Sublimation of refractory minerals in the gas envelopes of accreting rocky planets

Marie-Luise Steinmeyer; Peter Woitke; Anders Johansen

doi:10.1051/0004-6361/202245636

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A&A, 677 (2023) A181

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Issue		A&A Volume 677, September 2023


Article Number		A181
Number of page(s)		16
Section		Planets and planetary systems
DOI		https://doi.org/10.1051/0004-6361/202245636
Published online		26 September 2023

A&A, 677, A181 (2023)

Sublimation of refractory minerals in the gas envelopes of accreting rocky planets

Marie-Luise Steinmeyer¹, Peter Woitke² and Anders Johansen¹^,3

¹ Center for Star and Planet Formation, Globe Institute, University of Copenhagen, Øster Volgade 5–7, 1350 Copenhagen, Denmark
e-mail: steinmeyer_ml@yahoo.com
² Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria
³ Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 221 00 Lund, Sweden

Received: 7 December 2022
Accepted: 10 August 2023

Abstract

Protoplanets growing within the protoplanetary disk by pebble accretion acquire hydrostatic gas envelopes. Due to accretion heating, the temperature in these envelopes can become high enough to sublimate refractory minerals which are the major components of the accreted pebbles. Here we study the sublimation of different mineral species and determine whether sublimation plays a role during the growth by pebble accretion. For each snapshot in the growth process, we calculate the envelope structure and the sublimation temperature of a set of mineral species representing different levels of volatility. Sublimation lines are determined using an equilibrium scheme for the chemical reactions responsible for destruction and formation of the relevant minerals. We find that the envelope of the growing planet reaches temperatures high enough to sublimate all considered mineral species when M ≳ 0.4 M_⊕. The sublimation lines are located within the gravitationally bound envelope of the planet. We make a detailed analysis of the sublimation of FeS at around 720 K, beyond which the mineral is attacked by H₂ to form gaseous H₂S and solid Fe. We calculate the sulfur concentration in the planet under the assumption that all sulfur released as H₂S is lost from the planet by diffusion back to the protoplanetary disk. Our calculated values are in good agreement with the slightly depleted sulfur abundance of Mars, while the model over predicts the extensive sulfur depletion of Earth by a factor of approximately 2. We show that a collision with a sulfur-rich body akin to Mars in the moon-forming giant impact lifts the Earth’s sulfur abundance to approximately 10% of the solar value for all impactor masses above 0.05 Earth masses.

Key words: planets and satellites: formation / planets and satellites: atmospheres / planets and satellites: terrestrial planets / planets and satellites: composition

© The Authors 2023

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Protoplanetary disks surrounding very young stars contain several hundred Earth masses of millimeter-sized solids (Tychoniec et al. 2018; Carrasco-González et al. 2019). These pebbles are the result of coagulation and condensation of dust particles (Birnstiel et al. 2012; Estrada et al. 2016; Ros et al. 2019). However, protoplanetary disks become progressively depleted in solids as they evolve over a few million years (Ansdell et al. 2016). Primitive meteorites – the early Solar System materials – contain abundant millimeter-sized chondrules (Hewins 1997). Connelly et al. (2012) found that the ages of some chondrules overlap with the estimated age of Ca–Al-rich inclusions (CAIs) commonly considered as early Solar System condensates, which indicates that the solar protoplanetary disk was also rich in pebble-sized solids. However, some authors advocate for a time delay between the formation of chondrules and CAIs (Villeneuve et al. 2009). Nevertheless, these pebbles may be key to driving planetary growth.

Due to their small size range of 0.1–1 mm (Zhu et al. 2019), pebbles are influenced by the surrounding gas in the protoplanetary disk (Weidenschilling 1977). The drag of the gas increases the accretion cross section of pebbles onto a growing protoplanet compared to the pure gravitational cross section. Thus, accretion rates of pebbles are higher than the rate of accreting other planetesimals in the outer regions of the protoplanetary disk (Johansen & Lacerda 2010; Ormel & Klahr 2010; Lambrechts & Johansen 2012; Johansen & Bitsch 2019; Lorek & Johansen 2022). The rapid growth rates of pebble accretion explain how giant planets can reach the critical mass to enter runaway gas accretion during the life time of the disk (Bitsch et al. 2015; Tanaka & Tsukamoto 2019). The theory of pebble accretion has furthermore been applied to explain the high occurrence rate of the two most common type of exoplanets observed – super-Earths and mini-Neptunes (Bitsch et al. 2015; Venturini & Helled 2017; Bitsch 2019). Both of these planet types have massive cores that acquired a H–He atmosphere during their formation. However, super-Earths have either lost this primordial atmosphere or failed to accrete it, while mini-Neptunes kept their primordial atmospheres as a memory of their formation within the protoplanetary disk. An alternative explanation to pebble accretion is that these planets formed in massive protoplane-tary disks via the accretion of planetesimals (Chiang & Laughlin 2013).

Recent works have studied the formation of smaller rocky planets by pebble accretion (e.g Levison et al. 2015; Johansen et al. 2015; Lambrechts et al. 2019). Lambrechts et al. (2019) show that the pebble flux determines the outcome of pebble accretion in the inner region of a protoplanetary disk. While high pebble flux leads to formation of super-Earths, planets forming in a disk with a low pebble flux only grow to sizes between Mars and Earth. A possible cause for the reduction in the pebble flux is the presence of a giant outer planet (Weber et al. 2018; Liu et al. 2022). In the context of the Solar System, Johansen et al. (2021) show that pebble accretion provides a self-consistent picture to explain the masses and orbits of Venus, Earth and Mars, when taking into account that Earth only grew to its final mass after the moon-forming giant impact with an additional rocky planet. At the same time, there is evidence from isotopic studies that Earth formed from a combination of material from the inner and outer Solar System, which supports a pebble accretion scenario in order to efficiently provide the terrestrial planets with material drifting in from the outer Solar System (Schiller et al. 2018, 2020; Onyett et al. 2023). This interpretation is nevertheless challenged by Burkhardt et al. (2021) who propose that the terrestrial planets may have formed from material not represented by any meteorite samples.

An important aspect of pebble accretion is that it takes place during the lifetime of the protoplanetary disk. Protoplanets with a mass larger than a lunar mass acquire a hydrostatic spherical envelope during their growth phase (Ikoma & Hori 2012; Lee et al. 2014). Compared to the surrounding disk, the density inside the envelope of bodies more massive than Mars is increased by several orders of magnitude. At the same time, the dissipation of kinetic energy during the accretion process heats up the protoplanetary envelope from below. Pebbles that enter the envelope of a planet during pebble accretion can undergo different processes: (1) destruction by thermal ablation (Alibert 2017; Ali-Dib & Thompson 2020), (2) collisions leading to growth or fragmentation (Johansen & Nordlund 2020), and (3) sublimation if the temperature in the envelope is high enough (Alibert 2017; Brouwers et al. 2018; Brouwers & Ormel 2020).

The sublimation of pebbles leads to an enrichment of the envelope in elements heavier than H and He compared to the surrounding protoplanetary disk (Lambrechts et al. 2014). Previous work has studied the effect of such an envelope pollution for the formation of intermediate to high mass planets (Hori & Ikoma 2011; Venturini et al. 2015; Brouwers & Ormel 2020; Misener & Schlichting 2022). The enrichment of the envelope lowers the critical mass needed for the planet to enter runaway gas accretion (Hori & Ikoma 2011; Venturini et al. 2015). At the same time Brouwers & Ormel (2020) find that the contraction time scale of polluted planets is shorter than the one of unpolluted planets. Thus, the enrichment of the envelope helps to explain the formation of sub-Neptunes instead of gas giants (Lambrechts et al. 2014). Furthermore, envelope enrichment plays an important role in understanding the atmospheres and radii of these planets after the disk has dispersed (Misener & Schlichting 2022). Sublimation of pebbles may also important for volatile delivery to terrestrial planets formed by pebble accretion. Johansen et al. (2021) propose that both water ice and organics are destroyed either by direct sublimation or chemical reactions in the envelopes of protoplanets of a few lunar masses or higher. Part of the released volatiles are then lost due to disk recycling, which may explain the low water and carbon content of Earth.

Previously it has often been assumed that pebbles consist of a single Si-bearing mineral and that the sublimation temperature is given by the saturated vapor pressure of that mineral. In reality, pebbles in protoplanetary disks are a mixture of minerals including olivine, orthopyroxene, volatile and refractory organ-ics, water ice, troilite and metallic iron (Pollack et al. 1994). The formation of these minerals often involves gas–solid type reactions and thus cannot be represented by saturated vapor curves. In this paper, we study the sublimation of moderately-volatile to ultra-refractory minerals in the envelope of an accreting planet with a set of representative minerals. We calculate the sublimation temperatures based on an equilibrium scheme for chemical reactions responsible for destruction and formation of the representative minerals.

We focus on low-mass rocky planets and use a simple pebble accretion model to calculate the growth track of the planet. We then determine the envelope structure of the planet and the sublimation temperatures of the different mineral species for each snapshot in the growth track. We show that refractory minerals, such as forsterite (Mg₂SiO₄) and iron (Fe), sublimate in the envelope after the planet has reached M ≈ 0.15 M_⊕. The silicate sublimation region underlies a small radiative zone caused by the reduction in dust opacity. We propose that the region below the Mg₂SiO₄ sublimation line is slowly filled with SiO vapor that creates a mean-molecular weight barrier between the SiO zone and the upper layers. Convection will be suppressed by this mean molecular weight gradient (Leconte et al. 2017). The barrier further protects more refractory species, such as corundum (Al₂O₃), from moving into the outer envelope. Troilite (FeS) is likely to be the main sulfur carrier in protoplanetary disks (Pollack et al. 1994; Kama et al. 2019). The sublimation temperature of FeS is found to be approximately 720 K, significantly below the Mg₂SiO₄ sublimation temperature. Thus, FeS starts to sublimate earlier in the growth of the planet and farther out than silicates.

As a siderophile element, sulfur (S) is particularly interesting since it is one of the possible light components in the cores of Mars and Earth that complements iron and nickel (McDonough & Sun 1995; Stähler et al. 2021). In addition, both Earth and Mars are depleted in moderately volatile elements compared to the solar composition (Braukmüller et al. 2019; Yoshizaki & McDonough 2020) with Earth being much more depleted in S than Mars (Stähler et al. 2021). Once FeS reacts with H₂, gaseous H₂S and metallic iron are expected to form. The H₂S molecule is highly volatile and may easily be lost by diffusion across the Bondi radius. During the disk’s life time, recycling flows between the surrounding protoplanetary disk and the region around the Bondi radius then replace the H₂S-enriched gas with gas from the disk (Ormel et al. 2015; Kurokawa & Tanigawa 2018). After the disk dissipates, H₂S can also be lost during the atmospheric escape (Lammer et al. 2020a). Therefore, we find that the sulfur concentration in a planet formed by pebble accretion is a decreasing function of the mass of the planet. Our model agrees well with the moderate S depletion of Mars. For Earth the model underpredicts the depletion by a factor of approximately two. We nevertheless demonstrate that the moon-forming giant impact could raise the S concentration of Earth to the observed 10% level, because of the high S concentration of the smaller impactor. Thus, our main result is that the S content of Earth and Mars are well-explained in the pebble accretion framework of rocky planet formation.

