The Far Ultraviolet M-dwarf Evolution Survey. I. The Rotational Evolution of High-energy Emissions^*

To measure the NIR spectra of our sample targets, we used the TripleSpec (TSPEC) instrument (Wilson et al. 2004) on the ARC 3.5 m telescope at the Apache Point Observatory, as well as the similarly designed ARCoIRIS instrument ⁴ on the Blanco 4 m telescope at NOAO's Cerro Tololo Inter-American Observatory. A summary of these observations can be found in Table 1. Both instruments, which each have a 11 slit, provide R ∼ 3500 spectra continuously across NIR wavelengths in 5–6 echelle orders, with TSPEC spanning 0.95–2.46 μm and the updated ARCoIRIS design covering 0.80–2.47 μm.

Table 1. IR Data Observing Log

Name	UT Date	Instrument	Airmass	A0 Calibrator	Seeing	Exp. Time (s)	S/N a	Weather
GJ 4334	2017-11-22	TSPEC	1.08	HD 240290	2''	20	160	Windy, Clear
GJ 49	2017-11-22	TSPEC	1.15	HD 5031	2''	8	380	Windy, Clear
HIP 112312	2017-11-05	ARCoIRIS	1.17	HD 213044	18	8	280	Clear
LP 247-13	2017-11-22	TSPEC	1.02	HD 21038	2''	20	275	Windy, Clear
HIP 17695	2017-11-05	ARCoIRIS	1.24	HD 24003	18	8	360	Partly Cloudy
HIP 23309	2017-11-06	ARCoIRIS	1.38	HD 32507	13	10	500	Clear
CD-35 2722	2017-11-06	ARCoIRIS	1.38	HD 42681	13	8	390	Partly Cloudy
GJ 410	2019-04-30	TSPEC	1.04	HD101060	12	20–25	170	Cloudy
LP 55-41	2017-11-22	TSPEC	1.28	HD 32781	2''	25	120	Windy, Clear
G 249-11	2017-11-22	TSPEC	1.3	HD 32781	2''	25	180	Windy, Clear

Note.

^a This column reflects the typical signal-to-noise ratio in the H band of each spectrum.

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For all of our data, we took the same observing approach, using an ABBA slit nod sequence with short exposures (<30 s), mitigating sky emission-line variability, to provide clean sky subtraction from subsequent frames, and remaining on target over several tens of minutes in order to obtain a high signal-to-noise ratio (S/N) observation for each target (see Table 1). We also observed a nearby A0 star close in time and at a similar airmass, to provide a reference for flux and telluric calibration (Vacca et al. 2003).

We reduced the data using modified versions of Spextool (Cushing et al. 2004), one for TSPEC and a separate one for ARCoIRIS. ⁵ We summarize the data reduction procedure as follows. We first created the master flatfield for each observing night by median combining several dome lamp exposures and subtracting off the median thermal contribution from dome exposures taken with the lamps off. The thermal contribution is most significant in the K band. The wavelength calibration for each target was then determined from the median sky spectrum, created from each target's science observations. ⁶ Initial sky subtraction was performed by differencing AB nod pairs, from which we determined the object trace in each order. We extracted the spectrum in windows centered along the trace, applying the normalized flatfield and wavelength calibration. When variable cloud cover was evident, we also applied additional sky subtraction, removing a linear fit to the residual background. The spectra from individual frames were averaged together in order to increase the S/N, and then we used the similarly extracted A0 calibrator spectra to correct for telluric absorption and provide a flux calibration using xtellcorr (Vacca et al. 2003). We then merged the different echelle orders, averaging the spectra in overlapping wavelength regions to create the final spectrum of each target. The IR spectra of the FUMES sample are shown in Figure 1.

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We determine the NIR spectral type classifications for the FUMES sample using the composite spectral standards compiled from multiple stars by Newton et al. (2014). ⁷ These NIR spectra, taken with the NASA Infrared Telescope Facility (IRTF)/SpeX, are classified on the KHM system originally defined by Kirkpatrick et al. (1991, 1995, 1999) at red optical wavelengths. We first computed the H₂O–K2 index defined by Rojas-Ayala et al. (2012) as a K-band feature sensitive to spectral type, and used the calibration from Newton et al. (2014) (their Equation (2)) to convert this measurement to an initial type estimate. ⁸ We then used this classification to select a range of nearby spectral standards, spanning one type earlier and later, to compare against our observations in order to classify the spectra by eye using features across the entire infrared spectrum. This holistic approach helped mitigate potential feature mismatches introduced by metallicity differences between the targets and the standards by not relying on any single features in the spectra. We determined a single best type for each of the YJHK bands and took the median as our best classification.

Our NIR classifications for the FUMES sample are shown in Table 2, and we estimate spectral type uncertainties of half a subtype. We also include the literature optical spectral types for comparison, typically from the Palomar/Michigan State University (PMSU) survey (Reid et al. 1995; Hawley et al. 1996) for the field objects, but from additional sources for the younger stars. As discussed in Rojas-Ayala et al. (2012) and Newton et al. (2014), the literature optical spectral types are often dependent on metallicity across the M1–M4 range, and we consider the NIR classifications to be the more consistent metrics. We similarly report the NIR/optical spectral types for the literature sample in Table 2. These classifications will be important in Section 5.

Table 2. UV Sample ^a

Name	SpT	Lbol	Mass	Radius	Teff	Rot. Period b	References c
	Opt/NIR	(1031 erg s−1)	(M⊙)	(R⊙)	(K)	(days)
G 249-11	M4 d /M4	2.59 ± 0.07	0.237 ± 0.006	0.255 ± 0.010	$3277{\pm }_{68}^{71}$	52.76	14, 14, 12
HIP 112312 e	M4IV/M4.5	16.6 ± 0.5	$0.247{\pm }_{0.015}^{0.018}$	$0.688{\pm }_{0.016}^{0.015}$	$3173{\pm }_{22}^{25}$	2.355 ± 0.005	19, 14, 9
GJ 4334	M4.5/M5	3.68 ± 0.09	0.294 ± 0.007	0.307 ± 0.012	$3260{\pm }_{66}^{70}$	23.54	15, 14, 12
LP 55-41	M3.5 d /M3	8.10 ± 0.22	0.41 ± 0.01	0.416 ± 0.017	$3412{\pm }_{73}^{75}$	53.44	14, 14, 12
HIP 17695 f	M3/M4	11.5 ± 0.2	$0.435{\pm }_{0.014}^{0.011}$	$0.502{\pm }_{0.007}^{0.008}$	$3393{\pm }_{21}^{20}$	3.87 ± 0.01	1, 14, 9
LP 247-13	M2.7/M3.5	12.68 ± 0.35	0.495 ± 0.013	0.49 ± 0.02	$3511{\pm }_{76}^{79}$	1.289	16, 14, 5
GJ 49	M1.5/M1	18.7 ± 0.4	0.541 ± 0.015	0.534 ± 0.023	$3713{\pm }_{81}^{86}$	18.6 ± 0.3	15, 14, 4
GJ 410	M0/M0.5	21.3 ± 0.5	0.557 ± 0.015	0.549 ± 0.024	$3786{\pm }_{83}^{89}$	14.0	15, 14, 4
CD-35 2722 f , g	M1/M1	21.0 ± 0.3	0.572 ± 0.002	0.561 ± 0.003	$3727{\pm }_{6}^{8}$	1.717 ± 0.004	19, 14, 9
HIP 23309 e	M0/M0	$68.2{\pm }_{1.1}^{1.2}$	$0.785{\pm }_{0.010}^{0.009}$	0.932 ± 0.014	$3886{\pm }_{27}^{28}$	8.6 ± 0.07	19, 14, 9
Prox. Cen. h	M5.5/-	$0.5997{\pm }_{0.0075}^{0.0076}$	0.123 ± 0.003	0.147 ± 0.005	$2992{\pm }_{47}^{49}$	88.977	6, - , 13
GJ 1061	M5.5/-	0.628 ± 0.014	0.125 ± 0.003	0.152 ± 0.007	$2976{\pm }_{72}^{69}$	133	6, - , 2
GJ 1214	M4.5/M4	$1.343{\pm }_{0.034}^{0.036}$	0.181 ± 0.005	$0.204{\pm }_{0.0084}^{0.0085}$	$3111{\pm }_{66}^{69}$	125 ± 5	15, 11, 8
GJ 1132	M3.5/-	1.67 ± 0.05	0.194 ± 0.005	0.215 ± 0.009	$3196{\pm }_{69}^{72}$	129.15	6, - ,13
Gl 213	M4/M4	2.41 ± 0.04	0.218 ± 0.005	0.238 ± 0.009	$3334{\pm }_{63}^{67}$	170	15, 11, 2
GJ 628 h	M3.5/M3	4.21 ± 0.04	0.304 ± 0.007	0.319 ± 0.007	$3307{\pm }_{36}^{38}$	119.3 ± 0.5	15, 11, 18
Gl 581 h	M3/M2	4.56 ± 0.04	0.307 ± 0.007	0.310 ± 0.008	$3424{\pm }_{42}^{43}$	132.5 ± 6.3	15, 11, 17
YZ CMi	M4.5/M5	$4.35{\pm }_{0.14}^{0.15}$	0.316 ± 0.008	0.328 ± 0.013	$3293{\pm }_{71}^{74}$	2.7758 ± 0.0002	15, 11, 10
EV Lac	M3.5/M3	4.88 ± 0.15	0.320 ± 0.008	0.331 ± 0.013	$3370{\pm }_{70}^{75}$	4.3715 ± 0.0002	15, 11, 10
GJ 667C	M1.5/-	5.51 ± 0.13	0.327 ± 0.008	0.337 ± 0.014	$3443{\pm }_{71}^{75}$	103.9 ± 0.7	6, - , 17
GJ 876 h	M4/M3	5.01 ± 0.04	0.346 ± 0.007	0.372 ± 0.004	$3201{\pm }_{19}^{20}$	87.3 ± 5.7	15, 11, 17
Gl 821	M1/-	6.98 ± 0.09	0.355 ± 0.009	$0.363{\pm }_{0.014}^{0.015}$	$3512{\pm }_{69}^{73}$	107	15, - , 2
Gl 436 h	M2.5/M3	9.42 ± 0.11	0.425 ± 0.009	0.432 ± 0.011	$3477{\pm }_{44}^{46}$	44.09 ± 0.08	15, 11, 3
AD Leo	M3/M3	8.7 ± 0.1	0.426 ± 0.010	0.43 ± 0.02	$3425{\pm }_{68}^{69}$	2.2399 ± 0.0002	15, 11, 10
GJ 832	M1.5/-	10.6 ± 0.3	0.441 ± 0.011	0.442 ± 0.018	$3539{\pm }_{74}^{79}$	45.7 ± 9.3	6, - , 17
Gl 887 h	M0.5/	14.08 ± 0.22	0.48 ± 0.01	0.474 ± 0.008	$3672{\pm }_{34}^{36}$	33	6, - , 2
GJ 176 h	M2/M2	13.46 ± 0.12	0.485 ± 0.012	0.474 ± 0.015	$3632{\pm }_{56}^{58}$	39.3 ± 0.1	15, 11, 17
AU Mic e	M0/-	$37.7{\pm }_{0.7}^{0.8}$	$0.667{\pm }_{0.008}^{0.006}$	$0.798{\pm }_{0.013}^{0.014}$	$3619{\pm }_{25}^{23}$	4.86	6, - , 7

Notes.

