PLANET TRAPS AND PLANETARY CORES: ORIGINS OF THE PLANET–METALLICITY CORRELATION

The term "critical mass of planetary core" was coined in the early 1980s (e.g., Mizuno 1980) in the context of the core accretion picture in which the formation of gas giants consists of two parts: rocky planetary cores form via collisional growth by planetesimals, followed by gas accretion onto the cores (e.g., Pollack et al. 1996). At that time, purely hydrostatic calculations of gaseous envelopes around cores were performed which showed that hydrostatic envelopes cannot be maintained beyond the value of M_{c, crit}, and hence it was expected that efficient gas accretion onto the cores proceeds afterward. The subsequent work in which quasi-static evolution of the gaseous envelopes were examined confirmed that rapid gas accretion takes place when M_p > M_{c, crit} (Stevenson 1982; Bodenheimer & Pollack 1986; Ikoma et al. 2000).

The early calculations of this kind, by Pollack et al. (1996) constitute a milestone in this field. Two new insights into the formation of gas giants were developed: (1) rapid gas accretion occurs when the envelope mass becomes comparable to the core mass (the mass was newly referred to as the crossover mass), and (2) the rapid gas accretion follows slow, gas accretion called "phase 2." Phase 2 arises from the efficient accretion of planetesimals by cores. This is because the energy deposited by the infall of planetesimals in the envelopes makes it possible to maintain the hydrostatic envelopes for a long time ∼Myr (equivalently growing planetary cores satisfy the condition that M_c ≃ M_{c, crit} due to a higher value of $\dot{M}_{c} ({\sim} 10^{-6} \,M_{\oplus } \,{\rm yr}^{-1})$ (see Equation (1)). This postpones the onset of rapid gas accretion. These two features were confirmed in several more recent calculations and simulations (Hubickyj et al. 2005; Lissauer et al. 2009; Movshovitz et al. 2010). Since the crossover mass physically corresponds to the critical core mass (Ikoma et al. 2000; Shiraishi & Ida 2008), the results of Pollack et al therefore imply that the gas accretion can be determined largely by the behavior of the phase 2.

There is still some debate, however, about the existence of phase 2 (e.g., Ikoma et al. 2000; Fortier et al. 2007). For instance, Shiraishi & Ida (2008) pointed out that phase 2 is unlikely to occur when the dynamics of planetesimals around a planet is considered more realistically. Specifically, they show through the numerical integration of planetesimals' orbit around a planet that a coupling effect of gravitational scattering by the planet and gas drag damping ends up with the formation of a gap in planetesimal disks. In other words, the efficient planetesimal accretion assumed in Pollack et al. to be the primary driver of the phase 2, is unlikely to be established.

When the planetesimal infall through the envelopes becomes negligible as the energy input, the gravitational contraction of the envelopes acts as the dominant heat source for supporting the envelopes. If this is the case, the growth of planetary mass via gas accretion can be regulated well by the Kelvin–Helmholtz timescales (e.g., Ikoma et al. 2000; Ida & Lin 2004a).

In summary, while significant progress has been achieved that has led to better understanding of the onset of gas accretion and the evolution of envelopes surrounding planetary cores, some physical processes remain uncertain.

Based on the above discussion, we adopt a conservative treatment wherein the critical mass of planetary cores is used for initiating gas accretion onto the cores while the gas accretion is prescribed by the Kelvin–Helmholtz timescale (Ikoma et al. 2000; Ida & Lin 2004a, HP12). One advantage of this formulation is that it allows us to compare our results with those of the standard population synthesis calculations (e.g., Ida & Lin 2004a, 2008a; Ida et al. 2013), which incorporate such an approach.

The critical core mass (M_{c, crit}) depends on the rate of planetesimal accretion onto the core ($\dot{ M}_c$) and the grain opacity (κ) in envelopes around the cores (e.g., Ikoma et al. 2000; Ida & Lin 2004a; Hori & Ikoma 2011). This was written in HP12 as

$$\begin{equation} M_{c,{\rm crit}} \simeq 10 \,M_{\oplus } f_{c,{\rm crit}} \left(\frac{\dot{M}_c}{10^{-6}\,M_{\oplus } \;{\rm yr}^{-1}} \right)^{1/4}, \end{equation} \tag{ 1 }$$

where f_{c, crit} is treated as a free parameter. This parameter is directly related to the grain opacity in the envelopes (Ikoma et al. 2000; Ida & Lin 2004a); f_{c, crit} = (κ/1 cm² g⁻¹)^{0.2 − 0.3} whose variation is usually neglected. Many previous studies adopt f_{c, crit} = 1 as the canonical value in which both the abundance and the size distribution of dust grains in the envelopes are assumed to be similar to the interstellar medium (ISM; Pollack et al. 1996; Ikoma et al. 2000; Ida & Lin 2004a). In our study, the value of f_{c, crit} plays an important role as we investigate systems with lower metallicity, and we demonstrate later that a much lower value fits the extrasolar planet data better.

For notational convenience, we define and shall henceforth adopt a mass parameter that arises from the dependence of M_{c, crit} on the grain opacity, and that links to definitions used in our previous work, namely;

$$\begin{equation} M_{c,{\rm crit0}} \equiv 10 \,M_{\oplus } f_{c,{\rm crit}} = 10 \,M_{\oplus } \left(\frac{\kappa }{1 \;{\rm cm}^2 \;{\rm g}^{-1}} \right)^{0.2-0.3}. \end{equation} \tag{ 2 }$$

One might think that the isolation mass of cores (M_{c, iso}) which is the calculated core mass, assuming that all the planetesimals in the feeding zone are accreted onto the cores (see Equation (A6)), would be better for examining the planet–metallicity correlation. However, the value of M_{c, iso} is really useful only when the formation of cores occurs in-situ. In the case of migrating cores—in particular those caught in our moving planet traps—fresh planetesimals can always be introduced as the trap sweeps through the disk. It is not so obvious that an isolation mass is the correct choice in this situation. Hence M_{c, crit} is preferred in our models.

Gas accretion onto the cores starts once they achieve the value of M_{c, crit} (e.g., Mizuno 1980; Stevenson 1982; Ikoma et al. 2000). The gaseous envelopes of potential Jovian planets contract on the Kelvin–Helmholtz timescale, provided that the accretion of planetesimals by the cores is small (<10⁻⁶ M_⊕ yr⁻¹) at that time. A simplified version of this timescale has been derived from more detailed numerical simulations (Ikoma et al. 2000):

$$\begin{equation} \tau _{{\rm KH}} \simeq 10^{c} \;{\rm yr}\; \left(\frac{M_p}{M_{\oplus }} \right)^{-d}. \end{equation} \tag{ 3 }$$

It is likely that there are ranges in c (8 ≲ c ≲ 10) and in d (2 ≲ d ≲ 4). We adopt that c = 9 and d = 3 as fiducial values, following Ida & Lin (2004a, references herein). Note that τ_KH also depends on the grain opacity, κ, like M_{c, crit0} (see Equation (2)). Nonetheless, the dependency is neglected in this paper, because it can be incorporated into the variation of c, and d, both of which are the main parameters here. Also, note that τ_KH is sensitive to the computed mass (M_p) of planets. For instance, τ_KH = 8 × 10⁶ yr when M_p = 5 M_⊕, which roughly corresponds to the onset of gas accretion for our fiducial case (see below). When M_p = 10 M_⊕ the timescale goes down to 10⁶ yr. Our analysis will show that the lower critical core mass parameter fits the data better, which means that this model mimics the behavior of a drawn out phase 2 found by Pollack et al. (1996) but whose physical origin is different: for our case, a prolonged phase-2-like behavior originates simply from a longer Kelvin–Helmholtz timescale, and not with planetesimal accretion.

