Constraining the Physical Properties of Stellar Coronal Mass Ejections with Coronal Dimming: Application to Far-ultraviolet Data of Eridani

A CME is a volume of coronal plasma carrying its own entrained magnetic field that has been expelled from a star. This phenomenon occurs nearly daily on the Sun, even during solar minimum (Yashiro et al. 2004). CMEs often accompany solar flares, but both can occur in isolation. All solar flares of GOES class > X3.1 (> 3.1 × 10⁻⁴ W m⁻² peak 1–8 Å flux at 1 AU) have been accompanied by a CME (Yashiro et al. 2005; Harra et al. 2016).

A direct, measurable outcome of solar CMEs is coronal dimming. Ejected coronal material expands as it leaves the Sun, rapidly quenching its emission and leaving behind evacuated regions that appear dark against their surroundings (Hudson et al. 1996; Shiota et al. 2005). These are likely responsible for features identified as "coronal depletions," "transient coronal holes," and "coronal dimmings" throughout the history of observations of the solar corona (Hansen et al. 1974; Rust 1983; Hudson et al. 1996). Coronal dimmings were observed for some time as darkened areas in spatially resolved coronal observations before Woods et al. (2011) recognized that they produced a measurable decrease in disk-integrated emission as well (i.e., solar irradiance or Sun-as-a-star observations). Figure 1 illustrates this concept with spatially resolved and disk-integrated solar data from an example event on the Sun. Dimming is most pronounced in isolated emission lines at approximately ambient coronal temperatures (Woods et al. 2011; Mason et al. 2014), with a gradual return to a new quasi-quiescent state over the course of 3–12 hr (Reinard & Biesecker 2008).

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Coronal dimming is diagnostic of solar CMEs, suggesting it could be applied to stars. In a comparison of disk-integrated observables, Harra et al. (2016) found that only coronal dimming was a reliable indicator that a CME accompanied a stellar flare. Other observables tested, including flare duration, active region size, size of flare ribbons, thermal/non-thermal radiation ratios, and properties of chromospheric evaporation were not predictive of CMEs.

Veronig et al. (2021) applied the coronal dimming method to stars for the first time by analyzing a large set of archival extreme-ultraviolet (EUV) and X-ray observations. ¹³ They found 21 candidate CME events where coronal emission in at least one band dimmed by > 2σ following a flare, with depths ranging from 5% to 56% and durations of 1–10 hr. This maximum depth is larger than the largest dimmings that have been observed on the Sun ( ≈ 10%; Mason et al. 2019), consistent with the expectation that more active stars could produce more massive CMEs. The range of durations was similar between the Sun and stars, though stellar dimmings skewed toward shorter durations.

In this paper, we present our work to further develop the coronal dimming technique for use with stellar observations. We provide analytical tools to estimate event masses and accelerations/velocities, highlighting a key strength of the dimming technique: the ability to place limits on the mass of a flare-associated CME in the absence of a dimming detection. These tools can be used with any disk-integrated, emission-line-resolved data (Sun or star). Further, we show how coronal dimming can be applied to spectroscopic observations of stellar far-ultraviolet (FUV) emission, with results from an analysis we conducted of archival data of the active K2 dwarf Eridani ( Eri).

When a body of emitting plasma is ejected from the star, the fractional decline in the emission from the star (the maximum "dimming depth"), ${\delta }_{\max }$, is

$$\begin{eqnarray}&&{\delta }_{\max }=\displaystyle \frac{{F}_{\mathrm{CME}}}{{F}_{\mathrm{pre}}},\end{eqnarray} \tag{ 1 }$$

where F_CME is the flux from the plasma evacuated by the CME and F_pre is the disk-integrated flux of the star before any ejection occurred. Assuming the coronal material is optically thin at the wavelength of observation, its flux is

$$\begin{eqnarray}&&F={\int }_{V}G(T,{n}_{e}){n}_{e}{n}_{{\rm{H}}}{dV},\end{eqnarray} \tag{ 2 }$$

where V represents the volume of emitting material, n_e and n_H are the number densities of electrons and hydrogen, and G(T, n_e ) is a (known) function expressing the emissivity of the plasma, which varies with temperature and density. Assuming a constant temperature and a constant density throughout the emitting volume and setting n_e ≈ n_H ≡ n simplifies this to

$$\begin{eqnarray}&&F=G(T,n){n}^{2}V.\end{eqnarray} \tag{ 3 }$$

The expression n² V is related to the mass of the emitting plasma, m, since m ≈ μ nV, where μ is the mean mass of the ions. Using this to recast the numerator of Equation (1) in terms of the emitting mass yields

$$\begin{eqnarray}&&{\delta }_{\max }=\displaystyle \frac{{nG}(T,n)m}{\mu {F}_{\mathrm{pre}}}.\end{eqnarray} \tag{ 4 }$$

Every component of the above equation except m can be either measured or estimated. The pre-CME flux and dimming depth (or limit on depth) are direct observables. Density can be estimated using a density-sensitive line ratio, ideally one that includes the dimming line, but alternatively using a ratio of lines likely to be formed cospatially as indicated by their peak formation temperatures. The mean atomic weight can be taken to be that of the Sun, as variations in μ due to stellar metallicity differences contribute negligibly to the overall error budget. Emissivity can be retrieved from a database such as CHIANTI (Dere et al. 1997). The sharp peak in emissivity functions (Figure 2) means that, barring extreme slopes of the stellar emission measure distribution (EMD), the source of the line emission will be dominated by plasma with a temperature near the line's peak formation temperature. This is the case for Eri, where two disparate reconstructions of its EMD yield contribution functions for the Fe xxi and Fe xii lines that peak very near the peak of their emissivity functions (Figure 2).