^a Stellar properties are from J.S. Pineda et al. (2021, in preparation), quoting medians and the central 68% confidence interval, see Section 2.3. The horizontal division separates the new FUMES (top) targets from the literature stars (bottom). ^b Rotation periods are taken from the literature, typically from either photometric variations or long-term monitoring of periodic emission lines. Uncertainties are as reported in the literature if available. ^c References in order denote source for optical spectral type, infrared spectral type, and rotation period. ^d Optical spectral types were not available in the literature, so we used our NIR classification and Equation (3) from Newton et al. (2014) with the metallicity determined from their calibration of the equivalent widths of the Na doublet at 2.2 μm (their Equation (10)). ^e HIP 23309, HIP112312, and AU Mic are members of β Pic, which has a mean age of ∼24 Myr (Bell et al. 2015). ^f CD-35 2722 and HIP17695 are members of AB Dor, which has a mean age of ∼150 Myr (Bell et al. 2015). ^g CD-35 2722 is a binary with a substellar L4 companion with separation of ∼70 au (Wahhaj et al. 2011). ^h Star names listed with an asterisk have measured angular diameters from interferometric measurements (von Braun et al. 2011, 2014; Boyajian et al. 2012; Kane et al. 2017), incorporated in the radius estimates; see J.S. Pineda et al. (2021, in preparation).

References. (1) Alonso-Floriano et al. 2015; (2) Astudillo-Defru et al. 2017; (3) Bourrier et al. 2018; (4) Donati et al. 2008; (5) Hartman et al. 2011; (6) Hawley et al. 1996; (7) Küker et al. 2019; (8) Mallonn et al. 2018; (9) Messina et al. 2010; (10) Morin et al. 2008; (11) Newton et al. 2014; (12) Newton et al. 2016; (13) Newton et al. 2018; (14) This Work; (15) Reid et al. 1995; (16) Shkolnik et al. 2009; (17) Suárez Mascareño et al. 2015; (18) Suárez Mascareño et al. 2016; (19) Torres et al. 2006.

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A couple of objects deserve further attention. Since the Newton et al. (2014) standards do not include spectra for types earlier than M1, we also used the M0 and K7 standards within the IRTF spectral library (Rayner et al. 2009) when classifying GJ410 and HIP23309. The M1 star, CD-35 2722, has an L4 companion that is 5 mag fainter across the JHK bands (Wahhaj et al. 2011) at a separation of ∼70 au. This companion is unresolved in our spectroscopic observations, but the ∼1% contribution to the integrated NIR fluxes did not meaningfully distort the observed spectra. However, there was evidence for a slightly deeper 0.99 μm FeH feature than would be expected for an M1 dwarf, perhaps due to the strength of this feature in L-dwarf spectra (e.g., Kirkpatrick 2005). The young star HIP 112312 has an optical classification as a subgiant from Torres et al. (2006). Without subgiant standards with which to compare in the NIR, we cannot confirm this classification with our observations. For this star, we also compared the M-giant IRTF NIR standards (Rayner et al. 2009), finding that the HIP 112312 NIR spectrum largely agrees with the dwarf sequence except for deeper CO lines in the K band, which are very prominent in the M-giant spectra, reflecting the relatively low gravity of this young object. The other young FUMES stars (see Table 2) showed spectra consistent with the dwarf standards.

Our aim in this paper is to analyze the relation between FUV emissions and the physical properties of low-mass stars. To these ends, we desired self-consistent properties, mass, radius, bolometric luminosity, and effective temperature for each FUMES target, as well as any suitable additional targets found in the literature. Consistently determined properties are crucial to mitigate potential systematic effects introduced by relying on an assortment of eclectic literature determinations for these stellar properties. Although our IR spectra allowed us to utilize spectroscopic property calibrations (e.g., Mann et al. 2015; Newton et al. 2015; Terrien et al. 2015), such data were not uniformly available for both the FUMES and literature targets, and thus we rely instead on the largely photometric results from our companion paper summarized below (J.S. Pineda et al. 2021, in preparation). The corresponding stellar properties are shown in Table 2.

2.3.1. Field Stars

2.3.2. Young Stars

We used the Space Telescope Imaging Spectrograph (STIS) on HST to measure the FUV spectra of our FUMES sample through program HST GO-14640 (PI—Pineda). We show a summary of our observations in Table 3. Typically, data were taken using the G140L grating with the FUV-MAMA detector, providing a typical resolving power of ∼1000 across 1150–1730 Å. For the brighter targets, we used the echelle grating E140M instead, with similar wavelength coverage and a resolving power of ∼45,000. Although the lower resolution of the G140L grating does not permit measurement of the typical FUV line widths, and doublet lines are often blended, our focus in this study is simply the total flux in the strongest FUV lines, which we can measure using G140L for fainter targets than is possible at high resolution using STIS. All observations were taken in photon-counting TIME-TAG mode, providing high time resolution for our program. We focus on the quiescent emissions of the FUMES sample here, integrated across all exposures and orbits, after removing flares, with the time-resolved analysis to be presented in a follow-up paper.

Table 3. UV Data Summary

Name	UT Date	Grating	Aperture a	Orbits
GJ 4334	2017-09-20	G140L	02	2
GJ 49	2017-09-20	G140L	02	2
HIP 112312	2017-08-22	E140M	02	1
LP 247-13	2017-09-13	G140L	02	1
HIP 17695	2017-12-27	E140M	02	1
HIP 23309	2017-11-24	G140L	01	1
CD-35 2722	2017-09-26	G140L	02	1
GJ 410	2017-12-18	G140L	01	1
LP 55-41	2017-09-13	G140L	02	3
G 249-11	2017-09-10	G140L	02	3

Note.

^a Aperture denotes the width of the STIS slit.

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To extract the spectra from the FUV-MAMA photon-counting detector, we used the spectralPhoton routines from R. O. P. Loyd, previously used in the HST-STIS and HST-COS analyses of M-dwarf UV spectra (e.g., Loyd & France 2014; Loyd et al. 2018). ⁹ To summarize, the reduction procedure sums the photons in a narrow ribbon identified as the stellar trace, subtracting off a background count rate determined from the signal in offset regions with the same wavelengths. The reduction then uses the flux calibration from the full exposures to convert the photon counts to a calibrated spectrum. The advantage of spectralPhoton over the standard output from the STScI pipelines is that it allows for the definition of custom wavelength extraction regions, trace locations, and integrated time intervals. This was important because we found that, for the faint targets LP 55-41 and G 249-11, the standard pipeline products did not correctly identify the stellar trace. Additionally, some of the targets flared during our observations, which we removed by manually identifying when the flares took place and defining custom time intervals during the exposures for extraction of photons corresponding to the quiescent emission spectra.

The majority of our targets were observed in the wider 02 STIS slit, mitigating potential slit-loss effects that could affect the flux calibration by causing poor acquisition, target centering, or guiding. For one target, HIP 23309, observed with the narrower 01 slit for bright object protection considerations, the time series analysis showed a long-term trend in the photon counts over the course of the orbit (this effect was not seen in the data for GJ 410, the other program observation employing the 01 slit). To correct for this, we fit a third-order polynomial to the trend (in the 10 s binned light curve), in order to divide it out, and scaled to the average peak count rate. The quiescent spectrum was subsequently extracted from the appropriately scaled spectrum. For each target, we extracted spectra from each exposure (usually one per orbit) and then co-added them together, rebinning onto a grid of constant linear wavelength. For the echelle spectra, we also merged the wavelength regions in which the echelle orders overlap, averaging flux points falling within the same 0.05 Å bins, and preserving the integrated flux of the spectral bin. We show an example G140L spectrum of one of our targets in Figure 2, and an echelle E140M spectrum in Figure 3.

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By using STIS, we were able to observe Lyα in all of our objects, as well as all of the significant FUV emission lines He ii λ1640.4 Å, C ii λ λ1334.54, 1335.71 Å, C iii λ1175.7 Å, C iv λ λ1548.19, 1550.78 Å, Si iii λ1206.5 Å, Si iv λ λ1393.76, 1402.77 Å, and N v λ λ1238.82, 1242.806 Å. These lines span mean formation temperatures $\mathrm{log}T$ (K) = 4.5–5.2, probing the transition region of the stellar coronal atmosphere. ¹⁰ We focused on these lines because they are the most prominent ones in the data, being well-measured both in the FUMES targets and the literature sample.

To measure the target emission line fluxes, we used PyMC3 to fit the observed line shapes, typically with either a Voigt or Gaussian profile convolved with the instrument line spread function. ¹¹ Although we must assume a particular profile shape, these shapes are physically well motivated, and this approach has several advantages over simply summing the flux in the appropriate region for each line. At high resolution, we can measure the line widths, compare the emission core to the line wings, and examine centroid offsets, which are indicative of the stellar radial velocity. We can also simultaneously fit a continuum level below each line and incorporate the uncertainty in that estimate to our reported emission line fluxes. This effect was especially important in the G140L data, where there appeared to be some continuum level below many of the lines. For example, in the region around N v, some of the low-level extended Lyα wing emission could be seen. In the E140M data, no continua were evident. We quantify continuum levels in Section 3.3. Additionally, the line fitting is robust to potentially badly characterized data points, because it allows us to include a fit scatter term to the data in order to account for underestimated errors.