The paper is organized as follows: in Sect. 2, we describe our pebble accretion model, the model for the protoplanetary envelope structure and the calculation of the sublimation temperature of different mineral species. In Sect. 3, we present the resulting temperature profiles and location of the sublimation lines. In Sect. 4, we discuss the implications for the final composition of the planets. We compare the sulfur composition in our model to the one from the terrestrial planets in the Solar System in Sect. 5. In Sect. 6, we discuss the limitations of the model. Finally, we conclude with a summary of this paper in Sect. 7.

2 Model description

The radius of a protoplanet with mass M_p is given by $R_{P} = {(\frac{3 M_{p}}{4 π ρ_{p}})}^{1 / 3} .$ ${R_{\rm{P}}} = {\left( {{{3{M_{\rm{p}}}} \over {4\pi {\rho _{\rm{p}}}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}.$ (1)

We set the density of the protoplanet to ρ_p = 4050 kg m⁻³, which corresponds to the density of the uncompressed Earth (Hughes 2006).

The Bondi radius gives the characteristic length scale on which the protoplanet can bind gas from the surrounding protoplanetary disk to form an envelope (Ikoma & Hori 2012; Lee et al. 2014). It is defined as $R_{B} = \frac{G μ M_{p}}{k_{B} T} = \frac{G M_{p}}{c_{s}^{2}},$ ${R_{\rm{B}}} = {{G\mu {M_{\rm{p}}}} \over {{k_{\rm{B}}}T}} = {{G{M_{\rm{p}}}} \over {c_{\rm{s}}^2}},$ (2)

where µ is the mean molecular weight of the gas, k_B the Boltzmann constant, T is the temperature of the protoplanetary disk and c_s is the sound speed of the gas in the protoplanetary disk. The total mass of the protoplanet is thus the mass of the rocky protoplanet and the mass of the gas envelope enclosed in the Bondi radius. However, we assume that M_env ≪ M_p and ignore the mass contribution from the envelope in the following.

The Hill radius is given by $R_{H} = a {(\frac{M_{p}}{3 M_{*}})}^{1 / 3}$ ${R_{\rm{H}}} = a\,{\left( {{{{M_{\rm{p}}}} \over {3{M_*}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}$ (3)

and depends on the distance a to the central star with mass M_*. It gives the radius over which the gravitational force of the planet dominates over the tidal force of the central star. In this paper we take M_* = M_⊙ in all our calculations.

2.1 Pebble accretion

In order to calculate the growth track of the planet, we first need to assume a description of the surrounding protoplanetary disk. We use a classic α-disk model. In this model, the turbulent viscosity, which sets the radial gas accretion speed, is given by (e.g. Pringle 1981) $v = α c_{s} H,$ $v = \alpha {c_s}H,$ (4)

where α is determined from the mass accretion rates of protoplanetary disks to be in the range of 10⁻⁴ to 10⁻² (Mulders et al. 2017; Manara et al. 2023). Magnetohydrodynamic models of protoplanetary disk have shown that the disk temperature in the inner region is dominated by the irradiation of the central star (Mori et al. 2021). Therefore, we neglect the contribution of the viscous heating and describe the radial temperature profile by a power-law (Ida et al. 2016) $T (a) = 150 {(\frac{a}{au})}^{- ζ} K.$ $T\left( a \right) = 150\,{\left( {{a \over {{\rm{au}}}}} \right)^{ - \zeta }}\,{\rm{K}}{\rm{.}}$ (5)

The disk aspect ratio H/a, where H is the gas scale height, then follows the power law $\frac{H}{a} \propto {(\frac{a}{au})}^{- ζ / 2 + 1 / 2} .$ ${H \over a} \propto {\left( {{a \over {{\rm{au}}}}} \right)^{{{ - \zeta } \mathord{\left/ {\vphantom {{ - \zeta } 2}} \right. \kern-\nulldelimiterspace} 2} + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}.$ (6)

The sound speed c_s = HΩ is described by the power law $c_{s} = c_{s, 0} {(\frac{a}{au})}^{- ζ / 2},$ ${c_{\rm{s}}} = {c_{s,0}}\,{\left( {{a \over {{\rm{au}}}}} \right)^{{{ - \zeta } \mathord{\left/ {\vphantom {{ - \zeta } 2}} \right. \kern-\nulldelimiterspace} 2}}},$ (7)

where c_s,0 = 650 m s⁻¹ is the sound speed at 1 AU. We take ζ = 3/7 for the irradiated disk model (Chiang & Goldreich 1997; Ida et al. 2016).

We use the pebble accretion model from Johansen et al. (2019). The growth rate of the planet is given by $\begin{array}{l} \dot{M} = 2 \times {(\frac{St}{0.1})}^{2 / 3} \sqrt{G M_{*}} {(3 M_{*})}^{- 2 / 3} \\ \times \frac{ξ {\dot{M}}_{g} (t)}{2 π [χ St+ (3 / 2) α] c_{s,0}^{2} {au}^{ζ}} M^{2 / 3} {(\frac{a}{au})}^{ζ - 1} . \end{array}$ $\matrix{ {\dot M = 2 \times \,{{\left( {{{{\rm{St}}} \over {0.1}}} \right)}^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}}\sqrt {G{M_*}} {{\left( {3{M_*}} \right)}^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\, \times {{\xi {{\dot M}_{\rm{g}}}\left( t \right)} \over {2\pi \left[ {\chi {\rm{St + }}\left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\alpha } \right]c_{{\rm{s,0}}}^2\,{\rm{a}}{{\rm{u}}^\zeta }}}{M^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}}\,{{\left( {{a \over {{\rm{au}}}}} \right)}^{\zeta - 1}}.} \hfill \cr }$ (8)

The growth rate depends on a number of parameters. St is the Stokes number of the pebbles while $ξ = \frac{{\dot{M}}_{p}}{{\dot{M}}_{g}}$ $\xi = {{{{\dot M}_{\rm{p}}}} \over {{{\dot M}_{\rm{g}}}}}$ (9)

is the ratio of the inward pebble mass flux Ṁ_p to the gas mass flux Ṁ_g. Other important parameters are the negative logarithmic pressure gradient χ = β + ζ/2 + 3/2, the α-parameter, and the sound speed at 1 AU c_s,0. Here β = 15/14 is the logarithmic derivative of the gas surface density. The model takes the migration of the planet into account. This gives a growth track describing the distance from the star a for a given mass M with the shape $a (M) = a_{0} {(1 - \frac{M^{4 / 3} - M_{0}^{^{4 / 3}}}{M_{\max}^{4 / 3} - M_{0}^{^{4 / 3}}})}^{1 / (1 - ζ)},$ $a\left( M \right) = {a_0}\,{\left( {1 - {{{M^{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}} - M_0^{^{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}}} \over {M_{\max }^{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}} - M_0^{^{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 {\left( {1 - \zeta } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \zeta } \right)}}}},$ (10)

where M₀ is the initial mass of the protoplanet and M_max is the mass a planet would reach if it migrates all the way to a = 0. This maximum mass can be described in the form of a scaling law (Johansen et al. 2019) $\begin{array}{l} M_{\max} = 11.7 M_{\oplus} {\frac{{(St / 0.01)}^{1 / 2}}{[(2 / 3) (St / α) / χ + 1] / 2.9}}^{3 / 4} {(\frac{ξ}{0.01})}^{3 / 4} {(\frac{M_{*}}{M_{⊙}})}^{1 / 4} \\ \times {(\frac{k_{mig}}{4.42})}^{- 3 / 4} {(\frac{c_{s, 0}}{650 m s^{- 1}})}^{3 / 2} {(\frac{1 - ζ}{4 / 7})}^{- 1} {(\frac{a_{0}}{25 au})}^{(3 / 4) (1 - ζ} \end{array} .$ $\matrix{ {{M_{\max }} = 11.7{M_ \oplus }{{{{{{\left( {{{{\rm{St}}} \mathord{\left/ {\vphantom {{{\rm{St}}} {0.01}}} \right. \kern-\nulldelimiterspace} {0.01}}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {{{\left[ {{{\left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)\left( {{{{\rm{St}}} \mathord{\left/ {\vphantom {{{\rm{St}}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \mathord{\left/ {\vphantom {{\left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)\left( {{{{\rm{St}}} \mathord{\left/ {\vphantom {{{\rm{St}}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} {\chi + 1}}} \right. \kern-\nulldelimiterspace} {\chi + 1}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)\left( {{{{\rm{St}}} \mathord{\left/ {\vphantom {{{\rm{St}}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \mathord{\left/ {\vphantom {{\left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)\left( {{{{\rm{St}}} \mathord{\left/ {\vphantom {{{\rm{St}}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} {\chi + 1}}} \right. \kern-\nulldelimiterspace} {\chi + 1}}} \right]} {2.9}}} \right. \kern-\nulldelimiterspace} {2.9}}}}}^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}}}{{\left( {{\xi \over {0.01}}} \right)}^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}}}{{\left( {{{{M_ * }} \over {{M_ \odot }}}} \right)}^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times {{\left( {{{{k_{{\rm{mig}}}}} \over {4.42}}} \right)}^{{{ - 3} \mathord{\left/ {\vphantom {{ - 3} 4}} \right. \kern-\nulldelimiterspace} 4}}}\,{{\left( {{{{c_{s,0}}} \over {650\,{\rm{m}}\,{{\rm{s}}^{ - 1}}}}} \right)}^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}\,{{\left( {{{1 - \zeta } \over {{4 \mathord{\left/ {\vphantom {4 7}} \right. \kern-\nulldelimiterspace} 7}}}} \right)}^{ - 1}}\,{{\left( {{{{a_0}} \over {25\,\,{\rm{au}}}}} \right)}^{\left( {{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}} \right)\left( {1 - \zeta } \right.}}} \hfill \cr } .$ (11)

Here, a₀ is the starting location of the planet and we have assumed that M_max ≫ M₀.