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That each line only traces plasma within a narrow range of temperatures is a critical point. A mass estimate from line-dimming will only include plasma near the line's peak formation temperature. The estimate is blind to the ejection of plasma that was too cold or too hot to produce significant emission in the analyzed line. To make this limitation clear, we refer to the mass in Equation (4) as the "ejected emitting mass" (EE mass). We discuss the implications of this limitation further in Section 7.2.

Solving Equation (4) for mass gives

$$\begin{eqnarray}&&m=\displaystyle \frac{\mu {\delta }_{\max }{F}_{\mathrm{pre}}}{{nG}(T,n)}.\end{eqnarray} \tag{ 5 }$$

The inverse relationship with density might seem counterintuitive. A resolution comes from noting that varying only n means the dimming depth and hence the flux from the would-be CME, F_CME, are held fixed. Maintaining a constant F_CME while increasing n necessarily requires decreasing the mass, since the collisional excitations that produce radiation occur more frequently as n increases (up to the point that collisions are so rapid that they de-excite the ions before a radiative transition can occur).

Some readers might find it useful to recast the mass equation in terms of emission measure, EM = n² V, giving

$$\begin{eqnarray}&&m=\displaystyle \frac{\mu {\delta }_{\max }{\mathrm{EM}}_{\mathrm{pre}}}{n}.\end{eqnarray} \tag{ 6 }$$

However, we prefer Equation (5) because it makes the dependency on the assumed value of G(T, n) explicit.

Whereas our derivation yields $m\propto {\delta }_{\max }$, that of Mason et al. (2016) yields $m\propto {\delta }_{\max }^{1/2}$. The Mason et al. derivation was meant only for use with the Sun and assumes a fixed volume in which dimming occurs. Variations in dimming depth stem primarily from how effectively plasma is removed from that volume. Our assumptions are nearly the reverse and intended to be more general for use with other stars. We assume the dimming volume can vary, but the CME evacuates all plasma from that volume. Under this framework, variations in dimming depth stem primarily from the total volume that the CME evacuates.

The assumption that the CME plasma is fully evacuated does not strongly affect the estimated CME mass. Introducing an efficiency factor, f, into the above formalism, where f = 1 indicates full evacuation of the plasma, yields

$$\begin{eqnarray}&&m=\displaystyle \frac{1}{2-f}\displaystyle \frac{\mu {\delta }_{\max }{F}_{\mathrm{pre}}}{{nG}(T,n)}.\end{eqnarray} \tag{ 7 }$$

This lack of sensitivity to f results from a degeneracy between the volume of the evacuated plasma and the drop in density that, together, produce the observed dimming depth. If plasma is evacuated less efficiently (lower f), it must be evacuated from a larger volume to explain the observed drop in emission. Because the evacuated mass is the product of the drop in density and the volume, these factors nearly cancel. For a fixed dimming depth, as f → 0, the evacuated volume blows up, V → ∞ , and m asymptotes to 1/2 of its maximum value.

For the Sun, the question of which scaling is more accurate can be addressed by empirically fitting measurements of dimming depth for a population of CMEs with estimates of their masses made independent of dimming. Doing so for the data set from Mason et al. (2016), we found the empirical relationship to have an exponent of 0.84 ± 0.09, falling between the exponents derived here and by Mason et al. In magnetohydrodynamic (MHD) simulations of seven CMEs and corresponding dimmings, Jin et al. (2022) recovered a relationship with an exponent of 1.35 ± 0.25 between dimming depth and CME mass. A larger data set will help to resolve whether either or neither scaling is correct. This could be accomplished by future work using the automatically extracted events in Mason et al. (2019), vetted to include only bona-fide dimmings and cross-matched with the SOHO/LASCO CME catalog (Gopalswamy et al. 2009).

Dimming from a CME is not instantaneous. As the material being ejected moves away from the stellar surface into regions of lower plasma and magnetic pressure, it expands. The drop in density and any associated adiabatic cooling from the expansion reduce the rate of emission. This ties the rate of expansion to the rate of dimming, and it means that any functional form for dimming versus time will, implicitly or explicitly, make some assumptions about how the CME moves and expands with time. A CME does not expand into a vacuum, but rather into the ambient environment of the stellar wind. This affects its movement and expansion, hence the dimming signature. The global fields and wind properties of active stars differ from the Sun (e.g., Jeffers et al. 2017; Wood et al. 2021) and will affect CME dynamics (Alvarado-Gómez et al. 2018, 2019, 2020).

