The Formation of Mini-Neptunes

In this work, we use the term Neptunes to describe planets with masses between 10 and 20 M_⊕ and a H–He mass fraction of 0.1–0.25 (Helled et al. 2011; Nettelmann et al. 2013). We define a mini-Neptune with the same mass fraction of H–He (denoted as f_HHe), but for a total planet mass in the range of 0 < M_P ≤ 10 M_⊕. We define t_mini-Nept as the time spent by the protoplanet in the mini-Neptune region (i.e., with 0.1 ≤ f_HHe ≤ 0.25 and 0 < M_P ≤ 10 M_⊕).

We determine the likelihood of obtaining a mini-Neptune under different formation conditions, i.e., the mini-Neptune occurrence rate. This definition should capture the fact that if the core becomes critical while the disk is still present, a significant mass of gas is accreted and the planet will become a gas giant, not a mini-Neptune. In order to form a mini-Neptune, the growing protoplanet must spend a significant time in the mini-Neptune regime until the gaseous disk disappears.

As a result, our definition of the occurrence rate should include a certain prior of the disk lifetime. Observations show that disks have lifetimes of a few million years, and that they should not disappear in the first million years, regardless of the stellar mass. Then, at times 1–3 Myr, a sharp decrease in disk frequency occurs, and finally no disks are observed at ages around ∼10 Myr (Ribas et al. 2015). The fraction of stars with disks at different ages has been fit with an exponential decay of 2.5 Myr (Mamajek 2009) or 4 Myr (Pfalzner et al. 2014). Therefore, the cumulative distribution of the disk's lifetime is cdf = 1 − exp(−time/τ_disk), but is flat during the first ∼1 Myr if one accounts for the recent observations by Ribas et al. (2015). Consequently, we adopt the following probability density function of disk lifetime:

$$\begin{eqnarray}&&p(t)=\displaystyle \frac{1}{{\tau }_{\mathrm{disk}}}\displaystyle \frac{{e}^{-t/{\tau }_{\mathrm{disk}}}}{({e}^{-{t}_{0}/{\tau }_{\mathrm{disk}}}-{e}^{-{t}_{M}/{\tau }_{\mathrm{disk}}})}\end{eqnarray} \tag{ 1 }$$

if t₀ ≤ t ≤ t_M, or p(t) = 0 otherwise. t₀ is the minimum time at which the disk can disappear and t_M the maximum, and they are set to be 1 Myr and 10 Myr, respectively. Note that the integral between t₀ and t_M yields 1, as it must for any probability density function. The mini-Neptune occurrence rate is therefore defined as the integral of p(t) in the domain where ${t}_{\mathrm{mini} \mbox{-} \mathrm{Nept}}\ne 0$, which yields

$$\begin{eqnarray}&&{f}_{\mathrm{MN}}=\displaystyle \frac{{e}^{-{t}_{i}/{\tau }_{\mathrm{disk}}}-{e}^{-{t}_{f}/{\tau }_{\mathrm{disk}}}}{{e}^{-{t}_{0}/{\tau }_{\mathrm{disk}}}-{e}^{-{t}_{M}/{\tau }_{\mathrm{disk}}}},\end{eqnarray} \tag{ 2 }$$

where t_i is the time when the planet enters the mini-Neptune region if t_i ≥ t₀ or t₀, while t_f is the time when the planet exits the mini-Neptune region if t_f ≤ t_M or t_M.

It should be clear, however, that our calculated "occurrence rates" are given as a guidance for the likelihood of forming planets under given theoretical conditions.

Since low-mass, low-density exoplanets could have a wider range of H–He mass fractions, we also consider a second, less restricted definition of a mini-Neptune, assuming the same mass range, but allowing the mass fraction of H–He to be in the range of 0.05 ≤ f_HHe ≤ 0.5. As we show below, this definition (which we refer to as extended) yields occurrence rates typically higher by a factor of 2 with respect to the first, restricted definition.

Our baseline disk model is the standard minimum mass solar nebula (MMSN) of Weidenschilling (1977) and Hayashi (1981), where the initial surface density of solids Σ_s is given by the power law Σ_s = Σ₀ Z₀ (r/au)^−p, with Z₀ = 0.018 (the disk metallicity), p = 3/2, and Σ₀ = 1700 g cm⁻². A comparison with other disk parameters is discussed in Section 3.3.

