GLOBAL ENERGETICS OF SOLAR FLARES. IV. CORONAL MASS EJECTION ENERGETICS

The primary data set for this study consists of all solar GOES M- and X-class flare events observed with AIA/SDO during the first 3.5 years of the SDO mission (2010 June–2014 January), which amounts to 399 events. For each flare event, the start time t_start, the peak time t_peak, and end time t_end are extracted from the NOAA/GOES flare catalog. For the analysis of the EUV dimming profiles, we include a time margin of Δt_margin = 30 minute after the flare end, yielding a time window of [t_start, t_end + Δt_margin] for each event, during which we analyze AIA data with a cadence of Δt = 2 min, or shorter if necessary, amounting to n_t ≈ 10–200 time frames per event. We are using EUV images from SDO/AIA in all six coronal wavelengths, i.e., in 94, 131, 171, 193, 211, 335 Å. We select a field of view of FOV = 0.5 R_⊙ centered at the heliographic position of the flare site (obtained from the SolarSoft (SSW) latest events archive at http://www.lmsal.com/solarsoft/latest_events_archive.html, courtesy of Sam Freeland). The size of the FOV covers the initial expansion of the CME and the associated dimming region within a half solar radius, which is found to be sufficient to model the early phase of the CME origination where EUV data fitting is sensitive. Since the CME evolution is traced from 0.1 to 7.0 hr in various events, we correct for the solar rotation in the extraction of each time sequence of subimages, so that the chosen FOV tracks the rotating heliographic position of the flare, CME, and dimming region.

Previous analysis of EUV dimming during CMEs has been performed on soft X-ray or EUV images and thus is strongly wavelength-dependent. The set of coronal images obtained with AIA/SDO has the great advantage that essentially the entire relevant temperature range observed in the solar corona and in flares is covered, in a nominal range of T_e ≈ 0.5–20 MK. The DEM distribution of CMEs has been measured in a few previous studies only, using three temperature filters from EUVI/STEREO (Aschwanden et al. 2009a), or six filters from AIA (Cheng et al. 2012). Reconstructing a DEM distribution over the full coronal temperature range allows us to calculate the total mass of a CME in a wavelength-independent manner.

In order to determine the DEM distribution we apply the spatial-synthesis method, which fits single-Gaussian DEM functions to the EUV fluxes in each macro-pixel,

$$\begin{eqnarray}\displaystyle \frac{{dEM}(T,x,y)}{{dT}} & = & {\mathrm{EM}}_{p}(x,y)\\ & & \times \exp \left(-\displaystyle \frac{{[\mathrm{log}(T)-\mathrm{log}({T}_{p}(x,y))]}^{2}}{2{\sigma }_{T}^{2}(x,y)}\right),\end{eqnarray} \tag{ 1 }$$

characterized by the three parameters [EM_p, T_p, σ_p] for each macro-pixel position [x, y], where we choose a size of 32 × 32 pixels for a macro-pixel, which was found to be a good trade-off between computational speed and the spatial variation of thermal structures. In other words, using smaller macro-pixel sizes did not increase the accuracy of the total emission measure. The total DEM distribution function sampled over the entire FOV in the EUV dimming region of the CME is then obtained by integrating over the full temperature range and by summing over all macro-pixels,

$$\begin{eqnarray}&&{\mathrm{EM}}_{\mathrm{tot}}=\int \int \int \displaystyle \frac{{dEM}(T,x,y)}{{dT}}\ {dT}\ {dx}\ {dy}.\end{eqnarray} \tag{ 2 }$$

Note that the integration of the total emission measure EM_tot over the entire AIA temperature range of T_e ≈ 0.5–20 MK in principle conserves the particle number in this temperature range, except for plasma that cools below T_e ≲ 0.5 MK or heats above T_e ≳ 20 MK, and thus a decrease in the total emission measure is likely to be a consequence of adiabatic expansion (even at an insufficient speed to cause a CME, which is referred to a "confined flare" or "failed eruption"), rather than a plasma cooling effect. However we acknowledge that the DEM reconstruction at hot flare temperatures of T_e ≈ 10–20 MK using AIA data alone has a limited accuracy and may not capture the entire hot plasma that is detected with GOES or RHESSI (Ryan et al. 2014), which may affect the accuracy of the dimming-inferred CME masses.

This spatial-synthesis DEM code (Aschwanden et al. 2013) was cross-compared with 10 other DEM codes in a benchmark test of DEM codes and multithermal energies in solar active regions (Aschwanden et al. 2015) and was found to be among the three most accurate DEM codes tested therein. A similar DEM code based on Gaussian basis functions in each image pixel has recently been developed by the AIA Team also (Cheung et al. 2015).

The spatial-synthesis DEM distributions obtained during a flare event generally shows a low-temperature peak at T_e ≈ 1.5 MK in the preflare phase of the EUV dimming region. This temperature serves to estimate the initial density scale height λ in the dimming region,

$$\begin{eqnarray}&&\lambda \approx 50\ \mathrm{Mm}\times \left(\displaystyle \frac{{T}_{e}}{1\ \mathrm{MK}}\right).\end{eqnarray} \tag{ 3 }$$

which yields a density scale height of λ ≈ 75 Mm for a typical preflare temperature of T_e ≈ 1.5 MK in the EUV dimming region.

The only spatial information on the geometry of the CME available in EUV images is the projected area A_p of the EUV dimming region, which defines the CME footpoint area. We determine this area A_p of the dimming region by combining all macro-pixels that exhibit a significant emission measure decrease ΔEM_tot between the time t_max of the maximum emission measure EM_max (before the EUV dimming starts) and the time t_min of the minimum emission measure E_min afterwards (during the EUV dimming phase),

$$\begin{eqnarray}{\rm{\Delta }}{\mathrm{EM}}_{\mathrm{tot}} & = & {\mathrm{EM}}_{\max }-{\mathrm{EM}}_{\min }\\ & = & {\mathrm{EM}}_{\mathrm{tot}}(t={t}_{\max })-{\mathrm{EM}}_{\mathrm{tot}}(t={t}_{\min }).\end{eqnarray} \tag{ 4 }$$

In a gravitationally stratified atmosphere, the initial volume V of the dimming region corresponds to the product of the (unprojected) solar surface area A and the vertical scale height λ,

$$\begin{eqnarray}&&V=A\lambda ={L}^{2}\lambda ,\end{eqnarray} \tag{ 5 }$$

if we define a spatial length scale L by the square root of the area A,

$$\begin{eqnarray}&&A={L}^{2}.\end{eqnarray} \tag{ 6 }$$

The dimming area can have any arbitrary 2D shape, because the so-defined length scale L is just the equivalent linear size of a square-sized area. The measurement of the dimming area by summing over all dimmed pixels in the plane-of-sky, however, yields a projected area A_p that is foreshortened in the center-to-limb direction,

$$\begin{eqnarray}&&{A}_{p}=L\ {L}_{p},\end{eqnarray} \tag{ 7 }$$

where L_p is the projected length scale (in the center-to-limb direction), while L is the unprojected length scale (in the perpendicular direction). From the geometric diagram in Figure 2 we can see that the projected length scale L_p is related to the unprojected length scale L and vertical scale height λ by

$$\begin{eqnarray}&&{L}_{p}=L\cos (\rho )+\lambda \sin (\rho )=\displaystyle \frac{{A}_{p}}{L},\end{eqnarray} \tag{ 8 }$$

where ρ is the angle of the heliographic position from disk center, which is, according to spherical geometry,

$$\begin{eqnarray}&&\cos (\rho )=\cos (l)\,\cos \,(b),\end{eqnarray} \tag{ 9 }$$

with l being the heliographic longitude and b the heliographic latitude. We see that the projected length has the limits of L_p = L at disk center (ρ = 0), and L_p = λ at the limb (ρ = 90°), respectively. Inserting L_p = A_p/L from Equation (7) into Equation (8) leads to a quadratic equation of the length scale L, which has the algebraic solution,

$$\begin{eqnarray}&&L=\displaystyle \frac{-\lambda \ \sin (\rho )\pm \sqrt{{\lambda }^{2}{\sin }^{2}(\rho )+4{A}_{p}\cos (\rho )}}{2\cos (\rho )}.\end{eqnarray} \tag{ 10 }$$

With this analytical solution for L we can directly calculate the unprojected area A (Equation (6)) or volume V (Equation (5)) from the heliographic position (l, b), the projected area A_p (Equation (7)), and the density scale height λ (Equation (3)). It is worthwhile to use this exact solution, because the generally used approximation $L\approx \sqrt{{A}_{p}}$ overestimates the volume by a factor of L/λ ≈ 3 for typical solar conditions, which propagates as a similar error into the estimates of the electron density, mass, kinetic, and gravitational energy of the CME.

