The Escape of the Hydrogen-rich Atmosphere of Exoplanets: Mass-loss Rates and the Absorption of Stellar Lyα

The observations show that the radii of most exoplanets are smaller than 2 R_J (R_J is the radius of Jupiter; http://exoplanet.eu). Thus, we confined the planets to the range of radii less than 2 R_J. In addition, for exoplanets with high mass, their atmospheres can be compact and the escape of species is relatively difficult. We thus set the upper limit of the mass to 2 M_J (M_J is the mass of Jupiter). Finally, we further selected the sample by their gravitational potentials. The calculations of Salz et al. (2016b) found that the hydrodynamic escape of the atmosphere is difficult for exoplanets with high gravitational potentials (>4 × 10¹³ erg g⁻¹), which means that in order for hydrodynamic escape to occur, the exoplanets with high mass should have a large radius (or relatively low mean densities). Thus, the gravitational potential of the sample planets is smaller than 4 × 10¹³ erg g⁻¹.

According to Sheets & Deming (2014), planets with radii 0.0885–0.177 R_J are super-Earths, 0.177–0.354 R_J are mini Neptunes, and 0.354–0.531 R_J are super-Neptunes. In the investigation, we also classified the planets with different sizes. Specifically, the planets with radius smaller than 0.2 R_J are Earth-like planets, 0.2–0.6 R_J are Neptune-like planets, and 0.6–1.0 R_J are Saturn-like planets. Finally, the planets with radius larger than 1.0 R_J are Jupiter-like planets.

Koskinen et al. (2007) suggested that exoplanets with a separation smaller than 0.15 au will produce significant mass loss. Hence, we selected those planets with separation less than 0.1 au. The separations are in the range of 0.01–0.09 au for Earth-like and Neptune-like planets, 0.04–0.07 au for Saturn-like planets, and 0.02–0.08 au for Jupiter-like planets. The host stellar masses of each system are in the range of 0.08–1.105 M_⊙ (M_⊙ is the mass of the Sun) for Earth-like, 0.15–1.223 M_⊙ for Neptune-like, 0.816–1.3 M_⊙ for Saturn-like, and 0.8–1.46 M_⊙ for Jupiter-like planets. The stellar mass in some Earth-like and Neptune-like systems is relatively small and these systems account for a small proportion of each group.

Based on the conditions above, we selected 90 exoplanets from real systems (http://exoplanet.eu and https://exoplanetarchive.ipac.caltech.edu/), and some artificially made planets will be added to the real planets. For the sake of diversity, we investigated planets from Earth size to Jupiter size. We emphasize that it is not the goal of this paper to predict mass-loss rates and Lyα absorptions of real planets because accurate characterizations are difficult, due to the lack of some important physical inputs of the host stars. For example, the XUV spectral energy distributions cannot be easily obtained because they cannot be well determined from observations due to the obscuration by interstellar matter (ISM). In addition, the compositions of Earth-like planets can be different from the hydrogen-dominated atmosphere of giant planets. Here we assumed that they hold a hydrogen-rich envelope. The hydrogen in their atmospheres can originate from the dissociation of H₂O (Kasting & Pollack 1983; Guo 2019). Furthermore, the hydrogen-rich atmosphere can appear in the early phase of Earth-like planets due to the process of outgassing or accretion (Elkins-Tanton & Seager 2008). The observational signals in Lyα for Earth-like planets have been discussed by dos Santos et al. (2019) and Kislyakova et al. (2019).

The left panel of Figure 1 shows the mass and radius of the sample planets (in this panel, log ρ expresses the logarithm of the planetary mean density). One can see that our sample covers various exoplanets ranging from Jupiter-like planets to Earth-like planets. We ignored Earth-like planets with a large radius because such planets might be composed of gas rather than rock. As discovered by Lammer et al. (2003, 2009), the mass-loss rates are controlled essentially by the XUV flux and the mean density of the planets. Furthermore, the Lyα absorption by the planetary atmosphere can be estimated after the mass-loss rates are obtained. Therefore, we plot the sample in the F_XUV–$\rho$ diagram (Figure 1, right panel; for more details on F_XUV, see Section 2.2). The dark blue asterisks represent the planets in the radius–mass diagram. As we can see, most planets are located at the center of the F_XUV–$\rho$ diagram, and their mean density is similar to or lower than that of Jupiter. The planets with high density are located on the right side of the panel. In order to investigate the influence of F_XUV and $\rho$ on the mass-loss rates and the absorption of stellar Lyα by the planetary atmosphere on a larger F_XUV–ρ scale, we added many artificially made planets, which are represented by the light blue diamonds. Most of the artificial ones are made by changing the input F_XUV of the planets, and the rest of those planets are made by changing the planetary mass and radius. Most of the mock planets made by changing planetary masses and radii are Earth-like planets or Neptune-like planets, as the initial number of planets selected in this range is less than that of Jupiter-like planets. By modifying the masses and radii of those Earth-like to Neptune-like planets, the sample planets uniformly cover the F_XUV–$\rho$ diagram. In total, about 450 planets are included.

Zoom In Zoom Out Reset image size

In our solar system, the mean density of the rocky planets is higher than that of the gaseous planets. The distributions of the mean densities of our sample are similar to those of the solar system. In our sample, the mean densities of Jupiter-like and Saturn-like planets are in the range of 0.08–1.4 g cm⁻³. For the Neptune-like planets, the mean densities are in the range of 0.3–2.65 g cm⁻³. The smallest density of an Earth-like planet is 0.9 g cm⁻³, while the largest density is 8.2 g cm⁻³. Such a distribution means that the densities of most Jupiter-like planets are lower than those of Earth-like planets. Furthermore, the distributions of the gravitational potentials are different for different planets. For Earth-like and Neptune-like planets, their gravitational potentials are smaller than 3 × 10¹² erg g⁻¹. However, the gravitational potentials of Saturn-like and Jupiter-like planets cover a large range (1 × 10¹²–3 × 10¹³ erg g⁻¹) and are generally greater than those of planets with smaller sizes, although there is an overlap around 3 × 10¹² erg g⁻¹. The highest flux we set is 4 × 10⁵ erg cm⁻² s⁻¹, which corresponds to a planet orbiting the Sun at 0.003 au. The lowest value of the XUV flux is 2 × 10² erg cm⁻² s⁻¹, below which the occurrence of hydrodynamic escape can be difficult. Our calculations cover a large range in the F_XUV–ρ diagram. One can see from the F_XUV–ρ diagram that in any given level of fluxes, there are many planets with different mean densities. By investigating the dependence of the mass-loss rates on the XUV flux and mean density, we can further conclude the general trend of Lyα absorption and explore what factors determine the absorption levels.

