The MUSCLES Treasury Survey. IV. Scaling Relations for Ultraviolet, Ca ii K, and Energetic Particle Fluxes from M Dwarfs

Recent ultraviolet (UV) studies have shown that even optically inactive M dwarfs (i.e., those displaying Hα spectra in absorption only) display evidence of chromospheric, transition region, and coronal activity (France et al. 2013; Shkolnik et al. 2014; France et al. 2016) that may significantly affect heating and chemistry in the atmospheres of orbiting exoplanets (e.g., Segura et al. 2003, 2005; Miguel et al. 2015; Rugheimer et al. 2015; Arney et al. 2017). The Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary Systems (MUSCLES) Treasury Survey (France et al. 2016) observed seven nearby (d < 15 pc), optically inactive M dwarfs with known exoplanets using the Hubble Space Telescope (HST), Chandra, and XMM-Newton. All display quiescent UV emission lines and soft X-rays (SXRs) that trace hot (T > 30,000 K) plasma in the upper stellar atmosphere (Loyd et al. 2016). UV flares were observed from each M dwarf except GJ 1214, the faintest target.²¹

An M dwarf's far-UV (912–1700 Å) to near-UV (1700–3200 Å) spectrum is primarily composed of emission lines that form in the stellar chromosphere and transition region, with a few lines originating in the corona. There is comparatively little continuum emission due to the cool stellar photosphere (${T}_{\mathrm{eff}}\lt 4000$ K). The H i Lyα emission line (1215.67 Å) is prominent, composing 27%–72% of the total 1150–3100 Å flux (excluding 1210–1222 Å) for the seven MUSCLES M dwarfs (France et al. 2016; Youngblood et al. 2016). The extreme-UV (EUV; 100–912 Å) stellar spectrum is currently not observable in its entirety. No current astronomical observatory exists to observe the 170–912 Å spectral range, and the ∼400–912 Å range is heavily attenuated by neutral hydrogen for all stars, except the Sun. Thus, the EUV must be estimated from other proxies such as Lyα (Linsky et al. 2014) or SXRs (Sanz-Forcada et al. 2011; Chadney et al. 2015). For the M3 dwarf GJ 436, these two methods produce integrated EUV fluxes that agree within 30% (Bourrier et al. 2016; Youngblood et al. 2016).

Knowledge of an exoplanet's radiation environment is critical for modeling and interpreting its atmosphere and volatile inventory. Specifically, the spectral energy distributions (SEDs) of the host stars are essential, because molecular and atomic cross sections are strongly wavelength dependent. High-energy stellar flux heats upper planetary atmospheres and initiates photochemistry (e.g., Lammer et al. 2007; Miguel et al. 2015; Rugheimer et al. 2015; Arney et al. 2017). UV-driven photochemistry can produce and destroy potential biosignatures (O₂, O₃, and CH₄) and habitability indicators (H₂O and CO₂) in exoplanet atmospheres (Hu et al. 2012; Domagal-Goldman et al. 2014; Tian et al. 2014; Wordsworth & Pierrehumbert 2014; Gao et al. 2015; Harman et al. 2015; Luger & Barnes 2015). In particular, the ratio of far- to near-UV flux determines which photochemical reactions will dominate and thus the resultant planetary atmosphere. Compared to the Sun, M dwarfs have a far-UV/near-UV flux ratio (F(1150–1700 Å)/F(1700–3200 Å)) 100–1000 times larger, and thus most of the known O₂ false-positive mechanisms predominately impact planets orbiting M dwarfs. Accurately measuring the intrinsic Lyα flux is critical, because Lyα composes the majority of the far-UV flux. For an atmosphere with 0.02 bars of CO₂ similar to that simulated by Segura et al. (2007) with GJ 876's SED (France et al. 2012; Domagal-Goldman et al. 2014), a 20% increase (decrease) in the Lyα flux increases (decreases) the abiotic O₂ and O₃ column depths by nearly 30%.

M dwarfs are also prime targets for current and upcoming exoplanet searches and characterization efforts (see Scalo et al. 2007; Shields et al. 2016, for comprehensive overviews), due to their ubiquity in the solar neighborhood (Henry et al. 2006), high occurrence rates of small exoplanets (Dressing & Charbonneau 2015), and the larger transit and radial velocity signals their planets provide. The important UV region of an M dwarf's SED cannot yet be predicted by models, although semi-empirical modeling efforts are under way for individual stars (see Fontenla et al. 2016, and references therein). Thus, direct UV observations of individual stars are currently necessary to model and interpret planetary atmospheric observations, and the characterization of the high-energy SEDs of M dwarfs across a broad range of masses and ages is a community priority (e.g., Shkolnik & Barman 2014; France et al. 2016; Guinan et al. 2016; Bourrier et al. 2017).

Accurate UV spectral flux data are vital in understanding the salient physics and chemistry in an exoplanet atmosphere, as well as to break potential degenerate solutions in retrieval models. After HST and before future UV observatories begin operations, there will likely be a decade-long gap in UV observing capabilities. This gap may coincide with the majority of TESS's and CHEOPS's habitable zone (HZ) planet detections and subsequent study with the James Webb Space Telescope (JWST), so establishing a method of estimating UV spectral flux from ground-based proxies is imperative.

In the optical, the Ca ii resonance lines at 3933 Å (Ca ii K) and 3968 Å (Ca ii H), the Na i D resonance lines at 5890 and 5896 Å, Hα at 6563 Å, and the Ca ii infrared triplet (IRT) at 8498, 8542, and 8662 Å are known to be good indicators of stellar chromospheric activity (e.g., Walkowicz & Hawley 2009; Gomes da Silva et al. 2011; Lorenzo-Oliveira et al. 2016, and references therein). However, Hα and Na i D are only good indicators of stellar activity for very active stars (Cincunegui et al. 2007; Díaz et al. 2007; Walkowicz & Hawley 2009) and will not be considered here, because many of our target stars are optically inactive. The contrast between the Ca ii H and K absorption and emission cores is larger than for the Ca ii IRT, so we have limited the scope of this paper to include only one of the Ca ii resonance lines. We focus on the Ca ii K line at 3933 Å and ignore the Ca ii H line at 3968 Å, because ${\rm{H}}\epsilon$ is 1.6 Å redward and contaminates Ca ii H at low spectral resolution.

The Ca ii K line profile is a superposition of broad (>1 Å) absorption and narrow ($\lt 0.5$ Å) emission. In the cooler upper photosphere, Ca⁺ absorbs against the photospheric continuum, and in the hot upper chromosphere, Ca⁺ emits. In the cool photospheres and lower chromospheres of M dwarfs, Ca is mostly neutral, so the Ca⁺ absorption is much narrower than for solar-type stars (Fontenla et al. 2016). The Ca⁺ emission traces stellar activity and has historically been measured for main-sequence stars using the Mt. Wilson S-index (Vaughan et al. 1978; Wilson 1978) and ${R}_{\mathrm{HK}}^{\prime }$ (Noyes et al. 1984). The latter corrects the S-index for a spectral type dependence. Other methods involve isolating the emission core from the absorption by fitting and subtracting a non-LTE radiative equilibrium model to the observed spectrum (Walkowicz & Hawley 2009; Lorenzo-Oliveira et al. 2016).

For the Sun, Ca ii K is known to correlate well with far-UV emission lines over the 11 yr solar cycle and to trace short-timescale variation (e.g., flares; Tlatov et al. 2015). Ca ii K originates from solar features like plages and chromospheric network (Domingo et al. 2009 and references therein) and thus correlates well with the line-of-sight unsigned magnetic flux density (e.g., Schrijver et al. 1989). Ca ii has also been observed to correlate with H i Lyα for M dwarfs (Linsky et al. 2013) despite significant differences between the atmospheric structures of G and M dwarfs (e.g., Mauas et al. 1997; Fontenla et al. 2016) and potentially different dynamo mechanisms (e.g., Chabrier & Küker 2006; Dobler et al. 2006; Browning 2008; Yadav et al. 2015). Other known M dwarf scaling relations include SXR–Ca ii K, Hα–C iv, Hα–SXR, Hα–Mg ii, Ca ii K–Mg ii, and correlations between Ca ii K and various Balmer lines (Butler et al. 1988; Hawley & Pettersen 1991; Hawley & Johns-Krull 2003; Walkowicz & Hawley 2009). We improve on these past UV–Ca ii scaling relations by increasing the size and diversity (e.g., spectral type and magnetic activity) of the M dwarf sample and expanding to more UV emission lines.

Habitability studies of M dwarf exoplanets are beginning to include estimates of stellar energetic particles (SEPs; Segura et al. 2010; Ribas et al. 2016). SEPs can be accelerated by impulsive flares where particles pass along open magnetic field lines into interplanetary space and by shock fronts associated with coronal mass ejections (CMEs; Harra et al. 2016). SEP enhancements and steady-state stellar winds can contribute significantly to planetary atmospheric loss processes by compressing an exoplanet's magnetosphere (e.g., Cohen et al. 2014; Tilley et al. 2016) and stripping atmospheric particles. SEPs, like high-energy photons, can also catalyze atmospheric chemistry. Segura et al. (2010) found that without the inclusion of particles, large UV flares do not have a long-lasting impact on an HZ planet's O₃ column density. However, including SEPs, the expected NO_x production will deplete a planet's O₃ by ∼95%, requiring centuries for the O₃ column density to re-equilibrate after a flaring event ends. This can allow harmful (Voet et al. 1963; Matsunaga et al. 1991; Tevini 1993; Kerwin & Remmele 2007) or bio-catalyzing (Senanayake & Idriss 2006; Barks et al. 2010; Ritson & Sutherland 2012; Patel et al. 2015; Airapetian et al. 2016) UV radiation to penetrate to the surface.