We summarize the model profiles' fits to the data in Table 4, where the number in front of each profile shape indicates that multiple lines were fit simultaneously, and the continuum component is described by a line with an unknown slope and constant offset. When multiple lines are fit simultaneously, as in the C iv doublet, the widths of the Gaussian and Lorentzian components in the Voigt profile shape are constrained to be identical in each line. This reduces the number of free parameters in the fit, and is justified because each line comes from the same species, forming under similar temperature conditions with only a small difference in energy levels between the transitions. As priors in our Bayesian regression analysis, we used uniform distributions for the line flux, center, and when necessary, the continuum offset and slope. For the width parameters in the Gaussian and Lorentzian components, we used a weakly informative half-Cauchy distribution, with a characteristic width of 100 km s⁻¹. ¹² This choice had a negligible effect on the parameter posterior distribution, but greatly improved MCMC sampling efficiency and convergence relative to a log uniform prior within PyMC3.

We show examples of our fit line profiles in Figure 4, with the results for the total line fluxes shown in Table 5. The Lyα emission is the focus of Paper II (Youngblood et al. 2021), and we reproduce those results here. For the two targets with E140M data, we fit a single Gaussian line for Si iii, to be consistent with the approach for the G140L data, which required simultaneous fitting with Lyα (Youngblood et al. 2021). At the higher resolution, the low-flux C iii lines are somewhat blended but distinguishable, so we fit six components to the multiplet but only a single Gaussian feature at low resolution. The He ii multiplet, by contrast, remains unresolved in all the data sets, and we fit a single Gaussian for the combined emission accordingly. In general, for the He ii line fits, the best model scatter term was often larger than the fits for other lines of the same star, possibly indicating that the shape choice was not ideally suited to the data. For this unresolved multiplet, however, our model choice remains the simplest way to capture the line flux.

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Table 5. FUV Line Fluxes ^a

Name	Lyαb	He ii	C ii c	C iii	C iv	N v	Si iii d	Si iv
G 249-11	⋯	<0.012	<0.012	<0.014	0.013 ± 0.003	0.0065 ± 0.0012	⋯	<0.01
HIP 112312	$214{\pm }_{6}^{7}$	9.14 ± 0.48	3.97 ± 0.18	$3.01{\pm }_{0.47}^{0.53}$	$11.87{\pm }_{0.53}^{0.52}$	$1.90{\pm }_{0.13}^{0.14}$	$1.67{\pm }_{0.28}^{0.29}$	$2.20{\pm }_{0.14}^{0.13}$
GJ 4334	$7.03{\pm }_{1.16}^{3.93}$	$0.280{\pm }_{0.035}^{0.036}$	0.68 ± 0.03	0.15 ± 0.01	$1.62{\pm }_{0.06}^{0.07}$	0.337 ± 0.024	$0.211{\pm }_{0.018}^{0.017}$	0.36 ± 0.02
LP 55-41	⋯	<0.018	0.008 ± 0.002	<0.019	<0.034	0.014 ± 0.004	⋯	<0.003
HIP 17695	$110{\pm }_{3}^{2}$	$4.50{\pm }_{0.30}^{0.29}$	2.98 ± 0.14	$2.44{\pm }_{0.38}^{0.41}$	$8.88{\pm }_{0.39}^{0.40}$	1.74 ± 0.11	$1.82{\pm }_{0.18}^{0.19}$	$2.14{\pm }_{0.14}^{0.15}$
LP 247-13	$442{\pm }_{103}^{374}$	2.04 ± 0.13	1.77 ± 0.07	1.48 ± 0.081	3.57 ± 0.11	$0.918{\pm }_{0.052}^{0.056}$	$0.731{\pm }_{0.040}^{0.041}$	0.60 ± 0.04
GJ 49	$249{\pm }_{93}^{58}$	$1.38{\pm }_{0.07}^{0.08}$	1.84 ± 0.09	0.686 ± 0.035	2.28 ± 0.07	$0.674{\pm }_{0.031}^{0.032}$	$0.550{\pm }_{0.013}^{0.021}$	0.47 ± 0.02
GJ 410	$158{\pm }_{53}^{163}$	2.54 ± 0.17	2.68 ± 0.09	0.920 ± 0.099	$2.99{\pm }_{0.16}^{0.17}$	0.609 ± 0.051	$0.876{\pm }_{0.074}^{0.072}$	1.06 ± 0.06
CD-35 2722	$25.0{\pm }_{1.3}^{2.2}$	3.85 ± 0.18	$2.90{\pm }_{0.11}^{0.12}$	1.74 ± 0.09	4.49 ± 0.12	0.90 ± 0.05	1.32 ± 0.06	1.29 ± 0.05
HIP 23309	$54.0{\pm }_{1.9}^{2.0}$	$6.60{\pm }_{0.28}^{0.29}$	3.82 ± 0.09	$2.23{\pm }_{0.12}^{0.13}$	6.87 ± 0.19	1.54 ± 0.06	$2.01{\pm }_{0.08}^{0.09}$	2.42 ± 0.08

Notes.

^a All fluxes are as observed at Earth, in units of 10⁻¹⁴ erg s⁻¹ cm⁻², and the flux from multiplet lines are added together, unless indicated otherwise. Uncertainties reflect the central 68% confidence interval about the median. ^b Lyα fluxes as reconstructed in FUMES Paper II (Youngblood et al. 2021). For G249-11 and LP 55-41, the weak emission precluded a confident fit. ^c For HIP112312 and HIP17695 with E140M data, we report the flux of just the λ 1335 Å line. For the rest of the sample with G140L data, we report the total flux of the blended doublet, including ISM absorption. We estimate that these measurements should be corrected upward by ∼7% to account for the influence of the ISM (see Section 3.2). ^d Si iii fluxes are from FUMES Paper II (Youngblood et al. 2021), jointly fit with Lyα in G140L STIS data.

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For the E140M data, the C ii doublet is resolved, but at low resolution the two lines are blended, which impacts the flux measurements because one of the lines is attenuated by ISM absorption. We estimated the typical ISM attenuation of the C ii 1334 Å line by assuming typical ISM parameters from Redfield & Linsky (2004) and a stellar line approximated as a Voigt profile with a width of 30 km s⁻¹ (0.05 Å) for the Gaussian component and 20 km s⁻¹ (0.04 Å) for the Lorentzian component, based on our E140M spectra of HIP 17695's unattenuated C ii 1335 Å line. The range of C ii ISM parameters identified by Redfield & Linsky (2004) for stars inside 40 pc is 13.1–15.2 for the log column density, 2.9–6.6 km s⁻¹ for the Doppler b value, and a radial velocity of −30 to 30 km s⁻¹. We created 10,000 realizations of the attenuated 1334 Å flux, where the column density, b value, and radial velocity of the absorbers relative to the star's rest frame were drawn from uniform distributions with the ranges described previously. We found that the median attenuation for the 1334 Å line is 14%, with a 68% confidence interval range of 4%–45%. Assuming the 1334 and 1335 Å C ii lines are equally populated (a reasonable assumption for thermal equilibrium), total C ii fluxes derived from the low-resolution spectra where the two lines are blended should be corrected upward ∼7%, with a 68% confidence interval range of 2%–23%.

The STIS data for G249-11 and LP 55-41 were much fainter than the rest of the FUMES targets, and many of the typical emission lines were barely visible. We determined the 3σ upper limits for these lines by simply summing the flux in the region around the expected line center and computing the associated uncertainty in that flux. For the few lines that were measurable (C ii, C iv, and N v), we accommodated the lower S/N by only fitting single Gaussian line profiles in each case, one component for the blended C ii, and two components each for the C iv and N v doublets. These results are included in Table 5.

Although the FUV spectra of low-mass stars is dominated by discrete emission-line features, there is a weak underlying FUV continuum that is likely defined by the recombination edges of species like Si (Loyd et al. 2016; Peacock et al. 2019). As seen in Figures 2 and 3, continuum emission between the strong line features was evident in the low-resolution G140L spectra, but not apparent in the high-resolution E140M spectra. This is a consequence of the higher sensitivity and lower resolution of the G140L grating, making it easier to detect weak continuum levels. Compared to HST-COS medium-resolution gratings (G130M, G160M), the HST-STIS low-resolution grating (G140L) is slightly more sensitive at detecting faint continuum emission. ¹³ For the full FUMES sample, we quantify the FUV continuum levels apparent in our spectra following the work of Loyd et al. (2016).

As part of the MUSCLES program, they defined an ensemble of narrow bands (0.7–1 Å in width) interspersed between the prominent FUV emission lines, to assess the FUV continuum flux across 1300-1700 Å. We used these same bands (obtained from R. O. P. Loyd 2021, private communication) transformed to the radial velocity frame of our targets, to integrate the FUV continuum. ¹⁴ We were able to use the bands unaltered for the E140M data sets; however, for the lower-resolution G140L data, we visually verified that the bands did not overlap with any of the strong emission lines. This removed six narrow bands (∼6 Å) that encompassed parts of emission-line wings.

The integrated fluxes in these narrow bands are shown in Table 6, averaged over the ∼150 Å cumulative width of the passbands. For eight of the ten targets, including the two observed with the E140M grating, we measure a nonzero FUV continuum level, all except for G149-11 and LP 55-41. Although the integrated flux is usually insignificant in each narrow band, revealing no clear continuum shape, when adding up the emissions across all of the narrow passbands of the spectra, the overall continuum emission is detectable. For the quoted uncertainties of Table 6, we propagated the data uncertainty through the continuum flux summation.

Our results are broadly consistent with those of Loyd et al. (2016), accounting for the distance to each target; however, with this broader sample of stars, we see evidence for a wide array of continuum levels. Spectra with higher S/Ns are needed in order to verify the detection of these average continua and determine what defines their flux levels and spectral composition.

Our FUV emission line measurements are diagnostic of the stellar upper atmospheres and how they change with magnetic heating. Because they form in similar regions, the line strengths are strongly correlated, as has been well-illustrated in the literature (e..g, Youngblood et al. 2017). We build on these results by expanding the available UV samples using our new measurements with the FUMES targets. In addition to the FUMES data, we also analyzed the Mg ii NUV data for the literature sample, as representative of an additional atmospheric layer and the spectral region of interest.

