Metastable Helium Absorptions with 3D Hydrodynamics and Self-consistent Photochemistry. I. WASP-69b, Dimensionality, X-Ray and UV Flux Level, Spectral Types, and Flares

We conceptually divided a planet into four regions: (1) a dense core, (2) a convective inner atmosphere, (3) a quasi-isothermal outer atmosphere with equilibrium temperature T_eq, and (4) an outflowing region irradiated by high-energy photons (e.g., Rafikov 2006; Ginzburg et al. 2016; Owen & Wu 2016). The equilibrium temperature T_eq satisfies

$$\begin{eqnarray}&&{T}_{\mathrm{eq}}\simeq 886\,{\rm{K}}{\left(\displaystyle \frac{{L}_{* }}{{L}_{\odot }}\right)}^{1/4}{\left(\displaystyle \frac{a}{0.1\mathrm{au}}\right)}^{-1/2},\end{eqnarray} \tag{ 1 }$$

where L_* is the bolometric luminosity of the star and a is the semimajor axis of the planetary orbit. Our simulations will focus on regions 3 and 4, whereas the structure of regions 1 and 2 provides the correct boundary conditions crucial for correctly reproducing the measured mass and radius of the planet. We refer the reader to the Appendix for the details of how we set up the internal structures of our planet and resultant boundary conditions for our simulations.

We characterize the high-energy radiation SED of the host star with five different characteristic energy bins: (1) h ν = 2 eV for infrared, optical, and near-ultraviolet (NUV) photons; (2) h ν = 7 eV for “soft” far-ultraviolet (FUV) photons that can photoionize He*; (3) h ν = 12 eV for the Lyman–Werner band FUV photons that can photodissociate molecular hydrogen; (4) h ν = 20 eV for “soft” extreme-ultraviolet (soft EUV) photons that can ionize hydrogen but not helium; (5) h ν = 40 eV for hard EUV photons that ionize hydrogen and helium; and (6) h ν = 3 keV for the X-ray.

Our simulation combines ray-tracing radiative transfer, real-time nonequilibrium thermochemistry, and full hydrodynamics calculations (based on a higher-order Godunov hydrodynamic solver Athena++; Stone et al. 2020). The simulation is mostly based on our WD18 work, with a few modifications and improvements added for the higher dimensionality and the inclusion of He*.

The density distribution, temperature profile, and dynamics of the outflowing atmosphere all play a part in the He* observables. To capture the outflow dynamics accurately, simulations should include the gravity of the star and the planet and the effects of orbital motion: the centrifugal and Coriolis forces. Therefore, 3D simulations are required. Given its short orbital period and observed radial velocities (Anderson et al. 2014), we assume that WASP-69b is tidally locked and circularized. Our simulation is run in a corotating frame centered on the planet. We adopt a spherical polar coordinate (r, θ, ϕ) with θ = 0 pointing toward the host star and ϕ = 0 pointing in the direction of orbital motion.

The mesh covers the domain (r, θ, ϕ) ∈ [r_in, r_out] ⨂ [0, π] ⨂ [0, π]. Planet-specific radial boundaries r_in and r_out usually extend from the base of the quasi-isothermal layer to a relatively large radius (150 R_⊕ in this case) such that the density/opacity drops low enough. The radial grids are placed logarithmically to strategically capture the steep change of density, while latitudinal and azimuthal grids are spaced evenly. Reflecting boundary conditions are enforced at the r = r_in, ϕ = 0, and ϕ = π boundaries, while the r = r_out boundary is an outflowing boundary. The θ = 0, θ = π boundaries are polar wedges to avoid coordinate singularity. The whole mesh, with its polar axis always pointing toward the host star, corotates with the orbital motion and the rotation of the tidally locked planet.

In a 3D spherical polar mesh, the grids near the polar axis are narrow in the azimuthal direction ($\delta {x}_{\phi }\simeq {r}_{\mathrm{cc}}\sin {\theta }_{\mathrm{cc}}\delta \phi ;$ subscripts “cc” stand for “cell center”). This can result in highly nonunitary aspect ratios (δ x_θ ≃ r_cc δ θ) and a stringent Courant−Friedrichs−Lewy (CFL) condition. We thus introduce an adaptive “mesh coarsening” technique for the azimuthal grids near the poles. Without any violations of conservation laws, the effective aspect ratio of the high-latitude zones becomes close to 1 and the time-step constraints imposed by the CFL condition are not as severe (see also Nakamura et al. 2019; Müller et al. 2019). This helps to greatly speed up our model.

