How planets grow by pebble accretion - IV: Envelope opacity trends from sedimenting dust and pebbles

Astronomy & Astrophysics manuscript no. main ©ESO 2021
 June 8, 2021

 How planets grow by pebble accretion
 IV: Envelope opacity trends from sedimenting dust and pebbles
 M. G. Brouwers1 , C. W. Ormel2 , A. Bonsor1 , and A. Vazan3, 4

 1
 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA
 2
 Department of Astronomy, Tsinghua University, Haidian DS 100084, Beijing, China
 3
 Department of Natural Sciences, Open University of Israel, 4353701 Raanana, Israel
 4
 Astrophysics Research Center of the Open university (ARCO), The Open University of Israel, P.O Box 808, Ra’anana, Israel
 e-mail: mgb52@cam.ac.uk

 June 8, 2021
arXiv:2106.03848v1 [astro-ph.EP] 7 Jun 2021

 ABSTRACT

 Context. In the theory of pebble accretion, planets form by the subsequent accretion of solids (micron-sized dust and larger pebbles)
 and gas. The amount of nebular gas that a planet can bind is limited by its cooling rate, which is set by the opacity of its envelope.
 Accreting dust and pebbles contribute to the envelope opacity and, thus, influence the outcome of planet formation.
 Aims. Our aim is to model the size evolution and opacity contribution of solids inside planetary envelopes. We then use the resultant
 opacity relations to study emergent trends in planet formation.
 Methods. We design a model for the opacity of solids in planetary envelopes that accounts for the growth, fragmentation and erosion
 of pebbles during their sedimentation. It also includes a separate dust component, which can be both replenished and swept up by
 encounters with pebbles, depending on the relative velocities. We formulate analytical expressions for the opacity of pebbles and dust
 and map out their trends as a function of depth, planet mass, distance and accretion rate.
 Results. The accretion of pebbles rather than planetesimals can produce fully convective envelopes, but only in lower-mass planets
 that reside in the outer disk or in those that are accreting pebbles at a high rate. In these conditions, pebble sizes are limited by
 fragmentation and erosion, allowing them to pile up in the envelope. At higher planetary masses or reduced accretion rates, a different
 regime applies where the sizes of sedimenting pebbles are only limited by their rate of growth. The opacity in this growth-limited
 regime is much lower, steeply declines with depth and planet mass but is invariant with the pebble mass flux. Our results imply that
 the opacity of a forming planet’s envelope can not be approximated by a value that is constant with either depth or planet mass. When
 applied to the Solar System, we argue that Uranus and Neptune could not have maintained a sufficiently high opacity to avoid runaway
 gas accretion unless they both experienced sufficiently rapid accretion of solids and formed late.
 Key words. Planetary systems – Planets and satellites: composition – Planets and satellites: formation – Planets and satellites:
 physical evolution – Planet-disk interactions

 1. Introduction Besides these global effects, the accretion of smaller particles
 also has important consequences for a planet’s internal structure.
 Pebble accretion is a version of core accretion where planets So far, most inquiries have focused on compositional changes,
 grow primarily by intercepting a stream of sub-cm sized par- as the rapid vaporization of pebbles (Love & Brownlee 1991;
 ticles (Ormel & Klahr 2010; Lambrechts & Johansen 2012). McAuliffe & Christou 2006) naturally leads to the deposition of
 Previous works have mainly considered its dynamical aspects significant amounts of vapor, which can be stored in the deep in-
 and have demonstrated that it can provide a rapid channel for terior (Iaroslavitz & Podolak 2007; Lozovsky et al. 2017). This
 growth (Morbidelli & Nesvorny 2012; Chambers 2014), espe- naturally limits the size of central cores (Alibert 2017; Brouwers
 cially when pebbles are settled in the mid-plane and accretion et al. 2018) and results in a dense, polluted interior (Iaroslavitz
 proceeds in a 2D manner (Liu & Ormel 2018; Ormel & Liu & Podolak 2007; Venturini et al. 2016; Lozovsky et al. 2017;
 2018). A key feature of pebble accretion is that smaller parti- Bodenheimer et al. 2018). The traditional rigid core-envelope
 cles are naturally prevented from reaching the planet once it has structure of these planets is replaced by the natural emergence
 grown sufficiently large to perturb the surrounding disk (Mor- of a compositional gradient (Valletta & Helled 2020; Ormel
 bidelli et al. 2015; Bitsch et al. 2018; Eriksson et al. 2020). This et al. 2021; Vazan et al. 2020). This seems to be in line with
 has been suggested as a way of discerning gas and ice giant Juno’s measurements of Jupiter’s gravitational moments of iner-
 formation (Lambrechts et al. 2014), and of limiting the growth tia, which infer such a dilute core structure (Wahl et al. 2017;
 of super-Earths around low-mass stars (Liu et al. 2019; Chen Debras & Chabrier 2019; Debras et al. 2021).
 et al. 2020). Its drag-based mode of capture also introduces an
 asymmetry that might explain the structural prograde rotation of Before pebbles get to these inner regions, however, they must
 smaller solar system objects (Johansen & Lacerda 2010; Visser first sediment through the tenuous outer layers where their com-
 et al. 2020). bined surface area can contribute to the opacity. This influence
 of pebbles on the outer envelope is still largely unexplored, but
 Corresponding author: Chris Ormel (chrisormel@tsinghua.edu.cn) has important thermodynamic consequences. The more opaque
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A&A proofs: manuscript no. main