We consider in situ formation between 5 and 30 au. For the temperature profile we adopt a standard passive disk (see, e.g., Armitage 2010; Guilera et al. 2011), with T(r) = T₀ (r/au)^−1/2 and T₀ = 280 K. The protoplanet is assumed to be embedded in the gaseous disk during its growth, thus the outer temperature of the envelope matches the disk temperature at the position of the protoplanet.

For the disk scale height we assume a vertically isothermal disk. Using this, the definition of the disk scale height (H_gas = c_s/Ω) and the temperature profile described above, it can be shown that the aspect ratio (h = H/r) is given by h(r) = h₀ (r/au)^1/4, with ${h}_{0}=\sqrt{({k}_{B}\,\mathrm{au}/\mu {m}_{{\rm{H}}}{{GM}}_{* }){T}_{0}}$, with h₀ ≃ 0.036 for the chosen temperature profile, and the remaining symbols are in standard notation.

Besides temperature, the other necessary boundary condition to integrate the structure equations for the planetary envelope is the outer pressure. To obtain this we assume a classical power-law profile: P(r) = P₀ (r/au)^−q. The exponent q is derived from the ideal gas law and the condition that the column gas density is: ρ_gas = Σ_gas/H_gas. We get q = p + 7/4 and ${P}_{0}=\sqrt{({k}_{B}\,{{GM}}_{* }/\mu {m}_{{\rm{H}}}{\mathrm{au}}^{3}){T}_{0}{{\rm{\Sigma }}}_{0}}$, which for an MMSN disk is P₀ ≃ 36.5 dyn cm⁻².

In reality, the solids in the disk should have a size distribution, ranging from millimeter-dust to hundred-kilometer-sized planetesimals. This size distribution evolves as pebbles grow and convert into planetesimals (Dra̧żkowska et al. 2016), and as planetesimals fragment (Guilera et al. 2014). However, since the size distribution of solids and its time evolution is still poorly known, we consider a given size and concentrate on the effect of envelope enrichment assuming the accretion of just pebbles and just planetesimals, and compare the two. In the case of pebbles, we assume that envelope enrichment begins when the envelope mass is M_env = 10⁻⁶ M_⊕, whereas in the case of planetesimal enrichment, enrichment begins when M_env = 2 × 10⁻⁴ M_⊕ (see Table 4 of Venturini et al. 2016).

2.3.1. Pebble Accretion

For pebble accretion we use the prescription derived initially by Lambrechts & Johansen (2014) (hereafter LJ14), and adapted later to include the effect of disk turbulence (Morbidelli et al. 2015; Brasser et al. 2017). In this model, pebbles with Stokes numbers smaller than ∼1 drift toward the star and are intercepted by the growing embryo. The surface density of gas decays exponentially with time to mimic gas dissipation and is given by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{gas}}={{\rm{\Sigma }}}_{0}\,{(r/\mathrm{au})}^{-p}{e}^{-t/{\tau }_{\mathrm{disk}}},\end{eqnarray} \tag{ 3 }$$

where τ_disk is the disk's lifetime, which in our baseline model is taken as τ_disk = 3 Myr. The initial surface density of dust is simply Σ_s = Z₀ Σ_gas. The surface density of pebbles (Σ_P) can be found from the definition of the pebble flux: ${\dot{M}}_{F}=2\pi {{rv}}_{r}{{\rm{\Sigma }}}_{{\rm{P}}}$. The pebble flux can be computed, on the other hand, from the condition of equating the growth timescale with the radial drift timescale (see LJ14 for the details). The efficiency of pebble accretion depends on the pebble scale height: ${H}_{\mathrm{peb}}={H}_{\mathrm{gas}}\sqrt{\alpha /{\tau }_{{\rm{f}}}}$, where α is the viscosity parameter (Shakura & Sunyaev 1973) and τ_f is the Stokes number of the drifting pebbles, which can be computed assuming a dominant particle size as described in the simple dust growth model of Birnstiel et al. (2012) and LJ14.

If the Hill radius of the planet exceeds the pebble scale height, pebble accretion is 2D and is given by

$$\begin{eqnarray}&&{\dot{M}}_{z,2{\rm{D}}}=2{\left(\displaystyle \frac{{\tau }_{f}}{0.1}\right)}^{2/3}{R}_{{\rm{H}}}{v}_{{\rm{H}}}{{\rm{\Sigma }}}_{{\rm{P}}},\end{eqnarray} \tag{ 4 }$$

where R_H is the planet's Hill radius, v_H is the Keplerian velocity at a distance of the Hill radius from the center of the planet, and Σ_P is the surface density of pebbles at the position of the planet.