Our simulation includes a non-LTE reaction network that coevolves with the hydrodynamics (see WD18 for details), with the addition of the metastable state of neutral helium and all relevant reactions that populate and depopulate this state (see Oklopčić & Hirata 2018 and references therein). Our reaction network now has 26 thermochemical “species” that are considered important to the thermochemistry of photoevaporation, including He* and internal energy density, ³ and 135 reactions such as ionization, recombination, collisional (de-)excitation, photodissociation, and cooling and heating processes. The reaction network includes processes that are directly relevant to photoevaporation, including photoionization and photodissociation heating, the dust−gas thermal accommodation, recombination cooling, and collision−radiation cooling especially by the species containing carbon, oxygen, silicon, and sulfur (see, e.g., Tielens & Hollenbach 1985; Gorti & Hollenbach 2008; Gong et al. 2017; Wang & Goodman 2017; WD18). The ordinary differential equations (ODEs) of the thermochemical network were solved efficiently using the semi-implicit method specially optimized for the graphics processing units (GPUs). The resultant efficiency allows us to coevolve the hydrodynamics with the thermodynamics, rather than including thermodynamics as a post-processing step that is often done in the literature. Again, we refer interested readers to WD18 for more details.

We synthesize both the transmission line profiles (Nortmann et al. 2018) and the narrowband light curves (Vissapragada et al. 2020) of He* transitions using our simulations. At each wavelength λ and a particular orbital phase, the optical depth along a line of sight (LOS) is given by

$$\begin{eqnarray}\begin{array}{rcl}\tau (\lambda ;{\rm{\Phi }}) & = & \int | d{\boldsymbol{x}}| \ n(\mathrm{He}* ;\ {\boldsymbol{x}};{\rm{\Phi }})\\ & & \times \mathop{}\limits_{i}{\sigma }_{i}[\lambda ;{\hat{n}}_{\mathrm{LoS}}\cdot {\boldsymbol{v}}({\boldsymbol{x}};{\rm{\Phi }}),T({\boldsymbol{x}};{\rm{\Phi }})],\end{array}\end{eqnarray} \tag{ 2 }$$

where we have transformed from our planet-centered coordinate systems in the simulations to a star-centered coordinate system for the synthetic observations. Thus, x and v are the position and velocity vector measured from the host star. The integration goes along the designated LOS, and the summation index i runs over the three He* lines with different upper-state quantum number J. The cross section σ_i is assumed to be a Voigt profile that convolves the intrinsic Lorentzian profile (γ = A/4π, A = 1.0216 × 10⁷ s⁻¹; see Drake 2006) with a Gaussian profile from thermal broadening at temperature T( x ). This Voigt profile is shifted by the local bulk velocity including orbital motion and outflow kinematics and the projection onto the LOS ${\hat{n}}_{\mathrm{LoS}}\cdot {\boldsymbol{v}}({\boldsymbol{x}})$.

This integration is performed for all relevant LOSs that originate from the surface of the host star:

$$\begin{eqnarray}&&\epsilon (\lambda ;{\rm{\Phi }})=1-\int {\text{}}d{\rm{\Sigma }}\ S{{\rm{e}}}^{-\tau (\lambda ;{\rm{\Phi }})},\end{eqnarray} \tag{ 3 }$$

where (λ) is the relative extinction at wavelength λ, and S(λ) is the normalized surface brightness (∫dΣS(λ) = 1) of the star after accounting for limb darkening and stellar rotation. The integral runs through the entire projected stellar surface.

(λ; Φ) is the absorption-line profile as a function of wavelength and orbital phase (time). We mimic what observers often do in He* observations, i.e., time averaging (λ; Φ) over the entire transit event from nominal ingress to egress (t _II through t _III ). ⁴ The outcome Δ(λ) is a line profile of excess absorption to be compared with observations directly.

We report a number of summary statistics, including the equivalent widths (EWs) 〈W_λ 〉 ≡ ∫Δ(λ)d λ, the radial velocity shift of the absorption peak Δv_peak, and the FWHM of the absorption-line profile. These summary statistics help us to compare between models and observations efficiently and are less prone to measurement uncertainty, bad pixels, and other instrumental effects.

Finally, we integrated (λ; Φ) multiplied by a filter bandpass function over λ. The result is a transit light curve near the He* transitions. In this work, we use the bandpass function provided by Vissapragada et al. (2020) for a direct comparison with their results.

