LIFETIME AND SPECTRAL EVOLUTION OF A MAGMA OCEAN WITH A STEAM ATMOSPHERE: ITS DETECTABILITY BY FUTURE DIRECT IMAGING

The evolutionary model used in the present study is that described in Hamano et al. (2013), except for the calculations of radiative transfer. In the present study, we replace a gray radiative transfer code used in Hamano et al. (2013) with a newly developed, non-gray code to calculate thermal emission and reflection spectra from the planet. In the non-gray code, a two-stream formulation is used to calculate the radiative transfer in a plane-parallel atmosphere. We calculate planetary and stellar radiation separately in a wavenumber range from 0 to 30,000 cm⁻¹ with a spectral resolution of 0.01 cm⁻¹. The albedo of the planetary surface is fixed at 0.2. See the Appendix for details on the radiative transfer model and its validity.

A steam atmosphere is divided into 100 layers, which are equally spaced on a logarithmic scale of pressure. The pressure at the top of the atmosphere is fixed at 1 Pa in all calculations. The temperature structure is prescribed such that it follows an adiabatic lapse rate in all atmospheric layers, based on results of Abe & Matsui (1988). For atmospheric layers in which the temperature is above the critical temperature of water or the relative humidity is less than 1, the lapse rate is assumed to be a dry adiabat. For the other layers, the lapse rate is given by a pseudo-moist adiabat under the assumption that condensates are removed immediately as they form. For simplicity, the effects of clouds are ignored in the present study. Since we consider a pure steam atmosphere, the pseudo-moist adiabat overlaps the saturation curve of water in the present study. We apply a liquid–vapor saturation curve at temperatures above 273.16 K and an ice–vapor curve at temperatures below 273.16 K. The dry adiabat is calculated using the Peng–Robinson equation of state in a manner similar to Abe & Matsui (1988), but the heat capacity of water vapor in the ideal gas state is taken from Wagner & Pruß (2002) as a function of temperature in order to make the heat capacity applicable to higher-temperature conditions.

We use HITEMP 2010 (Rothman et al. 2010) as a spectral database to calculate the absorption coefficients for lines of water vapor. For all calculations, a Voigt line profile is used to consider the combined effect of Doppler and pressure broadening, using a computational code by Humlícěk (1982). Although no reliable model for water continuum is available for the high-temperature, high-pressure conditions considered herein, we adopt a MT_CKD 2.5 (Mlawer et al. 2012) model for its continuum together with a wavenumber truncation of 25 cm⁻¹, which is similar to a procedure used in recent studies on radiative transfer calculations of hot and thick steam atmospheres (Goldblatt et al. 2013; Kopparapu et al. 2013). As the number of line transitions in the HITEMP 2010 exceeds 10⁸, we generate a grid of absorption coefficients of water vapor and calculate the optical depth of each layer by interpolating the absorption coefficients.

We consider Rayleigh scattering by water vapor with a scattering cross-section per molecule of

$$\begin{eqnarray}&&{\sigma }_{\mathrm{sc}}={\displaystyle \frac{32{\pi }^{3}}{3}}{\displaystyle \frac{1}{{N}_{{\rm A}}^{2}}}{\left({\displaystyle \frac{{m}_{0}-1}{{n}_{0}}}\right)}^{2}{W}_{n}^{4},\end{eqnarray} \tag{ 1 }$$

assuming that the polarizability volume of water vapor does not depend on temperature or pressure. In the above equation, W_n is the wavenumber, N_A is the Avogadro constant, and n₀ and m₀ are the molar volume and the refractive index in the reference state, respectively. We adopt a refractive index of 1.000254 from Allen (1973) for D lines of Na at the standard temperature and pressure (0 °C, 1 atm).

Figure 1(a) shows the thermal emission spectrum from 0.33 to 100 μm with a surface pressure of 50 bar and a surface temperature of 2500 K. At this surface temperature, the peak wavelengths of the Planck function ${B}_{\lambda }$ and $\lambda {B}_{\lambda }$ are approximately 1.2 and 2 μm, respectively. The planetary surface emits strong radiation in the visible and near-IR. Although the hot planetary surface emits significant radiation, the outgoing thermal emission from the top of the atmosphere is greatly attenuated by absorption and scattering by the steam atmosphere. The stronger thermal radiation passes through the wavelength regions corresponding to the atmospheric windows (Table 1), at which the absorption of water vapor is relatively weak.

