FUMES. II. Lyα Reconstructions of Young, Active M Dwarfs

Our model is comprised of two components: the stellar emission component and the ISM absorption component. We tested different functions for the intrinsic stellar emission, including multiple, superimposed Gaussians, and found that a single Voigt profile in emission fits both the line core and the broad wings best. We use the astropy Voigt1D function, which is based on the computation from McLean et al. (1994). We assume no self-reversal, because past results have shown that the Lyα self-reversal of M dwarfs is small (Wood et al. 2005; Guinan et al. 2016), if present at all (Youngblood et al. 2016; Bourrier et al. 2017; Schneider et al. 2019). Given that the Lyα line center, the region in the spectrum where the self-reversal appears, is usually entirely hidden by the ISM and not well-constrained by the reconstruction, we assume no self-reversal is present. The Voigt emission line model component has four free parameters:

$$\begin{eqnarray}&&{F}_{\mathrm{emission}}^{\lambda }={ \mathcal V }(\lambda ,{V}_{\mathrm{radial}},A,{\mathrm{FWHM}}_{L},{\mathrm{FWHM}}_{G}),\end{eqnarray} \tag{ 1 }$$

where V_radial is the radial velocity of the emission line (km s⁻¹), A is the Lorentzian amplitude (erg cm⁻² s⁻¹ Å⁻¹; note that we parameterize it in all tables as log₁₀ A), and FWHM_L and FWHM_G (km s⁻¹) are the full width at half maximum values for the Lorentzian and Gaussian components, respectively. For use with Voigt1D, V_radial, FWHM_L , and FWHM_G are converted to Å. For the reconstructions on the E140M spectra where Lyα and Si III are not blended, Equation (1) is used, but for the G140L spectra where the two lines are blended, ${F}_{\mathrm{emission}}^{\lambda }$ = ${F}_{\mathrm{emission},{\rm{H}}\,{\rm\small{I}}}^{\lambda }$ + ${F}_{\mathrm{emission},\mathrm{Si}\,{\rm\small{III}}}^{\lambda }$.

We assume a single ISM absorbing cloud, as such low-resolution spectra (300 km s⁻¹) are not able to distinguish between ∼20–40 km s⁻¹ separated clouds. Youngblood et al. (2016) demonstrated that assuming a single-velocity ISM does not significantly impact the reconstructed Lyα flux. For the ISM component (used for Lyα only), we model the H I and D I absorption lines each as Voigt profiles with linked parameters, using the code lyapy ⁶ (Youngblood et al. 2016):

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{absorption}}^{\lambda } & = & { \mathcal V }(\lambda ,{V}_{{\rm{H}}{\rm\small{I}}},{\mathrm{log}}_{10}N({\rm{H}}\,{\rm\small{I}}),{b}_{{\rm{H}}{\rm\small{I}}})\\ & & \times \ { \mathcal V }(\lambda ,{V}_{{\rm{D}}{\rm\small{I}}},{\mathrm{log}}_{10}{\text{}}N({\rm{D}}\,{\rm\small{I}}),{b}_{{\rm{D}}{\rm\small{I}}}),\end{array}\end{eqnarray} \tag{ 2 }$$

where V_{H I} is the radial velocity (km s⁻¹) and is assumed to be the same for both H I (1215.67 Å) and D I (1215.34 Å) (V_{H I} = V_{D I} , so V_{H I} is the reported parameter). Here, log₁₀ N is the logarithm of the column density (cm⁻²), where N(H I) and N(D I) are linked by the parameter D/H, i.e., the deuterium to hydrogen ratio: N(D I) = N(H I) × D/H. The value of D/H is fixed to 1.5 × 10⁻⁵ (Linsky et al. 2006), so log₁₀ N(H I) is the reported parameter. The Doppler parameter b controls the width of the absorption line, and we link b_{H I} and b_{D I} so that b_{D I} = b_{H I} /$\sqrt{2}$. b_{H I} is the reported parameter. In order to reduce the number of free parameters for the G140L reconstructions, b_{H I} was fixed at 11.5 km s⁻¹, based on the standard T = 8000 K ISM (Redfield & Linsky 2004; Wood et al. 2004).