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The total emission measure EM_tot of the dimming region is related to the mean electron density n_e in a gravitationally stratified atmosphere by,

$$\begin{eqnarray}&&{\mathrm{EM}}_{\mathrm{tot}}=\int {n}_{e}^{2}({\boldsymbol{x}})\ {dV}={n}_{e}^{2}V={n}_{e}^{2}{L}^{2}\lambda ,\end{eqnarray} \tag{ 11 }$$

which yields an explicit expression for the mean electron density n_e in the CME source region,

$$\begin{eqnarray}&&{n}_{e}=\sqrt{\displaystyle \frac{{\mathrm{EM}}_{\mathrm{tot}}}{{L}^{2}\lambda }}.\end{eqnarray} \tag{ 12 }$$

Thus, by measuring the total emission measure EM_tot and the (projected) dimming area A_p at the heliographic location (l, b), and using the density scale height λ (Equation (3)), we can derive the center-to-limb angle ρ (Equation (9)), the unprojected length scale L (Equation (10)), and the mean electron density n_e (Equation (12)).

Armed with the measurements of the mean electron density n_e and the dimming volume V = L²λ, we can then directly calculate the mass of the CME, evaluated in the CME footpoint area at the time of the CME launch,

$$\begin{eqnarray}&&{m}_{\mathrm{cme}}={n}_{e}\ {m}_{p}\ V={n}_{e}\ {m}_{p}\ {L}^{2}\lambda .\end{eqnarray} \tag{ 13 }$$

Typically we find a density of n_e ≈ 10⁹ cm⁻³, a length scale of L ≈ 10¹⁰ cm, a scale height of λ ≈ 7.5 × 10⁹ cm in the coronal dimming regions, which yields a typical CME mass of m_cme ≈ 1.3 × 10¹⁵ g.

White-light coronagraph observations provide CME speeds at a distance of x > 2.2 R_⊙ (for LASCO/C2) from the Sun center, which is often close to a constant value, but the acceleration profile in the CME source region in the lower corona is unknown. Time sequences of AIA images, in contrast, provide kinematic parameters of acceleration a(t), speed v(t), and height x(t) of a CME all the way back to the initial origin of the CME. We attempt to measure these kinematic parameters by forward-fitting a kinematic model to the observed EUV dimming profile EM(t). The simplest kinematic model contains a short initial acceleration time τ_a that is temporally not resolved with our typical time cadence of Δt = 2 min. Since the altitude range of significant CME acceleration is estimated to be confined to h ≲ 0.1 R_⊙, the corresponding acceleration time interval is τ_a ≈ 2 min for a CME with a typical final speed of v ≈ 1000 km s⁻¹, so that the details of the acceleration time profile can be neglected for data fitting with a similar time resolution. Therefore, we set the acceleration time equal to the time resolution, i.e., τ_a = Δt, while the acceleration is zero before and afterwards.

However, for detailed modeling of the acceleration time profile see Appendix A for 4 simple models, which are used throughout our analysis. For sake of simplicity, we describe in this section only Model 1. Our simplest kinematic CME acceleration model is defined by,

$$\begin{eqnarray}a(t)=\left\{\begin{array}{ll}0 & {\rm{for}}\,{\rm{}}t\lt {t}_{0}\\ {a}_{0} & {\rm{for}}\,{\rm{}}{t}_{0}\lt t\lt {t}_{A}\\ 0 & {\rm{for}}\,{\rm{}}t\gt {t}_{A}\end{array}\right.\end{eqnarray} \tag{ 14 }$$

with $\tau ={t}_{A}-{t}_{0}$. From the acceleration profile a(t) (parameterized with 2 variables t₀ and a₀) we can then directly calculate the velocity profile v(t) by time integration, which yields

$$\begin{eqnarray}v(t) & = & {\displaystyle \int }_{{t}_{0}}^{t}a(t)\ {dt}\\ & = & \left\{\begin{array}{ll}0 & {\rm{for}}\,{\rm{}}t\lt {t}_{0}\\ {v}_{0}={a}_{0}(t-{t}_{0}) & {\rm{for}}\,{\rm{}}{t}_{0}\lt t\lt {t}_{A}={a}_{0}\tau \\ {a}_{0}\tau & {\rm{for}}\,{\rm{}}t\gt {t}_{A}.\end{array}\right.\end{eqnarray} \tag{ 15 }$$

Equally we can derive the height–time profile x(t) of the CME bulk mass by time integration of the velocity v(t),

$$\begin{eqnarray}x(t) & = & {\displaystyle \int }_{{t}_{0}}^{t}v(t)\ {dt}\\ & = & \left\{\begin{array}{ll}{h}_{0} & {\rm{for}}\,{\rm{}}t\lt {t}_{0}\\ {h}_{0}+(1/2){a}_{0}{(t-{t}_{0})}^{2} & {\rm{for}}\,{\rm{}}{t}_{0}\lt t\lt {t}_{A}\\ {h}_{0}+(1/2){a}_{0}{({t}_{A}-{t}_{0})}^{2} & \\ \quad +\,{a}_{0}\tau (t-{t}_{A}) & {\rm{for}}\,{\rm{}}t\gt {t}_{A}\end{array}\right.\end{eqnarray} \tag{ 16 }$$

where we allow for an integration constant h₀ that represents the initial height of the CME bulk mass. A good estimate of the initial height is the density scale height, h₀ ≈ λ = 50 Mm × (T_e/1 MK) as we determined from the preflare temperature T_e in Section 2.1.

The analytical expression of the velocity allows us also to calculate the kinetic energy E_kin(t) of the bulk mass of a CME anytime on their trajectory, using the mass m_cme (Equation (13)) and the CME velocity v (Equation (15)),

$$\begin{eqnarray}&&{E}_{\mathrm{kin}}(t)=\displaystyle \frac{1}{2}{m}_{\mathrm{cme}}\ {v}^{2}(t),\end{eqnarray} \tag{ 17 }$$

where the asymptotic velocity is ${v}_{\mathrm{cme}}(t\mapsto \infty )={a}_{0}({t}_{A}-{t}_{0})={a}_{0}\tau$ for our standard Model 1 (Section 2.5), while the corresponding values are ${v}_{\mathrm{cme}}=(1/2){a}_{0}\tau$ (Model 2), ${v}_{\mathrm{cme}}=(1/3){a}_{0}\tau$ (Model 3), and v_cme = a₀τ (Model 4), according to Equation (39) in Appendix A.

The key assumption in our CME modeling is the concept of radial adiabatic expansion, which provides a direct link between the height–time profile x(t) (Equations (16), (40), (41)) and the EUV dimming profile EM(t). We approximate the volume of the CME with a spherical shell that initially is aligned with the stratified atmosphere (Figure 3, top left), but then progressively expands in radial direction (Figure 3, top middle and right). The radial volume expansion V(t), starting from an initial volume V₀ with height h₀ can be parameterized as (Aschwanden 2009),

$$\begin{eqnarray}&&\displaystyle \frac{V(t)}{{V}_{0}}=\displaystyle \frac{{[{R}_{\odot }+h(t)]}^{3}-{R}_{\odot }^{3}}{{[{R}_{\odot }+{h}_{0}]}^{3}-{R}_{\odot }^{3}}.\end{eqnarray} \tag{ 18 }$$

For adiabatic expansion, no energy or mass is exchanged between the interior and the exterior of the (magnetically confined) CME volume, and thus conservation of the particle number dictates a reciprocal relationship between the expanding volume and the decreasing mean electron density inside the CME volume, i.e., n_e ∝ 1/V, which predicts the following evolution of the emission measure,

$$\begin{eqnarray}&&\mathrm{EM}(t)\propto {n}_{e}{(t)}^{2}V(t)\propto V{(t)}^{-1},\end{eqnarray} \tag{ 19 }$$

which, combined with the volume parameterization given in Equation (18) yields the following normalized emission measure ratio q_EM(t),

$$\begin{eqnarray}&&{q}_{\mathrm{EM}}(t)=\displaystyle \frac{\mathrm{EM}(t)}{{\mathrm{EM}}_{0})}=\displaystyle \frac{{[{R}_{\odot }+{h}_{0}]}^{3}-{R}_{\odot }^{3}}{{[{R}_{\odot }+h(t)]}^{3}-{R}_{\odot }^{3}}.\end{eqnarray} \tag{ 20 }$$

The normalized emission measure q_EM(t) varies in the range of [0, 1] between the launch of the CME and its propagation to infinite distance ($h(t)=\infty$). Including some constant emission measure EM_bg from a CME-unrelated background, which we define as a fraction q_bg of the maximum emission measure ${\mathrm{EM}}_{\max }={EM}(t={t}_{\max })$ during the flare/CME event, i.e., EM_bg = q_bg EM_max, we generalize Equation (20) to,

$$\begin{eqnarray}&&\mathrm{EM}(t)={\mathrm{EM}}_{\max }[{q}_{\mathrm{bg}}+(1-{q}_{\mathrm{bg}}){q}_{\mathrm{EM}}(t)],\end{eqnarray} \tag{ 21 }$$

which varies from EM(t₀) = EM_max at the initial time t = t₀ of CME launch asymptotically to ${EM}(t=\infty )={q}_{\mathrm{bg}}\ {\mathrm{EM}}_{\max }$ after the CME event.

