MASS–RADIUS RELATION FOR ROCKY PLANETS BASED ON PREM

Inside Earth, the density jump from the upper mantle to the lower mantle is 10% from 4 g/cc to 4.4 g/cc at 23.83 GPa. Earth's upper mantle consists of complex phases of Mg-silicates, including various polymorphs of olivine: (α) olivine, (β) wadsleyite, and (γ) ringwoodite (Bina 2003). The upper mantle to lower mantle transition occurs at a 670 km depth in the Earth (about 10.5% Earth radius). For more massive planets like Kepler-93b, this transition occurs at shallower depth (∼3% radius of Kepler-93b) for the same pressure.

Figure 1 shows the comparison of different EOS of Mg-silicates.

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Interestingly, the TFD beyond 1 TPa for MgO, SiO₂, MgSiO₃, and Mg₂SiO₄ are almost identical because they all have an average atomic weight of A = 20 and an average atomic charge of Z = 10. This coincidence simplifies the EOS of Mg-silicates, as it indicates that Mg/Si ratio does not matter toward the high-pressure end; it only matters toward the low-pressure end, which is captured in our EOS by using the PREM density variation in the upper-mantle pressure range.

According to Figure 1, except for the prominent density jump of 10% at the upper-lower mantle boundary, the other high-pressure phase transitions are only ∼1% level in density, and thus insignificant in mass–radius calculation (Unterborn et al. 2015). Therefore, in this paper we adopt the EOS of Mg-silicates mantle as follows:

• 1.  
  0 GPa–23.83 GPa: linear interpolation of the lower mantle density according to Appendix G of (Stacey & Davis 2008), which is taken from PREM. i.e., follow the green curve in Figure 1.
• 2.  
  23.83 GPa–3.5 TPa: BM2 EOS with ρ₀ = 3.98 g/cc, K₀ = 206 GPa, (error ∼1% in density). i.e., follow the black curve in Figure 1.
• 3.  
  $\gt 3.5$ TPa: TFD EOS of MgSiO₃ calculated using method in (Salpeter & Zapolsky 1967). i.e., follow the pink curve in Figure 1.

Figure 2 shows the comparison of different EOS of the Core.

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In the Earth, the density jump from liquid outer core to solid inner core is 5% from 12.2 g/cc to 12.8 g/cc at 328.85 GPa. The Rydberg EOS of the pure solid -Fe phase, which is experimentally determined from static compression up to 205 GPa (Dewaele et al. 2006; Wagner et al. 2012), is too high of a density (cyan curve in Figure 2) in the pressure region relevant to Earth, even including the temperature effect (green curve in Figure 2), even for the solid inner core. This is likely due to the presence of several weight percent of one or more light elements (McDonough 2014, pp. 559–577). The lighter elements could be S, Si, O, C, or a combination of them (Anderson & Ahrens 1994; Poirier 1994; Stixrude et al. 1997; Alfè et al. 2002; Badro et al. 2007; Fischer et al. 2012), but no agreement is reached as to which of these elements are the most important. The Rydberg EOS with or without temperature behaves poorly for extrapolation above ∼1 TPa as it overshoots and crosses over the TFD EOS, which serves as a lower bound for the density of Fe.

As such, we choose to only extrapolate the liquid outer core BM2 EOS to higher pressure until it asymptotically approaches the TFD EOS. The solid inner core is likely a secondary feature resulting from the crystallization of the liquid core. It currently comprises only a small fraction (≈3%) of the total mass of the Earth, and in more massive planets, which is the focus of this study, a solid inner core may not exist at all due to a higher heat content and a slower cooling rate. BM2 EOS behaves nicely as it asymptotically approaches the TFD EOS at very high pressure (∼12 TPa).

Therefore, in this paper we adopt the EOS of Fe core as follows:

• 1.  
  0 GPa–12 TPa: BM2 EOS with ρ₀ = 7.05 g/cc, K₀ = 201 GPa, (error ∼1% in density). i.e., follow the black curve in Figure 2.
• 2.  
  12 TPa: TFD EOS of Fe (A = 55.845 and Z = 26) calculated using method in (Salpeter & Zapolsky 1967). i.e., follow the pink curve in Figure 2.