Lastly, the temporal evolution of the gas accretion onto the star in a viscous α–disk is given by Hartmann et al. (1998) as ${\dot{M}}_{g} (t) = {\dot{M}}_{0} {(\frac{t}{t_{s}} + 1)}^{- (5 / 2 - γ) / (2 - γ)},$ ${{\dot M}_{\rm{g}}}\left( t \right) = {{\dot M}_0}\,{\left( {{t \over {{t_{\rm{s}}}}} + 1} \right)^{ - {{\left( {{5 \mathord{\left/ {\vphantom {5 {2 - \gamma }}} \right. \kern-\nulldelimiterspace} {2 - \gamma }}} \right)} \mathord{\left/ {\vphantom {{\left( {{5 \mathord{\left/ {\vphantom {5 {2 - \gamma }}} \right. \kern-\nulldelimiterspace} {2 - \gamma }}} \right)} {\left( {2 - \gamma } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2 - \gamma } \right)}}}},$ (12)

where γ = 3/2 − ζ and t_s is the characteristic time depending on the initial disk size R₁ and viscosity v₁ at R₁, $t_{s} = \frac{1}{3 {(2 - γ)}^{2}} \frac{R_{1}^{2}}{v_{1}} .$ ${t_{\rm{s}}} = {1 \over {3{{\left( {2 - \gamma } \right)}^2}}}{{R_1^2} \over {{v_1}}}.$ (13)

The gas accretion sets the gas surface density Σ_g of the inner regions of the disk to be ${\dot{M}}_{g} = - 2 π r u_{r} Σ_{g},$ ${{\dot M}_{\rm{g}}} = - 2\pi r{u_{\rm{r}}}{{\rm{\Sigma }}_{\rm{g}}},$ (14)

with the gas accretion speed $u_{r} = - \frac{3}{2} \frac{v}{a} = - \frac{3}{2} α c_{s} \frac{H}{a} .$ ${u_{\rm{r}}} = - {3 \over 2}{v \over a} = - {3 \over 2}\alpha {c_{\rm{s}}}{H \over a}.$ (15)

We calculate the mass evolution of the protoplanet by integrating Eq. (8) using a fourth-order Runge–Kutta method. At each time step we calculate the new position of the planet according to Eq. (10). Our model depends on a number of free parameters. For the pebble accretion growth track, we need to define the time t₀ when the protoplanet has the initial mass M₀. We also need to set the starting position a₀ of the planet. Furthermore, we need to assume a ratio between pebble and gas mass flux ξ. The lifetime of the protoplanetary disk t_life, the turbulence level α and the gas mass flux Ṁ_g as well as Stokes number of the pebbles are based on observational values. Table 1 shows an overview of the values we chose for the pebble accretion model. The resulting numerical growth track of the planet is shown in Fig. 1. The final mass of the planet at the end of the disk life-time is 0.68 M_⊕ and the final position a = 1 au. As Earth reaches its final mass after the moon forming giant impact, we will therefore treat our planet as an Earth-analogu (Lock et al. 2018). The calculation of the envelope structure is described in the next subsection.

Table 1

Parameters used for the pebble accretion model.

Fig. 1

Numerical growth track of the planet by pebble accretion starting at a₀ = 1.985 au and t₀ = 3.5 Myr. The initial mass of the planet is 0.001 M_⊕. The final mass of the planet is 0.68 M_⊕ at location of a = 1.0 au. The starting point of the planet is indicated by the square. The circle shows the final position of the planet when the disk dissipates and the planet stops migrating.

2.2 Envelope structure

We assume that the envelope of the protoplanet is spherically symmetric and in hydrostatic balance. In this case, the envelope structure is given by the standard structure equations of mass conversation, hydrostatic balance, and thermal gradient (e.g., Kippenhahn et al. 2013) $\frac{d m}{d r} = 4 π r^{2} ρ,$ ${{{\rm{d}}m} \over {{\rm{d}}r}} = 4\pi {r^2}\rho ,$ (16a) $\frac{d P}{d r} = - \frac{G M}{r^{2}} ρ,$ ${{{\rm{d}}P} \over {{\rm{d}}r}} = - {{GM} \over {{r^2}}}\rho ,$ (16b) $\frac{d T}{d r} = \nabla \frac{T}{P} \frac{d P}{d r},$ ${{{\rm{d}}T} \over {{\rm{d}}r}} = \nabla {T \over P}{{{\rm{d}}P} \over {{\rm{d}}r}},$ (16c)

where r is the distance to the center of the planet, and ρ, P, and T are the density, gas pressure, and temperature respectively. M is the mass of the planet, m is the integrated mass interior of r and ∇ ≡ dln T/dln P is the logarithmic temperature gradient. G is the gravitational constant. We assume an ideal gas with a mean molecular weight of 2.34 m_u corresponding to a solar mixture of H₂ and He and an adiabatic index of γ = 1.4. Here, m_u is the atomic mass unit. Beyond the Hill radius the stellar tides dominate over the gravitational pull of the planet and the assumptions of hydrostatic balance and spherical symmetry needed for Eq. (16) break down. Therefore, we set r_out = R_H.

The most important forms of energy transport in protoplanetary envelopes are radiation and convection. Convection occurs when the adiabatic temperature gradient $\nabla_{ad} \equiv \frac{γ - 1}{γ}$ ${\nabla _{{\rm{ad}}}} \equiv {{\gamma - 1} \over \gamma }$ (17)

is smaller than the temperature gradient given by radiative energy transport $\nabla_{rad} \equiv \frac{3 κ (T) P}{64 π G M_{p} σ T^{4}} L .$ ${\nabla _{{\rm{rad}}}} \equiv {{3\kappa \left( T \right)P} \over {64\pi G{M_{\rm{p}}}\sigma {T^4}}}L.$ (18)

Here, σ is the Stefan-Boltzmann constant, L = GM_pṀ/R_p the luminosity of the planet, and κ(T) the opacity in the envelope at a given temperature. We thus set the temperature gradient in Eq. (16c) to min(∇_ad, ∇_rad).

We approximate the opacity as a broken power law in the form of $κ = κ_{i} ρ^{a_{1}} T^{b_{i}},$ $\kappa = {\kappa _i}{\rho ^{{a_1}}}{T^{{b_i}}},$ (19)

where the parameters κ_i, a_i, and b_i are taken from Bell & Lin (1994). The dominant source of opacity depends on the temperature and density. For this work the main opacity sources are ice grains, metal grains, and gas molecules. The effect of different levels of opacity are shown in Appendix C.

In order to obtain the temperature and pressure profiles, we use a 4th order Runge–Kutta method to integrate Eq. (16) from the outer boundary to the surface of the planet. At the outer boundary, the density and the temperature is set to match the surrounding disk, T_out = T_d and ρ_out = ρ_d. The pressure at the outer boundary is given by the equation of state of an ideal gas law $P_{d} = \frac{ρ_{d}}{μ m_{u}} k_{B} T_{d} .$ ${P_{\rm{d}}} = {{{\rho _{\rm{d}}}} \over {\mu {m_{\rm{u}}}}}{k_{\rm{B}}}{T_{\rm{d}}}.$ (20)

Fig. 2

Natural logarithm of the supersaturation level S for the selected mineral species in a gas with solar composition as a function of the temperature calculated with the representative mineral approach at two different pressures. At temperatures where ln S > 0, the mineral species can exist, while if ln S < 0 the mineral will not form even if all the reactants are in the gas phase. The left plot with P = 0.1 Pa corresponds to typical conditions in the protoplanetary disk, while the right plot with P = 10⁴ Pa corresponds to the typical pressures in the inner envelope of an Earth-mass planet. The filled circles indicate the sublimation temperatures while the vertical dotted lines indicate the sublimation temperatures for the other pressure. Except for FeS the sublimation temperature increases with the pressure level. Mg₂SiO₄ and Fe sublimate at similar temperatures in the low pressure scenario. At a high pressure, Mg₂SiO₄ has a lower sublimation temperature than Fe.

2.3 Sublimation model

Pollack et al. (1994) find that the major dust species in dense molecular clouds and protoplanetary disks are olivine ((Fe,Mg)₂SiO₄), orthopyroxene ((Fe,Mg)SiO₃), metallic iron (Fe), troilite (FeS), volatile and refractory organics, and water ice (H₂O). We chose a set of minerals to represent the different sublimation behaviour: ultra-refractories (corundum Al₂O₃), silicates (forsterite Mg₂SiO₄), metals (Fe) and moderately volatile solids (troilite FeS). Tachibana et al. (2002) showed that enstatite (MgSiO₃) evaporates by forming a layer of Mg₂SiO₄ as an evaporation residue. Therefore, we chose to represent silicates by Mg₂SiO₄ instead of MgSiO₃ even though MgSiO₃ is the preferred carrier of condensed silicate in protoplanetary disks for the solar Si:Mg ratio of approximately unity (Gail 1998). We neglect the organics and water ice as the sublimation of these is discussed in Johansen et al. (2021).

The formation-destruction chemical reactions of these minerals are assumed to be completely set by four mutually independent reactions: ${Al}_{2} O_{3} (s) + 3 H_{2} (g) ⇌ 2 Al (g) + 3 H_{2} O (g)$ ${\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}\left( {\rm{s}} \right) + 3\,{{\rm{H}}_2}\left( {\rm{g}} \right)2\,{\rm{Al}}\left( {\rm{g}} \right) + 3\,{{\rm{H}}_2}{\rm{O}}\left( {\rm{g}} \right)$ (21) ${Mg}_{2} {SiO}_{4} (s) + 3 H_{2} (g) ⇌ Mg (g) + SiO (g) + 3 H_{2} O (g)$ ${\rm{M}}{{\rm{g}}_2}{\rm{Si}}{{\rm{O}}_4}\left( {\rm{s}} \right) + 3\,{{\rm{H}}_2}\left( {\rm{g}} \right){\rm{Mg}}\left( {\rm{g}} \right) + {\rm{SiO}}\left( {\rm{g}} \right) + 3\,{{\rm{H}}_{\rm{2}}}{\rm{O}}\left( {\rm{g}} \right)$ (22) $Fe (s) ⇌ Fe (g)$ ${\rm{Fe}}\left( {\rm{s}} \right){\rm{Fe}}\left( {\rm{g}} \right)$ (23) $FeS (s) + H_{2} (g) ⇌ Fe (s) + H_{2} S (g) .$ ${\rm{FeS}}\left( {\rm{s}} \right) + {{\rm{H}}_2}\left( {\rm{g}} \right){\rm{Fe}}\left( {\rm{s}} \right) + {{\rm{H}}_2}{\rm{S}}\left( {\rm{g}} \right).$ (24)

In reality, the destruction and formation of the minerals takes place via multiple different reactions that depend on the pressure and the composition of the surrounding gas. Additionally, we treat each reaction independently instead of minimizing the Gibbs free energy of the entire system. We benchmark our sim-plifled model against the chemical equilibrium code GGCHEM (Woitke et al. 2018) in Sect. 2.4.

