The Mass Fractionation of Helium in the Escaping Atmosphere of HD 209458b¹¹1Revised Manuscript on June, 7, 2022

Our model includes the equations of number density, velocity, and pressure for H, He, H⁺, He⁺ and e^-. In an ionized atmosphere composed of atoms and ions, photoelectrons produced by stellar XUV irradiation distribute their energy via collision with thermal electrons. Ions obtain energy from thermal electrons via Coulomb collisions, while hydrogen atoms exchange energy with thermal electrons and ions by elastic and inelastic collisions. Thus, the heating by stellar XUV radiation is included in the equation of electron pressure. The equations of multi-fluid HD models are listed as follows:

where subscripts $n$, $s$, and $e$ denote neutral, ion, and electron fluid, respectively. In Equations (1)-(6) $n$ is the number density, u is the velocity, $p$ is pressure. $S_{t}$ and $L_{t}$ are the production and loss of particles. $m$ is the particle mass, and $C$ is the artificial collision rate coefficient that is expressed from the collision frequency. $A$ is the particle mass in atomic mass units ($m_{amu}$), $a_{ext}$ is the acceleration of fluid produced by external forces, and $\gamma$ is the adiabatic index. On the left side of the Equation (3), the electric force $-\nabla p_{es}$ is treated as part of ion fluid pressure. Following Dong et al. (2017) $p_{es}$ can be expressed as

which is the consequence of quasi-neutrality when neglecting the momentum of the electron.

Including the equation of the electron pressure resulted in a numerical difficulty in using the Riemann solver. We thus modified the process of the calculation by solving the equations of atoms, ions, and electrons separately, which will be discussed in Section 2.2.

Our model also includes the impact of the ionization electrons, charge exchanges between neutral and ions, recombination of ions and electrons , and other collisions. All chemical reactions included in this model are listed in Table 1. The collision coefficients $C_{st}$ for two species $s$ and $t$ we used in Equations (1)-(6) are derived from collision frequency $\nu_{st}$ (Schunk & Nagy, 1980). Following the momentum of conservation, the $C_{st}$ is defined as

where $m_{amu}$ is the atomic mass unit, $s$ and $t$ are the two collision species. Then, we can get $\nu_{st}$ and $\nu_{ts}$ from $C_{st}$

It is convenient to include the collisional terms in momentum and energy equations. The collision coefficients are listed in Table 2. The ion-neutral collisions of the same element (.i.e, collisions between H and H⁺ and He and He⁺) are resonant collisions while the collisions between different elements (.i.e, collisions between H and He⁺ and He and H⁺) are nonresonant collisions.

In our model, the planetary and stellar tidal accelerations are included in the term of external forces. In this case $a_{ext}$ can be expressed as

where G is the gravitational constant, $M_{p}$ is the planet mass, $r$ is the altitude from center of the planet, $M_{*}$ is the stellar mass, $a$ is the semi-major axis. In Equation (10) the first and second terms denote the acceleration produced by the planet and star, respectively.

In the process of photoionization, electrons obtain most of the energy of photoelectrons, which will be transferred to other species through the collision processes. Therefore, the heating of the atmosphere is only included in the electron pressure equation. Here we use an average heating efficiency $\eta$ to represent the different heating efficiency of photons with different frequencies. The heating term can be written as

and

Where $\tau_{\nu}$ is optical depth, $\sigma_{\nu}^{H}$ and $\sigma_{\nu}^{He}$ are optical cross sections for H and He separately which are given by Ricotti et al. (2002) and $F_{\nu}$ is spectrum of irradiated stellar flux. We also included the $Ly\ \alpha$ cooling of hydrogen (Murray-Clay et al., 2009)

Because the mass of the electron is much smaller than other atoms and ions, we obtained the electron number density $n_{e}$ and velocity $u_{e}$ by the condition of quasi-neutrality (Tóth et al., 2012),

where $q_{s}$ is the electric charge of ion, $e$ is the electric charge of electron, $u_{+}$ is the mean velocity of all ion fluids and, $u_{e}$ is the velocity of the electron. Because the magnetic field is not included in this model, the effect of electric current is not considered.

