Estimating the Ultraviolet Emission of M Dwarfs with Exoplanets from Ca ii and Hα

M dwarfs (M_⋆ ≲ 0.6M_⊙) are excellent candidates for ongoing exoplanet detection and characterization efforts (Shields et al. 2016; Tarter et al. 2007). These are the most abundant stars in the stellar neighborhood (RECONS Survey ²³ ), making up ∼75% of the Galaxy's total stellar population (Reid & Gizis 1997; Bochanski et al. 2010). M dwarfs are known to harbor larger populations of small and terrestrial planets relative to solar-type stars (Batalha et al. 2013; Dressing & Charbonneau 2015). The signals from planets orbiting M dwarfs are larger and easier to identify for both the transit and radial velocity methods, the two most commonly used techniques for finding exoplanets. In addition, the long lifetime of an M dwarf (on the order of 10¹² years (Laughlin et al. 1997) for the lowest mass stars) provides ample time for planets to form and life to evolve. Despite concerns about extreme ultraviolet (EUV) emission, frequent energetic flares, coronal mass ejections, and possible tidal locking, which could alter the atmosphere of an exoplanet orbiting an M dwarf (see Shields et al. 2016 and references within), these smaller, dimmer stars remain a priority in current exoplanet detection efforts due to their many advantages.

The high-energy radiative environment around M dwarfs can significantly impact the upper atmospheric temperature and chemistry of the exoplanets orbiting them (Segura et al. 2005; Lammer et al. 2007; Miguel et al. 2015; Rugheimer et al. 2015; Arney et al. 2017). Quantifying the ∼1200–1700 Å (far-UV, FUV, hereafter) and 1700–3200 Å (near-UV, NUV) wavelength ranges of the ultraviolet emission of an M dwarf is essential to assessing any photochemical byproducts (e.g., hazes, biosignatures) in the atmospheres of its exoplanets (Shields et al. 2016; Meadows et al. 2018). In addition, knowledge of an M dwarf's FUV and NUV radiation environment allows for the estimation of EUV emission (Sanz-Forcada et al. 2011; Chadney et al. 2015; Linsky et al. 2013; France et al. 2018; Peacock et al. 2019). This is significant because EUV observations cannot be taken directly for stars other than the Sun; there is no current dedicated mission yet that observes 170–900 Å, and interstellar medium (ISM) attenuation makes observations between ∼500–911 Å infeasible for many stars.

M dwarfs have bolometric luminosities ranging from ∼10⁻⁴–10⁻¹ L_⊙, moving the location of cool exoplanets (including liquid-water habitable zone (HZ) planets) 5 to 100 times closer around an M dwarf than the comparable orbital separation from the Sun. Combining this with the relatively high UV fluxes of M dwarfs, the incident FUV fluxes are generally higher for cool planets (T_equil ≲ 1000 K) orbiting M dwarfs than they would be around stars of other spectral classes. Photochemical processes depend on the FUV to NUV flux ratio (France et al. 2013; Loyd et al. 2016) and can influence the formation of hazes (Morley et al. 2013, 2015; Crossfield & Kreidberg 2017; Libby-Roberts et al. 2020) as well as the abundances of key biosignatures (O₂, O₃, and CH₄) and habitability indicators (CO₂ and H₂O) in exoplanet atmospheres around M dwarfs (Hu et al. 2012; Tian et al. 2013; Domagal-Goldman et al. 2014; Wordsworth & Pierrehumbert 2014; Gao et al. 2015; Harman et al. 2015; Luger & Barnes 2015). For example, Lyα (Lyα = 1215.67), the brightest UV emission line in an M dwarf spectrum, has been shown to significantly alter the H₂O mixing ratios within the atmosphere of a mini Neptune orbiting its M dwarf host (Miguel et al. 2015). As H₂O and CO₂ in a terrestrial planet atmosphere absorb Lyα radiation from the M dwarf, they dissociate, creating free H and O atoms and OH and CO molecules, leading to catalytic cycles that can produce false-positive and false-negative biosignatures (Harman et al. 2015; Miguel et al. 2015; Rugheimer et al. 2015). UV radiation also leads to the formation of organic hazes in rocky planet atmospheres (Arney et al. 2017) as well as hazes in gaseous planets. Haze strongly affects an exoplanet's spectral features as well as habitability (Arney et al. 2018; Hörst et al. 2018), and an accurate UV spectrum is critical to haze formation modeling.

To analyze exoplanet atmospheres and identify potential false-positive biosignatures, it is essential to characterize the spectral energy distributions (SEDs) of M dwarf hosts, as the cross sections of important molecules and atoms are highly wavelength dependent and peak in the UV (Hu et al. 2012). Most stellar models do not extend beyond the photosphere (Husser et al. 2013; Allard et al. 2012; Hauschildt et al. 1999), thus excluding the primary regions of UV emission. Some progress has been made in including chromospheres, transition regions, and coronae for models of individual stars (Fontenla et al. 2016; Peacock et al. 2019), but those methods are not yet broadly applicable. As a result, SEDs must be directly observed to measure the FUV and NUV activity of a specific star. This requires dedicated space-based UV telescopes and underscores the importance of current observations by the Hubble Space Telescope (HST; France et al. 2016; Guinan et al. 2016; Loyd et al. 2018a; Ribas et al. 2017). However, there may be a gap in observing capability for UV characterization after HST stops UV observations in the coming years. As the Transiting Exoplanet Survey Satellite has completed its primary mission, the Characterizing Exoplanets Satellite has completed its commissioning, the James Webb Space Telescope (JWST) launch approaches, and the extremely large telescopes prepare to begin operations in the late 2020s, a lapse in UV spectral data would limit the characterization of exoplanets discovered with these instruments. Consequently, it is crucial to identify an alternative approach to estimating the UV emission of M dwarfs from optical data measured with ground-based observatories.

