Temporal Evolution of the High-energy Irradiation and Water Content of TRAPPIST-1 Exoplanets

A high-quality reference spectrum for the intrinsic stellar Lyα line of TRAPPIST-1 was built by B17 as the average of all spectra obtained in Visits 1, 2, and 3, excluding the spectral ranges contaminated by air glow emission or showing hints of flux variations. We first compared the Lyα line spectra obtained in each exposure of Visit 4 with this reference spectrum, searching for absolute flux variations over ranges covering more than STIS spectral resolution (about two pixels). As can be seen in Figure 4, the flux in the red wing of Visit 4 spectra is systematically higher than or equal to the reference spectrum, with a significant (>3σ) increase in orbits 2, 3, and 5. Similarly, the reference spectrum shows very little emission in the blue wing at velocities lower than about −160 km s⁻¹, whereas the flux in Visit 4 spectra is systematically higher than or equal to the reference in this range, with a marginal (>2σ) increase in orbits 2, 3, and 4. Therefore it comes as a surprise that Visit 4 displays an overall lower flux in the blue wing between about −120 and −55 km s⁻¹, with significant decreases in orbits 2 and 5 compared to the reference spectrum.

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To investigate the source of these differences, we studied the evolution of the Lyα line over the three-month span of our observations, averaging all spectra within each visit and integrating them in four complementary spectral bands (Figure 5). In agreement with the above spectral analysis, the flux in the symmetric wing bands (±[130; 250] km s⁻¹) did not vary significantly from Visit 1 to Visit 3 (top panels in Figure 5) but increased noticeably in Visit 4. This variation likely traces an increase in the emission of the intrinsic stellar Lyα line (Section 3.2). However, while we do not expect similar absolute flux levels in the observed wing bands because of interstellar medium (ISM) absorption in the red wing of the Lyα line (see B17), it is surprising that the relative flux increase in Visit 4 is much larger and more significant in the red wing than in the blue wing. The flux in the blue wing even shows a marginal decline at lower velocities ([−130; −50] km s⁻¹) from Visit 1 to Visit 4 (third panel in Figure 5), while the flux at velocities closer to the Lyα line core ([−50; −25] km s⁻¹) remained at about the same flux level (bottom panel in Figure 5).

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This comparison suggests that the shape of the line evolved between Visit 1 and Visit 4, with a change in the spectral balance of the Lyα line flux between the blue and the red wings. The search for absorption signatures possibly caused by the transit of TRAPPIST-1c in Visit 4 is made difficult by this evolution, since the spectra from Visits 1–3 cannot be used as a reference for the out-of-transit stellar line. We investigate this question in more detail in Section 3.3.

To further study the evolution of the stellar Lyα line, we sought to reconstruct its intrinsic profile at the time of Visit 4 using the same approach as in B17. We assumed a Gaussian line profile, which was absorbed by the ISM using the column density derived in B17. The model was then convolved by the STIS line-spread function (LSF) and compared with the average of all spectra in Visit 4. We excluded the pixels beyond ±250 km s⁻¹ from the fit, where the wings of the line become too faint. We also excluded the core of the line fully absorbed by the ISM and possibly biased by the air glow correction.

Contrary to the reconstruction performed in B17 on Visit 1–3 reference spectrum, in Visit 4, we found that it was not possible to fit the entire line profile well with our theoretical Gaussian model. Trying out different spectral ranges for the reconstruction, we found that a good fit was obtained when excluding the band [−190; −55] km s⁻¹, even in the range contaminated by the air glow (Figure 6). This best-fit line profile for Visit 4 peaks at about the same flux level as the reference spectrum in Visits 1–3 but displays much broader wings, which is consistent with the stability of the Lyα line core and the variability of its wings noted in Section 3.1. Assuming this best-fit model is correct, it would suggest that both wings of the Lyα line have increased similarly from Visits 1–3 to Visit 4, but that the band [−190; −55] km s⁻¹ is reabsorbed by an unknown source in this epoch. Alternatively, the intrinsic Lyα line of TRAPPIST-1 could have become asymmetric in Visit 4. Both scenarios would explain why the line profile observed in Visit 4 appears unbalanced between the blue and the red wing (see Section 3.1 and Figure 5). We further investigate this question in Section 3.3.

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The Lyα line arises from different regions of the stellar atmosphere, ranging from the low-flux wings of the line formed in the colder regions of the lower chromosphere to the core of the line, which is emitted by the hot transition region between the upper chromosphere of the star and its corona. M dwarfs display a lower chromospheric emission than earlier-type stars but equivalent amounts of emission from the transition region and the corona (see Youngblood et al. 2016 and references). This trend might be even more pronounced for late-type M dwarfs like TRAPPIST-1, since B17 suggested that this ultracool dwarf might have a weak chromosphere compared to its transition region and corona, based on its Lyα and X-ray emission (Wheatley et al. 2017) and the shape of its Lyα line. Interestingly, though, the broader wings of the TRAPPIST-1 Lyα line in Visit 4 might trace an increase in the temperature and emission of the stellar chromosphere.