As an order-of-magnitude approach, we assume a self-similar CME expansion into circumstellar space, akin to Howard et al. (1982). The CME grows in volume as $V\propto {(r/{R}_{\star })}^{3}$, where r is distance from the center of the star to the CME front. We treat the CME as a self-contained plasma such that the mass, m_CME = μ nV is constant, and the density drops as $n\propto {(r/{R}_{\star })}^{-3}$ as the CME expands outward. True solar CMEs are comprised of material in, around, and above the erupting filament, From Equation (3), the constant-mass assumption implies that the emission from the CME drops as

$$\begin{eqnarray}&&{F}_{\mathrm{CME}}={F}_{\mathrm{CME},0}{\left(r/{R}_{\star }\right)}^{-3},\end{eqnarray} \tag{ 8 }$$

and the corresponding dimming depth evolves as

$$\begin{eqnarray}&&\delta (t)={\delta }_{\max }\left(1-{\left(r/{R}_{\star }\right)}^{-3}\right).\end{eqnarray} \tag{ 9 }$$

The r⁻³ dependence means that most of the dimming occurs early in the CME's journey from the star. The dimming will have reached half maximum by the time the CME has expanded to a height of only ≈ 0.25 R_⋆. As a result, the early dynamics of the CME are responsible for most of the dimming, with the decline in flux likely tracing the CME's initial acceleration. Although we expect the CME acceleration to vary in both height and time, as an order-of-magnitude approach we assume the CME undergoes a constant acceleration. This mimics the fits to CME positions in the SOHO/LASCO catalog of solar CMEs (Gopalswamy et al. 2009), though those fits are to data above 2 R_⊙.

Assuming a constant acceleration implies that r(t) = R_⋆ +at²/2 and

$$\begin{eqnarray}&&\delta (t)={\delta }_{\max }\left(1-{\left(1+{{at}}^{2}/2{R}_{\star }\right)}^{-3}\right).\end{eqnarray} \tag{ 10 }$$

This enables a fit to a dimming lightcurve, where the measured flux evolves as

$$\begin{eqnarray}&&F(t)={F}_{\mathrm{pre}}(1-\delta (t)).\end{eqnarray} \tag{ 11 }$$

The fit takes three free parameters: the pre-dimming, pre-flare flux, F_pre; the maximum dimming depth, ${\delta }_{\max }$; and the CME acceleration, a.

In the LASCO Solar CME catalog, the constant-acceleration r(t) fits are used to extrapolate a uniform velocity at 20 R_⊙, v₂₀. We suggest this same convention be used to assign a CME velocity from a constant-acceleration dimming fit, and that it be fixed to 20 R_⊙ (rather than 20 R_⋆) to ensure consistent comparisons. Hence,

$$\begin{eqnarray}&&{v}_{20}=\sqrt{2a(20\,{R}_{\odot }-{R}_{\star })}\approx 6\sqrt{a\,{R}_{\odot }}.\end{eqnarray} \tag{ 12 }$$

For a CME undergoing a constant acceleration starting from near the surface of the Sun, the acceleration required to reach typical CME velocities at 20 R_⊙ are modest relative to surface gravity. Roughly speaking, a = 3 m s⁻² yields v₂₀ = 300 km s⁻¹ and a = 30 m s⁻² yields v₂₀ = 1000 km s⁻¹.

It is important to note the limitation of assuming a constant acceleration when inferring v₂₀. The true acceleration of a CME will vary with time as the forces ejecting it subside and it moves out of the stellar gravitational potential. The expansion of the CME as it leaves the star will be controlled by the pressure of the stellar wind plasma and magnetic field (e.g., Alvarado-Gómez et al. 2018), an environment that varies spatially and temporally. For active stars, ambient conditions will include stronger global fields and denser stellar winds (Wood et al. 2021; Reiners et al. 2022). The reader should treat v₂₀ as an indication of the magnitude of the velocity a CME could reach, rather than an accurate prediction of the velocity a CME will reach.

A useful metric for visual estimates is the time to half-depth, t_1/2. Under Equation (10), this is

$$\begin{eqnarray}&&{t}_{1/2}=\sqrt{{R}_{\star }/2a},\end{eqnarray} \tag{ 13 }$$

where we have made the approximation 2^1/3 − 1 ≈ 1/4. This makes it possible to quickly gauge what range of accelerations could produce a dimming that reaches most of its depth within an observational visit. A visit lasting ∼10 hr would accommodate CMEs from a solar-radius star with a ≳ 0.3 m s⁻², v₂₀ ≳ 90 km s⁻¹, very modest values for a solar CME.

This formalism yields a direct relationship between the CME speed at a given distance and the ratio of the dimming slope to the dimming depth, just as in the constant-speed formulation of Mason et al. (2016). Our formulation gives

$$\begin{eqnarray}&&\delta ^{\prime} ({t}_{1/2})\approx {\delta }_{\max }\sqrt{a/{R}_{\star }}\approx \displaystyle \frac{{\delta }_{\max }{v}_{20}}{7\sqrt{{R}_{\star }\,{R}_{\odot }}}.\end{eqnarray} \tag{ 14 }$$

A typical value from Mason et al. (2016) for $\delta ^{\prime} /{\delta }_{\max }$ is 0.0003. Our equation predicts a corresponding v₂₀ = 1500 km s⁻¹. This is near the upper range of the observed velocities (estimated by position tracking) of solar CMEs analyzed in that work and about a factor of 2 above the median. Although satisfactory for an order-of-magnitude approach, we believe this offset is an indication that CME accelerations typically decline as the CME leaves the star.

