TEMPORAL VARIATIONS OF X-RAY SOLAR FLARE LOOPS: LENGTH, CORPULENCE, POSITION, TEMPERATURE, PLASMA PRESSURE, AND SPECTRA

During a solar flare, it is believed that magnetic energy in the corona is converted to other forms of energy, causing the bulk motion and heating of surrounding plasma and accelerating particles up to non-thermal energies. If the energy is released in a low density coronal region, then it is possible that the majority of accelerated particles will leave the corona without losing energy collisionally until they reach the much higher densities of the underlying chromosphere. Here the particles will interact with the surrounding plasma and produce hard X-ray (HXR) bremsstrahlung emission. However, in most flares we also observe coronal X-ray emission at or near to the top of a loop (see Holman et al. 2011; Kontar et al. 2011a, as recent reviews), although with an interesting exception (Fleishman et al. 2011). Since bremsstrahlung emission grows with density, the low coronal densities can often make the observation of coronal loop emission challenging, especially if it is masked by brighter chromospheric X-ray emission (Krucker & Lin 2008). However, the observation of HXR coronal sources with instruments such as the Ramaty High Energy Solar Spectrometer Imager (RHESSI; Lin et al. 2002) are important since these X-ray loops are much closer to or may even contain the actual sites of energy release and primary electron acceleration (Xu et al. 2008; Kontar et al. 2011b; Bian et al. 2012; Guo et al. 2012). RHESSI creates X-ray images using Rotating Modulation Collimators (RMCs; Hurford et al. 2002; Schmahl et al. 2007). As the spacecraft spins, the grids placed in front of the detectors produce a modulation curve of the X-ray intensity. From this we can construct X-ray visibilities, which are equivalent to the visibilities used in radio astronomy (Schmahl et al. 2007). In the last few years, an imaging technique known as visibility forward fitting (VIS FWDFIT) has been successfully used to study the sizes of chromospheric footpoint sources (Kontar et al. 2008, 2010; Battaglia & Kontar 2011) and inferred that the cross-sectional sizes of HXR sources decrease with energy. This technique has the advantage over other imaging algorithms such as Clean and Pixon, etc. because it provides us with relatively accurate position and size measurements from the moments of the X-ray distribution, plus an error for each value. However, X-ray visibility fitting also has the disadvantage of limiting the number of sources and their shapes that can be studied using this technique. Therefore, careful comparisons with other imaging algorithms are required when using VIS FWDFIT.

Recently, a number of studies have used visibility forward fitting to study the sizes of coronal X-ray loop sources. Xu et al. (2008) and Guo et al. (2012) investigated how the lengths of X-ray coronal loops, that is, the direction parallel to the guiding magnetic field, varied with photon energy during the X-ray impulsive stage. For a number of different flares, they found that the loop lengths increased with energy with a form consistent with that of an initial extended acceleration region within the loop itself, plus an additional length proportional to the photon energy squared. This is due to higher energy electrons moving farther through the plasma before interacting. Kontar et al. (2011b) studied the length and width changes of one X-ray coronal loop with photon energy. They found that both the length and width increased with photon energy, with the loop width increasing proportionally to the photon energy. Increases in loop width, or corpulence, are more difficult to explain since the electrons are bound to the guiding field and cross field transport should be negligible. Kontar et al. (2011b) and Bian et al. (2011) inferred that the width increase could be due to the magnetic diffusion of field lines perpendicular to the direction of the field, caused by the presence of magnetic turbulence within the loop. Although, these observations indicate the usefulness of coronal loop spatial properties, the spatial evolution of loop sources has not yet been fully explored. Loop positions have been studied before, during and after the impulsive phase of the flare. Forbes & Acton (1996) observed in soft X-rays the changing locations of post flare loops, which was interpreted as the decrease in height that open field lines undergo after they have reconnected to form closed loops. They studied two long duration events near the limb and found that field loop shrinkage did occur and matched the shrinkage predicted by a simple model of the reconnecting field but found overall that the entire flare loop system grew with time. Sui & Holman (2003), Sui et al. (2004), Veronig et al. (2006), and Joshi et al. (2009) all noted a decrease in the altitude of loop top sources during the impulsive phase of the flare, until the peak X-ray emission and an increase in altitude after the impulsive phase. Sui & Holman (2003) also found evidence for an above the loop top source and interpreted the situation as the formation of a reconnection current sheet between the loop top source and the higher coronal source. Veronig et al. (2006) interpreted the decrease as a collapsing magnetic trap (Somov & Kosugi 1997; Karlický & Kosugi 2004). The contraction and expansion of the loop source has also been observed in other wavelengths of EUV (Liu et al. 2009; Joshi et al. 2009) and radio (Li & Gan 2005; Reznikova et al. 2010). Reznikova et al. (2010) found both changes in the radio loop span and height with time. More recently, Gosain (2012) looked for evidence of collapsing fields using Solar Dynamics Observatory (SDO/AIA and HMI) observations. The loops rose slowly and then moved into a collapse phase during the impulsive phase of the flare, where the loop tops contracted. Lower loops contracted earlier than higher loops and the loop contraction was interpreted as a reduction of magnetic energy as the system relaxed to a state of lower energy, i.e., relaxation theory (Taylor 1974).

