Semiempirical Modeling of the Atmospheres of the M Dwarf Exoplanet Hosts GJ 832 and GJ 581

The visible spectra were obtained with an REOSC spectrograph with the spectral resolution R = 13,000 on the 2.15 m Jorge Sahade telescope of the Complejo Astronómico El Leoncito (CASLEO). We follow the calibration procedure described in Cincunegui & Mauas (2004). The echelle spectrum of the target star is calibrated in flux with the long-slit spectrum of the same star. The spectra of GJ 832 and GJ 581 were obtained in September of 2012 and March of 2016, respectively. Each corresponding long-slit spectrum was obtained with the REOSC spectrograph in DS configuration on 2014 October and 2014 July, respectively. Following Cincunegui & Mauas (2004), we estimated a 10% error in the absolute flux in each echelle spectrum. Previously, Fontenla et al. (2016) used the long-slit spectra of GJ 1, an M1 dwarf, to calibrate the echelle spectra of GJ 832 in flux, arguing that since the stars are of similar spectral types, the associated calibration error may be around 10%. They noted, however, that it may be larger, since the differences between GJ 1 and GJ 832 were not precisely known. Indeed, this procedure resulted in a significant overestimate of the flux in the visible range (see Figure 2), prompting us to rebuild the model of GJ 832.

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The synthetic visible spectra are compared with observations in Figure 3. Hereafter, the plots comparing synthetic and observed spectra show synthetic spectra that have been smoothed to match instrumental resolution, unless specified otherwise. While there is no discernible continuum due to an extremely high density of molecular lines formed in the photosphere, we note that the intensity of the pseudo-continuum is set by the temperature at τ = 2/3. This value corresponds to the source of stellar emission because the stars are unresolved, and we observe their emergent flux at the mean angle of μ = 2/3, where μ is defined as in the radiative transfer equation. Pseudo-continua for GJ 832 and GJ 581 form at (P = 2.7 × 10⁵ dyne cm⁻², T = 3550 K) and (P = 4.5 ×10⁵ dyne cm⁻², T = 3450 K), respectively. The overall shapes and intensities of the pseudo-continua are well-reproduced by our models, but we note that there is likely a missing source of molecular opacity between 4000 and 4500 Å, leading to higher intensities in this wavelength range for both modeled stars, but especially for GJ 581. This can be attributed to lower temperatures in the upper photosphere of GJ 581 compared to GJ 832, leading to higher populations of molecular species.

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The synthesized chromospheric lines in the visible spectra generally match the observations well, particularly the Na i D doublet. In our models, this line forms at the base of the first chromospheric rise and is mostly affected by pressure and temperature at this layer of the atmosphere. This line was also well-matched in Fontenla et al. (2016), but due to the faulty calibration procedure described above, the continuum was too bright and the observed line profile was weaker in absorption and, accordingly, the temperature minimum occurred at a higher pressure than in the present model. The first chromospheric rise in the atmosphere of GJ 581 occurs at an even lower pressure (Figure 1) and lower temperature, hence the deeper absorption profile. This is consistent with previous work by Mauas (2000).

GJ 832 and GJ 581 have Hα in absorption, indicating that both stars are not very active as seen at optical wavelengths. We find good agreement between synthetic and observed Hα profiles in GJ 832 spectra (Figure 3). In GJ 581, however, the observed Hα profile is weak in absorption compared to GJ 832, while the synthetic profile shows neither emission nor absorption, possibly because it is filled in by overlapping molecular opacities. The relatively low (∼−5.7 for GJ 581 and ∼−5.1 for GJ 832) values of the activity indicator $\mathrm{log}R{{\prime} }_{{HK}}$, defined as the ratio between Ca ii H & K chromospheric emission and bolometric flux, also suggest that the stars are not very active (Astudillo-Defru et al. 2017; Melbourne et al. 2020).