Before analyzing our simulations, we ensured that quasi-steady states have been achieved. This usually involved running the simulations for many dynamical timescales; specifically, we set ${t}_{\mathrm{sim}}\gtrsim 30\,{\tau }_{\mathrm{dyn}}$. The dynamical timescale τ_dyn is estimated by the sound-crossing time of the Bondi radius:

$$\begin{eqnarray}&&{\tau }_{\mathrm{dyn}}\sim \displaystyle \frac{{{GM}}_{{\rm{p}}}}{{c}_{s}^{3}}\sim 1.2\times {10}^{3}\,{\rm{s}}\times \left(\displaystyle \frac{{M}_{p}}{10\,{M}_{\oplus }}\right){\left(\displaystyle \frac{T}{{10}^{4}{\rm{K}}}\right)}^{-3/2}.\end{eqnarray} \tag{ 4 }$$

Here c_s is the sound speed. For a typical T = 10⁴ K outflow we see for WASP-69b τ_dyn ∼ 10⁴ s. Moreover, we also check explicitly whether the simulation has settled down to a quasi-steady state by comparing key hydrodynamic/thermodynamic properties in neighboring dump files.

Our fiducial model for WASP-69b shows clear signs of a photoevaporative outflow. In Figure 1, we show 2D slices of the density, temperature, and LOS velocity distributions of our 3D simulations looking down the north pole of the planet. We see a T ≳ 10⁴ K hot ionized supersonic outflow originating at a wind base of r ≃ 13 R_⊕, which eventually accelerates to ∼ 23 km s⁻¹ when leaving the domain of our simulation. This outflow disperses the planet atmosphere at a mass-loss rate of $\dot{M}\simeq 5.5\times {10}^{-10}\,{M}_{\oplus }\,{\mathrm{yr}}^{-1}$. Since we assumed a constant high-energy radiation output from the host star, the mass-loss rates and the hydrodynamic/thermodynamic profiles remain nearly constant after reaching the quasi-steady state. We also note that we did not put in any stellar wind from the host star, as the current windless model fits the data reasonably well and is preferred by Occam’s razor. However, in a companion paper on WASP-107 we will show that stellar winds may generate Kelvin–Helmholtz instability that leads to fluctuations of a photoevaporative outflow.

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Which mechanisms control the population of the He* state? The bottom row in Figure 2 compares the rates of different (de)population processes along the two particular streamlines (thickened curves in Figure 1). We compute the rate of ionization, recombination, spontaneous decay, collisional excitation, and de-excitation, as well as an advection attenuation term ∣ v · ∇n(He*)∣. Along the representative streamline presented by the left column, the abundance of He* is determined by the relatively stiff balance between the recombination (He⁺ + e⁻ → He*) and the collisional de-excitation at small radii ≲ 30 R_⊕. As expected, these two processes are efficient at higher densities, consistent with the law of mass action. Photoionization of He* by soft FUV starts to take over the destruction channel of He*, where the density of free electrons declines at higher altitudes. The other channels have negligible importance, e.g., collisional excitation from 1¹ S to the metastable state is more than five orders of magnitude slower than recombination. In the right column of Figure 2, we show an interesting streamline that crosses in the shadow of the planet. The number density of He* soars in the shadow because the photoionization of He* by soft FUV vanishes here.

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Beneath the base of the photoevaporative outflow (r ≲ 13 R_⊕), the temperature gradient between the dayside and the nightside generates a slow “zonal” circulation (∼0.1 km s⁻¹). However, this region has little observational effect on the overall He** observables, which are mostly controlled by the much more extended low-density regions of the outflowing atmosphere. We will return to this point shortly. Moving to higher altitude, this day–night advection continues, amounting to a 2–3 km s⁻¹ blueshift at about 20–40 R_⊕. Going farther away from the planet, the Coriolis effect starts to shape the outflow streamlines into spiral curves, resulting in blueshifts on the leading edge and redshifts on the trailing edge. Considering that the outflow is still primarily radial, increments in the latitudinal velocity ∣Δv_θ ∣ after traveling through a radial distance Δr can be estimated by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{v}_{\theta } & \sim & 2\cos \phi \ {{\rm{\Omega }}}_{{\rm{K}}}\displaystyle \int {\text{}}{dt}\ {v}_{r}\simeq 2\cos \phi \ {{\rm{\Omega }}}_{{\rm{K}}}{\rm{\Delta }}r\\ & \simeq & 2.3\,\mathrm{km}\,{{\rm{s}}}^{-1}\ \cos \phi \ {\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{a}{0.05\mathrm{au}}\right)}^{-3/2}\left(\displaystyle \frac{{\rm{\Delta }}r}{10\,{R}_{\oplus }}\right).\end{array}\end{eqnarray} \tag{ 5 }$$

This estimation is confirmed by the velocity profile in the top panels of Figure 2: if we compare the values at r ≃ 40 R_⊕ and r ≃ 100 R_⊕, the difference in v_LOS (approximately equal to v_θ for this streamline) is Δv_LOS ∼ 11 km s⁻¹, and Equation (5) yields ∼ 10.8 km s⁻¹. This effect redistributes He* atoms in the velocity space and broadens the observed He* line profiles as we will see shortly.