In this paper, we study three flares with coronal X-ray loops to find how the emission lengths, widths and positions change with time at three energy ranges between 10–25 keV. From each flare spectra, we will also investigate how parameters such as emission measure and plasma temperature vary with time. Using a combination of imaging and spectroscopy parameters, we can infer how the X-ray loop volume, number density, thermal pressure, and energy density vary during the time evolution of the flare, allowing us to build a fuller picture and propose some explanations describing the processes occurring within the coronal loops.

The three events studied are: 2005 August 23 from 14:22:00 (Flare 1), 2002 April 14/15 from 23:58:00 (Flare 2), and 2004 May 21 from 23:40:00 (Flare 3). All three flares share similar characteristics: GOES M-class flares with similar light curves, strong coronal X-ray loop top emission and only relatively weak footpoint emission. Since the aim of our study is to examine how the properties of coronal loop emission vary with time, these events were chosen as they show loop emission throughout the rise, peak, and decay stages of X-ray emission and their spatial properties have been previously studied by Xu et al. (2008) and Kontar et al. (2011b). The coronal X-ray emission during each flare appears up to ∼25 keV and each source is a simple loop-like shape connecting HXR footpoints at 30–40 keV during certain time intervals. The main parameters of each flare—GOES class, date, observation time, peak time at 10 keV, and the time of footpoint appearance—are given in Table 1. Length variations of each of these coronal loops with photon energy were studied by Xu et al. (2008). Length and width changes with photon energy for Flare 2 were also studied by Kontar et al. (2011b). It should be noted that Flare 1 and Flare 3 show results similar to Flare 2 in Kontar et al. (2011b), where both the length and corpulence increase with energy as ∼² and ∼, respectively, but this paper will concentrate on the size, position, and spectral parameter changes with time. The light curves for each event are shown in Figure 5 (top row). Flares 1 and 3 have similar light curves, a simple shape with one peak. The light curve for Flare 1 is shown in Figure 5 (top row, left plot). Our study of this event begins at 14:22:00. At this time, emission from the 10 to 20 keV energy bands are slowly rising and reach a peak at ∼14:30:00. After this point, there is a gradual decrease in X-ray emission, which can be observed up until ∼14:50:00, where RHESSI enters into a night phase. In the 20–40 keV band we see a series of peaks between our observation range of 14:22:00 and 14:40:00. The light curve for Flare 3 is shown in Figure 5 (top, right plot). From the start of our study at 23:40:00, the X-ray emission is rising from 6–50 keV and peaks around 23:50:00. After this peak, the X-ray emission decreases. The light curve for Flare 2 is more complex and is shown in Figure 5 (top row, middle plot). During the observational time there are two main peaks in the light curve at ∼00:03:00 and ∼00:12:00, possibly more peaks, which are easiest to see in the 20–40 keV energy range.