The gas accretion rate onto the protoplanet from the surrounding disk is regulated by this Kelvin–Helmholtz timescale;

$$\begin{equation} \frac{{d M}_p}{dt} \simeq \frac{M_p}{\tau _{{\rm KH}}} \propto M_p^{d+1}. \end{equation} \tag{ 4 }$$

Thus, the growth rate of planets increases sharply with M_p.

There is one final mass that needs to be considered in this accretion picture, which is the mass scale at which the gas accretion finally runs away. The minimum possible mass for a core is M_{c, min} that can be derived from the condition τ_KH = τ_fin (see Equation (3)). Setting τ_fin = 10⁷ yr, which is the upper limit of the disk lifetime, we obtain

$$\begin{equation} M_{c,{\rm min}}=10^{(c-7)/d} \,M_{\oplus }. \end{equation} \tag{ 5 }$$

For our fiducial case (c = 9 and d = 3), we find M_{c, min} ≃ 4.6 M_⊕.

To summarize, gas accretion starts when M_c ⩾ M_{c, crit} (see Equation (1)), and the subsequent growth of planetary mass via the gas accretion is determined by the Kelvin–Helmholtz timescale (see Equations (3) and (4)). These prescriptions involve the three key physical quantities that are basic control parameters in our models; M_{c, crit0}, c, and d (see Table 2). The fundamental question addressed by this paper is how the value of the grain opacity defining M_{c, crit0} (see Equation (2)), affects the PFFs and so determines the planet–metallicity correlation (see Section 4). In pursuing this question, we investigate the effects of c and d (see Equation (3)) in Section 5 (see Table 4).

We obtained the data given in Figure 1 from the Extrasolar Planets Encyclopaedia (Schneider et al. 2011, http://exoplanet.eu/) and plot the mass of observed exoplanets as a function of metallicity for the three different populations in Figure 1. Table 1 summarizes the definition of these three populations. The identification of different exoplanetary populations was discussed in earlier work (Chiang & Laughlin 2013, HP13). Since our previous study already confirmed that the population of close-in (r ⩽ 0.5 AU) massive planets (the populations ending up either in Zone 1 or in Zone 2) are minor (HP13), we combine these two populations in this paper and refer to them as "hot Jupiters" (see Table 1). The classification of Jovian planets does not affect our conclusions.

Figure 1 clearly shows that the number of more massive planets gradually diminishes as one goes to lower metallicity, although there is a large scatter. For the low-mass planets, on the other hand, this trend is absent. These results were initially inferred from the radial velocity observations (e.g., Santos et al. 2004; Fischer & Valenti 2005; Bond et al. 2006; Udry & Santos 2007). Note that some observational biases may be present in Figure 1, since the observational data are obtained from the collection of various surveys. Nonetheless, the trend of the planet–metallicity correlation is evidently shown. Also note that planets with measured masses (the radial velocity method) as well as those inferred from the Kepler observations show similar patterns with metallicity.

We constructed evolutionary tracks of accreting planets in the mass–semimajor axis diagram by computing the time evolution of viscous disks with photoevaporation of gas in HP12. For the growth of planetary cores, we adopted the standard core accretion scenario (HP12, Ida & Lin 2004a, also see the Appendix) in which rocky planetary cores form via oligarchic growth (Kokubo & Ida 1998). Our prescription for the subsequent growth of planets by gas accretion was discussed in Section 2. In Figure 2, we give examples of 3 computed evolutionary tracks that feed the different populations in the mass–semimajor axis diagram.

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Planets are built in planet traps which naturally arise from some kinds of disk inhomogeneities. As the term implies, they act as barriers (technically as radii at which there is zero net torque exerted by the disk) for rapid type I planetary migration (Masset et al. 2006; Matsumura et al. 2007; Ida & Lin 2008b; Hasegawa & Pudritz 2010; Lyra et al. 2010, HP11). We considered three types of planet traps: dead zones, ice lines, and heat transition traps (HP11). The orbital evolution of planets forming in them is regulated by the movement of planet traps and subsequent type II migration. When planetary cores become more massive than the gap-opening mass, they drop out of their traps and switch to type II migration. Although it can be assumed that planet formation proceeds predominantly at three types of traps treated in isolation, the gravitational interactions between trapped planets will gradually become important for the later stage of disk evolution, an effect that is neglected in this work but will be treated in a future paper.

In building a theoretical evolutionary track (see HP12), we first compute the growth of the rocky core from the theoretical expression for $\dot{M_{c}}$ on the core accretion timescale τ_{c, acc} having to do with the collisions of planetesimals. One integrates the mass over time taking into account the fact that the accreting core is moving through the disk as it is carried by its trap. The actual formula for τ_{c, acc} and $\dot{M}_c$ are given by Equations (A1) and (A3) in the Appendix and follows Kokubo & Ida (2002, also see HP12). In this regime, the accretion rate is directly proportional to the surface density of the dust; $\dot{M_c} \propto \Sigma _d$, which as we will see in the next subsection, is related to the metallicity of the disk. This column density changes as the host planet trap moves radially inward through the evolving disk so that $\dot{M_c}$ changes with time. At the time in the numerical integration when the mass of the core exceeds the critical value given by condition (see Equation (1)), one has identified the critical core mass (M_{c, crit}).

In Figure 2, we mark the value of M_{c, crit} found for each of the tracks. Beyond this mass, the evolution of the planet is switched to gas accretion which is governed by the Kelvin–Helmoltz timescale given above (see Equations (3) and (4)). As discussed in Section 2, in order to achieve rapid gas accretion, one needs to have M_{c, crit} ⩾ M_{c, min} (see Equation (5)).

The protoplanet drops out of the trap motion and switches to subsequent type II migration when it becomes sufficiently massive to open a gap in the disk. We can see from Figure 2 that tracks that end up in the hot Jupiter zone must undergo slower gas accretion overall if they are to end up at such small orbital radii. This is a manifestation of type II migration where gap opening will significantly reduce the gas inflow onto the core. Tracks that end up in the exo-Jupiter zone are therefore less dominated by type II gap-opening effects, and hence accrete more rapidly to achieve higher masses. The final phase in the evolution arrives with the dissipation of the disk by photoevaporation.