Past research has demonstrated the effectiveness of using the Ca ii H&K resonance lines (3969 Å, 3934 Å) as indicators for stellar activity (e.g., Wilson 1963; Cincunegui et al. 2007a; Walkowicz & Hawley 2009). Ca ii H&K lines appear against the continuum as a superposition of broad Ca⁺ absorption (>1 Å) from the cool upper photosphere and lower chromosphere as well as narrow Ca⁺ emission (<−0.5 Å) from the hot upper chromosphere (Fontenla et al. 2016). Ca ii H&K emission has been studied in detail since the Mount Wilson observing program in the 1960s (Wilson 1968) through two stellar activity indices: the S index that includes both chromospheric and photospheric emissions, and ${R}_{\mathrm{HK}}^{{\prime} }$, a transformation of the S index that is normalized to the bolometric flux. This enables comparisons between different spectral types by excluding the contribution of photospheric emission to the measured S index. The S index is only a normalized measure of the line core emission and does not provide a value for the absolute energy emitted in the line, so it is essential to determine ${R}_{\mathrm{HK}}^{{\prime} }$ as well. The program at Mount Wilson defined the S index and ${R}_{\mathrm{HK}}^{{\prime} }$ exclusively using observations of F, G, and K stars, and until recently, the color indices used to calculate these parameters had not been well calibrated for M dwarfs. Cincunegui et al. (2007b) and Suárez Mascareño et al. (2015) used flux-calibrated observations to extend to M dwarfs the color correction factors needed to correct the S index for spectral type effects. Astudillo-Defru et al. (2016) reexamined both the S index and ${R}_{\mathrm{HK}}^{{\prime} }$ Ca ii H&K activity tracers for FGK stars and identified an improved method of accurately calculating the S index and ${R}_{\mathrm{HK}}^{{\prime} }$ for M dwarfs, which we follow in this paper.

The Hα (6562.8 Å) equivalent width (EW) and Hα luminosity normalized to the stellar bolometric luminosity (log₁₀ (L_Hα /L_bol)) were selected for this work as they are also a commonly used indicator for stellar activity, and measurements are widely available in the literature (Reid et al. 1995; Gizis et al. 2002; West et al. 2011; Douglas et al. 2014; Gaidos et al. 2014; Alonso-Floriano et al. 2015; Newton et al. 2017). Hα traces the top of the chromosphere (Mauas & Falchi 1994, 1996; Leenaarts et al. 2012). All M dwarfs appear to emit significant UV radiation (France et al. 2016), but they are still categorized in the literature as being either "active" or "inactive" depending on each star's Hα emission. For stars in the "inactive" regime (Hα EW > −1 Å, i.e., in absorption), including a significant fraction of the stellar sample used for this work, Hα has been shown to be nonmonotonic with stellar activity (Cram & Mullan 1985; Stauffer & Hartmann 1986) and therefore may not be a precise tracer of UV activity. However, Newton et al. (2017) showed log₁₀ (L_Hα /L_bol) to be significantly correlated with the stellar rotation period.

The correlation between M dwarf chromospheric optical and UV emission has been demonstrated previously (e.g., Hα-C iv, Hα-Mg ii, Ca ii K-Mg ii, and Ca ii K with several Balmer lines; Hawley & Pettersen 1991; Hawley & Johns-Krull 2003; Walkowicz & Hawley 2008; Youngblood et al. 2017). Youngblood et al. (2017) identified the relationships between Ca ii and nine UV spectral lines for 15 M dwarfs from 1200 to 2800 Å. In this paper, we expand the sample to 69 M dwarfs and increase the range of activity levels, spectral types, and ages of the stellar sample.

This paper is organized as follows: Section 2 details the M dwarf target selection and the observations used for the analysis in this work. Sections 2.3 and 2.4 discuss how measurements of each optical activity index were found with spectral analysis of the Hα and Ca ii H&K lines, respectively. Section 2 also includes a description of how the UV emission-line fluxes were determined. Section 3 describes the correlations found between each optical activity indicator and each UV emission line. We conclude with a discussion of these results in Section 4 and a summary of the findings in Section 5.