Each event was studied using the imaging algorithms of Clean (Högbom 1974; Hurford et al. 2002), Pixon (Pina & Puetter 1993; Metcalf et al. 1996), and VIS FWDFIT (Hurford et al. 2002; Schmahl et al. 2007). First, we studied each of our three events using only Clean and Pixon to confirm the loop shape of each coronal source and find the energy ranges over which a coronal source was present in each flare. Once we were confident that our chosen events only had a simple loop shape, we studied each event using VIS FWDFIT. This is important since the coronal source must have a loop-like shape so that VIS FWDFIT can effectively fit a curved elliptical Gaussian (loop) to the X-ray visibilities of the event and give realistic estimates with errors for the source parameters. VIS FWDFIT provides us with loop length FWHM (full width at half-maximum), loop width FWHM, and the (x, y) centroid position of the loop. It should also be noted that VIS FWDFIT gives the mean position of the loop shape, not the loop top centroid position, that is, the central position at the top of the loop. The coordinates for loop top position were obtained by extracting the coordinates of the central circular Gaussian that VIS FWDFIT uses to create the final loop shape along with a set of other circular Gaussians placed along the length of the loop. This is important as a loop that is very curved and approaching the shape of a ring will pull the shape centroid toward the ends of the loop, often masking small changes in loop position with time or energy. This was especially significant for the large loop shape of the 2005 August 23 event (Flare 1). In order to study changes in time at a specific energy range, we imaged the coronal sources of Flares 1, 2, and 3 over two minute intervals where possible and four minute intervals where the count rates were lower. The exact time bins used for each flare are shown in Figures 5 and 7. The energy ranges of 10–12 keV, 12–15 keV, and 15–20 keV were chosen for Flare 1 and Flare 2 and the energy ranges of 14–16 keV, 16–20 keV, and 20–25 keV were chosen for Flare 3. These energy ranges were chosen since VIS FWDFIT with a loop produced clear results with small errors at these energy ranges. Images for Flares 1, 2, and 3 are shown in Figures 1, 2, and 3, respectively. Figure 1 shows that Flare 1 is a limb event while Figures 2 and 3 show that both Flare 2 and Flare 3 are disk events (the main parameters of each flare are given in Table 1). Figures 1, 2, and 3 plot a background Clean image of the coronal source for each flare at one chosen time interval and over plot VIS FWDFIT contours for selected time intervals, one corresponding to the same time interval as the Clean image. Comparing the shape and size of the VIS FWDFIT contour with the Clean background image at the selected time interval for each flare shows good agreement between both algorithms. The Clean images here use a sigma beam width of 1.8. It should be noted that we also found the standard deviation from image cuts through the images of Clean, Pixon, and VIS FWDFIT and it was found that a Clean sigma beam width of ∼3.0 showed the best agreement with the standard deviations of Pixon and VIS FWDFIT. In terms of imaging, VIS FWDFIT also agrees well with the Clean and Pixon algorithms at all other times not shown in this paper. From Figure 3, it should be noted that the results for Flare 3 are probably less reliable than those of Flare 1 and Flare 2. From the position of the footpoints in the Clean image, it appears as through the southern "loop leg" is tucked underneath the observer's line of sight. This means that it is harder for VIS FWDFIT with a loop to fit it with a correctly shaped loop and usually fits it with a loop that is slightly too large or departs from a loop-shape. For each of the imaging time intervals for Flares 1, 2, and 3, energy spectra were created. The spectra of each flare at each time interval were fitted with a thermal component and a non-thermal component corresponding to thick target bremsstrahlung. The thermal fits provide us with the emission measure, EM, and the plasma temperature, T. We note all three flares show the emergence of footpoints in the 30–40 keV energy range. Spectra for Flares 1, 2, and 3 are shown in Figure 4 at three selected time bins corresponding to a rise, peak, and decay stage of X-ray emission for each flare.

