Living with a Red Dwarf: X-ray, UV, and Ca II Activity-Age Relationships of M Dwarfs

The vast majority of stars in the nearby stellar neighborhood are M dwarfs. Their low masses and luminosities result in slow rates of nuclear evolution and minimal changes to the star's observable properties, even along astronomical timescales. However, they possess relatively powerful magnetic dynamos and resulting X-ray to UV activity, compared to their bolometric luminosities. This magnetic activity does undergo an observable decline over time, making it an important potential age determinant for M dwarfs. Observing this activity is important for studying the outer atmospheres of these stars, but also for comparing the behaviors of different spectral type subsets of M dwarfs, e.g., those with partially vs. fully convective interiors. Beyond stellar astrophysics, understanding the X-ray to UV activity of M dwarfs over time is also important for studying the atmospheres and habitability of any hosted exoplanets. Earth-sized exoplanets, in particular, are more commonly found orbiting M dwarfs than any other stellar type, and thermal escape (driven by the M dwarf X-ray to UV activity) is believed to be the dominant atmospheric loss mechanism for these planets. Utilizing recently calibrated M dwarf age-rotation relationships, also constructed as part of the $\textit{Living with a Red Dwarf}$ program (Engle&Guinan 2023), we have analyzed the evolution of M dwarf activity over time, in terms of coronal (X-ray), chromospheric (Lyman-$\alpha$, and Ca II), and overall X--UV (5--1700 Angstrom) emissions. The activity-age relationships presented here will be useful for studying exoplanet habitability and atmospheric loss, but also for studying the different dynamo and outer atmospheric heating mechanisms at work in M dwarfs.


INTRODUCTION & BACKGROUND: STUDYING M DWARFS
Magnetic fields are theorized to exist around all cool, main sequence stars, as massive as late F dwarfs, down through the late M dwarfs (collectively referred to here as FGKM dwarfs).These fields are responsible for (or contribute to) a range of observable behaviors, which include heating the outer stellar atmosphere to 10 5 -10 6 K.These heated plasmas comprise the (if structured similar to the Sun) stellar chromospheres, transition regions, and coronae.The mechanism responsible for generating these magnetic fields is the dynamo effect, which involves contributions from both convective motions and stellar rotation, though in varying amounts depending on the mass of the star (described later).The combination of FGKM dwarf magnetic fields and stellar winds are also responsible for the spindown effect.First discovered just over 50 years ago (Skumanich 1972), this effect results in the lengthening of an FGKM dwarf's rotation period and the weakening of its dynamo as it ages.
Thus, the rotation periods and/or magnetic activity levels FGKM dwarfs can serve as age determinants.The age-rotation relationships we have constructed for M0-6.5 dwarfs as part of the Living with a Red Dwarf program have been described in a companion paper (Engle & Guinan 2023).This paper focuses on M dwarf activity-age relationships.
Several observable proxies exist for measuring a star's level of magnetic activity.Coronal emissions are regularly characterized by observations at soft X-ray energies.The soft X-ray band encompasses 0.1 -10 keV although, given the sensitivity of most current X-ray instruments and the narrow energy range over which quiescent FGKM dwarf coronae emit, most observed fluxes are measured in a somewhat narrower energy range (e.g., 0.2-3 keV, 0.2-5 keV).Emissions from the transition region or chromosphere are often characterized by a range of spectral features at UV and optical wavelengths.Even though these optical spectral features exist that probe plasmas with chromospheric temperatures, the majority of chromospheric emissions are observed in the UV range.For this reason, we will use the term X-UV to primarily refer to the outer atmospheric (coronal-chromospheric) emissions or activity of M dwarfs.The astrophysical importance of measuring these quantities has prompted widespread use in many studies (though this is a very active field with far too many studies to allow a comprehensive list, a selection of recent works involving rotation rates and/or atmospheric activity levels include (Wright et al. 2011;Stelzer et al. 2013; Barnes et al. 2016;Guinan et al. 2016;Youngblood et al. 2017;Suárez Mascareño et al. 2018;Curtis et al. 2020;Linsky et al. 2020;Magaudda et al. 2020;Johnstone et al. 2021;Youngblood et al. 2021;Marvin et al. 2023;Ibañez Bustos et al. 2023)).
For this study, we have constructed X-UV activity-age relationships for M0-M6.5 dwarfs (M dwarfs hereafter), using ages determined with the age-rotation relationships presented in Engle & Guinan (2023).These relationships are not only valuable for the information they provide about the stars themselves; they also delineate the space environments that M dwarfs create for any exoplanets orbiting them.
This information is invaluable for multiple reasons.First is that M dwarfs represent ∼75% of the nearby stellar inventory (Reylé et al. 2021) -studying M dwarfs allows us to understand the behavior of the universe's largest stellar population.Also, M dwarfs are less massive (M ≈ 0.6 -0.1 M ⊙ ), smaller (R ≈ 0.6 -0.1 R ⊙ ), and cooler (T eff ≈ 3900 -2850 K) than the Sun, and have much lower luminosities (L ≈ 0.06 -0.001 L ⊙ )1 .Their low masses result in slow core reaction rates and long, seemingly stable lifetimes (∼100 Gyr up to as long as ∼1 trillion (∼10 12 ) years (Choi et al. 2016)).While their bolometric luminosities are low, and experience little change over billions of years, the X-UV activity of M dwarfs is comparatively strong and highly variable.This can severely impact hosted exoplanets that would otherwise need to orbit near to the (low luminosity) star to maintain a temperate surface.As stellar X-UV emissions are substantial drivers of both photochemical reactions within, and loss of, planetary atmospheres, determining the evolution of these emissions with age has achieved a broader impact within the field, increasing the importance of such studies.
The paper is organized as follows.In Section 2, we briefly discuss the issues surrounding the potential habitability of planets orbiting M dwarfs.In Section 3, we present the data used for this study.In Section 4, the activity-age relationships will be provided and discussed.Finally, in Section 5, we will provide a brief summary of our results.