The accretion rate of pebbles transitions to the less efficient 3D fashion if

$$\begin{eqnarray}&&\displaystyle \frac{\pi {({\tau }_{{\rm{f}}}/0.1)}^{1/3}{R}_{{\rm{H}}}}{2\sqrt{2\pi }}\lt {H}_{\mathrm{peb}}.\end{eqnarray} \tag{ 5 }$$

In this case, the accretion rate is given by (Brasser et al. 2017)

$$\begin{eqnarray}&&{\dot{M}}_{z,3{\rm{D}}}={\dot{M}}_{z,2{\rm{D}}}\left(\displaystyle \frac{\pi {({\tau }_{{\rm{f}}}/0.1)}^{1/3}{R}_{{\rm{H}}}}{2\sqrt{2\pi }{H}_{\mathrm{peb}}}\right).\end{eqnarray} \tag{ 6 }$$

Therefore, when pebble accretion becomes 3D, its value depends on α, decreasing for higher disk viscosity. If the disk is turbulent (α > 10⁻³), the growth of the planet is typically so slow that gas accretion within the disk's lifetime is insignificant, and hence mini-Neptune formation is unlikely, as we show in Section 3.3.

For planetesimal accretion we use the high accretion rates given by Rafikov (2011). This rate is appropriate for small planetesimals (∼100 m–1 km) for a dynamically cold planetesimal disk (see Goldreich et al. 2004, for details)

$$\begin{eqnarray}&&{\dot{M}}_{z}=6.47{\rm{\Omega }}{P}^{1/2}{{\rm{\Sigma }}}_{s}{R}_{{\rm{H}}}^{2},\end{eqnarray} \tag{ 7 }$$

where Ω is the Keplerian frequency of the protoplanet, and P = R_c/R_H, R_c being the capture radius of the planet (computed following the prescription of Inaba & Ikoma 2003). Large planetesimals cannot be accreted efficiently since they are less affected by gas drag when passing through the planetary envelope (e.g., Pollack et al. 1996; Inaba & Ikoma 2003; Helled & Bodenheimer 2014). In addition, planetesimals' eccentricities and inclinations tend to increase during the planetary growth due to the perturbation of the embryo (Ida & Makino 1993; Ormel et al. 2010), which increases the relative velocities of planetesimals, reducing the accretion efficiency. Small planetesimals (∼100 m) can still be accreted due to the effect of gas drag, allowing for the formation of gas-rich objects before disk dispersal (Fortier et al. 2013). We therefore focus on planetesimal accretion rates that correspond to relatively small sizes. Nevertheless, we discuss the possibility of a lower accretion rate of solids (i.e., larger planetesimals) in Section 3.3.

The assumed opacity of the planetary envelope has a crucial role in determining the growth rate of the planet (Podolak 2003; Movshovitz et al. 2010). For our baseline model we assume gas opacities from Freedman et al. (2014), which account for the increase of gas opacity caused by the raise in metallicity. For the dust opacities we adopt the analytical model of Mordasini (2014), which includes dust growth and settling. Minimum values of these dust opacities in the outer layers of the envelope are rather low (∼0.001–0.01 cm² g⁻¹), hence we refer to this case as low dust opacity. Although these low opacity values are justified as well by other works (e.g., Movshovitz & Podolak 2008; Ormel 2014), higher dust opacities could be a result of the ongoing ablation of accreted planetesimals/pebbles. Indeed, the dust opacities of Mordasini (2014) were computed assuming a unique dominant size of grains. If ablation of solids is dominant in the upper atmosphere, most of the masses of solids could reside in smaller dust aggregates, and hence the opacity could be higher. Also, the recondensation of upstreaming water vapor (cloud formation) could increase the opacities. To account for this possibility, we also run calculations with dust opacities increased by a factor of 100 and we refer to this as the high dust opacity model.