Except for metallic iron, no corresponding molecules in the gas phase exists for the minerals on the left-hand-sides of Eqs. (21)–(24). Therefore, one cannot set up a simple sublimation-condensation equilibrium with a certain vapor pressure as has been done in previous works (Brouwers & Ormel 2020; Misener & Schlichting 2022). Instead, we follow Johansen & Dorn (2022) and calculate the sublimation temperature from the chemical equilibrium between the gas phases and the condensates. In this context, the sublimation temperature T_sub,j of the mineral species j is defined as the temperature at which the supersaturation level S_j equals to 1 (Nozawa et al. 2003), with ln S_j defined as $\ln S_{j} = - Δ G / (R_{gas} T) + \sum_{i} v_{i j} \ln (\frac{P_{i j}}{P_{std}}) .$ $\ln \,{S_{\,\,j}} = {{ - {\rm{\Delta }}G} \mathord{\left/ {\vphantom {{ - {\rm{\Delta }}G} {\left( {{R_{{\rm{gas}}}}T} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{R_{{\rm{gas}}}}T} \right)}} + \sum\limits_i {{v_{ij}}\,\ln \,\left( {{{{P_{ij}}} \over {{P_{{\rm{std}}}}}}} \right).}$ (25)

Here, P_ij are the partial pressures and of v_ij the stoichiometric coefficients of the reactants and product gas species. The pressure of the standard state is P_std = 10⁵ Pa and R_gas is the universal gas constant. The thermodynamic activity of solid minerals is set to unity. The Gibbs free energy of the reaction, ΔG, was calculated using the Gibbs free energies of the reactants and the products. We use the fit functions from Sharp & Huebner (1990) to calculate the of Gibbs free energy of each reaction from the thermodynamic data from the NIST/JANAF tables (Chase 1998).

The supersaturation levels of the different species as a function of temperature for typical pressures in a protoplanetary disk and in the inner envelope of a rocky planet are shown in Fig. 2. At temperatures where ln S > 0, the mineral species can exist, while for ln S < 0 the mineral species cannot be stable in a solar composition gas. In general, the sublimation temperature increases with pressure. The only exception is FeS where the sublimation temperature T_sub(FeS) = 720 K is independent of the pressure. We have assumed that all S in the gas phase is in the form of H₂S. This gives $P_{H_{2} S} = ϵ_{S} P$ ${P_{{{\rm{H}}_2}{\rm{S}}}} = {_S}P$ , where ϵ_S is the sulfur abundance. In addition, the main carrier of H in the gas phase is H₂ so $P_{H_{2}} \approx (ϵ_{H} / 2) P$ ${P_{{H_2}}} \approx \left( {{{{_S}} \mathord{\left/ {\vphantom {{{_H}} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)P$ . Therefore, the equilibrium constant of the H₂S reaction in Eq. (24) is $K (T) = \frac{P_{H_{2} S}}{P_{H_{2}}} = \frac{ϵ_{S}}{ϵ_{H} / 2} .$ $K\left( T \right) = {{{P_{{{\rm{H}}_2}{\rm{S}}}}} \over {{P_{{{\rm{H}}_2}}}}} = {{{_{\rm{S}}}} \over {{{{_{\rm{H}}}} \mathord{\left/ {\vphantom {{{_{\rm{H}}}} 2}} \right. \kern-\nulldelimiterspace} 2}}}.$ (26)

Here $P_{H_{2} S}$ ${P_{{{\rm{H}}_2}{\rm{S}}}}$ and $P_{H_{2}}$ ${P_{{{\rm{H}}_2}}}$ are the equilibrium partial pressures of H₂S and H₂ in the gas phase. The equilibrium constant and thus the sublimation temperature of FeS is clearly independent of the total pressure of the system, which is in accordance with experimental studies (Lauretta et al. 1997). Another noticeable feature from Fig. 2 is that at low pressures Mg₂SiO₄ and Fe have similar sublimation temperature, T_sub ≈ 1200 K with Fe starting to sublimate at slightly lower temperatures. In the high pressure case, on the other hand, Mg₂SiO₄ sublimates at a lower temperature than Fe.

Fig. 3

Total element abundances in the gas phase calculated with GGCHEM as a function of the temperature for five elements that represent refractory to moderately volatile minerals. We show the results at two different pressures, P = 0.1 Pa (left) and P = 10⁴ Pa (right). The dotted vertical lines show the sublimation temperature of our corresponding representative minerals. The sublimation temperatures agree well with the temperature at which gas phase molecules that contain the considered elements first appear.

2.4 GGCHEM benchmark test

We use the publicly available thermo-chemical equilibrium code GGCHEM¹ of Woitke et al. (2018) to benchmark the calculation of the sublimation lines in our representative mineral approach. GGCHEM solves the equilibrium for the speciation in the gas phase and the phase equilibrium for the condensed species based on the minimisation of the total Gibbs free energy of the system. We compare our model to a GGCHEM run with equilibrium condensation switched on. We consider 22 elements (H, He, C, N, O, Na, Mg, Si, Fe, Al, Ca, Ti, S, Cl, K, Li, Mn, Ni, Cr, V, W, Zr) with solar abundances taken from Lodders (2003) at two different pressures. The biggest difference between our model and complex thermo-chemical equilibrium codes such as GGCHEM is the number of considered species. For example, we assume that all Al is the form of Al(g) thus neglecting species such as A1O₂H or A1₂O that should exist in chemical equilibrium for a gas of solar composition.

In order to validate our approach we compare the sublimation temperatures of our representative model to the gas phase elemental abundances of Al, Mg, Si, Fe, and S. As Fig. 3 shows, the temperature at which the elements appear in the gas phase for the first time agree well with the sublimation temperature of the representative model. Therefore, we conclude that although our representative mineral approach is very simplified, it can be used as a good indicator of sublimation temperatures of the selected minerals. Further comparison of our model and GGCHEM is presented in Appendix A.

3 Envelope evolution

3.1 Pressure and temperature structure

Figure 4 compares the temperature and pressure profiles of the planet early in the growth phase when M = 0.1 M_⊕ to the envelope profiles of the final planet with M = 0.68 M_⊕. As the planet migrates inward from a = 1.985 au (R_H = 421 R_P) to a = 1.0 au (R_H = 214 R_P), the temperature and pressure in the disk increase. The relative pressure increases from P = 0.01 Pa at a = 1.985 au to P = 0.04 Pa at a = 1.0 au. The pressure in the envelope increases toward the surface of the planet. The maximum pressure for M = 0.1 M_⊕ is 26 Pa and ≈7300 Pa for M = 0.68 M_⊕. The temperature profile is roughly isothermal up to the Bondi radius, below which the temperature increases (Rafikov 2006; Piso & Youdin 2014). The surface temperature is ≈1000 K for M = 0.1 M_⊕ and ≈2800 K in case of M = 0.68 M_⊕. The kinks in the temperature profile at ≈1600 K and ≈1900 K in the high mass case are associated with the change in the opacity regime. The destruction of the main dust species at ≈ 1600 K in the Bell & Lin (1994) opacity scheme leads to a decrease in the opacity with temperature. Hence the radiative temperature gradient becomes smaller than the convective gradient and a radiative region develops. At ≈ 1900 K the main source of opacity changes again and the transport mechanism switches back to convective.

The evolution of the temperature profile in the inner envelope (r < 0.4R_H) during the whole growth phase is shown in Fig. 5. In the beginning the protoplanet has no significant envelope. Therefore the temperature close to the surface of the planet is similar to the temperature in the surrounding disk T_d = 111 K. Once the planet reaches a mass of ≈ 0.01 M_⊕, its Bondi radius becomes significantly larger than the planetary radius and the planet starts to bind an envelope. The energy released during the accretion process heats up this envelope as it continues to grow.

Figure 6 shows the energy transport mechanism in the envelope. The envelope is convective from the start. Once the planet grows to M ≈ 0.2 M_⊕ a radiative zone starts to develop close to the surface of the planet. As already discussed, this region is associated with the sublimation of silicate grains. For M > 0.5 M_⊕ a small convective region develops below the radiative region. In this region the opacity is dominated by gas molecules which increases the radiative temperature gradient above the adiabatic temperature gradient. We compare the influence of different levels of opacity in Appendix C

The evolution of the surface temperature is shown in Fig. 7. The temperature first increases sharply until the planet reaches a mass of ≈ 0.2M_⊕. The surface temperature continues to increase but less strongly. Once the planet has a mass M > 0.5 M_⊕, the increase in surface temperature with mass becomes steeper again. This change in temperature is again due to the transition of energy transport from convective to radiation as seen in Fig. 6.

Fig. 4

Temperature (left) and pressure (right) profiles in the envelope of a planet with M = 0.1 M_⊕ (purple curves) and M = 0.68 M_⊕ (orange curves). The distance to the surface of each planet is given in units of the Hill radius of the planets. The solid circle shows the position of the respective Bondi radius. The temperature and pressure in the envelope increase with the mass of the planet. Since the planet is migrating inward as it grows, the pressure in the surrounding disk increases while the Hill radius shrinks. The planet with M = 0.68 M_⊕ develops a radiative region in the inner envelope indicated by the dotted line style.

Fig. 5

Temperature map of the inner planetary envelope during the growth process after the planet has reached a mass of M = 0.01 M_⊕. The distance to the surface of the planet is given in units of the instantaneous Hill radius of the planet. The gray lines show the sublimation locations of the representative minerals. The white dotted line represents the Bondi radius. The temperature in the envelope starts to increase once the planet reaches a mass of 0.01 M_⊕. For the final planet the temperature in the inner is ≈ 2700 K. The sublimation line of FeS moves out to r ≈ 0.03 R_H. The sublimation lines of Mg₂SiO₄, and Fe lie close together. The sublimation temperature of A1₂O₃ is only reached close to the surface of a planet with M > 0.4 M_⊕.

Fig. 6

Type of energy transport in the envelope of the planet. The distance to the surface of the planet is given in units of the instantaneous Hill radius of the planet at the relevant mass. In orange regions the energy is transported via convection (mode 1), whereas in purple regions the radiative energy transport (mode 0) dominates. The black line shows the sublimation line of Mg₂SiO₄ from our chemical equilibrium model. Radiation as the main energy transport process only appears in the inner disk after the envelope close to the planet has reached the sublimation temperature of Mg₂SiO₄.

3.2 Sublimation lines

We do not take the change in gas composition due to the pebble sublimation into account. The main factor determining the sublimation temperature of a specific solid is thus the pressure level. Since the shape of the pressure profile does not change significantly as the planet grows, see Fig. 4, we do not expect the sublimation temperature to change significantly with planetary mass. Instead the location in the envelope where the sublimation temperature is reached will move outwards as the planet grows. In addition to the surface temperature, Fig. 7 shows the sublimation temperatures of the considered mineral species as a function of the mass of the planet. The sublimation temperatures are indeed relatively independent of the mass of the planet and increases from FeS to Mg₂SiO₄+Fe to Al₂O₃. FeS starts to sublimate at a planet mass of 0.05 M_⊕. Once the planet reaches ≈0.4 M_⊕ the surface temperature exceeds the sublimation temperatures of all considered minerals. Table 2 lists the sublimation temperature for each mineral as well as the mass of the planet when the surface temperature first reaches this temperature.