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Figure 1(c) shows the atmospheric pressures at which the optical depth is unity for absorption and scattering, respectively. In the far-IR and absorption bands of water vapor, the upper layers, in which the pressure is typically less than 10⁻³ bar, are responsible for the thermal emission spectrum, whereas the atmospheric windows reveal the deeper and hotter layers. At wavelengths longward of approximately 0.6 μm, the pressure level at which the optical depth of absorption is unity is much lower than that of scattering. In the near-IR wavelength region, therefore, the brightness temperature provides a good estimate of the atmospheric temperature at an optical depth of unity.

At the visible wavelength, the scattering process has a dominant role in attenuating the thermal emission. At shorter wavelengths, the deepest layer could contribute to the emergent spectrum because of the absence of strong absorption lines. The actual probed depths, however, depend on the atmospheric amount and continuum absorption of water vapor, the physics of which remain poorly understood. If the atmosphere is thin enough to allow the thermal radiation emitted from the surface to pass through the atmosphere, a Rayleigh scattering slope appears in the planetary spectrum at sufficiently optically thin wavelengths. Measuring the scattering slope can, in principle, provide information about the atmospheric column mass, the surface temperature and the surface albedo.

Figure 1(b) shows the spectral albedo for a stellar zenith angle ${\theta }_{\star }$ of ${\mathrm{cos}}^{-1}\left(1/\sqrt{3}\right)$. The spectral albedo is defined as the ratio of the upward diffusive radiation to the downward direct radiation at the top of the atmosphere. The bond albedo of the planet is obtained as a weighted mean of the spectral albedo by the Planck function with an effective stellar temperature.

In addition to the thermal radiation, the spectral albedo is higher within the atmospheric windows. Moreover, the spectral albedo varies greatly at the visible wavelengths, from approximately 1 at 0.4 μm to less than 0.1 beyond 0.9 μm. Therefore, in the near-IR, the reflected stellar light is expected to have a smaller contribution to the emergent spectrum. Although the spectral and resulting Bond albedos increase with stellar zenith angle, the difference is less than 0.2 throughout the visible wavelengths between ${\theta }_{\star }$ of 0° and 80°. In the present study, we use the spectral and Bond albedos with ${\theta }_{\star }$ of ${\mathrm{cos}}^{-1}\left(1/\sqrt{3}\right)$ degrees as a representative value. Spectra composed of both reflections and emissions are presented later herein under conditions that are consistent with the atmospheric evolution in the course of solidification of the magma ocean.

Energy and water budgets during planetary solidification are treated in the same manner as Hamano et al. (2013). Here, we briefly summarize our proposed model, focusing especially on the major changes made in the present study. For additional details, see Hamano et al. (2013). We assume an adiabatic temperature profile in the planetary interior to obtain the heat capacity of the whole silicate portion. The heat flux from the magma ocean is calculated as the difference between the outgoing planetary radiation and the net incoming stellar radiation. In order to reduce the computing time for the evolution calculations, the planetary radiation and albedo are precalculated on 882 surface-temperature–pressure grid points, from 1350 to 3000 K and from $5\times {10}^{-4}$ to 5000 bar, respectively. In order to follow the evolution of planet-to-star contrast, the thermal radiation from each band is also tabulated after averaging the high-resolution spectra with a resolving power of 100, which is a typical value for future direct imaging. We consider a Sun-like host star, the bolometric luminosity and XUV radiation of which evolve in a manner similar to solar analogs. We adopt a solar standard model by Gough (1981) for luminosity enhancement with stellar age and linearly extrapolate it after a stellar age of 4.567 Gyr. The stellar spectrum is assumed to be a blackbody spectrum with an effective stellar temperature of 5800 K.