ORBITING M DWARFS: CAN THEY HOST HABITABLE PLANETS?
Whether M dwarfs can host habitable planets is a question that has received considerable attention from the research community.For a thorough, recent review of the numerous factors influencing the habitability of planets orbiting G, K, and M dwarfs see Airapetian et al. (2020) and references therein.Although not the focus of this paper, since X-UV activity plays a prominent role in exoplanet atmospheric evolution/loss/retention, and habitability studies, we will provide a (very) short summary of the topic.Some of the major aspects governing potential M dwarf exoplanet habitability can be classified as: positives, negatives, and unknowns.
'The positives' mainly involve the simple statistics.M dwarfs represent the most numerous stellar component of observed space, live extremely long main sequence lifetimes with near-negligible changes in either size, optical luminosity, or surface temperature, and studies have thus far shown them to host terrestrial-size planets at a higher rate than more massive stars.All considered, even if the odds are overall very slim that M dwarfs can host habitable planets, there are so many to consider that perhaps some exoplanets have still managed to evolve with the necessary combination of characteristics that allow them to support habitable environments.It is also worth noting that M dwarf exoplanets are ideal targets for atmospheric composition studies via transmission spectroscopy, due to their relatively deep transits when compare to other more massive (luminous) spectral types.Though this last point is not a 'positive' for the actual habitability of the planet, it is a benefit when studying the planet's potential habitability and has also generated further interest in the field.
'The negatives' center on the planetary consequences of M dwarf magnetic activity.Due to their lower bolometric luminosities, their HZs are very close (∼0.07 -0.4 AU for M5 -M0 V stars) so that tidal locking and possible tidal heating may take place (Jackson et al. 2008).With the planet orbiting so near to its host star, it's exposed to more intense X-UV emissions, along with flares and coronal mass ejections (CMEs).The end result can be the stripping (erosion) of the planet's atmosphere and sterilization of its surface.
'The unknowns' primarily involve finer resolution of M dwarf activity, and at present almost all details about the planets themselves.The more intense X-UV radiation that potentialy habitable exoplanets around M dwarfs will receive has already been mentioned as an overall negative.However, there is the additional unknown component of the exact role that NUV radiation played in prebiotic chemistry on Earth (Ranjan et al. 2017).It is possible that M dwarfs are not luminous enough in the NUV to drive such chemistry, although their high flare frequencies could help compensate for this (Froning et al. 2019;Brown et al. 2023).All considered, much more research is still needed on these topics.The reader is again directed to Airapetian et al. (2020) for a more thorough discussion, but there are numerous paths along which both the formation and subsequent evolution of the planet and its atmosphere can proceed.M dwarfs have higher luminosities as they are still collapsing and evolving onto the main sequence.Some studies show this could force HZ planets into a runaway greenhouse environment, although (depending on conditions within the nebula from which the starplanet system formed) the planets could be protected from this.The strength/extent of the planet's magnetic field and whether it can offer adequate protection is another important unknown Ribas et al. (2016a), as is the amount of atmospheric replenishing that occurs due to volcanic outgassing.Related to this last unknown are the results of France et al. (2020), proposing that planets around older and less active M dwarfs (their study focused on GJ 699, aka Barnard's Star, age ≈ 9.5 Gyr and a target within this study as well) could develop 'second generation' atmospheres via outgassing and at last become habitable worlds.
Particularly important unknowns on the stellar side are robust delineations of both M dwarf flare frequencies and CMEs as they evolve over time.A comprehensive study by Wood et al. (2021) finds that "CMEs from M dwarfs may be much less common than generally thought, despite the high flare rate, so perhaps CME exposure is not as big a factor for habitability as often supposed."Even the habitability of a tidally locked planet can remain in question due to unresolved planetary characteristics such as cloud cover, atmospheric circulation, and surface arrangement.
Although the field is still some time away from precisely determining the conditions present in a wide range of exoplanet atmospheres, surfaces, and interiors, we have at last developed a method for determining the ages of the host stars (and thus their coeval planetary systems) and can more precisely track the X-UV evolution of M dwarfs over time.