Since C iv is a bright and readily accessible emission feature, we show in Figures 5–6 the correlation of each of the emission lines of Table 5 against C iv. We plot the luminosities of each feature as two-dimensional error ellipses (2σ contours) representative of the bivariate error distributions for the luminosities of each line pair. We further discuss the differences in undertaking this analysis in luminosity as compared to surface flux in Appendix A.1. Since each luminosity depends on the known parallax and its uncertainty, the luminosity determinations have correlated uncertainties, revealed by diagonally oriented ellipses. This correlation is more prominent when the parallax uncertainty dominates the luminosity measurements. Figures 5 and 6 show these data, with the filled shading of each shape indicating the effective temperature of the stars. The emission fluxes are not primarily determined by the stellar effective temperature, as both warmer and cooler objects in the FUMES and literature samples show high and low FUV luminosities, but instead by each object's activity regime, as will be discussed in Section 4.

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The correlations shown in Figures 5–6 further demonstrate that the C iv strength can be used to predict the quiescent emission of any of the other prominent transition region emission lines across four orders of magnitude in luminosity. To these data, we fit a power-law relation, using C iv as the predictor variable, within a Bayesian framework (Kelly 2007), accounting for uncertainties in both dimensions and incorporating an intrinsic scatter at fixed C iv luminosity. We defined the regression model for the line luminosities, L_y , as

$\begin{eqnarray}&&\mathrm{log}{L}_{y,i}| \gamma ,C,s\sim {{ \mathcal N }}_{i}(\gamma \mathrm{log}{L}_{x,i}+\mathrm{log}C,s),\end{eqnarray} \tag{ 1 }$

where x corresponds to C iv and y any of the other lines (e.g., N v, Si iv etc.), ${ \mathcal N }(\mu ,\sigma )$ denotes a normal distribution of mean μ and standard deviation σ, i denotes each star in the data set, and s is the intrinsic scatter of the power-law relation in log–log space. ¹⁵ The s parameter is fit along with the power-law model, with its marginalized posterior distribution describing the set of intrinsic scatters consistent with the uncertainty of the data and the posterior distributions on γ and C. The peak of that distribution is our best estimate of the representative intrinsic scatter in the line–line correlations. As an example, a scatter value s = 0.1 would suggest that an individual line luminosity could be predicted to within ∼25%, corresponding to the central 68% confidence interval, with a known C iv luminosity.

The probability distributions for each line luminosity pair were defined, in the case of FUMES targets, by their joint distribution created from the sampled line flux posteriors of the model fits (see Section 3.2) combined with the parallax measurement and its uncertainty. For literature data, we assumed Gaussian distributions for the line fluxes and combined with the parallax measurements to determine the joint luminosity distributions at each pair of data. The likelihood is defined by the product across the sample of stars with Equation (1). In log–log space, this is a linear model, and we used a uniform prior on $\mathrm{log}C$, the offset, a Cauchy distribution for the power-law exponent, γ, ¹⁶ and a half-Cauchy distribution with unity shape parameter for the scatter, s. These priors are minimally informative, and only marginally affect the posterior distributions in γ, s, and C, while improving Monte Carlo convergence efficiency. An example of the joint posterior distributions for these fits is shown in Figure 7. The results of these fits are indicated by the dashed lines in Figures 5–6, and we tabulate the fit parameters in Table 7.

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Table 7. Line Correlations with C iv ^a

Line	$\mathrm{log}{T}_{{\rm{f}}}$	Nsamp	γ	$\mathrm{log}C$	s
Mg ii	4.5	11	$0.89{\pm }_{0.15}^{0.14}$	4 ± 4	$0.42{\pm }_{0.08}^{0.12}$
Lyα	4.5	11	0.69 ± 0.09	$9.7{\pm }_{2.3}^{2.4}$	$0.35{\pm }_{0.07}^{0.10}$
C ii	4.5	15	1.02 ± 0.04	−1 ± 1	$0.15{\pm }_{0.03}^{0.04}$
Si iii	4.7	20	1.01 ± 0.04	−0.8 ± 1.0	$0.15{\pm }_{0.03}^{0.04}$
C iii	4.8	8	1.0 ± 0.2	−2 ± 5	$0.24{\pm }_{0.06}^{0.09}$
He ii	4.9	23	1.07 ± 0.04	−2 ± 1	$0.21{\pm }_{0.03}^{0.04}$
Si iv	4.9	21	0.98 ± 0.03	0.0 ± 0.9	$0.15{\pm }_{0.02}^{0.03}$
N v	5.2	22	0.89 ± 0.03	2.4 ± 0.7	$0.10{\pm }_{0.02}^{0.03}$

Note.

^a C iv has $\mathrm{log}{T}_{{\rm{f}}}=5.0$. Reported parameters correspond to the median of the marginalized posterior distributions, with uncertainties indicating the central 68% confidence interval.

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The line pair with the smallest intrinsic scatter is C iv–N v (∼0.1 dex). This is unsurprising, as these two lines are optically thin and form at similar temperatures high in the transition region. We thus considered whether there may be a trend in intrinsic scatter relative to the temperature of line formation (see Table 7). The data do not reveal any suggestive trends, although Mg ii and Lyα show the greatest scatter values and correspond to the lines with the lowest formation temperatures of those that we studied. On the contrary, C ii forms at temperatures similar to those of Mg ii and shows as small a scatter as the hotter Si iv transition region line. The larger scatter for Mg ii relation could be due to unaccounted for uncertainty associated with the ISM correction needed for the Mg ii flux estimates (Youngblood et al. 2016). This may also affect the Lyα correlation; however, the formation of Lyα, its central profile, and its broad emission (Youngblood et al. 2021) are likely more complicated (Peacock et al. 2019). Therefore, predicting the emission from the transition region C iv luminosity may not account for all of the processes impacting the Lyα flux.

The rest of the line–line correlations show consistent scatters of 0.1–0.2 dex at fixed C iv luminosity. These measurements represent the intrinsic scatter of FUV line emission across the population, as all of the observations were either taken simultaneously or closely spaced in time (France et al. 2016). If this is indeed due to intrinsic variation, it may define the limit to which FUV features can be used to predict one another. Additionally, this intrinsic variation may differ between the "inactive" and "active" subsamples; however, our samples in each are not large enough to fully investigate.

Canonical rotation–activity correlations with optical and X-ray emission (e.g., Pizzolato et al. 2003; Wright et al. 2011; Astudillo-Defru et al. 2017; Newton et al. 2017) have illustrated a strong rotational dependence, typically characterized by a power-law distribution, and a saturated regime of activity for the fastest rotators. Since Noyes et al. (1984), these relationships have been used to investigate the interplay between the internal convective motions and differential stellar rotation within an α–Ω magnetic dynamo, thought to operate in stars with partly convective interiors (e.g., Montesinos et al. 2001; Browning et al. 2006). Mediated by the tachocline interface between the radiative core and convective envelope, the dynamo sustains the magnetic field through cyclic regeneration of toroidal and poloidal field. Theory suggests that the magnetic fields—and hence the resulting activity tracers—should thus be mediated by the Rossby number, Ro = P/τ_c , the ratio between the rotational period and the convective turnover time, which characterizes the timescales for bulk motions of internal convection. Recent results have further extended these studies across the boundary between fully convective and partly convective interiors (Newton et al. 2017; Wright et al. 2018), with M-dwarf samples illustrating a continued rotational dependence on the activity tracers, despite the need for alternative dynamos for the lowest-mass stars, which lack that tachocline interface.

With the FUMES sample, we can now probe these same mechanisms with new indicators of the magnetic activity in the FUV. To our emission-line data, we fit a broken power law as a function of Ro:

$\begin{eqnarray}{L}_{y}/{L}_{\mathrm{bol}}=\left\{\begin{array}{ll}{\left({L}_{y}/{L}_{\mathrm{bol}}\right)}_{\mathrm{sat}}, & \mathrm{Ro}\leqslant {\mathrm{Ro}}_{c}\\ A{\left(\mathrm{Ro}\right)}^{\eta }, & \mathrm{Ro}\gt {\mathrm{Ro}}_{c},\end{array}\right.\end{eqnarray} \tag{ 2 }$

where ${({L}_{y}/{L}_{\mathrm{bol}})}_{\mathrm{sat}}$ denotes the strength of emission line, y, in the saturated regime relative to the star's bolometric luminosity, η indicates the slope of the unsaturated regime, and Ro_c is the critical Rossby number at which the activity transitions between regimes, with $A\equiv {({L}_{y}/{L}_{\mathrm{bol}})}_{\mathrm{sat}}{({\mathrm{Ro}}_{c})}^{-\eta }$ to ensure continuity. To compute the Rossby number, we use the known rotation periods (see Table 2) and the empirical calibration for the convective turn overtime, τ_c , as a function of mass from Wright et al. (2018). We discuss systematic effects of this choice of calibration in Appendix A.2. Our segmented regression fit further includes a scatter term, σ_L , describing intrinsic dispersion in the observed luminosities at a fixed Rossby number, such that

$\begin{eqnarray}&&\mathrm{log}{({L}_{y}/{L}_{\mathrm{bol}})}_{i}z| \,\{\beta \}\sim { \mathcal N }(\mathrm{log}[{L}_{y}/{L}_{\mathrm{bol}}({\mathrm{Ro}}_{i})],{\sigma }_{L}),\end{eqnarray} \tag{ 3 }$

where the index i indicates the data for each star, and $\{\beta \}=\{{\mathrm{Ro}}_{c},{({L}_{y}/{L}_{\mathrm{bol}})}_{\mathrm{sat}},\eta ,{\sigma }_{L}\}$ is the set of four model parameters. Within this Bayesian model, we employ as priors uniform distributions for $\mathrm{log}{({L}_{y}/{L}_{\mathrm{bol}})}_{\mathrm{sat}}$ and Ro_c , a zero-centered Cauchy distribution with unity shape parameter for the slope η (see footnote 17), and a half-Cauchy distribution with unity shape parameter for σ_L .