For the case of pebble accretion with low dust opacities, we find that mini-Neptunes are more likely to form when envelope enrichment is included. Figure 2 shows that if envelope enrichment is not included, the formation path of the planet crosses the parameter space of mini-Neptunes if the embryo grows at 20 au but for times longer than the disk's mean lifetime, yielding a mini-Neptune occurrence rate of 9% and 19% for the restricted and extended cases, respectively. At a = 5 au without enrichment and low opacities, the formation path only crosses the mini-Neptune regime by invoking the extended definition. In other words, for this case, only mini-Neptunes with f_HHe < 0.1 can be formed, but with a negligible occurrence rate (1%). When envelope enrichment is considered, the protoplanet crosses the mini-Neptune region, both at 5 and 20 au. However, at a = 5 au the crossover mass is reached so early (t_cross = 0.65 Myr) that the planet becomes a gas giant in ∼1 Myr, giving a mini-Neptune occurrence rate of zero. At a = 20 au the occurrence rate is 27% with the restricted definition and 41% with the extended definition.

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The fact that mini-Neptunes appear more easily when envelope enrichment is included can be understood as follows. When the volatiles are allowed to mix in the primordial H–He envelopes, the mean molecular weight increases, which translates into an increase in the envelope's density and self-gravity. The envelope is therefore more prone to contract, allowing accretion of H–He in larger amounts once enrichment begins. This leads to the formation of low-mass planets with larger fractions of H–He than in the case without envelope enrichment, and in addition, to relatively short formation timescales. It is important to note that for the case of pebble accretion analyzed in this section, α is set to 10⁻⁵ (laminar disk), which corresponds to high solid accretion rates. Hence, the timescale to form a gas giant is short, as summarized in Table 1. Results with other values of α are shown in Section 3.3.

When higher dust opacities are invoked for the envelope, because of the slower cooling, formation timescales are longer (see Table 2). In principle, this implies longer times spent in the mini-Neptune region, and therefore higher occurrence rates. The maximum occurrence rate of mini-Neptunes is found to be ∼79% (extended definition) for the case with envelope enrichment, and a = 20 au. Without envelope enrichment, mini-Neptune formation is very unlikely. Even when choosing different disk parameters (see Section 3.3), we find that typically envelope enrichment is required for the formation of mini-Neptunes with pebble accretion. The results for pebble accretion with low and high opacities are summarized in Tables 1 and 2, respectively. Since pebbles are small objects they are likely to enrich the envelope and to increase the opacity, and we expect the high occurrence rates for pebble accretion to be more appropriate.

The case of planetesimal accretion is rather different. As we discuss above, the classical in situ growth of a planet by planetesimal accretion is characterized by three phases of growth (Pollack et al. 1996). During phase-2, the planetary mass increases very slowly, and the heavy element mass at the beginning of this stage (known as the isolation mass, M_iso) strongly depends on the amount of solids in the embryo's vicinity. In other words, for more massive and/or metal-rich disks, the higher M_iso is. Indeed, Pollack et al. (1996) showed that ${M}_{\mathrm{iso}}\sim {a}^{3}{{\rm{\Sigma }}}_{s}^{3/2}$, with the exact relation being

$$\begin{eqnarray}&&{M}_{\mathrm{iso}}={\boldsymbol{C}}\,{({{\rm{\Sigma }}}_{0}{Z}_{0})}^{3/2}{\left(\displaystyle \frac{a}{\mathrm{au}}\right)}^{\tfrac{3}{2}(2-p)},\end{eqnarray} \tag{ 8 }$$

with ${\boldsymbol{C}}\,\approx \,0.0026$ to obtain M_iso in Earth masses.

For an MMSN disk profile M_iso ∝ a^3/4. The dependence of M_iso on the semimajor axis results in M_iso ≈ 1.5 M_⊕ at a = 5 au and M_iso ≈ 4.2 M_⊕ at a = 20 au (see Figure 3). Due to the relatively long timescale of phase-2, the value of M_iso plays a crucial role in determining whether a protoplanet can be in the mini-Neptune region. If the combination of a and Σ_s results in a M_iso < 10 M_⊕, then the protoplanet is likely to spend some time in the mini-Neptune region (see Section 3.3.2).

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As shown in Figure 2 and Table 1, with low dust opacities, at a = 5 au the occurrence rate of mini-Neptunes is extremely high when envelope enrichment is included (83% with the restricted definition and 100% with the extended one), and lower without envelope enrichment (33% and 76%). At 20 au and with envelope enrichment, the crossover time is shorter than that at 5 au (∼1.5 Myr), therefore the mini-Neptune occurrence rates are lower (see Figure 3 and Table 1).

When we use high dust opacities, M_iso is unchanged but the formation timescales are so long—particularly when neglecting envelope enrichment—that although the planetary mass remains low during the first 3 Myr of growth, the gas accretion rate is negligible. As a result, the H–He mass fraction is small in these cases, typically being less than 5% for t ≤ 3 Myr (see Figure 2), and mini-Neptunes are unlikely to form, unless envelope enrichment is invoked at large semimajor axes. For instance, for the enriched case at a = 20 au, the occurrence rate is found to be 66% with the restricted definition and 82% with the extended one.