As discussed above, the mass-loss rates are related to the properties of the planet and the XUV irradiation. It is difficult to obtain the accurate spectra of a star because the ISM can obscure the EUV emission of stars. Sanz-Forcada et al. (2011) fitted the observation for late-F to early-M dwarfs and found that the XUV luminosity (integrated XUV flux received at the planetary orbits) can be depicted by the empirical relation:

$$\begin{eqnarray}&&\mathrm{log}{L}_{\mathrm{euv}}=(29.12\pm 0.11)-(1.24\pm 0.15)\mathrm{log}t,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}{L}_{x}=\left\{\begin{array}{cc}6.3\times {10}^{-4}{L}_{\mathrm{bol}} & (t\leqslant {t}_{i})\\ 1.89\times {10}^{28}{t}^{-1.55} & (t\gt {t}_{i})\end{array}\right.,\end{eqnarray} \tag{ 2 }$$

with t_i = 2.03 × 10²⁰ ${{L}_{\mathrm{bol}}}^{-0.65}$, where t is the stellar age in Gyr, and L_euv and L_x (erg s⁻¹) are the luminosity in the EUV band and X-ray band, respectively.

This means that the integrated X-ray and EUV fluxes can be obtained if the age of the star is known. In this paper, we only choose the systems with given ages, and the ages are obtained from http://exoplanet.eu. Even though the ages are usually given with relatively large uncertainties, an accurate age is not necessary because the purpose of the present work is to explore the response of the mass-loss rates and the Lyα absorption depth to XUV flux. Thus, the age only provides an XUV flux estimation. In order to simulate the emission of the XUV spectra of the stellar corona, we used the software XSPEC-APEC in which the wavelength domain of 1–912 Å was used. The free parameters in APEC are metallicity and coronal temperature. We first set the metallicity to solar value (Asplund et al. 2009), so the spectra only depend on the coronal temperature. To obtain the XUV spectra of all stars, we calculated lots of spectra by using XSPEC-APEC. By comparing the theoretical and empirical ratios of L_x to L_EUV in a given age, the optimum coronal temperatures of the stars can be defined. Finally, the XUV spectra of all stars are obtained by this method. We inspected the profiles of those XUV spectra and found that the β indexes of all spectra are ∼0.9 (β = F_{1–400 Å}/F_{1–912 Å}; for details, see Guo & Ben-Jaffel 2016). Thus, the influence of the profiles of the XUV spectra is slight (we evaluated the variations of the mass-loss rates produced by the different profiles and found that the change is smaller than 5%.) We emphasize that the XUV fluxes obtained by Equations (1) and (2) can result in a deviation for the real planets due to the uncertainties of the age and the models. However, the empirical spectra can express the essential feature of those stars. Thus, as a theoretical exploration, the spectra can be used to describe the response of the atmosphere of the planet to the different levels of F_XUV. In the premise, we will explore how the mass-loss rates and the absorptions of Lyα are affected by the XUV flux of the stars and the properties of the planets.

We used the 1D hydrodynamic models (Yang & Guo 2018) to simulate the atmospheric structures of our selected planet sample. Compared to the early models of Guo (2013) and Guo & Ben-Jaffel (2016), there are two improvements in the model of Yang & Guo (2018). First, the former use the solar EUV (100–912 Å) as stellar radiation spectra, while in the latter model developed by Yang & Guo (2018), the radiation spectra expand to XUV (1–912 Å). The photoionization cross sections in X-ray are smaller than those in EUV, so the heating by X-rays is not remarkable compared with that by the EUV. However, the photoionization produced by X-ray can be important for heavy elements. Second, the former ones only include hydrogen, helium, and electrons, but the latter one also includes heavier elements such as C, N, O, and Si. To be specific, the latter model includes 18 kinds of particles, among which are seven kinds of neutral particles and 10 kinds of ions and electrons. In such a condition, the photochemical reactions include photoionization, photodissociation, impact ionization, recombination, charge exchange, and other important reactions (Table 1). For more details of the hydrodynamic model, the reader can refer to the papers above.