Equations (1)-(6) are solved using PLUTO (Mignone et al., 2007), and a third-order total variation diminishing (TVD) Runge-Kutta schemes method is implemented into PLUTO for the time integration. The integration of time from $t^{n}$ to $t^{n+1}$ is expressed as

where U is $n,\textbf{u}$ and $p$ for H, He, H⁺, He⁺ and e^-. The residual term H is the sum of flux terms (or advection terms) $\textbf{L}_{F}$ and source terms $\textbf{L}_{C}$ namely,

Note that all right-hand side terms in Equations 1-6 are included in $\textbf{L}_{C}$. We applied the semi-implicit method of García Muñoz (2007) to calculate H,

in which the Jacobian matrix $\frac{\delta\textbf{L}_{C}}{\delta\textbf{U}}$ is treated numerically (Tóth et al., 2012).

There are three kinds of fluids in our model, i.e., neutral fluids for H and He, ion fluids for H⁺ and He⁺ and e^- fluid. For the neutral fluids, the Riemann solver of PLUTO can solve them directly as in Tóth et al. (2012). For the ion fluids, the additional electric field force $-\nabla p_{es}$ impedes the use of the Riemann solver. The flux terms of the ion fluids can be expressed as

The terms in Equation (21) represent number density $n_{s}$, velocity $u_{s}$ and pressure $p_{s}$ flux terms of the ion fluids, respectively. In fact, the flux terms in the form

can be solved directly by the Riemann solver. By comparing the Equations (22) and (21), we cannot solve Equation (21) with the Riemann solver due to the additional term $p_{es}$. However, the Riemann solver can be applied if $p$ in Equation (22) is replaced by $p_{s}+p_{es}$. Under this condition, the flux terms are expressed as

Equation (23) can be solved by the Riemann solver and we can obtain the flux terms of $n_{s}$, $u_{s}$ and $p_{s}+p_{es}$. The integration of time in the Riemann solver is explicit, thus the flux terms of $n_{s}$ and $u_{s}$ in the time tⁿ⁺¹ only depend on the values at tⁿ. Thus, in the process of solving Equation (23) the correct values of $n_{s}$ and $u_{s}$ are obtained. At the same time, we used an incorrect flux term for $p_{s}$ in the pressure equation so that the value of $p_{s}$ obtained via the flux term of $p_{s}+p_{es}$ is discarded. In order to calculate $p_{s}$, we introduce a new density $n^{\prime}$ to replace the $n$ in Equation (22). The flux terms are modified again as

which are solved by the Riemann solver again. As expressed above, due to the explicit characteristic in the time integration we can obtain the $p_{s}$ in the process of solving Equation (24), and an artificial value of $n^{\prime}$ in Equation (24) is permitted in mathematics because $p_{s}$ is independent of $n^{\prime}$ when the Equation (24) is solved.

Nevertheless, a reasonable value for $n^{\prime}$ is necessary. The escape of ions is driven by both the pressure of ions and electrons. Thus they can be divided into two parts of which one is driven by the pressure of ions and the corresponding number density is denoted by $n^{\prime}$. Another is driven by the electron pressure with number density $n_{es}$. Finally, the number density is the sum of the two parts, namely, $n_{s}=n^{\prime}+n_{es}$. In fact, the escaping fluxes driven by $p_{s}$ and $p_{es}$ are proportional to the ratios of $p_{s}/p_{es}$, namely, $n^{\prime}/n_{es}=p_{s}/p_{es}$. Then the number density $n^{\prime}$ of the fluid driven by pressure $p_{s}$ is given by