In the case of planetesimal accretion, the results are very sensitive to the assumed density profile in the disk, due to the dependence on M_iso. It should be noted that M_iso is independent of a only for a disk with p = 2, which was the value chosen in Pollack et al. (1996). Observations typically infer values of p ∼ 1 (Andrews et al. 2010). For planetesimal accretion, the flatter the disk, the stronger the dependence of M_iso on the location of the embryo.

One difference between pebble and planetesimal accretion is that with pebbles the occurrence rate of mini-Neptunes is always higher when envelope enrichment is considered. With planetesimals, it is more difficult to draw a general conclusion. If phase-2 is so long that the mini-Neptune region is crossed at times larger than the disk's lifetime, then envelope enrichment favors the occurrence of mini-Neptunes, because it shortens phase-2 and allows the protoplanet to cross the mini-Neptune region within the disk's lifetime (see, e.g., the enriched and non-enriched cases at a = 5 au of Figure 3). On the other hand, if the duration of phase-2 is a few Myr when the envelope is pure H–He, envelope enrichment leads to formation timescales shorter than the disk's mean lifetime, promoting the formation of gas giants (simulations at a = 20 au and low dust opacities, see Figure 3).

Examples of the growth histories of the planets are shown in Figure 4. This figure shows the mass of the core (M_core), total mass of heavy elements (M_Z), the mass of H–He (M_HHe), and the total planetary mass (M_P) as a function of time. Shown are the cases for pebbles (top) and planetesimal (bottom) accretion for the two different formation locations at 5 au (left) and 20 au (right) for the low dust opacity case. Figure 5 shows the evolution of the envelope metallicity for the enriched cases. Since the envelope is thin at early times, the metallicity increases very rapidly when enrichment begins. Further details on the model can be found in Venturini et al. (2016).

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In this section we investigate the sensitivity of the results to the assumed disk parameters (α, Σ₀, Z₀, p). For the simulations presented in this section we used the baseline low dust opacities.

3.3.1. Pebbles

The growth of a planet via pebble accretion depends sensitively on the disk turbulence (Ormel 2017). Certainly, Figure 6(a) shows that for our baseline MMSN disk, the occurrence rate of mini-Neptunes depends drastically on the value of α. The more turbulent the disk is, the more spread the pebbles are vertically, and therefore the more difficult it is for the protoplanet to intercept them. For instance, for our baseline MMSN disk (Figure 6(a)) at a = 10 au and α = 10⁻³, the core can just grow from 0.01 M_⊕ to 0.05 M_⊕ during the disk's mean lifetime. This happens because for this case all the growth occurs in the 3D regime (Equation (5)), and for these particular disk's parameters the accretion rate of pebbles is of the order of 10⁻⁸ M_⊕ yr⁻¹. On the contrary, for low values of α (i.e., 10⁻⁵), the accretion rate is in the 2D form, and hence typically high (∼10⁻⁵–10⁻⁶ M_⊕ yr⁻¹). This is clear from the low values of crossover times shown in Figure 6, especially for 5 < a < 15 au.

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The most striking result from Figure 6 is that the formation of mini-Neptunes is much more likely when envelope enrichment is considered, similar to the results we get in Section 3.1. Not only is the inferred occurrence rate higher when envelope enrichment is included, but also the parameter space is much broader.

The detailed analysis of Figure 6 is not so intuitive, but the general trend is that if formation timescales are too short, the occurrence rate of mini-Neptunes is low and the most likely output is the formation of a gas giant. Also, if formation timescales are too long (for instance, the cases with t_cross > 10 Myr), the protoplanet accretes negligible amounts of H–He during the disk's lifetime. Thus, the formation of mini-Neptunes is unlikely, and the most typical output in these cases is lunar- to Earth-mass embryos.

We find that, other than the disk viscosity, the crucial variable that dictates the likelihood of obtaining a mini-Neptune is the amount of drifting pebbles during the disk's mean lifetime. For the simulations shown in the two panels of Figure 6, the amount of pebbles that drifted after 3 Myr is 140 M_⊕ for the MMSN disk and 180 M_⊕ for the baseline LJ14 disk (p = 1, Σ₀ = 500 g cm⁻², Z₀ = 0.01). For both disk models the formation of mini-Neptunes is likely, and occurs in a broad parameter space, as long as envelope enrichment is included.