Table 1.  The Coefficients of the Chemical Reactions

H2 + hν $\to$ ${{\rm{H}}}_{2}^{+}$ + e		Guo & Ben-Jaffel (2016)
H2 + hν $\to$ H+ + H + e		Guo & Ben-Jaffel (2016)
H + hν $\to$ H+ + e		Ricotti et al. (2002)
He + hν $\to$ He+ + e		Ricotti et al. (2002)
C + hν $\to$ C+ + e		Verner et al. (1996)
N + hν $\to$ N+ + e		Verner et al. (1996)
O + hν $\to$ O+ + e		Verner et al. (1996)
Si + hν $\to$ Si+ + e		Verner et al. (1996)
Si+ + hν $\to$ Si2+ + e		Verner et al. (1996)
H2 + M $\to$ H + H + M	1.5 × 10−9e−48,000/T	Baulch et al. (1992)
H + H + M $\to$ H2 + M	8.0 × 10−33(300/T)0.6	Ham et al. (1970)
${{\rm{H}}}_{2}^{+}$ + H2 $\to$ ${{\rm{H}}}_{3}^{+}$ + H	2.0 × 10 −9	Theard & Huntress (1974)
${{\rm{H}}}_{3}^{+}$ + H $\to$ ${{\rm{H}}}_{2}^{+}$ + H2	2.0 × 10−9	Yelle (2004)
${{\rm{H}}}_{2}^{+}$ + H $\to$ H+ + H2	6.4 × 10−10	Karpas et al. (1979)
H+ + H2(v ≥ 4) $\to$ ${{\rm{H}}}_{2}^{+}$ + H	1.0 × 10−9e−21,900/T	Yelle (2004)
He+ + H2 $\to$ HeH+ + H	4.2 × 10−13	Schauer et al. (1989)
He+ + H2 $\to$ H+ + H + He	8.8 × 10−14	Schauer et al. (1989)
HeH+ + H2 $\to$ ${{\rm{H}}}_{3}^{+}$ + He	1.5 × 10−9	Bohme et al. (1980)
HeH+ + H $\to$ ${{\rm{H}}}_{2}^{+}$ + He	9.1 × 10−10	Karpas et al. (1979)
H+ + e $\to$ H + hν	4.0 × 10−12(300/T)0.64	Storey & Hummer (1995)
He+ + e $\to$ He + hν	4.6 × 10−12(300/T)0.64	Storey & Hummer (1995)
${{\rm{H}}}_{2}^{+}$ + e $\to$ H + H	2.3 × 10−8(300/T)0.4	Auerbach et al. (1977)
${{\rm{H}}}_{3}^{+}$ + e $\to$ H2 + H	2.9 × 10−8(300/T)0.65	Sundstrom et al. (1994)
${{\rm{H}}}_{3}^{+}$ + e $\to$ H + H + H	8.6 × 10−8(300/T)0.65	Datz et al. (1995)
HeH+ + e $\to$ He + H	1.0 × 10−8(300/T)0.6	Yousif & Mitchell (1989)
H + e $\to$ H+ + e + e	2.91 × 10−8 $\left(\tfrac{1}{0.232+U}\right)$U0.39 exp(−U), U = 13.6/EeeV	Voronov (1997)
He + e $\to$ He+ + e + e	1.75 × 10−8 $\left(\tfrac{1}{0.180+U}\right)$U0.35 exp(−U), U = 24.6/EeeV	Voronov (1997)
H + He+ $\to$ H+ + He	1.25 × 10−15(300/T)−0.25	Glover & Jappsen (2007)
H+ + He $\to$ H + He+	1.75 × 10−11(300/T)0.75 exp(−128000/T)	Glover & Jappsen (2007)
O + e $\to$ O+ + e + e	3.59 × 10−8 $\left(\tfrac{1}{0.073+U}\right)$U0.34 exp(−U), U = 13.6/EeeV	Voronov (1997)
C + e $\to$ C+ + e + e	6.85 × 10−8 $\left(\tfrac{1}{0.193+U}\right)$U0.25 exp(−U), U = 11.3/EeeV	Voronov (1997)
O+ + e $\to$ O + hν	3.25 × 10−12(300/T)0.66	Woodall et al. (2007)
C+ + e $\to$ C + hν	4.67 × 10−12(300/T)0.60	Woodall et al. (2007)
C+ + H $\to$ C + H+	6.30 × 10−17(300/T)−1.96 exp(−170000/T)	Stancil et al. (1998)
C + H+ $\to$ C+ + H	1.31 × 10−15(300/T)−0.213	Stancil et al. (1998)
C + He+ $\to$ C+ + He	2.50 × 10−15(300/T)−1.597	Glover & Jappsen (2007)
O+ + H $\to$ O + H+	5.66 × 10−10(300/T)−0.36 exp(8.6/T)	Woodall et al. (2007)
O + H+ $\to$ O+ + H	7.31 × 10−10(300/T)−0.23 exp(−226.0/T)	Woodall et al. (2007)
N + e $\to$ N++ e + e	4.82 × 10−8 $\left(\tfrac{1}{0.0652+U}\right)$U0.42 exp(−U), U = 14.5/EeeV	Voronov (1997)
N+ + e $\to$ N + hν	3.46 × 10−12(300/T)0.608	Aldrovandi & Pequignot (1973)
Si + e $\to$ Si+ + e + e	1.88 × 10−7 $\left(\tfrac{1+\sqrt{U}}{0.376+U}\right)$U0.25 exp(−U), U = 8.2/EeeV	Voronov (1997)
Si+ + e $\to$ Si + hν	4.85 × 10−12(300/T)0.60	Aldrovandi & Pequignot (1973)
Si+ + e $\to$ ${\mathrm{Si}}_{2}^{+}$ + e + e	6.43 × 10−8 $\left(\tfrac{1+\sqrt{U}}{0.632+U}\right)$U0.25 exp(−U), U = 16.4/EeeV	Voronov (1997)
${\mathrm{Si}}_{2}^{+}$ + e $\to$ Si+ + hν	1.57 × 10−11(300/T)0.786	Aldrovandi & Pequignot (1973)
H+ + Si $\to$ H + Si+	7.41 × 10−11(300/T)−0.848	Glover & Jappsen (2007)
He+ +Si $\to$ He + Si+	3.30 × 10−9	Woodall et al. (2007)
C+ + Si $\to$ C + Si+	2.10 × 10−9	Woodall et al. (2007)
H + ${\mathrm{Si}}_{2}^{+}$ $\to$ H+ + Si+	2.20 × 10−9(300/T)−0.24	Kingdon & Ferland (1996)
H+ + Si+ $\to$ H + ${\mathrm{Si}}_{2}^{+}$	7.37 × 10−10(300/T)−0.24	Kingdon & Ferland (1996)

Download table as:  ASCIITypeset image

In the simulations, the metallicities of planets are consistent with those of their host stars. For those planets with unknown metallicities, we used the solar metallicity. In order to express the average of the energy received by the planet over the whole surface, the XUV fluxes are divided by a factor of 4. We choose the outer boundaries to be the host stars' radii for our sample. For the systems whose host stars' radii far exceed the planetary radii (such as an Earth-like planet), the calculated outer boundaries are set to 15 times the planetary radii. In order to cover the surface of the host stars, the results are extrapolated to the radii of their host stars.

In order to calculate the absorption of stellar Lyα by the atmosphere of the planets, the radiative transfer equations are as follows:

$$\begin{eqnarray}&&{F}_{\mathrm{out}}={F}_{\mathrm{in}}{e}^{-\tau },\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\tau ={\int }_{{Z}_{0}}^{Z}n\sigma {dl}.\end{eqnarray} \tag{ 4 }$$

Finally, the absorption depth by the planet and its atmosphere can be expressed as

$$\begin{eqnarray}&&\mathrm{Absorption}\,\mathrm{depth}=\displaystyle \frac{{F}_{\mathrm{in}}-{F}_{\mathrm{out}}}{{F}_{\mathrm{in}}}.\end{eqnarray} \tag{ 5 }$$

In Equations (3)–(5), F_in is the incident intrinsic stellar flux (note that it is different from stellar F_XUV.) F_out is the emergent flux due to the occultation by planets and the absorption by their surrounding atmospheres along the ray path. In the radiation transfer equations, the factor τ represents the optical depth, which is dependent on the atmospheric structure and directly pertains to the particles' number density n and cross section σ.