an envelope, the less heat is able to escape it and the less gas Pebble-pebble collisions Pebble-dust collisions
it can gravitationally bind. By regulating the pace of cooling,
the opacity is a key variable that directly affects the outcome of Coalescence ( < frag ) Erosion ( > erosion )
planet formation. The quantitative importance of the opacity is +
often understated in current formation models, where a lack of
physically motivated values is still a serious issue. It is common or
practice to adopt conveniently high ISM-like opacities or values
 Fragmentation ( > frag )
that are arbitrarily scaled down, with important consequences for
 peb = min( vlim, coal)
the outcome of these models. One of the reasons that envelope +
opacity values are currently poorly constrained, is that previous
grain growth models focused specifically on the formation of
Jupiter at 5 AU. In the most detailed of these, Podolak (2003);
 Dust
Movshovitz & Podolak (2008); Movshovitz et al. (2010) calcu- Dust production Equilibrium Dust sweep-up ( < erosion )
lated grain growth with numerical models that solve the Smolu-
chowski equation under the assumption that grains stick when
they collide. They found that grain growth effectively reduces
the dust opacity as a function of depth, with the opacity declin-
ing by around three orders of magnitude from the outer layers to Fig. 1. Qualitative sketch of our two-population opacity model. De-
the inner radiative-convective boundary (RCB). Simpler analyt- pending on the velocities of sedimenting pebbles, they can experience
 growth (coalescence), fragmentation and erosion. In the growth-limited
ical (Mordasini 2014) and numerical (Ormel 2014) models with regime, the pebble size in the interior is regulated by the collision and
a single characteristic grain size at a given height have been able sedimentation timescales (Rpeb = Rcoal ). In the velocity-limited regime,
to replicate this main result. However, because proto-planetary fragmentation and/or erosion restrict the pebble size below the growth
disks are thermodynamically very different across distances and potential (Rpeb = Rvlim < Rcoal ). In our model, dust grains of constant
masses, it is not clear that these results can be applied generally size are produced in collisions between pebbles or in high-velocity
to planets of varying sizes throughout the disk. pebble-dust encounters (erosion), while they are lost by sticking en-
 In addition, whether a planet is predominantly accreting sub- counters with slow-moving pebbles (dust sweep-up), leading to a local
cm pebbles or 100-km planetesimals is clearly important for the steady state in their abundance.
abundance of grains in the outer envelope. The assumption of
grain growth models that planetesimals deposit significant large findings to the formation of gas and ice giants in Sect. 5. We dis-
amounts of small grains in the upper layers is in conflict with cuss our model in relation to contemporary works, list potential
impact simulations, which predict that most of their mass is re- model improvements and contrast the scenarios of pebble and
leased close to the planet’s central cores, far below the RCB planetesimal accretion in Sect. 6. Finally, we conclude our work
(Brouwers et al. 2018; Valletta & Helled 2019, 2020). In con- in Sect. 7. In appendix A, we derive an expression for the non-
trast, Ali-Dib & Thompson (2020); Johansen & Nordlund (2020) isothermal critical metal mass that we use to study the onset of
have recently shown that pebbles are very susceptible to erosion runaway growth. In appendix B, we motivate our constant dust
by small dust grains when they enter planetary envelopes, es- size by comparing how far dust grains can travel before they col-
pecially when they are accelerated further by convective cells. lide. In the final appendix (C), we vary the limiting velocity, the
It was also pointed out by Ormel et al. (2021) that even without parameter in our model that represents the onset of fragmenta-
any size evolution, mm-sized pebbles can contribute a significant tion as well as erosion by micron-sized grains.
opacity to planetary envelopes if they accrete at a sufficiently
high rate.
 The total envelope opacity is a sum of the contributions of 2. Model description
solids and gas (κ = κs + κgas ). In this study, we formulate a model 2.1. Two-population approach
for the contribution of solids, which often dominates over the gas
during accretion. In Sect. 2, we develop a physical opacity model Growing proto-planets are supplied with a flux of solid particles
that accounts for the main processes that influence the evolution that enter their envelopes by two distinct mechanisms. The first
of solids in planetary envelopes. We consider a variation to the is the drag-assisted capture of sub-cm pebbles that drift towards
single-size approximation where we model the populations of the central star (Ormel & Klahr 2010; Lambrechts & Johansen
both small dust (κd ) and larger pebbles (κpeb ). The characteris- 2012), supplemented by the accretion of larger, km-sized plan-
tic size of the pebbles in our model is either determined by their etesimals (Alibert et al. 2018; Guilera et al. 2020). The second
growth from sticking collisions (coalescence) at low velocities or source is the population of small, micron-sized dust grains that
limited by fragmentation and erosion at high velocities, depend- are coupled to the nebular gas and that enter the planet’s enve-
ing on the local thermodynamic conditions. When pebbles travel lope when it is massive enough to bind one.
at speeds below the erosion velocity, they sweep up dust grains In two recent works, Ali-Dib & Thompson (2020); Johansen
without experiencing mass loss (Krijt et al. 2015; Schräpler et al. & Nordlund (2020) have shown that the interaction between dust
2018). We assume a constant dust size and calculate its steady and pebbles is crucial in evaluating their relative abundances, as
state abundance between dust sweep-up and production, which high-velocity collisions between the two can significantly erode
can occur as a by-product from collisions between pebbles or by the incoming pebbles and add to the dust population (Krijt et al.
erosion. In Sect. 3, we then formulate simple, physically moti- 2015; Schräpler et al. 2018). This erosion is a runaway process
vated analytical expressions for the opacity from solids, similar and is only halted at the pebble size where collisions switch to
to what was done by Mordasini (2014). We apply our opacity sticking and the dust abundance begins to drop. In our model, we
model across a wide parameter space to explore and map out the incorporate both dust and pebbles in a simple manner where each
resulting opacity as a function of distance, planet mass and ac- is represented with a single characteristic size. The pebble size is
cretion rate in Sect. 4. We then present the implications of these determined locally by collisions within the population (growth or
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M. G. Brouwers et al.: How planets grow by pebble accretion

Table 1. Descriptions and values of the default model parameters. The larger particles, the free-fall term instead dominates and gener-
disk conditions are adopted from the Minimum Mass Solar Nebula and ally, the total sedimentation velocity relative planet’s core is a
scale as T disk ∝ d−1/2 , ρdisk ∝ d−11/4 (Weidenschilling 1977b; Hayashi
1981).
 sum of the two:
 vsed = vfall + vgas . (3)
 Parameter Description Value
 Mp Total planet mass 5 M⊕ We consider quasi-static envelopes, where the grain transport
 Mc Central core mass 2 M⊕ timescale is assumed to be short relative to the timescale on
 which the envelope’s thermodynamic conditions change. This is
 ρc Central core density 3.2 g/cm3
 a good assumption in the outer layers, provided that the total en-
 ρ• Dust and pebble density 3.2 g/cm3
 velope mass is dominated by the planet’s polluted interior. If the
 Ṁpeb Pebble accretion rate 10−6 M⊕ /yr solids additionally enter in a spherically symmetric manner, their
 Ṁxy Gas accretion rate 10−7 M⊕ /yr volume density is given by:
 Rd Dust monomer radius 1 µm
 F Dust replenishment constant 0.1 Ṁs
 vfrag ρs = , (4)
 Fragmentation velocity 0.8 m/s 4πr2 vsed
 T vap Sublimation temperature 2500 K
 d Orbital distance 5 AU which represents radial mass conservation. We do not include
 M? Mass of the central star 1M large-scale convective motions (Popovas et al. 2018, 2019) in
 our model, which could locally transport clumps of solids and
 ρdisk,5AU Disk density at 5 AU 5 × 10−11 g/cm3
 alter their collision velocities (Ormel & Cuzzi 2007).
 T disk,5AU Disk temperature at 5 AU 150 K
 µxy Molecular weight nebular gas 2.34 mH
 ∇ad Adiabatic temperature gradient 0.31 2.3. Dust-pebble collisions
 2.3.1. Erosion at high velocities
fragmentation), as well as collisions with the dust (sweep-up and The sizes of pebbles that enter planetary envelopes are deter-
erosion). The size of individual dust particles is a parameter in mined by their growth, fragmentation and radial drift in the
our model, which we keep constant as motivated by the ongoing proto-planetary disk (i.e. Liu & Ji 2020; Drazkowska et al.
production and sweep-up of these smaller grains in the envelopes 2021). The observational evidence points towards typical sizes
(see Appendix B). Fig. 1 provides a qualitative overview of our between 100 µm − 1 cm (i.e. Birnstiel et al. 2010; Kataoka et al.
model, which we work out in the next subsections. 2016; Hull et al. 2018; Carrasco-González et al. 2019; Ohashi
 et al. 2020; Tazzari et al. 2020b), which generally decrease with
2.2. Sedimentation of solids distance to the central star (Tazzari et al. 2020a). During their
 capture by the planet, pebbles are subject to additional gravita-
Particles referred to as dust and pebbles are small enough to be tional acceleration and begin to collide with dust at increasing
effectively slowed down by gas drag and sink at speeds close velocities. It was pointed out by Ali-Dib & Thompson (2020);
to their terminal velocities (vfall ) relative to the local medium. Johansen & Nordlund (2020) that if this velocity crosses the
We adopt a simple, continuous two-regime approximation to threshold for erosion, the dust particles begin to chip off pebble
the drag force, where the transition from free molecular (Ep- material. This process was measured by Schräpler et al. (2018)
stein) to continuum flow (Stokes) is set at the canonical value to begin at:
lmfp /Rs = 4/9 and the terminal velocity is given by (Weiden- ! 1
schilling 1977a): Rd 1.62
 verosion = 2.4 m/s , (5)
 1 µm
 gRs ρ•
 !
 4Rs
vfall = max ,1 , (1) with an erosive mass loss per collision ∆merosion of:
 ρg vth 9lmfp
 !−0.62
 ∆merosion v
 !
 Rd
where g and ρg are the local gravitational acceleration and gas = 4.3 , (6)
density at a distance from the planet’s center r and the subscript md 10 m/s 1 µm
s refers to either population of solids (pebbles with size Rpeb and which requires a mass ratio below ∼ 10−2 (Krijt et al. 2015). The
dust with size Rd ), which share the same material density ρ• . proportionality of the erosion efficiency to the collision veloc-
We take the standard ideal gas expressions for the thermal ve- ity was also found in numerical investigations (Seizinger et al.
locity vth and the mean free path of molecules lmfp . The second 2013; Planes et al. 2017). The fragments that are produced are
component to the sedimentation velocity of solids is given by generally similar or smaller in size than the projectiles, which in
the downward flow of gas. Because we are mostly interested in our model are dust grains with mass md . Because the produced
the planet’s upper layers that only contain a fraction of the enve- fragments add to the dust population, the erosion of pebbles is
lope mass, we can approximate this gas velocity (vgas ) from mass a runaway process up to a characteristic size where they slow
conservation of the gas accretion rate ( Ṁxy ) as: down sufficiently by their increased susceptibility to gas drag. In
 our model with linear downward sedimentation, this is set by the
 Ṁxy condition that vpeb,fall = verosion (assuming vfall,d  vfall,peb ):
vgas = . (2)
 4πr2 ρg  ρg vth verosion
  (Epstein)
 gρ•
 