Direct measurements of an M dwarf's energetic particle output are not currently possible, but signatures of particle acceleration in the UV and radio should be detectable in principle. Coronal dimming, when EUV emission lines dim after part of the corona has been evacuated from a CME (Mason et al. 2014), was not observed by the Extreme UltraViolet Explorer (EUVE), likely due to insufficient sensitivity. Type II radio bursts that trace shocks associated with CMEs (Winter & Ledbetter 2015) are being searched for but have not yet been detected on other stars (Crosley et al. 2016), but other possible kinematic signatures of CMEs in observed M dwarf flares have been detected (e.g., Houdebine et al. 1990; Cully et al. 1994; Fuhrmeister & Schmitt 2004). Type III radio bursts are caused by the acceleration of suprathermal electrons from solar active regions and have been detected on the M3 dwarf AD Leo (Osten & Bastian 2006). Probing the astrosphere (analogous to the heliosphere) via high-resolution Lyα measurements allows for time-averaged measurements of the stellar mass-loss rate (Wood et al. 2005a), which includes the accumulation of impulsive events (CMEs) and the quiescent stellar wind. However, it is uncertain if the kilogauss surface magnetic fields of M dwarfs would allow the acceleration of particles into the astrosphere (Osten & Wolk 2015; Drake et al. 2016). Vidotto et al. (2016) find that rapidly rotating stars may have strong toroidal magnetic fields that could prevent stellar mass loss, and Wood et al. (2014) find that the scaling relation between SXR flux and mass-loss rate breaks down for stars with SXR surface flux >10⁶ erg cm⁻² s⁻¹, where a fundamental change in the magnetic field topology may occur. On the Sun in 2014 October, the large active region 2192 emitted many large flares, but no CMEs. Strong overlying magnetic fields likely confined the eruption (Sun et al. 2015; Thalmann et al. 2015). As more exoplanetary atmosphere models consider the important influence of stochastic flares (Lammer et al. 2007; Segura et al. 2010; Airapetian et al. 2016; Venot et al. 2016), improving constraints on the fluxes and energies of associated particles is necessary.

To estimate SEPs for any star, we must currently rely on solar relations between SEPs and flare emission (Belov et al. 2007; Cliver et al. 2012; Osten & Wolk 2015). However, traditional flare tracers in the SXR, UV, and U band (3660 Å) originate from thermally heated plasma and probably do not trace particle acceleration processes; CME particles appear to be drawn nonthermally from cooler plasma in the ambient corona (Sciambi et al. 1977; Hovestadt et al. 1981). Yet, correlations between flare tracers and SEPs have been detected, likely due to Big Flare Syndrome (Kahler 1982). Big Flare Syndrome explains positive correlations between flare observables that do not share an identified physical process; for a larger total energy release from a flare, the magnitude of all flare energy manifestations will statistically be larger as well. Big Flare Syndrome occurs because a multitude of energy transport mechanisms between layers of the solar atmosphere makes energy transport efficient. These mechanisms include thermal conduction, radiation, bulk convection, and electron condensation and evaporation. For example, Belov et al. (2007) found a correlation between the Sun's 1–8 Å (1.5–12.4 keV) SXR flux and the >10 MeV proton flux, both observed simultaneously by the Geostationary Operational Environmental Satellite (GOES) system. Although likely due to Big Flare Syndrome, correlations like this are still extremely useful, because they enable an empirical means of estimating particle fluxes from observations of photons—essential for the study of distant stars.

Not all flares produce SEPs, and there may be a fundamental difference between SEP flares and non-SEP flares. Belov et al. (2005) note that an arbitrary SXR flare has a <0.4% chance of being associated with a proton enhancement, although this estimate would likely increase if it were easier to confidently identify associated flares and SEPs, but the probability increases for larger flares. The GOES flare classification scheme (A, B, C, M, X) is based solely on the peak 1–8 Å SXR flux as observed from Earth (1 au), and each letter represents an increased order of magnitude from 10⁻⁸ W m⁻² to 10⁻³ W m⁻² (Table 1). For example, a C3.5-class flare has a peak 3.5 × 10⁻⁶ W m⁻² SXR flux at 1 au. Approximately 20% of C-class and ∼100% of X3-class flares occur with CMEs (Yashiro et al. 2006). Thus, estimating the GOES flare classification of an observed stellar flare is important for estimating the probability of associated SEPs.

To estimate the SEP flux during the great AD Leo flare of 1985 (Hawley & Pettersen 1991), Segura et al. (2010) used scaling relations for active M dwarfs between broadband near-UV and 1–8 Å flare flux (Mitra-Kraev et al. 2005) and solar scaling relations between 1–8 Å flare flux and >10 MeV proton flux (Belov et al. 2005, 2007). Much of the recent UV flare data of M dwarfs have been confined to the far-UV (Loyd & France 2014; France et al. 2016), where HST's COS and STIS spectrographs are most efficient, so we have developed a new method of particle flux estimation from far-UV emission line flares (Section 5).

Ideally, we would directly compare protons with a far-UV emission line directly accessible by HST, but high-cadence, disk-integrated, spectrally resolved far-UV observations of the Sun do not exist. To improve the method of SEP estimation from observed flares, we search for a potential correlation between energetic protons (>10 MeV) received at Earth and an EUV emission line, He ii at 304 Å, which has a similar formation temperature to two high-S/N lines observable in far-UV spectra (Si iv λλ1393, 1402 and He ii λ1640).

This paper is organized as follows. In Section 2, we describe our sample of M dwarfs and list the sources of the observations used in this work. We also describe the reductions performed. Section 3 describes the method we used to measure the Ca ii K equivalent widths of our sample of M dwarfs, and in Section 4 we present the UV–Ca ii scaling relations. In Section 5 we describe the UV–proton scaling relations and their application. We present a summary of the main findings of this work in Section 6.

We selected stars with HST UV spectra and ground-based optical spectra either obtained directly by us or available in the VLT/XSHOOTER or Keck/HIRES public archives. The 15 stars that meet this criterion (listed in Table 2) are all early to mid-M dwarfs, are nearby (d < 15 pc), and exhibit a broad range of rotation periods (2 to >100 days) and a broad range of ages (∼10 Myr to ∼10 Gyr). Nine of the 15 M dwarfs are known to host exoplanets.

Table 2.  The M Dwarf Sample

No.	Star	d	R	Spectral	${T}_{\mathrm{eff}}$	${P}_{\mathrm{rot}}$	W${}_{\lambda }$ (Ca ii K)	W${}_{\lambda ,\mathrm{corr}}$(Ca ii K)	Data	No. of
		(pc)	(R⊙)	Type	(K)	(days)	(Å)	(Å)	Includedb	Ca ii K Spectra
1	GJ 832a	5.0a	0.56f	M2f	3590f	45.7q	0.88 ± 0.09	0.73 ± 0.07	H, X, R	8
2	GJ 876a	4.7a	0.38g	M4g	3130g	96.7r	0.82 ± 0.15	0.17 ± 0.03	H, X, R, D	314
3	GJ 581a	6.2a	0.30z,h	M2.5h	3500h	132.5q	0.36 ± 0.08	0.25 ± 0.05	H	351
4	GJ 176a	9.3a	0.45g	M2.5g	3680g	39.5s	1.76 ± 0.27	1.76 ± 0.27	H, D	256
5	GJ 436a	10.1a	0.45aa,i	M3i	3420i	39.9q	0.58 ± 0.07	0.33 ± 0.04	H, D	257
6	GJ 667Ca	6.8a	0.46j	M1.5o	3450o	103.9q	0.44 ± 0.11	0.27 ± 0.06	H	39
7	GJ 1214a	14.6b	0.21k	M4.5o	2820o	>100t	1.16 ± 0.2	0.040 ± 0.007	X, M	10
8	AD Leo	4.7c	0.44f	M3f	3410f	2.2u	11.57 ± 1.61	6.31 ± 0.88	H, R	44
9	EV Lac	5.1a	0.36f	M4f	3330f	4.38v	14.86 ± 2.52	6.47 ± 1.10	H	24
10	Proxima Cena	1.3d	0.14l	M5.5l	3100l	83.5w	13.7 ± 5.93	2.50 ± 1.08	X, M, R	7
11	AU Mic	9.9a	0.83m	PMS M1m,p	3650p	4.85x	12.13 ± 2.17	11.44 ± 2.05	H, R	52
12	YZ CMi	6.0a	0.36f	M4f	3200f	2.78y	21.23 ± 4.67	5.92 ± 1.30	H, X	20
13	GJ 821	12.2a	0.37f	M2f	3670f	⋯	0.33 ± 0.06	0.32 ± 0.06	H	26
14	GJ 213	5.8a	0.25f	M4f	3250f	⋯	0.44 ± 0.12	0.15 ± 0.04	H	7
15	GJ 1132a	12.0e	0.21n	M4n	3270n	125n	1.40 ± 0.01	0.504 ± 0.005	M	1

Notes. Rotation period (${P}_{\mathrm{rot}}$) uncertainties are typically ≥10%, and effective temperature (${T}_{\mathrm{eff}}$) uncertainties are typically ±100–200 K unless determined from interferometry (then ±20–100 K). We have rounded all ${T}_{\mathrm{eff}}$ values to the nearest 10 K.

^aStars with known exoplanets. ^b(H) Keck/HIRES, (X) VLT/XSHOOTER, (R) CASLEO/REOSC, (D) APO/DIS, (M) Magellan/MIKE.