The CMF-dependent mass–radius relation of rocky planets can be fit to the following formula (for $0\;\leqslant$ CMF ≤ 0.4, $1{M}_{\oplus }\leqslant M\leqslant 8\;{M}_{\oplus }$). It agrees with the actual mass–radius curves in Figure 3 within ∼0.01 R_⊕ in radius. If we use it to calculate the CMF, it gives an uncertainty of ∼0.02.

$$\begin{eqnarray}&&\left(\displaystyle \frac{R}{{R}_{\oplus }}\right)=(1.07-0.21\cdot \mathrm{CMF}){\left(\displaystyle \frac{M}{{M}_{\oplus }}\right)}^{1/3.7}.\end{eqnarray} \tag{ 2 }$$

Equation (2) can be inverted to solve for CMF, given the mass and radius.

$$\begin{eqnarray}&&\mathrm{CMF}=\displaystyle \frac{1}{0.21}\left[1.07-\left(\displaystyle \frac{R}{{R}_{\oplus }}\right)/{\left(\displaystyle \frac{M}{{M}_{\oplus }}\right)}^{1/3.7}\right].\end{eqnarray} \tag{ 3 }$$

For comparison, this semi-empirical mass–radius formula is in agreement with the mass–radius relation of super-Earths presented in Valencia et al. (2006), which is a scaling law of $R\propto {M}^{0.267-0.272}$. This new formula takes one step further to articulate the dependence on the CMF, which is useful in differentiating among rocky planets with different CMFs (i.e., compositions).

For Earth, CMF${}_{\oplus }=\frac{1}{0.21}[1.07-{(1)/(1)}^{1/3.7}]$ = $0.07/0.21\quad =\quad 0.33$. For Kepler-93b, CMF${}_{{\rm{K}}93{\rm{b}}}=\frac{1}{0.21}[1.07-{(1.478)/(4.02)}^{1/3.7}]=0.26$. The uncertainties (δCMF) in CMF resulting from the uncertainties in mass and radius can be derived from Equation (3):

$$\begin{eqnarray}&&| \delta \mathrm{CMF}| \approx 5\times \sqrt{{\left|\displaystyle \frac{\delta r}{r}\right|}^{2}+{\left(\displaystyle \frac{1}{3.7}\right)}^{2}{\left|\displaystyle \frac{\delta m}{m}\right|}^{2}}.\end{eqnarray} \tag{ 4 }$$

To tell a 20% core mass apart from a 30% core mass, the radius needs to be measured to better than a 2% level, or equivalently, mass to a 6% level. For Kepler-93b, $\left|\frac{\delta r}{r}\right|=\frac{0.019}{1.478}=0.013$ and $\left|\frac{\delta m}{m}\right|=\frac{0.68}{4.02}=0.17$, so $| \delta \mathrm{CMF}| =0.2$. Therefore, CMF${}_{{\rm{K}}93{\rm{b}}}=0.26\pm 0.2$. Table 1, column 4, lists the CMF estimates of these exoplanets.

Assuming this population of dense exoplanets follows the same CMF distribution, we then apply a weighted least-square fit to Table 1.

Denote the weighted average of the CMF as $\bar{\mathrm{CMF}}$.

$$\begin{eqnarray}&&\bar{\mathrm{CMF}}=\displaystyle \frac{{\displaystyle \sum }_{i}\left(\displaystyle \frac{\mathrm{CMF}}{\delta {\mathrm{CMF}}^{2}}\right)}{{\displaystyle \sum }_{i}\left(\displaystyle \frac{1}{\delta {\mathrm{CMF}}^{2}}\right)}\end{eqnarray} \tag{ 5 }$$

Denote the standard deviation of the CMF as $\sigma \mathrm{CMF}$.

$$\begin{eqnarray}&&\sigma \mathrm{CMF}=\sqrt{1/\left({\displaystyle \sum }_{i}\displaystyle \frac{1}{\delta {\mathrm{CMF}}^{2}}\right)}\end{eqnarray} \tag{ 6 }$$

The result is $\bar{\mathrm{CMF}}=0.26$ and $\sigma \mathrm{CMF}=0.07$, indicating Earth-like composition ($\mathrm{CMF}\sim 0.3$) carries on up to at least ∼5 M_⊕.

