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

TL;DR

This study develops proxies for UV emission and energetic particle fluxes from M dwarfs using ground-based optical spectra and flare analysis, aiding habitability assessments of exoplanets around these stars.

Contribution

It introduces a method to estimate UV fluxes from Ca II K line measurements and a new technique to assess energetic particle fluxes during stellar flares, based on solar-like energy partition assumptions.

Findings

Ca II K line correlates with UV emission lines after spectral type correction.

Flares on GJ 876 can produce significant X-ray and proton fluxes impacting habitable zones.

Habitable zone planets around GJ 876 experience frequent large flares, much more than Earth.

Abstract

Characterizing the UV spectral energy distribution (SED) of an exoplanet host star is critically important for assessing its planet's potential habitability, particularly for M dwarfs as they are prime targets for current and near-term exoplanet characterization efforts and atmospheric models predict that their UV radiation can produce photochemistry on habitable zone planets different than on Earth. To derive ground-based proxies for UV emission for use when Hubble Space Telescope observations are unavailable, we have assembled a sample of fifteen early-to-mid M dwarfs observed by Hubble, and compared their non-simultaneous UV and optical spectra. We find that the equivalent width of the chromospheric Ca II K line at 3933 Angstroms, when corrected for spectral type, can be used to estimate the stellar surface flux in ultraviolet emission lines, including H I Lyman alpha. In addition, we address another potential driver of habitability: energetic particle fluxes associated with flares. We present a new technique for estimating soft X-ray and >10 MeV proton flux during far-UV emission line flares (Si IV and He II) by assuming solar-like energy partitions. We analyze several flares from the M4 dwarf GJ 876 observed with Hubble and Chandra as part of the MUSCLES Treasury Survey and find that habitable zone planets orbiting GJ 876 are impacted by large Carrington-like flares with peak soft X-ray fluxes >1e-3 W m-2 and possible proton fluxes ~100-1000 pfu, approximately four orders of magnitude more frequently than modern-day Earth.

Figures

Tables

Table 1. Table 1: The GOES classification scheme of solar flares a

Class	Peak flux at 1 AU		SXR Luminosity		Solar occurrence	Probability of	Expected peak $>$10 MeV
	(W m-2)	(erg cm-2 s-1)	(W)	(erg s-1)	rate (hr-1)b	CMEc	proton flux (pfu)d at 1 AU
X100	10-2	101	2.8$\times$1021	2.8$\times$1028	$<$2$\times$10-6 e	–	–
X10	10-3	100	2.8$\times$1020	2.8$\times$1027	2$\times$10-5 f	$\sim$100%	3300
X	10-4	10-1	2.8$\times$1019	2.8$\times$1026	0.002	80–100%	90
M	10-5	10-2	2.8$\times$1018	2.8$\times$1025	0.02	40–80%	2
C	10-6	10-3	2.8$\times$1017	2.8$\times$1024	0.15	$<$40%	$<$1f
B	10-7	10-4	2.8$\times$1016	2.8$\times$1023	$>$0.15g	–	–

Table 2. Table 2: The M dwarf sample

No.	Star	$d$	$R$	Spectral	$T_{\mathrm{eff}}$	$P_{\mathrm{rot}}$	Wλ    (Ca II K)	Wλ,corr    (Ca II K)	Data	No. of
		(pc)	($R_{\odot }$)	Type	(K)	(days)	(Å)	(Å)	included†	Ca II K spectra
1	GJ 832*	5.0 a	0.56 f	M2 f	3590 f	45.7 q	0.88$\pm$0.09	0.73$\pm$0.07	H, X, R	8
2	GJ 876*	4.7 a	0.38 g	M4 g	3130 g	96.7 r	0.82$\pm$0.15	0.17$\pm$0.03	H, X, R, D	314
3	GJ 581*	6.2 a	0.30 z,h	M2.5 h	3500 h	132.5 q	0.36$\pm$0.08	0.25$\pm$0.05	H	351
4	GJ 176*	9.3 a	0.45 g	M2.5 g	3680 g	39.5 s	1.76$\pm$0.27	1.76$\pm$0.27	H, D	256
5	GJ 436*	10.1 a	0.45 aa,i	M3 i	3420 i	39.9 q	0.58$\pm$0.07	0.33$\pm$0.04	H, D	257
6	GJ 667C*	6.8 a	0.46 j	M1.5 o	3450 o	103.9 q	0.44$\pm$0.11	0.27$\pm$0.06	H	39
7	GJ 1214*	14.6 b	0.21 k	M4.5 o	2820 o	$>$100 t	1.16$\pm$0.2	0.040$\pm$0.007	X, M	10
8	AD Leo	4.7 c	0.44 f	M3 f	3410 f	2.2 u	11.57$\pm$1.61	6.31$\pm$0.88	H, R	44
9	EV Lac	5.1 a	0.36 f	M4 f	3330 f	4.38 v	14.86$\pm$2.52	6.47$\pm$1.10	H	24
10	Proxima Cen*	1.3 d	0.14 l	M5.5 l	3100 l	83.5 w	13.7$\pm$5.93	2.50$\pm$1.08	X, M, R	7
11	AU Mic	9.9 a	0.83 m	PMS M1 m,p	3650 p	4.85 x	12.13$\pm$2.17	11.44$\pm$2.05	H, R	52
12	YZ CMi	6.0 a	0.36 f	M4 f	3200 f	2.78 y	21.23$\pm$4.67	5.92$\pm$1.30	H, X	20
13	GJ 821	12.2 a	0.37 f	M2 f	3670 f	–	0.33$\pm$0.06	0.32$\pm$0.06	H	26
14	GJ 213	5.8 a	0.25 f	M4 f	3250 f	–	0.44$\pm$0.12	0.15$\pm$0.04	H	7
15	GJ 1132*	12.0 e	0.21 n	M4 n	3270 n	125 n	1.40$\pm$0.01	0.504$\pm$0.005	M	1

Table 3. Table 3: UV emission line fluxes of the M dwarf sample (continued to Table  4 )

No.	Star	Si III	${\sigma }_{\mathrm{SiIII}}$	Ly$\alpha$ a	${\sigma }_{\mathrm{Ly}\alpha }$ a	Si II	${\sigma }_{\mathrm{SiII}}$	C II	${\sigma }_{\mathrm{CII}}$	Mg II	${\sigma }_{\mathrm{MgII}}$
1	GJ 832*	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 876*	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 581*	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 176*	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 436*	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 667C*	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 1214*	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-12 b	2.79E-13 b
9	EV Lac	3.86E-14	1.46E-15	2.75E-12 b	5.50E-13 b	2.49E-15	4.95E-16	3.76E-14	6.42E-16	–	–
10	Proxima Cen*	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-12 b	4.20E-13 b
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 1132*	–	–	2.64E-14	1.62E-14	–	–	–	–	–	–

Table 4. Table 4: UV emission line fluxes of the M dwarf sample – continuation of Table  3

No.	Star	Si IV	${\sigma }_{\mathrm{SiIV}}$	He II	${\sigma }_{\mathrm{HeII}}$	C IV	${\sigma }_{\mathrm{CIV}}$	N V	${\sigma }_{\mathrm{NV}}$	EUV c
1	GJ 832*	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 876*	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 581*	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 176*	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 436*	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 667C*	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 1214*	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 Cen*	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 1132*	–	–	–	–	–	–	–	–	2.31E-14

Table 5. Table 5: Correction factors for calculating W λ , corr subscript 𝑊 𝜆 corr W_{\rm\lambda,corr} (Equation  2 )

$T$	$x$($T$)/$x$(3680 K)	$T$	$x$($T$)/$x$(3680 K)
2400	1.1613$\times$10-3	3300	0.39680
2500	2.5175$\times$10-3	3400	0.53075
2600	5.6663$\times$10-3	3500	0.68191
2700	1.2907$\times$10-2	3600	0.85061
2800	2.9381$\times$10-2	3700	1.0425
2900	5.9956$\times$10-2	3800	1.2642
3000	0.11061	3900	1.5233
3100	0.18269	4000	1.8277
3200	0.27899

Table 6. Table 6: Fit parameters for UV surface flux and Ca II K equivalent width relations