We emphasize that all of this planetary evolution and movement occurs in disks whose properties are evolving at the same time, due to the viscous dissipation of disks as well as their relatively rapid dispersal by photoevaporation characterizing the final stage of disk evolution. In order to make this quantitative, we have focused on two key disk parameters: the disk accretion rate and lifetime parameters.⁶ In our improved formalism, these two parameters involve the time-evolution of the disk accretion rate that is given as (HP13, also see Tables 2 and 3)

$$\begin{eqnarray} \dot{M}(\tau) &\simeq & 3 \times 10^{-8}\, M_{\odot } \;{\rm yr}^{-1}\; \eta _{{\rm acc}} \left(\frac{M_*}{0.5\,M_{\odot }}\right)^2 \nonumber\\ && \times \left(1+ \frac{\tau }{ \tau _{{\rm vis}}} \right)^{-1.5} \exp \left(- \frac{\tau -\tau _{{\rm int}}}{\tau _{{\rm dep}}} \right), \end{eqnarray} \tag{ 6 }$$

where M_* = 1 M_☉ is the stellar mass, τ is a time, τ_vis = 10⁶ yr is the typical viscous timescale, τ_dep is the depletion timescale, and τ_int = 10⁵ yr is the initial time for our calculations to get started. The numerical factor in Equation (6) is set so as to well match the observed disk accretion rate for disks around classical T Tauri stars (M_* ≃ 0.5 M_☉) at τ = 10⁶ yr, if a factor η_acc is unity. Since the total disk mass is linearly proportional to $\dot{M}$ in our model, we can essentially change the disk mass by varying η_acc. We also assume that the depletion timescale can be written as

$$\begin{equation} \tau _{{\rm dep}} = \tau _{{\rm dep},0} \eta _{{\rm dep}}, \end{equation} \tag{ 7 }$$

where τ_{dep, 0} = 10⁶ yr.

Table 2. Key Quantities in This Work

Symbol	Meaning	Values
	Planetary growth by gas accretion
Mp	Planetary mass (the core mass + the envelope mass) computed along tracks
Mc, crit	The critical mass of planetary cores that can initiate efficient gas accretion (see Equation (1))
$\dot{M}_c$	Accretion rate of planetesimals by planetary cores (see Equation (A4))
Mc, crit0	A free parameter regulating Mc, crit (see Equations (1) and (2))	3 M⊕, 5 M⊕, 10 M⊕
κ	The grain opacity in accreting envelopes that links to Mc, crit0 (see Equation (2))	≃8 × 10−3, 0.06, 1 cm2 g−1
τKH	The Kelvin–Helmholtz timescale (see Equation (3))
(c, d)	A set of free parameters for τKH	(9,3)
Mc, min	The mass scale for gas accretion to finally run away within the disk lifetime (see Equation (5))	≃ 4.6 M⊕
	Disk model
$\dot{M}$	Disk accretion rate onto the central star (see Equation (6))
Σg	The surface density of gas ($\propto \dot{M}$)
Σd	The surface density of dust (=fdtgΣg)
[Fe/H]	Metallicity (see Equation (8))
fdtg	The dust-to-gas ratio (=fdtg, ☉ηdtg)
fdtg, ☉	The dust-to-gas ratio at the solar metallicity (=ηdtg, ☉ηice)	≃ 1.8 × 10−2
ηdtg, ☉	The dust-to-gas ratio at the solar metallicity within the ice line	≃ 6 × 10−3
ηice	A factor for increasing/decreasing fdtg, ☉ due to the presence of ice lines
τvis	The viscous timescale (see Equation (6))	106 yr
τint	The initial time for starting computations (see Equation (6))	105 yr

Notes. Note that free parameters here are only M_{c, crit0}, c, and d. For the other parameters, the given values are derived from the three free parameters, or our previous study confirmed that the results are insensitive to a specific choice of the values (HP13).

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Table 3. Input Parameters for the Statistical Analysis^a

Symbol	Meaning	Values
	Stellar parameters
M*	Stellar mass	1 M☉
R*	Stellar radius	1 R☉
T*	Stellar effective temperature	5780 K
	Disk mass parameters
ηacc	A dimensionless factor for $\dot{M}$ (see Equation (6))	0.1 ⩽ ηacc ⩽ 10
wmass(ηacc)	Weight function for ηacc modeled by the Gaussian function
ηdtg	A parameter for increasing/decreasing [Fe/H] (see Equation (8))	0.25 ≲ ηdtg ≲ 4
	Disk lifetime parameters
ηdep	A dimensionless factor for τdep (see Equation (7))	0.1 ⩽ ηdep ⩽ 10
wlifetime(ηdep)	Weight function for ηdep modeled by the Gaussian function

Note. ^aThese quantities are constrained by the observations, so that they are not free parameters.

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In our formalism, the disk evolution is therefore regulated by two quantities, τ_vis, and τ_dep. The disk lifetime is controlled largely by τ_dep. Following HP13, we adopt the exponential function for characterizing the end disk evolution stage. As discussed in HP13 (see its Appendix), the exact shape of the disk accretion, especially at the end stage of disk evolution, does not affect the resultant planetary populations (see Equation (9)). The most important disk parameter is the disk lifetime. In summary, we treat η_acc as a disk accretion rate parameter whereas η_dep is considered as a disk lifetime parameter.

Our model (HP12) succeeded in providing physical explanations for a number of observational features such as the observed mass–period relation (Udry & Santos 2007; Mayor et al. 2011).

The metallicity of the disk is readily parameterized in terms of the dust-to-gas ratio f_dtg. This can be written as f_dtg = f_{dtg, ☉}η_dtg (see Tables 2 and 3), where f_{dtg, ☉} = η_{dtg, ☉}η_ice is the dust-to-gas ratio at the solar metallicity, η_dtg is a factor for varying f_dtg, η_{dtg, ☉} is the dust-to-gas ratio at the solar metallicity within the ice line, and η_ice is a factor for changing f_{dtg, ☉} due to the presence of a water ice line. We set η_ice = 1 when planet traps are inside the ice line, η_ice = 4 when the traps are on the ice line, and η_ice = 3 when the traps are beyond the ice line (Pollack et al. 1994; Ida & Lin 2004a, HP13). Adopting the most likely value of f_{dtg, ☉}, which is ∼1.8 × 10⁻² (Asplund et al. 2009), we obtain that η_{dtg, ☉} = 6 × 10⁻³. Thus, the variation of the dust-to-gas ratio is regulated by η_dtg.

In our model, η_dtg is related to the dust density in disks: $\Sigma _d = f_{{\rm dtg}} \Sigma _g (\propto \eta _{{\rm dtg}} \dot{M})$, where $\Sigma _g (\propto \dot{M})$ is the surface density of gas in the disks. As shown by the previous studies (e.g., Ida & Lin 2004b; Mordasini et al. 2012), the efficiency of forming planetary cores in the core accretion scenario is sensitive to the value of Σ_d.