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Inserting the height–time profile h(t) = x(t) (Equation (16)) into the emission measure profile EM(t) (Equation (21)), we have then a complete model (Figure 4, bottom right) that can be fitted to the observed dimming curve EM_obs(t) obtained from the DEM method. Since the wavelength-dependent temperature effects are fully taken into account in the DEM fits, the emission measure profile EM(t) reflects directly the mass dependence of the expanding CME and is not affected by any temperature effects, such as radiative or conductive cooling. In other words, regardless of whether the plamsa inside the CME volume heats up or cools down, its number density is fully accounted for in our temperature-integrated DEM in the temperature range of T_e ≈ 0.5–20 MK.

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We can then forward-fit the model based on the Equations (16)–(21) and obtain the free variables a₀, t₀, and q_bg, as well as the kinetic energies E_kin of CMEs. Generally, this EUV dimming model fits the data well. Exceptions are large complex CME events with multiple expansion phases, which require multiple convolutions of expansion profiles. Other exceptions are the cases of "failed CME" events that produce insufficient energy to escape the solar corona.

In addition to the kinetic energy we calculate the gravitational potential energy E_grav that is required to lift a CME from the solar surface to infinity,

$$\begin{eqnarray}&&{E}_{\mathrm{grav}}(h)={\int }_{{R}_{\odot }}^{\infty }\displaystyle \frac{{{GM}}_{\odot }{m}_{\mathrm{cme}}}{{r}^{2}}{dr}=\displaystyle \frac{{{GM}}_{\odot }{m}_{\mathrm{cme}}}{{R}_{\odot }},\end{eqnarray} \tag{ 22 }$$

where r is the distance of the CME (centroid) to the center of the Sun, r_⊙ is the solar radius, M_⊙ is the solar mass, and G is the gravitational constant. The gravitational energy was found to be typically about an order of magnitude higher than the kinetic energy in the study of Vourlidas et al. (2000), and thus is an important quantity to include in estimates of the kinetic energy of CMEs.

It is also instructive to calculate the escape velocity, which we obtain from setting the kinetic energy equal to the gravitational potential energy, i.e., ${E}_{\mathrm{kin}}=(1/2){m}_{\mathrm{CME}}{v}_{\mathrm{esc}}^{2}={{GM}}_{\odot }{m}_{\mathrm{CME}}/{R}_{\odot },$ which yields,

$$\begin{eqnarray}&&{v}_{\mathrm{esc}}=\sqrt{\displaystyle \frac{2{{GM}}_{\odot }}{{R}_{\odot }}}\approx 618\ \mathrm{km}\ {{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 23 }$$

When applying the velocity model v(t) (Equation (15)) to the observed CME speeds v_cme, we have to be aware that a₀ represents the net acceleration rate, after subtracting the gravitational acceleration −g_grav,

$$\begin{eqnarray}&&{a}_{0}={a}_{\mathrm{LF}}-{g}_{\mathrm{grav}},\end{eqnarray} \tag{ 24 }$$

where a_LF represents the acceleration caused by the magnetic Lorentz force.

After we described the theoretical aspects of our EUV dimming model in the foregoing sections, we provide a brief summary of the numerical code with estimates of uncertainties. The numerical code executes the following tasks in sequential order:

• 1.  
  Data Acquisition: (see Section 2.1 on data setup). AIA/SDO images (Level 1.0) are read in all coronal wavelengths, decompressed, and processed with AIA_PREP to Level 1.5, which includes flat-fielding, removing of bad pixels, image coalignment between different wavelengths, rescaling to a common plate scale, and derotation (of the roll angle), so that the EW and NS axes are aligned with the horizontal and vertical image axes [x, y]. The coalignment of level 1.5 data is believed to be accurate to 0.5 pixels (SDO/AIA Data Analysis Guide). For our dimming analysis we rebin the data to macro-pixels (with a size of 32 × 32 full-resolution pixels), which yields ≈25² = 625 macro-pixel locations for the spatial analysis of EUV dimming regions, for each of the n_t = 10–200 time sequences in the n = 399 flare/CME events.
• 2.  
  DEM Analysis: The automated spatial-synthesis DEM code (Aschwanden et al. 2013) performs in each of the 625 macro-pixel locations a Gaussian forward-fit (with three parameters) to the six wavelength fluxes, which is then synthesized to a combined DEM function dEM(T, t)/dT for each time step and event. Bad DEM solutions, which occur mostly due to image saturation during the flare peak at too long exposure times, are interpolated from the DEM solutions of the previous and following time step. Bad DEM solutions are easily recognized from large χ²-values of the DEM goodness-of-fit criterion. The preflare temperature is evaluated from the first image taken at the flare start time t_start (defined by the NOAA flare catalog) and is always found around T_e ≈ 1.5 MK.
• 3.  
  EUV Dimming Time Profile: We find that every EUV dimming is preceded by a previous increase in the total emission measure EM_tot(t), and the largest amount of dimming occurs always at the location of the strongest previous emission measure increase. In order to determine the location with the most significant dimming thus requires the detection of the location with the maximum emission measure in time and space, i.e., EM_tot,max(x_max, y_max, t_peak). In order to improve the statistics we calculate the cross-correlation coefficients between this maximum emission measure time profile and the remaining 724 time profiles EM_tot(x, y, t), and combine all those emission measure profiles that have a cross-correlation coefficient of CCC ≥ 0.9. The summed emission measure profile EM_tot,sum(t), normalized to the total emission measure profile peak EM_tot(t_peak), is then also corrected for temporal outliers by interpolation. The temporal interpolation is found to be most effective in removing oscillatory fluctuations caused by the automated AIA exposure control, which is triggered by CCD saturation during flare peaks. The absolute uncertainty of the emission measure values in the time profile EM_tot(t), obtained with the spatial-synthesis DEM code, is conservatively estimated to be of order ≈10%, based on a benchmark test with simulated AIA data, which yielded an uncertainty in the peak emission measure of q_EM,p = 0.82 ± 0.08, and an uncertainty of the total (temperature-integrated) emission measure of q_EM,t = 1.00 ± 0.01 (Aschwanden et al. 2013). Since we apply multiple (n_model) model fits, the resulting uncertainty is ${\sigma }_{q}\approx 0.1/\sqrt{{n}_{\mathrm{model}}}$, which amounts to ≈5% for out set of four models (Appendix A).
• 4.  
  Forward-Fitting to Dimming Curve: The observed dimming curve of the total emission measure EM(t) includes data points from the rise time of the flare, while EUV dimming generally starts after the flare peak. Therefore we have carefully to define the time interval of fitting. Since the theoretical dimming curve (Figure 4 and Equations (20), (21)) predicts a monotonic decrease of the emission measure, we define the fitting time interval in the time range where the total emission measure decreases from 90% of the peak value down to 10% of the peak value (above the background level q_bg). Then we fit the four dimming models (specified in Appendix A and Figure 4), which are characterized by the four free parameters ${a}_{0},{t}_{0},{t}_{A},{q}_{\mathrm{bg}}$, but keep the acceleration time interval $\tau ={t}_{A}-{t}_{0}$ fixed at the time resolution Δt = 2 min, since we do not have a sufficient number of datapoints that are sensitive to extract a variation in the acceleration rate at this time resolution. The three-parameter dimming models (Equations (14)–(21)) are fitted to the observed time profiles EM_obs(t) with a Powell minimization method (Press et al. 1986, p. 294). The best-fit model based on least-squares fitting (with estimated uncertainties of ${\sigma }_{q}\approx 0.1/\sqrt{{n}_{\mathrm{model}}}$ yields then the acceleration profile a(t), velocity time profile v(t), and height–time profile x(t).
• 5.  
  CME Parameters: From the dimming profiles that show a decrease of the EUV emission measure above some threshold level (defined by the two-fold median value of emission measure differences before and after the dimming phase), we obtain the projected dimming area A_p in the source region of the CME, which yields, together with the heliographic position (l, b), the aspect angle ρ (Equation (9)), and with the preflare background temperature T_bg, the density scale height λ (Equation (3)), and the unforeshortened length scale L (Equation (10)). From the total emission measure EM_tot at the flare peak time (i.e., of the emission measure), we obtain then with the volume V = L²λ, the mean electron density n_e (Equation (12)) at the time of the CME onset, and the mass of the CME (Equation (13)). The asymptotic CME velocity v_cme is then computed from the best-fit parameters (a₀, τ) (Equation (15)), yielding also the kinetic energy E_kin (Equation (17)) for the CME. The gravitational potential energy E_grav (Equation (22)) follows directly from the CME mass m_cme (Equation (22)).
• 6.  
  Occurrence Frequency Distributions: Power law functions are fitted to the upper tails of the distributions, at x > x_max = x(N_max), where the lower fitting boundary is chosen at the maximum of the size distribution, $N(x)\propto {x}^{-\alpha }$. The uncertainty of the power law slope is estimated from ${\sigma }_{\alpha }=(\alpha -1)/\sqrt{(n)}$, with n the number of events contained in the fitted part of the size distribution, $n={\int }_{{x}_{\max }}^{\infty }N(x){dx}$ (Clauset et al. 2009).
• 7.  
  Comparison with LASCO: Compiling our list of 399 EUV dimming events with the existing LASCO CME catalog (Gopalswamy et al. 2009) (http://cdaw.gsfc.nasa.gov/CME_list), we compare the mass, speed, and kinetic energy of CMEs for identical events.