Water is assumed to be partitioned between the atmosphere and the magma ocean according to its solubility into basaltic melts. The incorporation of water into cumulates is also considered according to the partition coefficients of water between melts and solids and by assuming 1% of trapped interstitial melts. The mass loss rate of water is calculated from an energy-limited escape flux of hydrogen ${{\rm \Gamma }}_{{\rm H}}$ (Watson et al. 1981) with a heating efficiency η of 0.1, as follows:

$$\begin{eqnarray}&&{{\rm \Gamma }}_{{\rm H}}=4\pi {R}_{\mathrm{pl}}^{2}{\displaystyle \frac{{R}_{\mathrm{pl}}}{{{GM}}_{\mathrm{pl}}}}{\displaystyle \frac{\eta {S}_{\star }^{\mathrm{XUV}}}{4}}{\left({\displaystyle \frac{a}{1\;\mathrm{AU}}}\right)}^{-2},\end{eqnarray} \tag{ 2 }$$

where R_pl and M_pl are the planetary radius and mass, respectively, a is the orbital distance from the host star in AU, and G the gravitational constant. Here, ${S}_{\star }^{\mathrm{XUV}}$ is the XUV flux from the host star at 1 AU. The stellar XUV flux decreases rapidly with time, whereas the ratio of the X-ray luminosity to the bolometric luminosity remains nearly constant at $\sim {10}^{-3}$ for young active stars (Vilhu & Walter 1987; Stauffer et al. 1994; Pizzolato et al. 2003). Based on a power-law relation derived from observational data by Ribas et al. (2005) and Ribas (2009), we adopt the following expression for the evolution of ${S}_{\star }^{\mathrm{XUV}}$ as a function of stellar age τ in Gyr:

$$\begin{eqnarray}{S}_{\star }^{\mathrm{XUV}}=\left\{\begin{array}{cc}{S}_{\odot }^{\mathrm{XUV}}{\left({\displaystyle \frac{{\tau }_{\mathrm{sat}}}{{\tau }_{\odot }}}\right)}^{-1.23} & \mathrm{for}\ \tau \lt {\tau }_{\mathrm{sat}}\\ {S}_{\odot }^{\mathrm{XUV}}{\left({\displaystyle \frac{\tau }{{\tau }_{\odot }}}\right)}^{-1.23} & \mathrm{for}\ \tau \gt {\tau }_{\mathrm{sat}}\end{array}\right.\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&{\tau }_{\mathrm{sat}}\equiv 1.66\times {10}^{20}{L}_{\mathrm{bol}}^{-0.64},\end{eqnarray} \tag{ 4 }$$

where L_bol is the bolometric luminosity of the host star in $\mathrm{erg}\;{{\rm s}}^{-1}$, and ${\tau }_{\odot }$ and ${S}_{\odot }^{\mathrm{XUV}}$, which are the age and XUV flux at 1 AU of the current Sun, are fixed to 4.567 Gyr and $4.59\times {10}^{-3}\;{\rm J}\;{{\rm s}}^{-1}\;{{\rm m}}^{-2}$, respectively. The value of ${\tau }_{\mathrm{sat}}$ is approximately 0.066 Gyr for the solar standard model by Gough (1981). The mass loss rate of water is nine times as large as the mass loss rate of hydrogen under the assumption that all the dissociated oxygen atoms are consumed to oxidize the abundant surface magma (Gillmann et al. 2009; Hamano et al. 2013).

We consider terrestrial planets with the same mass and bulk composition as Earth. We start the evolution calculations from a very hot state with a surface temperature of 3000 K and stop the calculations when the surface temperature reaches the solidus temperature at the surface (approximately 1370 K) or the stellar age reaches 10 Gyr, which is comparable to the main sequence lifetime of the host star. The solidification is assumed to start at a stellar age of 0.05 Gyr (${\tau }_{0}$). We consider the initial total inventory of water ranging from 0.01 to 50 times the current ocean mass of Earth M_EO of $1.4\times {10}^{21}$ kg. This corresponds to a range in bulk water content from $2.3\times {10}^{-4}$ to 1.2 wt%. Furthermore, we consider the orbital distance from the star of between 0.4 and 1.5 AU.