Only the pressure of the electron remains to be calculated due to the assumption of quasi-neutrality. As manipulated above, we constructed the flux terms of $\rho_{e}^{\prime}$ and $\textbf{u}_{e}$ where $\textbf{u}_{e}$ is obtained by Equation (15). For the density of electrons, an arbitrary value is acceptable. Here we applied a similar method to define the $\rho_{e}^{\prime}$,

which is equal to the summation of all mass density driven by the electric field force $-\nabla p_{es}$. Finally, the electron pressure is obtained by

The verification of this Riemann split method is shown in Appendix A. The Riemann solver that we used in our work is HLL which is implemented in PLUTO. In fact, one can use different Riemann solvers for different requirements.

In order to test the collision of H and He, we ran a multi-fluid isothermal (setting polytropic exponent $\gamma=1.0001$) model and compared the result with the analytical result given by Hunten et al. (1987). Considering that this analytical result is suitable for an earth-like planet, we simulated the escape of a planet with the mass and radius. For simplicity, we run a two-fluid model constituted of neutral fluids of hydrogen and helium. The mass-loss rates of hydrogen fluid are modified by varying the temperature of the atmosphere. Following Hunten et al. (1987) the heavy species can be dragged by lighter particles through collisions if the mass of the heavy species is smaller than the crossover mass. The crossover mass and the escaping flux of helium are

and

where $m_{c}$ is crossover mass, $m_{H}$ and $m_{He}$ are the mass of H and He, $k_{B}$ is Boltzman constant, T is temperature, $F_{H}$ and $F_{He}$ are the escaping number density flux of H and He and $b$ is binary diffusion coefficient (Mason & Marrero, 1970). Meanwhile, $X_{H}=n^{0}_{H}/(n^{0}_{H}+n^{0}_{He})$ and $X_{He}=n^{0}_{He}/(n^{0}_{H}+n^{0}_{He})$ are the mixing ratio of H and He at the lower boundary. The escaping flux of He and crossover mass can be obtained after the escaping flux of H is given. It is clear from (29) that the He can be dragged by H only if $m_{c}>m_{He}$ or vice versa and the ratios of $F_{He}/F_{H}$ are functions of $m_{c}$. At the same time, we used our model to calculate the escaping fluxes of H and He, namely:

where $\dot{M}_{H,He}$ are the mass-loss rate of hydrogen and helium, $m_{(H,He)}$ are the masses of H and He, $R_{p}$ is the radius of the planet. Meanwhile, the $F_{He}/F_{H}$ of our model is given by

As shown in the left panel of Figure 1, the ratio of escape flux of helium to that of hydrogen in our model is consistent with that predicted by Hunten et al. (1987) quite well, which indicates the correctness of our model. The mass-loss rates increase roughly exonentially with the increase in temperature. The mixing ratio of helium can keep constant with the increase of the temperature as shown by the middle panel of Figure 1. Meanwhile, the velocities of helium are dragged higher by the fluid of hydrogen with the increasing hydrogen mass-loss rate as shown in the right panel of Figure 1. When the temperature drops to 380 K, the mixing ratio of helium drops dramatically. In this case, the escaping helium decreases to a fairly low level simultaneously. There are even some slight negative velocities at lower altitudes, probably due to the limitations of the code accuracy. In this extreme case, the $\frac{F_{He}}{F_{H}}$ predicted by Hunten et al. (1987)(blue dashed line in left panel of Figure 1) is zero, which is consistent with that of our model ($7\times 10^{-4}$).

We choose HD 209458b as a benchmark to explore the fractionation of He. The parameters of HD 209458 system are set as follows: the mass of the planet is $0.69M_{J}$, the radius $1.36R_{J}$, the mass of the star $1.119\ M_{\odot}$, the semi-major axes $a=0.04707$AU. They are the same as that of Lampón et al. (2020).