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Figure 5 shows the imaging parameters for Flare 1 (Column 1), 2 (Column 2), and 3 (Column 3): loop width FWHM (row 2), loop length FWHM (row 3), and loop-top radial position (row 4) for each imaging energy band. Above these plots, the light curve is plotted for each of the imaging energy bands to allow comparison with changes in the spatial parameters. The dashed lines drawn in each plot represent the time interval over which the peak X-ray emission occurs for our three energy bands. For each flare, in general, peaks in the light curve represent changes of width, length, and source position parameters with time. The last two rows of Figure 5 plot how the spectroscopic parameters vary with time for each flare. In general, for all three events, the emission measure rises throughout the observation time, either slowing or decreasing slightly during the last few minutes for each event. For each event, the plasma temperature peaks before the peak in X-ray emission and then slowly decreases after this point, as noted by Antiochos & Sturrock (1978). The plasma temperature decreases much slower than if it were decreasing by conduction only, suggesting prolonged energy release at the later decay stages of the flare.

3.1. Loop Corpulence

For all three flares, we see a general pattern for emitting loop width changes with time. Before a peak in X-ray emission, the source width for each flare tends to decrease and after a peak in X-ray emission, the source width for each flare increases. It is also notable that the rates of both width expansion and contraction (width change in time) are approximately the same as can be seen in Figure 6.

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Flare 1. For all energies plotted, the source width decreases until the peak X-ray emission at ∼14:30:00–14:32:00 UT and then increases after this point. The largest change occurs for the highest energy of 15–20 keV. This falls from 145 to 105 at 14:28:00–14:30:00 and then increases until 14:40:00 where the width peaks at ∼20'', producing a final larger loop corpulence than seen at the beginning of our observational time.

Flare 2. For the energies of 10–12 keV and 12–15 keV, the loop width decreases before the first peak in X-ray emission at ∼00:03:00. After this point the width increases before dropping at 00:12:00–00:13:00 where there is another X-ray peak in the light curve. After this peak, the width continues to grow again. Before 00:03:00, the width of the 10–12 keV source falls from around ∼10'' to ∼5'' and after 00:13:00 rises from ∼6'' to ∼13''. The 15–20 keV source also shows this pattern except there is a larger peak at 00:05:00 and then a more pronounced decrease in width until 00:12:00–00:13:00 UT.

Flare 3. Again, the change in loop width with time follows a similar pattern as Flares 1 and 2. For all three energies ranges considered, the loop width decreases from 23:42:00 to the peak in X-ray emission at 23:50:00. The loop width then increases after this time. The 14–16 keV and 16–20 keV sources ∼ fall from 7''–8'' at 23:42:00 to ∼5'' at 23:50:00 and then increase up to 8''–9'' at 23:58:00. The 20–25 keV source falls from 11'' to 5''. Note the missing data at the fourth and sixth time bins for the 20–25 keV energy range, as VIS FWDFIT was unable to fit successfully at these times for this energy range.

3.2. Loop Length

3.3. Loop Radial Position

From the loop width FWHM W and loop length FWHM L, we can infer the general changes in the emitting loop volume, V, for each flare over time, at each energy band. We assume that the volume of the loop is given by V = πW²L/4 that is, assuming a cylindrical loop. The changes in emitting loop volume with time for Flares 1, 2, and 3 are plotted in Figure 7 (second row). In general, for all three events, the source volume decreases before a peak in X-ray emission and increases after a peak in X-ray emission. The changes in plasma number density, thermal pressure and thermal energy density can all be calculated using combinations of loop volume, emission measure, and plasma temperature. The plasma number density, n, can be obtained via $n=\sqrt{{\rm EM}/V}$, the pressure, P, from P = nk_BT, where k_B is the Boltzmann constant and finally the energy density, E = 3nk_BT. The variation of these quantities with time for Flares 1, 2, and 3 are shown in Figure 7 (third, fourth, and fifth rows, respectively).