References. (a) van Leeuwen (2007), (b) Anglada-Escudé et al. (2013), (c) Jenkins (1952), (d) Lurie et al. (2014), (e) Jao et al. (2005), (f) Houdebine et al. (2016), (g) **von Braun et al. (2014), (h) **von Braun et al. (2011), (i) **von Braun et al. (2012), (j) Kraus et al. (2011), (k) Berta et al. (2011), (l) **Demory et al. (2009), (m) **White et al. (2015), (n) Berta-Thompson et al. (2015), (o) Neves et al. (2014), (p) McCarthy & White (2012), (q) Suárez Mascareño et al. (2015), (r) Rivera et al. (2005), (s) Robertson et al. (2015), (t) Newton et al. (2016), (u) Hunt-Walker et al. (2012), (v) Roizman (1984), (w) Benedict et al. (1998), (x) Vogt et al. (1983), (y) Pettersen (1980), (z) Henry et al. (1994), (aa) Hawley et al. (1996). **Interferometry measurements.

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Seven of the 15 stars are exoplanet host stars from the MUSCLES Treasury Survey (stars 1–7 in Table 2), and are weakly active with Hα absorption spectra and rotation periods greater than 39 days. These stars are all likely a few billion years old, and they range from M1.5 to M4.5 spectral type. We also included other weakly active M dwarfs, including MEarth planet host GJ 1132 (Berta-Thompson et al. 2015) and two stars from the "Living with a Red Dwarf" program (stars 13–15; Guinan et al. 2016). To increase the diversity in our sample, we included the well-known "flare" stars with HST observations (stars 8–12 in Table 2). These stars are highly active (Hα emission spectra), have short rotation periods (<7 days), and are likely young (<1 Gyr), with the exception of Proxima Centauri (${P}_{\mathrm{rot}}=83.5$ days, ∼5 Gyr old; see Reiners & Basri 2008; Davenport et al. 2016).

A goal of the MUSCLES Treasury Survey was to obtain ground-based optical spectra contemporaneous with the HST UV observations. Scheduling changes and weather did not allow for any truly simultaneous UV–optical observations, but several targets have spectroscopic data obtained with the Dual Imaging Spectrograph (DIS) on the ARC 3.5 m telescope at Apache Point Observatory (APO) or the REOSC echelle spectrograph on the 2.15 m telescope at Complejo Astronómico El Leoncito (CASLEO) within a day or two of the HST observations. We also gathered spectra of GJ 1132, GJ 1214, and Proxima Cen on the nights of 2016 March 7–9 using the MIKE echelle spectrograph on the Magellan Clay telescope. Because M dwarfs are prime targets of radial velocity exoplanet searches, there is a wealth of high-resolution Ca ii spectra in the public archives of major observatories, including VLT and Keck. The archival spectra compose the bulk of our Ca ii measurements.

APO/DIS (R ∼ 2500) and CASLEO/REOSC (R ∼ 12,000) spectra were reduced using standard IRAF²² routines. See details in Cincunegui & Mauas (2004) for CASLEO reductions and Cincunegui et al. (2007) and Buccino et al. (2014) for presentation of some of the Proxima Cen and AD Leo observations. Magellan/MIKE spectra (R ∼ 25,000) were reduced using the standard MIKE pipeline included in the Carnegie Python Distribution (CarPy). Science-level VLT/XSHOOTER (R ∼ 6000) data products were obtained from the ESO Science Archive Facility, and Keck/HIRES (R ∼ 60,000) pipeline-reduced spectra were obtained from the KOA archive. As we are interested in looking at as many spectra as possible and measuring only equivalent widths of the Ca ii K line, the pipeline-extracted spectra using MAKEE ²³ suit our purposes well. We did not use spectra where the automatic extraction failed or the wavelength calibration was incorrect. We also did not co-add the adjacent orders on which Ca ii K appears, but instead averaged the equivalent width measurements (Section 3) from each order to create one equivalent width measurement per echellogram.

The HST UV spectra were obtained either from the MAST online archive, including the MUSCLES High-Level Science Products²⁴ (HLSPs; Loyd et al. 2016), or the StarCAT portal.²⁵ GJ 1132's (star 15) STIS G140M spectrum was obtained on 2016 February 13. For the seven MUSCLES M dwarfs (stars 1–7), the UV line fluxes come from Youngblood et al. (2016) and France et al. (2016). The UV emission lines of stars 8–14 were directly measured from archival HST spectra or obtained from Wood et al. (2005b).

For AD Leo (star 8) and Proxima Centauri (star 10), we reconstructed the Lyα profiles using the methods described in Youngblood et al. (2016). Proxima Cen's reconstruction is included as part of a 5 Å–5.5 μm SED on the MUSCLES HLSP website.²⁶ The differences between our Lyα reconstructions and those presented in Wood et al. (2005b) are small, ∼20% in integrated Lyα flux for AD Leo and ∼4% in integrated Lyα flux for Proxima Cen. We also reconstructed the Lyα profile of GJ 1132 (star 15; $F(\mathrm{Ly}\alpha )=({2.64}_{-0.65}^{+2.58})\times {10}^{-14}$ erg cm⁻² s⁻¹) for the first time using the methods from Youngblood et al. (2016). Note that the Lyα error bars have been averaged for symmetry in Table 3. All UV fluxes used in this work are shown in Table 3.

Table 3.  UV Emission Line Fluxes of the M Dwarf Sample

No.	Star	Si iii	${\sigma }_{\mathrm{Si}{\rm{III}}}$	Lyαb	${\sigma }_{\mathrm{Ly}\alpha }$ b	Si ii	${\sigma }_{\mathrm{Si}{\rm{II}}}$	C ii	${\sigma }_{{\rm{C}}{\rm{II}}}$	Mg ii	${\sigma }_{\mathrm{Mg}{\rm{II}}}$
1	GJ 832a	2.55E–15	6.33E–17	9.50E–13	6.00E–14	7.65E–16	6.33E–17	3.78E–15	8.18E–17	1.84E–13	2.02E–15
2	GJ 876a	8.11E–15	1.01E–16	3.90E–13	4.00E–14	1.19E–15	7.06E–17	1.06E–14	1.26E–16	3.11E–14	9.31E–16
3	GJ 581a	2.94E–16	3.54E–17	1.10E–13	3.00E–14	7.93E–17	4.41E–17	4.81E–16	4.26E–17	1.88E–14	7.54E–16
4	GJ 176a	2.15E–15	5.84E–17	3.90E–13	2.00E–14	7.34E–16	6.49E–17	5.43E–15	9.66E–17	1.56E–13	1.53E–15
5	GJ 436a	5.15E–16	4.00E–17	2.10E–13	3.00E–14	1.87E–16	4.47E–17	1.09E–15	5.85E–17	3.84E–14	1.01E–15
6	GJ 667Ca	5.07E–16	3.91E–17	5.20E–13	9.00E–14	1.29E–16	6.13E–17	6.50E–16	4.63E–17	4.12E–14	1.10E–15
7	GJ 1214a	8.05E–17	2.77E–17	1.30E–14	1.00E–14	1.36E–17	2.09E–17	9.83E–17	3.00E–17	1.66E–15	2.70E–16
8	AD Leo	1.41E–13	8.88E–16	8.07E–12	2.00E–13	1.64E–14	2.58E–16	1.66E–13	4.83E–16	2.79E–12c	2.79E–13c
9	EV Lac	3.86E–14	1.46E–15	2.75E–12c	5.50E–13c	2.49E–15	4.95E–16	3.76E–14	6.42E–16	⋯	⋯
10	Proxima Cena	2.03E–14	7.57E–16	4.37E–12	7.00E–14	2.14E–15	2.08E–16	2.61E–14	2.79E–16	⋯	⋯
11	AU Mic	8.78E–14	3.45E–16	1.07E–11	4.00E–13	1.09E–14	8.97E–17	1.44E–13	1.81E–16	4.20E–12c	4.20E–13c
12	YZ CMi	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1.10E–12	1.42E–15
13	GJ 821	5.58E–16	6.56E–16	⋯	⋯	9.52E–17	9.52E–18	1.05E–16	2.89E–17	2.62E–14	2.69E–15
14	GJ 213	⋯	⋯	⋯	⋯	1.62E–16	1.62E–17	2.32E–16	2.32E–17	7.13E–15	2.30E–15
15	GJ 1132a	⋯	⋯	2.64E–14	1.62E–14	⋯	⋯	⋯	⋯	⋯	⋯

No.	Star	Si iv	${\sigma }_{\mathrm{Si}{\rm{IV}}}$	He ii	${\sigma }_{\mathrm{He}{\rm{II}}}$	C iv	${\sigma }_{{\rm{C}}{\rm{IV}}}$	N v	${\sigma }_{{\rm{N}}{\rm{V}}}$	EUVe
1	GJ 832a	3.33E–15	9.19E–17	6.81E–15	2.93E–16	7.64E–15	2.88E–16	3.51E–15	7.54E–17	8.65E–13
2	GJ 876a	8.44E–15	1.36E–16	5.51E–15	2.92E–16	2.29E–14	4.34E–16	1.07E–14	1.26E–16	3.43E–13
3	GJ 581a	4.44E–16	6.71E–17	7.22E–16	1.90E–16	1.84E–15	1.84E–16	5.34E–16	4.48E–17	1.01E–13
4	GJ 176a	2.30E–15	8.64E–17	6.73E–15	3.03E–16	1.23E–14	2.83E–16	3.10E–15	7.56E–17	3.57E–13
5	GJ 436a	6.81E–16	6.92E–17	1.50E–15	2.13E–16	2.39E–15	1.76E–16	9.56E–16	5.04E–17	1.90E–13
6	GJ 667Ca	8.25E–16	7.17E–17	1.94E–15	2.25E–16	2.97E–15	2.05E–16	7.16E–16	5.31E–17	4.73E–13
7	GJ 1214a	4.79E–17	3.74E–17	7.43E–17	1.44E–16	5.23E–16	1.21E–16	1.84E–16	3.58E–17	1.29E–14
8	AD Leo	1.74E–13	5.39E–16	3.02E–13	9.45E–16	5.66E–13	1.21E–15	1.41E–13	5.89E–16	8.16E–12
9	EV Lac	1.82E–14	9.76E–16	7.41E–14	1.58E–15	1.41E–13	1.96E–15	3.91E–14	9.03E–16	2.60E–12
10	Proxima Cena	2.72E–14	3.83E–16	3.08E–14	5.87E–16	1.38E–13	9.13E–16	4.00E–14	4.56E–16	3.86E–12
11	AU Mic	9.53E–14	1.63E–16	2.75E–13	3.50E–16	3.55E–13	3.85E–16	7.42E–14	1.82E–16	1.42E–11
12	YZ CMi	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
13	GJ 821	8.16E–17	4.29E–17	6.84E–16	2.53E–16	6.69E–16	1.90E–16	1.95E–16	1.95E–17	⋯
14	GJ 213	2.13E–16	2.13E–17	2.57E–16	2.57E–17	5.22E–16	1.55E–16	3.58E–16	3.58E–17	⋯
15	GJ 1132a	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	2.31E–14

Notes. All emission-line fluxes are as observed from Earth (erg cm⁻² s⁻¹). All flux measurements for stars 1–7 come from Youngblood et al. (2016) and France et al. (2016). The Lyα reconstructed flux for AU Mic also comes from Youngblood et al. (2016). All other measurements are from this work, except where noted.