Figure 7 shows the sensitivity of the results to the assumed disk mass. If we double the disk mass, the formation timescales are shorter because the flux of pebbles is larger, producing larger accretion rates of pebbles. In this case, a total amount of 365 M_⊕ of pebbles drift within 3 Myr and the likelihood of forming a mini-Neptune is much smaller than in the baseline LJ14 disk, yielding typical mini-Neptune occurrence rates of zero. On the other hand, for a disk with half of the mass of that in LJ14, the flux of pebbles diminishes, leading to a slower growth. In this case it is only possible to form mini-Neptunes by accounting for envelope enrichment, but the parameter space that leads to mini-Neptune formation is reduced compared to the baseline disk model. For this case, the total amount of pebbles that drifted after 3 Myr is 90 M_⊕, and it is only possible to form mini-Neptunes for very low-turbulent disks.

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We find that for Σ₀ = 250 g cm⁻² and Z₀ = 0.005 (same disk as in Figure 7(b), but half of the metallcity), the pebble mass that drifted after 3 Myr is only 30 M_⊕, and in this case, mini-Neptune formation does not occur, leaving lunar- to Earth-mass embryos. We find that in order to have a non-zero occurrence rate of mini-Neptunes, at least ∼50 M_⊕ of pebbles must drift during the disk's lifetime.

If instead of changing the disk's mass we just modify the disk metallicity, we find that the peak in mini-Neptune occurrence rate takes place for disk metallicities lower than solar for a typical disk as the baseline of LJ14, which contains an initial mass of solids of ∼100 M_⊕ between 5 and 100 au. The results are illustrated in Figure 8, for α = 10⁻⁵, where accretion is 2D for most of the growth. The fast drop to a mini-Neptune occurrence rate of 0% at Z₀ ≈ 0.004 (for a = 20 au) is because the accretion becomes 3D. The reason why the formation of mini-Neptunes does not require high metallicity is that the optimal accretion rate of solids to obtain mini-Neptunes is of ${\dot{M}}_{z}\sim {10}^{-6}$ M_⊕ yr⁻¹. This value is relatively low, so typical disks with metallicities equal or larger than solar are more prone to producing gas giants than mini-Neptunes.

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Figure 8 also shows the mini-Neptune occurrence rate for a disk with half the mass of the baseline of LJ14 (p = 1, Σ₀ = 250 g cm⁻²). In this case the peak in mini-Neptune occurrence rate shifts toward a bit larger disk metallicity (Z₀ ≈ 0.009) because when the total mass of the disk is reduced, the flux of pebbles diminishes, so a larger dust-to-gas ratio is required to have a solid accretion rate of ${\dot{M}}_{z}\sim {10}^{-6}$ M_⊕ yr⁻¹. Still, this disk metallicity is lower than solar, having an initial amount of solids of ∼50 M_⊕ between 5 and 100 au.

In summary, there is "sweet spot" for mini-Neptune formation in the case of pebble accretion, which is a disk with a total amount of drifting pebbles in the range of 50–300 M_⊕, or more precisely, for ${\dot{M}}_{z}\sim {10}^{-6}$ M_⊕ yr⁻¹. This condition is more likely to exist in low-metallicity disks, and low disk viscosity. If the disk is very turbulent (i.e., α > 10⁻³), the accretion rate of solids is too low, and gas accretion is unlikely to take place during the disk's lifetime, so in this case we are left with embryos of lunar-to-Earth mass.

3.3.2. Planetesimals

The results for planetesimal accretion for different disk models are shown in Figure 9. We corroborate that the only factors determining the occurrence rate of mini-Neptunes are M_iso and a. In other words, if we change Σ₀, Z₀, or p in order to have the same isolation mass at the same semimajor axis, we obtain exactly the same occurrence rate (of any type of planet). Figure 9 shows that there is a "sweet spot" for the formation of mini-Neptunes, and that the mini-Neptune "occurrence region" is shifted to smaller M_iso with envelope enrichment. This happens because gas accretion starts sooner when envelope enrichment is included. The general trends of the mini-Neptune occurrence rate can be understood as follows.