Figure 3 shows our synthetic observations of both the line profiles and light curves of WASP-69b (Nortmann et al. 2018; Vissapragada et al. 2020). We have binned the light-curve data from Vissapragada et al. (2020) for better clarity, and the uncertainty represents the standard deviation within each phase bin. Our fiducial model of WASP-69 seems to fit both the spectroscopic and photometric observations well simultaneously. In particular, the synthetic line profile reproduced the subtle blueshift of the peak absorption, the overall line depth, and the relative ratio between the lines of this triplet. Numerically, Nortmann et al. (2018) reported a net blueshift of −3.58 ± 0.23 km s⁻¹. This blueshift was based on fitting Gaussians to the observed line profiles; however, as we argued in the previous section, kinematic shift of the outflow introduces significant distortion of the spectral shape. Instead of fitting a Gaussian to our line profile, we compared our simulations directly to the line profile itself, which shows great agreement. We also use a different way to measure the blueshift: we report the blueshift of the peak absorption relative to a line-ratio-weighted average of the rest-frame line center for the two longer-wavelength transitions that are usually blended together.

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Nortmann et al. (2018) hinted at the possibility of a comet-like tail trailing behind WASP-69b. The basis of their suggestion is that additional He* absorption can still be seen ∼20 minutes after the nominal egress of the planet. The higher-precision, better temporally sampled photometric data from Vissapragada et al. (2020) nonetheless favor a symmetric transit. The symmetric transit shape (bottom panel of Figure 3) does not support an extended comet-like tail. Our simulation seems to be more consistent with Vissapragada et al. (2020); the photoevaporative outflow of WASP-69b in our fiducial model is largely symmetric between the leading and trailing edge, and hence it produces a more symmetric transit shape. We note that an extended comet-like tail will also introduce significant distortion to the He* line profile (see our companion paper on WASP-107b for the strong comet-like tail generated by strong stellar wind in that system). Here in the case of WASP-69b, our fiducial model produces a good fit to the resolved line profile (Nortmann et al. 2018), while it does not need to invoke a prominent comet-like tail.

Another point we would like to emphasize is that a significant part of the He* absorption for WASP-69 seems to be produced by an extended, optically thin (τ < 1) outer layer of the photoevaporative outflow. In Figure 4, we show the midtransit extinction ($1-{\exp }^{-\tau }$) at three different wavelengths near the He* transitions. The outer regions (tens of R_⊕) contribute significantly to the overall extinction thanks to their extended area and the slow decrease of He* number density in the outflow. Because of the unsaturated optical depth, the line ratios between the He* triplet are close to 1:3:5, i.e., their quantum degeneracies. More accurately, the line ratios are close to 1:8, as the longer two lines are blended by kinematics and thermal broadening. This suggests that the line ratios between the He* triplet can be a diagnostic of the number density in the outflow. If most of the He* absorption is due to the higher-density region, where one of the lines may saturate first, the line ratio will deviate from the quantum degeneracy ratio; this would tell us about the density of the outflow region in a model-independent way (see also discussions in Salz et al. 2018). This does not seem to be the case for WASP-69b, as most He* absorption happens in lower-density regions. Another feature revealed by Figure 4 is that the leading edge tends to be bluer and the trailing edge redder. This is consistent with the Coriolis effects (Section 3.1) and reiterates the importance of multidimensional studies to consistently understand the spectral profiles.

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It is worth noting that, due to the heavy computational cost of our 3D simulations, we could not afford to numerically fit the data with multiple simulation runs. Instead, our fiducial model serves as a validation of our self-consistent 3D hydrodynamic simulations: with reasonable assumptions of the planetary and stellar properties (such as the high-energy SED), we can at least qualitatively reproduce the various He* observables. The degree to which our simulation agrees with observations is quite encouraging if not remarkable. After this validation of the model, we stand at a position to perturb our fiducial model and investigate how the He* observables are impacted by various factors that control the underlying photoevaporative outflow and He* population in the following section.

To test how our model depends on the spatial dimensions of the simulations, we ran a 2D model with axisymmetry (about the ϕ-axis, i.e., N_ϕ = 1) while keeping all system parameters the same as the fiducial model. The Coriolis force is not captured in this 2D simulation while stellar gravity and orbital centrifugal forces are still involved. A reference 1D spherical symmetric model is also implemented using the θ = π/2, ϕ = π/2 radial line and removing the θ and ϕ components of the velocity.