^aStars with known exoplanets. ^bReconstructed Lyα fluxes. Asymmetric error bars from Youngblood et al. (2016) have been averaged. ^cWood et al. (2005b). ^eCalculated using the reconstructed Lyα fluxes (Table 3) and scaling relations from Linsky et al. (2014), which have been re-shown in Table 6.

A machine-readable version of the table is available.

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2.2.1. Interstellar Medium Corrections

We utilize time-series solar irradiance (disk-integrated) measurements from the Extreme ultraviolet Variability Experiment (EVE) suite of instruments on board the Solar Dynamics Observatory (SDO; Woods et al. 2012). We use the He ii (304 Å) irradiance from the 2010–2014 era from the MEGS-A channel (50–370 Å with 1 Å spectral resolution). MEGS-A operates at a 10 s cadence, but we use 1-minute averages in this work to reduce noise.

Time-series solar SXR irradiance measurements and in situ proton measurements come from the Geostationary Operational Environmental Satellite (GOES) system.²⁷ SXRs are measured in a 1–8 Å (1.5–12.4 keV) band. Note that we do not divide the GOES 1–8 Å flux by 0.7, the recommended correction factor, to obtain absolute flux units. This is because we utilize the GOES SXR flare classification scheme below, which operates on data that have not been corrected, and recent SXR observations with the MinXSS cubesat suggest that this correction factor is incorrect (Woods et al. 2017). We utilize the proton measurements from the >10 MeV and >30 MeV channels.

A total of 10 out of 15 members of our M dwarf sample have tens to hundreds of archival Ca ii K observations, and we derived our equivalent width measurements from all the spectra to avoid biases from stellar activity on all timescales (i.e., flares, rotation, stellar magnetic activity cycle). Equivalent width light curves are shown in Figure 2 for the 15 stars. Variability is observed, especially over the course of a few days, but no definitive signs of cyclic activity due to stellar rotation or the stellar dynamo are observed, nor are they ruled out. Approximately 20% spreads in the Ca ii K equivalent widths are typically observed over the course of a few days (see Figures 3 and 4) and are likely due to flares and rotation of starspots into and out of view. Figure 3 shows the evolution of the Ca ii K emission profile for GJ 876 over the course of 4 days and the corresponding change in the corrected Ca ii K equivalent widths.

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For each target, we computed a mean Ca ii K equivalent width with an uncertainty equal to the standard deviation of the measurements (Table 2). GJ 1132 has only one equivalent width measurement, so the photometric error bar was reported for that measurement. The averages from stars with few observations could be strongly skewed if they coincide with an unresolved episode of high activity. The targets with the fewest observations are GJ 832, GJ 1214, GJ 1132, Proxima Cen, and GJ 213 (see Section 2 and Table 2 for the list of sources for our optical spectra).

Figure 4 shows the demographics of our sample (stellar rotation period ${P}_{\mathrm{rot}}$ and effective temperature ${T}_{\mathrm{eff}}$) compared to the corrected Ca ii K equivalent widths and the observed scatter in those values. In general, the fast rotators (${P}_{\mathrm{rot}}\lt 7$ days) have the largest equivalent widths. Unsurprisingly, these stars are the well-known flare stars AD Leo, EV Lac, AU Mic, and YZ CMi, and they are also the youngest stars (<1 Gyr) in the sample. Between P_rot ∼ 40 and 85 days, most stars exhibit small equivalent widths, but Proxima Cen (star 10) and GJ 176 (star 4) are outliers. West et al. (2015) find that late M dwarfs (M5–M8) rotating faster than 86 days and early M dwarfs (M0–M4) rotating faster than 26 days exhibit greater optical activity (i.e., an Hα emission spectrum and greater Ca ii equivalent width; Walkowicz & Hawley 2009). Proxima Cen meets this optical activity criterion, but GJ 176 does not. However, even the "optically inactive" stars (Hα absorption spectrum) in our sample exhibit UV activity.

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The right panel of Figure 4 shows the scatter observed in the Ca ii K ${W}_{\lambda ,\mathrm{corr}}$ light curves normalized to ${W}_{\lambda ,\mathrm{corr}}$. The scatter ranges from ∼10% to 25% (although Proxima Cen's scatter is 43%), with no clear dependence on ${T}_{\mathrm{eff}}$ and a possible positive correlation with ${P}_{\mathrm{rot}}$, although our sample is sparse for ${T}_{\mathrm{eff}}\lt 3200$ K and ${P}_{\mathrm{rot}}$ > 60 days.

For our 15 M dwarf sample, we compare the average Ca ii K corrected equivalent widths (${W}_{\lambda ,\mathrm{corr}}$) reported in Table 2 with UV emission line surface fluxes. By using the corrected Ca ii K equivalent width (Section 3) and UV surface fluxes, we attempt to minimize spectral type dependences in our results. We note, however, that the uncertainties in the stellar radii (used to compute the stellar surface flux) are large, and there may be a systematic bias for the lowest-mass stars. Direct interferometric measurements yield vastly different radii for mid-M dwarfs of similar effective temperatures; in particular, directly measured radii are tens of percent larger than radii determined by indirect means (von Braun et al. 2014). The nine UV lines we used for comparison are listed with their formation temperatures in Table 5.

Table 5.  Fit Parameters for UV Surface Flux and Ca ii K Equivalent Width Relations

Transition Name	Wavelength (Å)	log ${T}_{\mathrm{formation}}$ a	α	β	ρ	n	σ
Si iii	1206.50	4.7	1.35 ± 0.26	3.21 ± 0.16	0.80	2.0 × 10−3	0.59
H i Lyα	1215.67	4.5 (line core)	0.88 ± 0.10	5.46 ± 0.06	0.95	1.1 × 10−5	0.18
		"active"	1.21 ± 0.16	5.20 ± 0.12	⋯	⋯	⋯
		"inactive"	0.10 ± 0.17	5.19 ± 0.08	⋯	⋯	⋯
Si ii	1260.42, 1264.74, 1265.00	4.5	0.99 ± 0.18	2.58 ± 0.11	0.82	6.7 × 10−4	0.42
C ii b	1335.71	4.5	1.29 ± 0.23	3.38 ± 0.14	0.82	6.0 × 10−4	0.54
Mg ii c	2796.35, 2803.53	4.5 (line core)	1.06 ± 0.13	4.88 ± 0.08	0.94	7.1 × 10−6	0.27
Si iv	1393.76, 1402.77	4.9	1.35 ± 0.24	3.22 ± 0.14	0.84	3.6 × 10−4	0.54
He ii	1640.4d	4.9	1.33 ± 0.15	3.53 ± 0.09	0.91	1.2 × 10−5	0.41
C iv	1548.19, 1550.78	5.0	1.31 ± 0.26	3.84 ± 0.16	0.81	8.8 × 10−4	0.56
N v	1238.82, 1242.8060	5.2	1.2 ± 0.26	3.36 ± 0.15	0.77	1.9 × 10−3	0.55
EUVe	100–912	⋯	0.80 ± 0.09	5.49 ± 0.06	0.94	1.5 × 10−5	0.18
		"active"	1.37 ± 0.18	5.10 ± 0.14	⋯	⋯	⋯
		"inactive"	0.12 ± 0.18	5.15 ± 0.08	⋯	⋯	⋯

Notes. All relations have the form log₁₀ ${F}_{{\rm{S}},\mathrm{UV}}$ = (α × log₁₀ ${W}_{\lambda ,\mathrm{corr}}$) + β, where ${F}_{{\rm{S}},\mathrm{UV}}$ is the surface flux of the UV emission line in erg cm⁻² s⁻¹ and ${W}_{\lambda ,\mathrm{corr}}$ is the Ca ii K equivalent width in Å. ρ is the Pearson correlation coefficient, n is the probability of no correlation, and σ is the standard deviation of the data points about the best-fit line (dex). The additional fit parameters for Lyα and EUV surface flux apply separately to the "active" (${W}_{\lambda ,\mathrm{corr}}\gt 1$) and "inactive" M dwarfs (${W}_{\lambda ,\mathrm{corr}}\lt 1$). Combining the active and inactive fit components, σ = 0.12 for Lyα, and σ = 0.13 for the EUV.

^aFormation temperatures are from the CHIANTI database (Dere et al. 1997; Landi et al. 2013). ^bDoes not include the 1334.54 Å line, due to significant ISM absorption. ^cFluxes corrected for 30% ISM absorption (see Section 2.2 and Youngblood et al. 2016). ^dAverage wavelength of the multiplet. ^eEUV fluxes calculated from Lyα fluxes using scaling relations from Linsky et al. (2014).