Ca ii H & K lines fit reasonably well in the case of GJ 832. Following the procedure in Fontenla et al. (2016), we merged the levels of Ca ii and Mg ii ions that have the same quantum numbers, except the total angular momentum, and assumed that the merged levels are in LTE with respect to one another. This method is useful for decreasing computing time, and only affects the relative fluxes of the H and K lines, hence the discrepancy between observed and synthesized Ca ii profiles. The combined fluxes of the two lines are similar between the synthetic and the observed spectra (see Table 2). In the case of GJ 581, the CASLEO spectrum does not have a high enough signal-to-noise ratio to discern the profiles of the Ca ii H & K lines. We therefore did not use Ca ii doublet as a constraint for the model of GJ 581. However, we report computed fluxes for these lines in Table 2. We note that the ratio between computed total Ca ii fluxes in GJ 832 and GJ 581 is similar to the ratio between equivalent widths of Ca ii K lines for these stars reported in Youngblood et al. (2017).

Table 2. Average Disk Intensities (erg cm⁻²Å⁻¹s⁻¹sr⁻¹) of Key Spectral Lines

Feature &	GJ 832			GJ 581
λ(Å)	Synthetic	Observed	Error a (%)	Synthetic	Observed	Error (%)
Ca ii H & K 3950	1.13E4	1.23E4	8.13	5.21E3	⋯	⋯
Mg ii h & k 2800	2.73E4	2.17E4 a	25.8	6.83E3	7.19E3 b	5.01
C ii 1334	84.3	133 b	36.8	43.9	62.2 b	29.4
C ii 1336	132	180	26.7	70.5	129	45.3
Si iv 1394	125	137	8.09	52.6	76.5	31.2
Si iv 1402	65.8	73.4	10.4	27.4	29.6	7.43
C iv 1548	201	323	37.8	325	297	9.43
C iv 1551	102	154	33.8	163	148	10.1
O iv 1401	7.92	11.6	31.7	2.38	7.37	67.7
N v 1239	121	138	12.3	138	90.5	52.5
N v 1243	60.2	67.1	10.3	69.2	50.2	37.8
Fe xii 1242	8.78	5.92	48.3	⋯	⋯	⋯
H i (Lyα) 1215.6	1.27E5	5.88E4 c	116	8.89E4	3.18E4 c	180

Notes.

^a Error is defined as percentage fractional difference, $\tfrac{| {I}_{\mathrm{observed}}-{I}_{\mathrm{synthetic}}| }{{I}_{\mathrm{observed}}}\cdot 100 \%$. This fractional difference exceeds measurement uncertainty in all cases except Si iv and C iv in GJ 581 spectra, where they are roughly equal (Youngblood et al. 2016). ^b Observed fluxes were increased by 30% to account for interstellar absorption. ^c Reconstructed Lyα flux from the MUSCLES portal.

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We compute contribution functions for several key lines, including the Ca ii doublet. The contribution function at a frequency ν is defined as ${f}_{\nu }=\tfrac{{c}^{2}}{2h\nu }\varepsilon {{\rm{e}}}^{-{\tau }_{\nu }}$, where ε is the total line emissivity, and h is the Planck constant. We use the dependence of the contribution function on temperature to make precise changes in narrow layers of the atmospheric profiles. In Figure 4, we show the contribution functions of the Ca ii lines. Both the emission peaks (H₂ and K₂ features—not to be confused with molecules) and the self-reversed cores (H₃ and K₃) are formed in the upper chromosphere and lower TR, thereby providing a useful diagnostic for this region. We note that the contribution function of Ca ii for GJ 581 is lower compared to GJ 832, which explains the relative strength of Ca ii doublet in the spectra of these stars.

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We used NUV spectral data from the MUSCLES Treasury Survey (Loyd et al. 2016), specifically, the spectra obtained with the G230L grating on the HST STIS (Space Telescope Imaging Spectrograph) instrument. All MUSCLES data products used in the present paper are version 2.2. For the computation of NUV spectra, we assumed plane-parallel geometry and used the same stellar radii as we did for the visible spectra. We also applied appropriate Doppler shift corrections of  ≈0.5 Å to the STIS G230L spectra of GJ 581 to match the Mg ii h & k profiles. The comparison between synthetic and observed NUV spectra is shown in Figure 5.