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Flare 1. As expected from the width and length results, the loop volume falls between 14:22:00 and the peak in the X-ray light curve at ∼14:30:00 and then rises after this time for all three energies. For all three energy bands, the decrease and increase in loop volume occurs at roughly the same rate. The largest decrease is for the highest 15–20 keV band, falling from ∼4.7 × 10²⁷ cm³ to ∼1.4 × 10²⁷ cm³ at 14:29:00 and then rising after this time to ∼4.7 × 10²⁷ cm³ at the last observational time. The 10–12 keV band falls from ∼2.4 × 10²⁷ cm³ to ∼1.3 × 10²⁷ cm³ at 14:31:00 and then rises back to ∼2.5 × 10²⁷ cm³ at the final observational time. The number density, thermal pressure, and energy density for all three energy bands tend to follow the same pattern, rising to a peak at some time after the peak X-ray emission and then slowly decreasing. For the 10–12 keV band, the number density rises from 1.5 × 10¹⁰ cm⁻³ at 14:22:00 to 6.0 × 10¹⁰ cm⁻³ at 14:35:00. It then falls to ∼5.5 × 10¹⁰ cm⁻³ at 14:38:00. The 12–15 keV and 15–20 keV bands follow similar patterns, peaking at ∼14:33:00. The pressure rises from ∼40 g (cm s²)⁻¹ at 14:22:00 and reaches 170 g (cm s²)⁻¹ at 14:31:00, the time where the X-ray emission peaks. After this time the pressure remains approximately constant at ∼160–170 g (cm s²)⁻¹ until the last observation time where it falls to ∼140 g (cm s²)⁻¹. The 12–15 keV and 15–20 keV energy bands peak at ∼14:33:00. The energy density is just the pressure multiplied by 3 and hence it follows the same pattern as pressure throughout the observed duration of the flare. The energy density peaks at 500 erg cm⁻³ in the 12–15 keV band at ∼14:33:00, after the peak X-ray emission.

Flare 2. For the 10–12 keV and 12–15 keV sources, the loop volume falls until the first peak at 00:03:00 and then increases until it reaches 00:12:00, drops at 00:13:00 and then increases again after this time. As for Flare 1, the number density, thermal pressure, and energy density all follow the same pattern for each energy band, rising to a time at or just after the peak in X-ray emission and then decreasing after this point. The highest number density, thermal pressure, and energy density for each energy band occurs at 00:12:00 to 00:14:00, just after the peak in X-rays at 00:11:00–00:13:00. The number density peaks at around 18 × 10¹⁰ cm⁻³ in the 12–15 keV band, while the thermal pressure and energy density peak at ∼500 g (cm s²)⁻¹ and 1500 erg cm⁻³, respectively, at this time and energy.

Flare 3. The loop volume again falls before the peak in X-ray emission at 23:50:00 and then rises again after this time, for all three energies. For this flare again, the number density, thermal pressure, and energy density all peak just after the peak in X-ray emission. At 23:50:00–23:52:00 the number density peaks at 23 × 10¹⁰ cm⁻³, the thermal pressure peaks at 750 g (cm s²)⁻¹, and the energy density at ∼2300 erg cm⁻³, in the 16–20 keV band.

Using visibility forward fitting, we have for the first time produced a dedicated study of how the spatial and spectral properties of three coronal X-ray loops changed during the temporal evolution of each flare. All X-ray loops exhibited similar changes in both their spatial and spectral properties and hence the results indicate that a common process is occurring for all three events; the emitting flaring loop volume is decreasing before the peak in X-rays and increases after the peak in X-rays is reached. Before the peak X-ray emission, the emitting lengths and widths of each coronal loop decreased with time, indicating that the X-ray emitting region of the loop volume was contracting (a reduction in loop corpulence and length) as the X-ray emission from the region grew. After the X-ray peak, the loop corpulence increased at approximately the same rate as during contraction stage. For Flares 1 and 3, with one peak in their light curves, once the minimum X-ray loop length was reached during the X-ray peak it remained approximately constant, at least within the errors of our results.