Transition name	Wavelength (Å)	log $T_{\mathrm{formation}}^a$	$\alpha$	$\beta$	$\rho$	$n$	$\sigma$
Si III	1206.50	4.7	1.35$\pm$0.26	3.21$\pm$0.16	0.80	2.0$\times$10-3	0.59
H I Ly$\alpha$	1215.67	4.5 (line core)	0.88$\pm$0.10	5.46$\pm$0.06	0.95	1.1$\times$10-5	0.18
		“active”	1.21$\pm$0.16	5.20$\pm$0.12
		“inactive”	0.10$\pm$0.17	5.19$\pm$0.08
Si II	1260.42, 1264.74, 1265.00	4.5	0.99$\pm$0.18	2.58$\pm$0.11	0.82	6.7$\times$10-4	0.42
C IIb	1335.71	4.5	1.29$\pm$0.23	3.38$\pm$0.14	0.82	6.0$\times$10-4	0.54
Mg IIc	2796.35, 2803.53	4.5 (line core)	1.06$\pm$0.13	4.88$\pm$0.08	0.94	7.1$\times$10-6	0.27
Si IV	1393.76, 1402.77	4.9	1.35$\pm$0.24	3.22$\pm$0.14	0.84	3.6$\times$10-4	0.54
He II	1640.4d	4.9	1.33$\pm$0.15	3.53$\pm$0.09	0.91	1.2$\times$10-5	0.41
C IV	1548.19, 1550.78	5.0	1.31$\pm$0.26	3.84$\pm$0.16	0.81	8.8$\times$10-4	0.56
N V	1238.82, 1242.8060	5.2	1.2$\pm$0.26	3.36$\pm$0.15	0.77	1.9$\times$10-3	0.55
EUVe	100–912	–	0.80$\pm$0.09	5.49$\pm$0.06	0.94	1.5$\times$10-5	0.18
		“active”	1.37$\pm$0.18	5.10$\pm$0.14
		“inactive”	0.12$\pm$0.18	5.15$\pm$0.08

Table 7. Table 7: Formulae for estimating extreme-UV fluxes from Ly α 𝛼 \alpha  and Ca II K

Flux in Wavelength
Band at 1 AU	Linsky et al. (2014)
(erg cm-2 s-1)	Table 5	This workb
$F$(100–200 Å)	10-0.491 $\cdot$ $F$(Ly$\alpha$)	104.97±0.06 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{0.88\pm 0.10}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)2.0
$F$(200–300 Å)	10-0.548 $\cdot$ $F$(Ly$\alpha$)	104.91±0.06 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{0.88\pm 0.10}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)2.0
$F$(300–400 Å)	10-0.602 $\cdot$ $F$(Ly$\alpha$)	104.86±0.06 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{0.88\pm 0.10}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)2.0
$F$(400–500 Å)	10-2.294 $\cdot$ $F$(Ly$\alpha$)1.258	104.57±0.08 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{1.11\pm 0.13}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)2.52
$F$(500–600 Å)	10-2.098 $\cdot$ $F$(Ly$\alpha$)1.572	106.49±0.09 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{1.38\pm 0.16}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)3.14
$F$(600–700 Å)	10-1.920 $\cdot$ $F$(Ly$\alpha$)1.240	104.85±0.07 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{1.09\pm 0.12}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)2.48
$F$(700–800 Å)	10-1.894 $\cdot$ $F$(Ly$\alpha$)1.518	106.39±0.09 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{1.34\pm 0.15}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)3.04
$F$(800–912 Å)	10-1.811 $\cdot$ $F$(Ly$\alpha$)1.764	107.82±0.11 $\cdot$ $W_{\lambda ,\mathrm{corr}}^{1.55\pm 0.18}$ $\cdot$ (4.65$\times$10-3$R_{\star }$)3.53

Table 8. Table 8: Correlations between UV flare fluence ( F 𝐹 F ) and the peak intensity ( I 𝐼 I ) and fluence ( F 𝐹 F ) of energetic protons during solar flares.

$x$	$y$	$\alpha$	$\beta$	$\rho$	$n$	$\sigma$
$F_{\mathrm{HeII},304}$	$I_{\ge 10MeV}$	1.06$\pm$0.21	2.42$\pm$0.17	0.83	2.2$\times$10-5	0.76
$F_{\mathrm{HeII},304}$	$I_{\ge 30MeV}$	0.85$\pm$0.20	1.62$\pm$0.16	0.80	6.5$\times$10-5	0.75
$F_{\mathrm{HeII},304}$	$F_{\ge 10MeV}$	1.20$\pm$0.26	2.23$\pm$0.21	0.84	1.0$\times$10-5	0.84
$F_{\mathrm{HeII},304}$	$F_{\ge 30MeV}$	1.01$\pm$0.25	1.37$\pm$0.20	0.82	3.6$\times$10-5	0.84
$F_{\mathrm{SiIV}}$	$I_{\ge 10MeV}$	1.06$\pm$0.21	3.34$\pm$0.25
$F_{\mathrm{SiIV}}$	$F_{\ge 10MeV}$	1.20$\pm$0.26	3.27$\pm$0.31
$F_{\mathrm{HeII},1640}$	$I_{\ge 10MeV}$	1.06$\pm$0.21	4.05$\pm$0.36
$F_{\mathrm{HeII},1640}$	$F_{\ge 10MeV}$	1.20$\pm$0.26	4.07$\pm$0.45

Table 9. Table 9: GJ 876 flare properties

Flare	Peak SXR flux (W m-2)		GOES	$>$10 MeV proton flux (pfu)
	1 AU	$⟨r_{\mathrm{HZ}}⟩$	class	1 AU	$⟨r_{\mathrm{HZ}}⟩$
GJ 876 flares:
2015 June 4	2.2$\times$10-5	6.6$\times$10-4	M2.2	8	240
2015 June 5	9.5$\times$10-5	2.9$\times$10-3	M9.5	80	2400
2015 July 7	3.8$\times$10-3	1.1$\times$10-1	X38	75	2300
Reference flares:
Carrington	4.5$\times$10-3	–	X45$\pm$5	–	–
Event 1859 a
2003 November 4	3.06$\times$10-3	–	X30.6	400	–
solar flare b
Great AD Leo	0.23	9	X2300	1.5$\times$106	5.9$\times$108
flare of 1985 c
DG CVn	60	6000	X600,000	–	–
2014 April 24 d

Table 10. Table 10: UV–UV emission line scaling relations

UV1, UV2	m	${\sigma }_m$	b	${\sigma }_b$	$\rho$	$n$	$\sigma$
Ly$\alpha$, 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, Ly$\alpha$	1.16	0.21	-3.83	1.29	0.87	1.1E-03	0.30
C II, Ly$\alpha$	1.71	0.27	-6.26	1.60	0.88	7.4E-04	0.40
Mg II, Ly$\alpha$	1.38	0.15	-2.68	0.84	0.96	1.1E-04	0.19
Si IV, Ly$\alpha$	1.71	0.25	-6.27	1.47	0.90	4.1E-04	0.38
He II, Ly$\alpha$	1.91	0.29	-7.26	1.71	0.96	1.6E-05	0.32
C IV, Ly$\alpha$	1.58	0.19	-4.86	1.15	0.92	2.0E-04	0.32
N V, Ly$\alpha$	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

Table 11. Table 11: UV emission line scaling relations with stellar rotation period

UV	m	${\sigma }_m$	b	${\sigma }_b$	$\rho$	$n$	$\sigma$
Si III	-1.40	0.29	5.37	0.44	-0.86	7.3E-04	0.44
Ly$\alpha$	-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