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We find a very good match between computed and observed NUV spectra for GJ 581 and a reasonably good match for GJ 832. The NUV continua of GJ 832 and GJ 581 form at (P ≈ 700 dyne cm⁻², T ≈ 2655 K) and (P ≈ 500 dyne cm⁻², T ≈ 2550 K), respectively. To get correct NUV intensities, we find it critical to include opacities of molecular species, such as CH, NH, OH, and SiH. Another important source of opacity is H₂, which in our models is calculated simultaneously with H⁻, ${{\rm{H}}}_{2}^{+}$, H⁺, and H⁰ assuming chemical equilibrium. This assumption, however, may not hold near the temperature minimum (Fontenla et al. 2015). We find that the opacity of H₂ affects the intensity of the NUV continuum. In particular, we used a higher ad hoc H₂ opacity for GJ 581 than for GJ 832, while leaving H₂ densities unmodified. We justify this by noting that the density of H₂ in the NUV formation region of GJ 581 exceeds that of GJ 832 (Figure 6). Other than the continuum, NUV spectra do not respond to changes in this parameter.

The Fe ii multiplets are also in excellent agreement with observations. As shown in Figure 7, these lines form in the upper chromosphere and, along with Mg ii doublet, can facilitate photolysis of HO₂, H₂O₂, and O₃ in exoplanetary atmospheres (Tian et al. 2014).

The Mg ii doublet is one of the brightest UV features in M dwarf spectra. These are optically thick lines that form in the chromosphere and TR over a wide range of temperatures: the line cores form around 10,000 K, while the wings form at lower temperatures in the chromospheric plateau. It is, therefore, a spectral feature that can be used to constrain the thermal structure of the chromosphere and lower TR. France et al. (2013) used high-resolution STIS E230H spectra of GJ 832 to show that the interstellar absorption removes 30%–35% of the intrinsic Mg ii flux. GJ 581 was not observed in this regime, but the H i column density of the ISM clouds in its line of sight (LOS) is intermediate between GJ 832 and GJ 667C (see Table 2 in France et al. 2013), both of which are predicted to have the Mg ii lines attenuated by 30%–35%. Further, the magnitudes of the radial velocities of GJ 832 and GJ 581 compared to their respective LOS ISM clouds are close to one another: 18 km s⁻¹ and 24 km s⁻¹, respectively (Redfield & Linsky 2008; Gaia Collaboration et al. 2018). Therefore, it is reasonable to assume that the Mg ii doublet in GJ 581 spectra is also attenuated by 30%–35%. Correcting for this interstellar absorption, our models reproduce the observed Mg ii fluxes well. The Mg ii lines show a strong central reversal in high-resolution computed spectra of both stars. This is because the peaks (h₂ and k₂) form at significantly lower temperatures compared to the cores (h₃ and k₃, Figure 7), where the line source function is decreasing with height. The central reversal is degenerate with the ISM absorption profile reported in France et al. (2013), meaning that said absorption for the Mg ii lines can be negligible .

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Mg i 2582 Å is a problematic line, especially in our model of GJ 832. This problem was also present in the GJ 832 model built by Fontenla et al. (2016). Mg i is an absorption line in both observed stellar spectra with a weak central emission, but, for GJ 832, the line core emission is too strong. As shown in Figure 7, Mg i is a line whose contribution functions differ between GJ 832 to GJ 581. In the GJ 832 model, the line forms at significantly higher pressures compared to GJ 581, which may explain why the fit is worse for GJ 832. The discrepancy between computed and observed Mg i line fluxes could be caused by inaccurate ionization balance or by an overestimate of magnesium abundance in the stellar atmospheres.

We use MUSCLES Treasury Survey spectra that were obtained with the G130M and G160M gratings of the HST Cosmic Origins Spectrograph (COS) instrument (Loyd et al. 2016). The combined coverage of G130M and G160M spans the range between 1150 and 1775 Å with spectral resolution between R = 12,000 and 20,000. We compare the synthesized Lyα profiles with the reconstructed STIS spectra from Youngblood et al. (2016).