The Type I planet is characterized by a sufficiently far distance from the host star, where the net incident stellar radiation is smaller than the radiation limit. Since the outgoing planetary radiation never falls below the radiation limit, the Type I planet is self-luminous over its solidification period. The minimum heat flux, which is given by the difference between the radiation limit and the net incident stellar flux, determines the overall solidification timescale, which is approximately 4 Myr in this nominal case (Figure 2(a)). The rapid solidification results in the monotonic growth of the steam atmosphere toward the end of the magma ocean period by keeping the degassing rate of water higher than its loss rate.

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Although the surface temperature during the magma ocean period is extremely high, the bolometric luminosity is not so large compared to that from the hot surface. Here, we define an effective emission temperature T_eff, using the planetary radiation F_pl, as follows:

$$\begin{eqnarray}&&{T}_{\mathrm{eff}}=\sqrt[4]{{F}_{\mathrm{pl}}/\sigma }.\end{eqnarray} \tag{ 6 }$$

The effective emission temperature never exceeds 800 K in this nominal case and is typically as low as 300 K over the solidification period. The thick steam atmosphere thus sustains the magma ocean longer, whereas the planet itself looks as faint as a planet with a cool or moderate surface temperature.

The most recognizable diagnostic sign of a hot surface is emission at the visible and near-IR wavelengths. Although the massive steam atmosphere strongly absorbs the thermal emission from hotter and deeper layers, thermal emission can leak through atmospheric windows. The third panel of Figure 2(a) shows the time variations of thermal radiation from atmospheric windows. In the early stages (less than 10⁴ years), the thermal radiation from all bands exceeds ${10}^{2}\;{\rm W}\;{{\rm m}}^{-2}\;\mu {{\rm m}}^{-1}$. These high fluxes, however, do not last long. At most of the atmospheric windows, the thermal fluxes rapidly decrease with time and fall below ${10}^{-2}\;{\rm W}\;{{\rm m}}^{-2}\;\mu {{\rm m}}^{-1}$ due to the decrease in the surface temperature and the growth of the steam atmosphere. The thermal radiation of the M, L, and K_s bands initially decreases with time as well, but eventually levels off.

Figure 3(a) shows the spectral evolution in the visible and near-IR. The planetary thermal radiation declines with time, and then, at wavelengths longer than approximately 2 μm, the thermal radiation becomes constant. This is a result of the same mechanism as the occurrence of the radiation limit of the steam atmospheres and indicates that, for the wavelengths at which water vapor has strong absorption, the emission level becomes as high as that at which the thermal structure is uniquely determined by the saturation curve of water. Since the thermal structure around the emission level remains the same, the thermal radiation from the M, L, and K_s bands becomes constant, as shown in Figure 2(a).

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In contrast to the Type I planet, the Type II planet receives a net stellar radiation that is larger than the radiation limit because of its proximity to the host star. The planet therefore would reach a radiative energy balance during the course of solidification. Early in its solidification, the outgoing planetary flux far exceeds the net stellar flux, and the planet is highly self-luminous, as in the case of the Type I planet (Figure 2(b)). In the case of the Type II planet, however, the planetary flux decreases with time and eventually balances the net stellar flux at approximately 0.6 Myr. If no water loss occurs, the radiative balance is perfectly achieved, and the magma ocean does not solidify. Actually, as the steam atmosphere becomes thinner with time due to the hydrodynamic escape, the planet emits slightly larger radiation than the net stellar radiation. The Type II planet thus slowly cools and solidifies, losing its atmosphere.

The solidification rate of the magma ocean is regulated by the loss rate of water such that a sufficient net loss of the steam atmosphere occurs so that the net outgoing flux becomes positive. The resulting solidification time is approximately 100 Myr in this nominal case, which is much longer than that of the Type I planet. The most striking feature is that the higher band fluxes continue over this long solidification period (third panel in Figure 2(b)). The thermal radiation from the atmospheric windows ceases to decrease at ∼0.6 Myr, at which radiative energy balance has been roughly achieved. The subsequent spectra do not vary with time at the opaque IR wavelengths or the atmospheric windows, which are relatively transparent (Figure 3(b)).