A Brief Discussion of Target Ages
We will begin by quickly summarizing the sources of M dwarf ages used in this study.The agerotation relationships of Engle & Guinan (2023) were used to determine most of the stellar ages used in this study.For a limited number of targets a reliable, previously determined age was available and this age was used instead.These targets are either members of star clusters/associations, belong to the Thick Disk or Halo populations of the Milky Way, or have a companion object for which an alternative age could be determined and applied to the M dwarf.The data initially displayed two evolutionary paths for M dwarf rotation periods (Guinan & Engle 2018), with the division occurring between spectral types M2 and M2.5.This division was confirmed as additional data were added to the relationships, and is likely caused by the M dwarfs on either side of this divide operating under different dynamo mechanisms.A similar spectral type division was also found in the activity measure data of Mullan & Houdebine (2020).For ages below ∼2.5-3 Gyr, a further subdivision between M2.5-3.5 and M4-∼6.5 stars is apparent in the cluster rotation rate data, and this subdivision is most likely due to the much longer pre-main sequence lifetimes of the M4-6.5 subset.
The age-rotation relationships of all M dwarf subsets show an inflection point, sharply dividing each relationship into a first (younger) and second (older) track.This is in agreement with previous studies (see Curtis et al. 2020), who referred to the end of the younger track as a stalling of the spindown effect, before resuming again along the older track.A proposed theory for why this occurs is that zero age main sequence M dwarfs initially have a strong differential rotation profile within their interiors.Along the younger track, the stellar interior is synchronizing and transfer of angular momentum from the interior compensates for that which is lost from the surface (Spada & Lanzafame 2020).The younger track ends with the re-synchronization of the stellar interior, when the star begins rotating as a solid body, and a new track of rotational evolution begins.

A Note on Target Spectral Types
The targets have been divided into subsets based on spectral type.As Magaudda et al. (2020) presented the largest recent database of M dwarfs with activity measures, their stellar parameters (derived using the empirical relationships of Mann et al. (2015)) are used for many of the targets in this study.However, the literature was also searched for spectral type measures of all stars from Magaudda et al. as a check.This literature search was also used for targets presented in this study that were not in Magaudda et al.. Preference was usually given to studies that either were more recent (e.g., included updated model atmospheres), employed spectral or SED fitting, or specifically focused on the determination of M dwarf parameters instead of a wider range of (or all) stellar types.For all tables presented here, if (depending on the spectral type source that was used) a target could have potentially been placed in the other M dwarf subset, that target's name will be in bold.