 
 
 When the solids are small enough, as is the case for the micron- 
 
 
 Rerosion = 
 
sized dust particles in our model, the local downward flow of gas (7)
 
  !1
dominates their total sedimentation velocity. This was described 3 ρg vth lmfp verosion 2
 
 
 
 (Stokes).
 
 
as the advection regime in the work by Mordasini (2014). For gρ•
 
 2
 

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A&A proofs: manuscript no. main

 material. In this sweep-up regime, Eqs. 9a - 9c still apply with
 10 cm ∆m = −md :
 101
 d Ṁd Ṁpeb md
 = , (10)
 1 cm dr lsweep,peb mpeb
 100
 sweep−up
Rpeb [cm]

 Now, the presence of pebbles acts to reduce the dust abundance,
 rather than increase it. In Sect. 3.2, we will discuss the scenario
 0.1 cm
 10 1 where the production of dust by erosion and pebble-pebble col-
 Rerosion lisions is in equilibrium with the sweep-up of dust, providing a
 steady state dust abundance.

 10 2
 0.90 0.92 0.94 0.96 0.98 1.00 2.4. Pebble-pebble collisions
 r/rB When considering straight downward sedimentation, collisions
 between two pebbles can occur either through Brownian motion
 Fig. 2. Erosive mass loss of pebbles that enter envelope of a growing or by differential settling (coalescence). The timescale of the lat-
 planet at its Bondi radius (rB ). The parameters of the planet are shown ter process decreases with increasing particle size, such that co-
 in Table 1. The three lines correspond to initial pebble entry sizes of 0.1
 alescence easily dominates their mutual collision rate when the
 cm (dotted), 1 cm (dashed) and 10 cm (solid). Due to the positive feed-
 back of erosive mass-loss on the dust abundance, the pebbles rapidly pebbles are larger than several microns (Boss 1998; Nayakshin
 converge to the size Rerosion (Eq. 7) where collisions with dust grains 2010; Mordasini 2014; Ormel 2014). As identical particles settle
 switch from causing mass loss by erosion to sticking (sweep-up). with the same speed, coalescence within a population formally
 arises from the parameterized width in the size distribution. In
 our model, we follow the works by Rossow (1978); Mordasini
 (2014) and set the typical mass ratio of collisions to 0.5. Al-
 In order to estimate how rapidly pebbles erode, we formulate the though a simplification, Krijt et al. (2016); Sato et al. (2016)
 radial equations of mass transfer and size evolution. The typical show that this is a good approximation to the average mass ra-
 distance that pebbles travel before they encounter a dust particle tio found in coagulation simulations that treat a complete size
 is given by (assuming Rd  Rpeb and vfall,d  vfall,peb ): distribution. When combined with Eq. 3, it leads to collisional
 md vsed,peb velocities in the range of 0.21 < vcol /vfall < 0.37 depending on
 lsweep,peb = , (8) the drag regime. In our model, we neglect this order unity dif-
 ρd σpeb vfall,peb ference in favor of a continuous collision velocity across drag
 regimes and approximate it as xR ≡ vcol /vfall ' 1/3.
 where σpeb = πR2peb is the pebble cross section. The equations
 for mass transfer and shrinkage follow as:
 2.4.1. Growth by coalescence at low velocities
 d Ṁpeb Ṁpeb ∆merosion
 = , (9a) When the collision velocities are sufficiently low, contact be-
 dr erosion lsweep,peb mpeb tween pebbles leads to sticking and growth (Dominik & Tielens
 d Ṁd d Ṁpeb 1997). To work out the rate of this growth, the typical travel dis-
 =− , (9b) tance between pebble collisions (lcol,peb ) can be related to their
 dr erosion dr local size, abundance and velocity as:
 dmpeb ∆merosion
 = , (9c) mpeb vsed,peb
 dr erosion lsweep,peb lcol,peb = . (11)
 ρpeb σpeb vcol,peb
 where Ṁpeb is the pebble mass flux. In this simple model, the in-
 Eq. 11 scales positively with pebble size, which indicates that
 crease in dust abundance from pebble erosion is essentially ex-
 smaller particles collide more often. By equating lcol,peb to the
 ponential and the pebbles shrink very rapidly, even without as-
 scale height H = kb T g /(µg g), which represents the characteris-
 suming any additional acceleration from convective eddies. As
 tic length scale in an envelope, it is possible to approximate the
 an example, we integrate Eqs. 9a-9c from the Bondi radius in-
 maximum sizes (Rcoal ) to which solids can grow as they sedi-
 wards for a nominal planet of 5 M⊕ at 5 AU with a dust disk-
 ment:
 to-gas ratio of 10−3 (see Table 1 for all default parameters). The
 resulting size evolution curves are shown in Fig. 2 for three dif-
  !1
 
 
  3xR H Ṁpeb vth ρg 2
 ferent pebble entry sizes. In all the runs, the pebbles quickly 
  (Epstein)
 16πGMp ρ2•
 
 
 
 erode down to the size where their velocity decreases below the 
 
 Rcoal = 
 
 (12)
 
 threshold for erosion and they are safe. In this manner, erosion 
  ! 13
 essentially cancels out the upper range of the initial pebble size 3 xR H Ṁpeb vth lmfp ρg
 
 
 