In addition to the Lyα line, the spectral range of the STIS/G140M grating covers the Si iii (1206.5 Å) and O v (1218.3 Å) transitions and the N v doublet (1242.8 and 1238.8 Å). We averaged our nine STIS spectra of TRAPPIST-1 to search for these stellar emission lines, and we detected the N v doublet (Figure 7). The two lines of the doublet were averaged and fitted with a Gaussian profile. Assuming that the width of the line is controlled by thermal broadening, we obtained a best-fit temperature on the order of 3 × 10⁵ K, which is close to the peak emissivity of the N v lines at 2 × 10⁵ K. We further derived a total flux in the doublet of about 7.3 × 10⁻¹⁷ erg s⁻¹ cm⁻². This line strength is consistent with the fits to the X-ray spectrum of TRAPPIST-1 by Wheatley et al. (2017), which predict N v line strengths of 1 × 10⁻¹⁷ and 100 × 10⁻¹⁷ erg s⁻¹ cm⁻² for two different models designed to span the possible range of EUV luminosities (the APEC and cemekl models, respectively). The dispersion is larger in the regions of the other stellar lines, and there is no clear evidence for their detection. More observations will be required to characterize the chromospheric and coronal emission from TRAPPIST-1.

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B17reported a hint of absorption in the blue wing of the Lyα line in Visit 3, possibly caused by a hydrogen exosphere trailing TRAPPIST-1c. The long-term evolution of the intrinsic Lyα line prevents us from comparing the different visits to one another, and we thus took the average of all spectra in Visit 4 as a reference to search for the presence of an exosphere in this epoch. We highlight the long-term variability in Figure 8, where we plotted the flux integrated in the same spectral bands as in Figure 5 but phase-folded over the TRAPPIST-1c orbital period. The spectrum measured in Visit 3 shows a lower flux level at high velocities in the wing bands compared to the average flux in Visit 4 but similar flux levels in other parts of the line, consistent with the changes in the intrinsic line shape discussed in Section 3.1. No significant deviations to the average spectrum were found in Visit 4, and in particular no variations that would be consistent with the transit of an extended exosphere surrounding TRAPPIST-1c. More observations will be required to assess the possible short-term variability in the Lyα line and to search for residual absorption signatures.

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If the Lyα line model derived in Section 3.2 corresponds to the actual intrinsic line of the star in Visit 4, what is the origin of the much lower flux observed in the band [−190; −70] km s⁻¹ (Figure 6)? Despite our careful extraction of the stellar spectrum (Section 2), it is possible that the stronger air glow in this epoch was overcorrected at some wavelengths. However, the air glow becomes negligible beyond about −100 km s⁻¹ and therefore cannot explain the lower flux observed at larger velocities. If confirmed, this feature might imply that colder hydrogen gas is moving away from the star at high velocities and is absorbing about half of the Lyα flux in this velocity range. This absorption is unlikely to originate from TRAPPIST-1c alone, as it occurs in all orbits of Visit 4 and displays no correlation with the planet transit. Radiative braking is less efficient around TRAPPIST-1 than around the M2.5 dwarf GJ 436 (Bourrier et al. 2015) because its radiation pressure is about three times lower and thus more than five times lower than stellar gravity (B17). This could lead to the formation of giant hydrogen exospheres around the TRAPPIST-1 planets even larger than the one surrounding the warm Neptune GJ 436b (Ehrenreich et al. 2015) and possibly extending both behind and ahead of the planets because of gravitational shear (Bourrier et al. 2015). Furthermore, the XUV spectrum of TRAPPIST-1 (Section 4.2) yields photoionization lifetimes for neutral hydrogen atoms ranging from ∼20 hr at the orbit of TRAPPIST-1b to nearly 600 hr at the orbit of TRAPPIST-1h (i.e., longer than the planetary period). Because of the very low Lyα and UV emission from TRAPPIST-1, neutral hydrogen exospheres could thus extend along the entire planetary orbits and could even cross the orbit of several planets. This suggests not only that some planets could be accreting the gas escaped from their companions, but also that a large volume of the TRAPPIST-1 system could be filled with neutral hydrogen gas, providing a possible explanation for the persistent absorption signature in Visit 4. Given that the evaporating planets sustaining this system-wide hydrogen cloud would be in different relative positions at a given epoch, the structure of the cloud and its absorption signature would be highly variable over time, which could explain why it was not detected in Visit 1. We note, though, that the hydrogen cloud would still have to be accelerated to very high velocities away from the star (possibly through charge exchange with the stellar wind; Holmström et al. 2008; Ekenbäck et al. 2010; Bourrier et al. 2016b) to explain the velocity range of the measured absorption.