X-UV Activity Measures
X-rays, and most wavelengths of UV radiation, are not able to reach the Earth's surface and thus require observations from space-based or high altitude (balloon-borne) observatories.As these are very high quality and well-calibrated instruments, the data they produce is second to none.Their observing time, however, is both limited and highly sought after (oversubscribed).Though oversubscription of these instruments is understood and unavoidable, it raises the difficulty in repeatedly measuring multiple targets over time which would greatly benefit the X-UV relationships.The main issue is that stellar coronal-chromospheric activity is variable over timescales of hours (flares), days to months (stellar rotation), years (magnetic activity cycles), and finally Myr to Gyr (weakening of the stellar magnetic dynamo due to spindown).Altogether, the amplitudes of variation involved can be two orders of magnitude or more for the most extreme cases.Single observations of sufficient exposure time can allow the observer to avoid contamination by stellar flares, which can cause some of the highest-amplitude variations.The issue still remains that, until a target has been observed multiple times over a significant time span, the observer is left with an incomplete measure of the target's mean activity level and this effect will likely propagate into higher uncertainties within the activity-age relationships.
The X-ray database currently has two advantages.First, measurements don't require high spectral (energy) resolution, allowing observations to be gathered more efficiently.Second, the German-US-UK collaborative Röntgen Satellite (RÖSAT) was designed for efficient X-ray observing and carried out numerous observations of stars in the 1990s, including an invaluable all-sky survey.Observations with modern X-ray missions (e.g.Chandra, XMM-Newton, Neil Gehrels Swift, and very recently eROSITA) can be added to this previous data if available and begin to better characterize the range of activity (and mean activity level) for each target.
The majority of UV activity from cool dwarfs can be attributed to emission lines.Specifically for M dwarfs, a single feature (the Lyman-α 1216Å [Lyα] line) is responsible for 50% of the total UV emissions or more (France et al. 2012).Due to its relative strength, this line has naturally become a high-priority target for observers, but a successful analysis requires data with medium to high spectral resolution and/or an instrument that can successfully mitigate geocoronal contamination.Interstellar absorption features of hydrogen and deuterium occur at the same wavelengths, and these need to be modelled and accounted for before the 'pure' stellar Lyα flux can be obtained.The instrument and analysis requirements result in a more limited database of reliable Lyα measures, and most of those presented here were obtained from the literature and measured in the previously described fashion by the MUSCLES and mega-MUSCLES HST observing programs (see France et al. 2016;Youngblood et al. 2016;Loyd et al. 2016;Froning et al. 2019;Wilson et al. 2021).
Fortunately, there are also a number of cool dwarf magnetic activity 'tracers' observable at optical wavelengths.As in the UV, this activity is observed through emission (sometimes absorption) lines.Though popular alternatives exist (e.g., Hα and Na i), we have selected the Ca ii H & K features for this study.The behavior of these features in cool stars has been studied for over 40 years (Vaughan & Preston 1980), and they occur at the short end of the optical spectrum (∼3930-3970Å), where M dwarf photospheric contributions will be minimal.Even so, a reliable and well-studied quantity (R ′ HK ) has been derived for these features which involves removing the photospheric contribution to allow an easier comparison of different spectral types.Still, a recent study by Marvin et al. (2023) found that offsets can occur depending on the exact methodology used over time.When applicable, the corrections of Marvin et al. were applied to target data before the representative value presented here was determined.After applying the correction, comparing the values determined in this study to those of Marvin et al.returned an average difference of ∆ log R ′ HK ≈ 0.031.The clear advantage of optical magnetic activity tracers like Ca ii HK is that they can be observed using ground-based instruments with lower oversubscription rates.Still, medium-to high-resolution spectroscopy is required and the faintness of the targets can be prohibitive for smaller telescopes, especially when observing M dwarfs at short optical wavelengths.Previous spectroscopic surveys were dedicated to such activity measures, but recent planet-hunting surveys (e.g., with the HARPS instrument) have produced large numbers of high quality spectra that have also been used for activity measures, many of which were included in this study (see Perdelwitz et al. 2021).