The M dwarfs analyzed in this work were chosen based on the availability of UV spectra from HST and Ca ii and/or Hα spectra from high-resolution optical spectrographs like HARPS, HIRES, and UVES. Many of the UV spectra came from recent HST guest observer programs, including the MUSCLES Treasury Survey (GO 13650; France et al. 2016; Loyd et al. 2016; Youngblood et al. 2016), the Mega-MUSCLES Treasury Survey (GO 15071; Froning et al. 2019), the Far Ultraviolet M-dwarf Evolution Survey (FUMES; GO 14640; J. S. Pineda et al. 2020, in preparation), the Habitable Zones and M dwarf Activity across Time (HAZMAT) survey (GO 14784; Loyd et al. 2018a), and the M Dwarf Stellar Wind survey (GO 15326; B. E. Wood et al. 2020, in preparation). Our targets range in spectral type from M0 to M9, and all but one target observed are within d < 60 pc of Earth. The sample covers a wide range of ages (∼0.01 to ∼10 Gyr): several stars in the TW Hya association are likely the youngest stars (∼10 Myr; Weinberger et al. 2012) and Barnard's Star and Kapteyn's Star are likely the oldest stars (∼10 Gyr; Kotoneva et al. 2005; Wylie-De Boer et al. 2010; Ribas et al. 2018). Figure 1 provides a visual of the range of radii, effective temperatures, and ${R}_{\mathrm{HK}}^{{\prime} }$; other details about each target may be found in Tables 1 and 2. Ages were gathered from the literature, where they were established through membership to young moving groups or clusters or galactic kinematics. In the absence of any age information, targets are assumed to be field age (∼5 Gyr).

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2.1. Optical Data

For the optical spectra, we used public archival data from three ground-based observatories, the High Accuracy Radial velocity Planet Searcher (HARPS) on the ESO 3.6 m telescope (Mayor et al. 2003), the Ultraviolet and Echelle Spectrograph (UVES) on the Very Large Telescope (Dekker et al. 2000), and the High Resolution Echelle Spectrometer (HIRES) on the Keck I Telescope (Vogt et al. 1994). HARPS (S1D) and UVES one-dimensional merged spectra were downloaded from the ESO archive. We also obtained new Keck/HIRES spectra of M dwarfs with available UV spectra but no Ca ii H&K spectra. We were awarded three half-nights (2019 March 1, 2019 July 7, and 2019 July 8). We used the HIRES blue arm with the C5 decker (1148 × 7'' slit), the KV370 filter, echelle angle = 0046918, cross-dispersion angle = 19523, and standard 2 × 1 binning. The nominal spectral resolving power of this mode is R = 37,000. We obtained new spectra covering wavelengths 3750–6720 Å for four M dwarfs (LP 5-282, G75-55, LP 247-13, GJ 3290), and we use the data products from the automatic pipeline MAKEE ²⁴ as were used for all archival HIRES spectra used in this work. For targets with no available optical spectra with sufficient signal-to-noise (S/N) from HARPS, HIRES, or UVES, we used single spectra available to our team from the Canada–France–Hawaii Telescope ESPaDOnS spectrograph ²⁵ (2MASS J02001277–0840516 and GJ 3997) and the Observatoire de Haute-Provence ELODIE spectrograph ²⁶ (2MASS J04223953+1816097 and 2MASS J04184702+1321585; Moultaka et al. 2004).

We did not flux-calibrate any of our optical spectra; the activity indicators used rely on normalizations to nearby continua, and we compared our measurements to overlapping samples from Newton et al. (2017) and Astudillo-Defru et al. (2016) to ensure our technique is in line with theirs. To account for the time variability of stellar activity levels with noncontemporaneous observations, we found the optical activity indices of each spectrum individually and took the S/N weighted average for the final result (described in Sections 2.3 and 2.4). Errors calculated for the optical parameters of our target sample do not take into account stellar variability, and not all stars were observed over multiple epochs. Measurements for a representative sample from our target list are shown in Figure 2.

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2.2. Ultraviolet Data

2.3. Hα Equivalent Widths and log₁₀ (L_Hα /L_bol)

We measured Hα EWs using the equation

$$\begin{eqnarray}&&\mathrm{EW}={\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\left(1-\displaystyle \frac{{F}_{\lambda }}{{F}_{c}}\right)d\lambda ,\end{eqnarray} \tag{ 1 }$$

where F_λ is the flux of each wavelength across the width of the line and F_c is defined as the average continuum from two ranges on either side of the line, 6500–6550 Å and 6575–6625 Å. For dλ, we use the average pixel width between each observed wavelength in a spectrum, as variations are negligible. We follow the method used by West et al. (2011) and Newton et al. (2016) and assign the bounds of integration to be λ₁ = 6558.8 Å and λ₂ = 6566.8 Å for all spectra. The EW-weighted means and uncertainties for all targets are listed in Table 2.

We validate our EW measurements by comparing with Newton et al. (2016) Hα EWs (Hα_N ) for 12 overlapping targets, finding the best-fit line Hα_N  = 1.05(±0.003)Hα + 0.04(±0.03). The slight departure from a precise 1:1 line is a result of the use of multiple instruments in our calculations, and no additional calibration was performed to match our measured values to Newton et al. When optical spectra were not available for Hα measurements, we used literature values from Riaz et al. (2006) and Malo et al. (2014a), who used the IRAF splot package to calculate Hα EWs for a sample of M dwarfs. Our calculated values are comparable to those found in the literature (Hα_lit). Fitting a least-squares linear regression, we found a best-fit line of Hα_lit = 0.98(±0.04)Hα − 0.34(±0.41), and therefore, we assume no significant variation in Hα values across both studies and this work.