Lyα absorption occurs in a hydrogen atom when an electron absorbs 10.2 ev energy and jumps from the n = 1 level to the n = 2 level, where n is the quantum number. The cross section of Lyα absorption can be evaluated via

$$\begin{eqnarray}&&{\sigma }_{12}=\displaystyle \frac{\pi {e}^{2}{f}_{12}{\phi }_{\nu }}{{m}_{e}c}\end{eqnarray} \tag{ 6 }$$

in cm², where e is the elementary charge of an electron, m_e is the electronic mass, f₁₂ is the oscillator strength, which is 0.4162 at 1215.67 Å (Mihalas 1978), and ϕ_ν is the Voigt profile, which combines Doppler and Lorentz profiles. The Voigt profile is related to the Voigt function H(a, u) through the following equations (Rybicki & Lightman 2004):

$$\begin{eqnarray}&&{\phi }_{\nu }={\left({\rm{\Delta }}{\nu }_{D}\right)}^{-1}{\pi }^{-\displaystyle \frac{1}{2}}H(a,u)\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&H(a,u)\equiv \displaystyle \frac{a}{\pi }{\int }_{-\infty }^{+\infty }\displaystyle \frac{{e}^{-{y}^{2}}{dy}}{{a}^{2}+{\left(u-y\right)}^{2}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&a\equiv \displaystyle \frac{{\rm{\Gamma }}}{4\pi {\rm{\Delta }}{\nu }_{D}},u\equiv \displaystyle \frac{\nu -{\nu }_{0}}{{\rm{\Delta }}{\nu }_{D}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\rm{\Gamma }}=\gamma +2{\nu }_{\mathrm{col}},{\rm{\Delta }}{\nu }_{D}=\displaystyle \frac{{\nu }_{0}}{c}\sqrt{\displaystyle \frac{2{kT}}{{m}_{a}}},\end{eqnarray} \tag{ 10 }$$

where a is the damping parameter, u is the frequency offset, ν₀ is the line center frequency, Δν_D (assuming no turbulence) is the Doppler width, and Γ is the transition rate. Here we set the damping parameter a to be 4.699 × 10⁻⁴.

3.1.1. The Dependence of $\dot{M}$ on F_XUV and $\rho$

We investigated 442 systems in which the number of Jupiter-like, Saturn-like, Neptune-like, and Earth-like planets is 151, 78, 104, and 109, respectively. The mass-loss rate predicted by the hydrodynamic model is defined as

$$\begin{eqnarray}&&\dot{M}=4\pi {r}^{2}\rho \upsilon .\end{eqnarray} \tag{ 11 }$$

Here, ρ and υ are the density and the velocity of the escaping particles.

In addition, the absorbed XUV irradiation is converted into heat and does work on the particles of the atmosphere to overcome the gravitational potential and supply the particles' kinetic and thermal energies. According to Erkaev et al. (2007) and Lammer et al. (2009), the mass-loss rates can be expressed as

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{\pi {F}_{\mathrm{XUV}}\eta {R}_{\mathrm{XUV}}^{2}}{{\rm{\Delta }}\phi +\tfrac{{\upsilon }_{{R}_{L}}^{2}-{\upsilon }_{{R}_{0}}^{2}}{2}+{c}_{p}({T}_{{R}_{L}}-{T}_{{R}_{0}})},\end{eqnarray} \tag{ 12 }$$

where R_XUV is the XUV absorption radius (at which the mean optical depth is 1). η is the heating efficiency, which is the ratio of the gas heating energy to the entire XUV energy input. Δϕ, ${\upsilon }_{{R}_{L}}^{2}$ − ${\upsilon }_{{R}_{0}}^{2}$ and c_p(${T}_{{R}_{L}}-{T}_{{R}_{0}}$) are the variations (from the lower atmosphere boundary to the Roche lobe) of the gravitational potential, and kinetic and thermal energies, respectively. R₀ and R_L are the locations of the atmosphere lower boundary and the Roche lobe. υ and T are the velocity and temperature of the particles. c_p is the specific heat at a constant pressure per unit mass. In the energy-limited loss approximation (Lammer et al. 2003; Erkaev et al. 2007; Lammer et al. 2009), the kinetic and thermal energies of the escaping particles are far smaller than the gravitational potential. Thus, the energy-limited equation is expressed as

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{\pi {F}_{{\rm{X}}{\rm{U}}{\rm{V}}}\eta {R}_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}}{{\rm{\Delta }}\phi }.\end{eqnarray} \tag{ 13 }$$

When the effect of stellar tidal force is included, ${\rm{\Delta }}\phi =\tfrac{{{GM}}_{P}}{{R}_{P}}K(\xi )$ (Erkaev et al. 2007). Therefore, the energy-limited equation can be further expressed as

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{3{\beta }_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}\eta {F}_{{\rm{X}}{\rm{U}}{\rm{V}}}}{4\,{\rm{K}}(\xi )G\rho },\end{eqnarray} \tag{ 14 }$$

where β_XUV is the ratio of the XUV absorption radius to the planetary radius. (Here we use the subscript "XUV" to distinguish it from the spectral index β.) ρ is the planetary mean density, and G is the gravitational constant. K(ξ) is the potential energy reduction factor due to the stellar tidal forces (Erkaev et al. 2007),

$$\begin{eqnarray}&&K(\xi )=1-\displaystyle \frac{3}{2\xi }+\displaystyle \frac{1}{2{\xi }^{3}}\end{eqnarray} \tag{ 15 }$$

with

$$\begin{eqnarray}&&\xi ={\left(\displaystyle \frac{{Mp}}{3{M}_{\star }}\right)}^{\tfrac{1}{3}}\displaystyle \frac{a}{{Rp}},\end{eqnarray} \tag{ 16 }$$

where M_p and M_⋆ are the planetary and stellar masses, a is the separation between the planet and its host star, and R_p is the planetary radius. By comparing Equations (12) and (14), one finds that the energy-limited formula can be revised by the terms of the kinetic and thermal energies. In this paper, Equation (12) is defined as the revised energy-limited formula. We will discuss the effect of the kinetic and thermal energies in Section 3.1.2.