Our derivation deviates in some important respects from the related derivations of Aschwanden et al. (2009) and Mason et al. (2016). Whereas Aschwanden et al. (2009) assume a functional form for the time-dependent dimming of a spatially resolved region in the solar coronae (a semi-Gaussian), we derive a functional form for the disk-integrated dimming based on the assumptions of constant acceleration and self-similar expansion. Equation (10) produces a functional form that resembles a logistic function as well as the semi-Gaussian dimming fits prescribed by Aschwanden et al. (2009). Mason et al. (2016) assume a constant velocity rather than a constant acceleration. Although little is known about the early acceleration of CMEs (Mason et al. 2016), the critical importance of the early dynamics of the CME motivate us to choose a constant acceleration rather than an instantaneous acceleration to a constant velocity. Both formulations yield a dependence of the CME speed on the ratio of dimming slope to depth. Assuming a constant-speed CME results in more rapid dimming, requiring unrealistically low velocities to produce the dimming slopes observed by Mason et al. (2016). This is not an issue for the Mason et al. work because the velocity–dimming slope relationship is empirically calibrated.

The takeaway result of this section is that one can use Equations (10) and (11) to fit a dimming lightcurve. This yields a dimming depth and CME acceleration. The acceleration can be used to estimate the CME speed at a given distance. The dimming depth, in conjunction with Equation 5 and estimates of plasma density (from an appropriate line ratio) and emission measure, yields an order-of-magnitude estimate of the mass of the emitting material that was ejected.

Following the 2015 February flare, flux from the Fe xxi line nearly disappears, dropping to roughly 20% of its pre-flare level (Figure 6). Could a CME be responsible? An 80% dimming is unheard of on the Sun, where the largest dimmings rarely exceed 10% in magnitude (Mason et al. 2019). Solar coronal dimmings also typically occur in cooler lines, near the ∼1 MK peak of the solar EMD, such as Fe xii (Mason et al. 2014). However, Eri's EMD peaks at hotter temperatures, and in 2015 exhibited a plateau out to 10 MK (Figure 3), suggesting that coronal dimming could be expected in hotter, higher ionization iron lines in comparison to the Sun, perhaps even Fe xxi. Simulations by Jin et al. (2020) indicate that dimming on more active stars could be shifted to higher-temperature lines relative to the Sun.

The 2015 flare on Eri began only 3.8 minutes after the start of the observations. This proximity to the start of the exposure suggests an alternative explanation for the Fe xxi dimming: that the initial "pre-flare" Fe xxi flux is elevated and its subsequent decay is simply a return to its original quiescent value rather than a dimming. An elevated flux could be due to an early response to the observed flare, preceding that of Si iv and C ii, or due to a larger, unseen flare that occurred prior to the start of the observing visit. To help evaluate the 2015 event, Figure 15 provides a closer look at each of the observed flares.

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Points challenging the CME explanation are that the peak of the flare in Fe xxi is subdued and its pre-flare value is significantly higher than in the 2017 epochs. Fe xxi flux increases 1.2–1.5× during the 2015 flare versus 2–4× in the 2017 flares. The onset of dimming offsets this peak, but only mildly. Accounting for it only boosts the flare increase to 1.7 ± 0.4.

Before the flare, the Fe xxi flux of the 2015 observations is elevated by about 3× relative to 2017 (Figure 15). This elevated flux is probably not an early increase due to the observed flare. The rise in C ii+Si iv flux clearly begins later in the exposure, and Fe xxi flux does not lead increases in this emission in later epochs. Further, on other stars transition region lines appear to brighten nearly simultaneous with the particle acceleration and heating from a flare (e.g., MacGregor et al. 2021). What is instead possible, and cannot be ruled out by the brief 3.8 minute baseline, is that a larger flare occurred prior to the start of the 2015 observations. In this scenario, the elevated Fe xxi emission and subsequent dimming are just the ongoing decay of the unseen flare. Although the C ii+Si iv emission does not appear elevated at the start of the observations, this is consistent with their more rapid decay following flares, clearly visible in the 2017 observations.

A hidden flare is statistically plausible. The probability of a hidden flare is roughly μ τ/α, where μ is the rate of flares producing peak fluxes at or above the observed level, τ is the flare decay timescale, and α is the power-law index of the (cumulative) flare frequency distribution (see Appendix A for a derivation). We estimated μ using the two largest flares in the other epochs of observation. Their Fe xxi flux peaks near the starting level of the 2015 epoch, meaning that flares exceeding this threshold occur at an approximate rate of 0.1 ks⁻¹. We estimated τ by fitting an exponential decay to the 2015 event (i.e., assuming the dimming is a flare decay), obtaining a value near 7 ks. We assumed α is similar to estimates made using solar and stellar events, which generally yield values between 0.5 and 1.5 (Aschwanden & Güdel 2021). The resulting probability that the observed decay is actually due to an unseen flare is greater than 10%.

What is odd is that, by the end of the 2015 observations, the Fe xxi flux has settled to a value well below the pre- or post-flare flux of the 2017 observing epochs, by about a factor of 3. This means that neither the pre- nor post-flare Fe xxi flux in 2015 is consistent with that of the later epochs. The dimming hypothesis requires that the baseline flux was 3× higher than in later epochs, whereas the hidden flare hypothesis requires that the baseline flux was 3× lower.