The C ii and Si iv doublets provide strong constraints on the lower-to-mid TR, particularly its temperature gradient and pressure, as they are formed around 30,000 K and 60,000 K, respectively (Figure 8). The relative weakness of the C ii and Si iv lines in GJ 581 compared to GJ 832 can be attributed to lower TR pressures in GJ 581. We note that the C ii 1334 line is subject to interstellar absorption and is therefore attenuated by ∼30%. Additionally, flares were not removed from GJ 832 co-added spectra observed with G130M. Doppler shifts in the COS spectra were corrected for: the majority of lines are redshifted by 0.05–0.1 Å, with the exception of Si iv lines in GJ 832 spectra that are blueshifted by 0.06 Å.

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Several bright FUV lines serve as constraints on the upper TR and the corona; these lines include C iv (formed around 100,000 K), O iv (160,000 K), and N v (180,000 K), which are all in good agreement with observed spectra (see Figure 9 and Table 2). Fe xii at 1242 Å is detected in the spectra of GJ 832 and provides a useful diagnostic for the lower corona, since it is formed at around 1.5 MK.

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The optically thick resonance line Lyα is the brightest UV feature in stellar spectra. Apart from driving photochemical processes in planetary atmospheres, it also back illuminates deeper layers of stellar atmospheres, thereby enhancing H₂ fluorescence and potentially altering the ionization balance of heavy elements (France et al. 2013; Fontenla et al. 2015; Kruczek et al. 2017). Stellar Lyα lines are strongly absorbed by the ISM; hence, it is important to obtain accurate profiles of this line. Our computed Lyα fluxes exceed those estimated by Youngblood et al. (2016; see Table 2 and Figure 10) by a factor of two to three, and the width of Lyα wings is likewise overestimated. In this sense, we were not able to make meaningful improvements upon the work of Fontenla et al. (2016). The wide wings may be due to errors in the bound-free opacities and recombination rates of several atomic species, such as C i (Fontenla et al. 2014), while the anomalously high fluxes likely indicate a problem with the heat balance in the TR. We also observe strong reversal cores in the Lyα profiles of both stars, which are not reported by Youngblood et al. (2016), although they are consistent with the previous SSRPM model of GJ 832 (Fontenla et al. 2016). These reversal cores, however, are not as strong as those computed using the PHOENIX models (Peacock et al. 2019a, 2019b). Observations of high radial velocity M dwarfs did not reveal such absorption cores (Guinan et al. 2016; Schneider et al. 2019). This points to a common underlying issue with the treatment of this line by different radiative transfer codes.

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The X-ray spectra are computed using spherical symmetry, since stellar coronae extend significantly beyond the chromospheres and are comparable in size to stellar radii. Loyd et al. (2016) used XMM-Newton data and the Astrophysical Plasma Emission Code (APEC) model fits to determine that the coronal emission from GJ 832 can be fit by two components, ${kT}={0.09}_{-0.09}^{+0.02}$ keV and ${kT}={0.38}_{-0.07}^{+0.11}$ keV (corresponding to approximately 1.04 × 10⁶ K and 4.41 × 10⁶ K, respectively), while in GJ 581, the X-ray flux is best fit by a single component with kT = 0.26 ± 0.02 keV (T ≈ 3.02 × 10⁶ K).

Synthesized X-ray spectra are shown in Figure 11. The disk-integrated X-ray luminosities of GJ 832 and GJ 581 are L_X,832 = 5.34 · 10²⁶ erg s⁻¹ and L_X,581 = 8.51 · 10²⁵ erg s⁻¹. The X-ray luminosity of GJ 832 was reconstructed by Sanz-Forcada et al. (2011) and Loyd et al. (2016) to be 6.02 · 10²⁶ erg s⁻¹ and 2.01 · 10²⁶ erg s⁻¹, respectively; thus, our value is intermediate (see Table 4). GJ 581 was not considered in Sanz-Forcada et al. (2011), but Loyd et al. (2016) obtained a value of 9.49 · 10²⁵ erg s⁻¹, which is reasonably close to our value. However, their inferred coronal temperatures of GJ 581 are lower than those in our models (see Figure 1). Also, unlike the present work and Loyd et al. (2016), they do not include X-rays below 5 Å in their analysis, but the contribution from this wavelength range to the overall X-ray luminosity is negligible.

