Reduced Water Loss due to Photochemistry on Terrestrial Planets in the Runaway Greenhouse Phase around Pre-main-sequence M Dwarfs

Terrestrial planets currently in the habitable zone around M dwarfs are estimated to have been in runaway greenhouse conditions for up to ∼1 Gyr due to the long-term pre-main-sequence phase of M dwarfs. These planets likely lose a significant portion of water during the pre-main-sequence phase owing to H2O photolysis followed by hydrogen and oxygen loss to space. However, the effects of H2O reproduction reactions and UV shielding by chemical products that reduce photolysis-induced water loss have yet to be estimated. Here, we apply a 1D photochemical model to a H2O-dominated atmosphere of an Earth-like planet around a pre-main-sequence M dwarf to estimate these effects. We find that water loss is suppressed by efficient H2O reproduction reactions and by UV shielding due to O2. The water loss rate decreases by several to several hundred times compared to that in previous studies, with the assumption that the water loss rate is limited by stellar X-ray and extreme-ultraviolet flux or hydrogen diffusion through the atmosphere. Our results imply that terrestrial planets currently in the habitable zone around M dwarfs are more likely to retain surface water than previously estimated.


Introduction
Planets currently in the habitable zone (HZ) around M dwarfs are thought to have experienced severe conditions that hinder the formation of habitable environments.M dwarfs are estimated to take up to 1 Gyr to settle onto the main sequence (MS), and the stellar luminosity during the pre-main-sequence (PMS) phase can be 1 or even 2 orders of magnitude greater than when they reach the MS (Baraffe et al. 1998).Provided they have sufficient water, these planets during PMS are expected to be in a runaway greenhouse condition due to higher stellar irradiation than the planetary radiation limit due to the strong greenhouse effect of H 2 O (e.g., Kasting 1988;Kasting et al. 1993;Kopparapu et al. 2013;Ramirez & Kaltenegger 2014;Luger & Barnes 2015).
In the runaway greenhouse condition, surface water is entirely evaporated due to the radiative heating, which may form a H 2 O-dominated atmosphere (e.g., Kasting 1988).This H 2 O-dominated atmosphere is gradually lost through H 2 O photolysis followed by the preferential escape of hydrogen to space over oxygen (Hunten 1973;Kasting & Pollack 1983;Kasting 1988;Kasting et al. 1993;Wordsworth & Pierrehumbert 2013); this leads to oxygen accumulation and oxidation in the planetary interior (Kasting & Pollack 1983;Chassefière 1996;Gillmann et al. 2009;Hamano et al. 2013;Wordsworth & Pierrehumbert 2014;Wordsworth et al. 2018).Previous studies that estimated the amount of water loss from a H 2 O-rich atmosphere on a terrestrial planet have focused mainly on hydrodynamic escape induced by vigorous stellar X-ray and extreme-ultraviolet (XUV) irradiation (Luger & Barnes 2015;Tian & Ida 2015;Tian 2015;Bolmont et al. 2017;Bourrier et al. 2017;Guo 2019;Johnstone 2020;Yoshida et al. 2022;García Muñoz 2023).Typically, the amount of water loss from terrestrial planets currently in the HZ around M dwarfs during PMS is estimated to be equivalent to that in several to several tens of times the terrestrial ocean (TO: 1.4 × 10 21 kg) assuming that H 2 O is fully photodissociated into atoms below the escape region due to intense stellar X-ray and UV irradiation.
Water loss from a H 2 O-dominated atmosphere, however, can be limited by the following processes: (1) H 2 O reproduction through various chemical pathways, such as H + HO 2 → H 2 O + O( 1 D); and (2) suppression of H 2 O photolysis due to UV shielding by chemical products such as O 2 , whose UV absorption wavelength (far-ultraviolet (FUV); 115-180 nm) overlaps with the photolysis wavelength of H 2 O (e.g., Keller-Rudek et al. 2013).However, as mentioned above, previous studies have assumed the efficient H 2 O photolysis without thoroughly investigating the impact of their effects on the water loss rate.In this study, we apply a 1D photochemical model based on PROTEUS (Nakamura et al. 2023) to a H 2 O-dominated atmosphere on an Earth-like planet around a PMS M dwarf to estimate the effects of these factors on the water loss rate.
This paper is organized as follows.In Section 2, we outline our photochemical model.In Section 3, we show the calculation results of how H 2 O reproduction reactions and UV shielding by chemical products affect the atmospheric profile and the water loss rate.In Section 4, we discuss the dependence of the water loss rate on the assumed O escape rate, the magnitude of photon flux, and the eddy diffusion coefficient.Additionally, we provide a proposition on the habitability of terrestrial planets around M dwarfs.
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Model Description
We focus on an Earth-like planet orbiting at 0.023 au around an M8 dwarf with the stellar properties of TRAPPIST-1.This planet currently receives the same bolometric stellar flux as Earth.Our 1D photochemical model solves continuity-transport equations considering production and loss by photochemical reactions for each chemical species (Nakamura et al. 2023 A1).
To consider the photolysis rate profiles of the chemical species, we use the stellar UV spectrum estimated for TRAPPIST-1 in the wavelength range from 115 to 1000 nm (Wilson et al. 2021; right panel in Figure A1), which covers the wavelength range affecting H 2 O photolysis and the other species considered in this study.As described in Chaffin et al. (2017), for simplicity, we neglect the stellar XUV absorption in the wavelength range below 115 nm and ionization reactions.
The temperature is assumed to decrease with altitude in the tropopause following the dry adiabat from a surface temperature of 2000 K and is a constant value in the stratosphere (Kasting 1988; middle panel in Figure A1).The temperature in the isothermal stratosphere is set to 400 K in the same way as Luger & Barnes (2015).
The molecular diffusion coefficient is taken from Hunten (1973; Figure A2).Particularly, for H, it is given by the following: where D H , T, and n a are the molecular diffusion coefficient of H, the temperature, and the number density of the background gas, respectively.Here, D H , T, and n a are expressed in m 2 s −1 , K, and m −3 , respectively.The eddy diffusion coefficient is set to 10 m 2 s −1 of an Earth-like value (Massie & Hunten 1981) regardless of altitude for the standard case.The dependency on the eddy diffusion coefficient of the water loss rate is discussed in Section 4.1.3.The upper boundary is set at the altitude where the pressure is lower than 10 −11 bar and is optically thin in the UV wavelength range considered in this study.For the upper boundary conditions, H and H 2 are assumed to escape to space.The escape of H is assumed to be diffusion-limited, with the effusion velocity given by the following: where h H is the scale height of H, h a is the mean scale height of the background gas except H, and z and α T are the altitude from the surface and the thermal diffusivity, respectively.H 2 is assumed to escape with the same effusion velocity as H, with the assumption that no fractionation occurs between H and H 2 .
The escape flux of hydrogen is given by where n H and n H 2 are the number density of H and H 2 at the upper boundary, respectively.The water loss rate is derived as follows: where R p is the planetary radius and m H O 2 is the mass of molecular water.The lower boundary is set at the surface.For simplicity, no supply or deposition is assumed for any species.
As the initial condition, the H 2 O vapor number density is set to be in hydrostatic equilibrium with a surface pressure equivalent to a mass of 1 TO (left panel in Figure A1).We obtain a quasi-steady state by numerical integration over 1 Myr, which is longer than the diffusion and chemical timescales.
Table 1 shows the parameters for the standard case, which are used to investigate the results in Section 3, and the parameter ranges in Section 4.1.