 
 (Stokes),
 
 distribution to more uniform pebble entry sizes, limited by their
 
 πGMp ρ2•
 
  4
 
 erosion at the Bondi radius.
 where we assumed that their total velocities are dominated by the
 2.3.2. Sweep-up at low velocities terminal component rather than by inward gas flow. In the regime
 where growth by coalescence provides the limit to their size, the
 If the pebbles are reduced in size below Rerosion , the relative pebbles are typically large enough (Rpeb & 100 µm) that this is
 velocities between the dust and the pebbles drop below verosion indeed a good assumption. The scaling of Eq. 12 with the pebble
 and dust can stick to the pebbles without dislodging additional accretion rate shows that a greater mass flux of pebbles leads to
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M. G. Brouwers et al.: How planets grow by pebble accretion

more collisions and faster growth. Its dependence on the planet’s 102 101
mass is more complicated. More massive planets have a stronger
gravitational pull, which increases the distance at which they can Stokes Epstein 100
bind gas and, therefore, extends the envelope (rB ∝ Mp ). This
lowers the gravitational acceleration at the planet’s outer bound- 101 Rvlim, B > Rcoal, B 10 1

 Rvlim, B [cm]
 Mp [M ]
ary as a function of mass (g ∝ Mp /rB2 ∝ Mp−1 ). Taken together, (growth)
the collision distance scales as lcol,peb ∝ Mp , which balances with 10 2
the increasing scale height H ∝ Mp and leads to a constant value Rvlim, B < Rcoal, B
of Rcoal across planetary masses. 100 (erosion / fragmentation) 10 3

2.4.2. Collisional fragmentation at high velocities 10 4
 10 1
At higher velocities, collisions between pebbles lead to bouncing 10 1 100 101
or even fragmentation. Collision experiments typically find that
silicate particles begin to fragment if their collision velocity is
 d [AU]
larger than 1 m/s (Wurm et al. 2005; Schäfer et al. 2007; Güttler Fig. 3. Characteristic pebble size limits at the Bondi radius, plotted
et al. 2010), and up to 10 m/s for water ice grains (Gundlach & with the parameters of Table 1. The colors indicate the pebble size
Blum 2015; Musiolik & Wurm 2019). This fragmentation crite- where vfall,peb = vlim = 2.4 m/s, the common limit set by erosion with
rion can be combined with the velocity for the onset of erosion micron-sized grains and fragmentation. The white dashed line separates
(see Eq. 5) to yield a limit on the terminal velocities of pebbles: the Stokes and Epstein regimes while the dotted line indicates the re-
   gions where either growth rate or erosion/fragmentation limits the local
vlim = min verosion , vfrag /xR . (13) pebble size. The plus sign marks the default planet mass (5 M⊕ ) and
 distance (5 AU).
Depending on the dust size and the pebble’s material properties,
either fragmentation or erosion can be the limiting factor to peb-
ble growth. For simplicity, we take a default dust size of 1 µm 3. Analytical opacity expressions
and a fragmentation velocity of 0.8 m/s, for which the two ve-
locity limits on the pebble’s terminal velocity are both equal to In this section, we formulate analytical expressions for the opac-
vfall,peb = vlim = 2.4 m/s. In appendix C, we vary the limiting ity contributions from pebbles and dust. For the pebbles, these
velocity within a wider range. We refer to the velocity-limited follow from the previously identified pebble size limits in differ-
pebble size as Rvlim and it can be found from Eq. 1 as: ent regimes (Rcoal , Rvlim ). The basic expression for the Rosseland
  ρg vth vlim mean opacity from solids (κs ) is:
  (Epstein)
 gρ• 3Qeff ρs
 
 
 
 κs = ,
 
 (16)
 
 
Rvlim =  4ρ• Rs ρg
 
 (14)
 
 ! 12
 ρ v v
 
 
 
  3 g th l mfp lim
 (Stokes). where the extinction efficiency Qeff can be approximated as
 
 
 gρ•
 
 2
 
 Qeff ' min 0.6πRs /λpeak , 2 , with the peak wavelength of the
 

The dependence of Rvlim at the Bondi radius is visualized in Fig. emitted photons by the local gas λpeak (cm) = 0.290/T g (Wien’s
3, with comparison to the limit set by the growth rate. In con- law). Laboratory experiments provide a more detailed temper-
trast to Rcoal , the value of Rvlim scales positively with the planet’s ature scaling of Qeff , as well as a factor ∼ 2 between differ-
mass. Hence, the sizes of pebbles in small envelopes are typi- ent species (Movshovitz & Podolak 2008; Bitsch & Savvidou
cally velocity-limited by erosion or fragmentation, whereas peb- 2021). We do not include these additional details here but do ac-
bles in more massive envelopes are typically only limited by count for the most important opacity difference between species,
their rate of growth. Close to the central star, the disk is rela- which is that solid particles are only present in layers of plane-
tively dense and pebbles of the same size sediment more slowly. tary envelopes that are sufficiently cool for the solids to escape
Consequently, pebbles erode or fragment down to the smallest sublimation. In the case of silicates, which we consider as the
sizes when they accrete onto low-mass planets that reside in the default composition of pebbles and dust, this threshold is posi-
outer disk. tioned deep inside planetary envelopes around ∼ 2500 K.
 Besides limiting the sizes to which pebbles can grow, colli-
sions between pebbles can also potentially result in production
of small dust as collisional by-products. In our two-component 3.1. Pebble opacity expression
model, we include this with a fractional dust production effi- The first opacity regime is the growth-limited regime, which ap-
ciency F, which can theoretically be between 0 (no dust produc- plies when pebbles are traveling relatively slowly through plan-
tion) and 1 (all growth beyond Rvlim is turned into dust). Physi- etary envelopes and experience sticking collisions, rather than
cally, F is a measure for the number of pebble-pebble collisions erosion or fragmentation. In this case, their size can be approx-
needed to completely grind down a pebble. The rate at which this imated by Rcoal (Eq. 12) and their opacity contribution is inde-
dust is produced is then given by the pebble collision rate as: pendent of their accretion rate. It follows the same equation in
 d Ṁd Ṁpeb both drag regimes:
 = −F , (15a)
 dr frag lcol,peb Qeff
 κcoal = , (17)
dmpeb mpeb xR Hρg
 =F . (15b)
 dr frag lcol,peb which was the main finding of Mordasini (2014). As indicated by
We will discuss reasonable values for the parameter F in Sect. Fig. 3 at the Bondi radius, this growth-limited regime is mainly
3.3. applicable in the envelopes of more massive planets, whose
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A&A proofs: manuscript no. main

larger envelopes allow pebbles to sediment more slowly. The re- from which their respective volume densities follow as
quired planetary size to enter this regime at the same accretion
 ρd
rate is an increasing function of distance from the central star, = F xR . (22)
as pebbles in the tenuous outer disk sediment faster than those ρpeb
closer in. When the conditions are such that the terminal veloc-
ity of pebbles exceeds vlim , the size to which solids can grow Eq. 22 implies that in a steady state between fragmenta-
becomes limited to Rvlim (Eq. 14) by either fragmentation or ero- tion/erosion and sweep-up, most of the radial mass flux is gen-
sion. In this regime, the pebble opacity is instead given by: erated by the larger sedimenting pebbles. But if F is near unity,
 the volume density of grains can nevertheless be comparable to
 3Qeff ρpeb that of the pebbles due to their slower sedimentation.
κvlim = (18a)
 4ρ• Rvlim ρg
 
  3Qeff Ṁpeb g 3.3. Dust opacity in steady state
 (Epstein)
 
 
 
 2 ρ g vth vlim (vlim + vgas )
 2
 
 
 
 
  16πr The dust opacity in steady state between dust production and
 = sweep-up is given by:
 
 (18b)
 
   21
 g
  
  Q Ṁpeb
 3Qeff,d ρd
 
   
  8πr2 (vlim + vgas )  ρ ρ3 v v l  (Stokes).
  