Our 1D multi-fluid hydrodynamic model has 1024 grids from 1-10 $R_{p}$. The stretched ratio $dr_{n}/dr_{n-1}$ is 1.0062 where $dr_{n}$ is the length of the $n$th grid. Thus, most grid points are accumulated in the lower regions of the atmosphere. The values at the outer boundary are extrapolated linearly. The values at the lower boundary are different for different species. For the neutral fluid, the number densities of H and He are fixed as $1.0\times 10^{13}\ cm^{-3}$ and $8.46\times 10^{11}\ cm^{-3}$. The ratio of $n_{H}/n_{He}$ is 0.0846, which is almost consistent with that of the Sun (Asplund et al., 2009). The number densities of ion fluid at the lower boundary are imposed to equal the value of the first grid point. We assumed that the temperatures at the lower boundary are the same for all fluids due to sufficient collisions, which is 1500 K and approximately equivalent to the effective temperature of HD 209458b. In such conditions, the pressure at the lower boundary is about 2 $\mu bar$, which is close to the homopause pressure given by García Muñoz (2007). The velocities of all fluids are given by equivalent extrapolation.

The spectral type of HD 209458 is similar to that of the Sun; therefore, we set a fiducial XUV flux (hereafter $F_{0}$) which is the same as that of the quiet Sun. The corresponding spectrum is obtained from Guo & Ben-Jaffel (2016)($F_{0}=2.62\ erg\ cm^{2}\ s$ at 1 AU in the wavelength range of 1Å-912Å), which is divided into 53 bins (1-912 Å). In fact, the activity of the star can result in a variation in the XUV flux. In addition, the heating efficient $\eta$ in our model is a free parameter. Therefore, we calculated many cases by varying the XUV flux and heating efficiency. These models are composed of three groups. In group A, $\eta$ is fixed at 0.2 which is close to that given by Shematovich et al. (2014) and $F_{XUV}$ ranges from 0.60-4.0 times fiducial value. In group B, $F_{XUV}$ is fixed at 1.80 $F_{0}$ and $\eta$ ranges from 0.10-0.5. Group C is similar to group B except the XUV flux is the fiducial value (Table 3).

Figure 2 shows the structures of the atmosphere for some cases of group A. The XUV fluxes are 0.60, 1.0, and 1.8 times F₀. The profiles of the number density are shown in the panels (a1)-(c1). We can see that the number density of He decreases faster than that of H with the increase of the altitude, which is due to the fact that the spectra shorter than 504$\AA$ occupied most of the XUV flux so that the atomic helium encounters a stronger ionization than H. It is clear from the panels (a2),(b2) and (c2) that the temperature first rises to about 8000 K and decreases dramatically. The decrease in temperature is caused mainly by the expansion of the atmosphere. The temperatures of all species have similar profiles until about 3 R_p. Beyond 3 R_p the decoupling is evident for atomic helium. The velocities of all species couple well for the case of 4 times F₀. A velocity decoupling between atomic helium and other species happens with the decrease of XUV flux. For the case of F_XUV=0.60F₀ the behavior of atomic helium is prominently different from other species. The velocity of atomic helium is even negative in the regions of $1.2R_{p}<r<3R_{p}$. In higher altitudes, the decoupling disappears due to the tidal forces of the host star. This negative velocity is not expected. We show the evolution of velocity with the integral time in Figure 3. It should be noticed that the velocities are almost unchanged after 200 integral time units, and the $|d\textbf{u}|$ converges to one-thousandth of the velocities after 400 integral time units. Thus, the calculation results of both velocity and $|d\textbf{u}|/\textbf{u}$ converged to the stable status. A possible reason for the negative velocity is that the neutral helium fluid is in the quasi-hydrostatic status rather than hydrodynamic status, and such calculations are not suitable for PLUTO. This case needs to be explored further.