Alternatively, this variation could have a stellar origin. This is also an intriguing possibility, because the Lyα line of TRAPPIST-1 was well approximated with a Gaussian profile in previous visits (B17) and the Lyα line profiles of later-type M dwarfs do not show evidence for strong asymmetries (e.g., Bourrier et al. 2015; Youngblood et al. 2016). The intrinsic Lyα line of TRAPPIST-1 might have become asymmetric in Visit 4 because of variations in the up-flows and down-flows of stellar hydrogen gas, or because of absorption by colder hydrogen gas at high altitudes in the stellar atmosphere. Filaments made of partially ionized plasma are, for the Sun, a hundred times cooler and denser than the coronal material in which they are immersed, and can thus be optically thick in the Lyα line (e.g., Parenti 2014). The large velocities of the putative absorber of the TRAPPIST-1 Lyα line might indicate that we witnessed the eruption of a filament that was expelled by a destabilization of the stellar magnetic field. Such eruptions can reach large distances and velocities (between 100 and 1000 km s⁻¹ for the Sun, e.g., Schrijver et al. 2008). In any case, our new observations of TRAPPIST-1 raise many questions about the physical mechanisms behind the emission of the Lyα line in an ultracool dwarf.

Despite these unknowns, our best-fit Gaussian profile can be used to estimate a conservative upper limit on the total Lyα irradiation of the planets at the epoch of Visit 4 (Section 4.2). More observations of TRAPPIST-1 at Lyα will be required to beat down the photon noise, to assess the effects of stellar variability, and to reveal absorption signatures caused by the putative planets' exospheres. The long-term variability of the TRAPPIST-1 Lyα line emphasizes the need for contemporaneous observations obtained outside and during the planet's transits.

We calculated mass loss rates from TRAPPIST-1 planets using an improved formalism based on the energy-limited formula (e.g., Lecavelier des Etangs 2007; Selsis et al. 2007; Luger & Barnes 2015):

$$\begin{eqnarray}&&{\dot{M}}^{\mathrm{tot}}=\epsilon \,{\left(\displaystyle \frac{{R}_{\mathrm{XUV}}}{{R}_{{\rm{p}}}}\right)}^{2}\,\displaystyle \frac{3\,{F}_{\mathrm{XUV}}({a}_{{\rm{p}}})}{4\,G\,{\rho }_{{\rm{p}}}\,{K}_{\mathrm{tide}}},\end{eqnarray} \tag{ 1 }$$

with ${F}_{\mathrm{XUV}}({a}_{{\rm{p}}})$ the XUV irradiation. The heating efficiency is the fraction of the incoming energy that is transferred into gravitational energy through mass loss. As in Bolmont et al. (2017) and Ribas et al. (2016), we calculated using 1D radiation hydrodynamic mass loss simulations based on Owen & Alvarez (2016). The efficiency varies with the incoming XUV radiation. For example, for today's estimated XUV flux (see next section and Table 4), we obtain values of 0.064, 0.076, 0.089, 0.099, 0.107, 0.112, and 0.115, respectively, for planets b to h. These efficiencies are on the same order as the 10% assumed by Wheatley et al. (2017) but larger than the 1% assumed by Bourrier et al. (2017).

Table 4.  High-energy Emission from TRAPPIST-1

Wavelength Domain	XUV ($0.5\mbox{--}100$ nm)		Lyα
	${L}_{\mathrm{XUV}}$ (erg s−1)	${\mathrm{log}}_{10}({L}_{\mathrm{XUV}}/{L}_{\mathrm{bol}})$	$\mathrm{Ly}\alpha$ (erg s−1)	${\mathrm{log}}_{10}(\mathrm{Ly}\alpha /{L}_{\mathrm{bol}})$
Lower estimate	$5.26\times {10}^{26}$	−3.58	$1.44\times {10}^{26}$	−4.15
Mean estimate	$6.28\times {10}^{26}$	−3.51	$1.62\times {10}^{26}$	−4.09
Upper estimate	$7.30\times {10}^{26}$	−3.44	$1.81\times {10}^{26}$	−4.05