The emission measure dimming curves EM_tot(t) are shown for 12 "simple events" in Figures 5–8, and for 12 "complex events" in Figures 9–12. What we mean by "simple" and "complex" events is depicted in Figure 13: a simple event consists of a short impulsive peak of the total emission measure, which is composed of a short rise time that is immediately followed by a short decay time interval, closely matching the theoretically predicted EUV dimming behavior after the emission measure peak time as displayed in Figure 4 (bottom panel). A measure of the simple EUV dimming behavior is the dimming ratio,

$$\begin{eqnarray}&&{q}_{\mathrm{dimm}}=\displaystyle \frac{({\mathrm{EM}}_{\max }-{\mathrm{EM}}_{\min })}{({\mathrm{EM}}_{\max }-{\mathrm{EM}}_{\mathrm{bg}})},\end{eqnarray} \tag{ 25 }$$

which is close to unity for simple events. The selection of 12 events displayed in Figures 5–8 have all high dimming ratios ofq_dimm ≈ 0.92–0.96 and acceptable goodness-of-fit values of χ² ≲ 1.2, but otherwise exhibit a wide variety of parameters. The typical CME speeds derived for simple events (see values labeled with v in bottom panels of Figures 5–8) occur in the range of v_cme ≈ 200–1700 km s⁻¹, which is consistent with the typical CME velocity range reported from white-light observations. These examples of "simple events" show a goodness-of-fit with reduced chi-square fits of χ ≲ 1.2, and thus demonstrate that the radial adiabatic expansion model (Section 2.6) yields reasonable CME speeds and acceptable fits. The spatial scales of simple events shown in Figures 5–8 vary in the range of L ≈ 80–250 Mm, and the associated flare durations show a range of D ≈ 0.1–0.5 hr. The variety of simple events shown in Figures 5–8 includes events observed near disk center (Figure 5 left and right; Figure 8 left and middle), near the limb (Figure 5 middle, Figure 6 right, Figure 8 right), events with CCD saturation at the flare peak time (Figure 7), and thus demonstrates that simple events do not occur at particular heliographic positions, and are retrieved even in the case of image saturation. In summary, the simple events reveal the following characteristics of EUV dimming: (i) the emission measure in the EUV dimming region increases by a large amount during the soft X-ray flare time interval, (ii) and drops by a large amount (≳90%) after the flare, and (iii) the dimming profile closely follows the theoretically predicted function that is expected for radial adiabatic expansion. In fact, we found a highly significant level of EUV dimming in all of the 399 analyzed flares, so there is no sensitivity issue in measuring the EUV dimming in M- and X-class flares.

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Let us go to the other extreme of "complex events," which we interpret as events that consist of multiple peaks that are convolved with each other (Figure 13 bottom panel) and may hamper the deconvolution of simple EUV dimming behavior. In Figures 9–12 we present a selection of 12 events that were selected in order of the longest flare durations (D ≳ 1.0 hr) and having acceptable fits (χ² ≲ 1.2). Long-duration flares are inherently more complex in the structure of their light curves, showing often multiple peaks that cannot easily be deconvolved from each other. Our simple model of a single dimming phase may not be adequate for such complex events, but we fitted the dimming curve by the same rules, in a fitting time interval that encompasses the 10%–90% range of the EUV dimming. In some sense, these examples shown in Figures 9–12 represent the worst cases, while the examples in Figures 5–8 represent the most ideal cases for fitting of our model. In the complex events we notice step-wise jumps in the emission measure, which are likely to be caused by CCD saturation and the automatic exposure control mode, data drop-outs during CCD saturation, multiple peaks, or gradual (rather than impulsive) transitions of emission measure increases to decreases (Figures 9–12). We notice also that the dimming decreases are significantly lower for complex events than for simple events, in the range of q_dimm ≈ 45%–85%. Moreover, the resulting CME velocities are significantly lower for complex events than for simple events, in the range of v_cme ≈ 30–120 km s⁻¹ for the 12 cases shown in Figures 9–12, which are partially caused by superimpositions of multiple dimming phases that cannot easily be deconvolved in the EUV dimming profiles.

The fact that the fitted CME speeds are systematically lower for complex events of longer duration than for simple events, may indicate a convolution bias of multiple dimming phases that are triggered at multiple times during the soft X-ray flare. The observed EUV emission measure time profile EM(t) in a complex event with duration D can be considered as a convolution of N elementary time profiles with duration d in simple events (Figure 13), which are related to each other by Poisson statistics for a random process,

$$\begin{eqnarray}&&D=\sqrt{N}d.\end{eqnarray} \tag{ 26 }$$

The CME speed v is proportional to the slope of the emission measure decrease dEM/dt, since,

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{EM}(t)}{{dt}}=\displaystyle \frac{d\mathrm{EM}(t)}{{dh}}\displaystyle \frac{{dh}}{{dt}}\approx \displaystyle \frac{{\rm{\Delta }}{EM}}{{\rm{\Delta }}h}\ v.\end{eqnarray} \tag{ 27 }$$

Low velocities determined with the EUV dimming method may therefore contain a convolution bias of multiple (spatially or temporally separated) dimming regions. Quantitative analysis of low-speed CME events may require more complex modeling with multiple dimming components, which is beyond the scope of this study.

We investigate statistical correlations of CME parameters that we measured from the 399 AIA events, which are presented in form of scatter plots in Figure 14, with the parameter ranges and distributions compiled in Table 1. We start with the geometrical parameters. In our "square-box" model of the CME source region (Equation (5)) we find a strong correlation between the observed projected area A_p and the inferred deprojected CME source area A, with $A\propto {A}_{p}^{1.5\pm 0.1}$ and a regression coefficient of R = 0.93 (Figure 14(a)), which is a statistical relationship that averages over all heliographic positions, while the exact relationship is given by Equations (5)–(10). The projection effect yields also ${A}_{p}\propto {L}^{1.3\pm 0.1}$ (Figure 14(b)), in contrast to the approximation A_p ≈ A ∝ L² that is expected when neglecting projection effects. Also the volume scales as $V\propto {L}^{1.98\pm 0.02}$ (Figure 14(c)), due to the constant vertical scale height in gravitationally stratified atmospheres, rather than the often used approximation V ≈ L³. In order to correct for projection or foreshortening effects one can use either these empirical scaling relationships, or the exact relationships given in Equations (5)–(10) if the heliographic position of the dimming region is known.

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Since the emission measure ${\mathrm{EM}}_{\mathrm{tot}}\propto {n}_{e}^{2}V$ scales with the volume V, we expect some correlation, which is indeed confirmed by the data (Figure 14(d)). A correlation of EM_tot ∝ V is expected since the preflare electron density has a very narrow distribution, i.e., n_e = (1.3 ± 0.6) × 10⁹ cm⁻³ and does not show any correlation with the CME source volume (Figure 14(e)). Similarly, the preflare electron temperature is found to have a very small variation also, i.e., T_e = (1.8 ± 0.3) MK (Figure 14(f)), which is not correlated with any spatial scale.

Owing to the small variation in electron density, the CME mass m_cme is expected to be highly correlated with the CME source volume V (Equation (13)), which indeed is the case, yielding m_cme ∝ V^1.07±0.06 with a regression coefficient of R = 0.95 (Figure 14(g)). Similarly, the CME kinetic energy (Equation (17)) shows the expected correlation E_kin ∝ v² (Figure 14(h)). However, neither the CME speed nor the CME kinetic energy exhibits any correlation with a spatial scale (Figures 14(j) and (k)), which indicates that the acceleration and speed occur in the vertical direction and are not physically coupled to the horizontal extent L of the CME, due to magnetic confinement. Also the dimming fraction is not correlated with the horizontal extent of the dimming region (Figure 14(i)), probably because the CME expansion occurs initially only in the vertical direction due to the magnetic confinement in the low plasma-β regions of the corona.