The locations of the sublimation lines of the different mineral species in the envelope during the growth process are also shown in Fig. 5. As expected the lines move outward in the envelope as the planet grows. All lines are located deep inside the Bondi radius of the planet. After the planet reaches a mass of M ≈ 0.2 M_⊕, the sublimation line of FeS moves to r > 0.01 R_H. FeS thus sublimates in the outer convective zone of the planet. Fe and Mg₂SiO₄ sublimate close to the radiative region and Al₂O₃ sublimates deep in the radiative region of the envelope.

Fig. 7

Sublimation temperature of the representative mineral species. The dotted gray line shows the temperature close to the surface as a function of the planetary mass. The filled circles indicate where the surface temperature first reaches the sublimation temperature of each mineral. The sublimation temperature is relatively independent of the planet mass. Hence, the order of the sublimation lines stays the same throughout the growth of the planet.

Table 2

Mass of the planet in Earth masses in our model when T_surf > T_sub for the first time.

4 Fate of sublimated pebbles

We have shown in the previous section that the envelope of a rocky planet becomes hot enough during its growth process to reach the sublimation temperatures of a set of minerals representing different levels of volatility from moderately volatile to ultra-refractory. Vapor from sublimated minerals can be lost from the planet while the planet is still embedded in the disk through so-called recycling flows. Hydrodynamic simulations found that the envelope of an embedded planet continuously exchanges gas with the disk (Ormel et al. 2015; Kurokawa & Tanigawa 2018). The recycling flows will typically penetrate the envelope down to the Bondi radius of the planet, below which the entropy drops significantly and the flow is stopped by the gas buoyancy (Lambrechts & Lega 2017; Kurokawa & Tanigawa 2018). Vapor from minerals that sublimate below the Bondi radius is brought up to the Bondi radius by upward flows in the convection cells and is then recycled back into the surrounding protoplanetary disk (Alibert 2017; Johansen et al. 2021; Wang et al. 2023). The turbulent diffusion coefficient of the convective motion can be written as D ~ α_convR_B c_s, where c_s is the sound speed in the envelope. From the hydrodynamical simulations of Popovas et al. (2018) we infer that the factor α_conv is on the order of unity. The time scale for vapor to be brought up to the Bondi radius is $t_{diff} = \frac{R_{B}^{2}}{D} = \frac{G M}{α_{conv} c_{s}^{3}} = 284.4 h (\frac{M}{M_{\oplus}}) {(\frac{T}{150 K})}^{- 3 / 2} .$ ${t_{{\rm{diff}}}} = {{R_{\rm{B}}^2} \over D} = {{GM} \over {{\alpha _{{\rm{conv}}}}c_{\rm{s}}^3}} = 284.4{\rm{h}}\,\left( {{M \over {{M_ \oplus }}}} \right)\,{\left( {{T \over {150\,{\rm{K}}}}} \right)^{{{ - 3} \mathord{\left/ {\vphantom {{ - 3} 2}} \right. \kern-\nulldelimiterspace} 2}}}.$ (27)

Here we have inserted the definition of the Bondi radius in Eq. (2) and α_comv = 1. The diffusion time scale of our final planet is thus ≈ 150 h. This is extremely short compared to the accretion time scale of the planet. Additionally, Ormel et al. (2015) find the time scale at which the gas in the envelope is replenished with gas from the surrounding disk to be much shorter than the typical life time of a protoplanetary disk which is a few million years (Kimura et al. 2016). Therefore, we can safely assume that the recycling of the envelope gas at the Bondi radius is an efficient process.

So far we have considered the sublimation based on chemical equilibrium considerations. However, the rate ds_j/dt at which a pebble of a specific mineral j sublimates also depends on the mass loss rate J and the time the pebble spends in the envelope of the planet, with $\frac{d s_{j}}{d t} = - \frac{J}{ρ_{j} / μ_{j}},$ ${{{\rm{d}}{s_j}} \over {{\rm{d}}t}} = - {J \over {{{{\rho _j}} \mathord{\left/ {\vphantom {{{\rho _j}} {{\mu _j}}}} \right. \kern-\nulldelimiterspace} {{\mu _j}}}}},$ (28)

where ρ_j is the density of the mineral and µ_j the mean molecular weight of the mineral. The net mass loss rate of a pebble of a specific mineral in the envelope of a planet is given by the difference between sublimation and recondensation (e.g., Richter et al. 2002) $J = \frac{α_{evap} (P_{sat} - P_{g})}{\sqrt{2 π μ_{g} m_{u} k_{B} T}} .$ $J = {{{\alpha _{{\rm{evap}}}}\left( {{P_{{\rm{sat}}}} - {P_g}} \right)} \over {\sqrt {2\pi {\mu _g}{m_u}{k_{\rm{B}}}T} }}.$ (29)

Here α_evap is the evaporation coefficient of the sublimation process, P_sat the saturated vapor pressure of the gas phase, P_g the partial pressure of the gas phase at the surface, µ_g the molecular weight of a key molecule on the gas phase side of the reaction, and m_u the atomic mass unit. Both the evaporation and recondensation flux depend on $P_{H_{2}}$ ${P_{{{\rm{H}}_2}}}$ which is a proxy for the total gas pressure (Tachibana & Tsuchiyama 1998; Richter et al. 2002, 2007; Mendybaev et al. 2021). At high pressures when the partial pressure is close to the saturated vapor pressure, recondensation can reduce the net mass loss rate significantly (Richter et al. 2002, 2007; Mendybaev et al. 2021). The evaporation coefficient takes into account that not only the rate at which gas particles interact with a surface is important for determining the net mass loss rate. Especially at low temperature, the timescale of surface processes can be long (Herbort et al. 2020). A low mass loss rate could make the mineral stable at temperatures much higher than what is predicted from equilibrium chemistry.

4.1 Silicates

Brouwers & Ormel (2020) study the sublimation of pure silicate (SiO₂) with an assumed constant sublimation temperature of 2500 K. They assume that once the planet reaches a certain mass, all accreted material sublimates in the envelope. Furthermore, they assume that if the envelope becomes saturated in the heavy vapor, such as SiO₂, the vapor will re-condense and rain out onto the planet. Instead, we consider Mg₂SiO₄ as a proxy for the sublimation behavior of silicates. The sublimation temperature of Mg₂SiO₄ from chemical equilibrium in our model is ≈1400 K at a pressure of ~10² Pa, which is significantly lower than the temperature chosen by Brouwers & Ormel (2020). Tsuchiyama et al. (1999) found that at temperatures and pressures needed for the sublimation of Mg₂SiO₄, T > 1400 K and P > 10² Pa, the sublimation rate depends only on the temperature and lies in the range of 10⁻¹¹ to l0⁻⁸ mol cm⁻² s⁻¹. This leads to a sublimation rate in the range of 10⁻² to 10 µm h⁻¹ indicating that Mg₂SiO₄ sublimation is a relatively fast process.

Therefore, we propose that the innermost envelope will build up a layer rich in gas phase SiO, Mg and H₂O once the sublimation temperature of Mg₂SiO₄ is reached in the envelope. This layer will be in equilibrium with the underlying magma ocean on the surface of the accreting planet. At the same time, recondensation of gas phase SiO, Mg and H₂O back onto the Mg₂SiO₄ plays an important rate at high total gas pressures (Richter et al. 2002, 2007; Mendybaev et al. 2021). As the envelope becomes saturated in SiO, the accreted pebbles will begin to move through this region without sublimating since evaporation and recondensation will cancel each other out. Thus, after the envelope is saturated in SiO, Mg₂SiO₄ will again be accreted onto the core of the planet without sublimation. However, if the temperature in the inner region of the envelope becomes higher than 1600 K, the pebbles will melt and rain out onto the magma oceans as liquid droplets (Monteux et al. 2016). In this picture, the SiO vapor is protected from diffusion to the upper envelope and hence from the recycling discussed above by the radiative region due to the change seen in Fig. 6. Additional protection comes for the mean molecular weight gradient between the saturated and unsaturated envelope. Therefore, we expect silicates to become part of the final planet despite the high temperatures reached in the envelope.

4.2 FeS

We chose FeS to represent the sublimation behavior of moderately volatile elements. Sulfur can be bound both in volatile form (as H₂S) and in moderately volatile form (as FeS). Kama et al. (2019) use measurements of the composition of the surface of young stars surrounded by protoplanetary disks to study the elemental composition of their disks. They find that almost all sulfur in a protoplanetary disk must be locked up in the moderately volatile mineral FeS since the surface of the young stars are S-poor. In chemical equilibrium the sublimation temperature of FeS is independent of pressure, as we have demonstrated in Sect. 2.3. For the abundances considered in this paper the sublimation temperature of FeS is ≈720 K. We show the dependence of T_sub,FeS on ϵ_S/ϵ_H in Appendix B. At higher temperatures H₂ reacts with the surface of FeS grain to form gaseous H₂S and solid Fe.

Tachibana & Tsuchiyama (1998) measured the reaction rates of the sublimation of FeS in conditions similar to the solar pro-toplanetary disk experimentally. They determine the evaporation coefficient to be $α_{H_{2} S} = 2.03 \times 10^{- 3} P_{H_{2}}^{0.106} \exp (- 940 / T) .$ ${\alpha _{{{\rm{H}}_2}{\rm{S}}}} = 2.03 \times {10^{ - 3}}\,P_{{{\rm{H}}_2}}^{0.106}\,\exp \,\left( { - {{940} \mathord{\left/ {\vphantom {{940} T}} \right. \kern-\nulldelimiterspace} T}} \right).$ (30)

Here P(H₂) is the ambient H₂ pressure measured in Pa. This leads to a FeS mass loss rate of $J {(S)}_{H_{2} S} = \frac{α_{H_{2} S} q P_{sat} (H_{2} S)}{\sqrt{2 π μ_{H_{2} S} m_{u} k_{B} T}},$ $J{\left( S \right)_{{{\rm{H}}_2}{\rm{S}}}} = {{{\alpha _{{{\rm{H}}_2}{\rm{S}}}}q{P_{\,{\rm{sat}}}}\,\left( {{{\rm{H}}_2}{\rm{S}}} \right)} \over {\sqrt {2\pi {\mu _{{{\rm{H}}_2}{\rm{S}}}}{m_u}{k_{\rm{B}}}T} }},$ (31)

where q is the proportion of the FeS surface area of the pebble and $μ_{H_{2} S}$ ${\mu _{{{\rm{H}}_2}{\rm{S}}}}$ is the molecular weight of FeS. We set q = 0.87. We define the saturated vapor pressure of H₂S as $P_{sat} (H_{2} S) = P_{std} \exp [Δ G (H_{2} S) / (R_{gas} T)] + \ln (\frac{P (H_{2})}{P_{std}}) .$ ${P_{\,{\rm{sat}}}}\left( {{{\rm{H}}_2}{\rm{S}}} \right) = {P_{{\rm{std}}}}\,\exp \,\left[ {{{{\rm{\Delta }}G\,\left( {{{\rm{H}}_2}{\rm{S}}} \right)} \mathord{\left/ {\vphantom {{{\rm{\Delta }}G\,\left( {{{\rm{H}}_2}{\rm{S}}} \right)} {\left( {{R_{\,{\rm{gas}}}}T} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{R_{\,{\rm{gas}}}}T} \right)}}} \right] + \ln \,\left( {{{P\left( {{{\rm{H}}_2}} \right)} \over {{P_{{\rm{std}}}}}}} \right).$ (32)

We assumed in Eq. (31) that P_sat (H₂S) ≫ P_υ(H₂S) during the sublimation, where P_υ is the partial pressure of H₂S at the surface of the pebble. The assumption quickly becomes valid at temperatures slightly above the sublimation temperature. Therefore, recondensation will no longer play a role.