The energy-limited equation hints that the mass-loss rates are a function of the XUV flux and the mean density. Figure 2 shows the dependence of the mass-loss rate on the properties of the exoplanet and the XUV flux. First, it is clear that the mass-loss rates increase with the increase of the XUV irradiation as expected. The mass-loss rates of Jupiter-like planets are on the order of magnitude of 10⁹–10¹⁰ g s⁻¹ when the integrated XUV flux is about 200 erg cm⁻² s⁻¹. In the case of F_XUV = 4 × 10⁵ erg cm⁻² s⁻¹, the mass-loss rates of Jupiter-like planets increase by a factor of ∼1000, which is on the order of magnitude of 10¹³ g s⁻¹. Such a trend is also found for the Saturn-like, Neptune-like, and Earth-like planets. For example, the mass-loss rates of the Earth-like planets vary almost linearly with F_XUV from 10⁸ to 10¹¹ g s⁻¹. Second, we note that the mass-loss rates of the planets decrease with the decrease of their sizes if the integrated flux is given. For the Saturn-like planets, the mass-loss rates decline by a factor of a few compared with those of Jupiter-like planets. However, for those smaller planets, the mass-loss rates can decrease by an order of magnitude or more. Such a behavior reflects the fact that the atmospheric escape of the planet is inversely proportional to the mean density.

Zoom In Zoom Out Reset image size

To clarify, we selected some planets of our sample to calculate the mass-loss rates on the different XUV fluxes. The corresponding value of the fluxes are 200, 400, 1000, 2000, 4000, 2 × 10⁴, 4 × 10⁴, 2 × 10⁵, and 4 × 10⁵ erg cm⁻² s⁻¹. We showed the correlation between the mean density and the mass-loss rate in Figure 3. Note that the real flux in the calculation is divided by a factor of 4 so that the mass-loss rates of a few Jupiter-like planets cannot be calculated in the situation with low XUV flux. The mass-loss rates of those planets are calculated by using the energy-limited equation (the value of ${\beta }_{\mathrm{XUV}}^{2}\eta$ is obtained from our fitting formulae; for details, see Section 4.1). Evidently, the mass-loss rates of the planets with lower density are higher than those of the planets with higher density. Thus, the hydrodynamic simulation confirmed the physical validity of the energy-limited assumption.

Zoom In Zoom Out Reset image size

However, we note that the mass-loss rates of different planets are slightly different even if the densities of the planets are the same. For example, at $\rho$ ≈ 0.28 g cm⁻³, the mass-loss rates of Jupiter-like planets are higher than those of Saturn-like planets although the XUV fluxes are same. Such behaviors also occur at $\rho$ ≈ 1 g cm⁻³ and $\rho$ ≈ 1.8 g cm⁻³. This reflects the fact that the energy available for driving the escape of the atmosphere is different for different planets. There are three factors that can result in the variations of the mass-loss rate. First, the heating efficiencies of different planets are different. Second, the effective areas of energy deposition are different for different planets. Third, the tidal forces of the host stars: the effect of tidal force has been discussed by Erkaev et al. (2007) who found that the tidal forces of stars can enhance the mass-loss rates. We will discuss the first two factors below.

3.1.2. The Heating Efficiency and the XUV Absorption Radius

A main factor in determining the mass-loss rates is the heating fraction of the XUV radiation. The net heating efficiency η is defined as

$$\begin{eqnarray}&&\eta =({H}_{{\rm{h}}{\rm{e}}{\rm{a}}{\rm{t}}}\text{-}{L}_{{\rm{c}}{\rm{o}}{\rm{o}}{\rm{l}}{\rm{i}}{\rm{n}}{\rm{g}}})/{{\rm{\Sigma }}}_{\nu }{F}_{\nu },\end{eqnarray} \tag{ 17 }$$

where F_ν is the input XUV energy at frequency ν, and H_heat and L_cooling are the radiative heating and cooling, respectively. We calculated the heating efficiency of our ∼400 planets and found that the heating efficiencies are insensitive to an individual parameter, but it is dependent on the product of the XUV flux and the gravitational potential (hereafter log(F_XUVGM_p/R_p)). The left panel of Figure 4. shows the heating efficiency with respect to log(F_XUVGM_p/R_p). The triangles represent the planets with gravitational potential lower than 1.5 × 10¹³ erg g⁻¹, and the crosses (plus signs) are planets with gravitational potential higher than 1.5 × 10¹³ erg g⁻¹. For planets with gravitational potential smaller than 1.5 × 10¹³ erg g⁻¹, we can see from this figure that the larger log(F_XUVGM_p/R_p) is, the higher the heating efficiency η. For the planets that concentrate on the range of 14 < log(F_XUVGM_p/R_p) < 16, the heating efficiencies of most planets are lower than 0.3. Only a few planets appear higher heating efficiency. When 16 < log(F_XUVGM_p/R_p) < 18, the heating efficiency rises from about 0.13 to 0.45. For planets with log(F_XUV GM_p/R_p) larger than 18, η varies in the range of 0.3–0.45. The appearance of η is, however, different for the planets with relatively large gravitational potential (the cross in Figure 4). In fact, the gravitational potential makes a separation to η. We can see from the left panel of Figure 4 that the heating efficiency is generally lower for planets with very large gravitational potential compared to those with relatively small gravitational potential. We further showed the dependence of η on the gravitational potential in the right panel of Figure 4. For the planets with relatively small gravitational potential, the heating efficiency can vary a factor of 9 and an increasing trend appears with the increase of gravitational potential. At the same time, the values of η show a transition around ∼1 × 10¹³ erg g⁻¹, above which the heating efficiency decreases with the gravitational potential. Such behavior causes lower mass-loss rates for some Jupiter-like planets. As shown in Figure 2, we found that there are some Jupiter-like planets for which the mass-loss rates deviate from the general trend. They are generally smaller than the mass-loss rates of other Jupiter-like planets and even smaller than those of some Neptune-like and Earth-like planets. The mass-loss rates are related to the mean densities. For such planets, their mean densities are around 1 g cm⁻³. The higher mean densities result in the decrease of a factor of a few for the mass-loss rates compared with those of Jupiter-like planets with lower mean densities. At the same time, these planets all have relatively large gravitational potentials (>1.5 × 10¹³ erg g⁻¹), which lead to relatively small heating efficiencies η (most are in the range of 0.05–0.1). Therefore, for these planets, the mass-loss rates could be about 10 times lower than those of other Jupiter-like planets. We also note that Salz et al. (2016b) found a rapid decrease of the evaporation efficiencies (by assuming a heating efficiency η = 1 in calculating the mass-loss rates predicted by the energy-limited equation, they defined the evaporation efficiency ${\eta }_{\mathrm{eva}}\,=\tfrac{{\dot{M}}_{\mathrm{model}}}{{\dot{M}}_{\mathrm{enengy} \mbox{-} \mathrm{limited}}}$ and found η = 1.2 η_eva) when the gravitational potential is higher than ∼1.3 × 10¹³ erg g⁻¹, which is similar to our results. However, we did not find the linear dependence of η on the gravitational potential when the values of the gravitational potential are lower than ∼1.3 × 10¹³ erg g⁻¹ (Figure 2 of Salz et al. 2016b). Our calculations based on a larger sample suggest that the heating efficiency can vary by a factor of a few even if the gravitational potentials of the planets are same.