We measure two temporal parameters of CMEs, namely the half-dimming time τ_dimm and the vertical CME propagation time τ_prop. We define the half-dimming time interval τ_dimm as the interval between the dimming onset time t₀ (from our fits of the dimming models) and the time when the emission measure drops to 50% of the (normalized) dimming fraction q_EM(t) (Equation (20)). The vertical CME propagation timescale τ_prop = λ/v is simply the vertical density scale height divided by the CME speed v. Both timescales have similar values in our sample of 399 events, i.e., τ_dimm = 0.5–33 minute, versus τ_prop = 0.3–54 minute (Table 1), and are highly correlated, with a scaling of ${\tau }_{\mathrm{prop}}\propto {\tau }_{\mathrm{dimm}}^{1.5\pm 0.3}$, and a linear regression coefficient of R = 0.80 (Figure 15(a)). However, none of these CME timescales is correlated to the horizontal length scale of the dimming region (Figures 15(b), (c)), which again suggests dominant vertical transport of the CME mass without horizontal diffusion.

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When we consider thermal timescales, such as the GOES rise time, decay time, or flare duration, which all are highly correlated among each other, we find that they are also correlated with the CME timescales. The best correlation among the three GOES timescales is found for the GOES decay time with the AIA half-dimming time (${\tau }_{\mathrm{dimm}}\propto {\tau }_{\mathrm{decay}}^{1.0\pm 0.3}$, with a regression coefficient of R = 0.69; Figure 15(f)), and the AIA CME propagation time (${\tau }_{\mathrm{dimm}}\propto {\tau }_{\mathrm{decay}}^{1.4\pm 0.4}$, with a regression coefficient of R = 0.72; Figure 15(i)). This is a remarkable result, because the correlated parameters are measured with different instruments (GOES and AIA), in different wavelength domains, and from different physical processes. The GOES decay time is supposedly determined by plasma heating and cooling timescales, while the AIA half-dimming timescale and propagation timescale is governed by plasma transport during an adiabatic expansion process. Our results indicate that the adiabatic EUV expansion time is correlated with the GOES flare duration or decay time, which is sensitive to plasma cooling by thermal conduction and radiative loss. This correlation is likely to be a consequence of multiple (concolved) flare episodes in complex events (Figure 13), which prolong both the overall dimming time and the overall flare duration or decay time.

How is the CME speed v related to temporal parameters? The scatter plots shown in Figures 15(j), (k), and 15l clearly show that the (final) CME speed is approximately reciprocally related to all timescales, the GOES flare duration, the AIA half-dimming time, and the AIA CME propagation time. This is expected for the AIA CME propagation time due to its definition, i.e., τ_prop = h₀/v, and as a consequence of the correlation of the CME dimming time (Figure 15(d)) or the CME propagation time (Figure 15(g)) with the GOES flare duration. Partially, these correlations are related to the convolution bias of "complex events," as depicted in Figure 13: The longer the flare duration, the longer the convolved gradient dEM/dt of the dimming profile, which implies a lower velocity v and a longer dimming half time ${\tau }_{\mathrm{dimm}}$.

The occurrence frequency distributions of the CME parameters measured with AIA are shown in Figure 16 and compiled in Table 2. These size distributions show power-law-like functions for most CME parameters, such as the length L (Figure 16(a)), the source volume V (Figure 16(b)), the flare duration D (Figure 16(c)), the total emission measure EM_tot (Figure 16(d)), the speed v (Figure 16(e)), the mass m (Figure 16(f)), the kinetic energy ${E}_{\mathrm{kin}}$ (Figure 16(g)), the gravitational potential energy E_grav (Figure 16(h)), and the total kinetic plus gravitational energy E_cme (Figure 16(i)). In contrast, the electron density n_e, the preflare electron temperature T_e, and the dimming fractions q_dimm (Figure 16(j)) have narrow distributions that steeply fall off without a power-law tail.

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The slopes of the power laws can be predicted from self-organized criticality (SOC) models, with some model modification for CME phenomena. The most fundamental prediction of SOC models is the scale-free probability conjecture, which states that the statistical probability of the occurrence rate N(L) of nonlinear energy dissipation avalanches scales reciprocally to the geometric scale L of the phenomena (Aschwanden 2012),

$$\begin{eqnarray}&&N(L)\propto {L}^{-d},\end{eqnarray} \tag{ 28 }$$

where L is the geometric length scale and d is the Euclidean space dimension, which for most real-world phenomena is generally d = 3. The distribution of CME length scales, i.e., $N(L)\propto {L}^{-(3.4\pm 1.0)}$ (Figure 16(a)), appears to be consistent with the scale-free probability conjecture, N(L) ∝ L^−d, within the uncertainties of the power-law fit.

From the length distribution $N(L)\propto {L}^{-3.4}$ and the theoretical CME volume scaling V ∝ L²λ ≈ L² (Equation (5), Figure 14(c)) we can directly predict the volume size distribution N(V), using L[V] ∝ V^1/2 and dL/dV = V^−1/2,

$$\begin{eqnarray}&&N(V)\propto N(L[V])\displaystyle \frac{{dL}}{{dV}}\propto {V}^{-2.2},\end{eqnarray} \tag{ 29 }$$

which is indeed consistent with the measured size distribution N(V) ∝ V^−2.2±0.5 (Figure 16(b)). Note the modification of standard models V ∝ L³, to V ∝ L² for CME source volumes in a gravitationally stratified atmosphere, which is a consequence of similar scale heights λ for all CMEs.

Using the approximation of m ∝ n_eV ∝ V for CME masses (due to the small variation of mean electron densities n_e) we expect that the size distribution of CME masses is identical to those of the volumes, which is indeed the case, i.e., $N(m)\propto {m}^{-2.2\pm 0.4}$ (Figure 16(f)). In the same vein we expect the same distribution for total emission measures, since ${EM}\propto {n}_{e}^{2}V\propto V$ (Equation (11)), which is indeed the case, i.e., $N({EM})\propto {{EM}}^{-2.4\pm 0.4}$ (Figure 16(d)).

It is not obvious how the length scale L is related to the event duration D for CMEs, since there is no significant correlation (Figure 15(b)). However, a theoretical prediction of the standard SOC model is that the spatio-temporal relationship is approximatedly a diffusive random walk that obeys the diffusion equation (Aschwanden 2012),

$$\begin{eqnarray}&&L=\kappa {D}^{1/2}\propto {D}^{1/2},\end{eqnarray} \tag{ 30 }$$

where κ is the diffusion coefficient, assuming to be uncorrelated with the event duration. Combining the two fundamental assumptions (Equations (28), (30)) predicts then the following size distribution for CME event durations D, using L ∝ D^1/2 and dL/dD ∝ D^−1/2,

$$\begin{eqnarray}&&N(D)\propto N(L[D])\displaystyle \frac{{dL}}{{dD}}\propto {D}^{-2.2},\end{eqnarray} \tag{ 31 }$$

which indeed is consistent with the observed distribution, $N(D)\propto {D}^{-2.5\pm 0.6}$ (Figure 16(c)). We use here the dimming time as a measure of the event duration, D ∝ τ_dimm, in order to compare EUV observables self-consistently, but the same result is obtained using GOES flare durations D.

Another fundamental relationship we found in our study is the relationship between the CME speed v and the CME event duration duration D, which scales with $v\propto {D}^{-1.6\pm 0.6}$ (Figure 15(j)). Combining this relationship with the distribution of event durations, we predict,

$$\begin{eqnarray}&&N(v)\propto N(D[v])\displaystyle \frac{{dv}}{{dD}}\propto {v}^{-1.8},\end{eqnarray} \tag{ 32 }$$

which is indeed consistent with the observed size distribution of CME speeds, i.e., N(v) ∝ v^−1.9±0.3 (Figure 16(e)). From this we can also estimate the size distribution of CME kinetic energies, using ${E}_{\mathrm{kin}}\propto {v}^{2}$, ${dv}/{{dE}}_{\mathrm{kin}}\propto {E}_{\mathrm{kin}}^{-1/2}$,

$$\begin{eqnarray}&&N({E}_{\mathrm{kin}})\propto N(v[{E}_{\mathrm{kin}}])\displaystyle \frac{{dv}}{{{dE}}_{\mathrm{kin}}}\propto {v}^{-1.4},\end{eqnarray} \tag{ 33 }$$

which matches the observed size distribution, i.e., $N({E}_{\mathrm{kin}})\propto {E}_{\mathrm{kin}}^{-1.4\pm 0.1}$ (Figure 16(g)).

For the gravitational energy E_grav, we expect a size distribution that is nearly identical to that of the masses, since E_grav ∝ m, which is indeed the case, i.e., $N({E}_{\mathrm{grav}})\propto {E}_{\mathrm{grav}}^{-(2.2\pm 0.5)}$ (Figure 16(h)), versus $N(m)\propto {m}^{-2.0\pm 0.4}$ (Figure 16(f)).

Finally, for the total CME energy, which is defined by the sum ${E}_{\mathrm{tot}}={E}_{\mathrm{kin}}+{E}_{\mathrm{grav}}$, we expect an approximate slope that is close to the average of the two power-law slopes, i.e., ${a}_{\mathrm{tot}}\approx ({a}_{\mathrm{kin}}+{a}_{\mathrm{grav}})/2\approx (1.6+2.2)/2\,=\,1.9\pm 0.3$, which matches the observed size distribution also, i.e., $N({E}_{\mathrm{tot}})\propto {E}_{\mathrm{tot}}^{-2.0\pm 0.3}$ (Figure 16(i)).