Figure 3 compares the synthesized line profiles and light curves with all three-dimensionality models. The 1D spherically symmetric model suffers from the loss of all nonradial kinematic information. It is clearly inconsistent with the observed line profile with no blueshift and limited kinematic broadening. The 2D axisymmetric test is able to capture the day–night advection. It shows good agreement with the observed line profile. The 3D fiducial model further modifies the line profile by including the Coriolis force. In the case of WASP-69b such a modification is quite subtle, which again testifies that the photoevaporative outflow on WASP-69b is largely symmetric between the leading and trailing edge. Again see our companion paper on WASP-107b for how this symmetry is broken by the inclusion of stellar winds.

The three models from 1D through 3D have almost identical EW (〈W_λ 〉 ∼ 3.1 Å) and light curves. This stresses the importance of spectrally resolving the He* line profiles that are seen to vary the most between dimensions. The mass-loss rates are again quite similar between 3D and 2D models at about $\dot{M}\simeq 5.5\,\times {10}^{-10}\,{M}_{\oplus }\,{\mathrm{yr}}^{-1}$. The mass-loss rate in our 1D model is off ($\dot{M}\simeq 6.9\times {10}^{-10}\,{M}_{\oplus }\,{\mathrm{yr}}^{-1}$) because it assumes perfect spherical symmetry. However, the streamlines in Figure 1 are clearly nonradial. We also compare our results with those from a 1D isothermal model (Oklopčić & Hirata 2018; Vissapragada et al. 2020) of 9.5 × 10⁻¹⁰ M_⊕ yr⁻¹ (≃3 × 10⁻³ M_Jup Gyr⁻¹) at an assumed temperature of 12,000 K. The results are in rough agreement with differences arising from more careful treatment of the hydrodynamics, thermodynamics, and radiative transfer.

Photoevaporative outflows are driven by high-energy radiation from the host star. Moreover, the population of He* states is also controlled by the critical balance high-energy photons of different energy bins. We examine the impact of high-energy radiation in each energy bin by perturbing the fiducial model. The amount of high-energy radiation a star outputs is variable depending on the evolution stage, activity, and spectral types of the host star. Direct measurements are also lacking, as the XUV measurements have to be performed in space. Therefore, in our models 69-1 to 69-5 (soft FUV for 69-1, LW for 69-2, soft EUV for 69-3, hard EUV for 69-4, X-ray for 69-5), we bump up the flux level in each high-energy bin by a whole order of magnitude to reflect the intrinsic variation in high-energy flux level. We summarize key He* observables in Table 3; we also show the synthetic line profiles and light curves in Figure 5 and the relative abundance of He* as a function of radius in Figure 6.

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The He* line profiles are controlled mainly by the FUV (adverse effect) and the EUV bands (positive effect); the LW and X-ray bands only play minor roles under the “typical” host star conditions. More specifically, the relative abundance of He* is suppressed by soft FUV because it photoionizes the He* states, thus significantly reducing its population and the He* absorption depth. In model 69-1 (×10 soft FUV flux), the overall mass-loss rate is enhanced by a few percent thanks to extra energy deposited into the atmosphere. However, the stronger soft FUV flux significantly lowered the number density of He* at a larger radial extent (r ≳ 20 R_⊕). The He* line profile depth and the light-curve depth are both reduced by about a factor of two (3.16 to 1.75 Å). The line width also decreased (FWHM from 17.3 down to 14.0 Å) because the high-altitude region with a higher velocity dispersion contributes less to the He* extinction.

In model 69-2 (×10 LW flux) the He* observables are mostly unaltered from the fiducial model. This is because the LW band is intrinsically narrow and thus only amounts to a very small fraction of the overall high-energy radiation flux. Moreover, most molecular H₂ is already dissociated at ∼ 10⁴ K in our simulations.

At higher EUV fluxes (models 69-3 and 69-4), the much faster outflows not only bring more He* into the exosphere but also spread them out in velocity space, effectively broadening the line profiles. This confirms our earlier picture that EUV flux is most effective in driving photoevaporative outflows (WD18).

Finally, X-ray seems to play a secondary role in photoevaporation and He* observables. Model 69-5 (×10 X-ray flux) has very similar observables to the fiducial model. Although X-ray photons are very energetic, they also have much smaller cross sections than EUV photons. As a result, X-rays penetrate deeper into the atmosphere, where collisional processes and dust particles quickly convert the X-ray energies to infrared radiation that escapes easily. This limits the heating potential of X-ray. We reached a similar conclusion in WD18.

Among the handful of reported He* detections, four out of six are planets around K-type hosts (Spake et al. 2018; Venzmer & Bothmer 2018; Allart et al. 2018; Salz et al. 2018; Ninan et al. 2020; Alonso-Floriano et al. 2019). Oklopčić (2019) confirmed that K stars, at least in 1D isothermal models, may be at the sweet spot of FUV/EUV flux balance that best promotes the He* population and thus observability. This section reevaluates such a claim with our 3D hydrodynamic simulation coupled with self-consistent thermodynamics and radiative transfer.