Results
The calculated density profiles of the H 2 O-dominated atmosphere are shown in Figure 1.H 2 O photolysis reaches a maximum at an altitude of ∼10 −7 bar (blue dashed line), which    3).The H 2 O photolysis rate exceeds the H 2 O reproduction rate at and above the photolysis peak (approximately above 10 6 bar as shown in Figure 1).Here, FUV in the wavelength range from 115 to 180 nm, whose photon flux accounts for 38.7% of the total photon flux from 115 to 230 nm, is mainly absorbed (Figure 3).Most H atoms at and above the photolysis peak are efficiently transported upward and escape to space.Above the photolysis peak, UV shielding by O 2 suppress H 2 O photolysis (left panel in Figure 2).O 2 is produced mainly at the photolysis peak by R1 and R30 (Figure A4), where the upward flux of O 2 reaches a maximum (right panel in Figure 2).Then, O 2 transported upward absorbs more than half of the FUV since the absorption cross section of O 2 exceeds that of H 2 O (Keller-Rudek et al. 2013), especially in the wavelength range from 140 to 160 nm (Figure 3).
On the other hand, the photolysis efficiency of H 2 O is significantly lower below the photolysis peak because of the effective H 2 O reproduction reactions, which occur mainly by OH and HO 2 (left panel in Figure 2).In this region, the photon flux in the wavelength range from 180 to 230 nm in nearultraviolet (NUV), which accounts for 61.7% of the total  photon flux from 115 to 230 nm, is mainly absorbed by H 2 O (Figure 3) without contributing to the water loss.
The hydrogen escape to space is regulated by the photolysis efficiency of H 2 O.The escape flux is 3.94 × 10 16 m −2 s −1 , which is equivalent to a water loss of 6.78 TO Gyr −1 .This value is lower than the energy-limited and diffusion-limited escape rates suggested by previous studies (e.g., Luger & Barnes 2015).This is due to the H 2 O photolysis limit resulting from H 2 O reproduction and UV shielding by O 2 , as described above.A detailed comparison of the escape rate with previous estimates is described in Section 4.3.We investigated the dependence of the water loss rate on the O escape flux imposed at the upper boundary.The escape efficiency of H is defined as the ratio of the H escape rate to the H production rate by H 2 O photolysis at and above the H 2 O photolysis peak.The water loss rate only slightly changes with the O escape flux (Figure 5).This occurs because O production by H 2 O photolysis above the photolysis peak dominates the O escape.The water loss rate minimally increases with a set O escape flux higher than ∼10 14 m −2 s −1 due to the more effective H 2 O photolysis at and above the photolysis peak.