In summary, most of the observed size distributions of CME parameters exhibit a power-law-like function, and are consistent with the scale-free probability conjecture and the random-walk assumption of a fractal-diffusive avalanche model of a slowly driven SOC system (Aschwanden 2012). However, two modifications are needed for CME phenomena, namely the CME source volume scaling V ∝ L² in a gravitationally stratified atmosphere, and the convolution bias of adiabatic expansion velocities, v ∝ D⁻¹, which scales reciprocally with the duration of flare/CME events. The predicted and observed power exponents of the size distributions of CME parameters and of the underlying parameter correlations are juxtaposed in Table 2.

While we described the quantitative CME results based on the EUV dimming analysis made with AIA data so far, we turn now to comparisons with white-light results from identical CME events observed with LASCO onboard SOHO. A LASCO CME catalog was generated that contains the mass, speed and kinetic energy of CMEs, which is publicly available at http://cdaw.gsfc.nasa.gov/CME_list, covering almost two decades of continuous LASCO observations, 1996 January–2015 July (Gopalswamy et al. 2009).

The first task is to identify identical events between the GOES flare list (used for identifying EUV dimming events with AIA) and the LASCO CME list, which is based on the delayed detection after the CME emerges from behind the C2/coronagraph occulter disk at a distance of 2.2 R_⊙ solar radii. Sometimes, LASCO events are detected with C3 only, which causes additional delays. The time delay between a CME-associated flare and the CME detection with LASCO thus depends on both the heliographic location (l, b) on the solar disk and the distance of first detection with LASCO (after passing the occulter disk at >2.2 R_⊙). Denoting the GOES flare peak time with t_peak, the time of first detection with LASCO with t_det,C2, and the distance from the Sun center at the time of first detection with x_det,C2, the detection delay is defined as,

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\det ,{\rm{C}}2}={t}_{\det ,{\rm{C}}2}-{t}_{\mathrm{peak}}=\displaystyle \frac{({x}_{\det ,{\rm{C}}2}-{x}_{\mathrm{flare}})}{{v}_{\mathrm{LE}}},\end{eqnarray} \tag{ 34 }$$

where v_LE is the average speed of the CME leading edge, and x_flare is the distance of the flare site from the Sun disk center. Implicitly we assume that the propagation of the CME leading edge is isotropic in the high plasma-β region of the upper corona and heliosphere. For the speed v_LE of the leading edge we use the linear fit to the height–time measurements at the first time of LASCO/C2 detection as listed in the LASCO CME catalog. The travel distance to first detection is nominally x_det,C2 = 2.2 R_⊙ for LASCO/C2, but can be larger when a data gap occurs before CME detection, or shorter when the flare occurs near the limb. The distance of the flare site from the Sun center depends on the heliographic longitude (l) and latitude (b) of the flare site and is given by the trigonometric relationship,

$$\begin{eqnarray}&&\cos (\rho )=\cos (l)\cos (b),\end{eqnarray} \tag{ 35 }$$

where ρ is the heliographic angle between the flare site and Sun center, which translates into a distance by,

$$\begin{eqnarray}&&{x}_{\mathrm{flare}}={R}_{\odot }\sin (\rho ).\end{eqnarray} \tag{ 36 }$$

The distribution of propagation distances $({x}_{\det ,{\rm{C}}2}-{x}_{\mathrm{flare}})$ is shown in Figure 17(a), ranging from 1.2 to ≳4.0 solar radii. The distribution of LASCO detection delays ${\rm{\Delta }}{t}_{\det ,{\rm{C}}2}\,=({t}_{\det ,{\rm{C}}2}-{t}_{\mathrm{start}})$ is shown in Figure 17(b), which varies in a range of ≈10–100 minute and has a median value of 48 min. The ambiguity of identical events detected with GOES and LASCO depends on the accuracy of LASCO velocity measurements, required to extrapolate the timing.

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In Figure 17 we show the distributions of various time delays, sampling all events from the LASCO catalog within a time window of ±2 hr with respect to the GOES flare start time. The time difference between the LASCO-extrapolated CME onset time t_LASCO and the GOES flare start time t_GOES shows a relatively broad distribution with a mean and standard deviation of Δt = −2 ± 46 min (Figure 17(c)). We are not sure how many of these LASCO events within this temporal coincidence margin are associated with the GOES and AIA-detected CME dimming events, but the fact that the AIA-inferred dimming time coincides with the GOES flare peak time within a much smaller time margin of Δt = 5 ± 7 min (Figure 17(e)) indicates that many of the LASCO events with delays ≳10 min may not be associated with the GOES and AIA-selected events.

A comparison of the occurrence frequency distributions of AIA and LASCO-inferred CME events is shown in Figure 18 (left panels), for the CME mass (top left panel), the CME speed (middle left panel), and the CME kinetic energy (bottom left panel), and the corresponding scatter plots between AIA and LASCO-inferred quantities are shown in the right-hand panels of Figure 18. In 218 events the CME mass and energy could be determined simultaneously with AIA and LASCO within a time window of ±1 hr. There is a overall agreement in the median values, but the corresponding quantities measured with both instruments scatter by up to two orders of magnitude. This leads us to conclude that the the parameters are differently defined for the two instruments. The method of measuring CME speeds with LASCO is designed to detect the leading edge of a CME, because this is the location where the largest density contrast occurs in white light, while the method of measuring speeds with AIA is weighted by the bulk plasma, which contributes most to the EUV emission measure. Since the leading edges are faster than the bulk speed, we suspect that LASCO obtains faster CME speeds than AIA, and consequently yields larger kinetic energies also. On the other hand, LASCO detects a number of CMEs with much lower masses than AIA, in the range of m_cme ≈ 10¹³–10¹⁵ g, which appear to be underestimates due to sensitivity issues in detecting white-light polarized brightness signals. We would expect that LASCO detections underestimate the mass and velocities of halo CMEs, but we are not able to find a center-to-limb effect in the LASCO data.

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Since the GOES fluxes are often used to characterize the magnitude of solar flares, we investigate the question of whether they are also suitable to express the magnitude of CMEs. Scatter plots of CME masses, speeds, and kinetic energies are compared with GOES 1–8 Å fluxes for both LASCO and AIA observations in Figure 19, which could be determined in 247 events with simultaneous GOES and LASCO coverage within a time window of ±1 hr. We find that the best correlation exists between the CME mass (as determined with AIA) and the GOES flux, with an almost linear relationship, i.e., $m\propto {F}_{{GOES}}^{0.8}$, with a linear regression coefficient of R = 0.67 (Figure 19(b)). The CME masses determined with LASCO reveal a similar correlation, i.e., $m\propto {F}_{{GOES}}^{0.7}$ (Figure 19(a)), but with a less significant linear regression coefficient of R = 0.32, indicating more random scatter in the uncertainties of the CME masses. A similar correlation between CME masses and GOES fluxes was found by Aarnio et al. (2011).

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There is also a weak correlation between the CME kinetic energies and the GOES fluxes (Figures 19(e) and (f)), which is a consequence of the mass dependence of kinetic energies. Nevertheless, these correlations reveal that a proportional amount of (magnetic) energy goes into plasma heating (as measured in the soft X-ray plasma with GOES) and into the kinetic energy of CMEs (as measured with AIA and LASCO).

Finally we compare the energy of CMEs with other forms of energies in the overall flare energy budget, such as with the dissipated magnetic energy E_mag in flares (Paper I and Figures 20(a) and (b)), the free energy E_free (Paper I and Figures 20(d) and (e)), the multithermal energy E_th (Paper II and Figures 20(g) and (h)), and the nonthermal energy E_nth (Paper III and Figures 20(j) and (k)). We compare the CME kinetic energy from LASCO data (Figures 20(a), (d), (g), (j)), the CME kinetic energy from AIA data (Figures 20(b), (e), (h), (k)), and the total (combined kinetic and gravitational) energy as measured from AIA data (Figures 20(c), (f), (i), (l)). From this comparison shown in Figure 20 we see that the CME kinetic energy, regardless of whether it is determined with LASCO or AIA, exhibits a large scatter with any form of other energies, while the total energy that includes both the kinetic energy and the gravitational potential energy of a CME combined reveals a much smaller scatter (down to a factor of three in the energy ratio). Therefore, it is imperial to include both the kinetic and gravitational energy in comparisons with other flare-generated energies, which is shown in the right column of Figure 20.