To model the observed (attenuated) profile, we multiply the emission and absorption models (Equations (1) and (2)) and convolve with the instrument line spread function (LSF) provided by STScI ⁷ for the appropriate grating and slit combinations to recover the true physical parameters and account for the non-Gaussian wings of the G140L LSF:

$$\begin{eqnarray}&&{F}^{\lambda }=({{ \mathcal V }}_{\mathrm{emission}}\times {{ \mathcal V }}_{\mathrm{absorption}}) \circledast \mathrm{LSF}.\end{eqnarray} \tag{ 3 }$$

To reconstruct the Lyα profiles, we used a likelihood-based Bayesian calculation and a Markov chain Monte Carlo (MCMC) method (emcee ⁸ ; Foreman-Mackey et al. 2013) to simultaneously fit the model (Equation (3)) to the observed spectra. We assume uniform (flat) priors for all parameters except for a logarithmic prior for the Doppler b value (Youngblood et al. 2016), and a Gaussian likelihood

$$\begin{eqnarray}&&\mathrm{ln}{ \mathcal L }=-\displaystyle \frac{1}{2}\displaystyle \displaystyle \sum _{i}^{N}\displaystyle \frac{{\left({y}_{i}-{y}_{\mathrm{model},i}\right)}^{2}}{{\sigma }_{{y}_{i}}^{2}}+\mathrm{ln}(2\pi {\sigma }_{{y}_{i}}^{2}),\end{eqnarray} \tag{ 4 }$$

where N is the total number of spectral data points y_i with associated uncertainties ${\sigma }_{{y}_{i}}$, and y_model,i corresponds to Equation (3). We maximize the addition of $\mathrm{ln}{ \mathcal L }$ and the logarithm of our priors with emcee. We used 50 walkers, ran for 50 autocorrelation times (∼10⁵–10⁶ steps), and removed an appropriate burn-in period based on the behavior of the walkers.

Tables 2–6 show all of our model parameters with the assumed priors (uniform or logarithmic) within a bounded range, and the 2.5, 15.9, 50, 84.1, and 97.5 percentiles as determined from the marginalized posterior distributions. We present the median (50th percentile) as the best-fit parameter values. The best-fit (median) and 68% and 95% confidence intervals on the reconstructed Lyα and Si iii fluxes were determined from the entire ensemble (i.e., a histogram of all the Lyα or Si iii fluxes from the MCMC chain). Often, the median parameter values do not create a self-consistent solution, so we obtain the best-fit models and reconstructed profiles from the median flux in each wavelength bin from the ensemble of models and reconstructed profiles. Figure 1 shows the best-fit model and reconstructed profile for the HIP 23309 data, and Figure 2 shows the marginalized and joint probability distributions of the fitted parameters for HIP 23309. Similar figures for the other stars are available in the figure set in the online journal.

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View figure set (8 panels)

Tarball of figure set images

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View figure set (8 panels)

Tarball of figure set images

In the rest of this section, we make note of any irregularities or the source of any constraints imposed on the reconstructions on a star-by-star basis. Most of the FUMES Lyα spectra were obtained with the low-resolution G140L STIS grating (λ/Δλ ∼ 1000), where the H i and D i ISM absorption lines are unresolved. Resolving the D i absorption is useful for constraining the ISM model parameters (column density, Doppler b value, and radial velocity), so we provide constraints on these parameter values with outside information when necessary to aid convergence to a best-fit solution. These constraints include stellar radial velocities from SIMBAD, predicted ISM radial velocities from the Local ISM Kinematic Calculator ⁹ (Redfield & Linsky 2008), predicted H i column densities for the Local Interstellar Cloud (LIC) ¹⁰ (Redfield & Linsky 2000), and measured H i column densities from nearby sightlines collated from Wood et al. (2005), and Youngblood et al. (2016, 2017).