We set up three additional models 69-F, 69-G, and 69-M, whose luminosities in different high-energy bins emulate typical F-type, G-type, and M-type main-sequence stars based on the compilation of Oklopčić (2019); see also Figure 7. We remind the reader that the fiducial model 69-0 has a K-star SED. The flux in each high-energy bin is summarized in Table 2). Broadly speaking, F-type and G-type stars output similar levels of soft and hard EUV fluxes to K-type stars; however, their FUV luminosities are significantly higher by a factor of ∼3000 and ∼130, respectively. For a typical M star, fluxes in all high-energy bands are lower by about one order of magnitude.

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He* observables of these models are summarized in Table 3 for their mass-loss rates and a few key diagnostics, etc. We show the line profiles and light curves in Figure 8 and the radial distribution of He* in Figure 6. With a soft FUV flux ∼3000 times higher than our fiducial model, F-type stars significantly suppress the population and observability of He* with an EW reduced by almost two orders of magnitude (3.16–0.04 Å). However, the mass-loss rate of the photoevaporative outflow is similar to that of the fiducial model. Again, photoevaporation is driven mostly by EUV, which has similar flux levels between F and K stars. On the other hand, for a typical M-star host, whose higher-energy flux levels are weaker in all bands, the mass-loss rate is reduced by one order of magnitude. Nonetheless, the EW of He* in the transmission spectrum only declines by a factor of 2 (3.16–1.4 Å). Again, this is because its much weaker soft FUV flux allows proportionally more He* to exist in the outflow (Figure 6). A G star represents some middle ground; its ∼130 times higher soft FUV flux suppresses the EW He* by a factor of 6.

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Table 3. Results of Various Models, Based on the Fiducial Model for WASP-69b

Model	Description	$\dot{M}$	〈Wλ 〉	Δvpeak a	FWHM b
		(10−9 M⊕ yr−1)	(10−2 Å)	(km s−1)	(km s−1)
69-0	3D fiducial	0.55	3.16	−2.1	17.3
69-0-2D	2D test (fiducial parameters)	0.56	3.10	−2.0	16.6
69-0-1D	1D test (fiducial parameters)	0.69	3.14	0.07	16.2
69-1	10×Flux for soft FUV (h ν = 7 eV)	0.61	1.75	−1.7	14.0
69-2	10×Flux for LW (h ν = 12 eV)	0.56	3.18	−2.1	17.4
69-3	10×Flux for soft EUV (h ν = 20 eV)	0.66	4.13	−2.4	24.2
69-4	10×Flux for hard EUV (h ν = 40 eV)	2.50	5.62	−3.8	48.1
69-5	10×Flux for X-ray (h ν = 3 keV)	0.55	3.12	−2.0	17.6
69-6	1/2 planet mass (Mp = 41.3 M⊕)	0.93	4.25	−2.1	17.2
69-F	Fiducial model with F-type host	0.70	0.04	−0.8	9.0
69-G	Fiducial model with G-type host	0.64	0.51	−1.5	10.3
69-M	Fiducial model with M-type host	0.05	1.40	−0.6	12.9

Notes. The values are time averages taken over the last 15 τ_dyn of the simulations. Fluctuations are negligible for all models in the table.

^a Shifts (positive values for redshifts and vice versa) of the right-hand-side peaks in the velocity space, compared to the line-ratio-averaged line center of λ = 10833.29 Å. ^b FWHM of the longer-wavelength peak in the velocity space.

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In summary, our findings suggest that K-star planet hosts are indeed favorable targets for He* observations, consistent with the suggestion of Oklopčić (2019). The high-energy SED, nonetheless, is expected to change significantly as a function of host star age and activity level. The suppression factors of He* around G- and M-type stars are often only a factor of a few. We encourage observers to keep them in their target list, particularly the young and active ones. We also predict that there will be more reports of He* detection around G- and M-type hosts soon. Another important point we would like to stress is that the depth of the He* line profile cannot be translated to the underlying mass-loss rate without knowing the high-energy SED of the host star. In other words, measuring the XUV SED of the host star directly is crucial for correctly interpreting the He* observations.

The mass-loss rate of photoevaporation depends quite strongly on the depth of the gravitational potential well of the planet. A shallower potential allows faster outflow with the same high-energy irradiation. In model 69-6 we adjusted the planet interior such that the planet mass is reduced by half while keeping the transit radius the same. This effectively lowers the surface gravity of the planet by a factor of two. The mass-loss rate in model 69-6 increases to $\dot{M}\simeq 9.3\times {10}^{-10}\,{M}_{\oplus }\,{\mathrm{yr}}^{-1}$, which is ∼ 70% larger than the fiducial model. This larger mass-loss rate can be decomposed into an increase in the terminal outflow velocity by ∼10% and an increase of the outflow density by ∼60%.