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According to these results we find that the total CME energy has a logarithmic mean ratio of E_cme/E_mag = 0.07 (with a scatter by a factor of 5.4; Figure 20(c)), but is virtually always lower that the dissipated magnetic energy in flares, as estimated from nonlinear force-free field modeling (Paper I). A similar ratio is found for the free magnetic energy, namely E_cme/E_free = 0.11 (with a scatter by a factor of 6.3; Figure 20(f)), which is a consequence of the fact that the dissipated magnetic energy was found to be a large fraction of the available free energy, at least in GOES M- and X-class flares (Paper I). The most interesting result is the ratio of the total CME energy with the multithermal flare energy (Paper II), for which we find the tightest correlation, with a mean logarithmi ratio of E_cme/E_th = 0.77 (with a scatter by a only a factor of 3.5; Figure 20(i)). This indicates that the energy that goes into a CME is most closely related to the heating of the flare plasma. This correlation is also consistent with the other correlation found between the CME mass and the GOES flux (Figure 19(b)). A relatively low ratio of the total CME energy to the nonthermal energy was found, E_cme/E_nth = 0.10 (with a scatter by a factor of ≈7; Figure 20(l)), calculated from the thick-target model of accelerated and precipitating electrons, based on a low-energy cutoff estimated with the warm-target model of Kontar et al. (2015).

The EUV dimming method developed here is an alternative method to the CME mass determination by Thomson scattering in white-light observations. Essentially, a reduction of the EUV brightness (or emission measure) anywhere on the solar disk or above the limb can be converted into a corresponding coronal mass loss, carried away by a CME, by an eruptive filament, or by drainage down to the chromosphere (coined "coronal rain"; Schrijver 2001). Early pioneering observations of coronal dimmings during energetic CMEs, using SOLWIND and LASCO/SOHO, are described, e.g., in Jackson & Hildner (1978), Howard et al. (1985), and Thompson et al. (2000). A statistical study found that 55% of EUV dimming events detected with SMM/CDS are associated with CMEs detected in LASCO data (Bewsher et al. 2008). Vice versa, 84% of CME events could be tracked to EUV dimming regions (Bewsher et al. 2008). EUV dimmings occur in 74% of flares associated with CMEs (Nitta et al. 2014). In contrast, so-called "stealth-CMEs" have been reported in a few cases (Robbrecht et al. 2009; Howard & Harrison 2013; Nieves-Chinchilla et al. 2013), which apparently leave the solar corona "with no trace left behind." In a study of 96 CME-associated EUV coronal dimming events using LASCO, the time profile of the EUV dimming was characterized by two phases, an initial rise and decay with similar timescales, followed by a flatter decay lasting several hours (Reinard & Biesecker 2008), which we model with a unified adiabatic expansion model in this study.

With the advent of STEREO, the 3D structure of CME source regions and associated EUV dimming could be observed with the EUVI/A+B imagers and modeled with stereoscopic methods (Aschwanden 2009, Aschwanden et al. 2009a, 2009b; Temmer et al. 2009; Bein et al. 2013). A key result was that the CME mass determined from EUV dimming agreed well with those determined with the white-light scattering method (m_EUVI/m_LASCO = 1.1 ± 0.3), and agreed also between the two STEREO spacecraft A and B (${m}_{A}/{m}_{B}=1.3\pm 0.6$) (Aschwanden et al. 2009a). Another benefit of stereoscopic observations is the determination of the 3D trajectory and de-projected CME speed and mass (Bein et al. 2013).

The most recent observations of EUV dimming use SDO/AIA images in multiple wavelengths, which allows for DEM analysis of the density and temperature structure of CMEs (Cheng et al. 2012; Mason et al. 2014). From such studies it was concluded that the EUV dimmings are largely caused by the plasma rarefaction associated with the eruption (Cheng et al. 2012), which we model in terms of adiabatic expansion in this study. Automated searches of flares, EUV dimmings, and EUV waves in SDO/AIA images are also now available (Kraaikamp & Verbeeck 2015).

In this study we developed a novel EUV dimming method that is capable to measure the timing, mass, acceleration, speed, propagation, and the kinetic energy of CMEs. The event selection in our statistical study encompasses all (N = 399) GOES M- and X-class flares during the initial 3.5 year period of AIA/SDO and we found that the EUV dimming method yielded a positive detection and measurements of CME parameters in 100% of the analyzed events. For the simultaneous detection of events with the white-light method using LASCO/C2, we find an upper limit of possible detections for 69% (with a time delay of ${\rm{\Delta }}t=({t}_{\mathrm{LASCO}}-{t}_{\mathrm{start}})=-2\pm 46\,\mathrm{minute}$ between the LASCO-extrapolated starting time at the coronal base and the GOES flare peak time; Figure 17(c)), and a lower limit of 30% (for a delay range of Δt = −0.3 ± 7.3 min that is equivalent to the delays of the AIA detections, with Δt = t_dimm −t_start = 5 ± 7 min; Figure 17(e)) between the onset of EUV dimming and the GOES flare peak time. Therefore, the EUV dimming method is found to be far more sensitive to detections of CMEs, and we can expect positive detections for weaker flares (of GOES C-class or lower). We know that there exist many CMEs that are associated with weaker flares than GOES M1.0 class, because the LASCO CME catalog shows many entries between the time intervals of M- and X-class events analyzed here. This is also consistent with the statistics of an EUVI/STEREO study, which included 185 flare events and where 31 EUV dimming events were identified, mostly C-class events, while only 11 events were of M-class (Aschwanden et al. 2009b).

On the other hand, however, there are a few cases of "stealth CMEs" known, which apparently were detected in white-light but not by EUV dimming (Robbrecht et al. 2009; Howard & Harrison 2013; Nieves-Chinchilla et al. 2013). It is conceivable that the larger temperature range of AIA (T_e ≈ 0.5–20 MK) could reveal EUV dimmings of stealth CMEs that are not detectable in the narrower temperature ranges of EIT/SOHO or EUVI/STEREO (Robbrecht et al. 2009; Howard & Harrison 2013), or that CME-associated flare sites were occulted behind the limb.

Since we find a large scatter between CME speeds determined with the white-light and the EUV dimming methods (Figure 18) we need to understand whether the differences result from different physical speed definitions or from uncertainties in the measurement methods.

CMEs observed in white light are detected from difference images in order to subtract out the unpolarized static white-light scattering component (of the entire Sun) from the polarized brightness produced by the CME. The features that are brightest in difference images are always where the steepest brightness gradients occur, which is generally at the leading edge of expanding CMEs, and therefore the inferred CME speed is actually the leading-edge speed v_LE, which is supposedly the fastest speed by definition. White-light observations of CMEs often show a "three-part" structure, consisting of (1) a bright front, (2) a dark cavity, and (3) a bright, compact core. The bright front corresponds to the leading edge. The position of the leading edge is then measured in terms of a height–time plot h(t), while the speed $v(t)={dh}(s)/{dt}$ is derived from various fitting functions. The speeds that are compiled in the LASCO CME catalog were derived with three different functions: a linear or first-order polynomial fit v₁, a quadratic or second-order fit v₂, and a speed v₃ evaluated at a distance of 20 solar radii. Comparing the 228 LASCO events with triple-velocity measurements at the earliest time of LASCO/C2 detection, out of the 399 GOES flare events analyzed here, we find ratios of ${v}_{1}/{v}_{2}=1.0\pm 0.2$, so we assess an uncertainty of ≈20% to the white-light CME leading-edge speed measurements. We have also to be aware that white-light speeds are projected speeds v_proj in the plane-of-sky, which should be corrected by $v={v}_{\mathrm{proj}}/\sin (l)$ at a heliographic longitude l, as was done for some stereoscopic measurements to derive the true CME speed and mass (Colaninno & Vourlidas 2009).

The speed measurements with AIA are conducted with a completely different method, based on the best fits of EUV dimming light curves EM_tot(t) (Equation (20)), which yields a best-fit height–time profile x(t), from which a speed v = dx(t)/dt is derived. Thus, the gradient in the dimming curve $d{\mathrm{EM}}_{\mathrm{tot}}/{dt}\propto v(t)$ is decisive for the speed measurements. Obviously, the largest contributions to the EUV dimming curve come from the locations with the largest emission measures, and thus it represents a bulk plasma speed v_bulk. Since the leading edge is by definition faster than the bulk plasma speed, v_LE > v_BP, we expect that the LASCO-inferred speed is faster than the AIA-inferred speed, which is indeed the case for the majority of events (Figures 18(d), 19(c), (d)). The slowest measured speeds amount to v_LE ≳ 100 km s⁻¹ for LASCO and to v_BP ≳  30 km s⁻¹ for AIA data. Earlier measurements with SOLWIND onboard P78-1 yield CME speeds of v_cme ≈ 100–1500 km s⁻¹ (Howard et al. 1985; Jackson & Howard 1993), or with LASCO v_cme ≈ 100–1300 km s⁻¹ (Chen et al. 2006).

There is also a time-dependence of the CME speed, but our models (Figure 4) indicate that the velocity function v(t) quickly reaches an asymptotic value at coronal heights, so that the speed is almost constant by the time it reaches a distance of x  ≳ 2.2 solar radii, where we compare LASCO/C2 and AIA-inferred CME speeds.

The CME masses determined with LASCO and AIA exhibit a large scatter up to two orders of magnitude (Figure 18(b)). We ask the question: what systematic errors affect the measurement of CME masses most?