GJ 4334—The fit had to be restricted to log₁₀ N(H I) > 17.8, because the fit preferred a log₁₀ N(H I) < 17.8 solution. The likelihood values are not higher at log₁₀ N(H I) < 17.8, but the parameter space is much more well-behaved (i.e., smoothly varying), which is likely why the fit prefers this parameter regime. With log₁₀ N(H I) restricted to lie between 17.8–19, the best-fit log₁₀ N(H I) = 18.03 is in agreement with the LIC model log₁₀ N(H I) = 18.04 prediction and measurements of nearby sightlines (log₁₀ N(H I) = 17.9–18.5).

HIP 17695—Despite the high spectral resolution obtained for this target, the fit is not consistent with probable log₁₀ N(H I) values (>17.5). The MCMC prefers the log₁₀ N(H I) value to be low (<17.0), which may be unphysically low based on knowledge of the local ISM (Wood et al. 2005), although a value <18.0 is justified based on literature measurements of nearby sightlines. We constrain the column density to be between 17.8–18.0, in agreement with the LIC model's predictions log₁₀ N(H I) = 17.93, and allow the MCMC to pile up near the lower boundary. We note that O v (1218.3 Å) is clearly detected in the Lyα red wing.

LP 247-13—We constrain the log₁₀ N(H I) parameter to be between 18.3–19.0 (the fit prefers <18.0) based on a previous measurement of log₁₀ N(H I) = 18.31 for a foreground star (Dring et al. 1997).

GJ 49—The fit reveals four different local maxima with no clear global maximum. We discard the solutions with a low log₁₀ N(H I) = 17.7 value and a high log₁₀ N(H I) = 18.7 value, because nearby sightlines indicate log₁₀ N(H I) = 18.0–18.3. We also rule out the solution with the >100 km s⁻¹ difference between V_{H I} and V_radial. With these restrictions on N(H I) and V_{H I} in place (see priors in Table 2), we ran the MCMC for the presented solution.

GJ 410—The posterior distribution for this star's fit is wide, as the 95% confidence interval spans a factor of 15 in Lyα flux. Nearby sightlines indicate log₁₀ N(H I) lies in the range of 17.6–18.6, and the solution's log₁₀ N(H I) = 18.32 is in agreement with this range.

The quality of the E140M reconstructions is high, but for many of our G140L reconstructions, >32% of the residuals lie outside of the ±1σ range (see Figure 1 and the extended figure set in the online journal). This indicates either that the data uncertainties are underestimated or that the model is misspecified. In general, the data appear to be fit well by the model, but a Durbin–Watson test (Durbin & Watson 1950) reveals some positive autocorrelation in the residuals. For half of our stars (HIP 23309, GJ 410, LP 247-13, HIP 112312), the Durbin–Watson statistic (dw) is between 1.5–1.8 (where 2 represents no autocorrelation and 0 represents perfect positive autocorrelation) and for the others (GJ 49, CD −35 2722, GJ 4334, HIP 17695) dw = 1.1–1.4. This autocorrelation of the residuals can be partially accounted for by a group of weak, unresolved emission lines present around 1190–1210 Å that are not included in our model. Based on detailed spectra of the Sun (Curdt et al. 2001) and prominent lines in high-quality M dwarf spectra like AU Mic (Pagano et al. 2000; Ayres 2010) and GJ 436 (dos Santos et al. 2019), these lines include S iii (1190, 1194, 1201, 1202 Å), Si ii (1190, 1193, 1194, 1197 Å), N i (1200, 1201 Å), Si iii (1206 Å), and H₂ (1209 Å). ¹¹ There are fewer unresolved emission lines in the blue wing of the Lyα line, including O v (1218 Å) and S i (1224, 1229, 1230 Å). This creates an apparent asymmetry (see the solar spectrum from Woods et al. (1995)), whereas our model is symmetric about the line center. We have included only the strongest of these adjacent emission lines (Si iii at 1206 Å) in our model, as the others are ill-constrained by our spectra.