Our fitting process using PyMC3 accounts for random uncertainty in both dimensions (Kelly 2007), and possible error correlations between the assumed mass and bolometric luminosity using the sampled posterior distributions from the work of J.S. Pineda et al. (2021, in preparation). The error correlations between M and L_bol are generally small for the field objects, but can be significant for the young stars (see J.S. Pineda et al. 2021, in preparation). We also included the scatter in the convective turnover time calibration from Wright et al. (2018) of 0.055 dex in $\mathrm{log}{\tau }_{c}$ at fixed mass. When using the literature periods, we further incorporated the quoted uncertainties in those measurements if available, or used a 10% Gaussian uncertainty if not; see Table 2 (E. Newton 2020, private communication). Except for the Lyα measurements of some stars, the random uncertainties in the rotation–activity data are typically dominated by the error on the Rossby number driven by the scatter in the τ_c calibration. This careful accounting of the known sources of random error enabled us to carefully examine systematic effects with this fitting (see Appendix A.2).

We applied these methods to eight UV emission lines, Lyα, Mg ii, C ii, Si iii, He ii, Si iv, C iv, and N v, for which the samples permitted a detailed rotation–activity fit. We show the best parameter results from the marginalized posteriors in Table 8, with the fit solutions plotted in Figure 8. We also show an example of the joint posterior distributions for this fitting in Figure 9, illustrating how the critical Rossby number and unsaturated regime slope are generally correlated in our rotation–activity parameterization of Equation (2).

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Table 8. Rotation–Activity Correlations ^a

Line	Nsamp	η	Roc	$\mathrm{log}{\left({L}_{\mathrm{line}}/{L}_{\mathrm{bol}}\right)}_{\mathrm{sat}}$	σL
Lyα	19	$-1.26{\pm }_{0.58}^{0.41}$	$0.21{\pm }_{0.11}^{0.16}$	$-3.58{\pm }_{0.15}^{0.16}$	$0.32{\pm }_{0.07}^{0.10}$
Mg ii	12	$-1.86{\pm }_{0.55}^{0.41}$	$0.20{\pm }_{0.08}^{0.11}$	$-3.99{\pm }_{0.15}^{0.14}$	$0.23{\pm }_{0.07}^{0.10}$
C ii	15	$-2.38{\pm }_{0.77}^{0.64}$	$0.24{\pm }_{0.11}^{0.12}$	−5.25 ± 0.16	$0.33{\pm }_{0.07}^{0.10}$
Si iii	20	$-2.08{\pm }_{0.46}^{0.41}$	0.20 ± 0.07	−5.37 ± 0.13	$0.32{\pm }_{0.06}^{0.08}$
He ii	24	$-2.19{\pm }_{0.33}^{0.30}$	0.19 ± 0.05	−4.92 ± 0.11	$0.27{\pm }_{0.04}^{0.06}$
Si iv	21	$-2.32{\pm }_{0.44}^{0.42}$	0.24 ± 0.07	−5.38 ± 0.14	$0.35{\pm }_{0.06}^{0.08}$
C iv	24	$-2.02{\pm }_{0.41}^{0.39}$	$0.18{\pm }_{0.06}^{0.07}$	−4.72 ± 0.15	$0.37{\pm }_{0.06}^{0.08}$
N v	23	$-1.84{\pm }_{0.40}^{0.37}$	0.19 ± 0.07	$-5.37{\pm }_{0.13}^{0.14}$	$0.33{\pm }_{0.06}^{0.07}$

Note.

^a Reported parameters correspond to the median of the marginalized posterior distribution, with uncertainties indicating the central 68% confidence interval.

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Because this regression analysis relies on stellar mass estimates for determining Ro, it may be potentially biased by our choice for the young stars to utilize the magnetic-model-based masses, which are larger than the non-magnetic-model masses (see J.S. Pineda et al. 2021, in preparation), with a correspondingly larger implied Ro. This concern applies only to the five young stars in our sample (see Table 2). However, of the five, only AU Mic (Ro = 0.11–0.17) appears to be near the transition between saturated and unsaturated regimes. HIP 112312, HIP 17695, and CD-35 2722 have Ro values well below 0.1 regardless of the model choice for the mass estimate, and HIP 23309 has an Ro = 0.32–0.39, larger than the literature values of Ro_c ∼ 0.15–0.20 (Newton et al. 2017; Wright et al. 2018). Because horizontal systematics within the saturated regime have no influence on the rotation–activity fits, and only one of the stars in a sample of ≳20 targets per emission line might be affecting the analysis, we consider this systematic effect in young star masses to minimally impact our results. Furthermore, these young stars do not appear to be outliers in our rotation–activity plots (Figure 8). Although the difference in Ro is small, only the result for Mg ii is likely to be impacted by the mass choice for AU Mic, because of the limited target sample for that line. Correspondingly, the rotation–activity fit parameters for Mg ii have larger uncertainties.

We further consider the quality of our fits by examining the residuals, as illustrated in Figure 10. The data residuals are consistent with being normally distributed about the best median rotation–activity relation, and they show no apparent dependence on stellar mass. There may be some hints of an excess of data points below the median fit at large Rossby numbers (∼1), but this sample is too small to be conclusive in this regard. This may be more evident in literature studies (e.g., Newton et al. 2017), which would indicate that, at slow rotation periods, the single power law may be overestimating the magnetic activity.

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With these considerations, for all of the emission lines, we deem the broken power law to accurately describe the rotation dependence of the UV activity indicators across the entire sample, with each line showing slightly different emissions levels in the saturated regime, consistent critical Rossby values of ∼0.2, and a range of slopes in the unsaturated regime spanning −1.3 to −2.4 (see Table 8). For our M-dwarf sample, the saturation levels of $\mathrm{log}{({L}_{\mathrm{line}}/{L}_{\mathrm{bol}})}_{\mathrm{sat}}$ in the C ii, Si iii, N v, and Si iv lines all exceed those measured by France et al. (2018) for an ensemble of FGKM stars, which were all in the range of −5.5 to −6. In the saturated activity regime, M-dwarf FUV emissions exceed those of warmer stars, relative to bolometric. For the power-law decay with decreasing rotation, all the lines except Lyα are consistent with a slope of −2, with Lyα corresponding to the shallowest slope in the rotation–activity analysis. While the error bar is large, because these data rely on model reconstructions of the emission flux, we consider whether those assumptions may be systematically impacting this result.

Our Lyα reconstructions (Youngblood et al. 2021) and those from the literature sample (Youngblood et al. 2016), typically assume a Voigt-like profile for the emission; however, some evidence and theory exist that suggest the ISM-obscured Lyα profile peak may show an absorption self-reversal like that observed in other chromospheric lines (e.g., Linsky et al. 1979; Redfield & Linsky 2002; Guinan et al. 2016; Youngblood et al. 2016; Peacock et al. 2019). Our Lyα reconstructions do not account for this effect, therefore we may be overestimating the Lyα flux. However, for this effect to impact the fitted slope in the rotation–activity unsaturated regime, the strength of self-reversals would also need to depend on the Rossby number. Theoretical Lyα line profiles from Peacock et al. (2019) for GJ 832, GJ 176, and GJ 436 also show strong core absorption due to non-LTE effects, such that the reconstructed fluxes are greater by a factor of ∼2. A systematic effect at this level or stronger between active and inactive M-dwarfs is needed to explain the shallow Lyα slope. However, more data are required in order to validate whether these theoretical profiles match those produced by nature. There is currently no evidence for such a systematic difference in Lyα core profiles between active and inactive M dwarfs.

If typical Lyα lines only show weak self-reversals in M dwarfs, as suggested empirically by Guinan et al. (2016) and Bourrier et al. (2017), then the shallow slope could indicate the persistence of Lyα emission at slow rotation rates even as other high-energy features decay more rapidly. As they are most prominent emission lines in the FUV and NUV, respectively, we consider the evolution of the ratio of Lyα to Mg ii emissions. We illustrate this in Figure 11, showing the emission ratio as a function of Rossby number. The shaded region denotes the ratio of the power-law fits in the two lines from Section 4.1 consistent with the uncertainties in the parameter estimates to the 1σ level. More data are needed to better refine rotation–activity relationships in both features, but the rise in ratio with angular momentum evolution indicates that the relative strengths of different portions of the high-energy spectrum changes over time. Evolutionary changes in the spectral illumination have implications for the prevalent photochemistry and atmospheric history of any exoplanetary systems orbiting low-mass hosts. For example, stronger FUV relative to NUV emissions may drive a build-up of photochemical ozone in planetary atmospheres (e.g., Gao et al. 2015; Harman et al. 2015). Although a full account of the relative contribution of Mg ii to the total NUV luminosity is necessary for a complete description, our data suggest that this effect may increases as stars spin down over time. Furthermore, such an evolutionary effect may be more prominent for early-M dwarfs but not as significant for late-M dwarfs (Schneider & Shkolnik 2018).

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Our results across all of the lines (Table 8) reveal that most of the best-fit slopes are around the canonical value of −2 implied by a distributed dynamo model (Noyes et al. 1984) in which dynamo amplification occurs throughout the convective region. By contrast, in an interface dynamo like that based on a tachocline, there are additional dependencies that lead to deviations away from this canonical slope of −2 (Parker 1993; Charbonneau & MacGregor 1997; Wright et al. 2011). To compare these results from the different lines, we plot our fit parameters for η and Ro_c as a function of line formation temperature in Figure 12. The UV lines are plotted at the temperatures listed in Table 7, to which we also add the Hα results of Newton et al. (2017) as representative of the chromosphere, and the X-ray results of Wright et al. (2011, 2018) corresponding to the corona using a nominal temperature of 10⁶ K, although X-ray flux contributions extend to higher temperatures. In Figure 12, we show the mean value representative of the transition region, taking the combined posteriors across all the lines in each parameter, yielding medians and central 68% confidence intervals of $\eta =-2.02{\pm }_{0.53}^{0.55}$ and ${\mathrm{Ro}}_{c}=0.20{\pm }_{0.07}^{0.09}$. Our mean result is consistent to within the uncertainties with both the chromospheric and coronal results in the literature.