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In the following sections, we consider a constant XUV flux and an evolving XUV flux. For the latter assumption, is computed accordingly. In the energy-limited formula, the parameter ${\left(\tfrac{{R}_{\mathrm{XUV}}}{{R}_{{\rm{p}}}}\right)}^{2}$ accounts for the increased cross-sectional area of planets to XUV radiation, while K_tide accounts for the contribution of tidal forces to the potential energy (Erkaev et al. 2007). Both are set to unity for these cool and small planets (Bolmont et al. 2017). The mean density of the planets was calculated using masses derived from TTV in Gillon et al. (2017; see Table 1). We consider here that the atmospheres of the planets are mainly composed of hydrogen and oxygen and compute their joint escape using the formalism of Hunten et al. (1987). The escape rates of both elements depend on the temperature of the thermosphere, the gravity of the planet, and a collision parameter between oxygen and hydrogen. As in Bolmont et al. (2017) and Ribas et al. (2016), we adopt a thermosphere temperature of 3000 K, obtained through our hydrodynamic simulations. We caution that more detailed models, including FUV radiative transfer, photochemical schemes, and non-LTE kinetics in the rarefied gas regions of the upper atmosphere, will be required to determine accurately the outflow properties. For example, the hydrodynamic outflow of hydrogen could drag water molecules, which are only slightly heavier than oxygen atoms, upward, and they would be photodissociated at high altitudes into more escaping oxygen and hydrogen atoms. Nonetheless, our assumptions likely maximize the XUV-driven escape (see Bolmont et al. 2017), and our estimates of the water loss should be considered as upper limits.

To calculate the planetary mass losses, we needed estimations of the XUV irradiation from TRAPPIST-1 over the whole history of the system. In a first step, we calculated the present-day stellar irradiation. We used the same value as in Bourrier et al. (2017) for the X-ray emission (5–100 Å ), studied by Wheatley et al. (2017). The stellar EUV emission between 100 and 912 Å is mostly absorbed by the ISM but can be approximated from semiempirical relations based on the Lyα emission. The theoretical Lyα line profile derived for Visit 4 (Section 3.2) yields an upper limit on the total Lyα emission of 7.5 ± 0.9 × 10⁻² erg s⁻¹ cm⁻² at 1 au. This is larger than the emission derived for previous visits (${5.1}_{-1.3}^{+1.9}\,\times {10}^{-2}$ erg s⁻¹ cm⁻², B17), in agreement with the increase in flux suggested by observations (Section 3.1). We chose to consider those two estimates as lower and upper limits on the present Lyα emission of TRAPPIST-1, and we used the Linsky et al. (2014) relation for M dwarfs to derive corresponding limits on the EUV flux. Table 4 gives our best estimate for today's range of fluxes emitted by TRAPPIST-1 at Lyα and between 5 and 912 Å. We computed the ratio of these two fluxes to the bolometric luminosity and obtained a value of log ${}_{10}({L}_{\mathrm{XUV}}/{L}_{\mathrm{bol}})$ between −3.39 and −3.73, which is about a factor of 2.5 lower than that estimated from the X-ray flux by Wheatley et al. (2017).

In a second step, we estimated the past stellar irradiation. We consider that when the planets were embedded in the protoplanetary disk, they were protected from irradiation and did not experience mass loss. In that frame, water loss began at the time when the disk dissipated, which we assume to be 10 Myr (Pascucci et al. 2009; Pfalzner et al. 2014; Pecaut & Mamajek 2016). We investigated two different scenarios, depending on our assumptions for the temporal evolution of the XUV emission after the disk dissipation:

• 1.  
  A constant ${L}_{\mathrm{XUV}}$ equal to today's range of emission. This assumption might be supported by the X-ray flux of TRAPPIST-1, which is consistent with a saturated emission typical of earlier-type M dwarfs, according to Wheatley et al. (2017).
• 2.  
  An evolving ${L}_{\mathrm{XUV}}$, considering the ratio ${L}_{\mathrm{XUV}}/{L}_{\mathrm{bol}}$ to be constant throughout the history of the star. The ratio was set to the present-day estimate of the star luminosities (see Table 4). We used the evolutionary models of Baraffe et al. (2015) to compute the evolution of the bolometric luminosity.

Here we set the stellar mass to its nominal value of 0.091 ${M}_{\odot }$. Considering the range allowed by the uncertainties would slightly change our water loss estimates, at the time the planet reached the HZ for the constant XUV flux prescription and at all ages for the evolving XUV flux prescription. We estimate that the difference would be less than 4% at an age of 8 Gyr.