The long-lived and constant thermal emission in the window regions occurs by attaining quasi-energy balance on the Type II planet. In the quasi-energy balance state, the outgoing planetary radiation roughly balances the incoming net stellar radiation. In order to satisfy this requirement, the amount of atmosphere must be regulated during the course of solidification. As the surface temperature decreases with time, the amount of atmosphere must decrease such that the planet maintains a planetary radiation that is nearly equal to, or, strictly speaking, slightly larger than, the net stellar flux (Figure 2(b)). Consequently, the resulting spectra do not change during a quasi-energy balance state, except at the shorter wavelengths, at which the opacity of the atmosphere is thin enough for the outgoing thermal emission to be sensitive to surface temperature (Figure 3(b)).

The above requirement for planetary radiation generally applies to planets in (quasi-)energy balance, irrespective of the atmospheric amounts and composition. Although additional opacity sources would increase the opacities at some wavelengths and would change the overall spectra, the planet in quasi-energy balance is still expected to emit high thermal radiation from the relatively transparent wavelengths.

Figure 4(a) shows the evolutionary paths of surface temperature and pressure (the ${p}_{{\rm s}}-{T}_{{\rm s}}$ path) during solidification for various initial inventories of water. The atmospheric pressure increases at early stages due to the high degassing rate. For an initial water inventory larger than 0.2–0.5 M_EO, the quasi-energy balance is attained at some particular surface temperature. The ${p}_{{\rm s}}-{T}_{{\rm s}}$ paths in the quasi-energy balance states are approximately independent of the initial water endowment, as long as the planets solidify during the main sequence of the host star.

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For a larger initial inventory of water, the planet reaches quasi-energy balance at a higher surface temperature and solidification requires a longer period of time. In particular, for the case in which the quasi-energy balance has been achieved during the course of solidification, the lifetime of the magma ocean is well approximated by the time required for total loss of the initial water inventory (Figure 4(b)). If the planet is endowed with water equivalent to more than 16 M_EO, the molten state would exist for over 10 Gyr, which is comparable to the lifetime of the G-type host star. In this case, the surface temperature gradually increases with time in accordance with the luminosity enhancement of the star.

Direct detection requires that the planets have a sufficiently high planet-to-star contrast. As described in the previous section, the time evolution of the thermal emission spectra is closely related to the atmospheric evolution during solidification, which strongly depends on the orbital distance from the star. Figure 5 shows the contrast evolution of the thermal radiation in near-IR windows for various orbital distances in the case of an initial water inventory of 5 M_EO. Although the total contrast consists of the thermal radiation and the reflected stellar light, in most cases, the thermal radiation is dominant in the near-IR band, especially for Type II planets (Section 4.3).

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The contrast evolution of Type I planets is different from that of Type II planets. For Type I planets, which are located beyond a_cr, the range of the planet-to-star contrast is approximately independent of the semimajor axis. The planet-to-star contrast is initially as high as 10⁻⁸–10⁻⁶, and then decreases on a common timescale of less than $\sim {10}^{6}$ years (Figure 6). This is because the magma-ocean lifetime of Type I planets is determined by the minimum heat flux, which is approximately independent of the orbital distance. In the case of a smaller inventory of water, the planet-to-star contrast would increase as the atmosphere becomes thinner, whereas the timescale for decreasing planet-to-star contrast becomes shorter. In the case of a larger inventory of water, the thermal radiation reaches the radiation limit earlier. This prolongs the duration of the magma ocean by a factor of at most 2.

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The contrasts of Type II planets first decrease with time as well, but then cease to decrease during the course of solidification (Figure 5). The minimum value of the contrast of Type II planets is higher than that of Type I planets. Since the planets are in quasi-radiative energy balance, they spend most of the solidification period at the minimum contrast level in this nominal case (Figure 6). The minimum contrast level is insensitive to the initial water inventory. As long as the planet solidifies on a shorter timescale than the main sequence lifetime of the host star, the minimum contrast level has the highest likelihood of detection. For the case in which the magma ocean is sustained for as long as the main sequence of the star, the planet-to-star contrast gradually increases with time from the minimum contrast level, because the planet must emit a larger planetary radiation from the near-IR band in order to maintain quasi-energy balance in accordance with the luminosity enhancement.