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However, there may be a trend in the unsaturated rotation–activity slope as a function of formation temperature. While this is largely a consequence of different values for the Hα and X-ray results, our UV data fall directly in between. As stars spin down in the unsaturated regime, the coronal X-ray emissions appear to decline rapidly, with deeper atmospheric layers showing a slower decay in their magnetic activity. The onset of activity decline with slower rotation begins first in X-rays, and only after some angular momentum evolution do the transition region and chromospheric features also begin to decline, as the best-fit Ro_c are greater for the deeper layers.

If these trends hold from analyses of larger UV samples, they will help reveal the role of magnetic heating processes in mediating the relationship between the internal magnetic dynamo and the emission features. While the rotation–activity relationships have been used to constrain stellar dynamos, this connection is indirect. The emission is necessarily a consequence of the nonthermal magnetic heating, and if the different layers exhibit distinct power-law slopes, then this effect of the heating processes needs to be accounted for when making dynamo inferences from rotation–activity relationships. The trend evident in the top panel of Figure 12 for the unsaturated slope suggests that, whether the dominant process is Alfvén wave heating or nanoflare reconnection, the decline in nonthermal heating with weaker average field strengths (e.g., Shulyak et al. 2017) as stars spin down takes place from the outermost layers inward. In other words, the magnetic heating processes persist more strongly in deeper atmospheric layers across angular momentum evolution, affecting the relative strengths of emission features formed in different layers of the atmosphere.

There are, however, some important caveats to the comparisons implicit in Figure 12. Between this work and the studies of Newton et al. (2017) and Wright et al. (2011, 2018), the samples are not identical. Our work and that of Newton et al. (2017) use similar mixed-age samples of M dwarfs, whereas the X-ray data of Wright et al. (2018) also include more massive stars, although normalizing by the bolometric luminosity and examining the rotation relationship in Ro is designed to account for these possible differences. More significantly, as we discuss in Appendix A.2, differences in the Ro calibration can impart systematic effects on the rotation–activity analyses. Newton et al. (2017) used the calibration of Wright et al. (2011), whereas we used that of Wright et al. (2018). Relative to our work, the Newton et al. (2017) results for critical Rossby number and unsaturated slope could be systematically greater than if they used the same calibration. The magnitude of this effect, however, can be small, and it depends on the stellar sample (see Appendix A.2). Moreover, there may have been issues with the analyses of Wright et al. (2011) leading to a much steeper power-law slope (Reiners et al. 2014). Nevertheless, it appears that, in the chromosphere, η is likely shallower than −2, whereas in the corona, it is steeper than −2, with the transition region value from our work right in the middle. Consistent methods and calibrations need to be applied across the different wave bands, to discern whether there are indeed rotation–activity trends as a function of where in the stellar atmosphere the emission originates.

Potential issues with a choice of calibration can be avoided by circumventing the Rossby number entirely. Reiners et al. (2014) argue for a more general approach in analyzing the rotation–activity relationships of low-mass stars, and conclude that the activity dependence can be described well by a combination of rotation period and radius, specifically L_X/L_bol = kP⁻² R⁻⁴, where k is a scaling constant. We consider how this relationship may apply to our UV data. In Figure 13, we show the He ii luminosity normalized by stellar bolometric luminosity against the stellar P⁻² R⁻⁴. Like the Rossby scaling, Figure 13 illustrates a potentially power-law decay in emission with a scatter of activity at a given abscissa. Although our sample is much smaller, there are hints of the mass dependence in activity using this scaling that was illustrated by Newton et al. (2017) with Hα data, in which more massive stars preferentially lie to the right of the locus of points in Figure 13. The mass dependence appears to increase the intrinsic scatter at fixed rotation in the activity beyond the ∼0.27 dex scatter evident in our fits to L_{He II} /L_bol versus Ro.

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While it is evident that rotation still plays a key role in the magnetic emission of fully convective stars, an empirical Rossby number may not be the most appropriate scaling to uncover the underlying relationships governing the dynamo action and possible differences from partly convective stars. It appears to work sufficiently well across the chromosphere, transition region, and corona; however, by defining Ro specifically to minimize scatter in the rotation–activity diagrams across this full range of stars, this procedure may be masking real differences in the evolution of magnetic activity between partly and fully convective objects (e.g., Magaudda et al. 2020). If these two stellar mass regimes did indeed show distinct rotation–activity correlations, i.e., statistically distinguishable rotational dependencies of their activity decay, the process of defining an empirical Rossby number across both samples would be averaging together these possible differences, since it presupposes that both samples follow the same power-law relation.

We thus show the evolution of FUV activity with age in Figure 14, focusing on the N v emission feature as representative of all of the other FUV lines (Sections 3.4 and 4). We chose the N v line for this analysis because it is tightly correlated with C iv, and existing literature relations allow us to also estimate EUV emission from their measurements; see below.

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For the five objects with moving group membership (see Table 2), we used the mean group age with error, and we used the gyrochronology relations for the rest of the sample. To assess the uncertainties on the ages from the gyrochronology calibration, we used the reported uncertainties of the parameters of the best-fit relations (Engle & Guinan 2018) and the period uncertainties as in Section 4.1, sampling each appropriately, assuming independently Gaussian distributed variables in computing the distribution for the age estimate. ¹⁷ Because the Engle & Guinan (2018) rotation–age relations are defined by two bins of spectral type ranges, for our sample, we used the appropriate relation corresponding to the optical spectral types listed in Table 2. However, some of our objects have spectral types, M1.5-M2, between the defined ranges for the rotation–age relations. For these objects, we combined the estimates from both the M0–M1 and M2.5–M6 relations in order to determine the age estimate and its uncertainty. Three of these stars were rapid rotators without known ages: AD Leo, EV Lac, and LP 247-13. Given the large uncertainty in the gyrochronology ages—especially early on, where the rotational evolution has yet to converge—we plot these age estimates as 3σ upper limits (triangles in Figure 14).

In Figure 14, we also shade the points according to their effective temperature. Based on these age estimates, Figure 14 illustrates how cooler, later M dwarfs persist with strong activity levels for a longer duration of their early lifetimes. Considering the mean level of $\mathrm{log}{({L}_{{\rm{N}}{\rm\small{V}}}/{L}_{\mathrm{bol}})}_{\mathrm{sat}}\sim -5.37$ and 0.33 dex scatter, some of the cooler objects may display near-saturation-level activity beyond an age of 1 Gyr. In contrast, the warmer M dwarfs, by this age, appear to have declined in their magnetic emissions by an order of magnitude relative to the saturated regime. Using GALEX photometry, Schneider & Shkolnik (2018) also found this divergence in UV evolution between early- and mid-M dwarfs.

Because of its importance to exoplanetary atmospheric heating, we also considered what our FUV evolutionary data imply for the extreme ultraviolet portion of the high-energy spectrum. To estimate the EUV from the FUV data, we used the empirical relation of France et al. (2018) between N v emission and the blue portion of the EUV passband, EUVb (90–360 Å). While not the full EUV spectrum, this is the portion for which available data and relatively low ISM attenuation have permitted any empirical estimates in the low-mass star regime. This EUVb band corresponds to about half of the total EUV energy from M-dwarf stars (Fontenla et al. 2016; France et al. 2018). In Figure 15, we plot this EUV evolution with the same ages as discussed above, and EUVb luminosities determined from the N v emission, incorporating the parameter uncertainties and 0.24 dex scatter in the France et al. (2018) relation. For young, active, M-dwarf stars, the EUV luminosity is ∼10^−3.5 relative to bolometric, with these emissions lasting for the first several hundreds of Myr of their lives—and perhaps longer.

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While these empirical relations for age, and difficult-to-detect portions of the spectrum, like the EUV, provide general estimates for the evolution of activity and high-energy emission, Figure 15 further illustrates the broad uncertainty inherent in using these estimates as inputs to exoplanetary evolution modeling. Refined rotation–age relations in the M-dwarf regime, and expanded samples of stellar EUV detections from missions like ESCAPE (France et al. 2019), will greatly expand the utility of these methods for exoplanetary applications.

Assuming the Rossby parameterization removes a significant mass dependence, we use the rotation–activity correlation regression fits together with the assumed rotation evolution in order to calculate the activity evolution of early- and mid-M dwarfs. In Figure 16, we plot separate curves for the evolution of N v emission with stellar age, for a 0.6 M_⊙ M dwarf, corresponding to the M0–M1 rotational evolution, and a 0.25 M_⊙ M dwarf, corresponding to the M2.5–M6 rotational evolution. The top panel of Figure 16 shows the expected median evolution for each mass star along with its central 68% confidence level. The uncertainties are determined at a given age by sampling the uncertainty distributions of the rotation–age relation, the calibration between mass and convective turn over time, and the joint posterior of the rotation–activity fit. The bottom panel of Figure 10 additionally includes the scatter at fixed Rossby number (see Section 4), to illustrate the range of likely emission values accounting for intrinsic scatter in the observed population. More data are required for further assessment; however, we expect this scatter to include the effects of rotational variability, activity cycles, and possible metallicity variations. While the nominal N v emission levels relative to bolometric are higher for lower-mass objects at the same age, there is little difference in the evolution within the uncertainties.

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With possibly distinct UV luminosity evolution between early- and mid-M dwarfs, as shown in Figure 16, we calculate the typical evolution of the UV illumination for exoplanetary systems using the median N v rotation–activity relation (Section 4), and the age–rotation evolution from (Engle & Guinan 2018). The average evolution for individual systems may vary by up to a factor of two. As a measure of the illumination histories, we assess how much quiescent UV energy a given exoplanet accumulates over time as

$\begin{eqnarray}&&{ \mathcal E }(t)={\int }_{0}^{t}{F}_{\mathrm{UV}}(\tau )d\tau ={\int }_{0}^{t}\displaystyle \frac{{ \mathcal R }(\tau ){L}_{\mathrm{bol}}(\tau )}{4\pi {d}^{2}}d\tau ,\end{eqnarray} \tag{ 4 }$

where F is the UV flux impinging on the planet at an average distance d from the host, and we integrate from 0 to an age t. The UV flux can be rewritten as a combination of the emission ratio evolution, ${ \mathcal R }(\tau )={L}_{y}/{L}_{\mathrm{bol}}$, determined from Section 4, and L_bol(τ), which accounts for changes in bolometric luminosity with stellar evolution. In evaluating Equation (4), we use evolutionary models (Dotter et al. 2008; Feiden 2016) for the bolometric luminosity, which can evolve substantially at young ages (see Figure 17) and allows a self-consistent metric across several Gyr. We compared the accumulated UV energies for planets around a 0.25 M_⊙ host and a 0.60 M_⊙ host with that of a planet around a Sun-like star.