In order to remove the stellar mass dependence of our Hα index (the Wilson–Bappu effect; Wilson & Vainu Bappu 1957; Stauffer & Hartmann 1986), we calculate log₁₀ (L_Hα /L_bol) using the methods of Douglas et al. (2014) and Newton et al. (2017). First, we subtract from our EW measurements the minimum Hα EW value for a star of a given mass using relations from Newton et al. (2017). We estimate the masses of our targets to accuracies of 10%–20% from various literature sources. For the few cases where literature masses were unavailable, we used BT-Settl isochrones ²⁷ (Allard et al. 2012) with solar abundances (Caffau et al. 2010) and the target's effective temperature (Table 2) to determine a mass. We then converted these corrected Hα EWs into L_Hα /L_Bol by multiplying each EW by a spectral type conversion factor (χ) determined by Douglas et al. (2014).

2.4. Ca ii H&K Indices

2.4.1. The S Index

The S index measures the flux ratio between the Ca ii H&K lines and the surrounding continuum, standardized by an instrumental calibration factor (Vaughan et al. 1978). Two triangular bandpasses are centered on the H and K lines with 1.09 Å FWHM, and two 20 Å top-hat bandpasses are centered at 3901 Å and 4001 Å, known as the V and R ranges, respectively (Figure 3). The S index is limited in comparing activity levels across spectral types; as the integrated continuum emission decreases for cooler stars, the S index will increase (Middlekoop 1982). In addition, the triangular bandpasses include contaminating photospheric emission in addition to the desired chromospheric emission.

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We follow the method in Lovis et al. (2011) and Astudillo-Defru et al. (2016) to allow for optimal validation of our measurements, as there is a significant overlap of our data set with theirs. Our measured values are compiled in Table 2. We calculate the S index for all of our target stars by determining the mean flux emitted across both 20 Å continuum regions and the 1.09 Å regions centered on each line instead of integrating over them,

$$\begin{eqnarray}&&S=\alpha \displaystyle \frac{\tilde{{f}_{H}}+\tilde{{f}_{K}}}{\tilde{{f}_{V}}+\tilde{{f}_{R}}},\end{eqnarray} \tag{ 2 }$$

where α ≈ 1 and $\tilde{{f}_{V}}$, $\tilde{{f}_{R}}$, $\tilde{{f}_{H}}$, and $\tilde{{f}_{K}}$ are the mean flux emitted in the 20 Å continuum regions (V, R) and the 1.09 Å line regions (H, K). To confirm the accuracy of our measurements, we fit a linear least-squares regression of our S-index values to the Mount Wilson values provided by Astudillo-Defru et al. (2016; 22 overlapping targets in total) to find the following relation: S = 1.05(±0.01) S_orig + 0.06(±0.01), with S representing the S indices found in their study and S_orig denoting those found in our work. For this comparison, we used S-index values found from all three ground-based spectrographs: HARPS, UVES, and HIRES. There were not enough overlapping observations obtained with these three spectrographs to quantify the differences between the measured S index of individual targets using multiple instruments.

2.4.2.  R^' _HK

Accurate characterization of the chromospheric activity levels is essential for this work. UV radiation originates in magnetically heated regions of the stellar atmosphere above the photosphere, meaning photospheric emission is not correlated with UV emission lines. Thus, we measure the ${R}_{\mathrm{HK}}^{{\prime} }$ index (Middlekoop 1982; Noyes 1984; Rutten 1984), which includes corrections for the photospheric flux in the continuum (V, R) and line (H, K) regions of the S index. ${R}_{\mathrm{HK}}^{{\prime} }$ has been well-defined for F, G, and K spectral types (Lovis et al. 2011) and has been previously extrapolated to later spectral types with less accuracy (as discussed in Astudillo-Defru et al. 2016). Past attempts to characterize ${R}_{\mathrm{HK}}^{{\prime} }$ for M dwarfs used the original B − V color index (Mittag et al. 2013), which is not best suited for M dwarfs due to their frequent lack of available B-band photometry in the literature and V band sensitivity to metallicity (Delfosse et al. 1998; Bonfils et al. 2013; Astudillo-Defru et al. 2016).

${R}_{\mathrm{HK}}^{{\prime} }$ is related to the S index by

$$\begin{eqnarray}&&{R}_{\mathrm{HK}}^{{\prime} }={R}_{\mathrm{HK}}-{R}_{\mathrm{phot}}=K{\sigma }_{\mathrm{SB}}^{-1}{10}^{-14}{C}_{\mathrm{cf}}(S-{S}_{\mathrm{phot}}),\end{eqnarray} \tag{ 3 }$$

where R_phot and S_phot are photospheric contributions to ${R}_{\mathrm{HK}}^{{\prime} }$ and S, respectively, σ_SB is the Stefan–Boltzmann constant, C_cf is the color correction factor, and K is a factor that transforms arbitrary fluxes to surface fluxes. Middlekoop (1982) and Rutten (1984) both calculated K, but 1.07 × 10⁶ erg cm⁻² s⁻¹ is the most recent value provided in Hall et al. (2007).

