Helium Absorption at 1083 nm from Extended Exoplanet Atmospheres: Dependence on Stellar Radiation

We model the abundance of excited metastable helium in the escaping exoplanet atmosphere using methods very similar to those described in Oklopčić & Hirata (2018). For completeness, here we give a brief overview of the theoretical model.

We assume a spherically symmetric and radially expanding planetary atmosphere, whose radial velocity (v) and density (ρ) profiles, as functions of altitude/radius (r), can be described by an isothermal Parker wind (Parker 1958; Lamers & Cassinelli 1999)² :

$$\begin{eqnarray}&&\displaystyle \frac{v(r)}{{v}_{s}}\exp \left[-\displaystyle \frac{{v}^{2}(r)}{2{v}_{s}^{2}}\right]={\left(\displaystyle \frac{{r}_{s}}{r}\right)}^{2}\exp \left(-\displaystyle \frac{2{r}_{s}}{r}+\displaystyle \frac{3}{2}\right)\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\rho (r)}{{\rho }_{s}}=\exp \left[\displaystyle \frac{2{r}_{s}}{r}-\displaystyle \frac{3}{2}-\displaystyle \frac{{v}^{2}(r)}{2{v}_{s}^{2}}\right].\end{eqnarray} \tag{ 2 }$$

The Parker wind model describes a radial outflow that starts with low velocities close to the planet, gradually accelerates and becomes supersonic at the radius (altitude) called of the sonic point (r_s), which is given by

$$\begin{eqnarray}&&{r}_{s}=\displaystyle \frac{{{GM}}_{\mathrm{pl}}}{2{v}_{s}^{2}},\end{eqnarray} \tag{ 3 }$$

where M_pl is planet's mass, G is the gravitational constant, and v_s is the sound speed in gas with temperature T and mean molecular weight μm_H (with k as the Boltzmann constant):

$$\begin{eqnarray}&&{v}_{s}=\sqrt{\displaystyle \frac{{kT}}{\mu {m}_{H}}}.\end{eqnarray} \tag{ 4 }$$

The gas density at the sonic point (ρ_s) can be calculated from the mass conservation equation

$$\begin{eqnarray}&&\dot{M}=4\pi {r}^{2}\rho (r)v(r),\end{eqnarray} \tag{ 5 }$$

assuming the value of the total mass-loss rate $\dot{M}$.

Our model atmospheres are assumed to be composed of atomic hydrogen and helium in 9:1 number ratio. The initial value of the molecular weight of the atmosphere is set to 0.9 m_H, but is then iteratively adjusted to the altitude-weighted mean value, taking into account the population of free electrons resulting from hydrogen ionization. In addition to atmospheric composition, the main free parameters of the model are the temperature of the thermosphere and the total mass-loss rate.

Assuming a steady-state (i.e., time independent) planetary wind, we calculate level populations of hydrogen and helium atoms for a gas streamline along the planet's terminator. At low densities typical of planetary thermospheres, atoms are not in local thermodynamic equilibrium, and hence we need to explicitly model the processes that affect level populations, such as photoionization, recombination, collisional transitions, and radiative decay.

We numerically solve Equation (13) from Oklopčić & Hirata (2018) to obtain the fraction of hydrogen in the neutral/ionized state and the density of free electrons. Then, we calculate the radial distribution of helium atoms in the singlet (ground) and triplet (excited) state by solving the following set of equations:

$$\begin{eqnarray}\begin{array}{rcl}v\displaystyle \frac{\partial {f}_{1}}{\partial r} & = & (1-{f}_{1}-{f}_{3}){n}_{e}{\alpha }_{1}+{f}_{3}{A}_{31}-{f}_{1}{{\rm{\Phi }}}_{1}{e}^{-{\tau }_{1}}\\ & - & {f}_{1}{n}_{e}{q}_{13a}+{f}_{3}{n}_{e}{q}_{31a}+{f}_{3}{n}_{e}{q}_{31b}+{f}_{3}{n}_{{{\rm{H}}}^{0}}{Q}_{31},\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}\begin{array}{rcl}v\displaystyle \frac{\partial {f}_{3}}{\partial r} & = & (1-{f}_{1}-{f}_{3}){n}_{e}{\alpha }_{3}-{f}_{3}{A}_{31}-{f}_{3}{{\rm{\Phi }}}_{3}{e}^{-{\tau }_{3}}+{f}_{1}{n}_{e}{q}_{13a}\\ & - & {f}_{3}{n}_{e}{q}_{31a}-{f}_{3}{n}_{e}{q}_{31b}-{f}_{3}{n}_{{{\rm{H}}}^{0}}{Q}_{31}.\end{array}\end{eqnarray} \tag{ 7 }$$

Here f₁ and f₃ mark the fractions of helium in the singlet and triplet state (relative to all helium atoms/ions), respectively, v is the radial gas velocity, n_e is the number density of electrons, and α and Φ are the recombination and photoionization rate coefficients, with optical depth τ calculated using flux-averaged cross sections. All these quantities are functions of altitude. Symbols q and Q mark various collision rate coefficients, while A₃₁ is the triplet-to-singlet radiative decay rate. More information on this calculation, including the references for the rate coefficients, can be found in Oklopčić & Hirata (2018).

