- Dana LongcopeDana LongcopeMontana State University, Department of Physics
A solar flare is a transient increase in solar brightness powered by the release of magnetic energy stored in the Sun’s corona. Flares are observed in all wavelengths of the electromagnetic spectrum. The released magnetic energy heats coronal plasma to temperatures exceeding ten million Kelvins, leading to a significant increase in solar brightness at X-ray and extreme ultraviolet wavelengths. The Sun’s overall brightness is normally low at these wavelengths, and a flare can increase it by two or more an orders of magnitude. The size of a given flare is traditionally characterized by its peak brightness in a soft X-ray wavelength. Flares occur with frequency inversely related to this measure of size, with those of greatest size occuring less than once per year. Images and light curves from different parts of the spectrum from many different flares have led to an accepted model framework for explaining the typical solar flare. According to this model, a sheet of electric current (a current sheet) is first formed in the corona, perhaps by a coronal mass ejection. Magnetic reconnection at this current sheet allows stored magnetic energy to be converted into bulk flow energy, heat, radiation, and a population of non-thermal electrons and ions. Some of this energy is transmitted downward to cooler layers, which are then evaporated (or ablated) upward to fill the coronal with hot dense plasma. Much of the flares bright emission comes from this newly heated plasma. Theoretical models have been proposed to describe each step in this process.
- Astronomy and Astrophysics
Observation and Overview
A solar flare is a sudden brightening of a small portion of the Sun’s surface, powered by the release of stored magnetic energy. In current practice, a flare is identified and classified as a peak in the Sun’s total brightness in the 1–8Å soft X-ray (SXR) band of NOAA’s GOES satellite (The Geostationary Operational Environmental Satellite of the US National Oceanic and Atmospheric Administration). During a strong flare the Sun’s brightness in this particular band can increase more than 100-fold as shown in Figure 1a. A peak exceeding , , or is categorized as a flare of class X, M, or C respectively. The example flare peaks at , designating it class M7. While describing flares in generality, the well-observed example shown in Figure 1 will be used for concrete illustration.
A flare includes simultaneous brightening, to some degree, in virtually every wavelength in the spectrum. The extreme ultraviolet (EUV) spectral lines shown in Figure 1b brighten by anywhere from 3% (Fe ix) to 1,000% (Fe xx) above pre-flare levels. The wide range is due mostly to the wide range of quiescent values on which the comparison is based. The Sun is brightest in the visible, around 5,000Å, and even a large flare will only brighten these wavelengths by about % (Kopp, Lawrence, & Rottman, 2005; Kretzschmar et al., 2010). Therefore, only extremely large flares are detectable as increases in overall visible brightness of the Sun, or a comparable increases in its bolometric luminosity. It is easier to detect a localized increase in visible surface brightness; a flare large enough to show such an increase is termed a white light flare.
Light curves of different wavelengths from a given flare tend toward one of two characteristic behaviors. Some radiation, such as microwaves, gamma rays, or the hard X-ray (HXR) curve shown in Figure 1d, originates from the flare’s footpoints, and has a brief, impulsive light curve. In general, these persist only for the initial 1–10 minutes, known as the flare’s impulsive (or rise) phase. Other radiation, such as the SXR and EUV curves from Figures 1a and 1b, originates from the coronal plasma, and evolves more gradually. These light curves rise during the impulsive phase, and then decay for times ranging from 10 minutes to over 10 hours, known as the gradual (or main) phase.
The time derivative of a coronal light curve, most commonly the GOES 1–8Å curve shown in Figure 1e, tends to resemble the more impulsive footpoint light curves. This resemblance is known as the Neupert effect (Dennis & Zarro, 1993; Neupert, 1968), and is taken as evidence that energy deposited in the lower atmosphere heats and ablates material upward into the corona. This ablation process is commonly called chromosphere evaporation (Canfield et al., 1980; Antonucci et al., 1999), even though it is related to familiar chemical evaporation only by analogy.
The gradually evolving coronal plasma has a temperature typically ranging from 1–30 MK. In the example from Figure 1, a ratio of SXR bands shows a temperature peaking at MK, and decreasing gradually thereafter. This cooling behavior is generally reflected in the light curves from progressively cooler ion species peaking at progressively later times (Aschwanden & Alexander, 2001; Qiu & Longcope, 2016; Warren, Mariska, & Doschek, 2013); compare the four curves in Figure 1b.
Emission during the impulsive phase from microwaves and HXR often appears to originate from a population of energetic electrons interacting with the coronal plasma or the footpoints (Krucker et al., 2008). Spectra show this population to have a non-Maxwellian (i.e., non-thermal) distribution of energies—typically a power law, above some lower cutoff, . The flare process evidently produces such a non-thermal electron population, at least during its impulsive phase.
Photons with energies above 1 MeV, that is, gamma-rays, are observed during the impulsive phases of some large flares (Murphy, 2007). These sometimes show evidence of spectral lines from nuclear processes, providing clear evidence of non-thermal ions of high energies. It is generally believed that ions are being accelerated in many, if not all, flares, even though gamma ray signatures can be observed in only the largest ones.
Flares, especially large ones, are frequently associated with the eruption of mass-loaded flux ropes, known as coronal mass ejections (CMEs). Several investigators have attempted to determine whether flares cause, or at least precede, CMEs or vice versa (Zhang, Dere, Howard, Kundu, & White, 2001). There are, however, numerous well-studied cases of flares occurring without CMEs (Chen et al., 2015b; Yashiro, Gopalswamy, Akiyama, Michalek, & Howard, 2005), called compact or confined flares, and of CMEs occurring without flares (Munro et al., 1979). This makes it clear that there can be no invariable cause–effect relation between these two distinct phenomena (Gosling, 1993). They are simply associated phenomena. Their association is most likely when the flare is large and has an extended gradual phase; such cases are known as eruptive flares or long-duration events.
In all cases, the flare is the component that produces the enhanced brightness, intimately related to the Sun’s lower atmosphere. The enhancement of EUV and X-rays increases the rate of ionization in the upper atmosphere of the Earth and other planets (Chamberlin, Woods, & Eparvier, 2008; Fuller-Rowell & Solomon, 2010). A flare thereby affects the ionosphere immediately, while a CME can have more varied effects at Earth, but only when the magnetized mass impacts its magnetosphere.
The brightness enhancement of a flare demands that it be associated with the release of energy beyond the Sun’s steady luminous radiation. It is relatively straightforward to compute the total radiative loss from the hot coronal plasma, since this is readily observed in EUV and SXR. The two SXR bands from GOES provide an estimate of temperature and emission measure from which total radiative losses can be computed. Doing so for the example in Figure 1 reveals a peak coronal radiation power, , roughly 20 times greater than the power in the narrow 1–8Å bandpass of GOES. Over its multi-hour duration the coronal plasma of this particular flare radiates ergs. These values are fairly average, and X flares will typically show coronal radiation up to and total energies up to a few times erg. More careful computations, made using the full differential emission measure of the coronal plasma, yield comparable values in general.
The coronal plasma is only part of the flare, so the power it radiates accounts for only a fraction of the flare’s total energy. That total has proven difficult to compute, although several serious attempts have been made to do so in particular cases (Emslie et al., 2004, 2005) or for collections (Emslie et al., 2012). Energy radiated from the lower atmosphere, while at lower temperature, will usually exceed, even far exceed, the coronal losses. This is, however, a much smaller fraction of the pre-flare losses at those wavelengths, and is thus far more difficult to measure. Other contributions cannot be quantified without a good model for the flare process: the power deposited in the chromosphere by non-thermal electrons can often exceed the losses to coronal radiation, but it is not entirely clear whether the deposited energy is ultimately radiated from the corona or chromosphere (and is therefore already counted) or is lost in some other way. A CME will also carry away energy, but its source may be from the flare, or from the much larger coronal volume it affects. In light of these factors, there is no simple relationship between the peak X-ray flux, or flare class, and the total energy powering a solar flare.