THE M DWARF ACTIVITY-AGE RELATIONSHIPS
Here we present the relationships constructed by the Living with a Red Dwarf (LivRed ) program, along with a discussion of the methods used to develop them, and their results.As was done previously for the rotation-age relationships, a two-segment linear equation was defined via numpy.piecewiseand then fit to each data set using scipy.optimize.least_squares.Within the sample of each activity index, an average 'scatter' value was calculated using the targets with multiple measures, and this value was used as the uncertainty for single-measure targets when fitting the data.This application of a larger uncertainty value than most targets' data would indicate assumes that all targets have variable activity levels, but only those with multiple measures have given an insight into what the average breadth of those variations should be.In Figs 2, 3, 4, 5, the lighter error bars show instances where this average scatter value has been applied, and the darker error bars show the uncertainty based on actual data of the target itself.
Activity-Age relationships have been constructed in the X-ray, UV, and for the short optical wavelength Ca ii emission line doublet.The Ca ii observational database is much richer than that of Lyα, and the two emission features form in plasmas of overlapping temperatures and stellar atmospheric regions (though the Lyα region extends to higher temperatures/altitudes, while the Ca ii regions extends to lower temperatures/regions - Vernazza et al. 1981;Hall 2008).These relationships serve two purposes: they provide additional age determinants for M dwarfs, as we will discuss, but they also delineate the average high energy (X-UV) fluxes that exoplanets orbiting such stars have been, are presently being, or will in the future be, subjected to.Further, thoroughly studied relationships exist between Ca ii, Lyα, and numerous other UV emission features (e.g., see Melbourne et al. 2020;Pineda et al. 2021 and references therein).These could allow any or all of the relationships (or combinations thereof) presented both here and in Engle & Guinan (2023) to be utilized in refining an M dwarf's age, and the high energy environment that an exoplanet orbiting it would be subjected to.
The X-ray relationships are plotted in Fig 2 (X-ray luminosity over time) and Fig 3 (the ratio of X-ray to bolometric luminosity over time), and the best-fitting parameters are presented in Table 1.We have endeavored to gather multiple measures of X-ray activity for each star whenever possible, to help mitigate scatter due to stellar variability.This does not eliminate the issue as many of the stars still only possess 1 or 2 measures.As further measures of the targets are carried out by current and future X-ray missions, it will be interesting to see how (or if) the remaining scatter in Figs 2 and 3 is reduced.M dwarf ages were calculated using the Age-Rotation relationships presented in Engle & Guinan (2023), unless there was an existing age determination (e.g., membership in a cluster or the Thick Disk/Halo populations).As seen in the plots, the sample is separated into the same 'early' and 'mid-late' subsets as for the rotation relationships, and the X-ray activity of both M dwarf subsets decreases by ∼2.5-3 orders of magnitude over the plotted age range (∼0.1 -12.5 Gyr).
The most notable difference between the M dwarf subsets is the substantially shorter 'saturation phase' of early M dwarfs, compared to the mid-late subset.Though many previous studies fixed the saturation phase to a flat level of activity, we instead opted to leave all parameters as free.In both subsets, there appears to be a slight decline in activity over the duration of the saturation phase, though it is within the scatter of the data.As with the age-rotation relationships, we advise restraint when using the first 'track' of the X-ray relationships to quote a precise age.Here, however, is where one aspect of the early subset relationship's usefulness is found.After ∼0.5 Gyr, early M dwarfs exit their saturation phase and X-ray activity levels begin to decline more apidly.This establishes a range of ages where, although the rotations of early M dwarfs are still evolving along the less reliable, first (re-synchronizing) track, their X-ray activity has shifted onto its second track, where more reliable ages can be determined.In effect, the X-ray relationship extends the range over which more reliable ages can be determined for early M dwarfs.Additional methods of age determination involving kinematics (Lu et al. 2021) and abundance (Carrillo et al. 2023) can also be employed to achieve results based on as many techniques as the data allows.This opportunity to determine M dwarf ages through multiple methods, or to compare the ages of each method for targets with high-quality data, holds great potential for future studies.
Although the mid-late subset appears to begin its saturation phase at a slightly lower X-ray luminosity than the early subset (∼28.9 vs. 29.2erg s −1 cm −2 , according to the fits [2], though within the scatter), it's important to note that both subsets show almost the same initial levels of X-ray activity when normalized according to bolometric luminosity (see Fig. 3).Also, the mid-late M dwarfs remain in their saturation phase for as long as ∼2.3 Gyr.This presents a far more extended period of time over which mid-late M dwarfs sustain enhanced levels of X-ray activity, making the ages determined via this method less reliable over a larger age-range, and any exoplanet orbiting a mid-late M dwarf must endure an initial phase of high X-ray activity that is ∼ 4 − 5× longer than it would be if the planet were orbiting an early M dwarf.The consequences of this phase on the planet's atmosphere, even potentially affecting its ability to retain elements heavier than H/He (see Ribas et al. 2016b;Shields et al. 2016), would be particularly relevant to M dwarfs and stands to benefit from additional theoretical work.
This dramatic difference in saturation phase length is also interesting in terms of stellar astrophysics.Early M dwarf X-ray activity begins to decline shortly after they arrive on the main sequence, but mid-late M dwarf X-ray activity remains saturated until they are nearly finished evolving along their first rotation track.This is likely a result of the different dynamo mechanisms at work within the two subsets, and the relative contributions of rotation/convection towards each one.As described in Mullan & Houdebine (2020), there are three stellar magnetic dynamos for which models exist -αΩ, α 2 , and α 2 Ω (where α represents the contribution due to convective turbulence (in the presence of rotation, giving rise to Coriolis force effects), and Ω represents angular velocity).The more massive early M dwarfs are theorized to operate under the αΩ dynamo -similar to the Sun -and this would explain why the spindown effect, and decreasing angular velocity, would quickly result in weakening magnetic fields and lower X-ray activity levels.The mid-late M dwarfs, however, generate magnetic fields via either the α 2 Ω or α 2 dynamo.Due to the diminished contribution of angular velocity, mid-late M dwarfs can evolve though > 2 Gyr of spindown, and associated angular velocity loss, without any substantial weakening of their magnetic fields.Eventually, though, rotation slows to the point where it sufficiently impacts the convective turbulence "α effect" mechanism, the magnetic field begins to weaken, and X-ray activity begins to decline.However, this is simply a qualitative description of how the dynamo models may explain the data and relationships as presented.A full, theoretical explanation is outside the scope of the current paper.