In calculating the color correction factor, C_cf, and R_phot, we follow Equations (9), (10), and Table 1 from Astudillo-Defru et al. (2016). We elect to use the coefficients provided corresponding to the V − K color index in the Johnson photometric system (Johnson 1966), as V-band fluxes are generally more available than I band fluxes. The relations between V − K, C_cf, and R_phot are

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{C}_{\mathrm{cf}} & = & -0.005{\left(V-K\right)}^{3}+0.071{\left(V-K\right)}^{2}\\ & & -\ 0.713(V-K)+0.973\end{array}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{R}_{\mathrm{phot}} & = & -0.003{\left(V-K\right)}^{3}+0.069{\left(V-K\right)}^{2}\\ & & -\ 0.717(V-K)-3.498.\end{array}\end{eqnarray} \tag{ 5 }$$

A small number of our stars have no V-band magnitudes available in the literature, and we calculate the V magnitude from V – K estimates provided by Pecaut & Mamajek (2013). Uncertainties are assumed to be 5% of the flux value; we assumed the same relative error for stars whose V magnitudes have no published uncertainties.

We analyzed the relation between each optical activity indicator and the luminosities of nine different UV lines each normalized by the bolometric luminosity (listed in Table 2). For log₁₀ (L_Hα /L_bol), the S index, and ${R}_{\mathrm{HK}}^{{\prime} }$, power laws were fitted to the data (including uncertainties) in log space using the same bootstrapping method described in Section 2. We resampled the data with replacement and found a best-fit line each time, then found the median and the 68% confidence interval among all the fits performed, shown in Figures 4–6. Tables 3–5 list the fitted power-law parameters for the median line and 1σ uncertainties as well as the Spearman rank-order correlation coefficients (ρ), and the standard deviations about the best-fit line.

Figure 4 shows the relation between log₁₀ (L_Hα /L_bol) and log (L_UV/log L_bol) for each line, and Table 4 describes the fit parameters. We find statistically significant, positive correlations between log₁₀ (L_Hα /L_bol) and log (L_UV/log L_bol) for all UV emission lines. There are two regions of stars, active (Hα in emission) and inactive (Hα in absorption). The active stars are clustered in the area of each graph in Figure 6 with log₁₀ (L_Hα /L_bol) > ∼−4, and the inactive stars are scattered along the rest of the best-fit lines. When analyzed separately, there was no significant correlation in either region; however, our sample spans a wide range of Hα values, which demonstrate a significant correlation when examined together. Mg ii has the weakest correlation with log₁₀ (L_Hα /L_bol) and the most uncertain best-fit line of all the UV emission lines, even when excluding the outlier GJ 676 A (Section 3.1) from the fit, due to a lack of stars in our sample that are active in both Mg ii and Hα. This is likely because of the recent HST brightness restrictions for M dwarfs that are strictest in the NUV regime, affecting the community's ability to collect Mg ii data for more active stars. We present in the caption of Table 4 an alternate fit that parameterizes the apparent flattening of L(Mg ii)/L_bol in the inactive regime. We have also removed the outlier LP 247-13 from the L(Lyα)/L_bol – log₁₀ (L_Hα /L_bol) fit because it drives the best-fit slope to a steep value that does not match the other active stars.

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Here we describe the comparison between Hα EW and log (L_UV/log L_bol), which is not shown because there is no apparent correlation. The targets are divided into two loci: an active and inactive regime. As the EW becomes more negative (more active), the UV luminosity plateaus at a nearly constant high value. For smaller EWs (EW ∼ 0; less active), there is a large spread of several orders of magnitude in normalized UV luminosity. Our sample has 52 M dwarfs from M0–M5.5 with Hα EWs concentrated near the threshold between active (<−1 Å) and inactive (>−1 Å) M dwarfs. Hα absorption lines deepen with increased activity before flipping to emission (Cram & Mullan 1985), leading to a nonmonotonic relation between stellar activity and Hα EW whereas UV emission scales monotonically with stellar activity. This mostly affects stars with Hα EWs ≳−1 Å, making Hα EW a poor activity indicator for inactive M dwarfs (Walkowicz & Hawley 2008).

The relationships between the log S index and log (L_UV/log L_bol) line emissions are shown in Figure 5. The S index includes photospheric contamination that leads to a wider spread of UV flux values at each S index, as the UV lines do not originate in the photosphere (e.g., Vernazza et al. 1981). Because of this, we did not expect a tight correlation with UV luminosity, but the Spearman rank-order correlation coefficients indicate a statistically significant positive correlation for each emission line. Parameters for the best-fit lines are shown in Table 5. The scatter about the S-index best-fit lines are greater than about the log₁₀ (L_Hα /L_bol) best-fit lines for all emission lines except Mg ii.

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Transforming the S index to ${R}_{\mathrm{HK}}^{{\prime} }$ removes the unwanted photospheric contribution that is present in the S index. Because the UV emission lines studied originate in regions of the stellar atmosphere that are above the photosphere and dominated by magnetic heating, we expected to find correlations with less scatter between log ${R}_{\mathrm{HK}}^{{\prime} }$ and log (L_UV/log L_bol) for each emission line. Figure 6 shows statistically significant correlations between log ${R}_{\mathrm{HK}}^{{\prime} }$ and each of the normalized UV emission-line luminosities. The scatter about the log ${R}_{\mathrm{HK}}^{{\prime} }$ best-fit lines is significantly smaller than for both log₁₀ (L_Hα /L_bol) and the S index, except for N v, where the log₁₀ (L_Hα /L_bol) has a smaller scatter by 0.07 dex.