In addition, Tachibana & Tsuchiyama (1998) find that for the pressure conditions found in planetary envelopes, the following reaction $FeS (s) ⇌ Fe (s) + \frac{1}{2} S_{2} (g)$ ${\rm{FeS}}\left( {\rm{s}} \right){\rm{Fe}}\left( {\rm{s}} \right) + {1 \over 2}{{\rm{S}}_{\rm{2}}}\left( {\rm{g}} \right)$ (33)

also plays a role for the sublimation of FeS. The evaporation coefficient of this reaction is (Tachibana & Tsuchiyama 1998) $α_{S_{2}} = 0.922 \exp (- 2220 / T) .$ ${\alpha _{{{\rm{S}}_2}}} = 0.922\,\exp \,\left( {{{ - 2220} \mathord{\left/ {\vphantom {{ - 2220} T}} \right. \kern-\nulldelimiterspace} T}} \right).$ (34)

The corresponding mass loss rate is $J {(S)}_{S_{2}} = \frac{2 α_{S_{2}} q P_{sat} (S_{2})}{\sqrt{2 π μ_{S_{2}} k_{B} T}} .$ $J{\left( S \right)_{{{\rm{S}}_2}}} = {{2{\alpha _{{{\rm{S}}_2}}}q{P_{{\rm{sat}}}}\,\left( {{{\rm{S}}_2}} \right)} \over {\sqrt {2\pi {\mu _{{{\rm{S}}_2}}}{k_{\rm{B}}}T} }}.$ (35)

with the saturated vapor pressure of S₂ $P_{sat} (S_{2}) = P_{std} \exp [Δ G (S_{2}) / (R_{gas} T)] .$ ${P_{{\rm{sat}}}}\left( {{{\rm{S}}_2}} \right) = {P_{{\rm{std}}}}\,\exp \,\left[ {{{{\rm{\Delta }}G\,\left( {{{\rm{S}}_2}} \right)} \mathord{\left/ {\vphantom {{{\rm{\Delta }}G\,\left( {{{\rm{S}}_2}} \right)} {\left( {{R_{{\rm{gas}}}}T} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{R_{{\rm{gas}}}}T} \right)}}} \right].$ (36)

S₂ is not stable at chemical equilibrium (Woitke et al. 2018). Equilibrium in the gas implies that S₂ will react with H₂ to form H₂S after the sublimation. The resulting sublimation rates for the two reactions are shown in Fig. 8. The total rate of FeS sublimation at the sublimation temperature given from the chemical equilibrium consideration is quite low, dFeS/dt ≈ 10⁻⁴ µ m/h, but increases strongly with temperature. This can also be seen in the plot on the left in Fig. 9 which shows the destruction time scale t_des of 10 and 100 µm FeS grains as a function of temperature. These two sizes are chosen to represent typical FeS grains found in ordinary chondrites (Kuebler et al. 1999). The destruction time scale at the equilibrium sublimation temperature is 8 × 10⁴ h for 10 µm grain and 8 × 10⁵ h for 100 µm grain. At T = 1000 K the destruction time scale decreases by two orders of magnitude.

Fig. 8

Sublimation rate of FeS as a function of temperature using the laboratory measurements of Tachibana & Tsuchiyama (1998). The orange line represents the contribution from reaction (24) and the purple line from Eq. (33). The black line is the total sublimation rate. The vertical dashed line indicates the sublimation temperature of FeS based on chemical equilibrium. The total sublimation rate increases strongly with temperature.

4.3 Settling time scales of FeS grains

We therefore calculate the time it takes for a 10 µm grain and a 100 µm grain to settle to the surface of the planet. As starting points we chose the equilibrium sublimation line of FeS and the location where T = 1000 K. The velocity of the grain is either determined by the terminal speed of the grain $υ_{t} = τ_{f} \frac{G M_{p}}{r^{2}}$ ${\upsilon _t} = {\tau _f}{{G{M_{\rm{p}}}} \over {{r^2}}}$ (37)

or the convection speed in the envelope (Ali-Dib & Thompson 2020) $υ_{c} = {(\frac{L}{4 π r^{2} (0.36 ρ_{g})})}^{1 / 3} .$ ${\upsilon _c} = {\left( {{L \over {4\pi {r^2}\left( {0.36{\rho _{\rm{g}}}} \right)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}.$ (38)

Here, τ_f is the friction time of the grain which depends on the grain size a_p, the local sound speed c_s, and the gas density ρ_g $τ_{f} = \frac{a_{p} ρ_{FeS}}{c_{s} ρ_{g}},$ ${\tau _f} = {{{a_p}{\rho _{{\rm{FeS}}}}} \over {{c_{\rm{s}}}{\rho _{\rm{g}}}}},$ (39)

where ρ_FeS is the density of the FeS grains. The plot on the right in Fig. 9 shows the resulting settling time scales of FeS grains. If grains settle with convection speed, the settling time scales are too short for the grains to sublimate. The settling time scales are longer if the settling velocity is determined by the terminal speed. Nevertheless, the destruction time scale at 720 K are roughly two orders of magnitude larger than the settling time scale. Only the settling time scale of a 10 µm FeS grain falls below the destruction time scale at 1000 K.

The destruction time scale in Fig. 9, however, is more likely to represent an upper limit for the actual destruction time scale of a settling grain. The temperature in the envelope increases sharply as the grain settles to deeper layers as seen in the temperature map of the envelope in Fig. 5. The resulting increase in mass loss rate could decrease the actual destruction time scale of FeS grains. Furthermore, we have assumed that the grains settle to the surface in a straight line, although the dynamics of pebbles in the envelope of an accreting planet can be quite complex (Morbidelli & Nesvorny 2012; Popovas et al. 2018; Takaoka et al. 2023). Instead of straight lines pebbles are more likely spiral onto the planet in a circumplanetary disk (Johansen & Lacerda 2010). These complex settling trajectories lead to an increase in the settling time scale. Therefore, we will assume in the following discussion that FeS sublimates either at the nominal 720 K (in case of slow orbit decay in a circumplanetary disk) or at an elevated temperature of 1000 K.

5 Implications for planet formation

We will now discuss how the sublimation of FeS in the envelope can affect the final composition of the planet. As part of the discussion, we will present a possible explanation of the sulfur content of Mars and Earth as a result of formation by pebble accretion. Since the envelope is convective, the newly formed H₂S molecules will quickly move up to the Bondi radius and are lost to the disk as part of the recycling flows discussed Sect. 4. The gaseous H₂S will not condense into ice grains in the envelope since the condensation temperature of H₂S is very low (≈80 K) compared to the temperatures in the envelope (Okuzumi et al. 2016). Even if H₂S would remain in the envelope during the disk life time, hydrodynamic simulations have shown that volatile molecules in the H₂-dominated envelope will also be lost during the atmospheric escape after the disk disperses (Lammer et al. 2020a,b; Erkaev et al. 2023).

5.1 Sulfur abundance as a function of planetary mass

We now calculate the S fraction of a planet under the assumption that once the surface temperature becomes hot enough to sublimate FeS all further sulfur will not be accreted onto the planet. The S abundance of a planet relative to composition of the pebbles in this case can be described as $f_{S} = {\begin{array}{l} 1 & if M \leq M_{S 1}, \\ M_{S 1} / M & if M > M_{S 1}, \end{array}$ ${f_S} = \left\{ {\matrix{ 1 \hfill & {\quad {\rm{if}}\,M\, \le \,{M_{S1}},} \hfill \cr {{{{M_{S1}}} \mathord{\left/ {\vphantom {{{M_{S1}}} M}} \right. \kern-\nulldelimiterspace} M}} \hfill & {\quad {\rm{if}}\,M\,{\rm{ > }}\,{M_{S1}},} \hfill \cr } } \right.$ (40)

where M_S1 is the mass when the temperature in the envelope first reaches the sublimation temperature of FeS. We use two different sublimation temperatures of FeS: T_s,1 = 720 K and T_s,2 = 1000 K with M_{S 1,1} = 0.04 M_⊕ and M_{S 1,2} = 0.09 M_⊕. We use the higher sublimation temperature to take into account that FeS might be stable up to higher temperatures due to the low sublimation rates as discussed in Sect. 4.2. Figure 10 shows the sulfur abundance of a growing planet normalized to the solar composition for both sublimation temperatures for two different pebble compositions according to Eq. (40). In the first case, we assume that all accreted material has solar composition. Compared to the most primitive CI chondrites and the solar composition, the other classes of chondrites already show a depletion in moderately volatile elements such as sulfur (Marrocchi & Libourel 2013; Braukmüller et al. 2018). In the second case, we therefore assume that the accreted material is already depleted in sulfur by 1/3 compared to the solar sulfur composition. In the equilibrium case the S abundance in the planet is reduced by 50% compared to the initial pebble composition once the planet reaches a mass of approximately 0.1 M_⊕; 90% depletion is reached once the planet has grown to ≈0.5 M_⊕. When considering a higher sublimation temperature instead, the planet is depleted by 50% compared to the initial pebble composition once it reaches a mass of ≈0.2 M_⊕ and by 90% once it reaches a mass of ≈0.9 M_⊕.

Fig. 9

Comparison of the destruction time scales as a function of temperature (left) and the settling time scales as a function of planetary mass (right) for FeS pebbles with a size of 10 µm (orange) and 100 µm (purple). The settling time scales are calculated from the point in the envelope where T = T₁ = 720 K (solid lines) and T = T₂ = 1000 K (dashed lines). The two temperatures T₁ and T₂ are also indicated in the left plot by the dotted and dashed line respectively. The settling time scale with convective speed is shown as black lines for the two temperatures. The settling time scales of both pebble sizes from the location of T₁ is too short compared to the destruction time scales of the pebbles. The settling time scale of a pebble from the T₂ location becomes longer than the destruction time scale of a 10 µm pebble at 1000 K once the planet reaches a mass of M ≈ 0.2 M_⊕.