We have tested adding a scatter term (f) to our model to account for underestimated data uncertainties, which is implemented by replacing the ${\sigma }_{i}^{2}$ terms in Equation (4) with ${\sigma }_{i}^{2}$ + f². We find that, for our higher-quality fits (e.g., HIP 23309), the fitted result was the same. For our lowest-quality fit (GJ 49), there was a large difference in the reconstructed flux, but the quality of the fit was not improved, as the structure in the residuals remained. Therefore, we do not present the fits with the scatter term in this work. We conclude that the model is missing a component, such as the weak emission lines and/or continuua in the line wings mentioned above.

GJ 49's reconstruction quality is the worst of our sample, and we note that its reconstructed Lyα flux should be interpreted with caution. We postulate that the reason this star's fit is so unconstrained is because of its Lyα line's narrow intrinsic width (see Section 4.3) and the high S/N of its spectrum. GJ 49's observed spectrum has higher S/N around the Lyα line than any of our other G140L spectra, and this precision increases the visibility of features not covered by our model. Other FUMES targets with wider intrinsic line widths (and lower S/N) may swamp the signals from unresolved emission lines and/or continuua. Despite large scatter in the residuals, GJ 49's Lyα and Si iii flux measurements appear to be consistent with other FUMES targets of similar rotation periods (Pineda et al. 2021).

Regarding our fitted radial velocity parameters, we note that the relative accuracy of the STIS MAMA's wavelength solution is reported in the STIS Instrument Handbook as 0.25–0.5 pixels (37–74 km s⁻¹ for the G140L grating; 0.8–1.6 km s⁻¹ for the E140M grating), and the absolute wavelength accuracy is 0.5–1 pixel (74–148 km s⁻¹ for G140L; 1.6–3.3 km s⁻¹ for E140M). We find that the quoted relative wavelength accuracy can easily describe the offsets between our fitted H i and Si iii radial velocities (accounting for the 68% confidence interval on those values). The quoted absolute wavelength accuracy can account for almost all of the offsets between the literature stellar radial velocities and our fitted radial velocities. The exception is GJ 4334, which has some disagreement in the literature over its radial velocity: −40 ± 4 km s⁻¹ from Newton et al. (2014), −16.5 ± 4.0 km s⁻¹ from Terrien et al. (2015), and −11.9 km s⁻¹ from West et al. (2015). This discrepancy is not large enough to account for the ∼200–300 km s⁻¹ offset between our fitted radial velocities and the literature values. However, GJ 4334's velocity difference between the fitted radial velocity and the fitted ISM radial velocity is in agreement with the velocity difference between the Newton et al. (2014) radial velocity and the ISM velocity (6.0 ± 1.4 km s⁻¹) predicted by Redfield & Linsky (2008), lending confidence to our fit and supporting the possibility that the absolute wavelength accuracy for GJ 4334's STIS observation is poorer than is typical.

To test the accuracy of the reconstructions based on the G140L spectra, we degraded the resolution of our E140M spectra (HIP 112312 and HIP 17695) to the resolution of the G140L spectra by convolving with the G140L LSF and rebinning to match the G140L dispersion. Tables 5 and 6 and show the results of the E140M (native resolution) and degraded resolution reconstructions for these two stars. There is substantial overlap between the native and degraded reconstructed Lyα fluxes at the 68% (for HIP 17695) and the 95% confidence interval (for both). The uncertainties with the G140L-quality reconstruction are much larger than for the E140M reconstructions, as expected. When comparing the individual fitted parameter values, we find that the G140L-quality reconstructions do not always agree with their higher-resolution counterparts. For HIP 17695, agreement between the individual fitted parameters is generally good, but this is not the case for HIP 112312. We provide confidence intervals for all of our G140L reconstruction parameters (Tables 2–4), but note that they should be interpreted with caution and may not reflect the true parameters that could be revealed with higher-resolution spectra. This may be because the G140L posterior distributions are generally very wide, and we report the median parameter values as the best-fit values, even though combining the median parameter values does not always yield a self-consistent best-fit to the data. However, this exercise in comparing E140M reconstructions with degraded resolution reconstructions shows that the reconstructed Lyα fluxes overlap within at least the 95% level.