The minimum contrast increases for the planet closest to the star, because this planet receives a larger net stellar flux and so must emit larger planetary radiation. Consequently, planets closer to a star are more favorable for direct imaging in terms of spectral contrast. On the other hand, the planet closest to the star also receives a stronger stellar XUV flux, which fuels hydrodynamic escape. Since the solidification time of Type II planets is approximately equal to the loss time of water, the occurrence rate of molten terrestrial planets is expected to be higher at orbital distances closer to a_cr.

As shown in Figures 2, 5 and 6, the Type II molten planets maintain relatively high contrasts in the near-IR band. The Type II planets with $a\lt 0.5$ AU are favorable targets for future ground-based direct imaging, because the contrasts of the K_s and L bands are above the levels of $\sim {10}^{-8}$ and $\sim {10}^{-7}$, respectively, for the entire life of the molten state. The duration with the contrast $\gt {10}^{-8}$ is approximated with the overall solidification time, which becomes longer with the initial water inventory, as shown in Section 4.2. Although these planets have smaller angular distances, in the near future, a 30–40 m ground-based telescope will resolve them within tens of parsecs. On the other hand, the near-IR contrast of the Type I planets declines below $\sim {10}^{-8}$ and $\sim {10}^{-7}$ on a short timescale of $\sim {10}^{5}$ years for the K_s and L bands, respectively. Young stars should be targeted when searching for Type I planets.

The thermal radiation in the near-IR band overwhelms the reflected stellar radiation for Type II planets. The contrast of the emission in the visible bands rapidly decreases to the 10⁻¹⁰ level by a few 10⁴ and $\sim {10}^{5}$ years for the Type I and Type II planets, respectively (see Figure 2). Therefore, with respect to the visible band, the reflected light should be considered in order to discuss the total contrast of the planets. These issues are discussed in Section 4.3.

For the Type II planets, the lifetime of the magma ocean strongly depends on how much water is acquired during planet formation. This suggests the possibility of placing constraints on the initial water inventory based on the observed occurrence rate of hot molten planets. Figure 7 shows an orbital region where molten Earth-sized planets potentially exist as a function of stellar age. The predicted lifetime of the magma-covered planets shows a sharp transition around a_cr. The Type I planets during the magma ocean period have less probability of being detected, whereas the probability of detecting molten Type II planets is greatly enhanced with initial water inventory.

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The calculated lifetimes of Type I and Type II planets allow us to estimate an expected number of G-type stars which host Earth-sized molten planets. For G-type stars whose distance from the solar system is less than the distance d, the expected number of G-type stars with molten planets ${N}_{{\rm D}}(d)$ can be estimated as follows,

$$\begin{eqnarray}&&{N}_{{\rm D}}(d)\sim {N}_{{\rm G}}(d)\cdot {P}_{\mathrm{EG}}\cdot {\displaystyle \frac{{\rm \Delta }{t}_{\mathrm{mol}}^{\mathrm{tot}}}{{\bar{\tau }}_{{\rm G}}(d)}},\end{eqnarray} \tag{ 7 }$$

where ${N}_{{\rm G}}(d)$ and ${\bar{\tau }}_{{\rm G}}(d)$ are, respectively, the number and average age of G-type stars within the distance d from the solar system, P_EG is the fraction of G-type stars with terrestrial planets in a range of planetary orbital distances considered, ${\rm \Delta }{t}_{\mathrm{mol}}^{\mathrm{tot}}$ is the total duration during which terrestrial planets are in a molten state. Let us consider the case when d is 10 pc. The total number of G-type stars is then around 30. The average age of G-type stars is assumed to be 5 Gyr. Using Kepler data, Petigura et al. (2013) reported that the fraction of GK-type stars which host Earth-sized planets is ∼0.12 for orbital distances of 0.05–0.42 AU, and ∼0.086 in the habitable zone. Here, we adopt P_EG of 0.1 for both Type I and Type II planets.