All observing epochs occur near minima in Eri's 2.92 yr activity cycle (Coffaro et al. 2020). X-ray observations by XMM-Newton simultaneous with the 2015 visit measured a broadband flux of 1.1 × 10⁻¹¹ erg s⁻¹ cm⁻², lower than measurements of 1.5 × 10⁻¹¹ and 1.4 × 10⁻¹¹ erg s⁻¹ cm⁻² that occurred −50 and +8 days from the 2017 March and 2017 August HST visits. The average Fe xii emission varied little across all three visits (Figure 15). On the Sun, activity cycles affect emission from hotter lines more, as indicated by an increasing variability in the solar EMD with increasing temperature. However, this does not seem to be the case on Eri (e.g., Coffaro et al. 2020).

The 2015 X-ray data begin just after the peak of the observed flare and mimic the decline of Fe xxi. The ratio between the peak and final values is ≈ 1.5, much less than the drop in Fe xxi emission (Loyd et al. 2018, Figure 9). Like Fe xxi, this drop in X-ray flux could also result from either a flare decay or coronal dimming.

If the CME explanation is correct, it could indicate that the 2015 event resulted from the ejection of plasma from a single dominant active region, emitting, outside of flares, 3× the flux of active regions at other epochs. In this picture, the magnetic reconnection that produced the flare in Si iv and C ii ejected the bulk of the Fe xxi-emitting plasma in that region, producing the deep Fe xxi dimming. Meanwhile, either the Fe xii-emitting plasma in that region was preserved, or what plasma was ejected made up a much smaller fraction of the star's disk-integrated Fe xii emission.

A third possibility for the dramatic drop in Fe xxi emission is that a dominant active region rotated out of sight over the stellar limb during the observation. However, the 11 day rotation period of Eri makes this unlikely, as the duration of the visit occupies only about 2.5% of this period. A bright region receding across the limb would have a redshift below 2.5 km s⁻¹ (Valenti & Fischer 2005), too low to test this theory given the instrument accuracy of 7.5 km s⁻¹ (Plesha et al. 2019).

That a single emission line could dim by as much as 80% following a CME is made more plausible by the recent work of Veronig et al. (2021). That work discovered dimmings as high as 56% in broadband X-ray emission from several stars and attributed those dimmings to CMEs. A 56% dimming in the bands they analyzed, which blend lines and continuum, could be deeper in an isolated line.

Like so many past CME candidates, the real cause of the 2015 dimming is simply inconclusive. However, this event hints at the potential power of searching for coronal dimming with FUV observations. Dedicated observations of this or other bright, active stars could capture many more flares with better pre-flare baselines and firmly establish whether near-total dimmings of the Fe xxi line do indeed occur, or if the 2015 event was more likely to be the result of an unseen flare.

The enticing Fe xxi signal should not overshadow what we consider the most valuable result of this analysis: actual constraints (in this case, upper limits) on the masses of emitting material that could have been ejected in association with flares on Eri. This is the subject of the next section.

On the Sun, the strongest flares in the three epochs of observation would likely each have been accompanied by a CME. The threshold above which flares on the Sun are always accompanied by CMEs is a peak soft X-ray flux exceeding roughly 3 × 10⁻⁴ W m⁻² (Harra et al. 2016). The peak soft X-ray fluxes of the three Eri flares were in the range of 10⁻⁴–10⁻² W m⁻² (GOES class X to X100) according to the empirical relationship between peak Si iv 1393,1402 Å and soft X-ray flux established by Youngblood et al. (2017). This range is driven by uncertainty in the empirical relationships for predicting the soft X-ray flux from peak Si iv flux at 1 au. The peak Si iv fluxes at 1 au of the three observed flares were 0.39, 0.25, and 0.44 erg s⁻¹ cm⁻¹. The solar flare–CME scaling of Aarnio et al. (2012) indicates that flares of this magnitude on the Sun could be associated with CMEs ranging from 10¹⁵–10¹⁸ g. This mass range would correspond to dimming depths ranging from undetectable to physically impossible (i.e., >100%, assuming a single emission line probed the full CME mass).

If the 2015 February event was caused by a CME, we estimate an EE mass of ${10}^{{15.2}_{-1.0}^{+1.2}}$ g for the Fe xxi dimming and 10^15.3±0.3 g for the Fe xii dimming (though the Fe xii dimming is statistically insignificant; see Section 5). The kinetic energy of this event, estimated by summing the Fe xxi and Fe xii masses and taking the velocity to be the v₂₀ values from the Fe xxi dimming, is ${10}^{{30.8}_{-0.3}^{+1.1}}$ erg. Figure 16 plots the confidence interval of these estimates from the MCMC samples in the context of the population of solar CMEs from the SOHO/LASCO catalog (Gopalswamy et al. 2009) through 2021 January (excluding those with data quality flags). In that plot, the second mode of MCMC samples above the main mode correspond to the sampler exploring fits near the upper limit of the prior on CME acceleration that effectively amount to a step function in dimming.