Solar flares almost always occur in active regions (ARs) of relatively strong, complex magnetic field (see Figure 2a). The most intense chromospheric emission is generally organized into elongated structures, called flare ribbons. In the prototypical case, called a two-ribbon flare, there is one ribbon in each of the AR’s magnetic polarities, and they are separated by the polarity inversion line (PIL; see Figure 2b). The ribbons generally move apart slowly over the course of the flare, providing evidence of progressing magnetic reconnection. There is also an apparent motion, especially early on, related to the formation and elongation of the ribbons (Fletcher, Pollock, & Potts, 2004; Qiu, 2009). Some ribbons can have very complex structure on their finest scales and this may undergo motions more disorderly than the foregoing description implies (Fletcher & Hudson, 2001).
Coronal wavelengths show loops of hot plasma tracing out field lines interconnecting the magnetic polarities. The loops often connect points on the opposing ribbons, thereby forming an elongated arcade: the flare arcade (see Figures 2c and 2d). The loops appear later in images from progressively cooler ions, consistent with the cooling plasma scenario (Aschwanden & Alexander, 2001; Warren et al., 2013). As the ribbons spread apart, the loops anchored to them appear to be rising upward. It is generally accepted, however, that individual loops are relatively stationary, or may even be contracting downward (Forbes & Acton, 1996). The apparent upward motion is thus ascribed to the appearance of new loops piling on top of older loops as magnetic reconnection proceeds.
When images are made from HXR emission, they typically show one or more concentrated sources. Often the sources fall at points along the ribbons, but not along the entire extended ribbon (Sakao, 1994). Such sources are attributed to electron deposition at the footpoint(s) of flare loop(s). The yellow contours on Figure 2b show one source on each ribbon, presumably from both footpoints of a single loop, such as the one appearing in Figure 2c. HXR footpoint sources from a single loop can appear of differing strength, as in Figure 2b. One interpretation is that the footpoint with stronger magnetic field mirrors a larger fraction of the precipitating electrons (Goff, Matthews, van Driel-Gesztelyi, & Harra, 2004; Sakao, 1994). The sources can appear to move along the ribbon, probably showing footpoints of different loops in succession. The directions and speed of this motion has been used to infer aspects of the magnetic reconnection, and the electron acceleration (Bogachev, Somov, Kosugi, & Sakao, 2005).
HXR images sometimes show a concentrated source between the footpoints, where the loop’s apex is expected to be (Masuda, Doschek, Boris, Oran, & Young, 1994). Flares at the limb show the source to be located just above the hottest loop visible in softer wavelengths, leading to the term above-the-looptop source for such features (see yellow contours in Figure 3a). Apparent motions of such sources has also been interpreted in terms of the time-dependance of the reconnection process (Sui & Holman, 2003).
The Standard Model Framework
The various observations have led to a model, or framework, of a generic flare. The model, shown in Figure 3b, is typically cast in terms of an eruptive flare (see Figure 3a), but most features are expected to have counterparts in compact flares. The earliest version is attributed to Carmichael (1964), Sturrock (1968), Hirayama (1974), and Kopp and Pneuman (1976), and is called the CSHKP model. Since then the basic model has been extended and augmented to accommodate new theoretical understanding and new observed features.
The model flare is initiated when an erupting flux rope (cyan circle in Figure 3b) pulls open the flux on either side of the PIL, creating a current sheet separating upward from downward open field. Reconnection occurs at some point in the current sheet, designated by a green ellipse labeled X. Open flux is swept inward, and reconnected to form closed field lines which are then swept out by long narrow outflow jets. Loop retraction stops abruptly at a point labeled termination, at the end of the jet. The fully retracted loop becomes the flare loop and its feet form the ribbons, which are seen end-on as magenta boxes in Figure 3b.
According to the foregoing, two-dimensional model, the outermost, or leading, edge of the flare ribbon anchors the separatrix (red curve in Figure 3b) which connects to the X-point. The amount of flux reconnected,, can be computed by integrating the vertical magnetic flux over which the ribbon appears to sweep (Forbes & Priest, 1984; Poletto & Kopp, 1986; Qiu, Lee, Gary, & Wang, 2002). Such measurements have become reasonably routine and provide reconnection rates typically peaking in the range in small flares to for the largest (Tschernitz, Veronig, Thalmann, Hinterreiter, & Pötzi, 2018).
The reconnection in the standard flare model reflects the structure found in studies of reconnection in generic contexts, and has been verified to some extent through observation (McKenzie, 2002). The diffusion region (green ellipse in Figure 3b) occupies only a small portion of the current sheet, thereby permitting reconnection of the fast, Petschek variety (Forbes, Priest, Seaton, & Litvinenko, 2013; Petschek, 1964). In this mode of reconnection the outflow jets are bounded by slow magnetosonic shocks, shown in dark red in Figure 3b, which accelerate, compress, and heat the jet’s plasma (Forbes & Priest, 1983). If accelerated to a speed above the local fast magnetosonic speed, there will be a fast magnetosonic shock at its termination (Forbes, 1986). Some evidence has been found for such a structure in radio observations (Aurass, Vršnak, & Mann, 2002; Chen et al., 2015a).
The model offers several possibilities by which the energy released by reconnection could generate a population of non-thermal electrons. Various theoretical models under study predict electron acceleration occurring within the diffusion region, in the outflow jet, or at the termination shock. Each of these possibilities offers an explanation for the population of non-thermal electrons observed in microwaves and HXRs, and each produces a distribution consistent with a power law. The electrons may be trapped near the termination, to produce an above-the-looptop source (shown in orange in Figure 3b), or they could precipitate along the flare loop to produce the footpoint sources.
Some of the energy released by magnetic reconnection will be guided downward by the magnetic field to the chromospheric flare ribbons. The energy could be transported by the non-thermal electrons or it could be transported through conventional thermal conduction, which is directed almost entirely along magnetic field lines. Once the energy reaches the feet, it will raise the temperature of the chromospheric plasma, and drive upward evaporation. This scenario nicely explains the Neupert effect where coronal emission appears as a response to the chromosphere. Spectroscopic measurements confirm the fast upflow of material within the flare ribbons (Antonucci & Dennis, 1983; Milligan & Dennis, 2009).
As reconnection proceeds, new loops will be moved through the outflow to lie atop the arcade. This will have the effect of moving the reconnection point upward and moving the separatrix, and hence the flare ribbons, outward. It will also produce a series of ever higher flare loops. Both effects are consistent with the observed evolution.
The Flare Population
While every flare is unique, there is a strong tendency for extensive quantities to scale together: a large flare is usually large in every measure (known as “big flare syndrome”; Kahler, 1982). It is therefore common to characterize a flare’s size is by a single measure, usually by its peak flux in GOES 1–8Å, designated here by . As was mentioned, no rigorous relation exists between this measure and any other characteristic of a flare. Nevertheless, is readily measured and scales with the flare’s size.
Flares occur at all sizes with frequency inversely dependent on size (Crosby, Aschwanden, & Dennis, 1993). Figure 4 summarizes the flare activity, as characterized by , over three solar cycles (from 1986 to 2016). Since flares are associated with ARs, they occur with highest frequency around solar maxima (1991, 2001, and 2014). At these times flaring rates can increase to over 8 C-class flares and one M-class flare per day, as shown in Figure 4b. These rates drop by more than two orders of magnitude during solar minimum. Averaged over all three cycles, M-flares occur at a mean rate of /day, or about 1,300 over an 11-year cycle. (There were 2,047, 1,441, and 681 M-flares in these three solar cycles, whose amplitudes clearly decrease.)
The average flaring rate over the entire three-cycle interval can be formed into a frequency histogram, shown in Figure 4d. This clearly shows a power-law behavior, . This means that on average an X-flare is less probable than an M-flare by a factor of . They occur at an average rate of /year, compared to 120 M-flares per year.
The power-law distribution of solar flares is reminiscent of the distributions of earthquakes energies, or avalanche sizes (Bak, Tang, & Wiesenfeld, 1988). This resemblance has led some investigators to propose analogous models to explain flare-size distributions (Aschwanden et al., 2018; Lu & Hamilton, 1991). It is also possible to use the empirical relationship along with observations of the current rate of small flares to forecast the likelihood of a larger flare in the near future (Wheatland, 2004).
Investigations reveal that the power-law distribution continues in the direction of smaller flares, in spite of the apparent roll-off evident in Figure 4d. That is caused by the systematic undercounting of small flares when the activity, and related background, rises. (Their absence is clear as light voids in the regions at solar maximum.) This confusion limit can be overcome using spatially-resolved measurements over more limited fields of view. Such measurements reveal a continuation to ever smaller flares, and perhaps still farther to yet-unobserved nano-flares.