 κd =
 
 (23a)
 
 
 • g th lim mfp 4ρ• Rd ρg
 Qeff,d Rpeb
In combination, the pebble opacity can be approximated analyti- = κpeb F xR . (23b)
cally from the expressions above by determining the appropriate Qeff,peb Rd
regime based on the characteristic sizes, which have to be evalu-
ated locally in the envelope: Eq. 23b is proportional to the pebble opacity, only differing from
 its trends due to the additional dependence on the pebble size.
 κ if Rvlim < Rcoal Because the opacity in planetary envelopes is generally far more
 
  vlim
 
 
κpeb =  variable than the pebble size, the opacity from dust in steady
 
 (19)
  κcoal if Rvlim > Rcoal .
 
  state generally follows a very similar trend to that of the pebbles.
 This makes it possible to evaluate the dust opacity, even though
For completeness, we also include the simplest scenario where the parameter F is largely unconstrained. In principle, it can vary
solids enter a planetary envelope with a constant size that re- between 0 − 1, with a value around unity more appropriate in the
mains unchanged during their sedimentation. While we follow erosion-limited regime where any pebble growth beyond the ero-
the more physically motivated expressions from Eq. 19 in the sion limit is converted into dust. Lower values where collisions
rest of this work, the scenario of a constant pebble size might convert a smaller fraction of their mass into dust are likely more
be applicable if pebbles are both sufficiently small to escape appropriate in cases where either fragmentation or growth (both
fragmentation and bounce rather than stick upon contact, as is pebble-pebble interactions) limits the pebble size. We will see in
the case if the pebbles are modeled as small molten chondrules, the next section that with our default value of F = 0.1, the dust
rather than dust agglomerates. Under this assumption, the opac- opacity is typically comparable to the pebble contribution. The
ity follows from Eq. 16, 3 as (see also the work by Ormel et al. biggest positive change with respect to the default model, there-
(2021)): fore, occurs for F = 1. For lower F, the opacity is not affected
 much because the pebble opacity then dominates and remains
 3Qeff Ṁpeb unchanged.
κpeb,cst = . (20)
 16πr2 ρ• ρg Rpeb vsed,peb
 4. Envelope opacity trends
3.2. Steady state between dust replenishment and In this section, we apply our model for the opacity of solids to
 sweep-up. a broad parameter space in order to investigate the its trends in
In previous grain growth models (Ormel 2014; Mordasini 2014), envelopes of planets throughout the disk. We first detail our en-
a large influx of small dust grains in the envelope’s outer lay- velope model and then evaluate the opacity of dust and pebbles
ers was found to quickly diminish due to mutual sticking col- as a function of depth (Figs. 4-6) for a range of planetary masses,
lisions. Hence, even when a significant fraction of the pebble distances and pebble accretion rates. After that, we consider the
mass is transferred to dust grains upon entry by efficient erosion opacity at the boundary between radiative and convective zones
(sect. 2.3.1), the smallest particles soon disappear from the en- to visualize the same trends in a single graph (Fig. 7).
velope, growing to the same limiting sizes as pebbles that we
discussed in the previous section. The diminishing of the small 4.1. Envelope structure
grains abundance is further hastened by the sweep-up of pebbles
below their erosion size. In order to maintain a dust population, We focus our modeling efforts on the outer envelope down to
therefore, it must be supplied by either continued erosion of peb- the polluted region, ulterior to which no significant sublimation
bles that grow beyond the erosion limit, or by fragmentation. If occurs and the opacity of solids is relevant. We refer to an ac-
that happens, the dust abundance has both a source and a sink companying paper by Ormel et al. (2021) and independent stud-
term, generating a steady state when they are equal and oppo- ies by Bodenheimer et al. (2018); Valletta & Helled (2020) for
site: detailed numerical models of polluted envelope interiors. The
 structure of the outer envelope is simple by contrast, as it shares
d Ṁd d Ṁd its gaseous composition with the surrounding disk and is unaf-
 =− , (21) fected by self-gravity prior to the onset of runaway accretion. In
 dr sweep−up dr frag

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M. G. Brouwers et al.: How planets grow by pebble accretion

 100 100 100 100
 (a) (a)
 10 1 10 1

 Rpeb / Rvlim

 Rpeb / Rvlim
Rpeb [cm]

 Rpeb [cm]
 Mp 10 1 10 1
 10 2 10 2
 d
 10 3 10 2 10 3 10 2
 103 100 100
 (b) 101 (b) d
 101
peb [cm2 / g]

 peb [cm2 / g]
 10 1 10 1
 10 1 10 1

 peb /

 peb /
 Mp
 10 2
 10 3 10 2
 10 3

 10 5 10 3 10 5 10 3
 103 100 100
 (c) 101 (c)
 101 d
 10 10
d [cm2 / g]

 d [cm2 / g]
 1 1
 10 1 10 1
 d/

 d/
 Mp 10 2
 10 3 10 2
 10 3

 10 5 10 3 10 5 10 3
 10 2 10 1 100 10 3 10 2 10 1 100
 r / rB r / rB
Fig. 4. Pebble growth tracks (a), their resulting pebble opacity (b) and Fig. 5. Pebble growth tracks (a), their resulting pebble opacity (b) and
produced dust opacity (c) for a standard set of model runs at 5 AU (see produced dust opacity (c) for a standard set of model runs at 5 AU (see
Table 1). The different lines indicate a range of planet masses, with Table 1). This figure is the same as Fig. 4, but now the mass is fixed
the arrow indicating a logarithmic progression from 0.5 − 20 M⊕ . The at the default 5 M⊕ and the planet’s distances from the star are varied
triangles indicate the location of the RCB, while the stars indicate the logarithmically from 0.1 − 30 AU.
depth where the ambient temperature exceeds the sublimation temper-
ature (2500 K) and the opacity from solids vanishes. The colors in the
top panel show the ratio of the pebble size in the model relative to Rvlim
(Eq. 14). The colors in the two lower panels show the relative value of
 The total envelope opacity is defined as the sum of the gas,
the indicated opacity to the total opacity. dust and pebble contributions (κ = κgas + κd + κpeb ). The gas
 opacity is often taken from lookup tables (i.e. Freedman et al.
 2008, 2014) that contain contributions from different gaseous
quasi-hydrostatic equilibrium, its structure equations read: species. We not not model this contribution here but choose to
 ∂m take a simple reference value of κgas as an analytical scaling of
 = 4πr2 ρg , (24a) the molecular opacity from Bell & Lin (1994):
 ∂r
 ∂Pg GMp ρg
 =− , (24b) 2
 ∂r r2 κmol = 10−8 ρg3 T g3 cm2 g−1 , (26)
 ∂T g ∂Pg T g
 = min ∇rad , ∇ad ,
 

 (24c)
 ∂r ∂r Pg
 which was also used in the first two papers in this series. A sim-
where ple molecular opacity scaling has the advantage that trends in the
 3κLPg opacity from solids, which we seek to characterize here, can be
∇rad = (25) more easily isolated in the results.
 64πσ̄GMp T g4
 We integrate the envelope structure equations 24a-24c from
is the radiative temperature gradient, which contains the gravita- the planet’s outer edge, which is taken as the minimum between
tional and Stefan-Boltzmann constants (G, σ̄) and is a function the Hill and Bondi radii. The outer boundary conditions are equal
of the luminosity L and the total Rosseland mean opacity κ. In to the local disk environment, for which we assume the simple
our model, we assume a constant (global) accretion luminosity temperature and density relations from the Minimum Mass So-
equal to L = GMc Ṁpeb /rc as is commonly done. However, we lar Nebula (see Table 1) (Weidenschilling 1977b; Hayashi 1981).
note the caveat that issued by Ormel et al. (2021) that this term The total opacity is explicitly included in the integration of the
can in reality vary substantially, as refractory material can be structure equations, and no iteration of the model is needed. We
absorbed before sinking to the core and a significant portion of use the ideal gas equation to relate the local density to the pres-
the gravitational energy can be processed in envelope mixing or sure and temperature, which is justified in the upper layers of the
used to heat the surrounding gas. atmosphere that are most important for the opacity calculation.
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A&A proofs: manuscript no. main