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The FUV continuum has only been detected in a handful of M dwarfs, and its contribution to the overall FUV emission is difficult to constrain. Loyd et al. (2016) put a lower boundary of 12.2% on the contribution of the FUV continuum for GJ 832 in the 1300–1700 Å wavelength range, as they did not account for continuum flux within emission lines. They were not able to detect the FUV continuum of GJ 581 in the same range, although they did not rule out that it could also be around 10%. Recently, Becker et al. (2020) reported a photometric UV continuum detection in TRAPPIST-1 spectra using the Swift UV/Optical Telescope uvw2 filter centered at 1928 Å. However, the uvw2 filter has an FWHM bandpass of 657 Å, so this photometric result includes emission lines and can only serve as a proxy for continuum intensity. Emission at these wavelengths can drive photolysis of water, carbon dioxide, and molecular oxygen in planetary atmospheres; hence, it is important to accurately quantify it.

The FUV continua in the observed spectra of GJ 832 and GJ 581 are at the noise level (Figure 9), but are prominent in the synthesized spectra. In Figure 13, we show MUSCLES spectra that have been binned to 5 Å bins to reduce the effect of noise, similarly binned synthesized spectra, and the FUV continuum in native resolution. In synthetic spectra, the ratio of FUV continuum to FUV overall emission between 1300–1700 Å is 57.2% for GJ 832 and 43.0% for GJ 581. These are true continua, i.e., in computing them, we do not take into account the many faint FUV lines that are blended with the continua due to instrumental broadening.

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Of significant concern is the apparent bound-free opacity edge just below 1350 Å associated with the first excited state of S i. It is spurious because it greatly exceeds observed fluxes (Figure 13), and because the intensity at wavelengths shorter than the edge is smaller than the intensity at longer wavelengths. One explanation for this reversal from the usual pattern is that this edge forms at heights where the temperature gradient is decreasing. The calculation of this continuum edge has been addressed in previous work (Fontenla et al. 2014). It is also worth noting that such reversed edges can sometimes be found elsewhere in the FUV continua, e.g., the Si i edge at 1682 Å in solar spectra (Fontenla et al. 2014), although an order of magnitude drop in flux is unusual. The abnormally strong S i continuum edge may be caused by issues with photoionization balance in our atomic database, but further investigation is needed to accurately diagnose the exact cause. To exclude the possibility that the S i edge is responsible for such high continuum-to-overall emission ratio, we computed the FUV continuum flux ratios in the range between 1450–1700 Å, and we found that the ratios were 54.4% for GJ 832 and 41.6% for GJ 581, i.e., they were only changed by 1%–3%. Such a high intensity of FUV continua necessitates further investigation, since it can have a significant impact on planetary atmospheres. This result appears to be consistent with the high ratios of radiative losses by the UV continuum in model chromospheres (both quiescent and active) of au Mic (Houdebine et al. 1996; Houdebine 2010). However, as far as we know, such high levels of FUV continuum emission in M dwarfs have not been previously reported, and, if verified, this result has important implications for planetary photochemistry.

The brightness temperature of the FUV continuum (T_b,FUV) in sunlike stars was found to be positively correlated with rotation speed (Linsky et al. 2012), and thus correlated with stellar activity. Whether this relation holds for M stars is hard to say due to their relatively low FUV flux at Earth. However, using our models, we calculated the brightness temperatures in the 1300–1700 Å range of the two stars and found that T_b,FUV is higher in the GJ 832 model by about 200 K. This suggests that a similar relation between FUV continuum brightness temperature and rotation speed may hold for M dwarfs, since the rotation periods of GJ 832 and GJ 581 are P₈₃₂ = 45.7 ± 9.3 d and P₅₈₁ = 132.5 ± 6.3 d (Suárez Mascareño et al. 2015). Linsky et al. (2012) also reported a correlation between FUV continuum flux and Si iv line intensities, but we do not observe such a correlation in our sample. The absence of this correlation among M dwarfs is consistent with previous work by France et al. (2018). Overall, it is necessary to model more stars to make robust inferences about FUV continua.