The distribution may also extend to extremely large flares, but small numbers make this unclear. Based on the power law, flares above X10 should occur at a rate of per year, which is 7.2% of the rate for flares X1–X10. This would amount to 19 in the 31-year sample. In actuality 16 flares were observed in this class, (10 in cycle 22 and 6 in cycle 23), which is within expectations for such a small sample. Thus we cannot rule out the extension of the power law to that and still higher levels. If it applies out to (i.e., X1000), we would expect one such super-flare every 300 years, on average.
This physics of flares shows that larger flares require larger active regions and the reconnection of more flux from them. Any of these factors may turn out to have an upper bound which would in turn place a limit on the possible size of a solar flare. Several efforts have been made to estimate that upper limit (Aulanier et al., 2013; Schrijver et al., 2012; Shibata et al., 2013), but no consensus has been reached.
Models and Theories
There have been many attempts to understand solar flares theoretically and to incorporate this understanding into models. Most efforts have tended to focus on one specific aspect, although a few notable efforts have attempted to combine multiple aspects into a single model. One focus has been on the large-scale dynamics of a solar flare, often including the CME. A second has been on the dynamics of the plasma flowing within the flare loop, with particular attention to the process of chromospheric evaporation. A final set of studies has focused on the generation of non-thermal particles and their propagation along the flare loop.
Triggering and Eruption
Under the standard model framework, flares and CME begin together, so their earliest phases are generally combined into a single flare/CME model. This initial evolution occurs on a very large scale and is almost always modeled using the single-fluid equations of MHD (see “Magnetohydrodynamics—Overview,” Priest, 2019). A successful model must explain the sudden onset of fast eruption after an extended period of slow evolution. In a number of models the eruption is initiated through a large-scale, current-driven MHD instability triggered when slow (quasi-static) evolution, driven from the lower boundary brings the system past the instability threshold. Every active region, and thus every flare, has a different geometry. Models typically use a simplified, generic geometry, in an effort to study flare evolution in general.
In one model (Biskamp & Welter, 1989; Hood & Priest, 1980; Mikic, Barnes, & Schnack, 1988) the initial magnetic field forms an equilibrium arcade across the PIL, which is sheared by slow motions of the lower boundary. This slow shearing causes a proportionately slow upward expansion of the equilibrium arcade. In some versions of this model there is a shear threshold beyond which equilibria are unstable (Kusano, Maeshiro, Yokoyama, & Sakurai, 2004). Once this threshold is crossed, the upward expansion becomes dynamic, rather than quasi-static. Other versions lack a genuine threshold, but instead exhibit an expansion of increasing speed until the system must behave dynamically, regardless of how slow the boundary moves (Mikic & Linker, 1994). In either case, the rapid upward expansion creates a current sheet at which reconnection occurs to form the erupting flux rope and thereafter the flare and CME.
Several other models assume a twisted (i.e., current-carrying) flux rope exists in equilibrium prior to eruption. The rope’s twist can be increased either by slow boundary motions or by reconnection (Amari, Luciani, Mikic, & Linker, 2000). Such equilibria are subject to large-scale instability for sufficient levels of twist. In one instability, the kink mode, the previously smooth axis of the flux rope develops a helical pitch, reducing the total magnetic energy. The equilibrium becomes unstable once the field lines wrap around the straight axis more by than a critical angle, generally – depending on the particular equilibrium (Hood & Priest, 1981).
A second instability concerns a twisted flux rope overlain by an external field required to balance the rope’s outward hoop force. The hoop force is a repulsion between the current in a section of the rope and its image current below the boundary. This force decreases as the flux rope moves upward, away from the boundary. A stable equilibrium requires that the overlying field, and the balancing force it supplies, decrease less rapidly than the hoop force, in order that a balance can always be achieved. If this is not the case, that is, if the overlying field deceases too rapidly with height, the net upward force will increase with height leading to a run-away expansion: an eruption. This is a form of lateral kink instability known as the torus instability, and it is triggered once the flux rope enters a region where the overlying field strength decreases sufficiently rapidly with height (Kliem & Török, 2006). Both kink and torus instabilities are considered viable mechanisms to produce a CME and associated flare (Démoulin & Aulanier, 2010). No consensus has yet been reached as to whether one is invariably responsible, or simply more frequently responsible, for observed eruptions.
In a separate class of models, called loss of equilibrium, the equilibrium contains a flux rope as well as one or two current sheets. Magnetic reconnection is assumed to occur at the current sheet, but slowly enough to drive quasi-static evolution (Moore & Sterling, 2006). This evolution can reach a point beyond which no neighboring equilibrium exists—a loss of equilibrium or a catastrophe—requiring rapid dynamical evolution to a new equilibrium (Forbes & Isenberg, 1991; Longcope & Forbes, 2014; Priest & Forbes, 1990; Yeates & Mackay, 2009). In one such scenario, called tether-cutting, reconnection occurs at a current sheet beneath the flux rope, causing it to rise slowly until equilibrium is lost and the flux rope rises dynamically: eruption (Moore, Sterling, Hudson, & Lemen, 2001). In an alternative scenario, called break-out, the initial equilibrium includes a horizontal current sheet above the flux rope, as well as the vertical sheet beneath it (Antiochos, Devore, & Klimchuk, 1999). Slow reconnection at the upper sheet causes the flux rope to rise slowly, until equilibrium is lost and it erupts dynamically. Under this scenario, the lower current sheet stretches and thins, first slowly under quasi-static evolution, and then more rapidly as a result of eruption. The latter phase is accompanied by rapid reconnection at the lower sheet, termed flare reconnection, which produces the flare itself according to the standard model.
Reconnection in a Flare
The flare itself is largely a consequence of the magnetic reconnection occurring at the current sheet beneath the erupting flux rope. At least in the standard model, the current sheet has global scale so reconnection there is typically modeled using MHD equations (the vertical structure evident in Figure 3a is more than 100 Mm long). In general such studies conform to results of more generic studies of reconnection in the MHD regime (Forbes & Priest, 1983). To obtain steady Petschek reconnection of the form depicted in the standard model, it is necessary that the reconnection electric field be somehow localized within the large-scale current sheet (Biskamp & Schwarz, 2001; Kulsrud, 2001). This localization will not occur if the magnetic induction equation includes only a uniform resistivity. For this reason, it is common for models to use a current-dependent anomalous resistivity (Magara, Mineshige, Yokoyama, & Shibata, 1996; Ugai & Tsuda, 1977). Doing so yields numerical results closely resembling the standard model cartoon, that is, Figure 3b, including a fast magnetosonic shock at the termination (Forbes & Malherbe, 1986), although exhibiting its own dynamics (Takasao, Matsumoto, Nakamura, & Shibata, 2015).
Other models have used uniform resistivity, or no resistivity at all, and cannot therefore have fast, steady reconnection. Instead they exhibit reconnection of an unsteady variety, including multiple, evolving magnetic islands (Karpen, Antiochos, & DeVore, 2012). Such reconnection is also found in more generic studies (Bhattacharjee, HuangYang, & Rogers, 2009; Loureiro, Schekochihin, & Cowley, 2007). These evolving islands have been invoked to explain certain features observed in the context of solar flares, such as supra-arcade downflows (McKenzie & Hudson, 1999; Savage, McKenzie, & Reeves, 2012).
Flare reconnection differs from more generic varieties owing to the significant role played by the cool chromosphere to which the reconnecting field lines are anchored. This layer is responsible for many of the observational signatures of a solar flare. Several numerical and analytic models have examined the role of field-aligned thermal conductivity by which energy may be transported to the chromosphere (Chen, Fang, Tang, & Ding, 1999; Forbes, Malherbe, & Priest, 1989; Yokoyama & Shibata, 1997). These exhibit a layer of hot plasma surrounding the outflow jet and chromospheric evaporation, but the most detailed studies of the latter process remain those using a class of one-dimensional flare loop models.