Dependence on the UV Photon Flux
The UV photon flux from M dwarfs in PMS can be 1 or 2 orders of magnitude greater than that in MS (Fleming et al. 2020;Johnstone et al. 2021).Here, we estimate the change in the water loss rate with time by applying the evolutionary track of the stellar UV flux; derived by Johnstone et al. (2021), this simply follows the same evolutionary track as that of the bolometric luminosity, assuming a stellar mass of 0.1 M e and a present age of 7.6 Gyr by referring to the properties of TRAPPIST-1 (Burgasser & Mamajek 2017), regardless of wavelength.The water loss rate minimally changes with the UV flux (Figure 6).Under an intense UV flux, eddy diffusion enhances the downward transport of H while relatively suppressing the upward transport and escape of H because the H 2 O photolysis peak altitude descends to lower regions.In addition, the increase in eddy diffusion at the photolysis peak enhances the upward supply of O 2 such that the UV shielding of O 2 in the region above the photolysis peak increases with the magnitude of the UV photon flux.

Dependence on the Eddy Diffusion Coefficient
In this section, we investigate the dependence of the water loss rate on the magnitude of the eddy diffusion coefficient.The water loss rate increases with the magnitude of the eddy diffusion coefficient, as shown in Figure 7.Under high eddy diffusion conditions, H 2 O is effectively transported upward; thus, the photolysis peak altitude increases, where the upward transport of O 2 and its associated UV shielding are suppressed, leading to the enhancement of H 2 O photolysis at the photolysis peak.Furthermore, a portion of the photon flux in the wavelength  range from 180 to 230 nm in NUV, where the absorption of H 2 O is weak, is absorbed at high altitudes.At high altitudes, molecular diffusion provides an efficient upward flux of H; thus, the photon flux not only in the FUV but also in a portion of the NUV is used to produce escaping H and water loss.In our model, the energy conservation is not considered accurately.We adopt a fixed temperature profile following the previous works using a 1D photochemical model (e.g., Chaffin et al. 2017).We have confirmed that uncertainty in the temperature profile has minor effects on the H 2 O photolysis rate and hydrogen escape rate; when varying stratospheric temperature from 400 to 600 K, the calculated water loss rate becomes 5.4 (TO Gyr −1 ), which is by a factor of 1.3 lower than the standard value.While temperature changes do not significantly affect the main results in this study, it is part of our future work to achieve self-consistent solutions by incorporating radiative transfer calculations.

Comparison with Previous Estimates
The calculated water loss rate in the standard settings is 6.78 TO Gyr −1 (standard value).The calculated water loss rates with the parameters provided in Section 4.1 are less than approximately twice the standard value.The standard value is 3.27 times lower than the diffusion-limited escape rate (22.2 TO Gyr −1 ) and 5.35-414 times lower than the energylimited escape rate under stellar XUV fluxes from 1.20 to 93.0 W m −2 , as described in Section 4.1.2.Here, the molar mixing ratio of O to H is assumed to be 1/2 at the homopause, with the binary diffusion coefficient for H and O adopted by Luger & Barnes (2015).The water loss rate limited by H 2 O photolysis is significantly lower than the energy-limited escape rate under the high XUV flux in the PMS phase.

