est mass planets in Fig. 6, the faster the metals are loaded into the planets (the shorter tz), the lower the mass of the cores assembled. The different behaviour shown by the planets of differ- ent initial mass in their response to a variance in tz is due to a fact that a number of factors, rather than just one, control 10 the core’s assembly rate inside the planet. The most impor- Number of planets tant of these factors are: the central density of the planet, its metal abundance, gas temperature, and the core’s lumi- nosity as this affects the convective flux and the convective 1 grain mixing which opposes grain sedimentation. Metallicity, [Fe/H] The weak dependence of the core’s mass on the metal loading time (and hence on the metallicity of the disc), Figure 5. Number of gas giant, super-Earth and Earth-like shown in Fig. 6, explains why the populations of the rocky planet groups versus host metallicity. Only planets left at sep- planets in Fig. 5 are broadly distributed over metallicities. arations 0.1

section. Collapse time [years]

4 5 6.2 Large separations 10 10 Metal loading time, tz [years] Figure 7 shows the metallicity correlations for planets sur- vived the disc migration phase at the outer a> 10 AU region of the star. For the Jovian mass planets in the figure, the planets that are stranded far from the host star are those Figure 6. Top: the mass of the solid core assembled inside the that migrate the slowest. Therefore, these fragments are not precollapse gas fragment as a function of the metal loading time, likely to be threatened by the tidal disruption, so can take tz, for non-migrating fragments of three different masses, as indi- as long as they need to contract. This is why the popula- cated in the legend. Bottom: The time it takes the fragments to tion of gas giants is now broadly distributed over the host contract from the initial state and collapse via H2 dissociation, t star metallicity, with just a shallow minimum at around the as a function of z. Solar metallicity. This is quite different from Fig. 5 where a strong positive corelation with metallicity was present. The Super Earth and the Earth-like planets in Fig. 7 are the products of tidal disruptions of precollapse gas frag- ments, so these tend to come from those fragments that

c 2008 RAS, MNRAS 000, 1–13 Planet-metallicity correlations 11

gas giant planets at low metallicities than at high metallic- ities, which is observationally not at all the case. 100 In contrast, formation of gas giants in the TD hypoth- esis is not preconditioned by the assembly of solid cores. As we found here, there is actually no simple relationship be- tween these two types of planets in general. Both the rocky Gas giants core dominated planets and the gas giants originate from Super Earths Earths the same source – the precollapse H2 dominated fragments 10 born by the gravitational instability of the outer massive disc – but the efficiency with which the different kinds of planets are assembled from this raw material is a function Number of planets of a number of factors. Our numerical population synthesis models allowed us 1 to study these issues in some detail in this paper. Focusing 1 first on the planets surviving the disc migration phase in Metallicity, [Fe/H] the inner 5 AU of the disc (§6.1), we found that increasing metallicity of the environment leads to a more abundant for- Figure 7. Number of gas giant, super-Earth and Earth-like mation of gas giant planets, as earlier found by Nayakshin planet groups versus host metallicity. Only planets found at sep- (2015b) for coreless gas fragments. This result is driven en- arations a> 10 AU are included in the figure. tirely by the pebble accretion accelerating collapse of the fragments at high metallicites and helping more of them to survive. were migrating faster (physically ”released to migrate” ear- lier, when their discs were more massive) than the gas giant However, formation of solid cores in our models does planets in the figure. The metal correlation of the population not correlate in a monotonic way with the metal content of of the Super-Earth planets in the outer 10 AU is not actu- the parent discs for either Earth-like or Super Earth plan- ally that different from that in Fig. 5, with similar physics ets. There are two primary reasons for this outcome. Firstly, driving it. higher pebble accretion rates on a gas fragment do not nec- Earth-like planets are however much less abundant at essarily make more massive cores. The core’s mass is equal to the integral of the core’s accretion rate over the duration high separations from the host, and especially at the highest t of time period when the core can grow, e.g., grow M˙ dt. metallicity bins. This is due to the fact that very few high 0 c metallicity fragments are disrupted in the outer disc. When The core’s accretion rate, M˙ c, increases with increasing peb- R they are disrupted, they tend to contain more massive cores ble accretion rate, but the duration of the core assembly because these cores had more time to grow by grain sedimen- phase, tgrow, decreases. For these reasons the core’s mass tation than cores disrupted in the inner disc (which would turns out to not depend sensitively on the pebble accretion usually migrate in much faster). This is another interesting rate (which is proxy for the disc metallicity), as could be example where predictions of TD do not follow the simple naively expected (see Fig. 6). ”more metals more cores” logic. Hopefully this prediction Secondly, in accord with the explanation offered for the may be observationally tested, although these relatively low observed gas giant planet correlation with the metal abun- mass planets are not likely to be found by direct imaging dance in TD, fewer precollapse fragments are disrupted by surveys in near future due to their exceedingly low lumi- the stellar tidal forces at higher metallicities. Thus, not only nosities. the cores are not necessarily much more massive at higher metallicities, but they are also less likely to be released from inside the overlying massive gas envelopes. This is the rea- son why the population of the rocky cores nose-dives at the 7 DISCUSSION highest metallicity bin in Fig. 5. This particular result is In this paper we argued for interpreting the metallicity corre- however preliminary and may weaken if we are to consider lations of observed exoplanets in the framework of the Tidal earlier generation of gas fragments that may migrate more Downsizing hypothesis for planet formation rather than in rapidly, so that a sizable fraction of precollapse gas frag- the context of the Core Accretion model. In the latter the- ments is tidally disrupted even at the highest metal abun- ory, the fact that massive cores form at all metallicities but dances (which is not currently the case in our models). In gas giants prefer metal-rich environments is odd. Once a that case there will probably be more rocky planets at the massive core is assembled, accreting a massive gas envelope high metallicity end in Fig. 5. on the top of that core is actually easier in metal-poor envi- Finally, in §6.2 we compared the metallicity correlations ronments. Lower opacities in the envelope allow the heat of of the simulated planets in the outer disc (a > 10 AU) to the core’s assembly to escape from the gas envelope faster, that found in the inner disc. Our models predict that gas accelerating its contraction and the eventual collapse. This giant planets of all metallicites may survive at these ”cold” can be seen from the well known result that the critical core regions of the disc, since tidal disruptions of precollapse frag- mass for atmosphere collapse in the CA theory increases (al- ments are far less likely there. One interesting coincidence though weakly) with opacity of the envelope (e.g., Stevenson here is the fact that the best well known system of giant 1982; Ikoma et al. 2000). So, if massive cores are equally gas exoplanets probably formed by GI of the disc, HR 8799 abundant at all metallicities, one would then expect more (e.g., Marois et al. 2008), is metal poor ([Fe/H] ≈−0.5).