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Figure 5 shows the correlations between ${W}_{\lambda ,\mathrm{corr}}$ and surface flux (${F}_{{\rm{S}},\mathrm{UV}}$) for nine UV emission lines and the EUV flux, which is derived from Lyα (see Section 4.2.1), with their power-law fits of the form log₁₀ F_S,UV = (α × log₁₀ ${W}_{\lambda ,\mathrm{corr}}$) + β (Table 5). We find statistically significant correlations between Ca ii K and all nine UV emission lines, which have formation temperatures ranging from 30,000 to 160,000 K. As expected, the correlations with the least scatter are for Mg ii and Lyα (0.27 and 0.18 dex, respectively), both optically thick lines with formation temperatures similar to Ca ii K. Ca ii has been previously found to correlate positively with Lyα (Linsky et al. 2013) and Mg ii (Walkowicz & Hawley 2009). Note that due to the large uncertainty in GJ 1214's Lyα flux (Youngblood et al. 2016), we do not include it in the fit, although its effect on the fit is small. GJ 1214 is an outlier from the fits for all nine UV lines, and GJ 876 is a significant outlier for all but Mg ii and Lyα. Of the seven MUSCLES M dwarfs, GJ 876 exhibited the most frequent and largest flares. These flares were observed in all lines except Mg ii and Lyα, which were observed on different HST visits from the other far-UV emission lines, so GJ 876's apparently elevated UV surface fluxes in Figure 5 may be due to these flares. There is no clear dependence on fit quality (as measured by the Pearson correlation coefficients or the standard deviation of the fit) with the formation temperature of the UV line.

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By eye, it appears that some of the Ca ii–UV correlations in Figure 5 may be better fit by two power laws with a break around ${W}_{\lambda ,\mathrm{corr}}\approx 1$, the transition in this data set between the "inactive" (${W}_{\lambda ,\mathrm{corr}}\lesssim 1$) and "active" (${W}_{\lambda ,\mathrm{corr}}\gtrsim 1$) M dwarfs. In the Lyα and EUV panels in Figure 5, we show two power-law fits in addition to the single power-law fit. As with the single power-law fits, GJ 1214 is excluded. The broken power-law fits indicate that for these two UV surface fluxes, the Ca ii–UV correlations become approximately constant at low stellar activity (Table 5). Greater study of the low-activity M dwarfs is necessary to determine whether the UV fluxes become approximately constant in that regime, as the apparent flattening of many of the correlations is largely due to a single star: GJ 1214.

4.2.1. Estimating the EUV Spectrum from Ca ii K

Predicting an M dwarf's Lyα flux is important not only because Lyα constitutes a major fraction of the far-UV flux, but also because it is a means for estimating the EUV spectrum, which currently cannot be observed for any star except the Sun. In Figure 5 and Table 5, we have determined the scaling relation between the total EUV flux (100–912 Å) and ${W}_{\lambda ,\mathrm{corr}}$ for our stars. The fit is very similar to the Lyα–Ca ii K best-fit line, because the EUV fluxes were derived from scaling relations with Lyα from Linsky et al. (2014).

In Table 6, we substitute our Ca ii–Lyα scaling relation into the Lyα–EUV scaling relations in Table 5 of Linsky et al. (2014) to allow the reader to directly reconstruct the EUV spectrum from a Ca ii K ${W}_{\lambda ,\mathrm{corr}}$ measurement. We have also re-shown the scaling relations from Linsky et al. (2014) in Table 6, although they have been simplified for brevity.

Table 6.  Formulae for Estimating EUV Fluxes from Lyα and Ca ii K

Flux in Wavelength
Band at 1 au	Linsky et al. (2014)
(erg cm−2 s−1)	Table 4	This Worka
F(100–200 Å)	${10}^{-0.491}$ · F(Lyα)	${10}^{4.97\pm 0.06}$ · ${W}_{\lambda ,\mathrm{corr}}^{0.88\pm 0.10}$ · (4.65 × 10−3 R⋆)${}^{2.0}$
F(200–300 Å)	${10}^{-0.548}$ · F(Lyα)	${10}^{4.91\pm 0.06}$ · ${W}_{\lambda ,\mathrm{corr}}^{0.88\pm 0.10}$ · (4.65 × 10−3 R⋆)${}^{2.0}$
F(300–400 Å)	${10}^{-0.602}$ · F(Lyα)	${10}^{4.86\pm 0.06}$ · ${W}_{\lambda ,\mathrm{corr}}^{0.88\pm 0.10}$ · (4.65 × 10−3 R⋆)${}^{2.0}$
F(400–500 Å)	${10}^{-2.294}$ · F(Lyα)${}^{1.258}$	${10}^{4.57\pm 0.08}$ · ${W}_{\lambda ,\mathrm{corr}}^{1.11\pm 0.13}$ · (4.65 × 10−3 R⋆)${}^{2.52}$
F(500–600 Å)	${10}^{-2.098}$ · F(Lyα)${}^{1.572}$	${10}^{6.49\pm 0.09}$ · ${W}_{\lambda ,\mathrm{corr}}^{1.38\pm 0.16}$ · (4.65 × 10−3 R⋆)${}^{3.14}$
F(600–700 Å)	${10}^{-1.920}$ · F(Lyα)${}^{1.240}$	${10}^{4.85\pm 0.07}$ · ${W}_{\lambda ,\mathrm{corr}}^{1.09\pm 0.12}$ · (4.65 × 10−3 R⋆)${}^{2.48}$
F(700–800 Å)	${10}^{-1.894}$ · F(Lyα)${}^{1.518}$	${10}^{6.39\pm 0.09}$ · ${W}_{\lambda ,\mathrm{corr}}^{1.34\pm 0.15}$ · (4.65 × 10−3 R⋆)${}^{3.04}$
F(800–912 Å)	${10}^{-1.811}$ · F(Lyα)${}^{1.764}$	${10}^{7.82\pm 0.11}$ · ${W}_{\lambda ,\mathrm{corr}}^{1.55\pm 0.18}$ · (4.65 × 10−3 R⋆)${}^{3.53}$

Note. F(Lyα) is the Lyα flux at 1 au (erg cm⁻² s⁻¹), and ${W}_{\lambda ,\mathrm{corr}}$ is the Ca ii K corrected equivalent width (Equation (2)).

^aSubstituted Lyα–${W}_{\lambda ,\mathrm{corr}}$ scaling relation from Table 5 into the relation from Linsky et al. (2014); R_⋆ in units of R_⊙; uncertainties propagated from fit uncertainties in Table 5.

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4.2.2. Estimating the Uncertainties on Derived UV Surface Fluxes

We identified 36 proton enhancements in the GOES >10 MeV and >30 MeV proton channels with an associated SXR (1–8 Å with GOES) and He ii λ304 flare. Our confidence levels for the proton–flare association varied, and we assigned each event a quality index (q) ranging from 1 (lowest confidence) to 4 (highest confidence). The number of events assigned to q = 1, 2, 3, and 4 is 7, 11, 7, and 11, respectively. Reasons for a low q include difficulty in identifying a proton enhancement's precursor flare and/or reliably measuring the properties of both fluxes. There can be too many candidate precursor flares, low S/N, difficulty in defining the beginning and end of the proton and flare events, and uncertainty in measuring background flux level. Figure 6 shows an example high-confidence event (q = 4) and a low-confidence event (q = 1). For the q = 1 example event, the onset of the >10 MeV protons after the photon flares is much more gradual than most events, and the >30 MeV proton flux shows no significant rise. The He ii λ304 light curve also shows many events, which may be confused, and estimation of the background level is challenging.

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We find that the proton, UV, and SXR flux errors are dominated by systematics in defining a background flux level to subtract and the duration of the flare. We applied a linear fit to the background around each event. For some events, this was straightforward, but for others, this method likely increases uncertainty in the measurement. We determined the beginning and end of the flare by where the light curve intersected the background level fit. To estimate the uncertainties in our measurements, we remeasured 4 of the 36 events three times, each time slightly changing the background fit and the beginning and end of the flare. We estimate the uncertainty in the background-subtracted flux to range from 10% to 300% for SXR, 10%–200% for He ii λ304, and 10%–30% for the protons. These ranges are a reflection of the varying quality factors: q < 3 events have the largest uncertainties, and q ≥ 3 events have the smallest uncertainties. If we consider just the fluence (no background subtraction) or the peak fluxes, the uncertainties drop to 10% for the SXRs and He ii λ304, indicating that background subtraction and not flare duration is the dominant uncertainty.

The 36 He ii λ304 flares have mean durations 5.3 ± 4.6 hr and on average peak a few minutes after the SXR flare peak, although there is large scatter in this average. When examining only the q ≥ 3 events, the average He ii λ304 peak occurs ∼5 minutes before the SXR peak. Kennedy et al. (2013) used He ii λ304 to trace the impulsive phase flares and found He ii λ304 to peak 1–4 minutes before the SXR peak, which traces the more gradual phase of the flare, and Milligan et al. (2012) found for an X-class flare that He ii λ304 peaked 18 s before the GOES SXRs. The associated proton enhancements begin within about 2 hr of the SXR peak and on average last for 3 days. Part of the scatter in the He ii λ304–proton relationships presented in Figure 7 is due to overlapping flares with contributing protons. In the duration of a proton enhancement, the same or another active region could flare again and accelerate protons that reach Earth. Many of the >10 MeV enhancements have many peaks over the course of several days, while the >30 MeV enhancements typically only have a rapid initial peak and a gradual decline. The peak proton fluxes between the two channels generally do not coincide temporally; the >10 MeV peak particle flux typically occurs after the >30 MeV peak. The energy-dependent arrival times of protons are not completely explained by differing speeds; it is thought that higher-energy protons are accelerated with electrons close to the solar surface and that either lower-energy protons are accelerated at a later time (farther from the solar surface) or their escape from the Sun is delayed owing to trapping by shocks (Krucker & Lin 2000; Xie et al. 2016).