Finally, the metallicity can be expressed in terms of η_dtg. Assuming that the value of the metallicity in disks is similar to their stellar metallicities, the metallicity [Fe/H]⁷ can be linked with η_dtg as follows (Ida & Lin 2004b):

$$\begin{equation} {[{\rm Fe}/{\rm H}]} \simeq { [{\rm m}/{\rm H}]} \equiv \log _{10} \frac{f_{{\rm dtg}}}{f_{{\rm dtg},\odot }} = \log _{10}(\eta _{{\rm dtg}}), \end{equation} \tag{ 8 }$$

where m represents a mixture of metals. This indicates that the effects of metallicities can be readily explored by varying η_dtg. Note that the ratio of [m/H] to [Fe/H] is generally considered as order of unity.

We adopt the same initial conditions as HP13 (also see HP12), wherein protoplanets of ∼0.01 M_⊕ start to grow at specific times τ and positions r. We take 100 times (and 100 positions) for each planet traps as the initial conditions of evolutionary tracks, so that the entire disk lifetime is covered by the tracks. This is equivalently that N_int = 300 for all the calculations (see Equation (9)). As confirmed in HP13, the number is large enough to get our results convergent.

For the value of the metallicity ([Fe/H]), we consider a wide range (−0.6 ⩽ [Fe/H] ⩽0.6). The variation in [Fe/H] leads to the change in η_dtg through Equation (8). Utilizing the above model with these initial conditions, we evaluate the PFFs for three zones as a function of metallicity.

In HP13, we developed a new statistical approach for computing the contributions of the various planet traps to the three major planetary populations that does not use a standard population synthesis approach. We partitioned the mass–semimajor axis diagram into five zones and calibrated PFFs in the zones. Here, we consider three zones which characterize the dominant populations (see Table 1 and Figure 2, also see HP13). The input parameters for the statistical analysis are summarized in Table 3.

Utilizing the calculations of evolutionary tracks of planets building at planet traps (HP12), we define the PFFs for tracks feeding into the ith zone as (HP13)

$$\begin{eqnarray} {{\rm PFFs}({\rm Zone\; i})} & \equiv & \sum _{\eta _{{\rm acc}}} \sum _{\eta _{{\rm dep}}} w_{{\rm mass}}(\eta _{{\rm acc}}) w_{{\rm lifetime}}(\eta _{{\rm dep}}) \nonumber \\ && \times \frac{N{({\rm Zone\; i}, } \eta _{{\rm acc}}, \eta _{{\rm dep}}),}{N_{{\rm int}}} \end{eqnarray} \tag{ 9 }$$

where w_mass and w_lifetime are both weight functions of η_acc and η_dep, respectively, N(Zone i, η_acc, η_dep) is the number of evolutionary tracks that end up in Zone i for a given set of η_acc and η_dep, and N_int = 300 is the total number of tracks that are considered in the calculations.

Briefly, each term in the weighted sum on the right hand side of Equation (9) involves the ratio, N(Zone i, η_acc, η_dep)/N_int, which is a planet formation efficiency for a specific set of η_acc and η_dep. This ratio is then summed (integrated in the continuum limit) over a wide range of both η_acc and η_dep with their weight functions to produce the PFF. Therefore the PFF represents the (integrated) planet formation efficiencies for a wide range of disk mass and lifetime. In this paper, we consider the ranges of η_acc (0.1 ⩽ η_acc ⩽ 10) and η_dep (0.1 ⩽ η_dep ⩽ 10), following HP13.

We adopt Gaussian distributions for both w_mass and w_lifetime. For w_mass, the peak value is obtained when $\dot{M} \simeq 1.7 \times 10^{-8} \,M_{\odot } \,{\rm yr}^{-1}$ (equivalently η_acc = 1). The standard deviation of w_mass is set as unity in the unit of M_☉ yr⁻¹, although we confirmed that its variation does not affect our results. The choice of these particular values is motivated by the observed disk accretion rate around classical T Tauri stars. For w_lifetime, the highest value is achieved when the disk lifetime is about 1.5 × 10⁶ yr (equivalently η_dep = 1.5) with the standard deviation being 3 in the unit of Myr. These two values are again selected to be consistent with the observations of disks which show that a disk fraction of stars can be well fitted by the exponential function with the e-folding time being 2.5 Myr (e.g., Williams & Cieza 2011). Thus, all the quantities in the weight functions are determined, so that the weight functions can fit to the observations of protoplanetary disks (see HP13). Gaussian functions are also useful, both because of their computational convenience as well as their reflection of possible Gaussian random noise in the initial conditions for disk formation in turbulent molecular clouds.

This approach enables one to compare the theoretical results with the observations without the standard population synthesis calculations being performed. As discussed in HP13, we have shown that the combination of planet traps with the core accretion scenario can account for the observations of exoplanets very well. Specifically, we have demonstrated that most formed gas giants tend to end up around 1 AU with fewer populations of the hot Jupiters. Also, our calculations have shown that a large fraction of low-mass planets in tight orbits can be populated by "failed" cores of gas giants and/or mini-gas giants.

We adapt our PFF approach to calculating the average critical mass of cores as a function of metallicity for a distribution of evolutionary tracks. In order to proceed, we calculate the mean mass of planetary cores (〈M_{c, crit}〉) that indeed initiate gas accretion when evolutionary tacks are computed, which is defined as

$$\begin{eqnarray} {\langle {M_{c,{\rm crit}}({{\rm Zone\; i}})}\rangle } & \equiv & \sum _{\eta _{{\rm acc}}} \sum _{\eta _{{\rm dep}}} w_{{\rm mass}}(\eta _{{\rm acc}}) w_{{\rm lifetime}}(\eta _{{\rm dep}}) \nonumber \\ && \times {\langle {M_{c,{\rm crit}}({{\rm Zone\; i}, } \eta _{{\rm acc}}, \eta _{{\rm dep}})}\rangle }, \end{eqnarray} \tag{ 10 }$$

where 〈M_{c, crit}(Zone i, η_acc, η_dep)〉 is the averaged critical mass of planetary cores that eventually fill out Zone i and that just start accreting gas during planetary growth for a certain set of η_acc and η_dep. We have shown in Figure 2 how these values are determined for each evolutionary track computed in our simulations (see the circle in each track).

Figure 3 (top) shows the computed PFFs as a function of metallicity for the hot and exo-Jupiters (see the dashed and solid lines, respectively). The results for the choices M_{c, crit0} = 5 M_⊕ and M_{c, crit0} = 10 M_⊕ are shown on the left and right panels, respectively. For both values of M_{c, crit0}, we find that the PFFs of the exo-Jupiters are much higher than those of the hot Jupiters. Thus, the results found in HP13 that exo-Jupiters should be much more frequent than hot Jupiters carries over to planets of all observed metallicities. We predict therefore that a higher population of Jovian planets around 1 AU is general, and is a reflection of the dynamics of planet traps in the metallicity range so far observed.