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The dimming nondetections related to the 2017 flares yield Fe xii EE masses of < 10^15.5 g. The Fe xxi EE masses are less constraining, at < 10^16.3 g.

In the context of solar CMEs, 10¹⁵–10¹⁶ g CMEs are unremarkable events (1–2 orders of magnitude below the largest observed) and occur once every few days to every few months (Figure 16 and Section 7.3). One might expect the EE masses to be well above the upper limit of solar CME masses given that Eri has a larger peak emission measure (Figure 2), the Fe xxi dimming is extreme, and the Fe xii dimming large in comparison to solar events. The relatively low EE masses in this context likely result from a combination of factors: plasma densities that are 2–3 orders of magnitude above the solar corona (Fludra et al. 1999), the exclusion of plasma beyond the range of plasma temperatures and densities probed by the lines, and the fact that solar CME masses grow as they leave the Sun prior to reaching the mass recorded in the SOHO/LASCO catalog (Gopalswamy et al. 2009).

The mass posterior in Figure 9 exhibits a cliff at its low-mass end. This is due to a near-linear decline in the emissivity, G(T, n), above n = 10¹³ cm⁻³ as collisional quenching of the (forbidden) Fe xxi 1354 Å radiative transition sets in. Because m ∝ (nG(T, n))⁻¹ (Equation (5)), an increase in n in this regime is effectively cancelled by a drop in G(T, n), yielding a constant mass and producing the apparent lower limit in the mass posterior of Figure 9.

Dimming analyses of the sort we present might be the only means to probe the allowable masses of events associated with any prominent stellar flare. However, it is important that we, as a community, recognize the assumptions and limitations of the method as we have presented it. We list those of which we are aware below. Many of these can likely be improved upon as the analysis techniques and observing strategies mature.

• 1.  
  The method is blind to plasma beyond the narrow range of formation temperatures of the analyzed line(s). The mass of plasma with temperatures in the gaps will be missing from the estimated total.
• 2.  
  The method is also blind to low-density, poorly emitting plasma (Appendix B). As a result, it likely probes mass in and around the erupting filament, but not mass above that is swept into the CME.
• 3.  
  The method assumes that emission from the non-ejected plasma remains at the pre-ejection level. Changes in this baseline will affect the inferred dimming depth.
• 4.  
  The method equates the density of the ejected plasma to a "disk-averaged" density inferred from disk-integrated observations. If the ejected plasma is overdense relative to the disk-average, the retrieved EE mass will overestimate the true EE mass.
• 5.  
  Density estimates from observations simultaneous with the dimming observations are not always possible. Changes in density between these epochs will affect the EE mass.
• 6.  
  The method assumes the emission in an analyzed line is dominated by plasma at at a temperature that maximizes that line's emissivity. If the plasma contributing most of this emission has a non-peak temperature, the retrieved EE mass will be an underestimate.
• 7.  
  The method assumes no plasma remains within the dimming region. If only a portion of the plasma is ejected, then the dimming region must be fractionally larger to account for the observed drop in emission. The net effect is that the ejected mass could drop to no less than half the original estimate (Section 2.1).
• 8.  
  The method assumes the CME has a constant mass. The mass of several solar CMEs has been observed to grow with time due both to mass added from behind and accumulated from ahead of the CME (e.g., DeForest et al. 2013; Veronig et al. 2019). For reference, Veronig et al. (2019) reported a 5× increase in the mass of a solar CME observed on 2012 September 27.

EE mass estimates are subject to statistical uncertainties as well. In the analysis of the Eri observations, the Fe xxi EE mass estimate has a range of 2 orders of magnitude because the density measurement we adopted from the literature has an equivalent range. In contrast, uncertainty in both density and dimming depth contribute to the uncertainty on the Fe xii EE mass for that event.

The assumed emissivity also introduces an error, though it is negligible in the Eri analysis. Figure 2 shows that the peak of the contribution functions is about 0.1 dex in temperature off from the peak of the line emissivities. Adopting the peak of the contribution function to estimate G(T, n) would increase the EE mass by about 30%.

The most important point when it comes to temperature is the hidden mass. For Eri, plasma from $\mathrm{log}(T)$ of 6.4 to 6.8 is effectively hidden (Figure 2): plasma within this range will emit less than 10% of the observed Fe xxi and Fe xii flux. An ejection of plasma at these temperatures would have little affect on the observed Fe xii and Fe xxi emission.

MHD simulations of CMEs from active stars will help to determine how well observables like coronal dimming and Type II radio bursts on Eri and other active stars can be expected to constrain CME properties. Ó Fionnagáin et al. (2022) simulated Type II radio burst signals from simulated CMEs on Eri and found that signals would remain in the LOFAR band for 20–30 minutes following an event. This could make Eri a great candidate for joint radio and FUV observations to cross-validate CME signals. We are currently adapting the Alfvén Wave Solar Model for simulations of CMEs and dimming signals on Eri and will describe the results in a follow-on publication. This will add additional context to the 2015 and 2017 HST observations.

The three observed flares and upper limits on the EE mass provide a basis to explore mass and angular momentum loss from Eri due to CMEs.