Photoionization of hydrogen and helium atoms plays a key role in controlling the triplet helium level population. The corresponding photoionization rates depend on the shape and intensity of incident stellar radiation field, particularly at wavelengths shortward of those associated with hydrogen ionization (λ = 912 Å), the ionization of the helium ground state (λ = 504 Å), and the ionization of the excited 2³S helium state (λ = 2600 Å).

In our analysis, we use stellar SEDs for main-sequence stars of different spectral types (see Figure 1) from the following sources:

• 1.  
  spectra of late-type stars (K1 to M5) from the MUSCLES survey (France et al. 2016; Loyd et al. 2016; Youngblood et al. 2016), versions 2.1 (GJ 876, GJ 436, HD 85512) and 2.2 (HD 97658, GJ 667C, HD 40307);
• 2.  
  for G2 type, we use the SORCE solar spectral irradiance data from the LASP Interactive Solar Irradiance Data Center,³ along with scaling relations between the Lyα flux and fluxes in extreme-ultraviolet (EUV) bands from Linsky et al. (2014) to fill the gap in the data between ∼400 and 1150 Å;
• 3.  
  for A7 and A0 types, we use the synthetic spectra for stars of effective temperatures of 7500 and 10,000 K from Fossati et al. (2018). The high-energy end of the spectrum for the A7 spectral type was constructed by taking the solar spectrum and multiplying it by a factor of 3.

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In order to isolate the effects of incident stellar radiation on the expected 1083 nm absorption from an extended planetary atmosphere, we keep all the parameters describing the planet (mass, radius, and orbital separation) and its escaping atmosphere (mass-loss rate, temperature, and composition) fixed.⁴ We consider a planet with properties similar to HAT-P-11b, one of the first exoplanets with an extended atmosphere detected via the helium 1083 nm line (Allart et al. 2018; Mansfield et al. 2018). The planet mass and radius are set to M_pl = 0.0736 M_Jupiter and R_pl = 0.389 R_Jupiter, and the orbital distance to the host star is 0.05254 au (Yee et al. 2018). For the model temperature and mass-loss rate, we choose T = 7300 K and $\dot{M}={10}^{10.6}$ g s⁻¹, which is a combination of parameters that, when irradiated with a K4-type stellar spectrum, produces an absorption signature that is well matched by the 1083 nm observations of HAT-P-11b by Mansfield et al. (2018) and Allart et al. (2018).

We irradiate the model planetary atmosphere with six different stellar spectra shown in Figure 1 (A0, A7, G2, K1, K6, and M3.5) and for each case we calculate the radial distribution of neutral and ionized hydrogen, as well as the distribution of helium atoms in singlet and triplet configurations. Planetary absorption signal in the 1083 nm line directly depends on the fraction of helium atoms in the triplet state, shown in Figure 2. The resulting triplet helium fractions span a wide range of values for different spectral types and show a fairly mild dependence on altitude for any given spectral type. The main conclusion of this analysis is that the triplet helium fraction is higher in late-type stars (K1, K6, and M3.5) than in stars of earlier spectral types (G2, A7, and A0). Although the exact value and the altitude-dependence of the triplet helium fraction depend somewhat on the assumed properties of the planet and its extended atmosphere, the trend shown here—higher triplet fraction in atmospheres around late-type stars—remains unaffected by these changes.

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What makes late-type stars particularly favorable for exciting helium atoms in their planets' atmospheres into the metastable triplet state is the hardness of their spectra, i.e., the relative flux intensity at EUV wavelengths, which are responsible for populating the metastable state (dashed line in Figure 1), and mid-UV wavelengths responsible for depopulating the metastable state (dotted line in Figure 1).