Flare Loop Models
Flare loop models consider, for the most part, the plasma dynamics in a static, closed magnetic loop. Magnetic evolution is neglected, or assumed to be complete, leaving a stationary curved tube. Plasma flows only along this static loop, with velocity parallel to the axis. The loop is assumed thin enough to reduce the problem to a single spatial dimension. Mass density, plasma velocity (parallel), and plasma pressure are all functions of axial position, and evolve in time as required by the conservation of mass, momentum, and energy. Assuming a static loop obviates the need for an equation governing magnetic field evolution, leaving a system of gas dynamic equations, rather than MHD. Restriction to a single spatial dimension permits numerical solutions to resolve scales as small as meters—scales which can develop in a flare’s low atmosphere (Fisher, Canfield, & McClymont, 1985a; MacNeice, Burgess, McWhirter, & Spicer, 1984).
The energy equation typically includes radiative transport, including optically thin losses from the corona and thermal conduction along the tube’s axis. It also includes an energy source term representing the magnetic energy released to produce the flare. In some versions the source term is an ad hoc function of space and time representing a generic dissipation, and typically concentrated around the loop top (Cheng, Oran, Doschek, Boris, & Mariska, 1983; MacNeice, 1986). In others it is taken to be the energy deposition from non-thermal electrons, which had originated at the loop top with a specified flux and energy spectrum (Emslie & Nagai, 1985; Fisher, Canfield, & McClymont, 1985b). Both versions have been extensively studied, and produce broadly similar evolution, largely conforming to observations.
Integrated (0D) Models
The one-dimensional gas dynamic equations can be simplified further by integrating them over the loop’s coronal section. This yields two ordinary differential equations governing the time evolution of total coronal mass and energy, or equivalently, average coronal density and temperature (Antiochos & Sturrock, 1978). A few assumptions are made about the spatial profiles of the primitive quantities, but the result is a robust zero-dimensional system based on global conservation laws. The equations may be solved numerically, or analytically after a few more assumptions; examples of each are shown in Figure 5. The former approach generally shows evolution in three phases, as illustrated in Figure 5c and 5d. Analytical approaches assume this three-phase evolution (Cargill, Mariska, & Antiochos, 1995).
During the heating phase (see Figure 5a) energy is added and the coronal temperature rises more rapidly than density can respond. This phase is particularly distinct if the duration of energy input is short, as in the example, otherwise it overlaps the next phase. In the next phase, coronal heat is transported to the chromosphere where it drives evaporation, carrying much of the energy back into the corona (i.e., by enthalpy flux). Evaporation thus increases the corona’s mass, but keeps its energy, and thus its pressure, largely constant: the evaporative phase proceeds along a line of constant , as shown by violet dashed lines in Figures 5a and 5c. The time scale for optically thin radiative losses, scaling inversely with the square of the density, is very long at the high temperatures and low densities found at the end of the heating phase. Evaporation thus proceeds on the shorter conductive time scale, until the coronal density has increased enough to make the radiative time-scale comparable. The corona then begins to lose energy through radiation—its final phase. It is no coincidence that the equality of these two time scales is also the condition for mechanical equilibrium, so the cooling occurs through a series of loop equilibria.
This final, radiative, phase is the longest and thus dictates the overall lifetime of the flare loop: s in Figure 5. Unless the heating persists throughout this phase, models generally find loop cooling times shorter than the gradual phases of flares with properties similar to simulated loops.1
Coronal emission, including the GOES 1–8Å band, peaks when the density peaks at the end of the evaporative phase. The zero-dimensional model can relate that emission peak to the total energy input and other parameters of the loop. Warren and Antiochos (2004) followed this approach to obtain an expression for peak emission , where is the total energy added to the loop in units of , and and are the loop’s full length and cross sectional area in units of cm and respectively. This would relate the observed SXR peak to the energy of a flare, provided the flare behaved as a single loop. However, a large flare is not well described as a single loop, so the above relation is only approximate.
Gas-Dynamic (1D) Models
Solutions to the full gas dynamic equations generally corroborate the conclusion of zero-dimensional models that radiative cooling occurs through a sequence of equilibria. Conversely, the processes of heating and evaporation are found to be very dynamic, with flow speeds at or above the sound speed, as shown in the s curves of Figure 6. These phases are therefore better studied using the full one-dimensional gas dynamic models (Mariska, Doschek, Boris, Oran, & Young, 1982; Nagai, 1980; Pallavicini et al., 1983).
Energy added to the chromosphere, either deposited by non-thermal electrons or conducted from the corona, creates a pressure peak that drives material upward as evaporation (see velocity plots in 6). This occurs in an ablative rarefaction wave with a shock at its front. The upflow speeds in the models are several hundred km/s, which is typically supersonic. Fisher, Canfield, and McClymont (1984) placed an upper bound on the evaporation speed of two to three times the isothermal sound speed. Longcope (2014) found that evaporation driven by thermal conduction flux , reached a velocity .
Energy deposited by non-thermal electrons results in chromospheric evaporation classified as either gentle or explosive. The electrons deposit energy in cooler denser chromospheric layers, where radiative loss is particularly effective and becomes more so as heating drives up the temperature there. It is therefore possible for the deposited energy to be immediately radiated with only minor effects; this is called gentle evaporation. Optically thin losses increase with temperature until peaking at a maximum volumetric rate at around 150,000 K. If the rate of deposition exceeds this maximum the radiation will be unable to compensate, allowing temperature and pressure to rise explosively: this is explosive evaporation. The low-lying pressure peak drives plasma upward (evaporation) as well as downward. This downward motion, essentially a back-reaction to the evaporation, is called chromospheric condensation, and has been observed (Canfield, Metcalf, Strong, & Zarro, 1987; Graham & Cauzzi, 2015).
Radiation cannot be assumed optically thin in deeper layers of the low chromosphere, or for wavelengths around very strong spectral lines. Accurate treatment of these cases requires an explicit treatment of the radiative transfer. This is currently the state of the art for flare modeling (Allred, Hawley, Abbett, & Carlsson, 2005; McClymont & Canfield, 1983), and is essential in order to accurately model the form of strong, optically thick spectral lines in a flare.
Investigators have encountered some difficulties modeling the full light-curves from a given flare as a single loop. A particularly vexing difficulty is that gradual phases tend to last longer than the radiative cooling time of a characteristic loop (Qiu & Longcope, 2016; Warren, 2006). This has led to the conclusion that a single flare consists of many distinct loops evolving independently after being energized at different times (Hori Yokoyama, Kosugi, & Shibata, 1997; Qiu et al., 2012; Reeves & Warren, 2002; Warren, 2006). This means the phases of flare evolution, impulsive and gradual, do not map simply onto the phases of flare loop evolution. Energy release is not restricted to the impulsive phase, but may continue through much or all of the gradual phase. This is consistent with many observations showing ribbons continuing their spreading motion during this phase (Longcope, Qiu, & Brewer, 2016). Nor is the gradual phase entirely equivalent to the cooling phase of a single loop. Figure 1b shows ample emission by 10 MK Fe xx throughout most of the gradual phase, so the plasma cannot be cooling everywhere.
This understanding has led to models capable of reproducing with reasonable fidelity most of a flare’s myriad light curves (Liu, Qiu, Longcope, & Caspi, 2013; Qiu, Sturrock, Longcope, Klimchuk, & Liu, 2013). The flare is synthesized from a set of loops, initiated in sequence, and the flare’s light curve is a super-position of the light curves of the loop sequence. The flare’s time scale is therefore set by the loop initialization sequence and not by the radiative cooling of a single loop. Moreover, the measured flux transfer rate, , reflects the rate of loop creation, rather than the X-point reconnection electric field that it would in steady models (Longcope, Des Jardins, Carranza-Fulmer, & Qiu, 2010; Longcope, Qiu, & Brewer, 2016).
Non-Thermal Particle Models
A population of particles can be described by its distribution function, , depending on particle energy , and the pitch-angle cosine . Coulomb collisions among charged particles will drive their distribution toward a Maxwellian, . This limiting form is achieved after a few dozen collisions. At particle densities typical of a flare () 1 keV electrons will collide with frequency Hz, and thereby remain approximately Maxwellian during a flare. The Coulomb collision frequency scales inversely with particle energy, , so electrons over keV collide rarely and can travel more than 100 Mm before colliding once. It is these electrons which travel unimpeded along the flare loop to deposit energy at the feet and create the flare ribbons and the footpoint HXR sources.