Fig. 10

Fraction f_S of S that directly reaches the planetary surface, normalized to the solar composition, for two different sublimation temperatures, T_sub = 720 K (purple, chemical equilibrium case) and T_sub = 1000 K (orange). The accreted material is assumed to be of solar composition (solid line) or depleted in S by a factor of 1/3 compared to solar composition (dashed line). We assume that the planet stops accreting S once it reaches the sublimation temperature of FeS at the bottom of the envelope. The gray horizontal bars represent the estimated bulk S concentrations of Mars and Earth. The filled circle shows the S concentration of a planet formed by giant impact (GI) of two protoplanets formed from solar composition pebbles and the unfilled circle for proto-planets that formed out of pebbles already depleted compared to solar.

5.2 Explaining the S content of Earth and Mars

Sulfur is one of the potential light components of the cores of Mars and Earth, besides H, C, and O. The core of Earth is estimated to contain ~ 1.7–1.9 wt% sulfur (Dreibus & Palme 1996; McDonough 2016). The bulk Earth (core+mantle+crust) contains 0.53–0.79 wt% S (Braukmüller et al. 2019). Compared to the solar S abundance of 8.6%, Earth is thus depleted by roughly 90% (Lodders 2003). The exact fraction of S contained in Mars is poorly constrained. Nevertheless, measurements by the InSight mission showed that the core of Mars is much less dense than Earth’s core with an estimated S fraction of around 9–15 wt% (Stähler et al. 2021; Khan et al. 2022). Almost all of the sulfur in Mars is thought to be in the core, with the abundance in the mantle of Mars being only at a level of 360 ± 120 ppm (Wang & Becker 2017). The depletion of Mars in S compared to the solar composition is thus in the range of 60–75%. Hence, both Earth and Mars are depleted in sulfur compared to the solar composition. Other moderately volatile elements show similar depletion patterns (Braukmüller et al. 2019; Yoshizaki & McDonough 2020).

In the classical picture of terrestrial planet formation, multiple theories exist to explain this S depletion. One theory states that the volatile elements such as S are lost from planetesimals due to internal heating and sublimative mass loss, e.g., during their differentiation (Hin et al. 2017; Wang et al. 2021; Hirschmann et al. 2021). However, Vesta and iron meteorites have much higher S abundances than Earth, which implies that S is likely not lost during planetesimal differentiation (Chabot 2004; Steenstra et al. 2019). Recently, Sossi et al. (2022) have proposed that the composition of the Earth is a result of accreting bodies that form at different locations in the solar protoplanetary disk. The composition of these bodies is then based on the which elements are condensed at the given formation temperature.

Johansen et al. (2021) argue that Solar System terrestrial planets form mainly via pebble accretion instead of planetesimal accretion. The pebbles are thought to have compositions similar to ordinary and carbonaceous chondrites depending on where in the protoplanetary disk they originate. We have shown that FeS sublimation in the envelope leads to a decrease in S concentration as the planet grows. We turn to Fig. 10 again and compare the estimated ranges of the sulfur abundance in Mars and Earth to the sulfur abundance in our model. In case of chemical equilibrium, the sulfur abundance predicted by our model is in great agreement with the estimated sulfur abundance of Mars. If the accreted material has a solar sulfur composition, our model matches the upper limit of the estimated sulfur abundance of Mars. In the depleted case, our predicted sulfur abundances matches the lower limit.

However, our model underestimates the S abundance of Earth by a factor 2–3. We therefore investigate the additional delivery of S to Earth by the moon-forming giant impact. A late delivery of S in the form of a giant impact is consistent with Huang et al. (2021), who find that the Mo/W ratio in the Earth’s mantle is can be explained by a late addition of sulfur to the Earth’s mass budget. For an Earth analog formed by a collision between two protoplanets with M₁ = xM_⊕, and M₂ = (1 − x)M_⊕, the final S concentration relative to the solar composition always lands at f_S = 0.1 and f_S = 0.06 for the depleted case. Similar to Mars, the case where the accreted pebbles have solar composition matches the upper limit of the estimated sulfur abundance of Earth, while the depleted case matches the lower limit.

In case of the higher sublimation temperature, the planet grows to a larger mass before it stops accreting sulfur. If the initial pebbles are already depleted in S, our Earth analog is depleted in S by 83% after the giant impact, which is close to the estimated sulfur depletion of Earth. However, the Mars analog in our model is depleted in S only by 47%, which is significantly less than the estimated depletion of Mars. Nevertheless, we overall demonstrate that the depletion in sulfur and other moderately volatile elements is a natural outcome for rocky planet formation by pebble accretion under the assumption that sublimated material will be lost form the growing planet.

6 Limitations of the model

An important simplification in this paper is that we do not take the change in the gas composition due to the sublimated minerals into account. Instead we use a constant element abundances, a constant mean molecular weight, and a constant adiabatic index. Sublimation of incoming pebbles increases the respective element abundances locally. Consequently, this effect should increase the condensation temperatures. Therefore, we expect the first pebbles to saturate the higher layers, and subsequent pebbles the deeper layers. This enrichment process is either stopped by upward diffusive transport, or when the elements in the pebbles finally survive until the surface. However, the main focus in this paper is the effect of the sublimation of FeS on the final sulfur abundance in a planet. FeS sublimates farther out in the envelope, where loss of vapor from the envelope means that the assumption of a constant mean molecular weight in the envelope is still appropriate.

In addition, we assume that all the heat from the accretion process is released at the surface of the planet. In reality, latent heat absorbed during the sublimation will cool the envelope and thus change the envelope profile (Misener & Schlichting 2022). Furthermore, we assumed that the luminosity contribution from the planet itself, e.g., due to radioactive decay, is small compared to the accretion luminosity. Johansen et al. (2023) find that radioactive heating only plays a role in the first 1–2 million years of the evolution of a planet and can therefore be neglected.

The opacity of an envelope is one of the key parameter determining the pressure profile of the envelope and consequently the sublimation temperature of the mineral species. However, opacity values are still poorly understood. In this paper we used the opacity power-law structure of Bell & Lin (1994). This approach assumes dust-to-gas ratio in the envelope of 0.01 with a dust size of 1 µm. The dominant source of opacity depends on the temperature and gas density. More realistic models of the opacity in the envelope of protoplanets are complex as they have to take into account the growth, fragmentation, erosion or possible destruction of pebbles in the envelope during the accretion process (Mordasini 2014; Ormel 2014; Brouwers et al. 2021; Bitsch & Savvidou 2021). Brouwers et al. (2021) find that, for example, for sufficiently high pebble accretion rate the opacities in the envelope is high enough to extend the convective region out to the Bondi radius of the planet. We test different levels of opacity in Appendix C.

7 Summary and conclusion

This paper studies the sublimation of different refractory mineral species in the envelope of rocky planets during pebble accretion. The mineral species are selected to represent ultra-refractory and refractory minerals, metals, as well as moderately volatile minerals. We use a simple pebble accretion model to create a growth track of a rocky planet with a final mass of 0.68 M_⊕ and a final position of 1 au. For each snapshot in the growth track we calculated the temperature and pressure profile of the surrounding envelope. Next we calculated the sublimation temperature of the selected mineral species based on the chemical equilibrium between the envelope gas and the incoming pebbles for each pressure profile. As a final step we identify the location of the sublimation lines in the different protoplanetary envelopes. Finally we discuss the fate of the sublimated material.

We show that the moderately volatile mineral FeS begins to sublimate early in the growth process once the planet has reached a mass of M = 0.05 M_⊕. After the sublimation, S is in the form of H₂S and is easily lost back to the protoplanetary disk by convection flows. We therefore expect the sulfur content of a rocky planet formed by pebble accretion to decrease with mass. We calculate the predicted sulfur content of a planet in our model for pebbles with a solar composition as well as for pebbles depleted in sulfur compared to the solar composition. The sulfur content of our Mars analog in both cases is in agreement with the estimated sulfur abundance of Mars. We also reproduce the estimated sulfur abundance of the Earth if we assume Earth has undergone a giant impact with a sulfur-rich impactor. Experimental data has shown that the sublimation rate of FeS is quite low at temperatures around the sublimation line at 720 K. Thus, we also calculated the sulfur content of a planet in the case FeS is stable until the planet has reached a mass of M = 0.09 M_⊕. This high sublimation temperature gives a poorer match to the S abundance of Mars. We therefore propose that the pebbles settle down relatively slowly through a circumplanetary disk and thus have ample time to fully sublimate at the equilibrium temperature of 720 K. Overall, we predict that rocky planets formed by pebble accretion have a decreasing concentration of moderately volatile elements with mass. The findings of this paper are hence in favor of a formation of terrestrial planets by pebble accretion.

More refractory minerals, for instance silicates, sublimate much closer to the planetary surface. We do not consider the fate of the silicates in detail in this paper. The sublimation of the refractory minerals is connected to the formation of a radiative region in the envelope, which protects the released material from being lost to the surrounding disk. Previous work has assumed that once the envelope is hot enough to sublimate silicon-bearing minerals, all incoming material will become part of the envelope (Brouwers et al. 2021). Instead we propose that once the envelope reaches the saturation pressure of silicates, all further accreted material will move through the envelope until it reaches the surface of the planet. However, future work is needed to better understand the fate of sublimated pebbles in the envelope. Taking into account pebble sublimation and the future evolution of the enriched envelope is a key step in understanding how the formation and composition of planets is connected.

Acknowledgements

We thank the anonymous reviewer, whose feedback helped us to improve the original manuscript. P.W. acknowledges funding from the European Union H2020-MSCA-ITN-2019 under Grant Agreement no. 860470 (CHAMELEON). A.J. acknowledges funding from the European Research Foundation (ERC Consolidator Grant 724687-PLANETESYS), the Knut and Alice Wallenberg Foun- dation (Wallenberg Scholar Grant 2019.0442), the Swedish Research Council (Project Grant 2018-04867), the Danish National Research Foundation (DNRF Chair Grant DNRF159) and the Göran Gustafsson Foundation. This paper makes use of the following Python3 packages: numpy (Harris et al. 2020) and matplotlib (Hunter 2007).

Appendix A Further comparison with GGCHEM

Figure A.1 compares the natural logarithms of the supersaturation level S of the selected minerals given by our representative model to the ones calculated GGCHEM. The biggest difference is that supersaturation level in GGCHEM does not reach values above unity. Overall however, there is a good agreement between the two models especially when it comes to the sublimation temperatures of the considered minerals. For GGCHEM we define the sublimation temperature as the highest temperature at which the condensate occurs. Figure A.2 compares the sublimation temperatures of the selected minerals given by GGCHEM to the sublimation temperatures in our model. The deviation between the two approaches is only 1-5%.

Fig. A.1

Comparison of the natural logarithm of the supersaturation level S from the representative mineral model (upper row) to the natural logarithm of the supersaturation level S given by GGCHEM for the different mineral species (lower row) in a gas with solar composition as a function of the temperature for two different pressure levels. The crosses in the upper row show the sublimation temperatures for P = 0.1 Pa while the circles show the sublimation temperatures for P = 10⁴ Pa, In the lower row, the vertical dotted lines indicate the sublimation temperatures from the representative mineral model.