The He* line profile depth responds to this increase of mass-loss rate sublinearly. In Figure 5, the line profile has ∼40% larger depth than that in the fiducial model, while the EW increases by about ∼30%. However, the He* line profile maintains a similar morphology with the peak velocity shift and the FWHM unchanged from the fiducial model (Table 3). In short, puffier, low surface gravity planets are more likely to undergo strong photoevaporative mass loss and should prove great targets for He* observations.

Photometric measurement of transits, compared to transmission spectra, features much lower costs of exposures and higher viability if one wants to study atmospheric dispersal processes over an ensemble of planets statistically. Our numerical experiments within and beyond this paper have indicated that the excess absorption by metastable helium atoms in the τ ≲ 1 outskirts plays an important role in casting the transmission features (e.g., Figure 4). The EW at a given orbital phase angle Φ can be reasonably approximated by (using Equations (2) and (3))

$$\begin{eqnarray}\begin{array}{rcl}{W}_{\lambda } & = & \displaystyle \int {\text{}}d\lambda \left[1-\displaystyle \int {\text{}}d{\rm{\Sigma }}\ S{{\rm{e}}}^{-\tau (\lambda ;{\rm{\Phi }})}\right]\\ & \simeq & \displaystyle \int S{\text{}}d{\rm{\Sigma }}\displaystyle \int {\text{}}d\lambda \ \tau (\lambda )\\ & \propto & \displaystyle \int {\text{}}d{\rm{\Sigma }}\mathop{\displaystyle \int }\limits_{\mathrm{LoS}}| {\text{}}d{\boldsymbol{x}}| n(\mathrm{He}* )\displaystyle \int {\text{}}d\lambda \displaystyle \sum _{i}{\sigma }_{i}(\lambda )\\ & \propto & \displaystyle \int {\text{}}{dV}\ n(\mathrm{He}* ).\end{array}\end{eqnarray} \tag{ 6 }$$

Here we ignored limb darkening and stellar rotation, and the last integration measure dV runs through the volume spanned by all LOSs over the projected stellar surface. Note that the integration ∫d λ σ(λ) is a constant regardless of thermal broadening or bulk velocity shifts. This expression is still valid under time averaging.

EWs of transmission spectra, which are equivalent to transit depths up to a conversion relation, are most relevant to stellar irradiation. Unless the FUV flux is too intense, it is a reasonable approximation for order-of-magnitude estimations about the total He*, in which the n(He*)/n_H ratio is mostly invariant throughout a relatively large range of radii (see Sections 4.2 and 4.3 and, e.g., Oklopčić 2019). If valid, these approximated expressions enable the EW as a good indicator of the materials that stay in front of the host star and a reasonable representative of the outflow as a whole. In addition, unless significant impact is coming from the host star, dimensionality effects mostly change the distribution of absorbers in the configuration and velocity space but are less relevant to the EW (e.g., Table 1). In these cases, the integrated EW can be consistently predicted by reasonable 2D or even 1D calculations and hence reduce the costs of numerical calculations once data-driven modeling is desired.

Admittedly, EW observation would still suffer from the losses in accuracy to constrain the physics of host stars. The first issue is the degeneracy of parameters. Roughly speaking, soft FUV photons (h ν > 4.8 eV) destruct metastable helium and lower the n(He*)/n_H ratio, while EUV photons enhance the EW mainly by launching stronger photoevaporative outflows. Stellar winds that are sufficiently strong can also concentrate materials (including He*) inside the cylinder above the projected stellar surface and thus raise the EW (we refer the reader to Wang & Dai 2021, hereafter Paper II, for more thorough discussions). The second issue is that the sublinear dependence of EW on stellar irradiation, mainly due to the interconnectivity of thermochemical networks, makes EW alone a less sensitive parameter. These issues can be partially solved by introducing other measurements (such as HST STIS for FUV in Loyd et al. 2016), to eliminate the degeneracy.

Spectral line shapes can be good representatives of photoevaporation structures. It should be noticed that, even though hard EUV photons (h ν > 24.6 eV) produce He⁺ as the most important progenitor to populate the 2³ S state, this effect is likely to saturate before the hard EUV flux becomes intermediately intense (e.g., Section 4.2) and thus does not affect the ratio n(He*)/n_He directly. EUV radiation affects the appearance of planetary outflows more by controlling the kinematics and shaping the spectral line. As the key component of launching photoevaporation, stronger EUV flux typically drives faster winds, spreading He* atoms wider in the velocity space and broadening the spectral lines. Combining this extra dimension of velocity information with the transit depths or EWs (Section 5.1; Equation (6)), reasonable estimations of atmospheric mass-loss rates can be established. Such estimations should be most robust for planets with FUV-weak K-type and M-type main-sequence stars. For host stars with excessive FUV irradiation, accounting for the FUV-dominated He* destruction by other observation measures is demanded (similar to Section 5.1).