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In their sample of 58 SXR and proton enhancement events, Cliver et al. (2012) found a statistically significant correlation (ρ = 0.52, n = 3 × 10⁻⁵), but our 36-event sample yields ρ = 0.18, n = 0.3 (Figure 8). If we force both samples to cover the same parameter space (enclosed in the vertical lines in Figure 8), the correlation coefficients become similar: ρ = 0.41, n = 0.003 for 50 of Cliver et al. (2012) events, and ρ = 0.35, n = 0.05 for 33 of our events. The different n values are due to the different sample sizes.

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We observe statistically significant correlations between He ii λ304 fluence (time-integrated flux with units J m⁻²) and proton fluence and peak proton flux (all fluence and flux values background subtracted; Figure 7). The data were fitted with power laws (log₁₀ x = α × log₁₀ y + β), and the parameters for the fits are presented in Table 7. The scatter in these fits is large (σ = 0.75–0.84 dex).

Table 7.  Correlations between UV Flare Fluence (F) and the Peak Intensity (I) and Fluence (F) of Energetic Protons during Solar Flares

x	y	α	β	ρ	n	σ
${F}_{\mathrm{He}{\rm{II}},304}$	${I}_{\geqslant 10\mathrm{MeV}}$	1.06 ± 0.21	2.42 ± 0.17	0.83	2.2 × 10−5	0.76
${F}_{\mathrm{He}{\rm{II}},304}$	${I}_{\geqslant 30\mathrm{MeV}}$	0.85 ± 0.20	1.62 ± 0.16	0.80	6.5 × 10−5	0.75
${F}_{\mathrm{He}{\rm{II}},304}$	${F}_{\geqslant 10\mathrm{MeV}}$	1.20 ± 0.26	2.23 ± 0.21	0.84	1.0 × 10−5	0.84
${F}_{\mathrm{He}{\rm{II}},304}$	${F}_{\geqslant 30\mathrm{MeV}}$	1.01 ± 0.25	1.37 ± 0.20	0.82	3.6 × 10−5	0.84
${F}_{\mathrm{Si}{\rm{IV}}}$	${I}_{\geqslant 10\mathrm{MeV}}$	1.06 ± 0.21	3.34 ± 0.25	⋯	⋯	⋯
${F}_{\mathrm{Si}{\rm{IV}}}$	${F}_{\geqslant 10\mathrm{MeV}}$	1.20 ± 0.26	3.27 ± 0.31	⋯	⋯	⋯
${F}_{\mathrm{He}{\rm{II}},1640}$	${I}_{\geqslant 10\mathrm{MeV}}$	1.06 ± 0.21	4.05 ± 0.36	⋯	⋯	⋯
${F}_{\mathrm{He}{\rm{II}},1640}$	${F}_{\geqslant 10\mathrm{MeV}}$	1.20 ± 0.26	4.07 ± 0.45	⋯	⋯	⋯

Note. The power-law fits (log₁₀ x = α × log₁₀ y + β) are based on the q ≥ 3 background-subtracted data (Figure 7). ρ is the Pearson correlation coefficient, n is the probability of no correlation, and σ is the standard deviation of the data points about the best-fit line (dex).

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The near identical formation temperatures of He ii λ304, Si iv, and He ii λ1640 suggest that the ratios of their fluxes are likely to be similar for all stars with metallicities similar to solar values. Thus, we estimated these ratios using synthetic spectra of the Sun (CHIANTI; Dere et al. 1997; Landi et al. 2013) and GJ 832 (Fontenla et al. 2016). For the Sun, ${F}_{\mathrm{Si}{\rm{IV}}}$/${F}_{\mathrm{He}{\rm{II}},304}=0.117$ and ${F}_{\mathrm{He}{\rm{II}},1640}$/${F}_{\mathrm{He}{\rm{II}},304}=0.0304$. For GJ 832, ${F}_{\mathrm{Si}{\rm{IV}}}$/${F}_{\mathrm{He}{\rm{II}},304}=0.137$ and ${F}_{\mathrm{He}{\rm{II}},1640}$/${F}_{\mathrm{He}{\rm{II}},304}\,=0.0291$. The ratios for these two stars are very similar, so we will use the GJ 832 ratios in the subsequent analysis. However, we note that the ratios could change by a factor of a few during a flare as estimated by comparing flux ratios of various lines for the active and quiet-Sun models listed in Table 1 of Fontenla et al. (2016). More M dwarf atmosphere models calculated at a range of activity levels would be valuable.

We apply the quiescence ratios to our proton–He ii λ304 relations (Table 7) to relate Si iv and He ii λ1640 flare flux to proton enhancements. Using one of the correlations listed in Table 7 and replacing ${F}_{\mathrm{He}{\rm{II}},304}$ with ${F}_{\mathrm{Si}{\rm{IV}}}$, we find that

$$\begin{eqnarray}&&\mathrm{log}\,{I}_{\gt 10\mathrm{MeV}}=(1.06\pm 0.21)\times \mathrm{log}\,{F}_{\mathrm{Si}{\rm{IV}}}+(3.34\pm 0.25),\end{eqnarray} \tag{ 3 }$$

where ${F}_{\mathrm{Si}{\rm{IV}}}$ is the background-subtracted Si iv (λλ1393, 1402) flare fluence (J m⁻²) as would be observed at 1 au from the star, and ${I}_{\gt 10\mathrm{MeV}}$ is the background-subtracted peak >10 MeV proton enhancement intensity (pfu; 1 pfu = 1 proton cm⁻² s⁻¹ sr⁻¹) as would also be observed at 1 au. If instead the proton fluence (${F}_{\gt 10\mathrm{MeV}}$, [pfu · s]) rather than peak proton flux is used, the relationship becomes

$$\begin{eqnarray}&&\mathrm{log}\,{F}_{\gt 10\mathrm{MeV}}=(1.20\pm 0.26)\times \mathrm{log}\,{F}_{\mathrm{Si}{\rm{IV}}}+(3.27\pm 0.31).\end{eqnarray} \tag{ 4 }$$

Similarly, we can derive ${I}_{\geqslant 10\mathrm{MeV}}$ and ${F}_{\geqslant 10\mathrm{MeV}}$ from ${F}_{\mathrm{He}{\rm{II}},1640}$:

$$\begin{eqnarray}\mathrm{log}\,{I}_{\gt 10\mathrm{MeV}} & = & (1.06\pm 0.21)\times \mathrm{log}\,{F}_{\mathrm{He}{\rm{II}},1640}\\ & & +(4.05\pm 0.36),\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}\mathrm{log}\,{F}_{\gt 10\mathrm{MeV}} & = & (1.20\pm 0.26)\times \mathrm{log}\,{F}_{\mathrm{He}{\rm{II}},1640}\\ & & +(4.07\pm 0.45).\end{eqnarray} \tag{ 6 }$$

When using Equations (3)–(6), it is important to note that not all far-UV flares will be accompanied by particle events. Less energetic flares are less likely to produce particles. The solar relation was quantified by Yashiro et al. (2006) using the GOES SXR classification scheme as the metric of flare strength. Specifically, they associated a probability of a CME with each GOES flare class. To relate this to far-UV data, we used our 36 events to fit an empirical power law between peak SXR intensity and peak He ii λ304  intensity (W m⁻²; Figure 9):

$$\begin{eqnarray}&&\mathrm{log}\,{I}_{\mathrm{SXR}}=(1.40\pm 0.09)\times \mathrm{log}\,{I}_{\mathrm{He}{\rm{II}},304}+(1.52\pm 1.62).\end{eqnarray} \tag{ 7 }$$

The Pearson correlation coefficient is 0.58 with n = 2.2 × 10⁻⁴ for the 36 events used in this analysis. The scatter about the best-fit line in Figure 9 is 0.52 dex, so the resulting classifications for M dwarf flares will be accurate within a factor of a few. This level of accuracy will be problematic for small flares ($\lt {\rm{X}}$ class), but less so for large flares, due to the ∼100% chance of associated proton enhancement (Table 1). Using our conversion ratio (${F}_{\mathrm{Si}{\rm{IV}}}$/${F}_{\mathrm{He}{\rm{II}},304}=0.137$) and assuming ${F}_{\mathrm{Si}{\rm{IV}}}$/${F}_{\mathrm{He}{\rm{II}},304}\,\sim {I}_{\mathrm{Si}{\rm{IV}}}$/${I}_{\mathrm{He}{\rm{II}},304}$, we can estimate the peak SXR flux from the peak Si iv and He ii λ1640 flux:

$$\begin{eqnarray}&&\mathrm{log}\,{I}_{\mathrm{SXR}}=(1.4\pm 0.1)\times \mathrm{log}\,{I}_{\mathrm{Si}{\rm{IV}}}+(2.7\pm 1.6),\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&\mathrm{log}\,{I}_{\mathrm{SXR}}=(1.4\pm 0.1)\times \mathrm{log}\,{I}_{\mathrm{He}{\rm{II}},1640}+(3.7\pm 1.6).\end{eqnarray} \tag{ 9 }$$

Approximately 20% of C-class flares (${F}_{\mathrm{SXR}}={10}^{-6}\mbox{--}{10}^{-5}$ W m⁻² at 1 au) and ∼100% of X3-class or greater flares (F_SXR ≥ 3 × 10⁻⁴ W m⁻² at 1 au) have associated CMEs (Yashiro et al. 2006; Table 1). Recall that ${F}_{\mathrm{SXR}}$ is the peak 1–8 Å flare intensity measured at 1 au from the star and is not corrected for pre-flare flux levels. An X-class or greater flare (≥10⁻⁴ W m⁻² at 1 au) corresponds to any Si iv flare with a background-subtracted peak intensity value ≥1.6 × 10⁻⁵ W m⁻² = 1.6 × 10⁻² erg cm⁻² s⁻¹ at 1 au and to any He ii λ1640 flare with a background-subtracted peak intensity value ≥3.2 × 10⁻⁶ W m⁻² = 3.2 × 10⁻³ erg cm⁻² s⁻¹ at 1 au.