 100 100 Bondi radius. If the pebble size is instead limited by erosion and
 (a) fragmentation, in which regime their sizes also scale positively
 with planet mass, the opacity from solids is an even steeper de-
 10 1

 Rpeb / Rvlim
 Mpeb
Rpeb [cm]

 clining function. Note that the pebble and dust opacity trends
 10 1 are almost identical because they are nearly proportional to one
 10 2 another (Eq. 23b).
 The difference in opacity between the low- and high-mass
 planets begins at around three orders of magnitude at the Bondi
 10 3 10 2 radius and these differences increase further with depth. Most of
 103 100 this increase is due to the transition from Epstein to Stokes drag
 (b) Mpeb that occurs in the more massive envelopes where larger pebbles
 101 enter a denser medium. With this, the difference at the Bondi
peb [cm2 / g]

 10 1 radius is extended further, to over six orders of magnitude be-
 10 1

 peb /
 tween the plotted values (0.5−20 M⊕ ) at the radiative-convective
 10 2 boundary (RCB).
 10 3

 10 5 10 3 4.3. Trend with orbital separation

 103 1.0 Next, we evaluate the importance of the planet’s distance to the
 (c) Mpeb central star in Fig. 5, where we plot five lines that indicate semi-
 101 0.8 major axes between 0.1-30 AU. The distance to the central star
d [cm2 / g]

 determines the local thermodynamic conditions of the surround-
 10 1 0.6 ing disk. In our model, we use the Minimum Mass Solar Nebula,
 d/

 where the temperature and density scale as T disk ∝ d−1/2 , ρdisk ∝
 0.4
 10 3 d−11/4 (Weidenschilling 1977b; Hayashi 1981). While the local
 0.2 conditions will also generally differ between different disk types
 10 5 and models, the general trend of lower densities and tempera-
 10 3 10 2 10 1 100 tures in the outer regions that receive less light is universal. This
 r / rB trend is important, because a more tenuous medium allows peb-
 bles of the same size to sediment faster. The velocity-limited
Fig. 6. Pebble growth tracks (a), their resulting pebble opacity (b) and sizes, therefore, decrease with distance from the star and peb-
produced dust opacity (c) for a standard set of model runs at 5 AU (see
 bles experience fewer collisions, reducing their rate of growth as
Table 1). This figure is the same as Fig. 4, but now the mass is fixed at
the default 5 M⊕ and the pebble accretion rates are varied logarithmi- well. This trend is reflected in Fig. 5, where the opacity of solids
cally from 10−7 − 10−4 M⊕ /yr. at the Bondi radius is seen to increase with orbital separation,
 regardless of the regime.
 The opacity trend with distance becomes more complex at
4.2. Trend with planet mass greater depth, where the opacity of solids at the RCB is seen
 to converge to similar values for the parameters in Fig. 5. In
In Fig. 4, we plot the pebble sizes (panel a) along with their these runs, the pebble sizes of planets in the outer disk increase
opacity (b) and the dust opacity (c) of our steady state model for with depth as the surrounding medium becomes more dense and
a range of masses that we vary logarithmically between 0.5-20 the same velocity limit allows for larger pebbles. As a result,
M⊕ . The other parameters in the runs share are the default val- the opacities also decline with depth until they are surpassed by
ues from Table 1. We first discuss the top panel, which shows the molecular contribution near the RCB, where the local peb-
two separate trends depending on the process that limits the peb- ble sizes become limited by growth with the parameters plotted
ble size. In the velocity-limited regime (Rpeb = Rvlim , yellow here. For smaller planets in the outer disk, or those that are ac-
color range), the pebble size is limited by either the erosion with creting pebbles at a higher rate, the pebble sizes at the RCB in-
small dust grains or by collisions with other pebbles. Because the stead remain limited by fragmentation and erosion. We will look
Bondi radius is positioned closer to the core in low-mass planets, into the distance trend of this velocity-limited opacity regime in
pebbles of the same size are able to sediment faster (g ∝ Mp−1 ) Sect. 4.5.
and the same terminal velocity limit leads larger pebbles in more
massive planets. Once the planet becomes massive enough that
 4.4. Trend with pebble accretion rate
Rvlim > Rcoal , the pebble size ceases to be velocity-limited and in-
stead becomes only limited by the rate of growth. In this growth Finally, we examine the opacity trend with the pebble accretion
regime, the pebble size is nearly invariant with a further increase rate in Fig. 6, where it is varied between 10−7 −10−4 M⊕ /yr. This
of the planet mass (see Sect. 2.4.1). probes both the variation in solid mass flux and accretion lumi-
 Regardless of the regime, we find that the opacity from solids nosity, which are proportional (L = GMc Ṁpeb /rc ) in our model.
steeply declines when planets grow more massive. This can be The effect on the pebble sizes (top panel) can be divided into
explained with two competing processes. As planets accrete ma- two regimes, depending on whether the pebble size is limited by
terial, the surface area of their Bondi sphere increases quadrati- growth (orange color range) or velocity (yellow color range). In
cally with mass and the same pebbles obscure a smaller fraction the former case, the pebble size scales positively with the ac-
of the envelope. Part of this trend is compensated by the slower cretion rate as an increased pebble mass flux increases the num-
sedimentation of pebbles in more massive planets but the net re- ber of collisions during their sedimentation and allows for faster
sult is that the pebble volume density scales as ρpeb ∝ Mp−1 at the growth. In this regime, the pebble opacity (per Eq. 17) is not
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M. G. Brouwers et al.: How planets grow by pebble accretion

 102 102 104
 (a) 10 7M /yr (b) 10 6M /yr
 102

 rcb [cm2/g]
 molecules
Mp [M ]

 Mp [M ]
 molecules 100
 101 101
 pebbles + dust
 10 2
 pebbles + dust
 Fully 10 4
 100 100 convective
 10 1 100 101 10 1 100 101
 d [AU] d [AU]
 102 102 104
 (c) 10 5M
 /yr (d) 10 4M /yr
 molecules 102
 pebbles + dust

 rcb [cm2/g]
Mp [M ]

 Mp [M ]
 100
 101 101
 pebbles + dust
 10 2
 Fully convective
 Fully convective 10 4
 100 100
 10 1 100 101 10 1 100 101
 d [AU] d [AU]
Fig. 7. Compilation grid of 106 runs whose colors indicate the opacity at the RCB as a function of the planet’s semi-major axis and mass at four
different pebble accretion rates. The white dashed lines divide the zones where different opacity contributions dominate. The red dotted lines mark
the parameter space where the entire envelope is convective.