Zoom In Zoom Out Reset image size

Furthermore, one needs to know the mean XUV absorption radius R_XUV in order to calculate the mass-loss rates predicted by the energy-limited equation. Here, R_XUV is defined as

$$\begin{eqnarray}&&{R}_{\mathrm{XUV}}=\displaystyle \frac{{\displaystyle \sum }_{\nu }{R}_{\nu }({\tau }_{\nu }=1){F}_{\nu }}{{F}_{\mathrm{XUV}}}.\end{eqnarray} \tag{ 18 }$$

In Equation (16), R_ν(τ_ν = 1) is the radius where the optical depth at frequency ν is unity. F_ν is the XUV flux at frequency ν. We calculated the XUV absorption radius using Equation (16) for all planets. The variations of R_XUV are shown in Figure 5. The upper panel is the R_XUV with respect to the planetary radii. For all the planets, R_XUV is in the range of 1.05–1.7 R_p. We found that the values of R_XUV are related to the the sizes of planets. For the Jupiter-like planets, R_XUV is mainly concentrated in a small range, 1.1–1.2 R_p. The values of R_XUV vary from 1.1 to 1.5 R_p when the sizes of planets are in the range of 0.2–0.8 R_J. For planets with radii less than 0.2 R_J, the R_XUV could reach 1.4–1.7 R_p. Compared to the Earth-like planets, the R_XUV of larger planets such as Jupiter-like planets could decrease by a factor of 1.5.

Zoom In Zoom Out Reset image size

The middle panel is R_XUV with respect to the planetary mass. R_XUV is basically negatively correlated to the planetary mass. For the planets with masses less than 0.1 M_J, R_XUV tends to be larger, especially for those less than 0.01 M_J. When the masses become higher than 0.2 M_J, the R_XUV is mainly in 1.1–1.2 R_p. Both the upper panel and the middle panel suggest that smaller planets such as Earth-like planets tend to have larger R_XUV, which means their XUV absorption radii are relatively far from the planetary surface. The lower panel of Figure 5 expresses how the R_XUV varies with the gravitational potential. It is clear that the absorption radii increase with the decrease of the gravitational potentials. For the planets with a high gravitational potential, the values of R_XUV are in the range of 1.1–1.2 R_p. The increase of R_XUV is dramatic when the gravitational potentials are lower than ∼10¹² erg g⁻¹. This is reasonable because for these planets, the gravitational potentials are so low that the atmospheres can expand to higher altitudes to absorb stellar XUV irradiation.

In Figures 6(a) and (b), we compare the mass-loss rates predicted by our hydrodynamic model (y-axis) and those from Equations (12) and (14) (x-axis). The mass-loss rates from Equations (12) and (14) are calculated using our β_XUV and η. The mass-loss rates in Figure 6(b) are corrected by the kinetic and thermal energies of the escaping atmosphere (Equation (12)). Different symbols represent the mass-loss rates of different types of planets. Before correcting for the kinetic and thermal energies of the escaping atmosphere (see Figure 6(a)), the hydrodynamic mass-loss rates are basically consistent with the energy-limited ones when the mass-loss rates are lower than a "critical value" for different types of planets. For Earth-like and Neptune-like planets, it is about 10¹⁰–10¹¹ g s⁻¹. For Saturn-like and Jupiter-like planets, it is about 10¹¹–10¹² g s⁻¹. By comparing with Figure 2, we found that the corresponding XUV levels of the critical mass-loss rates are about 2 × 10⁴–3 × 10⁴ erg cm⁻² s⁻¹ for Earth-like and Neptune-like planets and ∼4 × 10⁴ erg cm⁻² s⁻¹ for Saturn-like and Jupiter-like planets. Above the XUV radiation level, the hydrodynamic mass-loss rates of the planets are lower than the energy-limited ones, especially for smaller planets such as Earth-like and Neptune-like planets. The deviation of the two kinds of mass-loss rates comes from the kinetic and thermal energies of the escaping atmosphere. To specify, we show the influence of the kinetic and thermal energies in Figure 6(b). After correcting for the kinetic and thermal energies of the escaping atmosphere, the mass-loss rates of our model are generally consistent with those predicted by the revised energy-limited equation (Equation (12)) for all kinds of planets.