Type-I planets solidify on a timescale of the order of 10⁶ years, which shorter than typical time intervals of giant impacts at a late stage of formation. In this case, the total duration would be expressed as a product of an average number of giant impacts n_GI and the duration of a molten state after one collision ${\rm \Delta }{t}_{\mathrm{mol}}$. According to N-body simulations, Earth-like planets would experience 10–20 giant impacts during formation (Kokubo & Genda 2010; Stewart & Leinhardt 2012). We adopt n_GI of 10. Using a typical duration of a magma ocean of 2 × 10⁶ years, we obtain ∼0.01 for ${N}_{{\rm D}}(d=10\;\mathrm{pc})$ in the Type I orbital region. This value is common for Type I planets which formed with initial water inventory comparable to or exceeding the total amount of water of the modern Earth. The potentially favorable target for molten Type I planets would be young and other spectral type stars such as Fomalhaut. On the other hand, Type II planets have a wide range of solidification time after single impact, which strongly depends on the initial amount of water. In the extreme case that the solidification time exceeds the period of giant impact phase, which probably lasts until the stellar age of 0.1 Gyr, the total duration of a molten state is given by the duration of a molten state after single collision. In the Type II orbital region, the initial amount of water thus greatly affects the probability of detecting molten planets. Given that ${\rm \Delta }{t}_{\mathrm{mol}}^{\mathrm{tot}}$ does not exceed 10 Gyr, ${N}_{{\rm D}}(d=10\;\mathrm{pc})$ ranges from ∼0.01 to 6 in the case of Type II planets, considering the initial inventory of water of from one-tenth to several tens of times the Earth's ocean mass.

Here, we address the uncertainties in parameters which affect the loss time of water, the timing of the last giant impact, the heating efficiency and the EUV flux from the G-type host star. The giant impact phase is triggered by the dispersal of nebula gas and lasts until a stellar age of approximately 0.1 Gyr. Therefore, the last giant impact would occur between the stellar ages of 0.01 and 0.1 Gyr. The heating efficiency η in Equation (2) depends on the structure and details of the energy budget in the upper atmosphere and remains poorly constrained. The estimated values have a large spread and typically range from 0.1 to 0.5 (e.g., Watson et al. 1981; Chassefière 1996; Kulikov et al. 2007; Shematovich et al. 2014). The saturation level of XUV flux and its ending time ${\tau }_{\mathrm{sat}}$ remain quite uncertain. If we simply assume that the XUV-to-bolometric-luminosity ratio scales as the X-ray-to-bolometric-luminosity ratio, which is observationally estimated to be ${10}^{-3.2\pm 0.3}$ during the saturation phase for one solar-mass star (Pizzolato et al. 2003), the variation of the XUV saturation level is nothing more than a factor of 3 and ${\tau }_{\mathrm{sat}}$ ranges from 0.06 to 0.13 Gyr. Although the XUV flux could exhibit greater fluctuation for pre-main sequence stars, its effects might not be so significant because of its short duration relative to the subsequent evolution. Although no observational data are available at wavelengths between 360 and 921 $\AA$ due to strong absorption by the interstellar medium, the total XUV flux would not have profound uncertainty compared with that of the other parameters, because the estimated energy flux in this wavelength range is at most 10% to 20% (Ribas et al. 2005).

Although our model cannot predict the exact inventory of primordial water due to the numerous uncertainties described above, estimating the order of magnitude of the exact inventory of primordial water would be still useful. Our model suggests that, if the initial bulk content of water exceeds approximately 1 wt%, molten planets would be present over the main sequence of the host G-type star (Figure 7). Since the critical distance, which separates the Type I and Type II planets, originates from the presence of the radiation limit of steam atmospheres, detection of a critical distance would offer a diagnostic test to determine whether water is generally one of the major volatile species of extrasolar terrestrial planets. In the orbital region of Type II planets, the detection of hot signatures in the planetary spectrum will provide a lower limit for initial water inventory, which is sufficient to sustain the molten state for a given stellar age, whereas no detection will constrain its upper limit, so that the planet will lose most of its water and become solidified.