Lacking frequent collisions, the distribution function does not need to form a Maxwellian at high energies. Instead it is often observed to be better described by a power law, .2 The entire distribution function is most often written as a sum of a Maxwellian, called the thermal component, and a non-thermal component whose distribution function is a power law restricted to . The low-energy cut-off is formally required for normalizability, but more physically needed so that non-thermal collision rates are low enough to justify a departure from Maxwellian. The cut-off is generally expected at energies where the thermal component dominates the sum, and thus proves extremely difficult to constrain well by observation.
The lowest three moments of the distribution function correspond to density, fluid velocity, and pressure. A Maxwellian distribution is completely determined by these moments alone, but any other distribution requires more moments, or the entire function, to be specified. Fluid equations, such as assumed in previous sections (see also “Magnetohydrodynamics—Overview,” Priest, 2019), describe the evolution of the lowest three moments and can therefore be considered a valid, but partial, description of plasma evolution. Their description is reasonably complete provided energies are low enough, and collisions frequent enough, to keep the distribution close to Maxwellian, and thus fully described by its lowest moments.
When collisions are not frequent enough, the evolution of the distribution function must be followed using the Fokker–Planck equation. This includes effects of single-particle motion such as propagation along the field line, and mirroring from points of strong field (Parker, 1958; Rosenbluth, MacDonald, & Judd, 1957). It also includes a velocity–space diffusion arising from the average effect of random, high-frequency electric and magnetic fields. These high-frequency fields can arise from Coulomb collisions or from randomly phased plasma waves, of various kinds. The Coulomb contribution will, as mentioned above, cause the distribution function to relax toward a Maxwellian. The other contributions, however, can drive evolution in other directions, and can thus contribute to the creation of a non-thermal component.
Models of Particle Acceleration
The process of generating the non-thermal component from an erstwhile Maxwellian distribution is known as particle acceleration. A wide variety of models have been proposed for the acceleration process in flares. All are set within the standard flare model framework and produce distributions resembling power laws. It has thus proven difficult to reach a consensus on which mechanism is at work in a solar flare. This question remains open.
A number of models, collectively known as second-order Fermi or stochastic acceleration (SA) models, focus on the velocity-space diffusion from a spectrum of randomly-phased plasma waves. Waves of various kinds can be generated by the MHD turbulence expected within the reconnection outflow jet, featured in Figure 3b. Velocity diffusion is dominated by resonant interactions between the waves and the particles. This poses a challenge for SA models in general since plasma waves often have phase speeds much higher than thermal particles. Many investigators have, however, been able to show that resonances can occur with different wave modes under reasonable assumptions. Some even follow the evolution of the wave spectrum (Miller, Larosa, & Moore, 1996; Petrosian, Yan, & Lazarian, 2006). Stochastic acceleration therefore remains a viable explanation for high-energy flare particles.
A related model considers the effects of an MHD shock along with turbulence capable of repeatedly scattering the particles back to the shock (i.e., effective pitch-angle scattering). These elements combine in a process known as first-order Fermi or diffusive shock acceleration (DSA), which has been extensively studied and observed in other astrophysical and space plasmas (see Blandford & Eichler, 1987, for a review). It results in a power-law distribution whose index, , is related to the plasma compression ratio across the shock. The fast magnetosonic shock predicted at the termination point (see Figure 3) is an ideal location for DSA (Tsuneta & Naito, 1998; Mann, Aurass, & Warmuth, 2006), and some observations suggest acceleration is indeed occurring there (Chen et al., 2015a; Sui & Holman, 2003).
Charged particles can be temporarily confined either on closed field lines or between magnetic mirror points. As the magnetic field changes, it can add energy to the trapped particles through the betatron term, curvature-drift, or head-on reflection from a moving mirror point. A certain class of models invokes these effects to explain particle acceleration. The magnetic field strength will have a local minimum at the end of the outflow region. This can serve as a magnetic trap, and if it shrinks in size, the particles trapped there can gain energy. This is the basis of the collapsing trap model (Somov & Kosugi, 1997; Karlický & Kosugi, 2004). Alternatively, MHD turbulence in the outflow jet could feature closed magnetic islands, often called plasmoids in reconnection models (Loureiro, Schekochihin, & Cowley, 2007; Shibayama, Kusano, Miyoshi, Nakabou, & Vekstein, 2015). These islands will tend to evolve from elongated to circular, and in so doing accelerate the particles trapped within them (Drake, Swisdak, Schoeffler, Rogers, & Kobayashi, 2006).
Finally it is possible for the large-scale electric field, the defining feature of magnetic reconnection, to accelerate charged particles directly, in so-called direct acceleration. It has already been noted that reconnection is observed at rates Mx/s. If this occurred as a steady, large-scale electric field along an X-line, there would be a Volt drop along it—far more than observed in any electron. Simple as this seems, a detailed model faces several challenges. A plasma tends to screen out any electric field component parallel to the magnetic field, and undergoes a dramatic response if subjected to a field in excess of the so-called Dreicer field— V/m for a typical flare plasma. Moreover, the simplest scenario would predict all particles of a given charge to be accelerated in the same direction, in apparent contradiction to observations showing electron precipitation at both feet of a loop (see Figure 2b). Several investigators have produced models which overcome these challenges, demonstrating the viability of direct acceleration (Emslie & Hénoux, 1995; Holman, 1985; Litvinenko, 1996; Martens, 1988)
Models of Non-Thermal Particle Propagation
In most of these models, charged particles are energized within a certain region of the solar flare, such as the reconnection site, the outflow jet, or the termination shock. From there the particles propagate along magnetic field lines, until they have dissipated their energy and rejoined the thermal population. An electron with energy (in keV) and pitch-angle cosine will traverse a total column , before stopping. Electrons leaving the acceleration region roughly parallel () with keV will not stop until they have reached the chromosphere where (see the upper left axis of Figure 6). They will lose the vast majority of this energy at the very end of their journey, leading to chromospheric energy deposition.
Virtually all the energy lost by propagating particles goes into heating the background plasma: a colder thermal plasma. For electrons, a very small fraction (typically ) is converted to photons via bremsstrahlung, which thus provides the most direct diagnostic of the non-thermal electron population. (Ions lose a far smaller fraction to bremstrahlung, making their detection far more challenging.) An electron with energy can emit photons of energy . A single electron will emit a spectrum of bremsstrahlung photons before ultimately joining the thermal population; the complete process is known as thick-target emission. A power-law distribution of electrons, , will thereby produce a power-law distribution of photons, , with in this thick-target process (Tandberg-Hanssen & Emslie, 1988). Hard X-ray spectra from flare footpoints generally exhibit power laws with , corresponding to electron distributions with .
The coronal column will have little effect on the energy of the electrons propagating through it. Bremstrahlung emission under this condition, called thin-target emission, will reflect the distribution of energies at which electron are produced (accelerated). The resulting photon spectrum will therefore have a power-law index, , considerably softer than for thick-target emission (Tandberg-Hanssen & Emslie, 1988).
- Allred, J. C., Hawley, S. L., Abbett, W. P., & Carlsson, M. (2005). Radiative Hydrodynamic Models of the Optical and Ultraviolet Emission from Solar Flares. The Astrophysical Journal, 630, 573.
- Amari, T., Luciani, J. F., Mikic, Z., & Linker, J. (2000). A Twisted Flux Rope Model for Coronal Mass Ejections and Two-Ribbon Flares. The Astrophysical Journal, 529, L49.
- Antiochos, S. K., Devore, C. R., & Klimchuk, J. A. (1999). A Model for Solar Coronal Mass Ejections. The Astrophysical Journal, 510, 485.
- Antiochos, S. K., & Sturrock, P. A. (1978). Evaporative cooling of flare plasma. The Astrophysical Journal, 220, 1137.
- Antonucci, E., Alexander, D., Culhane, J. L., de Jager, C., MacNeice, P., Somov, B. V., & Zarro, D. M. (1999). Flare dynamics. In K. T. Strong, J. L. R. Saba, B. M. Haisch, & J. T. Schmelz (Eds.), The many faces of the sun: A summary of the results from NASA’s Solar Maximum Mission (p. 331). New York, NY: Springer.
- Antonucci, E., & Dennis, B. R. (1983). Observation of chromospheric evaporation during the Solar Maximum Mission. Solar Physics, 86, 67.