For a direct measure of UV activity, the Lyα emission line has been used, as it represents the majority of all UV flux emitted from cool stars, especially so for M dwarfs.The Lyα database consists of far fewer measures compared to the X-ray, though it is no less important as Lyα probes a different layer of the stellar atmosphere and allows accurate stellar wind measures.Obtaining these are crucial for gauging the likelihood that an exoplanet will be able to retain its atmosphere.Fig 4 plots the evolution of Lyα activity over time, using the same early and mid-late M dwarf subsets as for the X-ray relationships.As with the X-ray relationships, the best-fitting parameters for the Lyα data are provided in Table 1.The lack of data is immediately visible in the plots, and the increased uncertainty of the fits.The inflection points are more poorly defined, as is the saturation phase for the early subset where only 3 measures define this first track.Consequently, this diminishes the reliability of the fit for the second track, as well, though the current fit for this phase is not unreasonable for such a limited dataset.For the mid-late M dwarfs, a similar lack of data near the transition between the saturation and declining-activity phases leaves ambiguity in the determination of the inflection point.However, the fit does appear to track the data for the older stars, which makes the current estimate of ∼1.5 Gyr for the duration of the saturation phase a reasonable one.With the limited data currently available, the mid-late subset appears to have a slightly higher level of Lyα activity, relative to bolometric luminosity, when compared to the early subset.This agrees with Linsky et al. (2020), who also reported a trend of increasing activity with decreasing effective temperature.
The final activity index we have analyzed for this study is the log R ′ HK index, derived from the Ca ii H & K emission features at ∼3968.5Å and ∼3933.7Å,respectively.As this measure can be obtained from the ground, there is a greater availability of instruments and observing time.As a result, the Ca ii database is much more extensive than Lyα, and begins to rival the X-ray database, even though there haven't yet been any sensitivity-limited all-sky Ca ii surveys for M dwarfs as RÖSAT carried out in the X-ray regime.Due to the large-scale spectroscopic surveys searching for planets around low-mass stars that have come online in recent years, a particular advantage of the Ca ii database is that it contains a substantial sample of targets with time-series measures.Many of these surveys have determined or confirmed rotation periods for their M dwarf targets using the repeat measures of activity indices such as Ca ii.The evolution of M dwarf Ca ii activity over time is plotted in Fig. 5 (again, using the same M dwarf subsets as for the X-ray and Lyα relationships) and the best-fitting paramaeters are provided in Table 1.As Fig. 5 shows, the richer data sets allow for a better understanding of the levels of activity during the saturation phase.According to the fits, early M dwarfs appear slightly more active during both their saturation phase and at very old ages.Both subsets show a ∼1.5 order of magnitude decrease in Ca ii activity; less than the decrease they experience in X-ray activity.This phenomenon of chromospheric activity experiencing a less drastic decline with age has also been observed in other studies (Linsky et al. 2020;Loyd et al. 2021;Pineda et al. 2021).One potential theory put forth is that cool star chromospheres are acoustically heated, and the coronae are magnetically heated.This hints at a potentially interesting scenario where the chromospheres and coronae of early and mid-late M dwarfs are, when compared to each other, declining by the same relative amounts even though a) one subset is partially convective and the other is fully convective, and b the chromospheres are acoustically heated (driven by convection), yet the coronae of the two subsets should be heated by different convection/rotation contributions.In studying several FUV emission features, Loyd et al. (2021) further found that transition region activity levels may behave intermediate to those of the chromosphere and corona, perhaps indicating how the different heating contributions vary between the atmospheric layers.Linsky et al. (2020) found that FGKM stars all display similar levels of saturated coronal activity, but as they age they display an intensifying trend of inceasing coronal activity with decreasing effective temperature.As mentioned previously, Linsky et al. also reported a trend of increasing Lyα activity with decreasing effective temperature, but the trend was essentially consistent across stellar age and not as steep as the coronal activity trend for their oldest target group.Further, Linsky et al. proposed additional theories, including that flare heating could play a more prominent role in the atmospheres of lower mass stars, or that the higher surface gravity of M dwarfs could lead to different atmospheric structures, higher photospheric gas pressure and, in turn, stronger magnetic fields.
The chromospheric activity of early M dwarfs exits the saturation phase after ∼600 Myr, where coronal activity exits after ∼500 Myr.This difference is not conclusively determined and the chromospheric and coronal activity levels can be considered to exit the saturation phase at a similar age to each other.The mid-late M dwarf inflection point is not strictly defined by the fits, since the Ca ii database also suffers from a (albeit less substantial) lack of data near this age-range.For this subset, the Ca ii data show that chromospheric activity levels exit the saturation phase at ∼1.4 Gyr, where coronal activity levels take ∼2.3 Gyr to exit this phase.This is a more substantial difference, but again one that is owed heavily to the lack of near-inflection Ca ii data.Further data at this important age-range could reduce, possibly even eliminate, the current disparity between the lengths of the chromospheric and coronal saturation phases.Firmly establishing the chromospheric and coronal inflection points can help shed light on the outer atmospheric heating mechanisms at work in M dwarfs.It is encouraging to see that the chromospheric saturation phase lengths derived from Ca ii data are close to those obtained from the much more sparse Lyα data.
To better aid planetary atmosphere and habitability studies, we have converted our X-ray (log(L X /L bol )) relationships into cumulative X-ray to Ultraviolet (X-UV: ∼5 -1700Å) irradiance relationships.To do this, spectral energy distributions (SEDs) constructed by the MUSCLES and Mega-MUSCLES surveys (France et al. 2016;Youngblood et al. 2016;Loyd et al. 2016;Froning et al. 2019;Wilson et al. 2021) were obtained from MAST2 (France 2016) and used to determine integrated fluxes over the 5 -1700Å range.A linear relationship was determined to be: Using this equation, we were able to analyze the X-UV irradiances of M dwarfs over time and determine the segmented fit parameters shown in Table 1.These relationships have not been plotted, as they mirror those of the log(L X /L bol ) activity over time (save for the relative shifting/scaling of values).At present, since X-ray data were used as the basis for scaling, the saturation phases again last for ∼500 Myr and ∼2.3 Gyr for the early and mid-late M dwarfs, respectively.