Using the energy-limited model in Section 4.1 and our estimates for the XUV irradiation in Section 4.2, we calculated the mass loss from TRAPPIST-1 planets over time. In order to calculate the hydrogen loss, we used method (2) of Ribas et al. (2016), which consists in calculating the ratio between the oxygen and hydrogen fluxes as a function of the XUV luminosity. We consider here an infinite water reservoir. This allows us to consider that the ratio of hydrogen and oxygen remains stoichiometric even though the loss is not stoichiometric. This provides us with an upper limit on the mass loss (see Ribas et al. 2016 for a discussion on the effect of a finite initial water reservoir). The loss is given in units of Earth-ocean-equivalent content of hydrogen (referred to as $1\,{{EO}}_{H}$). In other words, the mass loss is expressed in units of the mass of hydrogen contained in one Earth ocean ($1.455\times {10}^{20}$ kg, with an Earth ocean mass corresponding to 1.4 × 10²¹ kg). For example, we estimate a current mass loss from planet b of $0.008\,{M}_{\mathrm{ocean}}$/Myr for the nominal XUV flux (Table 4). This corresponds to escape rates of oxygen and hydrogen from planet b of $2.9\times {10}^{8}$ g s⁻¹ and $4.3\times {10}^{7}$ g s⁻¹, respectively. The values for the other planets can be found in Table 5. Figure 10 shows the evolution of the hydrogen loss from the TRAPPIST-1 planets as a function of the age of the system in the two scenarios assumed for the evolution of the XUV flux. Table 6 gives the corresponding mass loss at different times of interest.

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Table 5.  Current Mass Loss Rate, Hydrogen Loss Rate, and Oxygen Loss Rate for the TRAPPIST-1 Planets

Planet	b	c	d	e	f	g	h
Mass loss (${M}_{\mathrm{ocean}}/$Myr)	$8.2\times {10}^{-3}$	$2.9\times {10}^{-3}$	$2.3\times {10}^{-3}$	$1.7\times {10}^{-3}$	$1.4\times {10}^{-3}$	$6.4\times {10}^{-4}$	$2.9\times {10}^{-4}$
Hydrogen loss (g s−1)	$4.3\times {10}^{7}$	$2.3\times {10}^{7}$	$1.3\times {10}^{7}$	$1.2\times {10}^{7}$	$1.1\times {10}^{7}$	$1.2\times {10}^{7}$	$4.3\times {10}^{6}$
Oxygen loss (g s−1)	$2.9\times {10}^{8}$	$1.0\times {10}^{8}$	$8.2\times {10}^{7}$	$5.7\times {10}^{7}$	$4.7\times {10}^{7}$	$1.5\times {10}^{7}$	$7.5\times {10}^{6}$

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Table 6.  Cumulative Hydrogen Loss (in EO_H) for Different Times (Table Corresponding to Figure 10)

			H loss (EOH)
	Planet	Mass	${T}_{\mathrm{HZ}}$	${T}_{\mathrm{HZ}}$	3 Gyr	8 Gyr
		(${M}_{\oplus }$)	(1.5 ${S}_{\oplus }$)	(0.84 ${S}_{\oplus }$)
${L}_{\mathrm{XUV}}$ evol	b	0.85	⋯	⋯	29.2–35.4	71.5–86.9
	c	1.38	⋯	⋯	15.2–17.6	38.1–44.1
	d	0.41	2.35–2.87	⋯	9.12–11.2	22.2–27.2
	e	0.62	1.20–1.46	2.05–2.49	7.98–9.57	19.6–23.4
	f	0.68	0.70–0.86	1.14–1.39	7.46–8.89	18.4–21.8
	g	1.34	0.31–0.37	0.58–0.67	7.79–8.46	20.1–21.7
	h	0.46	0.07–0.09	0.14–0.17	2.91–3.23	7.40–8.15
${L}_{\mathrm{XUV}}$ cst	b	0.85	⋯	⋯	25.3–30.7	67.5–82.1
	c	1.38	⋯	⋯	13.7–15.8	36.5–42.1
	d	0.41	1.17–1.44	⋯	7.80–9.57	20.8–25.5
	e	0.62	0.47–0.56	1.12–1.34	6.95–8.28	18.6–22.1
	f	0.68	0.21–0.25	0.45–0.54	6.52–7.70	17.4–20.6
	g	1.34	0.14–0.15	0.31–0.33	7.34–7.89	19.6–21.1
	h	0.46	0.02–0.02	0.05–0.06	2.69–2.94	7.18–7.86

Note. The parameter T_HZ is the age at which a planet enters the HZ (see Table 3). The two values given for each column correspond to the uncertainty coming from the different luminosity prescriptions (between low and high; see Table 4).