The actual spectrum from the planet is obtained as the superposition of its thermal emission and reflected stellar light. The thermal radiation generally overwhelms the reflected stellar radiation during the highly self-luminous stage. Figure 8 shows the planet-to-star contrast of the superimposed spectra of the planet in quasi-energy balance. According to the Lambert assumption, the planet–star contrast ${C}_{\mathrm{ps}}^{\mathrm{ref}}(\lambda )$ of the reflected light is given as follows:

$$\begin{eqnarray}&&{C}_{\mathrm{ps}}^{\mathrm{ref}}(\lambda )={\displaystyle \frac{2\phi (\beta )}{3}}\alpha (\lambda ){\left({\displaystyle \frac{{R}_{\mathrm{pl}}}{a}}\right)}^{2}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\phi (\beta )={\displaystyle \frac{\left[\mathrm{sin}\beta +(\pi -\beta )\mathrm{cos}\beta \right]}{\pi }},\end{eqnarray} \tag{ 9 }$$

where α is the spectral Bond albedo, and $\phi (\beta )$ is the Lambert phase function with respect to the phase angle $\beta =\angle (\mathrm{star}-\mathrm{planet}-\mathrm{observer})$. The overall superposed spectrum in the quasi-energy balance state is V-shaped with a bottom around 1 μm, except for the phase angle very close to the new phase (Figure 8). The dominant component is thermal emission at wavelengths longer than ∼1 μm and is reflected stellar light at the shorter wavelengths. This originates from the difference in the peak wavelength of the Planck function between the surface temperature and the stellar effective temperature. The peak wavelengths are approximately 1.4 μm for 2000 K and 500 nm for 5800 K.

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For $\beta =90^\circ$, the contrast of the reflection can be estimated as follows:

$$\begin{eqnarray}&&{C}_{\mathrm{ps}}^{\mathrm{ref}}\approx 3\times {10}^{-10}\left({\displaystyle \frac{\alpha (\lambda )}{1}}\right){\left({\displaystyle \frac{{R}_{\mathrm{pl}}}{{R}_{\oplus }}}\right)}^{2}{\left({\displaystyle \frac{a}{1\;\mathrm{AU}}}\right)}^{-2},\end{eqnarray} \tag{ 10 }$$

where ${R}_{\oplus }$ is the Earth radius. Because the albedo in the U–B band is ∼1, as shown in Figure 1(b) and the reflection dominates in this band, we roughly expect contrasts of $\sim {10}^{-10}$ and $\sim {10}^{-9}$ for the Type I and Type II planets, respectively. These contrasts are within the scope of the future space direct imaging survey. The contrasts decline steeply from the visible wavelengths to the near-IR wavelengths, which reflects the deep atmosphere of the planets.

The reflected light is bluer than that of the host star, whereas the thermal emission in the near-IR is redder than that of the host star. The middle panel in Figure 8 shows the color index of each band to the V band relative to the host star as a function of phase angle. As the phase angle increases, the color becomes redder for all bands because the contribution of the thermal emission increases. The color indexes of the I and R bands remain bluer than that of the host star, whereas the color indexes of the other spectral bands becomes redder than that of the host star. The color indexes could vary by approximately 6 for the J, H, K_s, and L bands during a complete cycle of orbital motion. These color changes are comparable to the color differences between rocky planets, such as Earth and Mars, and gaseous planets such as Uranus and Neptune.

Following Lupu et al. (2014), we consider the color–magnitude diagram of the molten planets. Figure 9 shows the evolutionary paths of the thermal emission in the H – K color and absolute H magnitude diagram. The molten planets are as faint as the planets in the solar system, but are distinctly redder in color. This is because the thermal emission, which is much redder than the stellar light, dominates the planetary spectrum for molten planets, whereas the color of the planets in the solar system mostly reflects the reflected light. We also estimate the effects of stellar light reflected by clouds on the H – K color by assuming that the cloud coverage is 0.5. In this case, as the thermal emission becomes fainter, the reflected light becomes more dominant in the near-IR spectrum. The reflected light could determine the color at the later stage for the planet at 1 AU, whereas the color is still much redder than that of the star during solidification of the planet at 0.7 AU, including the quasi-energy balance state. A self-consistent evolutionary calculation, including cloud formation, will be the subject of future research to examine actual color variations that occur during the course of solidification.

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