- Aschwanden, M. J., & Alexander, D. (2001). Flare Plasma Cooling from 30 MK down to 1 MK modeled from Yohkoh, GOES, and TRACE observations during the Bastille Day Event (14 July 2000). Solar Physics, 204, 91.
- Aschwanden, M. J., Scholkmann, F., B´ethune, W., Schmutz, W., Abramenko, V., Cheung, M. C. M., M¨uller, D., Benz, A., Chernov, G., Kritsuk, A. G., Scargle, J. D., Melatos, A., Wagoner, R. V., Trimble, V., & Green, W. H. (2018). Order out of Randomness: Self-Organization Processes in Astrophysics. Space Science Reviews, 214, 55.
- Aulanier, G., Démoulin, P., Schrijver, C. J., Janvier, M., Pariat, E., & Schmieder, B. (2013). The standard flare model in three dimensions. II. Upper limit on solar flare energy. Astronomy & Astrophysics, 549, A66.
- Aurass, H., Vršnak, B., & Mann, G. (2002). Shock-excited radio burst from reconnection outflow jet? Astronomy & Astrophysics, 384, 273.
- Bak, P., Tang, C., & Wiesenfeld, K. (1988). Self-organized criticality. Physical Review A, 38, 364.
- Bhattacharjee, A., Huang, Y.-M., Yang, H., & Rogers, B. (2009). Fast reconnection in high-Lundquist-number plasmas due to the plasmoid Instability. Physics of Plasmas, 16, 112102.
- Biskamp, D., & Schwarz, E. (2001). Localization, the clue to fast magnetic reconnection. Physics of Plasmas, 8, 4729.
- Biskamp, D., & Welter, H. (1989). Magnetic arcade evolution and instability. Solar Physics, 120, 49.
- Blandford, R., & Eichler, D. (1987). Particle acceleration at astrophysical shocks: A theory of cosmic ray origin. Physics Reports, 154, 1.
- Bogachev, S. A., Somov, B. V., Kosugi, T., & Sakao, T. (2005). The Motions of the Hard X-Ray Sources in Solar Flares: Images and Statistics. The Astrophysical Journal, 630, 561.
- Canfield, R. C., Brown, J. C., Brueckner, G. E., Cook, J. W., Craig, I. J. D., Doschek, G. A., Emslie, A. G., Henoux, J.-C., Lites, B. W., Machado, M. E., & Underwood, J. H. (1980). The Chromosphere and Transition Region. In P. A. Sturrock (Ed.), Solar flares. A monograph from Skylab Solar Workshop II (p. 231). Boulder: Colorado Associated University Press.
- Canfield, R. C., Metcalf, T. R., Strong, K. T., & Zarro, D. M. (1987). A novel observational test of momentum balance in a solar flare. Nature, 326, 165.
- Cargill, P. J., Mariska, J. T., & Antiochos, S. K. (1995). Cooling of solar flares plasmas. 1: Theoretical considerations. The Astrophysical Journal, 439, 1034.
- Carmichael, H. (1964). A Process for Flares. In W. N. Hess (Ed.), AAS-NASA Symposium on the Physics of Solar Flares (p. 451). Washington, DC: NASA.
- Chamberlin, P. C., Woods, T. N., & Eparvier, F. G. (2008). Flare Irradiance Spectral Model (FISM): Flare component algorithms and results. Space Weather, 6, S05001.
- Chen, B., Bastian, T. S., Shen, C., Gary, D. E., Krucker, S., & Glesener, L. (2015a). Particle acceleration by a solar flare termination shock. Science, 350, 1238.
- Chen, H., Zhang, J., Ma, S., Yang, S., Li, L., Huang, X., & Xiao, J. (2015b). Confined Flares in Solar Active Region 12192 from 2014 October 18 to 29. The Astrophysical Journal, 808, L24.
- Chen, P. F., Fang, C., Tang, Y. H., & Ding, M. D. (1999). Simulation of Magnetic Reconnection with Heat Conduction. The Astrophysical Journal, 513, 516.
- Cheng, C.-C., Oran, E. S., Doschek, G. A., Boris, J. P., & Mariska, J. T. (1983). Numerical simulations of loops heated to solar flare temperatures I. The Astrophysical Journal, 265, 1090.
- Crosby, N. B., Aschwanden, M. J., & Dennis, B. R. (1993). Frenquency distributions and correlations of solar x-ray flar parameters. Solar Physics, 143, 275.
- Démoulin, P., & Aulanier, G. (2010). Criteria for Flux Rope Eruption: Non-equilibrium Versus Torus Instability. The Astrophysical Journal, 718, 1388.
- Dennis, B. R., & Zarro, D. M. (1993). The Neupert effect - What can it tell us about the impulsive and gradual phases of solar flares? Solar Physics, 146, 177.
- Drake, J. F., Swisdak, M., Schoeffler, K. M., Rogers, B. N., & Kobayashi, S. (2006). Formation of secondary islands during magnetic reconnection. Geophysical Research Letters, 33, 13105.
- Emslie, A. G., Dennis, B. R., Holman, G. D., & Hudson, H. S. (2005). Refinements to flare energy estimates: A followup to “Energy partition in two solar flare/CME events” by A. G. Emslie et al. Journal of Geophysical Research, 110, A11103.
- Emslie, A. G., Dennis, B. R., Shih, A. Y., Chamberlin, P. C., Mewaldt, R. A., Moore, C. S., Share, G. H., Vourlidas, A., & Welsch, B. T. (2012). Global Energetics of Thirty-eight Large Solar Eruptive Events. The Astrophysical Journal, 759, 71.
- Emslie, A. G., & Hénoux, J.-C. (1995). The electrical current structure associated with solar flare electrons accelerated by large-scale electric fields. The Astrophysical Journal, 446, 371.
- Emslie, A. G., Kucharek, H., Dennis, B. R., Gopalswamy, N., Holman, G. D., Share, G. H., Vourlidas, A., Forbes, T. G., Gallagher, P. T., Mason, G. M., Metcalf, T. R., Mewaldt, R. A., Murphy, R. J., Schwartz, R. A., & Zurbuchen, T. H. (2004). Energy partition in two solar flare/CME events. Journal of Geophysical Research, 109, 10104.
- Emslie, A. G., & Nagai, F. (1985). Gas dynamics in the impulsive phase of solar flares. II - The structure of the transition region - A diagnostic of energy transport processes. The Astrophysical Journal, 288, 779.
- Fisher, G. H., Canfield, R. C., & McClymont, A. N. (1984). Chromospheric evaporation velocities in solar flares. The Astrophysical Journal, 281, L79.
- Fisher, G. H., Canfield, R. C., & McClymont, A. N. (1985a). Flare loop radiative hydrodynamics. V - Response to thick-target heating. The Astrophysical Journal, 289, 414.
- Fisher, G. H., Canfield, R. C., & McClymont, A. N. (1985b). Flare Loop Radiative Hydrodynamics. VI - Chromospheric Evaporation due to Heating by Nonthermal Electrons. The Astrophysical Journal, 289, 425.
- Fletcher, L., & Hudson, H. (2001). The Magnetic Structure and Generation of EUV Flare Ribbons. Solar Physics, 204, 69.
- Fletcher, L., Pollock, J. A., & Potts, H. E. (2004). Tracking of TRACE Ultraviolet Flare Footpoints. Solar Physics, 222, 279.
- Forbes, T. G. (1986). Fast-shock formation in line-tied magnetic reconnection models of solar ﬂares. The Astrophysical Journal, 305, 553.
- Forbes, T. G., & Acton, L. W. (1996). Reconnection and Field Line Shrink age in Solar Flares. The Astrophysical Journal, 459, 330.
- Forbes, T. G., & Isenberg, P. A. (1991). A catasrophe mechanism for coronal mass ejections. The Astrophysical Journal, 373, 294.
- Forbes, T. G., & Malherbe, J. M. (1986). A shock condensation mechanism for loop prominences. The Astrophysical Journal, 302, L67.
- Forbes, T. G., Malherbe, J. M., & Priest, E. R. (1989). The formation ﬂare loops by magnetic reconnection and chromospheric ablation. Solar Physics, 120, 285.
- Forbes, T. G., & Priest, E. R. (1983). A numerical experiment relevant to line-tied reconnection in two-ribbon ﬂares. Solar Physics, 84, 169.