The escape of helium plays a key role in simulating the absorption of helium triplets. As presented by Lampón et al. (2020), a low mixing ratio of helium (about 2%, which is fairly lower than the value of the Sun) is required in order to explain the absorption of helium triplet. Our simulations show that the mixing ratios of all models are lower than the solar value (0.086) (Table 3). Our results also indicate that a mixing ratio of 0.039 for He/H in the upper atmosphere of HD 209458b is the best-fit model for the absorption of helium triplets (Section 3.4). A difference from the results of Lampón et al. (2020) is that the low He/H ratio in the upper atmosphere is not set artificially but as a result of the mass fractionation of helium.

We compared the $F_{He}/F_{H}$ of our model with those of Hunten et al. (1987). The escaping flux of $F_{He}$ predicted by Hunten et al. (1987) can be obtained via Equations (28) and (29). It is convenient to apply the equations of Hunten et al. (1987). However, the temperature in a non-isothermal atmosphere needs to be defined firstly. In fact, the equations of Hunten et al. (1987) describe the friction of particles due to binary diffusion. Thus, here we used a pressure averaged temperature (Koskinen et al., 2013), which is calculated from the lower boundary to the altitude where the number density of atomic helium is equal to that of helium ions. The average temperature is defined as

where $T(p)=\frac{m_{H}T_{H}+m_{He}T_{He}}{m_{H}+m_{He}}$ is reduced temperature, $p_{1}$ is the pressure at the lower boundary, and $p_{2}$ is the pressure at the location of n(He)=n(He⁺). In fact, the altitude of n(He)=n(He⁺) denotes the location above which the dominant particles in the atmosphere are ions. At the same time, we also calculated the cases that the temperature is equal to that of the lower boundary.

The crossover mass increases with the increase of the XUV flux and heating efficiency (Colume 6 of Table 3). As a consequence, the fractionation of helium appears an opposite trend (panel (a) and (b) in Figure 4). One can see that the escaping fluxes of helium are dependent on temperature. The $F_{He}/F_{H}$ predicted by Equation (29) is higher than ours if T=$\overline{T_{p}}$ ($\overline{T_{p}}\sim$ 3000-4000 K, purple points in panels (a) and (b) of Figure 4) while those of T=1500 K are more consistent with ours (blue points in panels (a) and (b) in Figure 4). In fact, the collision rates between H and He increase with the increase in temperature as shown in Table 2. Thus, a higher temperature causes a larger crossover mass, and thus higher escaping flux for helium. The critical value of $F_{XUV}$ that is defined in the condition of $m_{c}=m_{He}$ is 0.87$F_{0}$ in group A. The equations of Hunten et al. (1987) indicate that the $F_{He}$ will decrease to zero if $m_{c}<m_{He}$. In fact, the condition of $m_{c}<m_{He}$ is controlled mainly by the escaping flux of hydrogen. Thus, one should notice that an uncoupled regime will occur if the mass-loss rates of hydrogen (or XUV radiation) is lower than the critical value. We also found that the $F_{He}/F_{H}$ ratios ultimately depend on the mass loss rates of hydrogen. The panel (c) of Figure 4 shows that the effect of the fractionation of helium increases with the decrease of $\dot{M}_{H}$ and the uncoupled regime occurs when the mass loss rate of hydrogen is smaller than $\sim 1.0\times 10^{10}$ (or the XUV radiation is lower than 1$F_{0}$), which is consistent with the trend predicted by Equation (29). In addition, there are some differences between ours and the analytical results of Zahnle & Kasting (1986) and Hunten et al. (1987) (pink and blue lines) in the uncoupled regime though both results are well consistent in the coupled regime. It should be noted that the degree of coupling is affected by other collisional processes in an atmosphere of partial ionization. Thus, we defined a general coupling factor,