We emphasize that full 3D hydrodynamic-thermochemical simulations are demanded to interpret the characteristics in spectra and light curves accurately. First, as was elaborated in WD18, the consistency of modeling photoevaporation requires multidimensional nonequilibrium thermochemical calculations, which are inevitable to account for an intertwined system of physical processes, including heating, cooling, chemical reactions, and even ray-tracing. Second, understanding temporal and spectral observations requires knowledge about the distribution of the concerned tracer (He*) in both configuration and momentum spaces. Simplified steady-state 1D spherical symmetric calculations have to assume isotropic geometry, but actual streamlines and stellar photons never move radially. The difference in hydrodynamic conditions (esp. gas temperature) between the day and night hemispheres causes latitudinal advection beyond 1D models. What is more, the outflows of close-in planets can be considerably modulated by stellar gravity and orbital motion, which cannot be appropriately handled even with 2.5D axisymmetric simulations. Moreover, there are various systems showing prominent gaseous tails in light curves, which denies the possibility of improper calculations that are simpler than 3D. We postpone the discussions about these complicated interactions to Paper II. Nevertheless, we still emphasize that 3D simulations should eventually reward us with consistent and thorough understandings of the physics that transmission spectrum observations should be able to bring about.

During one of the transits of WASP-69b in Nortmann et al. (2018), the He* line profile experienced a ∼30% drop in magnitude that lasted for about 20 minutes. This variability could well be instrumental in origin, but here we explore an alternative explanation that it is generated by a stellar flare on the host star.

Solar/stellar flares are associated with the surface magnetic activity of the host star. Their amplitudes can range below a percent to even orders of magnitude in extreme cases (“superflares”; see Günther et al. 2020, and references therein). The sudden rise of luminosity is often followed by exponential decays to nominal flux level on timescales of minutes or hours. To investigate the consequences of flaring events on He* observables, we first inject a simple flare model in which fluxes across all energy bands increase by a factor of 10 and 100, which then quickly decay to the quiescent state exponentially with a timescale of 500 s.

The temporal response of our fiducial model to these flares is shown in Figure 9. Independent of the energy injected, the common mode of response is as follows. Before the dynamics of the outflowing atmosphere can fully respond to the flares, the photoionization of He* by soft FUV photons reduces the number density of He* and causes a significant decrease in the EW. It is only after the dynamical timescale $t-{t}_{\mathrm{flare}}\sim {\tau }_{\mathrm{dyn}}\sim$ hours that the flare-generated surge of photoevaporative mass loss reaches the higher altitudes where most of He* absorption occurs. As a result, the EW of He* increases after the first hour or so and remains high for several hours. Looking at the mass-loss rate across the outer boundary of the simulation (bottom panel of Figure 9), it experiences several oscillations on dynamical timescales as the systems respond to the increased flux from the flare. The EW and other observables, however, are the spatially integrated quantities; thus, EW effectively smears out most of these oscillations and has a much smoother variation (top panel of Figure 9). Comparing with variability seen in WASP-69b (Nortmann et al. 2018), a flare that simultaneously raises all high-energy radiation does not appear to be a good explanation. This is because it should be observed as a decrease followed by an increase of He* absorption rather than the decrease only in the observations (Nortmann et al. 2018).

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We therefore explore a different flare model in which only the soft FUV band surges. We do not have observational support that flares of this kind exist. We explore this rather contrived scenario just out of curiosity. Remember from Section 4.2 that the soft FUV primarily suppresses the He* abundance by photoionization without significantly changing the overall kinematics. A soft-FUV-only flare may produce the observed decrease of He* line depths. We set up three extra simulation runs, again based on the fiducial model of WASP-69b. We put in soft FUV flares (10, 100 and 1000 times the nominal value) that start at the middle of the transits and last for 30 minutes. In Figure 10, we can see that the light curves respond to these soft-FUV-only flares quickly. In order to reproduce the ∼30% variation, a soft FUV flare between 10 and 100× the nominal level is required. This should be readily observable in the Ca ii H and K lines of the CARMENES spectra in Nortmann et al. (2018). Nevertheless, enhanced activity was not observed in the spectra during or preceding the observed light-curve variation (private communications, Nortmann). After all, stellar flares do not seem to be a viable explanation of the temporal variability seen in the He* line profiles of Nortmann et al. (2018); an instrumental effect is perhaps a better solution.

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