Zoom In Zoom Out Reset image size

The uncertainties of the mass-loss rates obtained from the energy-limited formula are mainly attributed to the unknown heating efficiencies and the absorption radii of the XUV irradiation. Many studies have applied fixed heating efficiencies for some types of planets. In our study, the heating efficiencies are different for different planets even if they are same type of planet. At the same time, the absorption radii also vary with the physical parameters of planets (Figure 5). In Figures 6(a) and (b), we used the heating efficiencies and the absorption radii of our hydrodynamic models to remove the uncertainties. However, the deviations are still evident for the energy-limited case. This highlights the importance of kinetic and thermal energies in using the energy-limited equation. As shown in Figures 6(a) and (b), a portion of the energy of the XUV irradiation is converted into the kinetic and thermal energies of the escaping atmosphere. Ignoring the kinetic and thermal energies will result in an overestimation of the mass-loss rates when the energy-limited formula is used. We further show the dependence of the kinetic and thermal energies to the planetary parameters in Figure 6(c). It is clear that the sum of the kinetic and thermal energies increases with the increase of log(F_XUVGM_p/R_p). The increasing trend is obvious for all types of planets, although there is a relatively large spread for Earth-like planets and Jupiter-like planets. Finally, we show the ratios of the kinetic and thermal energies to the gravitational potential in Figure 6(d). The ratios are smaller than unity when log(F_XUVGM_p/R_p) is smaller than 16. With the increase of log(F_XUVGM_p/R_p), the ratios become greater than unity for most Earth-like planets and Neptune-like planets, while the ratios are still smaller than unity for most Jupiter-like planets. The ratios of Saturn-like planets express mixed variations for the case of high log(F_XUVGM_p/R_p). The same trend also appears in Figure 6(a). The mass-loss rates of Jupiter-like planets predicted by the energy-limited formula are more consistent with those of the hydrodynamic model because the gravitational potential is dominant. For smaller planets, the deviation of the mass-loss rates becomes obvious with the increase of the mass-loss rate. Thus, the results obtained from the energy-limited formula should be revised by the kinetic and thermal energies of the escaping atmosphere, especially for those planets with high F_XUVGM_p/R_p.

We calculated the absorption of stellar Lyα using our sample and found that the excess absorption levels of Lyα could be higher if the planets have lower mean densities and relatively higher integrated F_XUV. To specify, Figure 7 shows the absorption depths in the F_XUV–ρ diagram. The x-axis is the planetary mean density and the y-axis is the integrated F_XUV. The absorption depths are divided into three or four levels depending on the absorption depth range. The absorptions of different depths are distinguished by different colored symbols. For the left and right panels of Figure 7, the velocity ranges are [−150, −50]∪[50, 150] km s⁻¹ and [−150, 150] km s⁻¹ from the Lyα line center, respectively. In the [−50, 50] km s⁻¹ range from the Lyα line center, the stellar Lyα could be contaminated by the ISM (dos Santos et al. 2019) and the geocoronal Lyα emission lines, so we exclude the range in the left panel of Figure 7.

Zoom In Zoom Out Reset image size

For the Lyα absorption depth, we can see from Figure 7 that no matter what wavelength range it is, the average absorption levels have a similar distribution trend. The regions of stronger Lyα absorptions are in the upper-left region of the F_XUV–ρ diagram where the planets receive higher irradiation and have lower mean densities. This indicates higher mass-loss rates around those planets. It is shown in the left panel of Figure 7 that for planets with lower mean densities, the decreasing F_XUV can cause the decrease of the absorption levels. For those planets with medium or high mean densities (>1 g cm⁻³), the dependence of Lyα absorption on the XUV flux shows a middle sensibility. Comparing with those planets with lower densities, one can see that there is a mixed region in which the absorption levels are variable over a large range. For instance, in the case of F_XUV = 10⁴ erg cm⁻² s⁻¹, the absorptions produced by the atmosphere vary with the mean densities of the planets from >5% and <1%. Finally, the absorption levels are very low for those planets with high densities (ρ > 1 g cm⁻³) and low XUV fluxes (lower than 10⁴ erg cm⁻² s⁻¹). In the lower-right region, almost all absorptions are lower than 1%. Generally, those planets with higher densities are Earth-like or Neptune-like planets.

It is also clear from the left panel of Figure 7 that the absorptions are dependent on XUV flux. Most absorption levels of the planets are still higher than 1% if the flux is higher than 2 × 10⁴ erg cm² s⁻¹. In the range of 10³–10⁴ erg cm² s⁻¹, the absorptions decrease with the increase of density. For those planets with high density, their absorptions can be a factor of a few smaller than those planets with lower density. Below 10³ erg cm² s⁻¹, only planets with low densities appear to have significant absorptions. Such behaviors can also be found in the right panel of Figure 7. Although the absorption of the line center is included, the distributions of Lyα absorptions reflect the same trend as shown in the left panel of Figure 7. The difference between the two cases is that the absorption levels in the right panel are far larger than those in the left panel because of the strong absorption in the line center. Kislyakova et al. (2019) modeled the in-transit absorption in Lyα for terrestrial planets with nitrogen- and hydrogen-dominated atmospheres under different levels of stellar irradiation. They also found deeper absorption for planets with higher XUV and smaller density, which is consistent with our results.

The absorption levels are proportional to the mass-loss rates. We showed the absorption of [−150, −50]∪[50, 150] km s⁻¹ in Figure 8. It is clear from Figure 8 that higher absorption levels correspond to higher mass-loss rates. For instance, the absorption depths of most planets can attain 1% if the mass-loss rates are higher than 10¹⁰ g s⁻¹. For the planets with mass-loss rates higher than 10¹¹ g s⁻¹, the absorption level can attain 10% or higher. In the cases where the mass-loss rates are lower than 10¹⁰ g s⁻¹, most planets appear to have a low absorption level except for a few planets. Finally, the absorption can be ignored if the mass-loss rates are lower than 10⁹ g s⁻¹.