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Our second major result apparent in this figure is that the formation of both Jovian planets is sensitive to the metallicity: gas giants are preferentially formed for disks with higher metallicities whereas their formation efficiencies are suppressed for low metallicity disks. This agrees very well with the sense of the observed planet–metallicity relation—massive planets are less frequently observed around stars of lower metallicity. This is a natural consequence of planet formation in the core accretion picture in our calculations. These results also agree with the findings of previous population synthesis studies (Ida & Lin 2004b; Mordasini et al. 2012).

A very interesting and new result is that the PFFs drop off very noticeably toward lower metallicities and remain rather constant above some particular values. For the case of M_{c, crit0} = 5 M_⊕ (the left panel), the PFFs of both the hot and exo-Jupiters drop rapidly at low metallicity, [Fe/H] = −0.2 to −0.4. For the case of M_{c, crit0} = 10 M_⊕ (the right panel), on the contrary, the PFFs of both of them decrease more gradually as the metallicity decreases. We explore this further in what follows.

Figure 3 (middle) shows the values of the critical core masses 〈M_{c, crit}〉 for hot and exo-Jupiters as a function of metallicity (see the dashed and solid lines, respectively). For comparison, 〈M_{c, crit}〉 for low-mass planets is also shown in the same panel (see the dotted line). The results show that 〈M_{c, crit}〉 for both kinds of Jovian planets is an increasing function of metallicity for both cases of M_{c, crit0} (see the left and right panels). Since the timescale of gas accretion onto planetary cores shortens for massive cores, they have more time to grow up to gas giants. This is the main reason why massive planets are preferentially formed for high metallicity stars.

The results in this panel also show that the value of 〈M_{c, crit}〉 of the exo-Jupiters is larger than that of the hot Jupiters. (see the middle in Figure 3). In our planet trap picture, the mass of the cores tends to increase with increasing the distance from the central star. This is achieved because planet traps are assumed to be effective until growing planetary cores open up a gap in their disks. The gap-opening mass is an increasing function of distance from the central star. It is therefore expected that 〈M_{c, crit}〉 of the hot Jupiters is smaller than that of the exo-Jupiters. In other words, the cores of the exo-Jupiters drop-out from their planet traps earlier because of their more massive cores, and hence they have more time to grow up to fully formed gas giants, compared with those of the hot Jupiters. Our results in fact support this trend.⁸

Another very interesting trend in these critical core mass curves is that the low-mass curve exceeds both Jovian planet curves at the lowest metallicities. As the metallicity increases, the rising exo-Jupiter curve intersects the low-mass curve and then flattens out beyond. At somewhat higher metallicity, the rising hot Jupiter curve then also intersects the low-mass planet curve. These intersection points are different for the two cases (see the left, middle panel in Figure 3). We define the corresponding values of the metallicity at these intersections as [Fe/H]_exo and [Fe/H]_hot, respectively—which we henceforth call transition metallicities, or TMs. In Figure 3, these TMs are denoted by vertical solid and dashed lines respectively. Note that these intersections in the 〈M_{c, crit}〉 domain are reflected by features in the upper panels where the rising PFFs cease their rapid increase at the corresponding TMs and begin to flatten out toward higher metallicity.

The values of the TMs for the different cases are easily read off the figure. For the case of M_{c, crit0} = 5 M_⊕ (see the left, middle plane), 〈M_{c, crit}〉 of the hot Jupiters intersects with that of low-mass planets at [Fe/H]_hot ≃ −0.1 (the vertical dashed line) and 〈M_{c, crit}〉 of the exo-Jupiters does at [Fe/H]_exo ≃ −0.2 (the vertical solid line). Note that in the observed range of metallicities we are modeling, there are two intersections for the case of M_{c, crit0} = 5 M_⊕, (see the left, top panel). For the case of M_{c, crit0} = 10 M_⊕ (right, middle panel), there is only one intersection at [Fe/H]_hot ≃ −0.5 for the hot Jupiters (the vertical dashed line).

Our results indicate that there are two important masses of planets for estimating the number of TMs that occur somewhere in the range [Fe/H] ≃ −0.2 to −0.4 in 〈M_{c, crit}〉–metallicity diagram. One of them is obviously the M_{c, crit0} parameter that regulates the onset of gas accretion onto planetary cores (see Equation (1)). The other is the minimum mass of planetary cores (M_{c, min}) that allows the cores to accrete enough amount of gas within a certain time τ_fin, so that they eventually become gas giants (see Equation (5)).

For the case M_{c, crit0} = 10 M_⊕, one finds that M_{c, min} < M_{c, crit0} (see Table 4). In this regime, gas giant formation proceeds efficiently even in the low metallicity environment. More physically, the timescale of the oligarch growth becomes longer for disks with lower metallicities. Nonetheless, the final mass of planetary cores can reach ≳ M_{c, min}. Therefore, gas accretion that takes place after the core formation could be done within the disk lifetime. This is because the timescale of gas accretion depends only on the mass of cores (see Equations (3) and (4)). In other words, the total timescale in which the formation of gas giants is complete can still be short, compared with the disk lifetime. Thus, 〈M_{c, crit}〉 of both the hot and exo-Jupiters becomes larger than that of low-mass planets even for disks with low metallicities, and hence there is only a TM at low values of [Fe/H] (see the right, middle panel).

For the case of M_{c, crit0} = 5 M_⊕, we obtain M_{c, min} ≃ M_{c, crit0}. Although we can apply the above argument for this case as well, it is expected that the formation of gas giants is not so efficient for disks with low metallicities. This occurs because M_{c, crit0} = 5 M_⊕(≃ M_{c, min}), so that the final mass of cores formed in low metallicity disks would be smaller than M_{c, min}. As a result, such cores cannot grow to gas giants. Thus, TMs for this case appear in higher metallicity environments than the case of M_{c, crit0} = 10 M_⊕, and two TMs exist (see the left, middle panel).

We now turn our attention to the metallicity dependence of the PFFs for the low-mass planets—as well as the total PFF which is the sum of all three populations. These plots are given in the bottom panels in Figure 3. It is evident that the values of the PFFs for the low-mass planets are rather high and quite insensitive to metallicity. This indicates that a large number of low-mass planets can be formed by the same mechanism that builds the cores of the gas giants, over the entire observed range of metallicity. This generalizes our conclusion in HP13 that such low-mass planets can be regarded as failed Jovian cores and/or mini-gas giants.

On closer examination, another feature of the PFFs of low-mass planets is that they descend toward a minimum and then rise again with increasing metallicity. This dip has a counterpart in the behavior of the PFFs of both the hot and exo-Jupiters (see the top panels), which undergo a strong rise in the PFF in this range of metallicities. Note that the total PFF for the sum of all populations shows a steady rise without any feature. Clearly the planetary cores that might have ended up in low-mass planets seed the growth of gas giants efficiently around TMs. For higher metallicity disks, both massive and low-mass planets can form simultaneously, so that the PFFs of low-mass planets get back to larger values.