For the white-light method, the mass is proportional to the scattered polarized light (Appendix B, Equation (45)). Summing up the polarized brightness from a coronagraph, all the CME mass below the coronagraph occulter height (h ≲ 2.2 R_⊙) is missing in the mass calculation (Figure 1), which is most dramatically underestimated during halo-CMEs, which propagate in the direction to the observer. The missing mass was estimated to be a factor of two for CMEs that are ≲40° from the plane-of-sky (Vourlidas et al. 2010). The LASCO CME catalog also lists many of the analyzed events as "poor" or "very poor events," which probably characterizes the data noise and related detection sensitivity in the polarized brightness difference images. The scatter plot of CME masses shown in Figure 18(b) indicates a low-mass cutoff of m ≳ 0.3 × 10¹⁵ g for AIA-inferred masses, while the LASCO-inferred masses extend down to m ≳ 10¹³ g, which are likely to be underestimates of the true masses. However, these underestimates do not exclusively apply to halo-CMEs, because we were not able to find a center-to-limb effect. Early studies, containing statistics of nearly 1000 CME events observed with the SOLWIND instrument on the P78-1 satellite, exhibit a range of m_cme ≈ 10¹⁴–10¹⁷ g for CME masses (Howard et al. 1985; Jackson & Howard 1993). A recent statistical study of 7741 CMEs detected with LASCO shows a distribution with a range of m ≈ 10¹²–10¹⁶ g, with the bulk of events in the range of m ≈ 10¹⁴–10^15.5 g (Aarnio et al. 2011).

What are the advantages and caveats of EUV dimming-derived CME masses? The EUV emission measure scales with the square of the electron density, EM ∝ n_e² V (Equation (11)), a property that already leads to a nonlinear amplification in the contrast of any detected EUV feature, in particular to EUV brightenings and dimmings with respect to the background EUV emission. Moreover, because no coronagraph is needed, the entire CME source region is visible to an EUV imager without any occultation effects (except behind the solar limb, as may happen for stealth CMEs). The CME mass is then directly obtained from the volume integral m_cme = n_em_pL²λ (Equation (13)). Some uncertainties in the CME mass determination may result from the inhomogeneity of the mass distribution inside the coronal region that corresponds to the CME initial source volume, which depends on the measurement of the projected dimming area A_p, which we estimate to be a factor of ≲2 in the CME mass estimate, being a relatively small correction for the observed variation of CME masses by two orders of magnitude (m ≈ (0.3–30) × 10¹⁵ g). In comparison, earlier measurements with SOLWIND on the P78-1 satellite, exhibit CME masses of m_cme ≈ 10¹⁴–10¹⁷ g.

The kinetic energy, E_kin = (1/2) m_cmev² (Equation (17)), is a combination of the CME mass and CME speed, and thus is subject to all of their uncertainties discussed above, mostly affected by the velocity, which has a square dependence. The obtained kinetic energies show indeed a large scatter between the LASCO and AIA-derived values, within ≈4 orders of magnitude (Figure 18(f)). The largest discrepancy comes from the different definitions of leading edge v_LE and bulk plasma speeds v_BP, which varies by about 1.5 orders of magnitude (Figure 18(d)), and causes with its square dependence a variation of 3 orders of magnitude in the kinetic energy. The range of kinetic energies obtained with both AIA and LASCO is E_kin ≈ 10²⁸–10³³ erg (Figure 18(f)). Earlier studies, containing statistics of nearly 1000 CME events observed with the SOLWIND instrument on the P78-1 satellite, exhibit a range of E_kin ≈ 10²⁹–10³² erg for CME kinetic energies (Howard et al. 1985; Jackson & Howard 1993).

The gravitational potential energy to lift a CME from the bottom of the corona to infinity depends only on the CME mass (Equation (22)). For solar gravity, the required escape velocity is v_esc = 618 km s⁻¹. Note that all CME velocities we quote are close to the asymptotic limit at large distances from the Sun, and thus represent roughly the sum of the escape velocity and the observed moving velocity. CMEs that are accelerated to a lower velocity than the escape speed, v_init < v_esc, produce so-called "failed CMEs," which initially move upward and fall back afterwards. In this study we do not distinguish between "failed/confined" and "eruptive" CME events, assuming that radial adiabatic expansion occurs for both phenomena (at least during the initial "explosive" phase), and thus our data set of 399 M and X-class flare events may contain both types, while the coincident 247 LASCO events are all eruptive events, since they are detected at large distances h ≳ 2.2 R_⊙.

For an estimate of the flare energy budget it is important to compute both the kinetic and the gravitational potential energy for each CME event. For the 399 analyzed CME events we find a range of E_kin = 3 × 10²⁷–10³³ erg for kinetic CME energies, and a range of E_grav = 2 × 10²⁹–6 × 10³¹ erg (Table 1). Actually, we find that the gravitational energy makes up a fraction E_grav/E_tot = 0.75 ± 0.28 of the total CME energy E_tot = (E_grav + E_kin) on average. Only for the most energetic CMEs is the kinetic energy larger than the gravitational energy, which is the case for 22% of the M- and X-class flares.

In an earlier study on the global energetics of flares and CMEs, the CME kinetic energy E_kin in the rest frame of the Sun, as well as in the solar wind frame, and the gravitational potential energy E_grav were calculated (Emslie et al. 2004, 2005, 2012). The gravitational energy was found to be about an order of magnitude lower than the kinetic CME energy, in contrast to the study of Vourlidas et al. (2000), where it was found that the center of mass accelerates for most of the events, while the CMEs achieve escape velocity at heights of around 8–10 R_⊙, and that the potential energy is greater than the kinetic energy for relatively slow CME events (which constitute the majority of events).

We are now in a position to study the global energetics of solar flares and CMEs, by combining the dissipated magnetic energies E_magn (Paper I), the available free (magnetic) energies E_free (Paper I), the multithermal energies E_th (Paper II), the nonthermal energies E_nth (Paper III), and the CME (kinetic and gravitational) energies from this study. In Figure 20 we present scatter plots between the different types of energies: the LASCO-inferred CME kinetic energies (Figure 20 left-hand panels) show a similar large scatter with the other forms of energy as the AIA-inferred energies do (Figure 20 middle panels). However, since the gravitational energy dominates the kinetic energy for most CME events, we have to compare the total (kinetic plus gravitational) CME energies, which we show for the AIA data (Figure 20 right-hand panels). Interestingly, the total CME energy exhibits a substantially smaller scatter with other forms of energies, about a factor ≈3 less than the kinetic energy (see the factors of the logarithmic standard deviations indicated in the top right corner of each panel in Figure 20), which substantially improves the correlations between different forms of energy that are dissipated and converted during flare/CME events.

For a test of the magnetic energy budget that can support a CME, the mean energy ratio is important. We find that the ratio of the total CME energy to the dissipated magnetic energy is (in the logarithmic mean) q_E = 0.07, with a standard deviation by a factor of 5.4 (Figure 20(c)), which corresponds to a range of q_e = 0.01–0.38. This is an important result that warrants that the dissipated magnetic energy (as estimated from the untwisting of helically distorted field lines by vertical currents; Paper I) is virtually always sufficient to support the launch of the observed CMEs. Also the available free energy shows a similar ratio, q_e = 0.11, with a standard deviation by a factor of 6.3 (Figure 20(f)), which corresponds to a range of q_e = 0.02–0.69. In principle, the free energy should be larger than the actually dissipated magnetic energy, and thus the difference must result from a combination of measurement uncertainties.

A most striking result is the relationship between total CME energies and thermal flare energies, which shows a much reduced scatter down to a factor of 3.5, which reveals a true correlation between these two forms of energies. The (logarithmic) mean ratio is q_e = 0.77, within a factor of 3.5 (Figure 20(i)). In other words there is about the same amount of energy that goes into a CME as is converted into thermal flare energy (as measured in soft X-rays). We find a relatively low ratio between the total CME energy and the nonthermal flare energy (Figure 20(l)), calculated from the thick-target model with a low-energy cutoff constrained by the warm-target model of Kontar et al. (2015).

If we envision a magnetic reconnection scenario with vertical eruption, such as described by Carmichael (1964), Sturrock (1966), Hirayama (1974), Kopp & Pneuman (1976), Tsuneta (1996), Tsuneta et al. (1997), and Shibata et al. (1995), the plasma heating by precipitating electrons and subsequent chromospheric evaporation is a direct consequence of the accelerated electrons in the reconnection region, behind a rising prominence and the CME, which explains the correlation between (multi-)thermal energies and total CME energies. In a previous study of 37 impulsive flare/CME events it was found that the CME peak velocity is highly correlated with the total energy in nonthermal electrons, which supports the idea that the acceleration of the CME and particle acceleration in the associated flare have a common energy source, most likely magnetic reconnection in the wake of an eruptive CME (Berkebile-Stoiser et al. 2012).