c 2008 RAS, MNRAS 000, 1–13 12 S. Nayakshin

8 CONCLUSIONS Cameron A. G. W., Decampli W. M., Bodenheimer P., 1982, Icarus, 49, 298 We believe the observed metallicity correlations of plan- Cha S.-H., Nayakshin S., 2011, MNRAS, 415, 3319 ets as a function of their size/radius give us a valuable Clarke C. J., 2007, MNRAS, 376, 1350 clue as to how planets form. It is highly significant that Fischer D. A., Valenti J., 2005, ApJ, 622, 1102 gas giant planets are found almost always around metal- Forgan D., Rice K., 2013, MNRAS, 432, 3168 rich stars (Gonzalez 1999; Fischer & Valenti 2005) but plan- Galvagni M., Hayfield T., Boley A., Mayer L., Roˇskar R., ets smaller than ∼ 4R⊕ or less massive than Neptune are Saha P., 2012, MNRAS, 427, 1725 abundant around stars of any metallicity (Sousa et al. 2008; Gammie C. F., 2001, ApJ, 553, 174 Buchhave et al. 2012). Gonzalez G., 1999, MNRAS, 308, 447 If solid cores are assembled first as a prerequisite to gas Hartmann L., Kenyon S. J., 1996, ARA&A, 34, 207 giant planet formation, then it is hard to see why there are Helled R., Bodenheimer P., 2011, ICARUS, 211, 939 massive cores in low metallicity environments but no gas Helled R., Bodenheimer P., Podolak M., et al., 2013, ArXiv giants since the lower the opacity of the envelope the easier e-prints it is to convert a massive core into a gas giant. Helled R., Podolak M., Kovetz A., 2008, Icarus, 195, 863 On the other hand, if gas fragments are the nurseries Helled R., Schubert G., 2008, Icarus, 198, 156 of massive solid cores (Helled & Schubert 2008; Nayakshin Ida S., Lin D. N. C., 2004, ApJ, 616, 567 2011), and if the fragments are preferentially destroyed in Ida S., Lin D. N. 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