where $F_{t}$ is the escape flux of particles $t$, $g_{r}$ is the acceleration of gravity and $X_{t}$ is the mixing ratio of particles $t$. This parameter can be further expressed as $\mathscr{C}_{st}(r)=m_{c}/m_{amu}$, which defines a relative crossover mass for general collisions and demonstrates how well the minor particles $s$ coupled with main particles $t$. In the equation of Hunten et al. (1987), the flux of helium will be cut off if $m_{c}<m_{s}$, which is similar to the behavior of $\mathscr{C}_{st}$. The regime of $\mathscr{C}_{st}<A_{s}$ means a decoupling of $s$ from $t$ . On the other hand, a large $\mathscr{C}_{st}$ reflects the tight coupling between two species. The panel (d) of Figure 4 shows the coupling factor $\mathscr{C}_{st}$ for some cases of group A. For the uncoupled cases (orange and red lines), most values of $\mathscr{C}_{H-He}$ of H-He collision are smaller than $A_{H_{e}}$, which means that the effect of the binary diffusion of hydrogen to helium at the bottom of the atmosphere is weak such that only minor helium can be dragged out of the gravitational well of the planet. We also notice that the $\mathscr{C}_{H^{+}-He^{+}}$ of the H⁺-He⁺ collision is small (orange and red dashed lines), but it is not negligible compared to $\mathscr{C}_{H-He}$. We thus further calculated the case of $\eta=0.2$ and $F_{XUV}=0.75F_{0}$ with an artificially low Coulomb collision rate ($10^{-3}$ of the original value) and found that the mass-loss rate of helium drop to almost zero as predicted by Zahnle & Kasting (1986) and Hunten et al. (1987). However, one should note that for the $F_{He}/F_{H}$ , a value of 0.2% predicted by our model is quite close to zero although the result of neglecting the Coulomb collision is closer. In addition, for the coupled cases (blue and purple lines of panel (d)), the coupling factors $\mathscr{C}_{H-He}$ are larger than the relative atomic mass of helium from the bottom of the atmosphere to 2 $R_{p}$, which means that the helium can be driven to high altitudes even if the Coulomb collision is neglected. In this situation, the results of Hunten et al. (1987) are almost consistent with ours. We can conclude from the above results that the coupling of particles in the lower atmosphere could be more important than those in the higher altitudes. However, we also emphasized that the conclusion should be explored with more examples.

For the case of $\eta=0.2$ and $F_{XUV}=0.60F_{0}$, the velocities of neutral helium are negative in the regions of 1.2-3.2 $R_{p}$ (see the panel (a3) of Figure 2) and the crossover mass calculated from our model is also smaller than the mass of atomic helium (Table 3). As shown in panel (a) of Figure 5, the escape of atomic helium seems to be separated by the regions of negative velocity. In the negative velocity region, the atomic helium can not escape because of gravity deceleration. In fact, in the lower altitudes of the atmosphere, the atomic helium can be in the state of quasi-hydrostatic equilibrium. Due to the negative velocities, no atomic helium can be transported to higher altitudes. Such behavior does not have to appear on the ions of helium. In the bottom of the atmosphere, the number of He⁺ increases rapidly so that they can be transported to higher locations. The atomic helium can be produced by the recombination of He⁺ and e^- and charge exchange between H and He⁺ as shown in panel (b) of Figure 5 so that there is a slight escape of atomic helium in the regions of higher than 3 $R_{p}$. In fact, there is a small circulation between He and He⁺. Since the charge exchange rate between H and He⁺ is greater than photoionization rate of He in the regions of 1.6-3.2$R_{p}$, some atomic helium will be produced (see panel (b) of Figure 5). As a consequence of negative velocities, the helium will flow toward the planet so that the inflowed helium will be ionized again and form a mass circulation from 1.2-3.2 $R_{p}$. This inflowed He leads to an accumulation of He in the region lower than 3.2$R_{p}$, while this accumulation also enhances the ionization of helium and leads to a superfluous He⁺ escape. As a consequence of the balance of these two antagonistic processes, the total mass-loss rate of He and He⁺ keeps a decent conservation as shown in panel (a) of Figure 5. Though $\mathscr{C}_{He+,H+}$ will grow to $10^{4}$ (which is fairly large compared to $A_{He^{+}}$) as shown in the panel (d) of Figure 4, the fractionation of helium seems to be affected little by the high coupling between He⁺ and H⁺ in high altitudes and keeps a low value due to insufficient collision between He and H in lower altitude.