Equations

$$W_{\lambda }=\frac{{\int }_{{\lambda }_1}^{{\lambda }_2}{F}_{\mathrm{observed}}d\lambda -{\int }_{{\lambda }_1}^{{\lambda }_2}{F}_{\mathrm{PHOENIX}}d\lambda }{{\overline{F}}_{\mathrm{continuum}}}.$$

$$W_{\lambda ,\mathrm{corr}}=W_{\lambda }\times \frac{x(T)}{x(3680K)}.$$

$$\log I_{\text{textgreater}10MeV}=(1.06\pm 0.21)\times \log F_{\mathrm{SiIV}}+(3.34\pm 0.25),$$

$$\log F_{\text{textgreater}10MeV}=(1.20\pm 0.26)\times \log F_{\mathrm{SiIV}}+(3.27\pm 0.31).$$

$$\log I_{\text{textgreater}10MeV}=(1.06\pm 0.21)\times \log F_{\mathrm{HeII},1640}+(4.05\pm 0.36),$$

$$\log F_{\text{textgreater}10MeV}=(1.20\pm 0.26)\times \log F_{\mathrm{HeII},1640}+(4.07\pm 0.45).$$

$$\log I_{\mathrm{SXR}}=(1.40\pm 0.09)\times \log I_{\mathrm{HeII},304}+(1.52\pm 1.62).$$

$$\log I_{\mathrm{SXR}}=(1.4\pm 0.1)\times \log I_{\mathrm{SiIV}}+(2.7\pm 1.6),$$

$$\log I_{\mathrm{SXR}}=(1.4\pm 0.1)\times \log I_{\mathrm{HeII},1640}+(3.7\pm 1.6).$$

$$W_{\lambda ,\mathrm{A}}=\frac{F_{\mathrm{A},\mathrm{S}}\times \frac{R_{\mathrm{A}}^2}{d_{\mathrm{A}}^2}}{x(T_{\mathrm{A}})\times \frac{R_{\mathrm{A}}^2}{d_{\mathrm{A}}^2}}=\frac{F_{\mathrm{A},\mathrm{S}}}{x(T_{\mathrm{A}})},$$

$$W_{\lambda ,\mathrm{B}}=\frac{F_{\mathrm{B},\mathrm{S}}\times \frac{R_{\mathrm{B}}^2}{d_{\mathrm{B}}^2}}{x(T_{\mathrm{B}})\times \frac{R_{\mathrm{B}}^2}{d_{\mathrm{B}}^2}}=\frac{F_{\mathrm{B},\mathrm{S}}}{x(T_{\mathrm{B}})}.$$

$$W_{\lambda ,\mathrm{corr}}=W_{\lambda }\times \frac{x(T)}{x(T_{\mathrm{ref}})}.$$

Full text

stars: low-mass — stars: chromospheres — Sun: flares

††journal: ApJ††facilities: HST (COS, STIS), ARC (DIS), CXO, Keck:I (HIRES), CASLEO:JST (REOSC), VLT:Kueyen (XSHOOTER), Magellan:Clay (MIKE), SDO (EVE), GOES.††software: IRAF, Astropy (Robitaille et al., 2013), IPython (Perez & Granger, 2007), Matplotlib (Hunter, 2007), Pandas (McKinney, 2010), NumPy and SciPy (van der Walt et al., 2011).

1 Introduction

1.1 UV and Ca II K emission from M dwarfs

Recent ultraviolet (UV) studies have shown that even optically-inactive M dwarfs (i.e., those displaying H$\alpha$ 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 7 nearby ($d$ $\textless$ 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\,\textgreater\,$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 target111The smaller flares from the MUSCLES M dwarfs show factor of $\textless$10 flux increases, which are below the S/N threshold of GJ 1214’s light curves (Loyd et al. 2017, in preparation)..

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 comparitively little continuum emission due to the cool stellar photosphere ($T_{\rm eff}$ $\textless$ 4000 K). The H i Ly$\alpha$ emission line (1215.67 Å) is prominent, comprising 27%–72% of the total 1150–3100 Å flux (excluding 1210–1222 Å) for the 7 MUSCLES M dwarfs (France et al., 2016; Youngblood et al., 2016). The extreme-UV (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 $\sim$400–912 Å range is heavily attenuated by neutral hydrogen for all stars, except the Sun. Thus, the extreme-UV must be estimated from other proxies such as Ly$\alpha$ (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 extreme-UV fluxes that agree within 30% (Ehrenreich et al., 2015; 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 to 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 Ly$\alpha$ flux is critical, because Ly$\alpha$ comprises 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 Ly$\alpha$ 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 and 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 underway 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).

Accurate UV spectral flux data is 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 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, H$\alpha$ 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, H$\alpha$ 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 & 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 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 ($\textgreater$1 Å) absorption and narrow ($\textless$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 (Wilson, 1978; Vaughan et al., 1978) and $R^{\prime}_{\rm HK}$ (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-year 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 Ly$\alpha$ 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, H$\alpha$–C IV, H$\alpha$–SXR, H$\alpha$–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.

1.2 Flares and energetic particles

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 a HZ planet’s O3 column density. However, including SEPs, the expected NOx production will deplete a planet’s O3 by $\sim$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 extreme-UV 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 Ly$\alpha$ 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 kilo-Gauss 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 $\textgreater$106 erg cm*-2* s*-1*, where a fundamental change in the magnetic field topology may occur. On the Sun in October 2014, the large active region 2192 emitted many large flares, but no CMEs. Strong overlying magnetic fields likely confined the eruption (Thalmann et al., 2015; Sun 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 non-thermally from cooler plasma in the ambient corona (Hovestadt et al., 1981; Sciambi et al., 1977). 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 $\textgreater$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 $\textless$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 $\times$ 10*-6* W m*-2* SXR flux at 1 AU. Approximately 20% of C-class and $\sim$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 $\textgreater$10 MeV proton flux (Belov et al., 2005, 2007). Much of the recent UV flare data of M dwarfs has 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 ($\textgreater$10 MeV) received at Earth and an extreme-UV 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 $\lambda\lambda$1393,1402 and He II $\lambda$1640).

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.

2 Observations & Reductions

2.1 The M dwarf sample

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 criteria (listed in Table 1.2) are all early-to-mid M dwarfs, nearby ($d$ $\textless$ 15 pc), and exhibit a broad range of rotation periods (2 to $\textgreater$100 days) and a broad range of ages ($\sim$10 Myr to $\sim$10 Gyr). Nine of the 15 M dwarfs are known to host exoplanets.

Seven of the 15 stars are exoplanet host stars from the MUSCLES Treasury Survey (stars 1–7 in Table 1.2), and are weakly-active with H$\alpha$ 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–M4.5 spectral type. We also included other weakly-active M dwarfs, including MEarth planet host GJ 1132 (Berta-Thompson et al., 2015b) 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 1.2). These stars are highly active (H$\alpha$ emission spectra), have short rotation periods ($\textless$ 7 days), and are likely young ($\textless$ 1 Gyr), with the exception of Proxima Centauri ($P_{\rm rot}$ = 83.5 days, $\sim$5 Gyr old; see Reiners & Basri 2008; Davenport et al. 2016).

2.2 M Dwarf UV and optical data

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 3.5m telescope at Apache Point Observatory (APO) or the REOSC echelle spectrograph on the 2.15m telescope at Complejo Astronómico El Leoncito (CASLEO) within a day or two of the HST observations. We also gathered spectra of GJ1132, GJ1214, 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 comprise the bulk of our Ca II measurements.

APO/DIS ($R$ $\sim$ 2500) and CASLEO/REOSC ($R$ $\sim$ 12,000) spectra were reduced using standard IRAF222IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation. 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 ($R$ $\sim$ 25,000) were reduced using the standard MIKE pipeline included in the Carnegie Python Distribution (CarPy). Science-level VLT/XSHOOTER ($R$ $\sim$ 6000) data products were obtained from the ESO Science Archive Facility, and Keck/HIRES ($R$ $\sim$ 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 MAKEE333http://www.astro.caltech.edu/$\sim$tb/makee/ suit our purposes well. We did not use spectra where the automatic extraction failed or the wavelength calibration was incorrect. We also did not coadd 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 Products444https://archive.stsci.edu/prepds/muscles/ (HLSPs; Loyd et al. 2016), or the StarCAT portal555http://casa.colorado.edu/$\sim$ayres/StarCAT/. 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 Ly$\alpha$ profiles using the methods described in Youngblood et al. (2016). Proxima Cen’s reconstruction is included as part of a 5 Å–5.5 $\mu$m SED on the MUSCLES HLSP website666https://archive.stsci.edu/prepds/muscles/. The differences between our Ly$\alpha$ reconstructions and those presented in Wood et al. (2005b) are small, $\sim$20% in integrated Ly$\alpha$ flux for AD Leo and $\sim$4% in integrated Ly$\alpha$ flux for Proxima Cen. We also reconstructed the Ly$\alpha$ profile of GJ 1132 (star 15; $F$(Ly$\alpha$) = (2.64${}^{+2.58}_{-0.65}$) $\times$ 10*-14* erg cm*-2* s*-1*) for the first time using the methods from Youngblood et al. (2016). Note that the Ly$\alpha$ error bars have been averaged for symmetry in Table 3. All UV fluxes used in this work are printed in Tables 3 and 4.