• 1.  
  Fixed a: for small M_iso, formation timescales are so long that gas accretion is negligible during the disk's lifetime. There is a threshold for which M_iso is large enough to allow gas accretion in the first 3 Myr. For high values of M_iso, phase-2 becomes so short that the planet becomes a gas giant, leading for the occurrence rate of mini-Neptune to drop for large M_iso.
• 2.  
  Fixed M_iso: the larger the semimajor axis, the longer the duration of phase-1 (Pollack et al. 1996). Thus, the sweet spot for mini-Neptunes (always during phase-2) occurs for times larger than 3 Myr as we increase a, which decreases the mini-Neptune occurrence rate.

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3.3.3. Lower Accretion Rates of Planetesimals

We find that Neptune-like planets are not a common output both for the pebble and planetesimal cases. With low dust opacities and planetesimal accretion, for all the models with isolation masses above the green shaded areas of Figure 9, either the duration of phase-2 is too short, or the amount of H–He when the planet crosses the 10 ≤ M_P ≤ 20 M_⊕ is too large. Hence, the most likely outcome is a gas giant, not a Neptune.

Indeed, with low dust opacities, as presented in Figure 9, the occurrence rate of Neptunes is zero, with the exception being cases with embryos located at 25 au, which reach a Neptune occurrence rate of 10%. A necessary condition to get a non-zero occurrence rate of Neptunes is to get 10 < M_iso < 15 M_⊕ (consistent with a surface density of solids equivalent to 2–3 MMSN) and a relatively long phase-2. This is the reason why formation is not possible with low dust opacities, because for the mentioned isolation masses, the longest derived duration of phase-2 is only 0.6 Myr. Assuming that the planet crosses the "Neptune region" during phase-2, this would yield a maximum occurrence rate of Neptunes of 20% (at 5 au).

The only way to have Neptune occurrence rates that are relatively large with planetesimal accretion is without envelope enrichment and high dust opacities, although the feasibility of such high dust opacities requires further investigations. When high dust opacities are invoked, a maximum occurrence rate of Neptunes of ∼40% is found for 5 < a < 10 au, p = 1, and Z₀ = 0.018, or p = 0.5 and Z₀ = 0.01; i.e., disks with masses of solids in the range of 200–400 M_⊕ between 5 and 30 au. Under similar conditions, Guilera et al. (2011) are able to form Uranus and Neptune simultaneously.

With pebble accretion there is no intermediate-mass long phase, so we find a very low occurrence rate of Neptunes, typically zero, or ∼10% in the best cases (see Tables 1 and 2). We find that the combination of high dust opacities and envelope enrichment slightly enhances the possibility of the formation of Neptunes (see Table 2, pebble accretion with enrichment at a = 5 au).

A possible mechanism to form Neptunes is the merging of mini-Neptunes, as proposed by Izidoro et al. (2015). It was shown that by colliding 3–6 M_⊕ embryos from a total mass of solids in embryos of 30–60 M_⊕, Uranus and Neptune can form. Our calculations show that mini-Neptunes form relatively easily under the same assumption of amount of solids. Therefore, we provide the seeds needed to form Neptunes. Thus, a key question that remains to be answered is what determines the formation of mini-Neptunes versus Neptunes in planetary systems.

Note that for the cases where we manage to form Neptune-like planets, the embryos are located at radial distances a < 10 au, which are smaller than the locations of Uranus and Neptune in our solar system. We therefore suggest that in the pebble accretion case Uranus and Neptune cannot form in situ as argued by Lambrechts et al. (2014). Instead, our results are consistent with the scenario in which the two planets formed at smaller radial distances followed by outward migration (Tsiganis et al. 2005).

Several studies explored the possibility of in situ formation of super-Earths and mini-Neptunes inside the iceline, assuming planetesimal accretion (Ikoma & Hori 2012; Bodenheimer & Lissauer 2014; Inamdar & Schlichting 2015). All of these investigations find that in situ formation is rather unlikely. The reason for this is illustrated in Figure 9: the isolation masses are extremely small. Indeed, we find that for planetesimal accretion in situ formation at a = 1 au, assuming 1 MMSN yields an isolation mass of M_iso ≈ 0.15 M_⊕. The envelope mass at this stage is hardly M_env = 2 × 10⁻⁷ M_⊕. Rocky planetesimals are expected to be fully disrupted before reaching the core when M_env ∼ 0.01 M_⊕ (Mordasini et al. 2006). Therefore, unless some fast formation mechanism to start with an Earth-mass embryo is invoked (Bodenheimer & Lissauer 2014), it is not possible to form a few Earth-mass planets with a considerable amount of H–He in mass inside the iceline if the dominant accretion process is that of planetesimals.