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The MUSCLES Treasury Survey observed several large flares from GJ 876 (d = 4.7 pc) with HST and Chandra in 2015 June and July (France et al. 2016; R. O. P. Loyd et al. 2017, in preparation; Figure 10). Due to an HST safing event, the Chandra and HST observations were not simultaneous as originally planned. On 2015 June 5, the SXR luminosity of GJ 876 was observed to increase by a factor >10 (Figure 10(a)), and on 2015 July 7, factor of ∼40–100 increases were observed in the far-UV lines, including Si iv (Figure 10(b)), and more modest increases (factor of ∼5) were observed in the far-UV continuum (France et al. 2016). He ii λ1640 was observed with a different grating than Si iv, so the two emission lines were not observed simultaneously, and no clear flare was observed in He ii λ1640 on 2015 July 8 (Figure 10(c)), although the C iv light curve indicates that GJ 876 did flare during this observation. In this section and Table 8, we characterize the magnitude of these flares with the GOES flare classification scheme and estimate the probability and magnitude of an associated particle enhancement received in GJ 876's HZ ($\langle {r}_{\mathrm{HZ}}\rangle =0.18$ au; Kopparapu et al. 2014). See Section 5.5 for a discussion on the limitations of these results.

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Table 8.  GJ 876 Flare Properties

Flare	Peak SXR Flux (W m−2)		GOES	>10 MeV Proton Flux (pfu)
	1 au	$\langle {r}_{\mathrm{HZ}}\rangle$	Class	1 au	$\langle {r}_{\mathrm{HZ}}\rangle$
GJ 876 flares:
2015 Jun 4	2.2 × 10−5	6.6 × 10−4	M2.2	8	240
2015 Jun 5	9.5 × 10−5	2.9 × 10−3	M9.5	80	2400
2015 Jul 7	3.8 × 10−3	1.1 × 10−1	X38	75	2300
Reference flares:
Carrington	4.5 × 10−3	⋯	X45 ± 5	⋯	⋯
Event 1859a
2003 Nov 4	3.06 × 10−3	⋯	X30.6	400	⋯
solar flareb
Great AD Leo	0.23	9	X2300	1.5 × 106	5.9 × 108
flare of 1985c
DG CVn	60	6000	X600,000	⋯	⋯
2014 Apr 24d

Note. $\langle {r}_{\mathrm{HZ}}\rangle =0.18$ au for GJ 876; 0.16 au for AD Leo; 0.1 au for DG CVn; 1 au for the Sun.

Reference. (a) Cliver & Dietrich (2013), (b) Kiplinger & Garcia (2004), (c) Segura et al. (2010), (d) Osten et al. (2016).

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Table 9.  UV–UV Emission Line Scaling Relations

UV1, UV2	m	${\sigma }_{{\rm{m}}}$	b	${\sigma }_{{\rm{b}}}$	ρ	n	σ
Lyα, Si iii	0.60	0.09	3.66	0.35	0.89	6.4E–04	0.23
Si ii, Si iii	0.82	0.04	−0.11	0.18	0.97	2.3E–07	0.15
C ii, Si iii	1.01	0.07	0.13	0.30	0.94	8.1E–06	0.27
Mg ii, Si iii	0.79	0.35	2.26	1.11	0.84	2.2E–03	0.36
Si iv, Si iii	1.01	0.04	−0.01	0.16	0.92	1.7E–05	0.29
He ii, Si iii	1.15	0.14	−0.2	0.6	0.95	1.6E–06	0.26
C iv, Si iii	1.03	0.04	0.49	0.19	0.96	7.5E–07	0.21
N v, Si iii	0.87	0.04	0.52	0.18	0.96	1.0E–06	0.20
Si ii, Lyα	1.16	0.21	−3.83	1.29	0.87	1.1E–03	0.30
C ii, Lyα	1.71	0.27	−6.26	1.60	0.88	7.4E–04	0.40
Mg ii, Lyα	1.38	0.15	−2.68	0.84	0.96	1.1E–04	0.19
Si iv, Lyα	1.71	0.25	−6.27	1.47	0.90	4.1E–04	0.38
He ii, Lyα	1.91	0.29	−7.26	1.71	0.96	1.6E–05	0.32
C iv, Lyα	1.58	0.19	−4.86	1.15	0.92	2.0E–04	0.32
N v, Lyα	1.43	0.21	−4.58	1.25	0.88	9.1E–04	0.35
C ii, Si ii	1.33	0.11	−0.05	0.38	0.93	4.1E–06	0.28
Mg ii, Si ii	1.57	0.27	0.54	0.78	0.86	6.0E–04	0.47
Si iv, Si ii	1.31	0.09	−0.12	0.30	0.92	1.1E–05	0.30
He ii, Si ii	1.57	0.14	−0.59	0.49	0.93	3.7E–06	0.32
C iv, Si ii	1.3	0.08	0.47	0.28	0.9	2.4E–05	0.33
N v, Si ii	1.18	0.10	0.23	0.36	0.92	1.0E–05	0.27
Mg ii, C ii	0.75	0.25	2.23	0.84	0.83	1.7E–03	0.39
Si iv, C ii	0.99	0.13	−0.10	0.60	0.98	1.6E–09	0.14
He ii, C ii	1.15	0.05	−0.41	0.23	0.94	1.3E–06	0.30
C iv, C ii	0.93	0.12	0.76	0.55	0.98	1.4E–08	0.17
N v, C ii	0.79	0.10	0.74	0.46	0.98	3.5E–09	0.15
Si iv, Mg ii	0.92	0.38	−1.04	1.78	0.82	2.1E–03	0.44
He ii, Mg ii	0.97	0.13	−1.10	0.61	0.96	4.1E–06	0.23
C iv, Mg ii	0.84	0.27	−0.15	1.31	0.85	8.4E–04	0.39
N v, Mg ii	1.27	0.51	−2.73	2.42	0.80	3.4E–03	0.53
He ii, Si iv	1.13	0.24	−0.17	1.07	0.94	1.9E–06	0.29
C iv, Si iv	0.93	0.06	0.89	0.26	0.96	1.1E–07	0.21
N v, Si iv	0.84	0.04	0.59	0.18	0.97	5.9E–08	0.17
C iv, He ii	0.82	0.16	1.02	0.80	0.95	6.3E–07	0.25
N v, He ii	0.67	0.13	1.07	0.64	0.92	7.4E–06	0.29
N v, C iv	0.85	0.06	0.08	0.29	0.99	4.9E–10	0.11

Note. Scaling relations are power laws of the form log₁₀ F_{S, UV1} = m× log₁₀ F_{S, UV2} + b, where ${F}_{{\rm{S}}}$ is the surface flux in erg cm⁻² s⁻¹, ρ is the Pearson correlation coefficient, n is the probability of no correlation, and σ is the standard deviation of the data points about the best-fit line (dex).

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Table 10.  UV Emission Line Scaling Relations with Stellar Rotation Period

UV	m	${\sigma }_{{\rm{m}}}$	b	${\sigma }_{{\rm{b}}}$	ρ	n	σ
Si iii	−1.40	0.29	5.37	0.44	−0.86	7.3E–04	0.44
Lyα	−0.92	0.22	6.97	0.31	−0.86	7.5E–04	0.29
Si ii	−0.85	0.13	3.93	0.18	−0.88	3.8E–04	0.28
C ii	−1.37	0.29	5.49	0.44	−0.86	8.0E–04	0.44
Mg ii	−1.18	0.15	6.52	0.23	−0.95	3.0E–05	0.25
Si iv	−1.32	0.30	5.31	0.45	−0.84	1.1E–03	0.43
He ii	−1.26	0.19	5.52	0.27	−0.93	4.5E–05	0.30
C iv	−1.40	0.32	6.03	0.49	−0.82	1.9E–03	0.49
N v	−1.31	0.32	5.38	0.48	−0.81	2.5E–03	0.47

Note. Scaling relations are power laws of the form log₁₀ F_{S, UV} = m;× log₁₀ P_rot + b, where ${F}_{{\rm{S}}}$ is the surface flux in erg cm⁻² s⁻¹ and ${P}_{\mathrm{rot}}$ is the rotation period in days, ρ is the Pearson correlation coefficient, n is the probability of no correlation, and σ is the standard deviation of the data points about the best-fit line (dex).

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Table 1 provides comparison points for solar flares, and in Table 8 we note the GOES classification and other parameters for well-known solar and stellar flares to give context for the GJ 876 flares. The Carrington event of 1859, arguably the largest solar flare ever recorded, has been calculated to be an X45 (±5) class flare (Cliver & Dietrich 2013). The November 4 solar flare of the Halloween storms in 2003, probably the largest flare observed during the space age, is estimated to be an X30.6-class flare (Kiplinger & Garcia 2004; note that the GOES detectors saturate for events between X10 and X20). For the great AD Leo flare of 1985 (Hawley & Pettersen 1991), Segura et al. (2010) estimated that an HZ planet ($\langle {r}_{\mathrm{HZ}}\rangle =0.16$ au) would have received peak >10 MeV proton and SXR fluxes of 5.9 × 10⁸ pfu and 9 W m⁻², respectively. The peak SXR flux at 1 au (0.23 W m⁻²) would make this an X2300-class flare. Osten et al. (2016) found that the superflare observed from the young M dwarf binary system DG CVn was equivalent to an X600,000-class flare (60 W m⁻² at 1 au, or 6000 W m⁻² at 0.1 au, the approximate HZ distance). We note, however, that there is evidence that SXR–proton scaling relations should break down for such large events (>X10-class; Hudson 2007; Drake et al. 2013).