2.2.1 Interstellar medium corrections

5/9 of our UV emission lines (Ly$\alpha$, Mg II, C II, Si II, and Si III) are affected by absorption from the interstellar medium (ISM), and here we describe our attempts to mitigate the effect of ISM absorption on our measurements. All the reported Ly$\alpha$ fluxes have been reconstructed from the wings of the observed line profile using the technique described in Youngblood et al. (2016) or Wood et al. (2005b) for EV Lac. The Mg II fluxes were corrected uniformly for a 30% ISM absorption, assuming a typical log10 $N$(Mg II) $\sim$ 13 for stars within 20 pc (Redfield & Linsky, 2002). Many of the Mg II observations are not sufficiently resolved to allow for a profile reconstruction, and we do not apply a correction factor scaling either with distance or H i column densities measured along the line-of-sight, because the Mg+ abundance varies in the local ISM. C II 1334 Å is the most significantly-impacted of the C II $\lambda\lambda$1334, 1335 doublet, and so we excluded its contribution from the reported C II fluxes.

We do not apply ISM corrections to the Si II (see Redfield & Linsky 2004 for a discussion of Si II absorption in the local ISM) or Si III fluxes, noting that the ISM’s effect on Si III is likely small. The intrinsic narrowness of the M dwarf emission lines may mean that for some sightlines, the ISM absorption coincides with the stellar emission line, but for others the ISM absorption is shifted away from the emission, and the line’s flux is not attenuated. Ca+ from the ISM can significantly attenuate Ca II H & K, but only for distant stars ($d$ $\textgreater$ 100 pc; Fossati et al. 2017). Because all of our targets are within 15 pc, Ca II ISM absorption is likely negligible, and we do not apply a correction.

2.3 Solar X-ray, UV, and proton data

We utilize time-series solar irradiance (disk-integrated) measurements from the Extreme ultraviolet Variability Experiment (EVE) suite of instruments onboard 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-second cadence, but we use one-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) system777https://www.ngdc.noaa.gov/stp/satellite/goes/index.html. 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 this correction factor is incorrect (Woods et al., 2017). We utilize the proton measurements from the $\textgreater$10 MeV and $\textgreater$30 MeV channels.

3 Ca II K Equivalent Widths

To isolate the chromospheric Ca II K emission line so we can compare it to the UV emission line fluxes, we must correct for the absorption by subtracting a radiative equilibrium model. This is particularly important for the low resolution spectra, which typically cannot resolve the emission cores. Only the CASLEO/REOSC spectra are flux calibrated, so to be consistent with the rest of the spectra, we normalized the flux in the isolated Ca II K emission core to nearby continuum, making this measurement akin to an equivalent width. Busà et al. (2007), Marsden et al. (2009), and Walkowicz & Hawley (2009) used this “residual” equivalent width technique in their analyses of Ca II lines. Note that we do not use the widely-used S-index (Wilson, 1978; Vaughan et al., 1978) or $R^{\prime}_{\rm HK}$ index (e.g., Noyes et al. 1984), because for our medium-resolution spectra the H$\epsilon$ emission line is blended with the Ca II H emission core line.

The residual equivalent width $W_{\lambda}$ is given by:

[TABLE]

$F_{\rm observed}$ is the observed Ca II K profile, $F_{\rm PHOENIX}$ is the radiative equilibrium PHOENIX model (Husser et al., 2013) scaled to $F_{\rm observed}$ in the Ca II K absorption wings, $\lambda_{1}$ and $\lambda_{2}$ are the interactively-chosen bounds of integration around the narrow chromospheric Ca II K emission, and $\overline{F}_{\rm continuum}$ is the average observed flux in the continuum region (before subtraction of the PHOENIX model) from 3937–3940 Å. We selected the 3937–3940 Å region for continuum normalization because it is close to the Ca II K line (3933 Å), and the spectral response curve of the various CCDs we are using are unlikely to vary significantly over $\sim$7 Å. Walkowicz & Hawley (2009) and Rauscher & Marcy (2006) normalize their effective equivalent widths using continuum regions 3952.8–3956.0 Å and 3974.8–3976.0 Å. We do not believe there is a significant difference between these continuum choices, because neither are truly continuum due to the high density of absorption lines in this spectral region.

The PHOENIX models were retrieved from the Husser et al. (2013) grid using literature values for $T_{\rm eff}$ (Table 1.2) and assuming log${}_{10}~{}g$ = 5 and [Fe/H] = 0 (solar). However, for the 7 MUSCLES targets, we use the PHOENIX models incorporated into the high-level science products available on MAST. See Loyd et al. (2016) for details about the parameters used to retrieve these models. The PHOENIX models have resolution $R$ = 500,000 around Ca II K (Husser et al., 2013) and were convolved with a Gaussian kernel to match the resolution of the spectra from the various instruments used (Section 2). The PHOENIX models were also shifted in velocity space to the rest frame of each target before fitting. We scaled the PHOENIX spectra to match 3 points in the broad Ca II K absorption wings.

The M dwarfs in our sample have a broad range of effective temperatures, so we must account for a dependence on spectral type in this residual equivalent width measurement (see Appendix A for more details on the following description). Stars of earlier spectral type have brighter continuum fluxes than later type stars, affecting the normalization of the emission core’s flux. Walkowicz & Hawley (2009) restricted their Ca II analysis to a single spectral subtype (M3 V) to avoid the issue of a spectral type dependence. To correct the residual equivalent widths ($W_{\rm\lambda}$ from Equation 1) for spectral type dependence, we normalized to the $W_{\rm\lambda}$ value for a reference star parameterized by $T_{\rm eff}$ = 3680 K, log $g$ = 5, and [Fe/H] = 0. The normalization or “correction” factor is the ratio of the star’s average continuum flux and the reference star’s average continuum flux. Both continuum averages come from the PHOENIX models. The corrected residual equivalent width is given by

[TABLE]

$x$($T$) is the average continuum flux value from 3936.9–3939.9 Å from a PHOENIX model for a star with temperature T, and $x$(3680 K) is the PHOENIX model’s average continuum flux value for the reference star. The reference star’s effective temperature corresponds to GJ 176 and was chosen arbitrarily. To find the continuum values, $x(T)$, we used a grid of PHOENIX models from $T_{\rm eff}$ = 2400 K to $T_{\rm eff}$ = 4000 K, all with log10 $g$ = 5 and [Fe/H] = 0 (solar). We fit a cubic spline function to the average flux values in the 3936.9–3939.9 Å region to determine the correction factor $x(T)$/$x({\rm 3680K})$ (Table 5, Figure 1). The uncertainty in $W_{\rm\lambda,corr}$ introduced by the correction factor has two roughly equivalent sources: the uncertainty in $T_{\rm eff}$ and the assumption of log10 $g$ = 5 and [Fe/H] = 0 for all our stars. Uncertainties in $T_{\rm eff}$ are likely 100–200 K, and this translates to a $\sim$20% uncertainty in the correction factor at the high temperature end ($T_{\rm eff}$ = 3700 K), to $\sim$70% uncertainty around $T_{\rm eff}$ = 3000 K, and $\sim$100% uncertainty at the low temperature end ($T_{\rm eff}$ = 2400 K). The surface gravity of our M dwarf sample ranges from $\sim$4.75–5.0, and given the coarseness of the PHOENIX grids, log10 $g$ = 5 is a good assumption. The metallicity ranges from -0.5 to 0.5, but [Fe/H] = 0 is valid for most of the M dwarfs in our sample. Examining the average flux values in the 3936.9–3939.9 Å region for PHOENIX models sampling a range of surface gravity and metallicity values, we find that the dispersion in $x$($T$)/$x$(3680 K) values for a given temperature is of similar magnitude to the dispersion in $x$($T$)/$x$(3680 K) values (assuming log10 g = 5 and [Fe/H] = 0) between $\Delta T_{\rm eff}$ = 200 K bins (Figure 1).