Although similar behavior is observed for different values of M_{c, crit0} (see the left and right bottom panels), there are some quantitative differences. The dip in the PFFs is more prominent for the case of M_{c, crit0} = 10 M_⊕ than the case of M_{c, crit0} = 5 M_⊕. This can be understood by the same argument developed in Section 4.3, namely, that when M_{c, crit0} = 10 M_⊕, the final mass of planetary cores becomes massive enough to speed up the subsequent gas accretion process. As a result, gas giants can form efficiently even in relatively low metallicity disks. We find that the metallicity dependence of the total PFFs is insensitive to the value of M_{c, crit0} (see the thick line in the bottom two panels). Therefore, such efficient formation of gas giants for the case of M_{c, crit0} = 10 M_⊕ lowers the PFFs of low-mass planets more than the case of M_{c, crit0} = 5 M_⊕.

Note that the overall value of the PFFs for the low-mass planets is comparable to or slightly lower than those for the Jovian planets. The recent observations, on the contrary, infer that the low-mass planets are the most dominant populations. Given that both Kepler and radial velocity surveys are weighted toward Sun-like stars, which makes our model applicable to these observations, one explanation might be that there are a number of mechanisms for forming the low-mass planets (see Section 6.3). However, we note that this may still be a consequence of the effect of stellar masses on the statistics. We need to perform similar calculations along the initial mass function (IMF) and to examine how different the resultant PFFs are for different stellar types. This is because low-mass stars, that tend to generate more low-mass planets due to low-mass disks, occupy the dominant contribution in the standard IMF. This kind of approach may be useful for microlensing surveys which tend to focus on low-mass stars.

The important conclusion from all of this is that the dominant planetary populations switch from low-mass planets to massive ones at the TMs (see Figure 3).

We now examine how our PFFs and critical core masses are altered by changing the values of M_{c, crit0}, c and d, which control the efficiency of planet formation (see Equations (1) and (3)). The different model parameters are given in Table 4. We choose the values of M_{c, crit0}, c and d, so that M_{c, crit0} satisfies the condition M_{c, min} ≃ M_{c, crit0} (see Table 4). As discussed in Section 4.3 and will be shown below, TMs appear in the range [Fe/H] ≃ −0.2 to −0.4 when this condition is met. Note that this was confirmed by performing more than 30 runs.

Figure 4 shows some of our results. The upper, left panel shows the results for the case M_{c, crit0} = 3 M_⊕, c = 8, and d = 2.5 (Run A), the upper, right panel is for M_{c, crit0} = 10 M_⊕, c = 9, and d = 2 (Run B), the lower, left panel is for M_{c, crit0} = 10 M_⊕, c = 10, and d = 3 (Run C), and the lower, right panel is for M_{c, crit0} = 10 M_⊕, c = 10, and d = 3.5 (Run D).

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The resultant PFFs show the same trends for a wide range of M_{c, crit0}, c and d (see the top figure in each panel), as those observed for the case of M_{c, crit0} = 5 M_⊕, c = 9, and d = 3 shown in Figure 3. The shape of the PFFs for the two Jovian class planets has the same general structure. The exo-Jupiters always dominate the hot Jupiter population and after a rather steep rise from lower metallicity, both PFFs roll over into very gently increasing values beyond the exo-TM. The PFFs for the low-mass planets all have dips and drop significantly around [Fe/H] ≃ −0.2 to −0.4, and rise again to near their original values at higher metallicity. This is the hallmark of the planet–metallicity relation. The other general result is that at low metallicity, the low-mass planets dominate the total population for all models. It is impressive how robust these relations are over such a wide range of the basic parameters of the model.

The obvious difference between the models in our parameter study is the behavior of the TMs. For Run A, two TMs appear around [Fe/H]_hot ≃ 0.1 and [Fe/H]_exo ≃ −0.2 for the hot and exo-Jupiters, respectively. For Run B, there is only one TM at [Fe/H]_{hot, exo} ≃ −0.2. For Run C, only one TM exists around [Fe/H]_{hot, exo} ≃ −0.4. For Run D, there are two TMs located around [Fe/H]_hot ≃ −0.4 and [Fe/H]_exo ≃ −0.6 for the hot and exo-Jupiters, respectively. The TMs focus on properties of the cores which give us important additional information than the behavior of the PFFs. Why do the positions (in metallicity) and numbers of TMs change?

First consider the positions. These are affected by the sensitivity of 〈M_{c, crit}〉 for both Jovian planets, to the value of M_{c, crit0}. As shown in Figures 3 and 4, the magnitude of 〈M_{c, crit}〉 for both Jovian planets is almost linearly proportional to the value of M_{c, crit0}. As an example, the value of 〈M_{c, crit}〉 for the exo-Jupiters at solar metallicity is approximately equal to the value of M_{c, crit0}. On the contrary, the value of 〈M_{c, crit}〉 for the low-mass planets is quite insensitive to the value of M_{c, crit0}. Our results show that 〈M_{c, crit}〉 ≃ 2–4 M_⊕ for various values of M_{c, crit0}. As a result, the locations of the TMs generally sweep toward lower metallicity with increasing M_{c, crit0}.

What about the difference in the number of the TMs? There is only one TM for Runs B and C. Note that for these two cases, the curves for 〈M_{c, crit}〉 as a function of metallicity of both the hot and exo-Jupiters almost completely line up with one another. In this case, it is obvious that both intersect the low-mass curve at one metallicity—hence there is one TM. Why do the curves line up in these cases? Based on a large number of similar simulations (which are not shown here), we found that this is a consequence of a large value of M_{c, crit0}(= 10 M_⊕). As discussed below, this match is attributed to the cloud coupling of planetary migration with planetary growth.

As discussed in Section 4.2, 〈M_{c, crit}〉 of the hot Jupiters is generally smaller than that of the exo-Jupiters. Most of our results obviously support the trend (see the middle panel in Figures 3 and 4). For the case of M_{c, crit0} = 10 M_⊕, however, the critical mass curves for the hot Jupiters are raised up to the values of the exo-Jupiter curves for Runs B and C, (but not the two other cases). In this more massive regime, the final mass of planetary cores can become larger than the gap-opening mass. This leads to switch of core migration from trapped type I to type II, even as the formation of the planetary cores still proceeds. As a result, the distribution of cores that is generated by planet traps can be washed out, making it easier for the core mass curves of these populations to line up with one another.

It is interesting to investigate further the coupling of migration with planet formation, especially focusing on type II migration as well as gas accretion onto cores. When the results of Run C are compared with those of Run D, there is the difference in the behavior of 〈M_{c, crit}〉 for both Jovian planets, although the value of M_{c, crit} itself is the same for these two cases. For the case of Run C, 〈M_{c, crit}〉 for both Jovian planets lines up entirely (only one TM exists) whereas for Run D, the value is different for different populations (two TMs exist).