We measure the peak flux for the large GJ 876 SXR flare observed at ∼45 ks in Figure 10(a) to be (9.72 ±1.28) × 10⁻¹³ erg cm⁻² s⁻¹ in the 0.3–10 keV bandpass (1.25–41 Å). A total of 11% ± 1.3% of the flare flux was emitted at energies 1.5–10 keV (1.25–8 Å), similar to the GOES long channel (1–8 Å), so we find that this flare is equivalent to an M9.5-class flare (error range: M7.8–X1.1; Table 1). X1-class flares have an 80%–100% chance of associated energetic particles from the Sun (Yashiro et al. 2006), with an estimated peak proton flux of ∼80 pfu at 1 au (Cliver et al. 2012). The SXR and proton fluxes received in GJ 876's HZ will be ∼30 × larger: 2.8 × 10⁻³ W m⁻² and 2400 pfu, respectively. Veronig et al. (2002) find that the Sun emits X-class flares roughly every month, but flares that unleash SXR fluxes of 10⁻³ W m⁻² on Earth occur only approximately once every 5 yr.

The smaller GJ 876 SXR flare observed at ∼7 ks in Figure 10(a) is estimated to be equivalent to an M2.2-class flare (Table 1). This smaller flare has a 40%–80% chance of associated energetic particles (Yashiro et al. 2006) with an estimated peak proton flux of ∼8 pfu at 1 au. M-class flares are emitted by the Sun about every other day (Veronig et al. 2002), but flares that are a factor of 30 larger in SXR and proton fluxes, as would be experienced in GJ 876's HZ, occur only a few times per year.

GJ 876's 10³¹ erg UV flare (Δλ = 400–1700 Å) observed on 2015 July 7 with HST (France et al. 2016) emitted 1.2 × 10²⁹ erg in the Si iv emission line over ∼25 minutes (Figure 10(b)). Using Equation (3) and the fluence at 1 au, we find that the peak proton flux received at 1 au during the flare was ∼75 pfu, or ∼2300 pfu at 0.18 au. Using the SXR–UV scaling relation (Equation (8)) and the observed Si iv flare peak, we find that this flare was X38-class with an estimated error of a factor of ∼3 (X13–X114). Note that the peak proton flux calculated for this 2015 July 7 Si iv flare (∼2300 pfu at 0.18 au) is similar to the peak proton flux calculated for the 2015 June 5 SXR flare (∼2400 pfu at 0.18 au), but that the GOES classifications are different: M9.5 compared to X38, or a factor of 40 difference in SXR flux.

In total, Chandra observed three M- to X-class flares in 8.25 hr, and HST observed six comparable flares (within an order of magnitude) in 12.35 hr, including observations from the MUSCLES pilot survey (France et al. 2013). Thus, we estimate that GJ 876 emits flares of this magnitude at a rate of ∼0.4–0.5 hr⁻¹. From Veronig et al. (2002), the Sun's rate of M-class flares is ∼0.02 hr⁻¹, a factor of ∼20 less frequent than GJ 876. However, note that these M-class flares are effectively 30× stronger (i.e., X10-class) in GJ 876's HZ at 0.18 au. The Sun emits X10-class flares approximately once every 5 yr, so planets in the GJ 876 HZ are receiving SXR flare fluxes ≥10⁻³ W m⁻² associated with proton fluxes ∼10³ pfu about four orders of magnitude more frequently than Earth.

In Figure 11, we show the effect of frequent X-class flares with associated proton enhancements on the O₃ column depth in an Earth-like atmosphere with no magnetosphere (Segura et al. 2010; M. A. Tilley et al. 2017, in preparation). For comparison to the M- to X-class GJ 876 flares analyzed in this section, we show the dramatic responses of O₃ after the great AD Leo flare (Segura et al. 2010) and the Carrington event (see Table 8). To represent the GJ 876 flares discussed in this section, we scaled the SED of the great AD Leo flare (Hawley & Pettersen 1991) down in intensity and duration to match flares of the dM4e star GJ 1243 (Hawley et al. 2014). This resulted in ∼X1-class flares with ∼4-minute durations. Peak proton fluxes were assigned as 1.2 × 10³ pfu (a typical value for the flares discussed in this section), and we provide two cases distinguished by their flare frequency: 0.08 hr⁻¹ (Case A) and 0.5 hr⁻¹ (Case B; similar to GJ 876's observed flare frequency). Both the Case A and Case B flares turn off after a period of 40 months, but via extrapolation, we show that for Case A, the O₃ column approaches near-complete depletion at approximately 10¹³ s (318 kyr) of similar, constant stellar activity. Case B shows near-complete O₃ depletion after only approximately 5 × 10⁹ s (160 yr). Given the several-gigayear period of high activity during the evolution of early to mid-M dwarfs, both scenarios indicate massive, and likely complete, O₃ depletion. This suggests that planetary surfaces in the HZ would be bathed in stellar UV flux. A detailed analysis of atmospheric effects will be presented in the upcoming work by M. A. Tilley et al. (2017, in preparation).

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Our new method of proton flux estimation due to stellar flares has important limitations, but it will be useful until advancements are made in the indirect detection of particles from stellar eruptive events (e.g., coronal dimming or Type II or III radio bursts; Crosley et al. 2016; Harra et al. 2016) or in our understanding of particle acceleration under kilogauss magnetic field strengths. The scaling relations are statistical and are relatively inaccurate for individual flares.

The first caveat to the method is that we are necessarily assuming that particle acceleration in M dwarf atmospheres is similar to that in the Sun. This could be a poor assumption as M dwarfs have different atmospheric structure and stronger surface magnetic fields. Magnetic processes are ultimately responsible for flares and particle acceleration. Also, fast-rotating M dwarfs have extremely large surface magnetic fields with photospheres possibly saturated with active regions. Overlying magnetic fields could prevent the acceleration of particles away from the stellar atmosphere (e.g., Vidotto et al. 2016). This phenomenon was observed on the Sun in 2014 October when the large active region 2192 emitted many X-class flares, which have a >90% probability of an associated CME, but no CMEs were ejected. Strong overlying magnetic fields were observed and have been cited as a possible explanation for the lack of associated CMEs (Sun et al. 2015; Thalmann et al. 2015).

We should be careful when extrapolating solar-based SXR–particle scaling relations to large energies. There is evidence for a break in the SXR–proton power law around X10-class flares (see Lingenfelter & Hudson 1980; Hudson 2007, and references therein). It is unclear whether only the expected proton flux flattens out for increasingly large SXR flares (>X10), or whether the frequency of >X10-class flares also breaks from the expected power-law frequency distribution (e.g., Veronig et al. 2002). This uncertainty is partly due to the rarity of these energetic events, but also because the GOES SXR detectors saturate around 10⁻³ W m⁻². Drake et al. (2013) find that extrapolating SXR–CME scaling relations to energies applicable to highly active solar-type stars would require CMEs to account for ∼10% of the total stellar luminosity, a likely unreasonable fraction. Those authors conclude that the SXR–CME scaling relations must flatten out at some point.

Predicting proton flux even for the Sun is not an easy feat, because the mechanisms of particle acceleration are not well understood. SXRs and UV photons during flares correlate with particle flux, and Kahler (1982) proposes that such correlations are manifestations of "Big Flare Syndrome," which describes how the energy of an eruptive event can power numerous physical processes that are not directly linked. Thermally heated plasma (T > 80,000 K) emits short-wavelength photons that we observe as a flare, but particles are accelerated by a nonthermal process, and the ambient corona is thought to be the major contributor to CME mass (Sciambi et al. 1977; Hovestadt et al. 1981). For our purposes of estimating proton fluxes from M dwarfs and their effects on orbiting planets, a correlation between photons and particles without direct causation is valuable.

Another limitation is our use of the GJ 832 synthetic spectrum from Fontenla et al. (2016) to estimate He ii λ304 flare flux from far-UV Si iv or He ii λ1640 flare flux. The model is applicable to the quiescent star, and flux ratios between emission lines likely change during a flare. Comparing flux ratios between the active and quiet-Sun models listed in Table 1 of Fontenla et al. (2016), the ratios of emission lines formed at different temperature change by a factor of a few during a flare. In a future work, we will improve the flux ratio relations using flare atmospheres.

We chose He ii λ304 as a proxy because of its similar formation temperature to Si iv λλ1393,1402 (and He ii λ1640), but we note that the electrons responsible for the collisional excitation of He ii λ304 (40.8 eV above ground) must have significantly higher energies than the thermal electrons at ${T}_{\mathrm{form}}$ = 80,000 K (k_B ${T}_{\mathrm{form}}=6.9$ eV) that collisionally excite Si iv (8.8 eV above ground; Jordan 1975). Higher-energy electrons could diffuse down through the transition region, as recombination/ionization timescales are much longer than dynamical timescales (e.g., Shine et al. 1975; Golding et al. 2014, 2016). Also, the He ii line fluxes receive some contribution from recombination, and this becomes more important during flares.

Another challenge in the analysis is quantifying the probability that any ejected particles will intersect the exoplanet. There are many unknowns here, including the interplanetary magnetic field topology that charged particle trajectories will follow (Parker 1958); the opening angle of the accelerated particles; the planet's cross section, which may be larger than $\pi {R}_{{\rm{p}}}^{2}$ owing to the presence of a magnetosphere; and the direction of the particle ejection with respect to the planet's orbital plane (see Kay et al. 2016). A thorough treatment of this issue is beyond the scope of this work.