Our STIS G140L reconstructed spectra of M dwarfs show their broad, ∼500–1000 km s⁻¹ Lyα wings in detail (Figure 3). As demonstrated in Ayres (1979), the widths of chromospheric emission lines like Ca II H&K, Mg II h&k, and Lyα are predominantly controlled by the stellar temperature distribution rather than chromosphere dynamics or magnetic heating. This explains the remarkable Wilson–Bappu correlation between absolute stellar magnitude and FWHM for the Ca II H&K emission cores (Wilson & Bappu 1957) and other chromospheric emission lines (McClintock et al. 1975; Cassatella et al. 2001) across many orders of magnitude of stellar bolometric luminosity. In Figure 4, we show that our data support a similar correlation (ρ = 0.72; p = 0.0015) between bolometric luminosity and Lyα FWHM, albeit over a much smaller parameter space than that explored by Wilson & Bappu (1957).

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Ayres (1979) notes that stellar magnetic activity (e.g., due to nonradiative heating) does play a role in the widths of chromosphere emission lines, with greater activity corresponding to wider lines, in addition to the stronger influences of stellar effective temperature, surface gravity, and elemental abundance compared to hydrogen. Ayres (1979) and Linsky (1980) present a linear model of chromospheric emission line width as a function of chromospheric heating (i.e., activity as measured by the flux of a chromospheric emission line), effective temperature, surface gravity, and elemental abundance. To determine which stellar properties are most responsible for our observed Lyα widths, we construct a linear model based on our observations. We select surface gravity, Si III luminosity as a fraction of bolometric luminosity (a general "activity" proxy), and effective temperature as predictor variables. Because we are examining a hydrogen line, we do not include a metallicity term. We scale each variable (by subtracting the mean and dividing by the standard deviation), construct a correlation matrix, and calculate the eigenvalues and eigenvectors via principal component analysis (PCA) (Table 7). Only the first two principal components (PCs) or eigenvectors have eigenvalues >1 or are correlated significantly (∣ρ∣ > 0.5; p < 0.05) with any of the predictor variables; therefore, we only include PC₁ and PC₂ in the linear model of Lyα width.

We perform a multiple linear regression to relate our previously determined PCs to a response variable, the Lyα full width at 20% maximum flux (FW_20%), a term that is analogous to W(K₁) from Ayres (1979). Regression coefficients are reported in Table 7. Simplifying the linear model expressions into the original unscaled predictor variables rather than PCs, we find that, for the Lyα emission line:

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{log}}_{10}{\mathrm{FW}}_{20 \% } & = & -0.29{\mathrm{log}}_{10}g+0.09{\mathrm{log}}_{10}\displaystyle \frac{L(\mathrm{Si}\,{\rm\small{III}})}{L(\mathrm{bol})}\\ & & +\ 2.13{\mathrm{log}}_{10}{T}_{\mathrm{eff}}-5.54,\end{array}\end{eqnarray} \tag{ 5 }$$

where FW_20% is in Å, g is in cm s⁻², L(Si III)/Lbol is unitless, and T_eff is in K. There are some similarities in the coefficients between this paper's Equation (5) and Equation (8) from Linsky (1980) (log W(K₁) = −0.25 log g + 0.25 log F + 1.75 log T_eff + 0.25 log A_met, where F is the scaled nonradiative heating rate and A_met is the metal abundance), such as the sign and magnitude of each coefficient being roughly the same. Dissimilarities are likely due to the differences in terms (F and A_met) and parameter ranges in the sample stars. In this analysis, the stars used have log₁₀ g between 3.9–5.2, log₁₀ L(Si III)/Lbol between −7.5 and −5.0, and T_eff between 3000–3900 K. The observed range of FW_20% values is 0.6–3.8 Å.