EUV spectra are shown in Figure 14, where we display contributions from lower (T < 10⁵ K) and upper (T > 10⁵ K) component models separately. The integrated EUV luminosities between 100 and 912 Å in our models are $\mathrm{log}{L}_{\mathrm{EUV},832}\,=27.25$ erg s⁻¹ and $\mathrm{log}{L}_{\mathrm{EUV},581}=26.53$ erg s⁻¹ (see Table 4). Youngblood et al. (2016) used Lyα-EUV empirical relations (Linsky et al. 2014) to obtain EUV luminosities of these stars; their estimates—$\mathrm{log}{L}_{\mathrm{EUV},832}=27.40$ erg s⁻¹ and $\mathrm{log}{L}_{\mathrm{EUV},581}\,=26.67$ erg s⁻¹—are very close to ours. However, we find that their SEDs in this wavelength range differ from ours by up to an order of magnitude, especially in the case of GJ 581 (Figure 15). We find that the lower components of model atmospheres contribute 24.1% (logL = 26.63 erg s⁻¹) and 33.3% (logL = 26.05 erg s⁻¹) of the EUV flux in GJ 832 and GJ 581, respectively. For GJ 581, we find that the flux below 400 Å is underestimated, whereas above 400 Å, the intensity is higher than that predicted by the empirical relation. A similar trend is seen in synthetic spectra of GJ 832 but to a lesser extent. The upper component model of GJ 581 contributes a significant fraction of flux above 400 Å, which is not the case for GJ 832. Additionally, the discrepancy could be caused by the errors in Lyα reconstruction procedure, which could propagate and cause errors in reconstructed EUV fluxes.

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Sanz-Forcada et al. (2011) described an empirical relation between EUV and X-ray fluxes and estimated EUV luminosities for a series of stars, including GJ 832. Their estimated EUV flux, $\mathrm{log}{L}_{\mathrm{EUV},832}=27.83$ erg s⁻¹ (Table 4), exceeds both the one reported in Youngblood et al. (2016) and the one in the present work. However, as discussed in Section 3.4, they used a higher value of X-ray flux, which also increases EUV intensity. Using the computed X-ray flux from our model and their scaling relation, we obtain $\mathrm{log}{L}_{\mathrm{EUV},832}=27.71$ erg s⁻¹ and $\mathrm{log}{L}_{\mathrm{EUV},581}=27.18$. While our computed EUV luminosities are lower, we note that they are well within the margin of error of these estimates (see Equation (3) in Sanz-Forcada et al. 2011). Additionally, the wavelength ranges they considered are different from ours: X-rays in their work extend from 5–100 Å (as opposed to 1–100 Å in the present work), and EUV covers an additional wavelength band between 912 and 920 Å. Accounting for these differences could improve the agreement somewhat. Another empirical relation, based on FUV line fluxes, was described in France et al. (2018) with the use of archival EUVE data. They managed to estimate the fluxes of both GJ 832 and GJ 581 in the 90–360 Å range, and these estimates agree reasonably well with our calculations (Table 4).

The upcoming work by Duvvuri et al. (submitted) describes a DEM formalism and applies it to a series of stars, including GJ 832. They obtain $\mathrm{log}{L}_{\mathrm{EUV},832}=26.83$ erg s⁻¹, but their DEM method does not take into account flux contributions due to various continua, of which the most prominent one, the H i Lyman continuum, contributes 14%–15% ($\mathrm{log}{L}_{832}\,=26.42$ erg s⁻¹ and $\mathrm{log}{L}_{581}=25.67$ erg s⁻¹) of the total EUV flux in our spectra. This is similar to what Woods et al. (2009) found while analyzing solar irradiance reference spectra, where this fraction is also 15%.