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In order to compare with the lower-mass objects, the UV emission history for Sun-like stars required a treatment different from our approach with M dwarfs. Using a sample of G dwarfs, Ribas et al. (2005) determined the EUV evolutionary history of Sun-like stars as a simple power-lay decay with time across 0.1–6.7 Gyr. Since results from France et al. (2018) suggest that the N v emission and EUV of solar- and lower-mass stars are related by a simple multiplicative factor, we consider the Sun-like N v emission to follow the same power-law defined by Ribas et al. (2005). We thus scaled their result to the Sun's current N v luminosity from Duvvuri et al. (2021) using the solar minimum quiescent solar irradiance spectrum of Woods et al. (2009):

$\begin{eqnarray}&&{F}_{{\rm{N}}{\rm\small{V}},\odot }=0.105\,{\tau }^{-1.23},\mathrm{erg}\ {{\rm{s}}}^{-1}\ {\mathrm{cm}}^{-2},\ 0.1\lt \tau \lt 6.7\ \mathrm{Gyr},\end{eqnarray} \tag{ 5 }$

where τ is in units of Gyr. We take the current age of the Sun to be 4.6 Gyr, and this flux is evaluated at a distance of 1 au. Although the EUV bands from Ribas et al. (2005) and France et al. (2018) are different, this conversion allows a representative evolutionary history for the Sun-like N v emissions, which we can compare with the activity–age evolution determined for M dwarfs in Section 5.1. Because Equation (5) is only applicable to a particular age interval, to extrapolate to younger ages, we consider the UV emissions at τ = 0.1 Gyr to correspond to a saturated level relative to bolometric that is constant for τ < 0.1 Gyr, $\mathrm{log}{L}_{{\rm{N}}{\rm\small{V}}}/{L}_{\mathrm{bol}}=-5.75$, set by the model bolometric luminosity at τ = 0.1 Gyr. We then let the bolometric luminosity change according to the nonmagnetic Dartmouth stellar models for τ < 0.1 Gyr (e.g., Dotter et al. 2008).

We show our results in Figure 18 as the accumulated N v emission experienced by a planet at a fixed distance of 1 au from its host, for the three planet host cases, 0.25 M_⊙, 0.60 M_⊙, and 1.0 M_⊙, starting at 1 Myr, the lower limit of the evolutionary models. The absolute N v illumination is typically higher around the higher-mass objects than around the lower-mass stars. The shaded bands in Figure 18 propagate the uncertainty in the median evolution (as in Figure 16) through the integral of Equation (4). We do not include the intrinsic scatter, because we focus here on the cumulative average history from the rotation–activity analysis. Our treatment, however, excludes uncertainty from unknown systematics with the bolometric luminosity evolution, although at least with regards to the role of magnetism, its contribution may be minor (see Figure 17). Ribas et al. (2005) did not report uncertainties for their EUV evolution in Sun-like stars, and we thus do not include it either.

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In Figure 19, we also show the ratios (${{ \mathcal E }}_{0.60}/{{ \mathcal E }}_{\odot }$, ${{ \mathcal E }}_{0.25}/{{ \mathcal E }}_{\odot }$, and ${{ \mathcal E }}_{0.25}/{{ \mathcal E }}_{0.60}$) of the accumulated UV energies, propagating the uncertainty in the M-dwarf median rotational evolution. The left-side axes give the result for planets at equal distances around each star, and the right-side axes indicate the same ratio but for the respective habitable zones. This further assumes planets at that distance have remained there throughout history. The curves of the right axis are thus a constant multiple of those of the left axis, as in

$\begin{eqnarray}&&{\left.\displaystyle \frac{{{ \mathcal E }}_{a}}{{{ \mathcal E }}_{b}}\right|}_{\mathrm{HZ}}=\displaystyle \frac{{\int }_{0}^{t}{{ \mathcal R }}_{a}\,{L}_{\mathrm{bol},a}\,d\tau }{{\int }_{0}^{t}{{ \mathcal R }}_{b}\,{L}_{\mathrm{bol},b}\,d\tau }\times \displaystyle \frac{{d}_{\mathrm{HZ},b}^{2}}{{d}_{\mathrm{HZ},a}^{2}},\end{eqnarray} \tag{ 6 }$

where the habitable zone distances are determined by the 5 Gyr bolometric instellations. The right-side axes in Figure 19 thus indicate the relative accumulated UV energies for planets in the respective field-age habitable zones of each star.

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In Figure 19, we illustrate that, in the low-mass star regime, at a constant distance, exoplanets orbiting higher-mass objects experience a greater absolute accumulation of UV energy. However, within the respective habitable zones around each host, the planets orbiting the lower-mass star intercept much more UV energy. For example, relative to the Earth around the Sun, the same planet in the habitable zone around the 0.25 M_⊙ star will accumulate ∼20 times more UV energy by the time it has reached field age, which is ∼2 times more than the same planet in the habitable zone around the 0.60 M_⊙ star. Within the known uncertainty of the median rotational evolution, these figures can vary by factors of a couple. The curves of Figure 19 reflect at early times the differential bolometric luminosity evolution modulated by activity–age evolution at late times, with modulations of the evolution dictated by the age at which each star begins its power-law decay of UV activity. The bottom panel of Figure 19 also shows the results using both magnetic (dashed line) and nonmagnetic (solid line) models to illustrate that similarity of the luminosity evolution, as well as the robustness of our results to this potential systematic effect.

These results were derived specifically using N v emissions; however, the line is broadly representative of the entire FUV band, given the strong correlations between emission lines (Section 3.4). Thus, although the absolute energy levels determined from Equation (4) and shown in Figure 18 will vary between different choices of UV features, the ratios of Figure 19 are broadly representative of total FUV emissions. Moreover, to the extent that the N v emission is directly proportional to EUV emissions (France et al. 2018), the ratios of Figure 19 translate exactly to the relative exposure of planets to EUV emissions around each stellar host. This is a key result, given the importance of EUV fluxes for exoplanetary atmospheric heating and escape.

In Figure 20, we further show the importance of different epochs in the total accumulated UV exposure for exoplanetary systems. Relative to the cumulative quiescent UV energy experienced at an age of 5 Gyr, planets around low-mass hosts reach total energetic exposures exceeding ∼50% by ages of 800, 600, and 250 Myr, respectively, for hosts of mass 0.25, 0.60, and 1.0 M_⊙. These results quantify the importance of the first Gyr of stellar lifetimes in their total energetic input to exoplanetary systems. These early ages clearly need to be taken into account when considering the evolutionary histories of planetary systems and their responses to high-energy emissions.

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The FUV emission lines directly probe the transition region of the stellar coronal atmosphere. Our spectroscopic data showed how the different features change with rotation, directly implying how the atmospheric structure changes across time, between active and inactive low-mass stars. These data can thus be used to directly assess those changes through stellar chromospheric and coronal models (e.g., Fontenla et al. 2016; Peacock et al. 2019). The comparison of the rotation–activity relationships across Hα, the FUV, and X-rays, however, suggests more significant changes in the corona with spin-down relative to the changes in the deeper layers of the atmosphere. These differences must be a direct consequence of how the nonthermal heating processes change with rotation rate. Models of the magnetic heating process itself in M dwarfs must be able to account for these evolutionary effects as well as their significance at different layers of the atmosphere.

This aspect of nonthermal chromospheric/coronal heating is an important consideration in making dynamo inferences, because the observed rotation–activity relationships in different wave bands, often used for this endeavor, are mediated by the magnetic heating and are not directly defined by the dynamo processes. The multiwavelength rotation–activity relationships need to be reconciled with more homogeneous methodologies, including the possibly different behaviors in different upper atmospheric layers, in order to provide definitive conclusions with respect to dynamo theory. Crucially, while our analysis shows that a Rossby scaling works well for normalizing the rotation–activity relationships in both partly and fully convective M dwarfs, the empirical scaling may be masking real differences in these populations. This effect is potentially evident in alternative scalings; however, larger samples are required to examine these differences in the UV, to compare with other wave bands (e.g., Magaudda et al. 2020).

The stellar high-energy spectrum largely defines the prevalent photochemistry and mass-loss history of exoplanetary systems. While these effects have often been investigated in the context of individual nearby planets around low-mass stars, the present-day observations of the planetary atmosphere have been shaped by the cumulative history of these stellar emissions. Our spectroscopic FUV data provide the rotational evolution for M-dwarfs directly, which can be transformed using rotation–age gyrochronology relationships. Although the latter remain uncertain for M dwarfs, their improvement will greatly improve our assessments of this evolutionary history. Employing literature scaling relations using FUV emission features then enables estimates across the high-energy spectrum, including the EUV (e.g., France et al. 2016, 2018). To understand exoplanetary atmospheres, this stellar emission history needs to be taken into account.

Of particular importance is the likely difference between early- and mid- to late-M dwarfs, with regard to how long they persist in exhibiting near saturation level activity. By old field ages, planets in similar orbits around these two different kinds of hosts will likely have experienced different histories in high-energy radiative environments (e.g., Luger & Barnes 2015). Moreover, changes in the relative significance of FUV or NUV emissions over time will influence the prevalent exoplanetary atmospheric molecules that are observable today. Our results enable a way to account for these effects across different emission features and wave bands when considering new exoplanetary systems.

Using our rotation–activity correlation fits (Section 4), assuming no residual mass dependence, we can predict the most prominent FUV emission features in quiescence from a known rotation period and mass to generally within 0.3 dex of intrinsic scatter. This scatter pertains to the sample population and is likely a consequence of the combined effects of activity cycles, rotation variations in visible active regions, metallicity differences, and/or the stochastic nature of magnetic heating. As an example, we imagine a 0.4 M_⊙ star with a 60 day rotation period, with 3% uncertainty on the mass and 5% in the rotation period. Accounting for scatter in the Rossby number calibration and our best-fit parameters, our rotation–activity correlations would predict a mean value of $\mathrm{log}({L}_{{\rm{C}}{\rm\small{IV}}}/{L}_{\mathrm{bol}})\,=-6.21\pm 0.16$. With improved rotation–age relations, our data will enable a more comprehensive assessment of the high-energy radiative input to exoplanetary systems across time.