As is the case for Ca ii, stellar activity appears to be a minor factor in the width of Lyα, as also indicated by the lack of correlation between FW_20% and L(Lyα)/L_bol (ρ = −0.06, p = 0.82) or L(Si III)/L_bol (ρ = 0.29, p = 0.27). The more dominant factors are surface gravity and effective temperature, as indicated by the correlation coefficients between FW_20% and T_eff (ρ = 0.50; p = 0.05) or g (ρ = −0.51; p = 0.04). From Figure 3, we find that, in general, the M dwarfs with larger Lyα wing flux values tend to be more active. The "inactive" MUSCLES M dwarfs (as determined by optical activity indicators such as Ca II; see France et al. (2016)) have the narrowest profiles, and Proxima Centauri has a surprisingly narrow profile given its known levels of moderate activity (Robertson et al. 2013, 2016; Davenport et al. 2016; Howard et al. 2018). For example, compare Proxima Centauri's log₁₀ L(Si III)/L_bol = −6.2 to the −7.2 to −7.5 values for the inactive MUSCLES M dwarfs GJ 832, GJ 581, and GJ 436. However, as discussed, these line widths are dominated by stellar structure, and in general, M dwarfs with lower surface gravity (i.e., younger ones) tend to be more active.

The electron density in the line forming region (the chromosphere for the Lyα broad wings) is a main factor in controlling the width of the Lyα line (Gayley 1994). We estimate chromosphere electron density values, which are a valuable constraint for stellar models, using the formalism from Gayley (1994) that explicitly relates the surface flux density of the Lyα broad wings to chromospheric electron density and other stellar properties:

$$\begin{eqnarray}&&{F}_{\mathrm{wing}}({\rm{\Delta }}\lambda )\approx \displaystyle \frac{{F}_{\mathrm{peak},\odot }}{{\rm{\Delta }}{\lambda }^{2}}\,{\left(\displaystyle \frac{{n}_{e}}{{n}_{e,\odot }}\right)}^{2}\,\displaystyle \frac{{g}_{\odot }}{g}\,\displaystyle \frac{{T}_{\mathrm{chromo}}}{7500\,K}\,\displaystyle \frac{{J}_{2c,\odot }}{{J}_{2c}},\end{eqnarray} \tag{ 6 }$$

where F_wing(Δλ) is the Lyα surface flux density at Δλ Å from line center, F_peak,⊙ is the peak solar Lyα flux (∼3 × 10⁵ erg cm⁻² s⁻¹), n_e is the chromospheric electron density, g is the surface gravity, T_chromo is the chromospheric temperature, and J_2c is the Balmer continuum flux. Each parameter is normalized to the solar (⊙) value. Stars with larger electron densities and hotter chromospheres will have broader wings, but the wing intensity is diminished for stars with greater surface gravity and greater Balmer continuum flux.

Figure 5 shows the Lyα surface flux densities of the FUMES targets, the MUSCLES M dwarfs (France et al. 2016), Proxima Centauri and AU Mic (Youngblood et al. 2017), and the Sun (SORCE/SOLSTICE; McClintock et al. 2005), plotted against surface gravity. Lines of constant electron density are drawn on the plot using Equation (6). We assume T_chromo and J_2c are both equivalent to solar values (T_chromo = 7500 K; J_2c = 1.7 × 10⁵ erg cm⁻² s⁻¹ Å⁻¹ sr⁻¹). For stars with known chromospheric electron densities, the Gayley (1994) approximation works well. The Sun's electron density log₁₀ n_e = 11 cm⁻³ (Song 2017), and GJ 832's log₁₀ n_e = 10 cm⁻³ (Fontenla et al. 2016), are both in agreement with the gray curves in Figure 5.