Fig. A.2

Comparison of the sublimation temperature calculated with GGCHEM (y-axis) and the representative mineral approach (x-axis). The crosses are for P = 0.1 Pa and the solid circles for P = 10⁴ Pa. The solid black line indicated the case where T_sub (GGCHEM) = T_sub (repr.minerals). The actual sublimation temperatures lie very close to this line.

Appendix B Sublimation temperature of FeS

Figure B.1 shows the sublimation temperature of FeS for H₂S/H₂ ratios in the range of 10⁻⁶ to 10⁻⁴. The sublimation temperature increases with increasing amount of H₂S from 558 K to 783 K. The H₂S/H₂ ratio based on the abundances from Lodders (2003) is 3.6 × 10⁻⁵ resulting in the sublimation temperature of ≈720 K.

Fig. B.1

Sublimation temperature of FeS as a function of the H₂S/H₂ ratio. The sublimation temperature increases with the amount of H₂S. The dotted black line indicates the H₂S/H₂ ratio used in the paper.

Appendix C The effect of the envelope opacity

The opacity of the envelope determines the pressure and temperature profile. In the main body of this paper, we use the opacity power-law from Bell & Lin (1994). To further test the influence of the opacity, we compare the envelope profiles where κ = 0.1 × κ_BL, κ = 1 × κ_BL, and κ = 10 × κ_BL· The level of opacity in an envelope determines how fast the heat generated by the accretion can radiate away. Thus, we start by comparing the temperature in the envelope for the different levels of opacity, see Fig. C.1. It can clearly be seen in Fig. C.1 that the temperature in inner envelope for κ = 10 × κ_BL is higher than in standard case. The final planet reaches a temperature of ≈3300 K close to the surface. The temperature in the low opacity case and the standard case are very similar. As a consequence, the sublimation location of the minerals are further our in the envelope in the case of κ = 10 × κ_BL·

Fig. C.1

Temperature maps of the inner planetary envelope during the growth process for the different opacities. The gray lines shows the sublimation locations of the different minerals. The temperature in the envelope increases with the opacity level and the sublimation locations move outwards.

The biggest difference between the different levels of opacity is the dominant form of energy transport, which is shown in Fig. C.2. For both κ = 1 × κ_BL and κ = 10 × κ_BL the envelope is mostly convective with a small radiative region in the inner envelope starting once the planet grows toM ≈ 0.2 M_⊕. The radiative region is smaller for κ = 10 × κ_BL· The inner convective region starts to appear once the planet has reached a mass of ≈0.35 M_⊕. For κ = 0.1 × κ_BL however, the envelope is mainly radiative with a convective region in the outer envelope in the beginning of the growth process and an inner convective region oce the planet has reached a mass of M ≈ 0.5 M_⊕.

Fig. C.2

Type of energy transport in the envelope of the planet for the three different levels of opacity. Convection is the main transport of energy in orange regions (mode 1) while in purple regions energy is transported by radiation (mode 2). The sublimation line of Mg₂SiO₄ is indicated by the black lines. In the low opacity case (top panel) the envelope is dominated by radiative energy transport. In the standard opacity level and high opacity level only a small radiative zone exits beneath the Mg₂SiO₄ sublimation line.

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¹

https://github.com/pw31/GGCHEM

All Tables

Table 1

Parameters used for the pebble accretion model.

In the text

Table 2

Mass of the planet in Earth masses in our model when T_surf > T_sub for the first time.

In the text

All Figures

	Fig. 1 Numerical growth track of the planet by pebble accretion starting at a₀ = 1.985 au and t₀ = 3.5 Myr. The initial mass of the planet is 0.001 M_⊕. The final mass of the planet is 0.68 M_⊕ at location of a = 1.0 au. The starting point of the planet is indicated by the square. The circle shows the final position of the planet when the disk dissipates and the planet stops migrating.
In the text

Fig. 2

Natural logarithm of the supersaturation level S for the selected mineral species in a gas with solar composition as a function of the temperature calculated with the representative mineral approach at two different pressures. At temperatures where ln S > 0, the mineral species can exist, while if ln S < 0 the mineral will not form even if all the reactants are in the gas phase. The left plot with P = 0.1 Pa corresponds to typical conditions in the protoplanetary disk, while the right plot with P = 10⁴ Pa corresponds to the typical pressures in the inner envelope of an Earth-mass planet. The filled circles indicate the sublimation temperatures while the vertical dotted lines indicate the sublimation temperatures for the other pressure. Except for FeS the sublimation temperature increases with the pressure level. Mg₂SiO₄ and Fe sublimate at similar temperatures in the low pressure scenario. At a high pressure, Mg₂SiO₄ has a lower sublimation temperature than Fe.

In the text

Fig. 3

Total element abundances in the gas phase calculated with GGCHEM as a function of the temperature for five elements that represent refractory to moderately volatile minerals. We show the results at two different pressures, P = 0.1 Pa (left) and P = 10⁴ Pa (right). The dotted vertical lines show the sublimation temperature of our corresponding representative minerals. The sublimation temperatures agree well with the temperature at which gas phase molecules that contain the considered elements first appear.

In the text

Fig. 4

Temperature (left) and pressure (right) profiles in the envelope of a planet with M = 0.1 M_⊕ (purple curves) and M = 0.68 M_⊕ (orange curves). The distance to the surface of each planet is given in units of the Hill radius of the planets. The solid circle shows the position of the respective Bondi radius. The temperature and pressure in the envelope increase with the mass of the planet. Since the planet is migrating inward as it grows, the pressure in the surrounding disk increases while the Hill radius shrinks. The planet with M = 0.68 M_⊕ develops a radiative region in the inner envelope indicated by the dotted line style.

In the text

Fig. 5

Temperature map of the inner planetary envelope during the growth process after the planet has reached a mass of M = 0.01 M_⊕. The distance to the surface of the planet is given in units of the instantaneous Hill radius of the planet. The gray lines show the sublimation locations of the representative minerals. The white dotted line represents the Bondi radius. The temperature in the envelope starts to increase once the planet reaches a mass of 0.01 M_⊕. For the final planet the temperature in the inner is ≈ 2700 K. The sublimation line of FeS moves out to r ≈ 0.03 R_H. The sublimation lines of Mg₂SiO₄, and Fe lie close together. The sublimation temperature of A1₂O₃ is only reached close to the surface of a planet with M > 0.4 M_⊕.

In the text

Fig. 6

Type of energy transport in the envelope of the planet. The distance to the surface of the planet is given in units of the instantaneous Hill radius of the planet at the relevant mass. In orange regions the energy is transported via convection (mode 1), whereas in purple regions the radiative energy transport (mode 0) dominates. The black line shows the sublimation line of Mg₂SiO₄ from our chemical equilibrium model. Radiation as the main energy transport process only appears in the inner disk after the envelope close to the planet has reached the sublimation temperature of Mg₂SiO₄.

In the text

Fig. 7

Sublimation temperature of the representative mineral species. The dotted gray line shows the temperature close to the surface as a function of the planetary mass. The filled circles indicate where the surface temperature first reaches the sublimation temperature of each mineral. The sublimation temperature is relatively independent of the planet mass. Hence, the order of the sublimation lines stays the same throughout the growth of the planet.

In the text

Fig. 8

Sublimation rate of FeS as a function of temperature using the laboratory measurements of Tachibana & Tsuchiyama (1998). The orange line represents the contribution from reaction (24) and the purple line from Eq. (33). The black line is the total sublimation rate. The vertical dashed line indicates the sublimation temperature of FeS based on chemical equilibrium. The total sublimation rate increases strongly with temperature.

In the text

Fig. 9

Comparison of the destruction time scales as a function of temperature (left) and the settling time scales as a function of planetary mass (right) for FeS pebbles with a size of 10 µm (orange) and 100 µm (purple). The settling time scales are calculated from the point in the envelope where T = T₁ = 720 K (solid lines) and T = T₂ = 1000 K (dashed lines). The two temperatures T₁ and T₂ are also indicated in the left plot by the dotted and dashed line respectively. The settling time scale with convective speed is shown as black lines for the two temperatures. The settling time scales of both pebble sizes from the location of T₁ is too short compared to the destruction time scales of the pebbles. The settling time scale of a pebble from the T₂ location becomes longer than the destruction time scale of a 10 µm pebble at 1000 K once the planet reaches a mass of M ≈ 0.2 M_⊕.

In the text

Fig. 10

Fraction f_S of S that directly reaches the planetary surface, normalized to the solar composition, for two different sublimation temperatures, T_sub = 720 K (purple, chemical equilibrium case) and T_sub = 1000 K (orange). The accreted material is assumed to be of solar composition (solid line) or depleted in S by a factor of 1/3 compared to solar composition (dashed line). We assume that the planet stops accreting S once it reaches the sublimation temperature of FeS at the bottom of the envelope. The gray horizontal bars represent the estimated bulk S concentrations of Mars and Earth. The filled circle shows the S concentration of a planet formed by giant impact (GI) of two protoplanets formed from solar composition pebbles and the unfilled circle for proto-planets that formed out of pebbles already depleted compared to solar.

In the text

Fig. A.1

Comparison of the natural logarithm of the supersaturation level S from the representative mineral model (upper row) to the natural logarithm of the supersaturation level S given by GGCHEM for the different mineral species (lower row) in a gas with solar composition as a function of the temperature for two different pressure levels. The crosses in the upper row show the sublimation temperatures for P = 0.1 Pa while the circles show the sublimation temperatures for P = 10⁴ Pa, In the lower row, the vertical dotted lines indicate the sublimation temperatures from the representative mineral model.

In the text

	Fig. A.2 Comparison of the sublimation temperature calculated with GGCHEM (y-axis) and the representative mineral approach (x-axis). The crosses are for P = 0.1 Pa and the solid circles for P = 10⁴ Pa. The solid black line indicated the case where T_sub (GGCHEM) = T_sub (repr.minerals). The actual sublimation temperatures lie very close to this line.
In the text

	Fig. B.1 Sublimation temperature of FeS as a function of the H₂S/H₂ ratio. The sublimation temperature increases with the amount of H₂S. The dotted black line indicates the H₂S/H₂ ratio used in the paper.
In the text

	Fig. C.1 Temperature maps of the inner planetary envelope during the growth process for the different opacities. The gray lines shows the sublimation locations of the different minerals. The temperature in the envelope increases with the opacity level and the sublimation locations move outwards.
In the text

Fig. C.2

Type of energy transport in the envelope of the planet for the three different levels of opacity. Convection is the main transport of energy in orange regions (mode 1) while in purple regions energy is transported by radiation (mode 2). The sublimation line of Mg₂SiO₄ is indicated by the black lines. In the low opacity case (top panel) the envelope is dominated by radiative energy transport. In the standard opacity level and high opacity level only a small radiative zone exits beneath the Mg₂SiO₄ sublimation line.

In the text

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