A challenge to estimating CME-driven mass loss is that events spanning orders of magnitude in mass and occurrence rate could contribute. This is certainly true on the Sun. Figure 17 plots the distribution of observed CME masses from the SOHO/LASCO catalog. Summing the masses of all events with mass estimates (including those with quality flags) yields a CME mass-loss rate of 6 × 10⁻¹⁶ M_⊙ yr⁻¹. Small events contribute the most to CME mass loss on the Sun. Events with masses less than 3% of the largest observed (i.e., < 5 × 10¹⁵ g) contribute more than 50% of the summed mass of all events in the LASCO catalog.

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Assuming a power-law distribution of CME masses can enable an estimate of cumulative CME mass loss incorporating events over a range of masses. More sophisticated treatments are also possible (Odert et al. 2017). Portions of the observed solar CME mass distribution follow a power law reasonably well (Figure 17). For a power law of the form

$$\begin{eqnarray}&&\nu {(\gt m)=\mu (m/{m}_{\mathrm{ref}})}^{\alpha },\end{eqnarray} \tag{ 15 }$$

where ν gives the rate of events with mass > m and μ is the rate of events with mass > m_ref, events within the range m_min and m_max account for a cumulative mass loss of

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{\alpha \mu {m}_{\mathrm{ref}}}{\alpha +1}\left[{\left(\displaystyle \frac{{m}_{\max }}{{m}_{\mathrm{ref}}}\right)}^{\alpha +1}-{\left(\displaystyle \frac{{m}_{\min }}{{m}_{\mathrm{ref}}}\right)}^{\alpha +1}\right].\end{eqnarray} \tag{ 16 }$$

For α ≈ − 2, as on the Sun, small events dominate and $\dot{M}$ is not sensitive to m_max. Integrating the power law from where it begins to deviate significantly at ${m}_{\min }=2\times {10}^{15}$ g to ${m}_{\max }\to \infty$ yields a mass-loss rate within a factor of a few of the sum of the cataloged solar CMEs.

For Eri, taking m_ref < 3 × 10¹⁵ g based on dimming fits to the three prominent flares within the 10.6 hr of observations (μ ≈ 7 day⁻¹), and adopting solar values for the remaining parameters, implies $\dot{M}\lt {10}^{-14}$ M_⊙ yr⁻¹, or $\dot{M}\lt 0.6$ ${\dot{M}}_{\odot }$ (where we take ${\dot{M}}_{\odot }$ to be 2 × 10⁻¹⁴ M_⊙ yr⁻¹ for consistency with Wood et al. 2005). We emphasize that this estimate is confined only to Fe xii-emitting plasma and only to flare-associated CMEs. The total mass-loss rate of Eri, incorporating CMEs and the stellar wind, has been estimated from the astrospheric absorption method to be $\dot{M}=30$ ${\dot{M}}_{\odot }$ (Wood et al. 2005). Our results suggest that CMEs contribute only negligibly to Eri's total mass loss, similar to the present-day Sun. This conflicts with the MHD simulations of Ó Fionnagáin et al. (2022) that, coupled with flare statistics, indicate CMEs could shed mass from Eri at up to half the rate of the stellar wind. For any statement about Eri's mass loss rate to progress beyond conjecture, constraints across a wide range of CME rates and masses are needed.

Observationally calibrated scalings between CME mass and flare energy provide another path to constraining the CME mass-loss rate (Odert et al. 2017). For the Sun, Aarnio et al. (2013) measured ${m}_{\mathrm{CME}}\propto {E}_{\mathrm{flare}}^{0.63}$. The sub-linear relationship is consistent with solar CME masses following a steeper distribution (α ≈ −2) than solar flare energies (α ≈ −1; Aschwanden et al. 2016). ¹⁷ Empirical flare–CME scalings for stars could allow future work to fold long-baseline observations of a star's flares from missions like TESS into determining the cumulative effects of stellar CMEs. Calibrating such a scaling for Eri or other stars will require measured CME masses across a range of flare energies, so is not possible with the present data.

Alvarado-Gómez et al. (2018, 2019) modeled CMEs triggered beneath global dipolar magnetic fields ranging from 75 to 1400 G representative of active stars (Donati & Landstreet 2009; Reiners et al. 2022) and found such global fields could prevent incipient CMEs from escaping. Strong magnetic fields are typical of low-mass stars in particular. The relationship between average field strength and Rossby numbers implies that, for M stars, even those that have spun down with age to rotation periods of 100 days (Ro ≈ 1; Wright et al. 2018) will exhibit average field strengths of 100 G and above (Reiners et al. 2022).

Magnetic confinement could explain why it has proven difficult to detect the exceptional CMEs one might expect around stars with exceptionally energetic flares (Drake et al. 2013). Evidence of strong fields confining incipient CMEs has been observed on the Sun (Thalmann et al. 2015), and a reanalysis of purported stellar CME detections in the literature suggests these events, though they ultimately escaped, spent much of their kinetic energy breaking free of overlying fields (Moschou et al. 2019).