Regarding growth with pebbles, the simple model of LJ14 is probably not valid inside the iceline. First, the dust properties might vary since the composition and the sizes of the grains change. Second, inside the iceline the solids tend to pile up (Dra̧żkowska et al. 2016), so the assumption of a continuous flux of pebbles is not applicable. Concerning the first challenge, Morbidelli et al. (2015) showed that the growth of an embryo with pebbles inside the iceline is extremely inefficient as long as icy pebbles disrupt when crossing the iceline and release small dust grains. A Mars-size object is expected to grow from accretion of silicate dust grains (just like with planetesimal accretion), but not super-Earth mass planets. On the other hand, Dra̧żkowska et al. (2016) found that rocky planets could form in situ by pile-ups of solids (up to 1000 M_⊕ at 1 au). These objects could accrete some H–He and be enriched afterward by silicate-grains, albeit gas accretion inside the iceline is challenging, given that hot gas tends to flow rather than stay bound to the planet (Ormel et al. 2015).

In all the results presented above, the disk's mean lifetime was set to be 3 Myr and the planetary occurrence rates were computed assuming the disk disappears after 1–10 Myr. In this section we explore the dependence of the emergence of mini-Neptunes on these parameters. We consider two cases: reducing the disk mean lifetime by half or increasing it by a factor of 2. If we consider τ_disk = 1.5 Myr instead of 3 (and same t₀ and t_M), then for the baseline enriched case and a = 20 au with pebble accretion, the occurrence rate of mini-Neptunes rises from 41% to 84%. This happens because the flux of pebbles (which is proportional to exp(-time/τ_disk)) decays more slowly for shorter disks lifetimes. Therefore, the solid accretion rate is smaller and the protoplanet can spend more time in the mini-Neptune region before becoming a gas giant. The opposite occurs when we set τ_disk = 6 Myr: the occurrence rate of mini-Neptunes falls to 23%.

For the same baseline case for planetesimals, i.e., enriched at a = 20 au, the occurrence rates increase in the same direction as with pebbles: for τ_disk = 1.5 Myr, f_MN = 27% instead of 15%; and f_MN = 10% τ_disk = 6 Myr.

Note that for the cases where τ_disk = 1.5, if we narrow the range of possible disks lifetimes to 1–3 Myr, then the occurrence rate of mini-Neptunes is 100% for pebble accretion and 37% for planetesimal accretion (for baseline model at a = 20 au, enriched cases). According to Ribas et al. (2015), systems with stellar masses larger than ∼2 M_⊙ show no presence of disks for ages larger than 3 Myr. In this context, the formation of mini-Neptunes would be favored for such systems.

As we illustrate with the examples of this section, it is clear that the likelihood of obtaining a mini-Neptune depends on the disk's lifetime. Therefore, constraining disk lifetimes from observations will allow us to provide more accurate predictions of mini-Neptune occurrence rates from formation models. Vice versa, when combined with statistics of mini-Neptune occurrence rates, the models presented here could be used to constrain the origin of the observed systems. Note, however, that due to the possibility of migration, tracking the origin of the planet is not trivial.

Although in situ formation of mini-Neptunes is rather unlikely (see Section 3.5), most of the detected exoplanets orbit within about 1 au from the central star. As a result, it is important to test whether mini-Neptunes are a common outcome when migration is also included. In order to do so, one needs to couple the formation models with consistent migration rates and the feedback on the disk, which is beyond the scope of this work. Nevertheless, by invoking simple timescale arguments we can estimate the effect of migration. Type-I migration timescales are typically of τ_mig ∼ 10⁵ years (e.g., Tanaka et al. 2002). This is so short that planets that survive must have started to migrate late in the lifetime of the disk, when most of the solids had already either accreted to form larger objects or were lost into the star. Even if small planetesimals/pebbles exist, the planet is unlikely to accrete many solids during this short timescale. While this argument suggests that the planet is not expected to accrete many solids, it can still accrete gas. Nevertheless, our simulations show that for the favorable cases of mini-Neptune formation, when solid accretion stops, the timescale to accrete H–He is of the order of 10⁶ years. This timescale is in fact a lower bound because there is no heating from the bombardment of solids that prevents the planet from a fast cooling and contraction. We therefore conclude that the mini-Neptunes formed beyond the iceline are unlikely to accrete substantial H–He while migrating inward, remaining as mini-Neptunes.