We also examined the effect of PHOENIX model parameters on $W_{\rm\lambda,corr}$ during the model subtraction and the effect of interactively choosing the integration limits. Using a single Keck/HIRES spectrum and three PHOENIX model spectra, we measured $W_{\rm\lambda,corr}$ 30 times and compared to the “true” measurement for that spectrum and its photometric error bar (the true measurements are shown in Figure 2 and used to calculate the equivalent width values reported in Table 1.2). We find that the PHOENIX model chosen for subtraction has a negligible effect on the equivalent width measurements, but that the uncertainty introduced from differences in the interactively-chosen integration bounds is slightly larger than the photometric uncertainty. We do not correct the photometric error bars on the individual equivalent width measurements, and the error bars reported in Table 1.2 come from the dispersion in the many equivalent width measurements made for each target (described fully in Section 4.1).

4 UV–Ca II Relation

In this section we present scaling relations between Ca II K and far- and near-UV emission lines, as well as the total extreme-UV flux (100–912 Å). See Appendix B for a presentation of scaling relations between the far- and near-UV emission lines themselves.

4.1 Ca II K variability

10/15 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 lightcurves are shown in Figure 2 for the fifteen 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 four days and the corresponding change in the corrected Ca II K equivalent widths.

For each target, we computed a mean Ca II K equivalent width with an uncertainty equal to the standard deviation of the measurements (Table 1.2). 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 1.2 for the list of sources for our optical spectra).

Figure 4 shows the demographics of our sample (stellar rotation period $P_{\rm rot}$ and effective temperature $T_{\rm eff}$) compared to the corrected Ca II K equivalent widths and the observed scatter in those values. In general, the fast rotators ($P_{\rm rot}$ $\textless$ 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 ($\textless$ 1 Gyr) in the sample. Between $P_{\rm rot}$ $\sim$ 40–85 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 H$\alpha$ 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 (H$\alpha$ absorption spectrum) in our sample exhibit UV activity.

The right panel of Figure 4 shows the scatter observed in the Ca II K $W_{\rm\lambda,corr}$ lightcurves normalized to $W_{\rm\lambda,corr}$. The scatter ranges from $\sim$10%–25% (although Proxima Cen’s scatter is 43%) with no clear dependence on $T_{\rm eff}$ and a possible positive correlation with $P_{\rm rot}$, although our sample is sparse for $T_{\rm eff}$ $\textless$ 3200 K and $P_{\rm rot}$ $\textgreater$ 60 days.

4.2 UV–optical correlations

For our 15 M dwarf sample, we compare the average Ca II K corrected equivalent widths ($W_{\rm\lambda,corr}$) reported in Table 1.2 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 6.

Figure 5 shows the correlations between $W_{\rm\lambda,corr}$ and surface flux ($F_{\rm S,UV}$) for nine UV emission lines and the extreme-UV flux, which is derived from Ly$\alpha$ (see Section 4.2.1), with their power law fits of the form log10 $F_{\rm S,UV}$= ($\alpha$ $\times$ log10 $W_{\rm\lambda,corr}$) + $\beta$ (Table 6). We find statistically significant correlations between Ca II K and all nine UV emission lines, which have formation temperatures ranging from 30,000 K to 160,000 K. As expected, the correlations with the least scatter are for Mg II and Ly$\alpha$ (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 Ly$\alpha$ (Linsky et al., 2013) and Mg II (Walkowicz & Hawley, 2009). Note that due to the large uncertainty in GJ 1214’s Ly$\alpha$ 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 Ly$\alpha$. 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 Ly$\alpha$, 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.

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 $W_{\rm\lambda,corr}$ $\approx$ 1, the transition in this dataset between the “inactive” ($W_{\rm\lambda,corr}$ $\lesssim$ 1) and “active” ($W_{\rm\lambda,corr}$ $\gtrsim$ 1) M dwarfs. In the Ly$\alpha$ and extreme-UV 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 6). Greater study of the low-activity M dwarfs is necessary to determine if 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 Extreme-UV Spectrum from Ca II K

Predicting an M dwarf’s Ly$\alpha$ flux is important not only because Ly$\alpha$ constitutes a major fraction of the far-UV flux, but it is also a means for estimating the extreme-UV spectrum, which currently cannot be observed for any star except the Sun. In Figure 5 and Table 6, we have determined the scaling relation between the total extreme-UV flux (100–912 Å) and $W_{\rm\lambda,corr}$ for our stars. The fit is very similar to the Ly$\alpha$–Ca II K best-fit line, because the extreme-UV fluxes were derived from scaling relations with Ly$\alpha$ from Linsky et al. (2014).

In Table 7, we substitute our Ca II–Ly$\alpha$ scaling relation into the Ly$\alpha$–EUV scaling relations in Table 5 of Linsky et al. (2014) to allow the reader to directly reconstruct the extreme-UV spectrum from a Ca II K $W_{\rm\lambda,corr}$ measurement. We have also reprinted the scaling relations from Linsky et al. (2014) in Table 7, although they have been simplified for brevity.

4.2.2 Estimating the Uncertainties on Derived UV Surface Fluxes

Here we estimate the uncertainties in the UV fluxes estimated from Ca II K using the presented scaling relations (Table 6 and 7). Important sources of error include the non-simultaneity of our UV and optical observations, the uncertainties in the Ca II K and UV measurements, the uncertainties in the stellar radii and distances (although the uncertainties in distances for these nearby stars are small) that are used to calculate surface flux, and the uncertainties in the stellar effective temperatures, surface gravities, and metallicities used to calculate the equivalent width correction factors. The error bars on the $W_{\rm\lambda,corr}$ values come from the dispersion in the many equivalent width measurements made for each target (with the exception of GJ 1132, which has only one measurement), and these error bars range from $\sim$10%–25%, although Proxima Cen’s $W_{\rm\lambda,corr}$ error bar is close to 45%.

Of the nine UV emission lines, Ly$\alpha$ and Mg II have the least scatter about the best-fit line (0.18 and 0.27 dex, respectively). This is important, because these two emission lines comprise the majority of the far-UV and near-UV emission line flux, respectively, for the M dwarfs. Considering $F$(emission line)/$\sum F$(nine emission lines) for the nine stars with measurements for all nine emission lines, $F$(Ly$\alpha$)/$\sum F$(nine emission lines) = 65%–91% and $F$(Mg II)/$\sum F$(nine emission lines) = 6%–27%. The other emission lines comprise smaller percentages of the total emission line flux: $\sim$0.1%–5% each. For example, the scatter in the Si III–Ca II K scaling relation is the largest ($\sigma$ = 0.59 dex), but $F$(Si III)/$\sum F$(nine emission lines) = 0.09%–1.7%.

Due to the limited number of extreme-UV observations of M dwarfs, quantifying the true uncertainty in the calculated extreme-UV spectrum is challenging. Here we estimate the uncertainty and list all the main sources of error. Linsky et al. (2014) used Extreme UltraViolet Explorer (EUVE) observations (100–400 Å) of six M dwarfs with HST/STIS Ly$\alpha$ observations, including AU Mic, Proxima Cen, AD Leo, EV Lac, and YZ CMi. The 400–912 Å spectra were provided by semi-empirical models (Fontenla et al., 2014). Scaling relations for eight $\sim$100 Å bandpasses in the extreme-UV were derived, with dispersions of 13–24%. Linsky et al. (2014) describe three sources of uncertainty in their technique: errors in the extreme-UV fluxes, errors in the reconstructed Ly$\alpha$ fluxes, and errors associated with stellar variability (the extreme-UV and Ly$\alpha$ observations were not simultaneous). The observed dispersion (13–24%) in the scaling relations is surprisingly small given the expected magnitudes of the three uncertainty sources. Linsky et al. (2014) attribute this to the avoidance of EUVE observations containing flares.