explicitly dependent on the mass flux, aside from the extinction The results of these runs are shown in Fig. 7, ordered into
efficiency which is higher for larger pebbles in these conditions. four panels that correspond to different pebble accretion rates,
The second effect is that a higher luminosity increases the local which typically increase as planets grow and capture pebbles
radiative temperature gradient, which alters the conditions of the more efficiently (Ormel & Klahr 2010; Lambrechts & Johansen
local medium, ultimately affecting the opacity as well. 2012; Liu et al. 2019). The opacity trends reflect the discussions
 When the accretion rate is increased beyond ∼ 10−5 M⊕ /yr of the previous subsections but also contrast the opacity of solids
for the plotted default parameters, the pebble size becomes and molecules, which generally follow a dichotomy based on the
velocity-limited at the fragmentation-erosion barrier. At this planet’s distance and mass. Hot molecules contribute most of
point, any additional material no longer increases the pebble size the envelope’s opacity in the warm and dense inner disk, where
and directly increases their volume density, allowing the opacity solids sediment slowly and coalesce to form larger agglomerates.
to rise sharply. The increased opacity in these runs with high Because the molecular opacity increases with both temperature
pebble accretion rates turns the envelopes almost entirely con- and density, this provides a clear contrast based on orbital sep-
vective (see upper two curves), with RCB locations at or close to aration. The molecular opacity at the RCB is not very sensitive
the Bondi radius. to planet mass, as can be seen from the nearly vertical opacity
 contours. In contrast, the opacity from solids declines steeply as
 planets grow more massive and particles sediment more slowly
4.5. Description of three opacity regimes due to the higher densities. Fig. 7 shows that the total envelope
 opacity can be divided into three regimes:
In order to visualize the broad opacity trends more clearly, we
also evaluate a large 2D grid of envelope opacities at the RCB 1. The molecular opacity regime applies to higher-mass planets
(κrcb ) as a function of their distance to the star and planetary in the inner disk that are accreting pebbles at a low rate. In
mass. While the opacity throughout the entire radiative zone is this regime, the opacity at the RCB is almost independent of
relevant for the envelope’s structure, its value at the boundary be- planet mass and steeply declines with orbital separation.
tween the radiative and convective zones is generally considered 2. The growth-limited solids opacity regime applies to higher-
the most important. This is both because most of the radiative mass planets in the outer disk that are accreting pebbles at a
portion of the envelope’s mass is contained near the RCB and low rate. In this regime, the RCB opacity is almost indepen-
because it controls the rate at which the convective interior cools dent with distance and declines steeply with planet mass. The
(i.e. Lee & Chiang 2015; Ginzburg et al. 2016). opacity is invariant to an increase of the mass flux, as any ad-
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A&A proofs: manuscript no. main

 ditional mass just adds to pebble growth. However, the RCB 15.0
 opacity scales positively with the accretion luminosity. Including pebbles + dust grain-free
3. The velocity-limited solids opacity regime applies to lower- 12.5
 mass planets that are accreting pebbles at a sufficiently high
 10.0 10 5 M /yr

 Mz, crit [M ]
 rate. It is typically characterized by fully convective en-
 velopes, with the plotted RCB located at the Bondi radius.
 A higher pebble accretion rate increases the volume density 7.5
 of solids in this regime and greatly increases the parameter 10 6 M /yr
 range where it applies (to include larger Mp and smaller d). 5.0
One caveat to this third regime is that if the entire outer envelope 2.5
is convective, the envelope density remains relatively low and as 10 7 M /yr
 0.0
pointed out by Johansen & Nordlund (2020), the envelope can 10 1 100 101
then locally become radiative at the depth where solids subli-
mate (2500 K in our model). At this point, which roughly coin-
 d [AU]
cides with the dissociation temperature of molecular hydrogen, Fig. 8. Trends in the critical metal mass as a function of distance.
only the molecular contribution to the opacity remains. If there The different lines correspond to a variation in solids accretion rates:
are sufficient free electrons to ionize the hydrogen and produce 10−7 M⊕ /yr (dotted), 10−6 M⊕ /yr (dashed), 10−5 M⊕ /yr (solid), with
H− , this becomes the dominant opacity component (Lee & Chi- proportional gas accretion Ṁxy / Ṁpeb = 5 and the remaining parameters
ang 2015). The situation in this regime is further complicated by set by Table 1. Planets with grain-free envelopes (gray lines) exhibit
the balancing effect of a compositional gradient that also begins a downward trend with orbital separation because the molecular opac-
to form at these temperatures (Bodenheimer et al. 2018; Müller ity scales positively with temperature. The opacity generated by peb-
et al. 2020), in which the Ledoux criterion (Ledoux 1947) has bles and dust (black lines) increases the critical mass significantly in
to be used to evaluate stability against convection, rather than the outer disk, especially at higher accretion rates.
the Schwarzschild criterion. The thermodynamic consequences
of this transition region are an active topic of investigation (Val- defined as the total mass of solids accreted at the onset of run-
letta & Helled 2020; Ormel et al. 2021) and are outside the scope away accretion, then becomes an explicit function of both T rcb
of this opacity study, where we focus specifically on the opacity and κrcb :
of solids in the outer envelope. We provide a broader discussion
 ! 16 !7 !8
of this potential zone in Sect. 6. κrcb d 108 T vap 27
 Mz,crit ≈ 5.5 M⊕ (27)
 0.01 g cm−2 AU 2500 K
 ! 61 !1 !− 126
5. Implications for giant planet formation Ṁpeb Mc 2 T rcb 972
 .
Planets transition from slow to runaway gas accretion when they 10−6 M⊕ yr−1 M⊕ T disk
reach a critical point where their self-gravity begins to exceed In line with the findings by Ormel et al. (2021), we assume core
the envelope’s pressure support. A transparent envelope is char- growth to be uniformly limited to 2 M⊕ , with the rest of the
acterized by more efficient radiative energy transport in the outer solids being absorbed in the deep envelope. We plot the resultant
regions, which results in faster cooling and more rapid gas ac- critical metal masses as a function of orbital separation for three
cretion. As a result, both analytical and numerical works (e.g., accretion rates in Fig. 8. The black lines include the opacity from
Mizuno 1980; Stevenson 1982; Pollack et al. 1996; Movshovitz dust and pebbles, contrasted with the gray lines that assume en-
et al. 2010; Lee & Chiang 2015) have shown that the mass at tirely grain-free envelopes.
which planets transition to runaway growth is positively linked The general shape of the curves in Fig. 8 reflects the opac-
to the envelope’s opacity. In this section, we apply our model to ity trends described in the previous section. Molecules dominate
examine this trend across the proto-planetary disk. the opacity of envelopes in the hot inner disk, where the critical
 mass is seen to decrease with orbital separation. In the grain-free
5.1. Trends in the critical metal mass curves, this downward trend continues towards the outer disk
 where the critical mass reaches very low values of only a few
In a previous work (Brouwers & Ormel 2020), we showed that M⊕ . Accounting for the opacity of pebbles and dust breaks this
the traditional criterion of a critical core mass becomes mean- trend in the intermediate disk, where solids begin to dominate
ingless if the growth of the core is limited. For this reason, we the opacity of the envelopes. This halts the decline of the critical
introduced and derived an analogous criterion called the critical mass, which then instead flattens towards the outer disk. Besides
metal mass, which measures the total metal content of a planet at its variation with distance, the critical metal mass is also posi-
the onset of runaway gas accretion. It shares the opacity depen- tively dependent on the solids accretion rate. This is both due to
dence with previous expressions for the critical core mass, but is the increased luminosity and the increased opacity from solids,
additionally an explicit function of the core mass itself. One limi- although the two are linked thermodynamically (see Sect. 4.4).
tation of our previous work was that we modeled envelopes with We note that in reality, a pebble-accreting planet would not expe-
an isothermal radiative layer, which is a good assumption for rience a single pebble accretion rate during its evolution, but one
grain-free envelopes but invalid for envelopes with higher opac- that generally increases as it gains in mass and is able to capture
ities due to the presence of solids. In order to correctly account pebbles more efficiently.
for thermodynamic changes in the radiative part of the envelope, Physically, the trends in the critical metal mass translate to
we slightly modify the analytical structure model of Brouwers similar trends in the occurrence rates of gas giant planets. Data
& Ormel (2020) in Appendix A to be applicable to envelopes from the Kepler mission has provided good exoplanet abun-
with non-isothermal radiative regions. The critical metal mass, dance statistics up to about 1 AU in semi-major axis (Borucki
Article number, page 10 of 15