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Planets b and c never reach the HZ, so they are expected to have lost water via the runaway greenhouse mechanism (Section 4) throughout the full lifetime of the star. If TRAPPIST-1 is 3 Gyr old (the lower estimate of the age given in Luger et al. 2017), planet b potentially lost more than 20 EO_H and planet c more than $10\,{{EO}}_{H}$. If planets b and c formed with an Earth-like water content, they are likely dry today, whatever the assumptions on the XUV flux and the age of the star. Alternatively, they might have formed as ocean planets (Léger et al. 2004), in which case a loss of 20 Earth oceans for planet b would represent only 0.5% of its mass. Currently this scenario is not favored by the planet formation model proposed by Ormel et al. (2017), which excludes water fractions larger than about 50%, and by the densities of planets b and c derived observationally by Gillon et al. (2017) and Wang et al. (2017), although we note that they still allow for a significant water content.

If we consider that the water loss only occurs during the runaway phase, planets d to f lost less than $4\,{{EO}}_{H}$ before reaching the HZ, and planets g and h lost less than $1\,{{EO}}_{H}$. In that scenario, the outer planets of the TRAPPIST-1 system might thus still harbor substantial amounts of water, especially planets e to h if the low densities derived by Wang et al. (2017) are confirmed. What if hydrodynamic water loss continued once the planets reached the HZ? After 3 Gyr, we estimate that planet g and those closer in would have lost more than $7\,{{EO}}_{H}$. After 8 Gyr, they would have lost more than $20\,{{EO}}_{H}$ (Table 6). Interestingly, the relation obtained by Guinan et al. (2016) for M0–5 V dwarf stars yields an age of about 7.6 Gyr for TRAPPIST-1, using the X-ray flux obtained by Wheatley et al. (2017) in the ROSAT band (0.14 erg s⁻¹ cm⁻² at 1 au). This age is at the upper limit of the range derived by Luger et al. (2017), and if confirmed it suggests that all TRAPPIST-1 planets have lost substantial amounts of water over the long history of the system. Refined estimates of the planet densities will, however, be necessary to determine whether they still harbor a significant water content. We also note that our estimates for the water loss once planets are in the HZ (Table 6) are probably upper limits. If the planet is able to retain its background atmosphere, the tropopause is expected to act as an efficient cold trap, preventing water from reaching the higher parts of the atmosphere (e.g., Wordsworth & Pierrehumbert 2014; Turbet et al. 2016). In that case, water escape would be limited by the diffusion of water through the cold trap. However, if the background pressure is low, the water vapor mixing ratio increases globally in the atmosphere (Turbet et al. 2016), and hydrogen escape is no longer limited by the diffusion of water. The farther out the TRAPPIST-1 planets, the more likely they would have been able to sustain an important background atmosphere, thus protecting the water reservoir once in the HZ.

While the radii of the TRAPPIST-1 planets are known to a good precision, their masses remain very uncertain. We set today's XUV luminosity to the nominal estimate in Table 4, and we investigated the effect of changing the value of the planetary mass within the range of uncertainty estimated by Gillon et al. (2017; see Table 1). For planet h, we investigated a range of compositions from 100% ice to 100% iron, which corresponds to masses between $0.06\,{M}_{\oplus }$ and $0.86\,{M}_{\oplus }$. Figure 11(b) shows that for most planets the uncertainty on their mass dominates the final uncertainty on the mass loss, compared to the effect of varying the XUV irradiation with its uncertainty (see also Luger & Barnes 2015). This is because hydrogen loss is not only inversely proportional to the planetary mass (see Equation (1)) but also linked to the ratio of the escape fluxes of hydrogen and oxygen (${r}_{{\rm{F}}}={F}_{{\rm{O}}}/{F}_{{\rm{H}}}$), which is a function of the crossover mass (see Hunten et al. 1987; Bolmont et al. 2017). For a given value of the XUV luminosity and an infinite initial water reservoir, the hydrogen mass loss is described by a polynomial in ${M}_{{\rm{p}}}$ of the form $\alpha /{M}_{{\rm{p}}}+\beta {M}_{{\rm{p}}}$, where α and β are constants depending, for example, on the radius of the planet (we refer to Equations (8) and (9) of Bolmont et al. 2017 to derive these values). The mass flux decreases with increasing planetary mass, and the ratio of the flux of oxygen over the flux of hydrogen decreases with increasing planetary mass. For high masses, the mass flux is lower (gravity wins over cross section), but the mass loss mainly occurs via hydrogen escape. This results in a high hydrogen escape rate. For low masses, the outflow is a mixture of hydrogen and oxygen, meaning less hydrogen escapes for an equal energy input, but the overall mass flux is higher. For a low-enough mass, the increased overall mass flux does more than compensate, and this also results in a high hydrogen loss. The hydrogen loss is therefore high for very low mass and very high mass planets with a minimum for intermediate mass. Figure 11(a) shows this dependence of the hydrogen flux with mass for each planet of the system. For all planets but planet b, the minimum loss is achieved for an intermediate mass in the range allowed by the observations (this range is displayed as a thicker line on the graph). The lower curves delimiting the colored areas in Figure 11(b) correspond to the hydrogen loss calculated for this intermediate mass for all planets but b. For planet b, the minimum hydrogen loss is obtained for the highest mass allowed by the observations.