- Forbes, T. G., & Priest, E. R. (1984). Reconnection in Solar Flares. In D. Butler & K. Papadopoulos (Eds.), Solar terrestrial physics: Present and future (p. 35). Washington, DC: NASA.
- Forbes, T. G., Priest, E. R., Seaton, D. B., & Litvinenko, Y. E. (2013). Indeterminacy and instability in Petschek reconnection. Physics of Plasmas, 20, 052902.
- Fuller-Rowell, T., & Solomon, S. C. (2010). Flares, coronal mass ejections, and atmopsheric responses. In C. J. Schrijver & G. Siscoe (Eds.), Heliophysics II. Space storms and radiation: Causes and effects (p. 321). Cambridge, UK: Cambridge University Press.
- Goff, C. P., Matthews, S. A., van Driel-Gesztelyi, L., & Harra, L. K. (2004). Relating magnetic ﬁeld strengths to hard X-ray emission in solar ﬂares. Astronomy & Astrophysics, 423, 363.
- Gosling, J. T. (1993). The solar ﬂare myth. Journal of Geophysical Research, 98, 18937.
- Graham, D. R., & Cauzzi, G. (2015). Temporal Evolution of Multiple Evaporating Ribbon Sources in a Solar Flare. The Astrophysical Journal, 807, L22.
- Hirayama, T. (1974). Theoretical Model of Flares and Prominences. I: Evaporating Flare Model. Solar Physics, 34, 323.
- Holman, G. D. (1985). Acceleration of runaway electrons and Joule heating in solar ﬂares. The Astrophysical Journal, 293, 584.
- Hood, A. W., & Priest, E. R. (1980). Magnetic instability of coronal arcades as the origin of two-ribbon ﬂares. Solar Physics, 66, 113.
- Hood, A. W., & Priest, E. R. (1981). Critical conditions for magnetic instabilities in force-free coronal loops. Geophysical and Astrophysical Fluid Dynamics, 17, 297.
- Hori, K., Yokoyama, T., Kosugi, T., & Shibata, K. (1997). Pseudo–Two- dimensional Hydrodynamic Modeling of Solar Flare Loops. The Astrophysical Journal, 489, 426.
- Kahler, S. W. (1982). The role of the big ﬂare syndrome in correlations of solar energetic proton ﬂuxes and associated microwave burst parameters. Journal of Geophysical Research, 87, 3439.
- Karlický, M., & Kosugi, T. (2004). Acceleration and heating processes in ay collapsing magnetic trap. Astronomy & Astrophysics, 419, 1159.
- Karpen, J. T., Antiochos, S. K., & DeVore, C. R. (2012). The Mechanisms for the Onset and Explosive Eruption of Coronal Mass Ejections and Eruptive Flares. The Astrophysical Journal, 760, 15.
- Kliem, B., & Török, T. (2006). Torus Instability. Physical Review Letters, 96, 255002.
- Klimchuk, J. A., Patsourakos, S., & Cargill, P. J. (2008). Highly Eﬃcient Modeling of Dynamic Coronal Loops. The Astrophysical Journal, 682, 1351.
- Kopp, G., Lawrence, G., & Rottman, G. (2005). The Total Irradiance Monitor (TIM): Science Results. Solar Physics, 230, 129.
- Kopp, R. A., & Pneuman, G. W. (1976). Magnetic reconnection in the corona and the loop prominence phenomenon. Solar Physics, 50, 85.
- Kretzschmar, M., de Wit, T. D., Schmutz, W., Mekaoui, S., Hochedez, J.-F., & Dewitte, S. (2010). The eﬀect of ﬂares on total solar irradiance. Nature Physics, 6, 690.
- Krucker, S., Battaglia, M., Cargill, P. J., Fletcher, L., Hudson, H. S., MacKinnon, A. L., Masuda, S., Sui, L., Tomczak, M., Veronig, A. L., Vlahos, L., & White, S. M. (2008). Hard X-ray emission from the solar corona. Astronomy and Astrophysics Review, 16, 155.
- Kulsrud, R. M. (2001). Magnetic reconnection: Sweet-Parker vs. Petscheck. Earth, Planets and Space, 53, 417.
- Kusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T. (2004). The Trigger Mechanism of Solar Flares in a Coronal Arcade with Reversed Magnetic Shear. The Astrophysical Journal, 610, 537.
- Litvinenko, Y. E. (1996). Particle Acceleration in Reconnecting Current Sheets with a Nonzero Magnetic Field. The Astrophysical Journal, 462, 997.
- Liu, W.-J., Qiu, J., Longcope, D. W., & Caspi, A. (2013). Determining Heating Rates in Reconnection Formed Flare Loops of the M8.0 Flare on 2005 May 13. The Astrophysical Journal, 770, 111.
- Longcope, D. W. (2014). A Simple Model of Chromospheric Evaporation and Condensation Driven Conductively in a Solar Flare. The Astrophysical Journal, 795, 10.
- Longcope, D. W., Des Jardins, A. C., Carranza-Fulmer, T., & Qiu, J. (2010). A Quantitative Model of Energy Release and Heating by Time-dependent, Localized Reconnection in a Flare with a Thermal Looptop X-ray Source. Solar Physics, 267, 107.
- Longcope, D. W., & Forbes, T. G. (2014). Breakout and tether-cutting eruption models are both catastrophic (sometimes). Solar Physics, 6, 2091.
- Longcope, D. W., Qiu, J., & Brewer, J. (2016). A reconnection-driven model of the hard X-ray loop-top source from ﬂare 2004-Feb-26. The Astrophysical Journal, 833, 211.
- Loureiro, N. F., Schekochihin, A. A., & Cowley, S. C. (2007). Instability of current sheets and formation of plasmoid chains. Physics of Plasmas, 14, 100703.
- Lu, E. T., & Hamilton, R. J. (1991). Avalanches and the distribution of solar ﬂares. The Astrophysical Journal, 380, L89.
- MacNeice, P. (1986). A numerical hydrodynamic model of a heated coronal loop. Solar Physics, 103, 47.
- MacNeice, P., Burgess, A., McWhirter, R. W. P., & Spicer, D. S. (1984). A numerical model of a solar ﬂare based on electron beam heating of the chromospheres. Solar Physics, 90, 357.
- Magara, T., Mineshige, S., Yokoyama, T., & Shibata, K. (1996). Numerical Simulation of Magnetic Reconnection in Eruptive Flares. The Astrophysical Journal, 466, 1054.
- Mann, G., Aurass, H., & Warmuth, A. (2006). Electron acceleration by the reconnection outﬂow shock during solar ﬂares. Astronomy & Astrophysics, 454, 969.
- Mariska, J. T., Doschek, G. A., Boris, J. P., Oran, E. S., & Young, T. R., Jr. (1982). Solar transition region response to variations in the heating rate. The Astrophysical Journal, 255, 783.
- Martens, P. C. H. (1988). The generation of proton beams in two-ribbon ﬂares. The Astrophysical Journal, 330, L131.
- Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., & Ogawara, Y. (1994). A loop-top hard X-ray source in a compact solar ﬂare as evidence for magnetic reconnection. Nature, 371, 495.
- McClymont, A. N., & Canfield, R. C. (1983). Flare loop radiative hydrodynamics. I - Basic methods. The Astrophysical Journal, 265, 483.
- McKenzie, D. E. (2002). Signatures of Reconnection in Eruptive Flares. In P. C. H. Martens & D. Cauffman (Eds.), Multi-wavelength observations of coronal structure and dynamics—Yohkoh 10th anniversary meeting (COSPAR Colloquia Series, p. 155). Elsevier.
- McKenzie, D. E., & Hudson, H. S. (1999). X-Ray Observations of Motions and Structure above a Solar Flare Arcade. The Astrophysical Journal, 519, L93.
- Mikic, Z., Barnes, D. C., & Schnack, D. D. (1988). Dynamical evolution of a solar coronal magnetic ﬁeld arcade. The Astrophysical Journal, 328, 830.
- Mikic, Z., & Linker, J. A. (1994). Disruption of coronal magnetic ﬁeld arcades. The Astrophysical Journal, 430, 898.
- Miller, J. A., Larosa, T. N., & Moore, R. L. (1996). Stochastic Electron Acceleration by Cascading Fast Mode Waves in Impulsive Solar Flares. The Astrophysical Journal, 461, 445.