The atmosphere of GJ 832 has been approximated by one-dimensional models before. Fontenla et al. (2016), whose approach we revise in the present work, used SSRPM to synthesize panchromatic spectra of this star, and their integrated X-ray and EUV fluxes are virtually indistinguishable from ours (Table 4). This is expected, as the primary difference between their model and ours lies in the photosphere and lower chromosphere, which do not significantly affect the short-wavelength emission. More recently, Peacock et al. (2019b) synthesized the EUV spectra of GJ 832 using a one-dimensional PHOENIX model that did not include temperatures above 200,000 K. They argued that the omission of the corona only significantly affects EUV emission below 200 Å. In order to test whether our models support this claim, we built a separate model for each star that extends from T = 10⁵ K to T = 2 · 10⁵ K and computed EUV spectra emerging from this layer. We find that the EUV contribution from temperatures above 2 · 10⁵ K is 72.6% ($\mathrm{log}{L}_{832}=27.11$ erg s⁻¹) and 52.4% ($\mathrm{log}{L}_{581}=26.24$ erg s⁻¹) in GJ 832 and GJ 581, respectively. Fractions of EUV emission from the three components in 100 Å bins are shown in Figure 16. In our models, coronal EUV emission is significant, especially below 500 Å. We caution, however, that splitting upper atmosphere models into separate components can produce erroneous results, as population densities at different heights are coupled in these highly non-LTE environments. Peacock et al. (2019b) originally reported the integrated EUV flux of $\mathrm{log}{L}_{\mathrm{EUV},832}=27.31$ erg s⁻¹, but this value was later revised and is now $\mathrm{log}{L}_{\mathrm{EUV},832}=27.85$ erg s⁻¹ (S. Peacock 2020, personal communication). This value is very close to that reported in Sanz-Forcada et al. (2011; Table 4). Their integrated EUV flux is higher than in the present work, and would be significantly higher when coronal emission is included. Their obtained SED is intermediate between ours and that reconstructed by Youngblood et al. (2016) in most bands, except below 200 Å, where the contribution from corona is significant, and between 400 and 600 Å (Figure 15). Overall, the reasonably close agreement between different EUV reconstruction methods inspires confidence in each of them, but it is important to also accurately characterize the SED in the EUV range.

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GJ 832 and GJ 581 have similar photospheres (Figure 1), which explains why their visible spectra are also comparable. The difference in lower photospheric temperature gradients is not significant; we find that it does not affect visible spectra. The lower chromospheres of the two stars are different; however, GJ 581 has a lower temperature minimum (2300 K as opposed to 2650 K for GJ 832), which is constrained by the K₁ flux of Ca ii H & K and the k₁ flux of the Mg ii h & k doublets.

The first chromospheric rise is significantly less steep in the model of GJ 581 and occurs at lower pressures. While the pressure is relatively easy to infer using NUV spectral diagnostics, the steepness of the chromospheric rise is somewhat poorly constrained due to very low fluxes of potential diagnostic lines. Si ii (1524 Å) is a strong line that can be used as a diagnostic for this region in the solar atmosphere (Vernazza et al. 1981), but we find that in the atmospheres of GJ 832 and GJ 581, the Si ii doublet forms in the upper chromosphere between 15,000 and 20,000 K. The first chromospheric rise may be responsible for FUV continuum formation (Fontenla et al. 2015), although the temperature minimum region may also play a role (Houdebine 2010; Fontenla et al. 2014).

The upper chromosphere and lower TR are very well constrained by bright NUV and FUV emission lines. The base of the TR is at a slightly lower pressure in GJ 581 (0.686 dyne cm⁻²) than in GJ 832 (0.836 dyne cm⁻²), and the TR itself is less steep—at 100,000 K, the pressure is 0.682 dyne cm⁻² in GJ 581 and 0.835 dyne cm⁻² in GJ 832. A steeper TR of GJ 832 could indicate stronger magnetic heating. Other indicators of higher magnetic activity in GJ 832 compared to GJ 581 include brighter chromospheric and TR lines, such as Ca ii, Mg ii, Lyα, and N v (Youngblood et al. 2017), and more rapid rotation (e.g., West et al. 2015). The activity indicator $\mathrm{log}R{{\prime} }_{{HK}}$ of GJ 832 exceeds that of GJ 581 by ∼0.6, which also suggests that GJ 832 is the more active star (Melbourne et al. 2020; Astudillo-Defru et al. 2017).