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In the end, the uncertainty on the hydrogen loss comes mainly from the uncertainty on mass rather than the uncertainty on the prescribed luminosity (although we note that our calculations assume an unlimited water supply, yielding upper limits on the water losses; see Section 4.3). The uncertainty on the hydrogen loss thus ranges from about 80% for planets b and e to ∼50% for planets d and h, ∼20% for planet c and g, and only 4% for planet f. Table 7 gives the upper and lower estimates of hydrogen loss for the mass range for each planet and for the different times considered in this study. Due to the relative precision of the mass of planet f, the mass loss is relatively well constrained with our model, with a loss of less than 0.5 EO_H before reaching the HZ and less than $20\,{{EO}}_{H}$ at an age of 8 Gyr. Given that the low density of planet f is compatible with a nonnegligible water content (Gillon et al. 2017), this could indicate that water loss may not have been a very efficient process or that the planet formed with a large fraction of its mass in water.

Table 7.  Cumulative Hydrogen Loss (in EO_H) for Different Times

		Mass	H loss (EOH)				Uncertainty
	Planet	range	${T}_{\mathrm{HZ}}$	${T}_{\mathrm{HZ}}$	3 Gyr	8 Gyr	Range
		(${M}_{\oplus }$)	(1.5 ${S}_{\oplus }$)	($0.84\,{S}_{\oplus }$)			%
${L}_{\mathrm{XUV}}$ cst	b	0.13–1.57	⋯	⋯	18.8–160	50.3–429	79
	c	0.77–1.99	⋯	⋯	14.5–19.2	38.6–51.3	14
	d	0.14–0.68	1.04–3.16	⋯	6.97–21.0	18.6–56.0	50
	e	0.04–1.20	0.49–5.21	1.17–12.5	7.26–77.6	19.4–207	83
	f	0.50–0.86	0.23–0.25	0.48–0.55	6.99–7.88	18.7–21.0	6
	g	0.46–2.22	0.12–0.19	0.28–0.43	6.60–10.3	17.6–27.5	22
	h	0.06–0.86	0.02–0.05	0.05–0.13	2.59–6.84	6.92–18.3	45

Note.The parameter T_HZ is the age at which a planet enters the HZ (see Table 3). The two values given for each column correspond to the uncertainty coming from the masses (for the mean estimation of the XUV luminosity, see Table 4).

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We note that for most planets, the estimates of the masses from Wang et al. (2017) are more precise but are consistent compared to Gillon et al. (2017). However, for planet f, the mass ranges obtained by these different studies are incompatible, and using the mass from Wang et al. (2017) would lead to a higher mass loss than what is shown on Figure 11(b). More data are needed to refine our measurements of the masses of the planets of the system.

The atmospheric mass loss can be limited by the amount of hydrogen formed by photodissociation of water molecules. We computed the rate of hydrogen production driven by the FUV part of the spectrum, which is taken to be restricted to the Lyα emission (as in Bolmont et al. 2017). We note that water molecules could further be dissociated through impact with high-energy electrons, in particular those produced by the ionization of water (considering the high X-ray emission of TRAPPIST-1 and the large ionization cross section of water in the XUV; Heays et al. 2017). Figure 12 shows the hydrogen loss for the nominal irradiation (Table 4) and planet masses, and the hydrogen quantity available due to photodissociation of water for two different efficiencies: ${\epsilon }_{\alpha }=1$ (each photon leads to a dissociation; see Bolmont et al. 2017) and 0.2. We obtained the following results for a constant XUV luminosity:

• 1.  
  For planets b and c, photodissociation is not the limiting process for high efficiencies. The hydrogen loss is limited by the hydrodynamic escape as computed in the previous section;
• 2.  
  For planets d to g, photodissociation is the limiting process whatever the efficiency: the rate of hydrogen formation by photodissociation is below the escape rate of hydrogen.

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Photodissociation of water also becomes the limiting process for planets b and c if ${\epsilon }_{\alpha }\simeq 0.60$. Because of various processes, such as photon backscattering or recombination of hydrogen atoms, only a fraction of the incoming FUV photons actually result in the loss of a hydrogen atom. As a result, we do not expect photodissociation to be more efficient than 20% (Bolmont et al. 2017). It should be the limiting process for all planets, and the mass losses estimated in Section 4 can be considered as upper limits. The quantity of hydrogen available from photolysis assuming a 20% efficiency is given in Table 8 for each planet, to be compared with the hydrodynamic hydrogen loss. We note that the photolysis process does not depend on the mass of the planets (only on their radii, known to a high precision for the TRAPPIST-1 planets), so these results are more robust than the hydrogen loss estimates, which highly depend on the mass of the planets.