The scatter in our own Ca II K–Ly$\alpha$ relationship is surprisingly small ($\sigma$ = 0.18 dex) given the uncertainties listed in the first paragraph of this subsection. The $W_{\rm\lambda,corr}$ error bars range from $\sim$10%–30%, and Youngblood et al. (2016) find that the uncertainties in the reconstructed Ly$\alpha$ fluxes range from $\sim$5%–20% for moderate-to-high S/N observations. GJ 1214 was the lowest S/N observation and had a $\sim$100% uncertainty in the reconstructed flux, but was not included in the Ca II K–Ly$\alpha$ fit.

Starting with a $W_{\rm\lambda,corr}$ measurement with an assumed 30% uncertainty and using the Ca II K–Ly$\alpha$ scaling relation from Table 6, we find that the propagated uncertainty in the calculated Ly$\alpha$ flux is unchanged, indicating that the uncertainty in the Ca II K equivalent width dominates. Using the calculated Ly$\alpha$ flux to estimate the extreme-UV spectrum and adding the 30% uncertainty in quadrature with the 24% dispersion from Linsky et al. (2014) yields an uncertainty of $\sim$40% for the resulting extreme-UV flux.

We have assumed no additional uncertainty due to the variations in metallicity of our target stars. Metallicity variations likely have the largest impact on the H i Ly$\alpha$–Ca II relations, but it appears that this effect is negligible compared to other sources of scatter for our near-solar metallicity (-0.5 $\leq$ [Fe/H] $\leq$ 0.5) target stars. The metallicity effect could become significant for metal-poor stars ([Fe/H] $\textless$ -1), where the relative abundance of Ca with respect to H is approximately an order of magnitude less than for solar-metallicity stars, and we caution against applying these correlations to stars with any physical parameters beyond the bounds of our 15 star sample. The addition of metal-poor M dwarfs into the sample would allow for a determination of the effect of metallicity on these UV–optical correlations.

5 Energetic proton estimation from UV flares

HST has observed dozens of spectrally and temporally resolved far-UV flares from M dwarfs (Loyd & France 2014; France et al. 2016; Loyd et al. 2017, in preparation), which can be used to constrain the time-dependent energy input into the upper atmospheres of orbiting exoplanets. Energetic particles from stellar eruptive events are not frequently included in these energy budgets, because there are no observational constraints for stars other than the Sun. Existing solar correlations between SXRs and protons detected near Earth (i.e. Belov et al. 2007; Cliver et al. 2012) cannot be directly applied to HST’s UV flares, because we do not know the energy partition between stellar UV emission lines and SXRs during flares. Thus we have developed a new scaling relation between energetic protons detected near Earth and UV flares from the Sun.

Ideally, we would determine the relationship between energetic protons detected by the GOES satellites and far-UV spectra of the Sun, because this would be directly comparable to the flares detected by HST. However, there are no disk-integrated, high-cadence solar observations of UV emission lines within HST’s STIS or COS nominal far-UV spectral ranges (1150–1700 Å). We elected to use an extreme-UV emission line from high-cadence solar irradiance measurements as a proxy for far-UV emission lines. SDO/EVE measured the solar spectral irradiance of the Sun from 50 – 370 Å in the MEGS-A channel at 10 second cadence from 2010–2014. The only high-S/N ion with a formation temperature similar to those accessible by HST STIS/COS in the far-UV is He II at 304 Å (log $T_{\rm formation}$ = 4.9; CHIANTI; Dere et al. 1997; Landi et al. 2013). The far-UV flare tracers Si IV (1393, 1402 Å; log $T_{\rm formation}$ = 4.9) and He II (1640 Å; log $T_{\rm formation}$ = 4.9) observed by HST have similar formation temperatures to He II $\lambda$304. After determining the relationship between $\textgreater$10 MeV proton flux and He II $\lambda$304, we use the semi-empirical model of GJ 832 (M1.5 V; Fontenla et al. 2016) to scale He II $\lambda$304 flux to Si IV and He II $\lambda$1640 flux. In Section 5.4, we estimate the proton flux from a Si IV flare observed with HST from the M4 dwarf GJ 876, and in Section 5.5, we discuss the limitations of applying these proton scaling relationships to flares on M dwarfs.

5.1 The solar flare and proton enhancement sample

We identified 36 proton enhancements in the GOES $\textgreater$10 MeV and $\textgreater$30 MeV proton channels with an associated SXR (1–8 Å with GOES) and He II $\lambda$304 flare. Our confidence levels for the proton–flare association varied, and we assigned each event a quality index ($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 $\textgreater$10 MeV protons after the photon flares is much more gradual than most events, and the $\textgreater$30 MeV proton flux shows no significant rise. The He II $\lambda$304 lightcurve also shows many events, which may be confused, and estimation of the background level is challenging.

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 lightcurve intersected the background level fit. To estimate the uncertainties in our measurements, we re-measured four 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%–300% for SXR, 10%–200% for He II $\lambda$304, and 10%–30% for the protons. These ranges are a reflection of the varying quality factors: $q$ $\textless$ 3 events have the largest uncertainties and $q$ $\geq$ 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 $\lambda$304, indicating background subtraction and not flare duration is the dominant uncertainty.

The 36 He II $\lambda$304 flares have mean durations 5.3 $\pm$ 4.6 hours and on average peak a few minutes after the SXR flare peak, although there is large scatter in this average. When examining only the $q$ $\geq$ 3 events, the average He II $\lambda$304 peak occurs $\sim$5 minutes before the SXR peak. Kennedy et al. (2013) used He II $\lambda$304 to trace the impulsive phase flares, and found He II $\lambda$304 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 $\lambda$304 peaked 18 s before the GOES SXRs. The associated proton enhancements begin within about 2 hours of the SXR peak, and on average last for 3 days. Part of the scatter in the He II $\lambda$304–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 $\textgreater$10 MeV enhancements have many peaks over the course of several days, while the $\textgreater$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 $\textgreater$10 MeV peak particle flux typically occurs after the $\textgreater$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 lower-energy protons are either accelerated at a later time (farther from the solar surface) or their escape from the Sun is delayed due to trapping by shocks (Krucker & Lin, 2000; Xie et al., 2016).

In their sample of 58 SXR and proton enhancement events, Cliver et al. (2012) found a statistically significant correlation ($\rho$ = 0.52, $n$ = 3$\times$10*-5*), but our 36 event sample yields $\rho$ = 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: $\rho$ = 0.41, $n$ = 0.003 for 50 of Cliver et al. (2012) events, and $\rho$ = 0.35, $n$ = 0.05 for 33 of our events. The different $n$ values are due to the different sample sizes.

5.2 Solar UV–proton scaling relations

We observe statistically significant correlations between He II $\lambda$304 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 $x$ = $\alpha$ $\times$ log10 $y$ + $\beta$), and the parameters for the fits are presented in Table 8. The scatter in these fits is large ($\sigma$ = 0.75–0.84 dex).

The near identical formation temperatures of He II $\lambda$304, Si IV, and He II $\lambda$1640 suggests 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, $F_{\rm SiIV}$/$F_{\rm HeII,304}$ = 0.117, and $F_{\rm HeII,1640}$/$F_{\rm HeII,304}$ = 0.0304. For GJ 832, $F_{\rm SiIV}$/$F_{\rm HeII,304}$ = 0.137, and $F_{\rm HeII,1640}$/$F_{\rm HeII,304}$ = 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 $\lambda$304 relations (Table 8) to relate Si IV and He II $\lambda$1640 flare flux to proton enhancements. Using one of the correlations listed in Table 8, and replacing $F_{\rm HeII,304}$ with $F_{\rm SiIV}$, we find that:

[TABLE]

where $F_{\rm SiIV}$ is the background-subtracted Si IV (1393, 1402 Å) flare fluence (J m*-2*) as would be observed at 1 AU from the star, and $I_{\rm\textgreater 10MeV}$ is the background-subtracted peak $\textgreater$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*>10MeV*, [pfu $\cdot$ s]) rather than peak proton flux is used, the relationship becomes

[TABLE]

Similarly, we can derive $I_{\rm\geq 10MeV}$ and $F_{\rm\geq 10MeV}$ from $F_{\rm HeII,1640}$:

[TABLE]

and

[TABLE]

5.3 UV-based GOES flare classification

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 $\lambda$304 intensity (W m*-2*; Figure 9):

[TABLE]

The Pearson correlation coefficient is 0.58 with $n$ = 2.2 $\times$ 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 ($\textless$X class), but less so for large flares due to the $\sim$100% chance of associated proton enhancement (Table 1). Using our conversion ratio ($F_{\rm SiIV}$/$F_{\rm HeII,304}$ = 0.137) and assuming $F_{\rm SiIV}$/$F_{\rm HeII,304}$ $\sim$ $I_{\rm SiIV}$/$I_{\rm HeII,304}$, we can estimate the peak SXR flux from the peak Si IV and He II $\lambda$1640 flux:

[TABLE]

and

[TABLE]

Approximately 20% of C-class flares ($F_{\rm SXR}$ = 10*-6*–10*-5* W m*-2* at 1 AU) and $\sim$100% of X3 or greater class flares ($F_{\rm SXR}$ $\geq$3 $\times$ 10*-4* W m*-2* at 1 AU) have associated CMEs (Yashiro et al. 2006; Table 1). Recall that $F_{\rm SXR}$ 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 ($\geq$10*-4* W m*-2* at 1 AU) corresponds to any Si IV flare with a background-subtracted peak intensity value $\geq$1.6 $\times$ 10*-5* W m*-2* = 1.6$\times$ 10*-2* erg cm*-2* s*-1* at 1 AU, and to any He II $\lambda$1640 flare with a background-subtracted peak intensity value $\geq$3.2 $\times$ 10*-6* W m*-2* = 3.2 $\times$ 10*-3* erg cm*-2* s*-1* at 1 AU.

5.4 Application to observed flares from GJ 876

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; 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 $\textgreater$10 (Figure 10a), and on 2015 July 7, factor of $\sim$40–100 increases were observed in the far-UV lines, including Si IV (Figure 10b), and more modest increases (factor of $\sim$5) were observed in the far-UV continuum (France et al., 2016). He II $\lambda$1640 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 $\lambda$1640 on 2015 July 8 (Figure 10c), although the C IV lightcurve indicates GJ 876 did flare during this observation. In this section and Table 5.4, we characterize the magnitude of these flares with the GOES flare classification scheme and estimate the probability and magnitude of an associated particle enhancement received in GJ 876’s HZ ($\langle r_{\rm HZ}\rangle$ = 0.18 AU; Kopparapu et al. 2014). See Section 5.5 for a discussion on the limitations of these results.

Table 1 provides comparison points for solar flares, and in Table 5.4 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 a X45 ($\pm$5) class flare (Cliver & Dietrich, 2013). The November 4 solar flare of the Halloween storms in 2003, probably the largest flare observed during the space age, is estimated to be X30.6-class (Kiplinger & Garcia 2004; note that the GOES detectors saturate for events between X10–X20). For the great AD Leo flare of 1985 (Hawley & Pettersen, 1991), Segura et al. (2010) estimated that a HZ planet ($\langle r_{\rm HZ}\rangle$ = 0.16 AU) would have received peak $\textgreater$10 MeV proton and SXR fluxes of 5.9$\times$108 pfu and 9 W m*-2*, respectively. The peak SXR flux at 1 AU (0.23 W m*-2*) would make this an X2300-class flare. Osten et al. (2016) found that the superflare observed from the young M dwarf binary system DG CVn was equivalent to an X600,000-class flare (60 W m*-2* at 1 AU, or 6000 W m*-2* at 0.1 AU, the approximate HZ distance). We note, however, that there is evidence that SXR–proton scaling relations should break down for such large events ($\textgreater$X10-class; Hudson 2007; Drake et al. 2013).

We measure the peak flux for the large GJ 876 SXR flare observed at $\sim$45 ks in Figure 10a to be (9.72$\pm$1.28)$\times$10*-13* erg cm*-2* s*-1* in the 0.3–10 keV bandpass (1.25–41 Å). 11$\pm$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 $\sim$80 pfu at 1 AU (Cliver et al., 2012). The SXR and proton fluxes received in GJ 876’s HZ will be $\sim$30$\times$ larger: 2.8$\times$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 five years.

The smaller GJ 876 SXR flare observed at $\sim$7 ks in Figure 10a is estimated to be equivalent to a 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 $\sim$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 a year.

GJ 876’s 1031 erg UV flare ($\Delta\lambda$ = 400–1700 Å) observed on 2015 July 7 with HST (France et al., 2016) emitted 1.2$\times$1029 erg in the Si IV emission line over $\sim$25 minutes (Figure 10b). Using Equation 3 and the fluence at 1 AU, we find that the peak proton flux received at 1 AU during the flare was $\sim$75 pfu, or $\sim$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 $\sim$3 (X13–X114). Note that the peak proton flux calculated for this 2015 July 7 Si IV flare ($\sim$2300 pfu at 0.18 AU) is similar to the peak proton flux calculated for the 2015 June 5 SXR flare ($\sim$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–X class flares in 8.25 hours, and HST observed six comparable flares (within an order of magnitude) in 12.35 hours, including observations from the MUSCLES pilot survey (France et al., 2013). Thus, we estimate GJ 876 emits flares of this magnitude $\sim$0.4–0.5 hr*-1*. From Veronig et al. (2002), the Sun’s rate of M-class flares is $\sim$0.02 hr*-1*, a factor of $\sim$20 less frequent than GJ 876. However, note that these M-class flares are effectively 30$\times$ stronger (i.e. X10-class) in GJ 876’s HZ at 0.18 AU. The Sun emits X10-class flares approximately once every five years, so planets in the GJ 876 HZ are receiving SXR flare fluxes $\geq$10*-3* W m*-2* associated with proton fluxes $\sim$103 pfu about four orders of magnitude more frequently than the 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; Tilley et al. 2017, in preparation). For comparison to the M–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 5.4). 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 $\sim$X1-class flares with $\sim$4 minute durations. Peak proton fluxes were assigned as 1.2$\times$103 pfu (a typical value for the flares discussed in this section), and we provide two cases distinguished by their flare frequency: 0.08 hr*-1* (Case A) and 0.5 hr*-1* (Case B; similar to GJ 876’s observed flare frequency). Both the Case A and Case B flares turn off after a period of 40 months, but via extrapolation, we show that for Case A, the O3 column approaches near-complete depletion at approximately 1013 s (318 kyr) of similar, constant stellar activity. Case B shows near-complete O3 depletion after only approximately 5$\times$109 s (160 yr). Given the several Gyr period of high activity during the evolution of early-to-mid M dwarfs, both scenarios indicate massive, and likely complete, O3 depletion. This suggests 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 Tilley et al. (2017), in preparation.

5.5 Limitations of the method

Our new method of proton flux estimation due to stellar flares has important limitations, but 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; Harra et al. 2016; Crosley et al. 2016) or in our understanding of particle acceleration under kG 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 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 October 2014 when the large active region 2192 emitted many X-class flares, which have a $\textgreater$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 (Thalmann et al., 2015; Sun 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 and references within Hudson 2007). It is unclear if only the expected proton flux flattens out for increasingly large SXR flares ($\textgreater$X10), or if the frequency of $\textgreater$X10-class flares also break 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 $\sim$10% of the total stellar luminosity, a likely unreasonable fraction. Those authors conclude 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$$\textgreater80,000 K) emits short-wavelength photons that we observe as a flare, but particles are accelerated by a non-thermal 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 $\lambda$304 flare flux from far-UV Si IV or He II $\lambda$1640 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 $\lambda$304 as a proxy because of its similar formation temperature to Si IV $\lambda\lambda$1393,1402 (and He II $\lambda$1640), but we note that the electrons responsible for the collisional excitation of He II $\lambda$304 (40.8 eV above ground) must have significantly higher energies than the thermal electrons at $T_{\rm form}$ = 80,000 K (k_{B}$$T_{\rm 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 $\pi R_{\rm p}^{2}$ due to the presence of a magnetosphere, or 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.