We used our results to fit the excess absorption of He in 10830 Å. The absorption is produced by the metastable He(2³S) in the atmosphere of HD 209458b. To fit the observations, we calculated the population of He(2³S) and the transmission spectrum of He 10830 Åaccording to Yan et al. (2022). The population of He(2³S) was obtained by using a nonlocal thermodynamic model, which solves the rate equilibrium of He(2³S). In the model, the near- and far-ultraviolet spectrum that ionizes the He(2³S) population is taken from the stellar atmosphere model of Castelli &Kurucz (2003). When modeling the transmission spectrum, we assume that the geometry of the stellar line is in plane-parallel and that of the planetary atmosphere is in 1D spherical. Besides, the stellar He 10830 Å intensity is assumed to be limb-darkened with an Eddington approximation. In fact, an one-dimensional hydrodynamic model combined with three-dimensional population calculation is self-consistent to some extent because the contribution to the He 10830 Å mainly comes from the lower-middle atmospheres. In this situation, the assumption of 1D hydrodynamics causes relatively reliable results because the asymmetry of the atmosphere increases with the altitudes. For more details, please refer to the paper of Yan et al. (2022). In order to fit the transmission spectrum of He10830, the spectral line is shifted -1.8km/s (-0.065 Å) toward the blue side because our 1D model can not simulate the blue shift of about -1.8km/s detected by Alonso-Floriano et al. (2019).

We compared the profile of the spectral line of our models with that of observation. We used $\chi^{2}$ method to match the observed spectral line. The best-fit parameters are determined from the results of group A-C. One can see that the best-fit case is $\eta=0.20$ and $F_{XUV}=1.80F_{0}$ (Figure 6). In this case, the mass loss rate is $2.01\times 10^{10}g\ s^{-1}$ and $F_{He}/F_{H}$ is $0.039$. Though around the -13km/s (-0.47 Å) blue shift of transfer spectrum, we didn’t fit the observation signal. While in Yan et al. (2022), the model fitting with transfer observation of Wasp-52 b didn’t show this difference. The signal is hard to explain even with 3D MHD model of Khodachenko et al. (2021a). We thought this signal needs a more detailed physical modeling and research. Our results show that the depth of absorption increases with the increase of $F_{XUV}$ roughly linearly. However, for the case with $\eta=0.20$ and $F_{XUV}=0.60F_{0}$ the escaping flux of helium is about $10^{-4}F_{H}$ so that the absorption depth at the line center is less than 0.001, which means that it is difficult to detect the observable signals in such atmosphere.

An increase of $\eta$ will cause more hydrogen to escape, thus also more escaping helium (see Table 3). In this situation, our model predicts that a deeper absorption at the line center of 10830Å occurs if the $F_{XUV}$ is fixed, but the heating efficiency is higher. As shown in panel (a) of Figure 7, however, the absorption at the line center approaches saturation with the increase of $\eta$. The reason is that He $2^{3}$S is produced mainly via the recombination reaction of He⁺ and e^- (Oklopčić & Hirata, 2018). In this situation, the production of He⁺ is mainly controlled by the ionization of XUV irradiation because a higher heating efficiency does not increase the ionization degree but only slightly modifies the recombination rate via the variations of the temperature profiles. This explains why the number densities of He increase with the increase of $\eta$ (panel (c) of Figure 7), but the number densities of H${}_{e}^{+}$ and He $2^{3}$S are insensitive to $\eta$ in the regime of higher $\eta$ (panels (b) and (d) of Figure 7).