In order to demonstrate that both wavelength bands play a role in controlling the population level of excited helium atoms, we take the spectrum of a K1 star and artificially change its high-energy and low-energy part of the spectrum separately. The results are shown in Figure 3. The upper panels show the effects of changing the high-energy end of the spectrum (λ < 1000 Å) by multiplying it by factors of 0.1, 0.5, 2, and 10. As shown in the upper right panel, the higher the EUV and X-ray (i.e., XUV) flux, the higher the population level of metastable triplet helium. Bottom panels show the effects of changing the spectrum at wavelengths λ > 1000 Å by factors of 0.01, 0.1, 10, and 100. In this case, the population of helium atoms in the triplet state is maximized when the mid-UV flux is minimized. In conclusion, stars with higher levels of XUV flux (due to stellar activity, for example)⁵ and lower levels of flux in the mid-UV part of the spectrum (due to lower effective temperature) are most efficient at exciting helium atoms into the triplet metastable state.

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To provide a physical explanation for this trend, we look at the contribution of each individual term in the helium triplet balance equation (Equation (7)). For the sake of clarity, in Figure 4 we only show the magnitudes of the most significant terms: recombination into the triplet state (first term on the right side of Equation (7)), photoionization of the metastable state (third term), depopulation of the metastable triplet helium state via collisions with electrons (sum of the fifth and sixth terms), and collisions with hydrogen atoms (seventh term). Recombination is the main populating mechanism of triplet helium, while the remaining processes shown in Figure 4 depopulate the metastable triplet helium state.

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In sufficiently dense planetary atmospheres irradiated by hard spectra, such as those of K-type and cooler stars, helium recombination into the triplet state is balanced by collision-induced triplet-to-singlet transitions. Using the temperature dependence of the rate of helium recombination into the triplet state, ${\alpha }_{3}\approx 2\times {10}^{-13}{({10}^{4}/{\rm{T}})}^{0.8}$ cm³ s⁻¹ (Osterbrock & Ferland 2006), Equation (7) can be simplified in the following way:

$$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & \approx & \displaystyle \frac{(1-{f}_{1}){\alpha }_{3}}{{q}_{31a}+{q}_{31b}}=7\times {10}^{-6}{\left(\displaystyle \frac{{10}^{4}{\rm{K}}}{{\rm{T}}}\right)}^{0.8}(1-{f}_{1})\\ & \approx & 7\times {10}^{-6}{\left(\displaystyle \frac{{10}^{4}{\rm{K}}}{{\rm{T}}}\right)}^{0.8}\left(1-{e}^{-{{\rm{\Phi }}}_{1}{e}^{-{\tau }_{1}}r/v}\right).\end{array}\end{eqnarray} \tag{ 8 }$$

For systems in this regime, the triplet helium fraction is almost insensitive to variations in the mid-UV part of the spectrum and increases with increasing EUV flux, which controls the ground-state photoionization rate (Φ₁). This dependence is confirmed by the triplet helium fractions in atmospheres irradiated by K1 or harder spectra shown in Figure 3: green, yellow, and red lines in the bottom right panel practically overlap each other because lowering the mid-UV flux below the K1 level has almost no effect once the system reaches the regime described by Equation (8).

Planetary atmospheres irradiated by spectra softer than K1, such as those around G- and earlier-type stars, or atmospheres in which the density of free electrons is too low for collisions to balance helium recombination, recombination is balanced by direct photoionization of the metastable state, driven by stellar flux at mid-UV wavelengths. In that case, Equation (7) reduces to:

$$\begin{eqnarray}\begin{array}{rcl} & & \ {f}_{3}\approx \displaystyle \frac{(1-{f}_{1}){n}_{e}{\alpha }_{3}}{{{\rm{\Phi }}}_{3}{e}^{-{\tau }_{3}}}\\ & \approx & 2\times {10}^{-13}{\left(\displaystyle \frac{{10}^{4}{\rm{K}}}{{\rm{T}}}\right)}^{0.8}\displaystyle \frac{\left(1-{e}^{-{{\rm{\Phi }}}_{1}{e}^{-{\tau }_{1}}r/v}\right)}{{{\rm{\Phi }}}_{3}{e}^{-{\tau }_{3}}}{n}_{e}\ [{\mathrm{cm}}^{3}\ {{\rm{s}}}^{-1}].\end{array}\end{eqnarray} \tag{ 9 }$$

In this regime, the triplet helium fraction increases with increasing EUV flux (which governs the ground-state photoionization rate, Φ₁) and decreases with increasing mid-UV flux (which governs the photoionization rate of the metastable state, Φ₃). Our calculations of f₃ for planets irradiated by spectra softer than K1, shown in Figure 3 (the yellow and red lines in top panels and blue and purple lines in bottom panels) confirm these trends.