Implication for the Habitability of Planets around M Dwarfs
Previous studies that investigated the evolution of pure H 2 O atmospheres in Earth-like planets orbiting around M dwarfs indicated that several to several tens of times the amount of the terrestrial ocean can be lost during the PMS (e.g., Luger & Barnes 2015).These results suggested that the planets currently in the HZ around M dwarfs should have dry surface environments.
In contrast, our results showed that water loss is significantly limited by H 2 O reproduction reactions and by UV shielding due to photochemically produced O 2 .Our results suggest that if a planet formed with several tens of times the terrestrial ocean around an M dwarf, surface water can be retained after the end of the runaway greenhouse phase, which lasts up to 1 Gyr.Although this study considers only the H 2 O-dominated phase, accumulated O 2 and carbon-bearing species such as CO 2 may also suppress H 2 O photolysis.A hydrogen-rich atmosphere possibly formed during accretion can survive during the longterm PMS phase due to the suppression of hydrodynamic escape by radiative cooling effects of H 2 O and other radiatively active species, which may also inhibit water loss (Yoshida et al. 2022).A cold trap of H 2 O in the upper atmosphere can prohibit water loss near the end of the runaway greenhouse phase (e.g., Kasting 1988).A long-lived magma ocean during the PMS phase may also provide a barrier against water loss through the dissolution of water in the magma (Dorn & Lichtenberg 2021;Moore et al. 2023).These water-protecting processes may have led to the formation of habitable environments on some terrestrial planets around M dwarfs after the end of the runaway greenhouse phase even though these planets experienced longlasting severe conditions.

Summary
We investigated the water loss from H 2 O-dominated atmospheres on an Earth-like planet around an M dwarf in the runaway greenhouse phase by applying a 1D photochemical model.According to our results, the reproduction reactions of H 2 O occur as efficiently as H 2 O photolysis below the H 2 O photolysis peak.Moreover, photochemically produced O 2 suppresses H 2 O photolysis by providing UV shielding above the photolysis peak.As a result of these processes, the water loss rate is reduced to 6.78 TO Gyr −1 , which is several to several hundred times lower than those reported in previous studies.Our results suggest that habitable environments can be formed after the end of the runaway greenhouse phase on some terrestrial planets currently in the HZ around M dwarfs, and this probability is greater than that suggested by previous estimates.Figures The chemical reactions considered in this study are listed in Table A1. Figure A1 shows the model input: the initial density profile, the temperature profile, and the incoming stellar UV spectrum.Figure A2 shows the diffusion coefficient profiles in the calculated H 2 O-dominated atmosphere with the standard parameters.
Figure A3 shows the calculated density profiles of all species and the optical depth profiles in the entire wavelength range in our model.Figure A4 displays the reaction rate profiles of the reactions providing O 2 at the photolysis peak.OH provided by the photolysis of R1 reacts with O to produce O 2 via R30 at the photolysis peak.

A.2. Profile of H above the Photolysis Peak
This section explains the reason why the number density profile of H above the photolysis peak does not follow its scale height but follows the O scale height as shown in Figure 1.
Here we assume that this region consists of only H and O with an isothermal temperature.Following Catling & Kasting (2017), the molecular diffusion flux of H is given by the following: Above the photolysis peak, O remains stationary (the right panel in Figure 2) with the following density profile: where h O is the scale height of O and z 0 is the reference altitude.Substituting Equation (A3) in Equation (A2), the density profile of H is given by the integration of Equation (A2) with respect to z as follows: ( ) does not exceed β.When Φ H = 0, the density profile of H follows its scale height.However, when the molecular diffusion flux of H is significant as in our calculation, i.e.,

Figure 1 .
Figure 1.Density profiles of the main species: H, H 2 , H 2 O, O, O 2 , and OH.The dashed horizontal line represents the peak altitude of H 2 O photolysis.The blue-shaded horizontal line represents the altitude of the homopause.
Figure 2 shows the profiles of the main photochemical reaction rates (left panel) and the diffusion fluxes of H, O 2 , and O (right panel).The photolysis efficiency of H 2 O (violet dashed line in the left panel) is defined as the ratio of the net H 2 O photolysis rate to the H 2 O photolysis rate at each altitude; here, the net H 2 O photolysis rate is calculated by subtracting the H 2 O reproduction rate from the H 2 O photolysis rate.The atmospheric pressure at which the optical depth becomes unity depending on the UV wavelength is shown in Figure 3.The photon flux in the wavelength range from 115 to 230 nm is involved in H 2 O photolysis (green line in Figure

Figure 2 .
Figure 2. Left panel: profiles of the main photochemical reaction rates.The violet dashed line represents the photolysis efficiency of H 2 O, which is defined as the ratio of the net H 2 O photolysis rate to the H 2 O photolysis rate at each altitude.Right panel: profiles of the diffusion fluxes of H, O 2 , and O.The dashed and solid lines correspond to the upward and downward fluxes, respectively.