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For each optical activity index, all of the individual UV lines' power-law slopes are consistent with each other within 1σ uncertainties. For ${R}_{\mathrm{HK}}^{{\prime} }$, S index, and log₁₀ (L_Hα /L_bol), the weighted averages are, respectively, 1.57 ± 0.06, 1.89 ± 0.07, and 0.96 ± 0.15. The Mg ii fit dominates the uncertainty in the log₁₀ (L_Hα /L_bol) average power-law slope. Excluding Mg ii, the weighted average power-law slope for log₁₀ (L_Hα /L_bol) becomes 0.95 ± 0.04. We find no correlation between the power-law slope and line formation temperature for any of the fits. The Spearman correlation coefficient indicates a strong positive correlation with values for each UV emission line ranging between 0.76 ≤ ρ ≤ 0.85. The probability of no correlation (n) is <10⁻⁶ for each fit. The standard deviations about the best-fit lines are 0.31 ≤ σ ≤0.61 dex. Comparing these parameters to Youngblood et al. (2017), we see that our standard deviations about the best fit are generally larger, indicating more scatter. This is likely due to our larger and more diverse sample of stars in addition to using a different Ca ii activity indicator (log ${R}_{\mathrm{HK}}^{{\prime} }$). Additionally, we analyzed the relationship between the residuals around our log ${R}_{\mathrm{HK}}^{{\prime} }$ best fit for each UV line and stellar effective temperature and found a shallow, statistically significant negative correlation. This indicates that the hotter stars' UV luminosities are generally overpredicted by the best-fit line and cooler stars are generally underpredicted.

3.1. Outliers

Are our presented UV–optical scaling relations precise enough for photochemical and atmospheric escape models of exoplanets? In this section, we analyze the expected UV precision for a range of typical log₁₀ ${R}_{\mathrm{HK}}^{{\prime} }$ values and measurement precisions. We focus on ${R}_{\mathrm{HK}}^{{\prime} }$ because those relations had the least scatter. Using a star with log₁₀ ${R}_{\mathrm{HK}}^{{\prime} }$ = −4.5 ± 0.30 (∼7% uncertainty) as an example, the precision of our UV luminosity estimates (σ_L /L) for each emission line ranges from factors of 2.27–4.65 (0.36–0.67 dex), depending on the specific emission line. We evaluated this precision by comparing the calculated error and the predicted average luminosity value. The uncertainties of each predicted UV emission-line luminosity are dominated by the error of the best-fit line intercept and the error on log₁₀ ${R}_{\mathrm{HK}}^{{\prime} }$. From the standard deviations about the best-fit lines (Table 3), we estimate that using our scaling relations allows one to approximate the individual L_UV/L_bol of the nine UV emission lines examined in this work within a factor of ∼2–4 (0.31–0.61 dex) for a typical M dwarf UV spectrum. Underscoring the utility and impact of these correlations is the fact that the parameter space for the L_UV/L_bol of our target stars spans almost three orders of magnitude.

Rugheimer et al. (2015) examined the effect of variations in UV spectra on modeled exoplanet spectra and found that a factor of ≲10 UV flux variations propagate to 10%–30% level changes in the depths of spectral features from simulated directly imaged Earth-like planets. Depending on the precision of observed reflection spectra, using our ${R}_{\mathrm{HK}}^{{\prime} }$ scaling relations could be suitable for photochemical modeling purposes. For determining atmospheric escape rates from exoplanets, obtaining accurate EUV fluxes of M dwarfs is notoriously challenging due to a dearth of EUV spectra, and much of the exoplanet community relies on scaling relations between the EUV and other spectral regions like the FUV and X-ray (Sanz-Forcada et al. 2011; Linsky et al. 2013; Chadney et al. 2015; France et al. 2018). For some exoplanets needing atmospheric escape modeling, only an optical spectrum of the host star may be available, and here we estimate the suitability of our optical–FUV scaling relations for extrapolating to the EUV. Bolmont et al. (2017) showed that in the low- and high-EUV flux regimes, water-loss rates on an Earth-like planet orbiting TRAPPIST-1 increase at the same rate as EUV flux. However, in the moderate EUV flux regime, water-loss rates increase more slowly than a 1:1 relation with increasing EUV flux, indicating that uncertainties in the incident EUV flux up to a factor of 10 may be acceptable in this regime. However, determining where this moderate EUV regime is may depend on the particular star and simulated planet. Thus, it is unlikely that our optical–FUV scaling relations can be propagated into other FUV–EUV scaling relations (e.g., Linsky et al. 2013; France et al. 2018) and retain a sufficiently small level of uncertainty that would not dominate over the escape model's uncertainties. Based on these examples of models that consider the impact of absolute UV flux on exoplanets, we conclude that the precision provided by this work's scaling relations may be sufficient for photochemical modeling needs, but not atmospheric escape modeling. Further work is needed to demonstrate the impact of UV spectrum uncertainties on photochemical models, and will be addressed in an upcoming paper (D. J. Teal et al. 2020, in preparation).