The simulations of Alvarado-Gómez et al. (2019) produced a sustained dimming in a band at 171 Å (dominated by Fe ix emission) regardless of whether the simulated event escaped or not. This implies that coronal dimming on active stars could be a result of "failed" CMEs that do not actually escape the star. On the Sun, flares lacking CMEs produced no dimming signature in the analysis of 42 flares, nine without CMEs, by Harra et al. (2016). Six of these were from active region NOAA12912, where strong overlying fields likely stymied incipient CMEs (Thalmann et al. 2015). Veronig et al. (2021) also analyzed a sample of solar flares as a precursor to their search for coronal dimmings in stellar data. Within the 64 solar flares they analyzed, they found that of the 16 events without an accompanying CME, two produced spurious dimming signatures. More work is needed to understand what factors result in a CME false-positive through dimming. However, even if coronal dimming can result from eruptions that do not ultimately escape, it remains a valuable observable. In this case, the absence of dimming would indicate no eruption, and certainly no CME, occurred. Further, rates of confined eruptions, potentially probed by dimming, could provide some sense of the rate of more powerful eruptions that do escape, even if they are too infrequent to easily observe.

For Eri, CME confinement is possible, but uncertain. The maximum strength of Eri's (dominantly poloidal) surface magnetic field was 20–33 G near the 2015 observations (Jeffers et al. 2017), versus 8 G for analogous observations of the Sun in 2013 (Vidotto 2016). At the 75 G field strengths modeled by Alvarado-Gómez et al. (2018), only eruptions associated with flares of GOES class X30 or greater could escape confinement. This falls within the range of uncertainty on the GOES class of the Eri flares we analyzed, making it unclear whether associated eruptions should have been capable of breaking free of an overlying field. The 2015 February event, if a CME, had a kinetic energy consistent with the distribution of kinetic energies of solar CMEs for events of similar mass (Figure 16), in contrast with the more extreme events analyzed by Moschou et al. (2019). Note, however, that the v₂₀ value we used to estimate the 2015 CME candidate's kinetic energy might not be sensitive to confinement, since this could slow the CME after the initial acceleration probed by dimming and used to estimate v₂₀.

The ideal coronal dimming observations would incorporate emission lines tracing a broad range of formation temperatures, with emphasis on those near the peak of the stellar EMD. The benefit of this coverage would be minimizing the potential for ejected mass to be hidden from a dimming analysis. CME mass constraints for Eri in particular would benefit from observations of lines formed between $\mathrm{log}T$ of 6.4 to 6.8.

With sufficient coverage of coronal emission lines and sufficient S/N, it would be possible to construct EMDs for dimming events themselves. Coupled with measurements of plasma density from lines tracing a similar range of temperatures, one could construct a distribution of mass versus temperature, dm/dT, and integrate this distribution to obtain a total mass. Future work could test such an approach on observations of solar CMEs.

Observations of coronal (irradiance) dimming on the Sun rely on EUV emission lines, particularly those near 200 Å, such as observed by SDO/EVE (Mason et al. 2016). From 100–350 Å, the EUV band contains many bright Fe lines spanning a range of ionization states and formation temperatures. No astrophysical observatories currently operate that can access 100–350 Å emission. The former Extreme Ultraviolet Explorer (EUVE), which operated from 1992 to 2001, could observe this range. Veronig et al. (2021) included these archival data in their comprehensive search for coronal dimmings, identifying a single candidate dimming from the EUVE data on AB Dor.

X-ray observations are dominated by coronal emission. Effective areas are 1–3 orders of magnitude below those of HST/COS, but because the emission is dominantly coronal, wide bands can be integrated to achieve higher S/N. For the star EK Dra, which has a nearly identical flux in Fe xxi 1354 Å emission as Eri (Ayres & France 2010), the analysis by Veronig et al. (2021) yielded an S/N of $\approx 130/\sqrt{\mathrm{hr}}$ in a 0.2–12 keV band and $\approx 60/\sqrt{\mathrm{hr}}$ in a narrower 1–2 keV band. In comparison, median S/Ns were 17/$\sqrt{\mathrm{hr}}$ in Fe xxi and 40/$\sqrt{\mathrm{hr}}$ in Fe xii for the 2017 HST/COS observations of Eri. The higher S/N of broadband X-ray searches comes at the cost of being able to isolate emission lines in order to enable mass estimates through the process outlined in Section 2. FUV spectroscopy with the HST also offers simultaneous measurements of 10–100× brighter chromospheric (e.g., C i) and transition region (e.g., Si iv) lines that can be used to closely trace flares.

For dimming observations seeking mass constraints, an important consideration is how to estimate plasma densities. In the FUV, observing both the 1242 Å and 1349 Å components of Fe xii enables a simultaneous density measurement of the Fe xii-emitting plasma. Unfortunately, no additional bright components of Fe xxi are available in the FUV that could enable a direct measurement of its source plasma density. Coupling HST observations with contemporaneous X-ray observations of a density-sensitive doublet like Ne ix (Ness et al. 2002) will be important to obtain mass estimates of ejected Fe xxi-emitting plasma.

When using HST for FUV measurements of coronal emission, COS provides the best combination of sensitivity and wavelength coverage. However, for many active stars, especially M dwarfs, the potential for flares to violate detector count rate limits is a concern. Using the Space Telescope Imaging Spectrograph (STIS) enables the observation of brighter targets in exchange for lower sensitivity. The STIS G140M mode provides a throughput of 1.7% at 1350 Å (Branton & Riley 2021) versus COS G130M's 4.0% (Hirschauer 2021).