In Section 3.4, we correlated FUV line flux measurements, illustrating tight relationships between different emission features probing distinct temperatures of the transition region. In the literature, these kinds of correlations have been expressed similarly, but instead of luminosity, they have been expressed using the surface flux, normalizing the luminosities by the stellar surface area (e.g., Wood et al. 2005; Youngblood et al. 2017). While this attempts to normalize the emissions accounting for the area of the emitting region to enable comparisons across different kinds of stars, the inclusion of the radii introduces additional uncertainty and increases the error correlations, as the radius uncertainty will dominate the error budget relative to the parallax and flux measurements. ¹⁸ In Figure 21, we illustrate this effect using the same data for C iv and Si iv that comprise the upper right panel of Figure 5, but transforming the emission measurements to surface flux with the radius determinations from Table 2. The representative 2σ error ellipses now are all highly inclined, revealing the strong correlations between the uncertainties in each quantity.

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The power-law slope estimated in such line–line correlations using the surface flux should be identical to the luminosity approach used in this work; however, the uncertainties on such estimates do not accurately reflect the nature of the underlying data if they do not account for this correlated error, and they are less precisely constrained when accounting for the actual radii uncertainty in the analysis. This additional uncertainty may obscure intrinsic scatter in the analyzed correlations. We therefore recommend luminosity as a currently more robust choice when defining predictive relations between stellar emission features. However, whenever surface fluxes are necessary, a careful accounting of possible correlations should be included.

In Section 4, we analyzed the rotation–activity relation of M-dwarfs in FUV emission lines with the FUMES and literature samples. We used the empirical calibration of Wright et al. (2018) to estimate the convective turnover time from the stellar mass in computing the Rossby number, Ro = P/τ_c , for each star. This empirical calibration is based on minimizing the scatter in the X-ray rotation–activity correlation of low-mass stars. Wright et al. (2011) details the typical procedures used in developing this kind of empirical calibration. Since the Ro is generally closely related to the internal dynamo action (although see Reiners et al. 2014), the use of this kind of calibration enables a dynamo comparison among stars with convective interiors, from F- to M-type stars. However, changes in the kind of magnetic dynamo that generates field in fully convective stars, for example α² instead of α–Ω (e.g., Browning 2008), suggest that there is no physical reason a single such calibration should work across that full range. Moreover, with relatively deeper convective zones, a single representative value for the timescale of convective motions is likely an increasingly poor approximation with decreasing stellar mass. Accordingly, although the empirical calibration reduces the X-ray rotation–activity scatter, it may not be representative of the dynamo behavior in the fully convective regime.

Nevertheless, using these relations provides a means to compare to literature results and test how the X-ray-calibrated relation applies to the activity at other wavelengths, as we have done in Section 4. As detailed in this appendix, we further investigated how those results are impacted by the choice of empirical calibration for the convective turnover time as a function of stellar mass. In Figure 22, we plot three such literature calibrations from Wright et al. (2011), Núñez et al. (2015), and Wright et al. (2018), including the scatter about those relationships, as reported in those works or obtained via private communication (0.064 dex, Nuñez, A.). The Núñez et al. (2015) calibration uses the same data as Wright et al. (2011), but assumes the canonical value for the best-fit slope (−2) instead of the best-fit value from Wright et al. (2011), and the Wright et al. (2018) calibration, which sits in between the other two, updates the Wright et al. (2011) result with more fully convective stars and a slightly different functional form.

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We refit the rotation–activity data presented in Section 4, using the same methods—but also using the two additional literature calibrations in order to determine their impact on the best-fit parameters. These results are shown in Table 9 for the critical Rossby number, Ro_c , and in Table 10 for the slope of the unsaturated regime, η. The effect of the different calibrations is most readily illustrated by comparing the results using Wright et al. (2011) versus Núñez et al. (2015), as they have a greater separation in τ_c –M space (see Figure 22). The Núñez et al. (2015) calibration gives higher convective turnover times at a given mass than that of Wright et al. (2011). Consequently, the best-fit Ro_c is systematically smaller using Núñez et al. (2015) than it is when using the Wright et al. (2011) calibration—higher τ_c corresponds to smaller Ro. Between the two calibrations as applied to our data, this yielded a systematic offset of ∼0.04–0.05 in Ro_c ; see Table 9.

Table 9. Critical Rossby Systematics: Ro_c ^a

Line	Wright et al. (2011)	Núñez et al. (2015)	Wright et al. (2018)
Lyα	$0.25{\pm }_{0.12}^{0.20}$	$0.19{\pm }_{0.10}^{0.14}$	${\bf{0.21}}{\pm }_{0.11}^{0.16}$
Mg ii	$0.23{\pm }_{0.09}^{0.14}$	$0.19{\pm }_{0.08}^{0.11}$	${\bf{0.20}}{\pm }_{0.08}^{0.11}$
C ii	$0.28{\pm }_{0.14}^{0.15}$	$0.20{\pm }_{0.09}^{0.10}$	${\bf{0.24}}{\pm }_{0.11}^{0.12}$
Si iii	0.23 ± 0.08	0.18 ± 0.06	0.20 ± 0.07
He ii	0.22 ± 0.06	$0.16{\pm }_{0.04}^{0.05}$	0.19 ± 0.05
Si iv	0.28 ± 0.08	0.22 ± 0.06	0.24 ± 0.07
C iv	0.22 ± 0.08	0.17 ± 0.06	${\bf{0.18}}{\pm }_{0.06}^{0.07}$
N v	0.22 ± 0.09	0.18 ± 0.06	0.19 ± 0.07

Note. Bold values indicate results used in analyses of Section 4.

^a Reported parameters correspond to the median of the marginalized posterior distribution, with uncertainties indicating the central 68% confidence interval. The reference for each column indicates the source used for the calibration of the mass-dependent convective turnover time.

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Table 10. Unsaturated Slope Systematics: η ^a

Line	Wright et al. (2011)	Núñez et al. (2015)	Wright et al. (2018)
Lyα	$-1.21{\pm }_{0.54}^{0.39}$	$-1.25{\pm }_{0.56}^{0.42}$	$-{\bf{1.26}}{\pm }_{0.58}^{0.41}$
Mg ii	$-1.72{\pm }_{0.51}^{0.37}$	$-1.86{\pm }_{0.58}^{0.44}$	$-{\bf{1.86}}{\pm }_{0.55}^{0.41}$
C ii	$-2.25{\pm }_{0.72}^{0.62}$	$-2.25{\pm }_{0.70}^{0.57}$	$-{\bf{2.38}}{\pm }_{0.77}^{0.64}$
Si iii	$-1.94{\pm }_{0.41}^{0.39}$	$-2.10{\pm }_{0.42}^{0.41}$	$-{\bf{2.08}}{\pm }_{0.46}^{0.41}$
He ii	$-2.11{\pm }_{0.29}^{0.27}$	$-2.15{\pm }_{0.32}^{0.30}$	$-{\bf{2.19}}{\pm }_{0.33}^{0.30}$
Si iv	$-2.21{\pm }_{0.41}^{0.40}$	$-2.31{\pm }_{0.42}^{0.39}$	$-{\bf{2.32}}{\pm }_{0.44}^{0.42}$
C iv	$-1.92{\pm }_{0.39}^{0.37}$	$-2.04{\pm }_{0.39}^{0.37}$	$-{\bf{2.02}}{\pm }_{0.41}^{0.39}$
N v	$-1.76{\pm }_{0.38}^{0.37}$	$-1.88{\pm }_{0.37}^{0.36}$	$-{\bf{1.84}}{\pm }_{0.40}^{0.37}$

Note. Bold values indicate results used in analyses of Section 4.

^a Reported parameters correspond to the median of the marginalized posterior distribution, with uncertainties indicating the central 68% confidence interval. The reference for each column indicates the source used for the calibration of the mass-dependent convective turnover time.

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This effect on the best-fit critical Rossby number is relatively intuitive, given the direct impact on the convective turnover time between calibration choices. However, we also observe a systematic difference in the best-fit slope from the rotation–activity analysis when changing between empirical calibrations. With our data, larger assumed convective timescales (smaller Ro) yielded systematically steeper slopes (more negative) for the unsaturated regime; see Table 10. Although small, this effect is generally evident across the eight different lines we analyzed. To illustrate this systematic effect, we show representative ellipses for the joint posterior distributions of Ro_c –η across four FUV lines in Figure 23. The error ellipses for each individual line shift to the left (smaller Ro_c ), and down (steeper η) when changing calibrations from Wright et al. (2011) to Núñez et al. (2015). Because these fit parameters are correlated, it is perhaps unsurprising that systematic effects would appear in both Ro_c and η; however, the systematic shift is not in the same direction, as the critical Rossby number and slope posteriors are anticorrelated rather than correlated.

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We attribute this systematic effect to the nonlinearity of the calibrations as applied to individual samples. If the choice of empirical calibration scaled the assumed Rossby numbers of all of the stars in the same way, we could expect the best-fit slope of the unsaturated regime to remain constant. A comparison of the calibrations for τ_c (M) (see Figure 22), shows that this is generally not the case. Thus, depending on the sample of stars, some objects shift in Ro space more than others. A large number of fully convective stars in the sample would likely increase the magnitude of this systematic effect on the best-fit slopes between the calibrations of Wright et al. (2011) and Núñez et al. (2015), as that is where those functions largely diverge. It is therefore difficult to estimate the extent of this systematic effect on the slopes without doing the entirety of the analysis with multiple calibrations for τ_c for each sample of stars. This makes it somewhat more difficult to draw comparisons across the literature, as methods have been updated and evolved over time, with different stellar samples. Future comparisons across wave bands and samples will greatly benefit from homogeneous analysis methodologies. The framework presented in this paper for the rotation–activity work (Section 4) accounts for known uncertainties across all available data, as well as possible correlations, and it includes a measure of the intrinsic scatter within the regression fit. The presence of these systematics effects, especially when using a quantity as uncertain as the convective turnover timescale, also supports the argument in favor of finding simpler descriptions that capture the relevant physics for characterizing the dependence of activity on stellar physical and rotational properties, as discussed in Reiners et al. (2014). We tested some of those methods in Section 4.4.