Lyα alone represents 75%–90% of the 1200–1700 Å flux for typical M dwarfs (e.g., GJ 832, GJ 876, GJ 176; France et al. 2013), so by estimating (or directly measuring and reconstructing) the Lyα line, one can account for the majority of the FUV flux from an M dwarf. However, not accounting for the remaining SED across the FUV might significantly change results from photochemical models given the strong wavelength dependence of photoabsorption cross sections of key atmospheric molecules. Here we estimate the percentage of non-Lyα FUV flux made up by the seven FUV lines ²⁹ (excluding Lyα) we analyzed (Si ii, Si iii, Si iv, C ii, C iv, He ii, N v). An important limitation in our ability to characterize the FUV spectra of M dwarfs is the extremely faint FUV continuum (photospheric and chromospheric), which is well below the COS and STIS instrument background levels in almost all cases. Loyd et al. (2016) detected weak FUV continuum emission in three of seven of the MUSCLES M dwarfs (GJ 832, GJ 876, GJ 176) by integrating across multiple line-free bandpasses and estimated that the continuum emission comprises at least 10% of the 1307–1700 Å FUV flux region. D. Tilipman et al. (2020, in preparation) created high-resolution synthetic FUV spectra of GJ 832 and GJ 581 and found that the percentage of FUV emission between 1300 and 1700 Å comprised by continuum is 57% and 43%, respectively. Our ability to develop scaling relations for estimating the FUV continuum of M dwarfs depends on more sensitive observations of these stars as well as model stellar atmospheres that accurately treat the upper atmosphere (Fontenla et al. 2016; Peacock et al. 2019; D. Tilipman et al. 2020, in preparation). Linsky et al. (2012) showed significant correlations between the FUV continuum of G dwarfs (measured over 1382–1392 Å) and stellar rotation period as well as Si iv flux, so similar correlations likely exist for M dwarfs.

By measuring the mean flux density in two line-free regions (1337–1351 Å and 1374–1392 Å) from our spectra of GJ 832, GJ 876, and GJ 176, we estimate that weak FUV emission lines and continuum comprise 20%–50% of the non-Lyα 1200–1700 Å flux, while our seven FUV lines (excluding Lyα) comprise 30%–50%. Other weak-to-moderate intensity emission lines that we did not include in our study comprise any remaining flux. This means that 50%–70% of the non-Lyα FUV flux is unaccounted for by our scaling relations. A direct FUV spectrum may be best to accurately characterize the 1200 – 1210 + 1222 – 1700 Å FUV flux. However, if the weak forest of emission lines and FUV continuum is below the detector sensitivity, as is the case for essentially all but the brightest M dwarfs observable with HST, using the scaling relations to scale a high-S/N M dwarf spectrum could be an appropriate substitute for a direct FUV spectrum with HST. Similarly, our scaling relations do not account for flux from faint emission lines in the NUV or any chromospheric continuum. In the NUV, Mg ii at 2796, 2802 Å is by far the brightest emission line, but there is a forest of much fainter Fe II lines as well as other atomic species that can be difficult to measure with HST depending on the brightness of the target. More work is needed to assess the impact of excluding faint but numerous emission lines from UV spectral inputs on photochemical models of exoplanet atmospheres.

We also examine whether the scaling relations change based on the stellar age given that magnetic dynamos, which are ultimately responsible for the presence of these emission lines, change over a star's lifetime. We subdivided each plot of ${R}_{\mathrm{HK}}^{{\prime} }$, S index, and log₁₀ L_Hα /L_bol as a function of UV activity into groups of stars <0.1, 0.1–1 Gyr (inclusive), and >1 Gyr as shown in Figures 7–9. As expected, the oldest stars are typically the least active and the youngest are usually the most active. The youngest stars are clumped in the high activity region of the plots with no apparent linear relationship between optical and UV activity, whereas the intermediate- and field-age stars span a range of activity levels. We note that we have no stars in our sample with log₁₀ ${R}_{\mathrm{HK}}^{{\prime} }$ values >−3.8, which align with the finding in Astudillo-Defru et al. (2016) that activity saturates around log₁₀ ${R}_{\mathrm{HK}}^{{\prime} }$ = −3.5. Stelzer et al. (2013), Shkolnik & Barman (2014), and France et al. (2018) demonstrated a similar saturation limit with UV luminosity, although this limit varies depending on the emission line. We conclude that these young stars are saturated or nearly saturated in UV and Ca ii luminosity, which explains the lack of correlation among them. However, we expect that a similarly narrow range of emission strength for older stars would also show a lack of correlation.

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We have extended and improved upon previous efforts to determine useful scaling relations between the optical and UV spectra of M dwarfs. Through empirical analysis of four standard optical activity indices (Hα EW and log₁₀ (L_Hα /L_bol), Ca ii S index, and ${R}_{\mathrm{HK}}^{{\prime} }$), we have determined a new method of estimating the UV luminosity of M dwarfs when UV data are not available. The main findings are outlined below.