M. G. Brouwers et al.: How planets grow by pebble accretion

et al. 2010, 2011; Batalha et al. 2013), which show a general pled to the gas to pass through (Pinilla et al. 2016; Bitsch et al.
increase in the abundance of giant planets with distance in this 2018; Weber et al. 2018; Haugbølle et al. 2019) and provide the
range (Howard et al. 2012; Dong & Zhu 2013; Santerne et al. required opacity to prevent rapid cooling. Our results indicate
2016). When combined with data from radial-velocity measure- that this solution essentially faces the same problem as continu-
ments (Mayor et al. 2011), it shows a break in the giant planet ous pebble accretion: Regardless of whether the mass is supplied
occurrence rate between 2-3 AU, with a declining power law be- by pebbles or planetesimals, the opacity from solids is insuffi-
yond this value (Fernandes et al. 2019). While other factors are cient to prevent runaway growth without a high accretion rate.
certainly at play in planet formation across the proto-planetary In fact, the required planetesimal accretion rate is even higher
disk, these trends are generally well matched by the critical metal than the equivalence in pebbles because the abundance of small
mass predicted by our opacity model. solids is insufficient to reach the velocity-limited regime and the
 envelope opacity thus remains low.
 The implication is that regardless of whether the ice giants
5.2. Implications for the formation of Uranus and Neptune
 became isolated from the pebble flow, their formation requires
Far out in the disk, one of the key theoretical challenges is to ex- the constant accretion of material at high rates if they formed
plain how Uranus and Neptune accreted their substantial masses in-situ. This introduces a timescale problem, as they must then
of ∼ 15 M⊕ without entering runaway growth (Helled & Bo- have formed in the window where they had enough time to ac-
denheimer 2014; Helled et al. 2020). In the model developed by crete their observed masses, but not too early such that they ac-
Lambrechts et al. (2014), it is suggested that the accretion of creted too much material and became gas giants. Accurately es-
solids continued throughout the disk lifetime and that the heat timating the duration of this formation window requires more
from impactors kept the envelope stable during its evolution. detailed models the ice giant’s interiors, which are currently not
However, the planet’s cold birth environments and low molec- yet well constrained (Helled et al. 2020; Vazan et al. 2020). In
ular opacity means that their accretion luminosity is easily radi- addition, it is important to account for the cooling of the solar
ated away, causing grain-free envelopes to implode into gas gi- nebula over time. The cooling of the surrounding disk has the
ants before they have grown beyond a few Earth masses. In order same general effect as increasing the planet’s distance from the
to prevent runaway accretion, the planets must either form when central star (reduced T disk , ρdisk ), which we showed increases the
most of the nebula is gone (Lee & Chiang 2016; Ogihara et al. envelope opacity contribution of solids. In a future work, we plan
2020), or their envelopes must still be sufficiently opaque that to quantitatively estimate the required accretion rate and forma-
their heat does not escape so quickly. The common approach to tion window of these ice giants.
this problem is to assume that the envelopes contain a large ISM-
like opacity from small dust grains. In line with previous works
on coagulation (Movshovitz & Podolak 2008; Movshovitz et al. 6. Discussion
2010; Mordasini 2014; Ormel 2014), our results indicate that 6.1. Comparison with contemporary works
such a high opacity is not realistic when grain growth determines
the pebble sizes. It is possible, however, for planetary envelopes In this work, we focused on the growth and destruction of solids
in the outer disk to attain these high opacities if the pebble sizes as they travel through planetary envelopes, ignoring their evo-
are velocity-limited by fragmentation or erosion. As we show in lution prior to accretion. A contrasting approach was taken in a
Fig. 7, this requires the planet to continuously accrete pebbles at contemporary work by Bitsch & Savvidou (2021), who modeled
a sufficiently high rate. the size distribution of solids in the disk similar to Savvidou et al.
 Alternatively, planets begin to carve partial gaps around (2020), with the assumption that their distribution and opacity
their orbit when they grow beyond several Earth masses in the disk mid-plane also apply to the envelope’s interior. We
(Paardekooper & Mellema 2004, 2006). When they continue to present evidence that the opacities at the RCB are actually very
grow, they locally invert the pressure gradient in the disk and different from the disk due to a typically much higher gas tem-
stop accreting pebbles, a criterion which is known as the pebble perature and density, as well as the significant size evolution of
isolation mass (Morbidelli et al. 2015; Morbidelli & Nesvorny solids in the envelope. Larger pebbles fragment or face erosion
2012; Ataiee et al. 2018). As shown by Bitsch et al. (2015, 2018), when they enter planetary envelopes, while the lower end of the
it is important to account for the cooling of the Solar nebula accreted size distribution experiences significant growth during
and the reduced disk scale height over time. In these simula- sedimentation. Effectively, we predict much smaller envelope
tions, the pebble disk scale height is found to reduce sufficiently opacities in the growth-limited regime than Bitsch & Savvidou
in the first few Myr that both Uranus and Neptune can reach (2021), although high opacities remain possible in the velocity-
their isolation mass and stop accreting pebbles. This introduces limited regime. Nevertheless, while we argue that the sizes of
a new issue however, as ceasing the supply of accretion heat solids in the envelope are disconnected from their size distribu-
can again easily trigger cooling and rapid gas accretion. Unlike tion in the disk, modeling their size evolution in the disk remains
the close-in super-Earths and mini-Neptunes which can be pre- important for physical estimates of their accretion rates and effi-
vented from accreting gas by entropy advection (Ormel et al. ciency.
2015; Cimerman et al. 2017; Kurokawa & Tanigawa 2018; Ali- An alternative approach to modeling the envelope opacity is
Dib & Thompson 2020; Moldenhauer et al. 2020), the ice giants presented by Chachan et al. (2021), which is based on the work
will rapidly accrete gas due to their Kelvin-Helmholtz contrac- by Lee & Chiang (2015). The authors argue that the sublima-
tion. The solution offered by Alibert et al. (2018); Guilera et al. tion of dust leads to a second radiative zone at the sublimation
(2020) is that the reduction in pebble accretion can be compen- front, with an inner boundary around 2500 K where dissociation
sated by the accretion of planetesimals, which could naturally of H2 provides a new opacity source in the form of H− . The ap-
form at the surrounding pressure bump. In a similar vein, Lam- pearance of this inner radiative zone is based on the assumption
brechts & Lega (2017); Chen et al. (2020); Bitsch & Savvidou that the envelope’s outer layers are convective due to a high dust
(2021) reason that while mm-cm pebbles are easily blocked by a opacity. The requirement of free elections to form H− means that
local pressure bump, small dust (≤ 200 µm) is sufficiently cou- the opacity studied by Chachan et al. (2021) is proportional to
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