Zoom In Zoom Out Reset image size

The (revised) energy-limited equation is a convenient way to evaluate the planetary mass-loss rates (Lammer et al. 2003). In fact, Figures 2 and 3 can be explained to a certain extent by the energy-limited assumption. However, it is not easy to evaluate the planetary mass-loss rates by using Equations (12) and (14) because the values of β_XUV and η must be specified. For convenience, one only needs to know the product of ${\beta }_{{\rm{XUV}}}^{2}$ and η in evaluating the mass-loss rates. Therefore, we fitted the parameter ${\beta }_{{\rm{XUV}}}^{2}$η for the planet with gravitational potential smaller than 1.5 × 10¹³ erg g⁻¹. The planets are classified into four categories by their gravitational potentials. We used different functions to fit the values of ${\beta }_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}$η and showed the fit in Figure 9. The fit of ${\beta }_{{\rm{XUV}}}^{2}$η to log(GM_pF_XUV/R_p) can be expressed as

$$\begin{eqnarray*}{\beta }_{\mathrm{XUV}}^{2}\eta =\left\{\begin{array}{ll}7.980(\pm 2.315)-1.091(\pm 0.292)\theta +0.0383(\pm 0.0092){\theta }^{2} & {{GM}}_{p}{F}_{\mathrm{XUV}}/{R}_{p}\lt 1.5\times {10}^{12}\\ -1.901(\pm 0.0537)+0.135(\pm 0.0033)\theta & 1.5\times {10}^{12}\leqslant {{GM}}_{p}{F}_{\mathrm{XUV}}/{R}_{p}\lt 5\times {10}^{12}\\ -0.581(\pm 0.1218)+0.061(\pm 0.0072)\theta & 5\times {10}^{12}\leqslant {{GM}}_{p}{F}_{\mathrm{XUV}}/{R}_{p}\lt 1\times {10}^{13}\\ -1.232(\pm 0.0192)+0.093(\pm 0.0110)\theta & 1\times {10}^{13}\leqslant {{GM}}_{p}{F}_{\mathrm{XUV}}/{R}_{p}\lt 1.5\times {10}^{13},\end{array}\right.\end{eqnarray*}$$

where θ = log(GM_pF_XUV/R_p).

Zoom In Zoom Out Reset image size

When GM_p/R_p*F_XUV is given, the values of ${\beta }_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}$ η decrease with the increase of gravitational potential, except for the third group. We found that the distributions of the ${\beta }_{{\rm{XUV}}}^{2}$ η of the third group are higher than those of adjacent groups. The gravitational potentials of these planets (5 × 10¹² erg g⁻¹–10¹³ erg g⁻¹) are in an intermediate range, but the slope of this line is the smallest. To explain this issue, we checked the dependence of the XUV absorption radius β_XUV and heating efficiency η on the gravitational potential (see Figures 4 and 5). For planets with gravitational potentials smaller than 1.5 × 10¹² erg g⁻¹, β_XUV is in the range of 1.1–1.7. Their values of η vary from 0.05 to 0.45. Thus, ${\beta }_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}$ η covers a broad range. For planets with gravitational potentials higher than 1.5 × 10¹² erg g⁻¹, β_XUV is in a small range (1.03–1.3). However, the heating efficiencies are different for different planets. For the second and fourth groups, the distributions of the heating efficiency are similar to those of the first group. Due to the smaller β_XUV, their ${\beta }_{{\rm{XUV}}}^{2}$ η are smaller than those of the first group. Furthermore, β_XUV will decrease with the increase of gravitational potentials so that the values of the ${\beta }_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}$ η of the fourth group are smaller than those of the second group. However, the planets in the third group (most planets in the third group are Jupiter-like planets) are close to the transitional regions of the heating efficiency (see the right panel of Figure 4), where the heating efficiencies maintain higher values (0.3–0.45). Compared with the second and fourth groups, their XUV absorption radii are similar. Thus, the high heating efficiency of the third group caused the high values of ${\beta }_{{\rm{X}}{\rm{U}}{\rm{V}}}^{2}$ η and the smaller slope of the third line.

In the simulations, the atmospheric outer boundaries of planets are set to be equal to the stellar radii. However, the real boundaries cannot be determined well because the atmosphere of the planet can be constrained to a few or tens planetary radii by the stellar wind. For planets with the size of Jupiter, the total pressure of the stellar wind can be balanced at about a few planetary radii by the ram pressure and the thermal pressure of the planet (Murray-Clay et al. 2009). Thus, the outer boundaries can roughly be denoted by the radius of the host star. However, the calculated outer boundaries of the Earth-like planets could be far larger than their real boundary or magnetosphere, due to the decrease of planetary pressure with the increase of radius. Here we inspected the dependence of the absorption of Lyα on the atmospheric outer boundaries using some planets of our sample. The effect of the outer boundary on the Lyα absorption is shown in Figure 10. In the left panel of Figure 10, we investigated the cases of Jupiter-like and Saturn-like planets. The right panel shows the dependence of the absorption on the outer boundary for Neptune-like and Earth-like planets.

Zoom In Zoom Out Reset image size

As we can see, both absorptions increase with the increasing atmospheric boundaries. For most Jupiter-like planets, the increase is prominent with the increase of outer boundary so that the maximum absorption can reach 30%–50%. For planets with the size of Neptune and Earth, the absorptions produced within 10 R_p are smaller than 10% (even the absorptions of some Earth-like planet are negligible). With the increase of the radii, the absorption of some planets can reach 15%. If the real boundary or magnetosphere of Earth-like planets are smaller than the radii of their host stars, this hints that the absorptions of Earth-like planets in Figure 7 can be overestimated. In addition, the absorptions of some Earth-like planets are almost insensitive to the outer boundary. As shown in the right panel of Figure 10, the increase of a factor of a few in the outer boundary only results in a few percent increase in Lyα absorption. Compared to the case of Jupiter-like planets, it is obvious that detecting the absorption signals of Lyα in Earth-like planets is not easy.

In our work, the mechanism of atmospheric escape is thermal escape due to intense stellar XUV radiation. We account for the photochemical interactions of planetary particles. However, other aspects such as the influence of the interplay between the stellar wind and planetary wind and the impact of planetary magnetism are not taken into consideration. Moreover, this work is based on 1D simulations of hydrodynamic atmospheric escape. For tidally locking planets, the results may lead to some deviation. Therefore, in our future study, we are going to take these factors into account. Furthermore, in this paper, we only focus on the absorption of Lyα by the atmosphere of the planet. It is the first step toward predicting the observable signals because we do not include the influence of charge exchange and the extinction of the ISM. In addition, an accurate XUV spectra is also needed if the observable signals of some special targets need to be known. Finally, the excess absorption depth of stellar Lyα by the planetary atmosphere we predicted here can provide some clues for future observations.