The difference can be understood by the combined effects of the gas accretion and type II migration. If the gas accretion proceeds quickly (equivalently for a small value of M_{c, min}, e.g., see Run D), gas giant formation is completed earlier. Then the inertia of such fully formed planets becomes important for slowing down type II migration significantly. This eventually minimizes the effects of type II migration, and hence the final distribution of massive planets is affected largely by the distribution of planetary cores generated by planet traps. Consequently, 〈M_{c, crit}〉 of the hot Jupiters tends to be smaller than that of the exo-Jupiters (see the lower right panel in Figure 4). When planetary cores accrete gas slowly (equivalently for a large value of M_{c, min}, e.g., see Run C), they need more time to form gas giants. This, in turn, maximizes the importance of type II migration. Eventually, the effects of planet traps are washed out and 〈M_{c, crit}〉 of the hot Jupiters becomes comparable to that of the exo-Jupiters (see the lower left panel in Figure 4).

Thus, similar results in the PFFs can be obtained for a wide range of M_{c, crit0}, c, and d. Nonetheless, such degeneracies may be able to be resolved if the position and the number of TMs in the 〈M_{c, crit}〉–metallicity diagram are examined carefully. This occurs because planet formation is intimately coupled with planetary migration.

Having demonstrated that the planet–metallicity relation is well described by the trend in the PFFs, what value for M_{c, crit0} is most compatible with the data? It turns out that the distributions of hot and exo-Jupiters as a function of metallicity can provide an important and rather robust constraint, as we now show.

We first discuss the observations of exoplanets. In Figure 5, the observational data are replotted to show the distributions of hot and exo-Jupiters as a function of metallicity in the left and right panels respectively. We here focus in particular on the radial velocity data for the following reasons (also see Section 6.5). First, exoplanets at r > 1 AU are currently detected mainly by the radial velocity methods. Thus, picking up only the radial velocity data enables us to use the observational data that are consistent at least on the detection method for both the hot and exo-Jupiters. Second, the resultant populations are in good agreement with the current knowledge of the statistical properties of gas giants: the hot Jupiters are rare and the exo-Jupiters are dominant in the population of massive planets. Figure 5 implies that the hot Jupiters may be more sensitive to metallicity than the exo-Jupiters. While it seems that the observational data are still not large enough to examine this trend statistically, some clues about it have recently been reported (Beaugé & Nesvorný 2013; Adibekyan et al. 2013).

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Figure 5 also shows, by means of vertical lines, the TMs for both the hot and exo-Jupiters ([Fe/H]_hot and [Fe/H]_exo) above which the PFFs for them will increase significantly (see values in Table 4). In particular, we select the cases of M_{c, crit0} = 3 M_⊕, c = 8, and d = 2.5 (see the dashed line), of M_{c, crit0} = 5 M_⊕, c = 9 and d = 3 (see the solid line), and of M_{c, crit0} = 10 M_⊕, c = 10, and d = 3 (see the dotted line). We note that the dashed line for the exo-Jupiters lines-up with the solid one completely (see the right panel).

Since the TM values mark the metallicities above which the populations of hot and exo-Jupiters begin to dominate the low-mass planets, it is clearly evident that the choice of the value M_{c, crit0} = 10 M_⊕ for the hot Jupiters puts the TM far to the left of the distribution. For the exo-Jupiter population, this value puts the TM in the midst of the distribution. On the other hand, the choice of a value M_{c, crit0} = 3 M_⊕ puts the TM for hot Jupiters in the midst of the population, with a significant amount of power at the lower metallicities. This is also problematic. It is clear that the optimal choice for this parameter is M_{c, crit0} = 5 M_⊕. For the hot Jupiters, this places the TM just to the left of the rise in the population of observed population of hot Jupiters, and at a reasonable TM for the exo-Jupiters as well. Our results therefore indicate that the case, M_{c, crit0} = 5 M_⊕, c = 9 and d = 3 is the most consistent match with the currently observed metallicity distributions derived for hot and exo-Jovian planets detected by radial velocity observations.

It is particularly interesting that the observations pick out the value; M_{c, crit0} = 5 M_⊕. This best fit model is the fiducial case shown on the left panel of Figure 3. If we examine the PFF for the exo-Jupiter curve in the top panel of this figure, we note that at solar metallicity, the most probably core mass of such a planet is,

$$\begin{equation} {\langle {M_{c,{\rm crit}}}\rangle } \simeq 5 \,M_{\oplus }, \end{equation} \tag{ 11 }$$

which is the value of our parameter, M_{c, crit0}. Therefore our best fit critical core mass parameter—given the available data on the distributions of metallicities of Jovian planets—is indeed behaving like a physical critical core mass in our model. This shows that our model is internally self-consistent.

If our own Jupiter can be regarded as a typical member of the exo-Jupiter population, then our model predicts that it likely has a core mass of the order of 5 M_⊕. Can this be justified?

On the theoretical side, we recall that M_{c, crit0} is directly related to the opacity in the planetary envelope (Equation (2), also see Hori & Ikoma 2010). A value of 5 M_⊕ implies that the grain opacity in the atmosphere of forming Jupiters must have a characteristic value that is more than one order of magnitude lower than the canonical value; κ ≃ 0.06 g cm⁻². Recently, Movshovitz & Podolak (2008) have shown through numerical simulations that the resultant opacity in planetary atmospheres is very unlikely to depend strongly on the metallicity of accreting materials. This suggests that the growth and subsequent settling of dust grains in the envelope play an important role in lowering the grain opacity there. Such effects are in fact confirmed by Movshovitz et al. (2010), wherein more complete calculations of planetary atmospheres coupled with grain growth and settling there were undertaken. This detailed study implies that the grain opacity can be reduced to ⩽2% of the conventional value (equivalently a factor of ⩾ 50). Note that the difference in the value of κ may not be crucial because the dependence of κ in Equation (2) is very shallow. In fact, the recent population synthesis calculations suggest that, while it is very likely that the grain opacity in the planetary atmosphere is much lower than the ISM value, it is very difficult to constrain κ sharply due to such dependence (Mordasini et al. 2014).

In addition, our results show that the PFFs of low-mass planets for the case of M_{c, crit0} = 5M_{c, crit0} are less sensitive to those for M_{c, crit0} = 10M_{c, crit0} (see Figures 3 and 4). It is interesting that 〈M_{c, crit}〉 of low-mass planets for both cases of M_{c, crit0} is very similar, which is about 2–4 M_⊕ (see Figures 3 and 4). This indicates that it may be difficult to derive invaluable constraints on the core mass of Jovian planets by examining the mass of super-Earths and hot Neptunes.

Detailed numerical simulations of planetary envelopes show that the accretion of solid materials can proceed efficiently in the final, runaway gas accretion phase (Pollack et al. 1996; Hubickyj et al. 2005; Movshovitz et al. 2010). Based on the orbital integration of planetesimals around a planet done by Shiraishi & Ida (2008), the additional accumulation of the solids can become about serval M_⊕. Thus, a 5 M_⊕ core mass for the Jupiter estimated by our model is obviously the lower limit.