This is backed up by recent studies of disintegrated planet debris in polluted white dwarf spectra (Jura & Young 2014; Xu et al. 2014; Wilson et al. 2015). These studies show that the accreted extrasolar planet debris generally resemble bulk Earth composition (>85% by mass composed of O, Mg, Si, Fe), similar Fe/Si and Mg/Si ratio, and are carbon-poor. This indicates formation processes similar to those controlling the formation and evolution of objects in the inner solar system (Jura & Young 2014).

In our solar system, evidence suggests that rocky bodies were formed from chondritic-like materials (cf. Lodders & Fegley 2010). Current planet formation theory suggests that the solar nebula was initially heated to very high temperatures to the extent that virtually everything was vaporized except for a small amount of presolar grains (Ott 2007). The nebula then cools to condense out various elements and mineral assemblages from the vapor phase at different temperatures according to the condensation sequence (White 2013). Fe–Ni metal alloy and Mg-silicates condense out around similar temperatures of 1200–1400 K (depending on the pressure of the nebula gas) according to thermodynamic condensation calculation (Lodders 2003). Oxygen, on the other hand, does not have a narrow condensation temperature range, as it is very abundant and it readily combines with all kinds of metals to form oxides which condense out at various temperatures as well as H, N, and C to form ices condensing out at relatively low temperatures (Lewis 1997). As supported by the polluted white dwarf study, we expect other exoplanetary systems to follow similar condensation sequence as the solar system in a H-dominated nebular environment for the major elements: Fe, Mg, Si, and O (Jura & Young 2014).

In our solar system, the primitive CI Carbonaceous Chondrites have Fe:Mg:Si = 0.855:1.047:1 (McDonough & Sun 1995). If all this Fe forms the core, the CMF ≈ 0.38 is the upper limit. If some Fe is incorporated into the mantle either as metal or oxides, the CMF becomes less. The solar ratio of Fe/Si is representative for the stars in the solar neighborhood, which is a tight distribution centered at 1, while the Mg/Si = 1 seems to tend toward the lower end of the distribution centered at 1.34 (Gilli et al. 2006; Grasset et al. 2009). A Mg/Si ratio higher than 1 could produce more olivine (Mg₂SiO₄) or more MgO to affect the mineralogy of the upper mantle (Bond et al. 2010; Delgado Mena et al. 2010; Pagano et al. 2015). However, it does not affect high-pressure EOS much. As we have pointed out earlier in Section 2 of EOS, the TFD EOS of MgO, SiO₂, MgSiO₃, and Mg₂SiO₄ will converge above ∼1 TPa due to their identical average atomic weight of 20 and their atomic charge of 10. Therefore, for more massive planets, the effect of Mg/Si tends to be smaller. The dispersion expected from the variation in Mg/Si and Fe/Si ratios cause approximately 2% difference in radius (Grasset et al. 2009; Dressing et al. 2015).

Oxygen is more readily available than Fe, Mg, or Si (Lodders 2003) because it is richly produced in nuclear synthesis of massive stars and the chemical evolution of the Galaxy (Pagel 1997), so there is usually enough O to combine with Mg and Si to form Mg-silicates as well as to oxidize some Fe-metal. If we compare bodies in our solar system, the core mass fractions of Earth and Venus (Rubie et al. 2007, pp. 51–90) are around 0.3, the core mass fraction of Mars is estimated to be 0.2 (McSween 2003), and the core mass fraction of Vesta is smaller, about 0.17 (Ruzicka et al. 1997; Ermakov et al. 2014). The core mass fractions of asteroid parent bodies of iron meteorites are even smaller (Petaev & Jacobsen 2004). As such, there seems to exist a trend of increasing CMFs from smaller objects toward bigger objects in the solar system. This trend seems to turn flat around 1 M_⊕.

These dense exoplanets between 2 and 5 M_⊕ so far appear to agree with the mass–radius relation with a CMF ≈ 0.26, suggesting that they are like the Earth in terms of their proportions of mantle and core. But their surface conditions are utterly different as they are much too hot. This is due to the observational bias that currently it is much easier for us to detect close-in planets around stars. The fact that we now see so many of them suggests there may be abundant Earth-like analogs at proper distances from their stars to allow the existence of liquid water on their surfaces.