6 Summary

In this paper, we have developed methods for estimating the high-energy radiation and particle environments of M dwarfs for use when direct observations are unavailable. We have empirically determined scaling relations that can be used to estimate the UV spectra of M dwarfs from optical spectra and the energetic particle flux from UV flares. The main results of this work are summarized as follows.

Time-averaged Ca II K (3933 Å) residual equivalent width correlates positively with stellar surface flux of nine far- and near-UV emission lines, including H i Ly$\alpha$ (Table 6). The presented Ca II K and UV scaling relations allow for the UV spectrum of any M dwarf to be approximated from ground-based optical spectra. 2. 2.

We present a scaling relation between Ca II K equivalent width and the extreme-UV spectrum for M dwarfs based on the Ly$\alpha$–extreme-UV scaling relations presented in Linsky et al. (2014) and the Ca II K–Ly$\alpha$ scaling relation determined in this work (Table 7). We estimate that the Ca II-based extreme-UV fluxes are accurate within 40%. 3. 3.

We present a new method to estimate the energetic ($\textgreater$10 MeV) proton flux emitted during far-UV emission line flares (specifically Si IV $\lambda\lambda$1394,1402 and He II $\lambda$1640; Table 8). The UV–proton scaling relations are derived from solar irradiance observations and in situ proton measurements near Earth. 4. 4.

We present methods to estimate the GOES flare classification corresponding to far-UV Si IV and He II flares (Equations 8 and 9). This is important for estimating the probability that any far-UV flare will have an accompanying proton enhancement. Larger solar flares have a higher probability of associated particle enhancements. 5. 5.

We analyze several flares observed with Chandra and Hubble from the M4 dwarf GJ 876 as part of the MUSCLES Treasury Survey, and place the flare properties in context with solar flares (Section 5.4). We find that planets in GJ 876’s HZ experience large, Carrington-like flares (soft X-ray flux $\textgreater$10*-3* W m*-2*) and particle enhancements approximately four orders of magnitude more frequently than Earth. Flare activity at the level observed on GJ 876 is predicted to lead to complete stripping of O3 from an Earth-like planet on timescales between 102 and 105 years.

M dwarfs are currently a prime target for exoplanet searches and characterization efforts with current and near-future technologies including JWST, but important questions have been raised about these planets’ potential habitability. JWST will be able to characterize only a few M dwarf exoplanets, and we need to ensure that HZ terrestrial planets orbiting nearby M dwarfs with the maximum probability of hosting a detectable atmosphere are chosen for JWST target selection. The stellar host’s UV SED and energetic particle flux are important drivers of chemistry, heating, and mass loss, but we do not have the space telescope resources for reconnaissance of every nearby M dwarf system, and energetic particles cannot be measured directly across interstellar distances. In this paper, we have provided tools to estimate UV SEDs from comparatively easy-to-obtain Ca II measurements and energetic particle fluxes from observed UV and X-ray flares.

The data presented here were obtained as part of the HST Guest Observing programs #12464 and #13650 as well as the COS Science Team Guaranteed Time programs #12034 and #12035. This work was supported by NASA grants HST-GO-12464.01 and HST-GO- 13650.01 to the University of Colorado at Boulder. This work is also based on data from the Chandra X-ray Observatory (ObsIDs: 17315 and 17316) and supported by CXO grant G05-16155X. Observations were also obtained with the Apache Point Observatory 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium. This research has made extensive use of the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. The ESO Science Archive Facility was also used to access VLT/XSHOOTER science products. This paper includes data gathered with the 6.5-m Magellan Telescopes located at Las Campanas Observatory, Chile. AY thanks Don Woodraska for assistance with the GOES and SDO/EVE data, Juan Fontenla for providing the GJ 832 synthetic spectrum, and Lauren W. Blum for helpful discussions. ERN acknowledges support from an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1602597, and AS acknowledges the support of UNAM-PAPIIT project IN109015.

Appendix A Equivalent Width Correction

The following exemplifies how a spectral type dependence is introduced in Ca II K equivalent width measurements, and how it can be removed. See Figure 12 for examples on how the PHOENIX models used in Section 3 change as a function of spectral type.

Two stars, A and B, have different spectral types ($T_{\rm A}$ $\textgreater$ $T_{\rm B}$ and $R_{\rm A}$ $\textgreater$ $R_{\rm B}$) and are observed at different distances, $d_{\rm A}$ and $d_{\rm B}$, from Earth. These stars have the same Ca II emission core surface flux (erg cm*-2* s*-1*), $F_{\rm A,S}$ = $F_{\rm B,S}$, and therefore should have the same Ca II K equivalent widths, $W_{\rm\lambda}$. From flux-calibrated Ca II spectra, we perfectly isolate the emission core from the observed line profile by fitting and subtracting a radiative equilibrium model, and we integrate over the residual Ca II K emission, finding $F_{\rm A,S}$ $\times$ $\frac{R_{\rm A}^{2}}{d_{\rm A}^{2}}$ and $F_{\rm B,S}$ $\times$ $\frac{R_{\rm B}^{2}}{d_{\rm B}^{2}}$ for Star A and Star B, respectively.

To convert the observed fluxes into equivalent widths, we normalize to nearby continuum, which has an average surface flux $x$($T$). Because Star A is warmer, $x$($T_{\rm A}$) $\textgreater$ $x$($T_{\rm B}$), and the resulting equivalent widths are not equal:

[TABLE]

and

[TABLE]

To correct the equivalent widths so that they are equal for Star A and Star B, we multiply by $x$($T$) scaled to a reference value $x$($T_{\rm ref}$). The “corrected” equivalent width is given by:

[TABLE]

Figure 12 shows radiative equilibrium model spectra (PHOENIX; Husser et al. 2013) for 3 spectral subtypes from our sample – 3100 K, 3300 K, and 3500 K. The cooler spectral subtypes have narrower Ca II absorption, as well as fainter continuum.

Appendix B UV–UV correlations

Here we present scaling relations between the nine far- and near-UV emission lines presented in Tables 3, 4, and 6, and between these UV emission lines and the stellar rotation periods presented in Table 1.2. See Section 2.2 for a discussion of the UV spectra, including Ly$\alpha$ reconstructions and other corrections made for ISM absorption. All observed fluxes have been converted to surface fluxes using the stellar radii and distances presented in Table 1.2.

In Figures 13–16, we show the positive correlations between UV surface fluxes ($F_{\rm S,UV1}$, $F_{\rm S,UV2}$) with their power law fits of the form log10 $F_{\rm S,UV1}$ = m $\times$ log10 $F_{\rm S,UV2}$ + b. We find statistically significant correlations for all 36 combinations of the nine UV emission lines. The Pearson correlation coefficients range from 0.80–0.99, and the scatter about the best-fit lines ranges from 0.11–0.53 dex.

In Figure 17, we present the negative correlations between stellar rotation period ($P_{\rm rot}$) and the nine UV emission lines with their power law fits of the form log10 $F_{\rm S,UV1}$ = m $\times$ log10 $P_{\rm rot}$ + b. All of the correlations are statistically significant, although our M dwarf sample is divided between two widely-separated populations: $P_{\rm rot}$ $\textless$ 5 days and $P_{\rm rot}$ $\textgreater$ 39 days. Within the slowly-rotating population or the fast-rotating population, most of the correlations are not statistically significant.

Youngblood et al. (2016) presented scaling relations between Ly$\alpha$ and Mg II, Si III, C IV, and $P_{\rm rot}$ for a smaller sample of M dwarfs. The shallow correlations (m $\textless$ 0.15) found between Ly$\alpha$ and Si III, C IV were not statistically significant and were consistent with zero, likely due to the small sample size and the small range of explored parameter space. This work’s power law exponents between Ly$\alpha$ and Si III, C IV are m = 0.60$\pm$0.09 and m = 0.63$\pm$0.08, respectively, compared to the m = 0.07$\pm$0.31 and m = 0.13$\pm$0.35 values found by Youngblood et al. (2016). For Ly$\alpha$ and Mg II, our power law exponent m = 0.73$\pm$0.08 is consistent with the m = 0.77$\pm$0.10 value from Youngblood et al. (2016), as is our exponent for the Ly$\alpha$ and $P_{\rm rot}$ relation: m = -0.92$\pm$0.22 compared to m = -0.86$\pm$0.16.