The coronal models are also qualitatively similar: they are both characterized by temperatures and pressures that are essentially constant with height (Figure 1). The corona of GJ 581 is hotter than that of GJ 832, which may explain the possible excess EUV emission above 400 Å. The coronal temperature in our model of GJ 581 also exceeds previous estimates.

There are several challenges to synthesizing accurate stellar spectra using SSRPM. First, the atomic and molecular data should be updated. The versions of CHIANTI and NIST databases used in the present work are not fully up to date, which may cause inaccuracies in our computed spectra. More importantly, our databases lack some molecular opacity data, e.g., CO and H₂ opacities. H₂ is a major opacity source in the NUV, and H₂ populations are currently computed in chemical equilibrium. The agreement between computed and observed fluxes between 4000 and 4500 Å could be improved by adding VO, CaH, and FeH molecular transitions. The missing UV opacity sources may result in enhanced ionization of neutral heavy species, which in turn leads to a further decrease of UV opacity (Fontenla et al. 2015). So long as the photodissociation opacity of H₂ remains an ad hoc parameter, the fitting of UV fluxes will remain an imprecise procedure. Additional opacity sources might alter the Lyα wings and the region between 4000 and 4500 Å. The bound-free opacity edge at 1350 Å, which produces abnormally high fluxes between 1350 and 1450 Å, must be corrected in order for us to study the importance of the FUV continuum. Atomic abundances used in the present work are identical to solar values, but until more precise and reliable measurements of stellar metallicities are available, this will continue to be a potential source of error.

Another avenue to improve the fidelity of synthesized spectra is to build multicomponent models for each star to account for inhomogeneity across stellar surfaces. M dwarfs are known to have active regions and starspots (e.g., Howard et al. 2020), and with the recent launches of TESS and Gaia, it is becoming increasingly feasible to use photometry for estimating starspot coverage. Computing weighted averages of stellar surface fluxes should improve the fits between computed and observed spectra, inform our understanding of stellar thermal structures (especially in the chromosphere and corona), and provide an additional empirical constraint. However, this approach will still not account for important three-dimensional effects occurring at the boundaries of starspots, such as enhanced downward Lyα flux due to Wilson depressions (Fontenla et al. 2015). This particular effect can increase H₂ fluorescence, ionize heavier elements, and raise core fluxes for such lines as Mg ii h & k, Ca ii H & K, and Lyα.

Coronal emission is currently poorly constrained in our models, as we can only rely on a handful of faint emission lines and estimates of coronal temperatures. This introduces the possibility of degeneracy in T–P profiles of the upper atmospheres. We have not yet quantitatively studied the extent to which different coronal structures affect EUV emission, as it is currently computationally expensive to synthesize X-Ray and EUV spectra via SSRPM. Considering that most of the EUV emission is produced in the upper TR and the corona, it is important to have an accurate model of the corona for each target star. Further observations of the stars, especially if they are concurrent with UV observations, could improve model accuracy significantly. This would be especially useful for refining the coronal structure of GJ 581. Future EUV missions, such as the proposed ESCAPE, will be invaluable, as presently there are virtually no empirical constraints on EUV fluxes of M dwarfs. Additionally, it is important to constrain elemental abundances in stellar coronae to accurately predict coronal emission. For instance, when GJ 832 and GJ 581 coronal emissions were computed assuming solar coronal abundances (Feldman 1992), in which the fraction of iron is almost five times what it is in the photosphere due to the first ionization potential (FIP) effect, the X-ray emission increased by a factor of three to four. This is unlikely to be realistic, as M dwarf coronae are expected to have the inverse FIP effect (Wood et al. 2012), but it shows that coronal emission is sensitive to elemental abundances in the corona.