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We find that LP 247-13, a 625 Myr M2.7V star, has a chromospheric electron density similar to that of the Sun. All of the FUMES targets ("active" stars) have electron densities larger than those of the "inactive" M dwarfs from the MUSCLES survey, except for GJ 176. We note that GJ 176 is the least "inactive" of the MUSCLES stars, as it is the most rapidly rotating (P_rot = 39.5 day, per Robertson et al. (2015)) and is possibly younger than 1 Gyr based on its large X-ray luminosity (Guinan et al. 2016; Loyd et al. 2018).

The presented Lyα reconstructions are the first based on λ/Δλ ∼ 1000 spectra with the ISM H i absorption completely unresolved. Using the STIS G140L mode provides some observational advantages, including avoiding prohibitively long exposure times of higher-resolution STIS modes for M-dwarf targets deemed too hazardous for the COS instrument (Bright Object Protections ¹² ). Based on the six M dwarfs presented here, we find that the precision of Lyα reconstructions performed on STIS G140L spectra can range from 5% to 100% at the 68% confidence level (Figure 6). At the 95% confidence level, the precisions range from ∼10% to a factor of nine. There appears to be no dependence of these precisions on the S/N of the observed spectrum; we note that all G140L Lyα emission lines were detected at high S/N (90–250 integrated over the line). Rather, our three G140L targets with the largest reconstructed flux uncertainties (GJ 4334, GJ 49, and GJ 410) are also the G140L targets with the lowest surface flux in the Lyα wings, or in other words, the narrowest profiles. We hypothesize that, for narrow profiles (Lyα surface flux at ±2 Å ≲10⁴ erg cm⁻² s⁻¹ Å or FW_20% < 2.5–3.0 Å), the spectrum does not provide enough spectrally resolved information for the fit to distinguish between solutions with large flux and large ISM columns versus those with small flux and small ISM column solutions. The higher-resolution E140M grating results in reconstructed flux precisions of approximately 2%–4% at the 68% confidence level for high S/Ns (we note that the two stars with E140M observations, HIP 112312 and HIP 17695, have a line-integrated S/N = 70–90). However, for lower S/N spectra, Youngblood et al. (2016) found uncertainties up to 150% in E140M reconstructions of K dwarfs (HD 97658, HD 40307, HD 85512) with S/N = 20–30 integrated over the line. The precision found by Youngblood et al. (2016) with the STIS G140M grating (λ/Δλ ∼ 10,000) is 5%–30% for medium to high S/Ns, and can be a factor of ∼2 for low S/Ns (e.g., GJ 1214, S/N = 4 integrated over the line). Thus, STIS G140L spectra can produce reconstructed Lyα fluxes for young, active M dwarfs with precisions comparable to G140M spectra, but the precision is much lower than what is obtainable with high S/N G140M or E140M spectra.

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Adopting FW_20% > 2.5 Å as the threshold between precise and imprecise Lyα reconstructions with G140L, Equation (5) may be useful for guiding future observers toward whether or not G140L is suitable for a Lyα reconstruction for a particular M dwarf. Surface gravity and effective temperature, two of the three stellar parameters in Equation (5), are readily available in the literature for many M dwarfs. The third parameter, L(Si III)/L(bol), is not available for most M dwarfs, but can be estimated from the stellar rotation period (Pineda et al. 2021) or common activity indicators like ${R}_{{HK}}^{{\prime} }$ or L(Hα)/L(bol) (Melbourne et al. 2020).

Figure 6 shows how the observed Lyα fluxes compare to the reconstructed (intrinsic) fluxes. The observed fluxes were obtained simply by integrating over the observed, ISM-attenuated Lyα profiles. In some cases, the observed Lyα fluxes are only 10%–50% less than the reconstructed fluxes, while in others they are a factor of a few to an order of magnitude less. The dominant factor in the flux differences is the column density of the ISM absorbers and the radial velocity of the ISM absorbers relative to the stellar radial velocity. A small radial velocity offset between the star and ISM, along with larger column densities, will result in larger flux differences between observed and reconstructed fluxes. Figure 6 may give readers a sense of whether or not performing a reconstruction on G140L Lyα spectra is worthwhile for their science goals.