Figure 3 .
Figure 3. Atmospheric pressure where the optical depth becomes unity and the absorption cross sections of H 2 O and O 2 as a function of the UV wavelength.The green and orange lines correspond to the absorption cross sections of H 2 O and O 2, respectively.The dashed line represents the atmospheric pressure at which H 2 O photolysis reaches a maximum.

Figure 4
photon flux from 115 to 230 nm, is mainly absorbed by H 2 O (Figure3) without contributing to the water loss.The hydrogen escape to space is regulated by the photolysis efficiency of H 2 O.The escape flux is 3.94 × 10 16 m −2 s −1 , which is equivalent to a water loss of 6.78 TO Gyr −1 .This value is lower than the energy-limited and diffusion-limited escape rates suggested by previous studies (e.g.,Luger & Barnes 2015).This is due to the H 2 O photolysis limit resulting from H 2 O reproduction and UV shielding by O 2 , as described above.A detailed comparison of the escape rate with previous estimates is described in Section 4.3.Figure4summarizes the efficiency of H 2 O photolysis in a H 2 O-dominated atmosphere calculated with the standard parameters.H 2 O photolysis is significantly suppressed by efficient H 2 O reproduction reactions below the photolysis peak and by the subsequent UV shielding due to O 2 .As a result, only 13% of the total photon flux involved in H 2 O photolysis contributes to H 2 O photolysis at and above the photolysis peak and subsequent water loss.

Figure 4 .
Figure 4. Efficiency of photolysis-induced water loss in a H 2 O-dominated atmosphere.

Figure 5 .
Figure 5. Dependence on the O escape flux.The upper panel shows the water loss rate (blue line) and the escape efficiency of H (orange line).The lower panel shows the fraction of the photon flux in the wavelength range from 115 to 230 nm absorbed by O 2 in the region above the H 2 O photolysis peak (blue line), that absorbed by H 2 O in the region below the photolysis peak (orange line), and that directly contributes to the water loss (green line).

Figure 7 .
Figure 7. Dependence on the eddy diffusion coefficient.The upper panel shows the water loss rate (blue line) and the escape efficiency of H (orange line).The lower panel shows the fraction of the photon flux in the wavelength range from 115 to 230 nm absorbed by O 2 in the region above the H 2 O photolysis peak (blue line), that absorbed by H 2 O in the region below the photolysis peak (orange line), and that directly contributes to the water loss (green line).

Figure 6 .
Figure 6.Change in the water loss rate following the evolutionary track of the stellar UV flux derived by Johnstone et al. (2021).The upper panel shows the water loss rate (blue line) and the evolutionary track of bolometric luminosity (violet line).The middle panel shows the escape efficiency of H.The lower panel shows the fraction of the photon flux in the wavelength range from 115 to 230 nm absorbed by O 2 in the region above the H 2 O photolysis peak (blue line), that absorbed by H 2 O in the region below the photolysis peak (orange line), and that directly contributes to the water loss (green line).
Φ H has a maximum value called the diffusion-limited flux:


, the first term on the right side of Equation (A4) rapidly decreases to zero with altitude and the second term becomes dominant such that the density profile of H forms a structure similar to that of O.

Figure A1 .
Figure A1.Left panel: initial density profile composed of H 2 O vapor in hydrostatic equilibrium with a surface pressure equivalent to a mass of 1 TO.Middle panel: fixed temperature profile based on Kasting (1988).Right panel: input UV spectrum estimated for TRAPPIST-1 in the wavelength range from 115 to 1000 nm (Wilson et al. 2021).

Figure A2 .
Figure A2.Diffusion coefficient profiles.The molecular diffusion coefficients are fromHunten (1973), and this eddy diffusion coefficient is an Earth-like value(Massie & Hunten 1981).The blue-shaded horizontal line represents the altitudes of the homopause.

Figure A3 .
Figure A3.Left panel: the density profile of the considered species: H 2 O, H, OH, H 2 , O( 1 D), O 3 , O 2 , O, HO 2 , and H 2 O 2 .Right panel: optical depth profiles in the wavelength range from 115 to 1000 nm.

Figure A4 .
Figure A4.Reaction rate profiles for the reaction providing O 2 at the photolysis peak; H 2 O photolysis R1 followed by R30.