- Milligan, R. O., & Dennis, B. R. (2009). Velocity Characteristics of Evaporated Plasma Using Hinode/EUV Imaging Spectrometer. The Astrophysical Journal, 699, 968.
- Moore, R. L., & Sterling, A. C. (2006). Initiation of Coronal Mass Ejections. In E. Robbrecht & D. Berghmans (Eds.), Solar Eruptions and Energetic Particles (Vol. 165, p. 43). Washington DC: American Geophysical Union Geophysical Monograph Series.
- Moore, R. L., Sterling, A. C., Hudson, H. S., & Lemen, J. R. (2001). Onset of the Magnetic Explosion in Solar Flares and Coronal Mass Ejections. The Astrophysical Journal, 552, 833.
- Munro, R. H., Gosling, J. T., Hildner, E., MacQueen, R. M., Poland, A. I., & Ross, C. L. (1979). The association of coronal mass ejection transients with other forms of solar activity. Solar Physics, 61, 201.
- Murphy, R. J. (2007). Solar Gamma-Ray Spectroscopy. Space Science Reviews, 130, 127.
- Nagai, F. (1980). A model of hot loops associated with solar ﬂares. I - Gas- dynamics in the loops. Solar Physics, 68, 351.
- Neupert, W. M. (1968). Comparison of Solar X-Ray Line Emission with Microwave Emission during Flares. The Astrophysical Journal, 153, L59.
- Pallavicini, R., Peres, G., Serio, S., Vaiana, G., Acton, L., Leibacher, J., & Rosner, R. (1983). Closed coronal structures. V - Gasdynamic models of ﬂaring loops and comparison with SMM observations. The Astrophysical Journal, 270, 270.
- Parker, E. N. (1958). Suprathermal Particle Generation in the Solar Corona. The Astrophysical Journal, 128, 677.
- Petrosian, V., Yan, H., & Lazarian, A. (2006). Damping of Magnetohydrodynamic Turbulence in Solar Flares. The Astrophysical Journal, 644, 603.
- Petschek, H. E. (1964). Magnetic ﬁeld annihilation. In W. N. Hess (Ed.), AAS-NASA Symposium on the physics of solar flares (p. 425). Washington, DC: NASA
- Poletto, G., & Kopp, R. A. (1986). Macroscopic electric ﬁelds during two-ribbon ﬂares. In D. F. Neidig (Ed.), The lower atmospheres of solar flares (p. 453). National Solar Observatory.
- Priest, E. R. (2019). Magnetohydrodynamics – Overview. In B. Foster (Ed.), Oxford Research Encyclopedia of Physics. Oxford, UK: Oxford University Press.
- Priest, E. R., & Forbes, T. G. (1990). Magnetic ﬁeld evolution during prominence eruptions and two-ribbon ﬂares. Solar Physics, 126, 319.
- Qiu, J. (2009). Observational Analysis of Magnetic Reconnection Sequence. The Astrophysical Journal, 692, 1110.
- Qiu, J., Lee, J., Gary, D. E., & Wang, H. (2002). Motion of Flare Footpoint Emission and Inferred Electric Field in Reconnecting Current Sheets. The Astrophysical Journal, 565, 1335.
- Qiu, J., Liu, W.-J., & Longcope, D. W. (2012). Heating of Flare Loops With Observationally Constrained Heating Functions. The Astrophysical Journal, 752, 124.
- Qiu, J., & Longcope, D. W. (2016). Long Duration Flare Emission: Impulsive Heating or Gradual Heating? The Astrophysical Journal, 820, 14.
- Qiu, J., Sturrock, Z., Longcope, D. W., Klimchuk, J. A., & Liu, W.-J. (2013). Ultraviolet and Extreme-ultraviolet Emissions at the Flare Footpoints Observed by Atmosphere Imaging Assembly. The Astrophysical Journal, 774, 14.
- Reeves, K. K., & Warren, H. P. (2002). Modeling the Cooling of Postﬂare Loops. The Astrophysical Journal, 578, 590.
- Rosenbluth, M. N., MacDonald, W. M., & Judd, D. L. (1957). Fokker-Planck Equation for an Inverse-Square Force. Physical Review, 107, 1.
- Sakao, T. (1994). Characteristics of solar ﬂare hard X-ray sources revealed with the hard X-ray telescope aboard the Yohkoh satellite. PhD thesis, University of Tokyo.
- Savage, S. L., McKenzie, D. E., & Reeves, K. K. (2012), Re-interpretation of Supra-arcade Downﬂows in Solar Flares. The Astrophysical Journal, 747, L40.
- Schrijver, C. J., et al. (2012). Estimating the frequency of extremely energetic solar events, based on solar, stellar, lunar, and terrestrial records. Journal of Geophysical Research, 117, A08103.
- Shibata, K., et al. (2013). Can Superﬂares Occur on Our Sun? Publications of the Astronomical Society of Japan, 65, 49.
- Shibayama, T., Kusano, K., Miyoshi, T., Nakabou, T., & Vekstein, G. (2015). Fast magnetic reconnection supported by sporadic small-scale Petschek type shocks. Physics of Plasmas, 22, 100706.
- Somov, B. V., & Kosugi, T. (1997). Collisionless Reconnection and High-Energy Particle Acceleration in Solar Flares. The Astrophysical Journal, 485, 859.
- Sturrock, P. A. (1968). A Model of Solar Flares in IAU Symp. 35: Structure and Development of Solar Active Regions, 471.
- Sui, L., & Holman, G. D. (2003). Evidence for the Formation of a Large- Scale Current Sheet in a Solar Flare. The Astrophysical Journal, 596, L251.
- Takasao, S., Matsumoto, T., Nakamura, N., & Shibata, K. (2015). Magnetohydrodynamic Shocks in and above Post-ﬂare Loops: Two-dimensional Simulation and a Simpliﬁed Model. The Astrophysical Journal, 805, 135.
- Tandberg-Hanssen, E., & Emslie, A. G. (1988). The physics of solar flares (Cambridge Astrophysics Series). Cambridge, UK: Cambridge University Press.
- Tschernitz, J., Veronig, A. M., Thalmann, J. K., Hinterreiter, J., & Pötzi, W. (2018). Reconnection Fluxes in Eruptive and Conﬁned Flares and Implications for Superﬂares on the Sun. The Astrophysical Journal, 853, 41.
- Tsuneta, S., & Naito, T. (1998). Fermi Acceleration at the Fast Shock in a Solar Flare and the Impulsive Loop-Top Hard X-Ray Source. The Astrophysical Journal, 495, L67.
- Ugai, M., & Tsuda, T. (1977). Magnetic ﬁeld line reconnection by localized enhancement of resistivity. I. Evolution in a compressible MHD ﬂuid. Journal of Plasma Physics, 17, 337.
- Warren, H. P. (2006). Multithread Hydrodynamic Modeling of a Solar Flare. The Astrophysical Journal, 637, 522.
- Warren, H. P., & Antiochos, S. K. (2004). Thermal and Nonthermal Emission in Solar Flares. The Astrophysical Journal, 611, L49.
- Warren, H. P., Mariska, J. T., & Doschek, G. A. (2013). Observations of Thermal Flare Plasma with the EUV Variability Experiment. The Astrophysical Journal, 770, 116.
- Wheatland, M. S. (2004). A Bayesian Approach to Solar Flare Prediction. The Astrophysical Journal, 609, 1134.
- Yashiro, S., Gopalswamy, N., Akiyama, S., Michalek, G., & Howard, R. A. (2005). Visibility of coronal mass ejections as a function of ﬂare location and intensity. Journal of Geophysical Research, 110, A12S05.
- Yeates, A. R., & Mackay, D. H. (2009). Initiation of Coronal Mass Ejections in a Global Evolution Model. The Astrophysical Journal, 699, 1024.
- Yokoyama, T., & Shibata, K. (1997). Magnetic Reconnection Coupled with Heat Conduction. The Astrophysical Journal, 474, L61.
- Zhang, J., Dere, K. P., Howard, R. A., Kundu, M. R., & White, S. M. (2001), On the Temporal Relationship between Coronal Mass Ejections and Flares. The Astrophysical Journal, 559, 452.
2. Measurements tend to work with the particle flux distribution, , whose power law is traditionally written . The power in the latter is .