• 1.  
  Time-averaged ${R}_{\mathrm{HK}}^{{\prime} }$, S index, and log₁₀ (L_Hα /L_bol) correlate positively and significantly with the normalized UV luminosity (L_UV/L_bol) of nine far- and near-UV spectral lines (see Tables 3–5). The scatter about the best-fit lines is lowest for ${R}_{\mathrm{HK}}^{{\prime} }$ (0.31–0.61 dex) and highest for the S index (0.58–0.84 dex). The scatter around the log₁₀ (L_Hα /L_bol) best fit lines ranges from 0.42–0.68 dex, excluding Mg ii. No statistically significant correlation was found between L_UV/L_bol and Hα EW.
• 2.  
  The luminosity of individual UV emission lines normalized to stellar bolometric luminosity can be estimated with ${R}_{\mathrm{HK}}^{{\prime} }$ within a factor of ∼2–4 (0.31–0.61 dex; Table 3). This implies that the scaling relations defined in this study can be a useful substitute for direct UV observations of M dwarfs.

The results presented here address important problems in the characterization of cool (T_eq  ≲ 1000 K) exoplanets and the search for habitable exoplanets. M dwarfs are excellent targets for finding and characterizing small and/or cool exoplanets. However, their UV spectra can have significant and misleading effects on the composition of exoplanet atmospheres. The UV–${R}_{\mathrm{HK}}^{{\prime} }$ scaling relation developed in this paper provides an alternative method for completing photochemical analysis of exoplanet atmospheres without needing observations from space-based telescopes with valuable and limited resources. This will allow for efficient follow-up on exoplanet discoveries, which is essential given the number of current and upcoming dedicated exoplanet missions. In addition, this work will help determine which planets may be the most amenable for further study so that the outcome of observations on major missions like JWST can be maximized.

Based on observations with the NASA/ESA Hubble Space Telescope obtained from MAST at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Data used were obtained as parts of GO # 13650, 14784, 14640, 13020, 14462, 14767, 12361, 11616, 12011, 9090, 15071, 15326, and 15190. This research also relied on the European Southern Observatory (ESO) Archive Facility for HARPS and UVES science products, and the Keck Observatory Archive (KOA) operated by the W. M. Keck Observatory for HIRES data. This work made use of spectral data retrieved from the ELODIE archive at Observatoire de Haute-Provence (OHP, http://atlas.obs-hp.fr/elodie/) and is based in part on data products available at the Canadian Astronomy Data Centre (CADC) as part of the CFHT Data Archive as well. CADC is operated by the National Research Council of Canada with the support of the Canadian Space Agency. This work was supported by a NASA Keck PI Data Award, administered by the NASA Exoplanet Science Institute. Data presented herein were obtained at the W. M. Keck Observatory from telescope time allocated to the National Aeronautics and Space Administration through the agency's scientific partnership with the California Institute of Technology and the University of California. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This research also made use of the SIMBAD database, operated at CDS, Strasbourg, France. K.M. thanks Nicola Astudillo-Defru for helpful correspondence. She also acknowledges support from the NASA Internship program, the Universities Space Research Association (USRA) undergraduate scholarship awards, the National Space Grant Foundation's John Mather Nobel Scholarship Program to present this research, and the Bruce M. Babcock '62 Travel Research Fellowship to complete observations in Waimea, Hawaii. A.Y. and S.E.L. acknowledge support by an appointment to the NASA Postdoctoral Program at Goddard Space Flight Center, administered by USRA through a contract with NASA. R.O.P.L. and E.S. gratefully acknowledge support from NASA HST Grant HST-GO-14784.001-A for this work. P.C.S acknowledges support from DLR 50 OR 1901. The authors thank the anonymous referee for the review of this paper.

Data were graciously made available through the ESO archive from the following ESO programs: 072.C-0488(E), 183.C-0437(A), 198.C-0838(A), 077.C-0364(E), 191.C-0873(D), 191.C-0873(B), 191.C-0873(A), 082.C-0718(B), 183.C-0972(A), 085.C-0019(A), 091.C-0034(A), 090.C-0421(A), 191.C-0873(F), 191.C-0873(E), 095.C-0718(A), 192.C-0224(A), 191.C-0505(A), 192.C-0224(H), 192.C-0224(B), 089.C-0904(A), 095.D-0291(A), 088.C-0506(A), 095.C-0437(A), 082.C-0218(A), 180.C-0886(A), 093.C-0343(A), 076.C-0010(A), 074.C-0037(A), 60.A-9036(A), 192.C-0224(G), 192.C-0224(C), 096.C-0876(A), 097.C-0390(B), 099.C-0225(A), 68.D-0166(A), 075.C-0202(A), 099.C-0205(A), 075.C-0321(A), 082.D-0953(A), 099.C-0880(A), 096.C-0258(A), 089.C-0207(A), 077.C-0012(A), 079.C-0046(A), 080.D-0151(A), 276.C-5054(A), 086.D-0062(A), 081.D-0190(A), 089.C-0732(A), 093.C-0409(A), 095.C-0551(A), 096.C-0460(A), 092.C-0721(A), 192.C-0852(M), 098.C-0366(A), 088.C-0662(B), 089.C-0497(A), 076.C-0155(A), 495.L-0963(A), 074.B-0639(A).

Facilities: HST (COS - , STIS) - , Keck:I (HIRES) - KECK I Telescope, ESO (HARPS - , UVES) - , CFHT (ESPaDOnS) - Canada-France-Hawaii Telescope, OHP (ELODIE). -

Software: IPython (Pérez & Granger 2007), IRAF, Matplotlib (Hunter 2007), NumPy (van der Walt et al. 2011), Pandas (McKinney 2010), SciPy (Virtanen et al. 2020).