As discussed above, the hot Jupiters are produced as a consequence of the orbital migration induced by planet–disk interactions in our model. In fact, the first discovery of the massive planet around the Sun-like star (Mayor & Queloz 1995) invokes the importance of gas-induced planetary migration (Lin et al. 1996). On the other hand, the recent observations of the Rossiter–McLaughlin effect have detected close-in massive planets whose orbital planes are strongly misaligned with those of their host stars (e.g., Triaud et al. 2010; Winn et al. 2010). It is suggested that such misalignment can be understood by planet–planet interactions with the aid of convergent gas-induced migration and/or the Kozai mechanisms combined with stellar tides (e.g., Fabrycky & Tremaine 2007; Nagasawa et al. 2008; Chatterjee et al. 2008). In addition, the recent exoplanet observations infer that the formation of the hot Jupiters in higher metallicity disks may be more efficient than our results predict (e.g., Santos et al. 2000; Fischer & Valenti 2005; Dawson & Murray-Clay 2013). Furthermore, the hot Jupiters are subject to the exposure of the strong irradiation of the host stars which can considerably affect the composition and the structure of the atmosphere of the planets (e.g., Seager & Deming 2010). In order to fully understand origins of the hot Jupiters, it is therefore important to consider additional physical processes that will occur after the dissipation of gaseous disks in which gas giant formation (including the hot Jupiters) predominately proceeds.

How do low-mass planets form? Our results on the critical core mass of massive planets of 5 M_⊕ have important implications for low-mass planets in tight orbits such as super-Earths.

Currently, there are probably two main ideas to understand the formation of low-mass planets: one is that they form in situ, the other is that they are formed in the outer part of disks and migrate to the current position. Since the former scenario considers mergers of embryos and planetesimals as the main assembly process to form super-Earths (e.g., Hansen & Murray 2013; Chiang & Laughlin 2013), the finally formed planets are predicted to be scaled-up version of rocky planets like our Earth. The latter picture emphasizes that such low-mass planets can form by the same mechanism as gas giants, and that therefore they are essentially failed cores of gas giants and/or mini-gas giants (e.g., Rogers et al. 2011, HP12, HP13). Note that the in situ scenario requires unusually large amounts of solids in the inner part of disks, so that it is very likely that some kind of migration would be needed to transport embryos of planets from the outer to the inner part of disks (e.g., Ogihara & Ida 2009; McNeil & Nelson 2010; Kretke & Lin 2012).

The statistical investigation presented in this paper can provide additional insight into the failed core scenario. We predict that any core mass beyond 5 M_⊕ will have an increasing gaseous component, since gas accretion takes place efficiently when M_c > M_{c, crit}(≃ 5 M_⊕). As a result, we can expect that super-Earths beyond 5 M_⊕ will not be composed of pure solid cores, but will have a greater admixture of gas as one goes from 5–10 M_⊕. This may give a clue to understand the recent results of the Kepler observations which derive a new empirical relation between planetary mass and radius that implies that the composition of super-Earths may change from solid materials to gaseous ones around the radius of planets, R ≃ 1.5 R_⊕ (equivalently the mass of planets, M_p ≃ 5 M_⊕ using the relation) (Weiss & Marcy 2014; Marcy et al. 2014).

Note that low-mass planets in tight orbits are anticipated to be influenced by stellar irradiation like hot Jupiters, which affects the abundance and the volume of the atmosphere (Rogers et al. 2011; Lopez et al. 2012; Lopez & Fortney 2013). Thus, it would also be important to consider the evolution of such planets to interprets their current properties more accurately.

Combining our results with the above discussion, we suggest the following hierarchy for planets formed in protoplanetary disks.

• 1.  
  Massive planets (M_p ⩾ 10 M_⊕): cores (≃ (f + 5) M_⊕) + massive envelopes (⩾5 M_⊕).
• 2.  
  Intermediate-mass planets (5 M_⊕ ⩽ M_p ⩽ 10 M_⊕): cores (≃ 5 M_⊕) + low-mass envelopes (<5 M_⊕).
• 3.  
  Low-mass planets: only cores (M_p ⩽ 5 M_⊕).

Note that f in the core mass of massive planets arises from the additional capture of solid materials during the runaway gas accretion (see Section 6.1).

In the hierarchy, the difference between gas giants like the Jupiter and sub-giants like the Neptune originates from the timing of disk dissipation. It determines the timescale during which growing planets can accrete the disk gas. Thus, when planets are formed in the later phase of disk evolution, they tend to be sub-giants. The regime of super-Earths covers both the intermediate and the low-mass planets. This implies that they can be regard as both failed cores of gas giants as well as massive Earths. It is therefore expected that the composition of them would be diverse. Interestingly, Buchhave et al. (2014) have recently examined the Kepler data as a function of metallicity, and inferred that there are likely to be three types of exoplanets with different compositions. Although more observational data would be needed, our proposed hierarchy roughly corresponds to their classification, by converting the planetary size to the mass using an empirical mass–radius relation. More detailed studies are obviously required for examining the proposed hierarchy.

There are two caveats for our results. The first is related to the state of the observational data sets that we have employed. The currently available observations of exoplanets, especially around low metallicity stars, may still be highly biased even for the most successful radial velocity methods. Thus, it is very important to conduct intensive surveys of exoplanets, especially orbiting around stars with [Fe/H] ≃ −0.2 to −0.4, to create a more complete sample of the distribution of massive planets at these values of [Fe/H]. This would be invaluable information for our model, since it would allow us to better constrain the parameter values, using the approach that we have outlined here. As a concrete example, if more data were to fill out the metallicity distribution in Figure 5 for the hot Jupiters toward lower values of the metallicity, this would tend to push the value of M_{c, crit0} toward a higher values like 10 M_⊕. In addition, more data for metal-poor stars may clarify the difference in the distribution of massive planets: the observed hot Jupiters show a sharp transition in Figure 5 whereas, for the exo-Jupiters, neither the computed PFFs nor the observed ones show such a sharp trend.

The second has to do with our theoretical evolutionary tracks. We have adopted an approximation, in common with the Ida & Lin models that gas accretion onto critical cores occurs at a rate that is governed by the Kelvin–Helmholtz timesscale (see Equations (3) and (4)). We find that the resulting masses of our Jovian planets are underestimated by this approach. We hasten to note that this is not likely to affect any of the conclusions related to the PFFs as is shown by Figure 4 in HP13. There we found that even if the amount of accreted gas onto planetary cores changes drastically, the high-mass regime did not significantly affect the PFFs. There is some sensitivity however to the low-mass planets, whose PFFs are an overestimate if accretion rates are increased. We note that other authors have adopted what is surely an extreme upper limit to the planet accretion rate—namely the full disk accretion rate (e.g., Mordasini et al. 2009). In that limit, there is of course no problem to form very massive Jupiters, although the low-mass planets are entirely missed. More theoretical work is needed to formulate a more accurate description of the physics of late-phase gas accretion (also see Hasegawa & Ida 2013, for a recent discussion).