To investigate how the population of excited helium atoms depends on the magnitude of the received stellar flux, we place our model planetary atmosphere on various orbital distances from a star of K1 spectral type, thereby changing the magnitude of the radiation flux received by the planet. As before, we keep all other stellar and planetary parameters fixed. However, changing the planet–star separation and the level of incident flux could affect the planet's mass-loss rate; we later investigate (separately) how varying $\dot{M}$ changes the helium level population.

In Figure 5 we show how the resulting fraction of helium atoms in the triplet state depends on the planet's distance from the star, ranging between 0.01 and 0.11 au. The triplet fraction, especially at lower planetary altitudes, decreases at larger orbital separations. Consequently, we do not expect prominent absorption signals at 1083 nm in exoplanets orbiting main-sequence stars at distances greater than ∼0.1 au. At very short distances (≲0.03 au), the fraction of ionized helium in the extended atmosphere increases, thereby reducing the fraction of helium in the (neutral) triplet state at high planetary altitudes.

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This dependence on flux magnitude can help us understand why the triplet helium fraction shown in Figure 2 is not maximized in planets around stars with the hardest spectra, i.e., M-dwarfs. As shown in Figure 1, at a fixed orbital distance, the EUV flux of M-stars in the MUSCLES sample is at least an order of magnitude lower than in K-stars. If the EUV flux of M-dwarfs in the MUSCLES sample is representative of high-energy radiation typical for M-stars, then planets irradiated by M-star spectra have to be much closer to their host stars to achieve the same (or higher) population level of excited helium as planets around K-stars.

Changing the level of incident stellar flux may, at the same time, affect the basic properties of planetary winds, such as their mass-loss rates. Even though these changes would in reality occur simultaneously, here we treat them separately in order to more easily isolate the effects of each model parameter. In Figure 6 we show the fraction of helium atoms in the metastable triplet state in T = 7300 K atmospheres at 0.05 au from a K1 host star, with mass-loss rates varying between 10⁹ and 10^11.5 g s⁻¹. For low mass-loss rates, $\dot{M}\lesssim {10}^{10}$ g s⁻¹, the density of the planetary wind (and consequently the density of free electrons) is low and the atmosphere has not reached the regime dominated by electron collisions, described by Equation (8); instead, the triplet helium fraction is better described by Equation (9). Therefore, as $\dot{M}$ increases, so does the overall atmospheric density, the density of electrons, and the triplet helium fraction. For intermediate values of the mass-loss rate (∼10¹⁰−10¹¹ g s⁻¹), the triplet helium fraction, especially in the part of the atmosphere closest to the planet, is maximized. At high mass-loss rates, $\dot{M}\gtrsim {10}^{11}$ g s⁻¹, as the density of the planetary wind increases, the hydrogen atoms begin to self-shield and the inner region of the atmosphere becomes devoid of free electrons. The lack of free electrons reduces the rate of helium recombination, and consequently, the population of helium atoms in the triplet state close to the planet.

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Stellar spectra, particularly in the EUV and mid-UV wavelength range, play a crucial role in our modeling of helium level populations and the resulting 1083 nm absorption signals from extended exoplanet atmospheres. Direct observations of the EUV flux are currently not possible for stars other than the Sun. Scaling relations and models connecting the EUV flux and other observables, such as X-ray and Lyα fluxes, have been developed (e.g., Sanz-Forcada et al. 2011; Linsky et al. 2014). Here we investigate the level of uncertainty caused by the differences between two methods of EUV flux reconstruction.

The red line in Figure 7 (top panel) shows the synthetic EUV spectrum of an M-dwarf GJ 3470 calculated using a coronal model based on X-ray and UV spectroscopic data, following Sanz-Forcada et al. (2011). The green line shows the synthetic EUV spectrum of the same star, obtained using the Lyα flux and semiempirical relations from Linsky et al. (2014). Both EUV spectra are from Bourrier et al. (2018). Above λ = 1100 Å, we use the spectrum of a different M1.5 star (GJ 667C) from the MUSCLES data.

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As in the previous sections, we irradiate a model atmosphere with these two stellar SEDs. For simplicity, we keep all planetary and stellar parameters fixed on values for HAT-P-11b, as described before. In each case, we calculate the population level of triplet helium, shown in the middle panel of Figure 7, and the resulting transit depth at 1083 nm, shown in the bottom panel. Different methods of EUV flux reconstruction can lead to considerable differences in the predicted triplet helium level population and transit depth at 1083 nm. This highlights the importance of obtaining—through direct observations and modeling—reliable stellar spectra at UV wavelengths for stars of various spectral types and levels of activity.