CONCLUSIONS
The Living with a Red Dwarf (LivRed ) program has constructed activity-age relationships for M dwarfs using three different tracers of magnetic activity, along with a broad X-ray to Ultraviolet band (X-UV: 5-1700Å).Multiple measures of each star's activity levels were gathered whenever possible to help mitigate scatter due to stellar variability, and many of the ages were calculated using the age-rotation relationships presented in Engle & Guinan (2023) and shown in Fig. 1.
The stars were divided into two subsets, as previously observed (Guinan & Engle 2018; Mullan & Houdebine 2020) and utilized when constructing the age-rotation relationships -what we have called the "early" [M0-2] and "mid-late" [M2.5-6.5]subsets.Both subsets show similar drops in relative activity levels as they evolve.The most significant difference between the two subsets' activity levels is the duration of their initial saturation phases.Early M dwarfs experience a coronal (Xray) and chromospheric (Ca ii) saturation phase of ∼500-600 Myr.Mid-Late dwarfs, by contrast, experience a coronal saturation phase of ∼2.3 Gyr and a chromospheric saturation phase that is currently estimated to last ∼ 1.4-1.5 Gyr, but is not as well constrained by either Ca ii or Lyα data.As proposed by other studies in the literature, the disparity in coronal vs. chromospheric activity behaviors for mid-late M dwarfs could potentially indicate that the atmospheric plasmas are being heated by different mechanisms, and further study along these lines is encouraged.
For both subsets, however, the extended initial periods of enhanced X-UV activity will be important to account for when analyzing the potential habitability of any planets orbiting M dwarfs.For example, the publicly available VPlanet3 software has a number of useful routines dealing with exoplanet habitability calculations (do Amaral et al. 2022), but the default prescription for highenergy stellar evolution is that of Ribas et al. (2005) which was constructed for solar-type G dwarfs and uses a saturation phase length of 100 Myr.This can significantly underestimate the saturation phase of M dwarfs, and the data and relationships presented here will help future studies to better account for such differences.2023).With the exception of cluster or galactic population members, these relationships were used to calculate the stellar ages used in this study and presented in the Tables.Each row of plots is devoted to a specific subset of M dwarfs.The 'early' M dwarfs, M0-2 dwarfs, are plotted in the top row.The 'mid-late' M dwarfs are plotted in the middle row.For the 'older' track, M2.5-6.5 dwarfs are plotted together since they have all settled onto a common evolutionary path.However, only M2.5-3.5 dwarfs are plotted on the 'young' track due to the large differences in pre-main sequence lifetimes that are encountered within this mass-range.Consequently, a third subset was created and shown in the bottom row.Here, only M4 (and later) dwarfs are plotted.As seen, there is a large difference between the young tracks of the middle and bottom rows, but any difference between the older tracks is well within the uncertainties of the fits.Figure 2. The evolution of M dwarf X-ray activity over time.Early M dwarfs are plotted on the left, and mid-late M dwarfs on the right.Lighter-colored error bars indicate when the 'average' uncertainty value was applied to a target for fitting purposes.A two-component, segmented linear model was applied to each subset and is plotted.Each subset shows an initial "saturation phase" where high activity levels are sustained for a period of time, before an inflection point is reached.After this, the activity decreases at an accelerated rate.It is worth noting that many studies assume a constant level of activity during the saturation phase, where our fits indicate that a slight decrease occurs.The saturation phase is found last ∼470 Myr for the early M dwarfs and ∼2.1 Gyr for the mid-late M dwarfs.Figure 3.The evolution of M dwarf X-ray activity over time is again plotted, but now the ratio of X-ray to bolometric luminosity is used to better normalize the stars.Again, lighter-colored error bars indicate when the 'average' uncertainty value was applied to a target for fitting purposes.As revealed by the segmented fits to the data, both subsets begin their saturation phases at essentially equal activity levels (log(L X /L bol ) ≈ −3), and decline by ∼2.5 orders of magnitude.The length of the saturation phase is found to be ∼450 Myr for the early M dwarfs, but ∼2.3 Gyr for the mid-late M dwarfs.) The reliability of the fits is impacted by the lack of data, particularly for the early M dwarfs.The current data show saturation phase lengths of ∼720 Myr for early M dwarfs and ∼1.5 Gyr for mid-late M dwarfs.Again, however, these quantities should not be regarded as highly reliable due to the lack of data.Instead, we would advise using the Ca ii relationship (Fig. 5) as a more reliable indicator of chromospheric activity over time.The Ca ii emission line cores form within the stellar chromosphere and can thus serve as a proxy for cooler UV features, such as Lyman-α.Given the richer dataset, the Ca ii relationships are also more reliable than those for Lyman-α.Both stellar subsets are observed to decline by ∼1.5 orders of magnitude, a notably less drastic decline than shown for X-ray activity, with saturation phase lengths of ∼500 Myr for early M dwarfs and ∼1.4 Gyr for mid-late M dwarfs.The early M dwarf saturation phase length can be considered comparable to the X-ray, given the uncertainties.The mid-late M dwarfs show a considerable difference in saturation phase lengths at present.It's important to note, however, that although the Ca ii dataset is far richer than Lyman-α, one age-range that is not well-covered occurs right near the inflection point.Further data at this vital age-range will be necesary to reliably pin down the chromospheric saturation phase length in comparison to the coronal, as shown by the X-ray data.Another notable feature of these relationships is that early M dwarfs appear to exhibit higher relative levels of chromospheric activity at both young and old ages.However, much of this difference can be accounted for by the uncertainties and scatter of the fits and data.Measured in this study using data from: A = APT, M = MEarth, R = RCT, S = Skynet, Z = ZTF ) -K19:

Figure 1 .
Figure1.M dwarf age-rotation relationships from the Living with a Red Dwarf program(Engle & Guinan 2023).With the exception of cluster or galactic population members, these relationships were used to calculate the stellar ages used in this study and presented in the Tables.Each row of plots is devoted to a specific subset of M dwarfs.The 'early' M dwarfs, M0-2 dwarfs, are plotted in the top row.The 'mid-late' M dwarfs are plotted in the middle row.For the 'older' track, M2.5-6.5 dwarfs are plotted together since they have all settled onto a common evolutionary path.However, only M2.5-3.5 dwarfs are plotted on the 'young' track due to the large differences in pre-main sequence lifetimes that are encountered within this mass-range.Consequently, a third subset was created and shown in the bottom row.Here, only M4 (and later) dwarfs are plotted.As seen, there is a large difference between the young tracks of the middle and bottom rows, but any difference between the older tracks is well within the uncertainties of the fits.

Figure 4 .
Figure4.The evolution of Lyman-α activity over time is plotted.(Again, lighter-colored error bars indicate when the 'average' uncertainty value was applied to a target for fitting purposes.)The reliability of the fits is impacted by the lack of data, particularly for the early M dwarfs.The current data show saturation phase lengths of ∼720 Myr for early M dwarfs and ∼1.5 Gyr for mid-late M dwarfs.Again, however, these quantities should not be regarded as highly reliable due to the lack of data.Instead, we would advise using the Ca ii relationship (Fig.5) as a more reliable indicator of chromospheric activity over time.

Figure 5 .
Figure 5. Evolution of Ca ii HK emission over time, as given by the log R ′ HK index.(Once again, lightercolored error bars indicate when the 'average' uncertainty value was applied to a target for fitting purposes.)TheCa ii emission line cores form within the stellar chromosphere and can thus serve as a proxy for cooler UV features, such as Lyman-α.Given the richer dataset, the Ca ii relationships are also more reliable than those for Lyman-α.Both stellar subsets are observed to decline by ∼1.5 orders of magnitude, a notably less drastic decline than shown for X-ray activity, with saturation phase lengths of ∼500 Myr for early M dwarfs and ∼1.4 Gyr for mid-late M dwarfs.The early M dwarf saturation phase length can be considered comparable to the X-ray, given the uncertainties.The mid-late M dwarfs show a considerable difference in saturation phase lengths at present.It's important to note, however, that although the Ca ii dataset is far richer than Lyman-α, one age-range that is not well-covered occurs right near the inflection point.Further data at this vital age-range will be necesary to reliably pin down the chromospheric saturation phase length in comparison to the coronal, as shown by the X-ray data.Another notable feature of these relationships is that early M dwarfs appear to exhibit higher relative levels of chromospheric activity at both young and old ages.However, much of this difference can be accounted for by the uncertainties and scatter of the fits and data.

Table 2 .
Early M Dwarf X-ray Data

Table 2
continued on next page

Table 3 .
Mid-Late M Dwarf X-ray Data

Table 3
continued on next page

Table 4 .
Early M Dwarf Ca

Table 5 .
Mid-Late M Dwarf Ca

Table 5
continued on next pageTable 5 continued on next page Table 5 (continued)

Table 6 .
Early M Dwarf Lyα Data

Table 7 .
Mid-Late M Dwarf Lyα Data