Table 8.  Hydrogen Loss and Hydrogen Production (in EO_H) for Different Times

			H loss (EOH)
	Planet	Mass	${T}_{\mathrm{HZ}}$	${T}_{\mathrm{HZ}}$	3 Gyr	8 Gyr
		(${M}_{\oplus }$)	(1.5 ${S}_{\oplus }$)	($0.84\,{S}_{\oplus }$)
${L}_{\mathrm{XUV}}$ evol	b	0.85	⋯	⋯	32.4–13.0	79.5–28.6
	c	1.38	⋯	⋯	16.5–6.54	41.2–14.4
	d	0.41	2.62–0.65	⋯	10.2–1.76	24.8–3.88
	e	0.62	1.33–0.39	2.28–0.55	8.79–1.44	21.6–3.18
	f	0.68	0.79–0.21	1.27–0.29	8.19–1.08	20.1–2.38
	g	1.34	0.34–0.12	0.63–0.19	8.13–0.85	20.9–1.87
	h	0.41	0.08–0.02	0.16–0.03	3.07–0.20	7.78–0.43
${L}_{\mathrm{XUV}}$ cst	b	0.85	⋯	⋯	28.1–9.30	75.0–24.8
	c	1.38	⋯	⋯	14.8–4.69	39.4–12.5
	d	0.41	1.31–0.19	⋯	8.70–1.26	23.2–3.37
	e	0.62	0.51–0.07	1.23–0.17	7.63–1.03	20.4–2.76
	f	0.68	0.23–0.02	0.50–0.05	7.12–0.77	19.0–2.06
	g	1.34	0.14–0.01	0.32–0.03	7.62–0.61	20.3–1.62
	h	0.41	0.02–<0.01	0.05–<0.01	2.82–0.14	7.52–0.37

Note. The two values given for each column correspond to the quantity of hydrogen lost (calculated for the nominal estimate of the XUV luminosity in Table 4 and the nominal masses) and the quantity of hydrogen available from photolysis (in bold, assuming an efficiency of 20%).

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Water is contained within a rocky planet's mantle in the form of hydrated minerals, as unbound fluids, or in melt. This water can be released to the surface through volcanic activity. Such outgassing processes are very different during the early stages of planet evolution (<10–100 Myr; see Solomatov 2007), where the surface and a significant fraction of the planet's mantle could be molten (magma ocean phase), and at a later stage, where the largest fraction of the planet's mantle is solid. In the following sections, we will discuss the amounts of outgassed water during the magma ocean phase of the TRAPPIST-1 planets, assuming they are rocky, and we compute the water outgassing rates for the later stage of subsolidus convection as a function of time.

5.2.1. Outgassing from a Magma Ocean

5.2.2. Outgassing after the Magma Ocean Phase

5.2.3. Methods

5.2.4. Results

We find that after the magma ocean phase, the TRAPPIST-1 planets can outgas significant amounts of water, especially the more massive ones. We show in Figure 13—while accounting for significant uncertainties in structure and model parameters—the range of minimal planet ages where outgassing can occur as a function of planet mass, assuming that the system is not older than 8 Gyr. Within this domain, we also plot the solution for our standard model (in pink) without and with consideration of the density crossover at 12 GPa. We see in Figure 13 that the minimal ages during which outgassing can occur vary largely (mainly modulated by core fraction and Nusselt–Rayleigh parameter uncertainty). However, our standard model shows a robust behavior, independent of density crossover, suggesting that planets formed from more massive refractory parent bodies will be able to outgas much longer. Figure 14 (showing the outgassed amount of water, in Earth oceans, for 500 ppm and 0.4 weight % of water, respectively) also exemplifies that planets with more massive refractory parent bodies can outgas more water and outgas that water at much later times in their evolution.

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Combining this finding with Figure 9, which shows that planets within the orbits of TRAPPIST-1d and TRAPPIST-1h enter the HZ within about 100 Myr to a few hundred million years, and considering that the largest atmospheric loss processes occur before entering the HZ, suggest that especially planets farther away from TRAPPIST-1 and planets that are more massive could deliver up to one or two ocean masses of water after they entered the HZ. This emphasizes that late-stage geophysical outgassing might be a critical component helping to sustain habitable environments within the TRAPPIST-1 system.