Similar to previous studies (Antiochos & Sturrock 1978; Siding & Spicer 1980; Gunkler et al. 1984; McTiernan et al. 1999), spectroscopy for each event showed that the plasma temperature initially grew but began to decrease before the peaks in X-ray emission and emission measure. The emission measure for each flare generally grew with dips observed during the final observational times of Flares 2 and 3. At the same time, the number density, thermal pressure, and energy density of the plasma also increased as the X-ray emission grew. The plasma temperature decreased much slower than by thermal conduction only, even during the X-ray decay phase, suggesting at later stages additional energy release is required (e.g., Kołomański et al. 2011) to explain the long lasting X-ray loop emission. For the limb event (Flare 1), we do see a decrease in loop altitude before the peak in X-ray emission of roughly 2'' ≃ 1.4 Mm, which is comparable in magnitude to the decrease in loop width but overall, the largest changes occurred for the X-ray loop volume, not the X-ray loop position. Decreases in loop altitude before the peak in X-ray emission have been well noted before and are often referred to as coronal implosion, loop contraction or field line shrinking.

Sui et al. (2004) also studied the loop position changes of Flare 2. From their observations, they did conclude that the loop was indeed decreasing in altitude before the first peak in the light curve at 00:04:00. They concluded that the loop decreased by ∼2'' over a 4 minute period, which is consistent with our radial distance results for Flare 2. However, no altitude decrease before the larger peak in X-ray emission at 00:12:00 was noted, where again we see decreases in loop length and loop width.

It has been suggested that a decrease in loop altitude may be an indication of collapsing magnetic trap acceleration/heating (Takakara & Kai 1966; Somov & Kosugi 1997; Priest & Forbes 2002; Karlický & Kosugi 2004; Veronig et al. 2006; Grady et al. 2012). Although this may be the case for other events, we do not believe that the collapsing magnetic trap is the prime solution to our observations, first due to overall larger loop volume changes over relatively small position changes. Veronig et al. (2006) observed the coronal loop of a GOES X-class flare and found downward velocities as large as ∼14 km s⁻¹ in 10–15 keV band and ∼29 km s⁻¹ in the 15–20 keV band prior to the peak in X-ray emission. This is not what we observe for these events. For Flare 1, we only observe an average downward velocity in the 10–20 keV band of ∼4 km s⁻¹, which is generally comparable to the decrease in loop corpulence during this period. Average changes in loop radial position for Flares 2 and 3 are also ∼4 km s⁻¹. More convincingly, in a collapsing magnetic trap model, simple compressive heating during the contraction stage would imply that NT∝1/A (Maetzler et al. 1978; Emslie 1981), where N is the number of particles in the region, T is the plasma temperature, and A is the cross-sectional area of the region, given by πW², where W is the loop width. Figure 8 plots log NT against log A for each flare and shows this not to be the case. Straight lines fits to both the contraction and expansion phases show that the gradients are either greater or less than −1.

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Observations by Kontar et al. (2011b) showed how the loop corpulence of the 2002 April 14/15 event (Flare 2) increased with energy during two times corresponding to the first rise and first decay stages of our observations and suggested that the presence of magnetic turbulence (the diffusion of field lines) was the cause of the energetic width increases. Hence, the suggestion that magnetic turbulence in the region can account for the energetic changes in loop corpulence may also be able to account for the extra energy in the loop, since both Kontar et al. (2011b) and Bian et al. (2011) found the energy density of magnetic fluctuations to be significant and could be comparable to that of the flaring plasma and higher than the energy density of non-thermal particles. Flares 1 and 3 also show similar length and width increases with energy at a single time range, also suggesting the presence of magnetic turbulence.

Many observations (Li & Gan 2005; Liu et al. 2009; Joshi et al. 2009; Reznikova et al. 2010; Gosain 2012) of loop height and length changes have been explained in terms of a reduction in magnetic pressure. Usually the reduction in magnetic pressure is referred to as Taylor relaxation but this only refers to a special case where the resulting field is linear force free (Taylor 1974). The reduction in magnetic pressure could also account for the reduction of loop corpulence or cross-sectional area, as shown in simulations by Janse & Low (2007) and hence the observed trends of number density and pressure. Liu et al. (2009) studied coronal implosion of one coronal source. They explained the reduction in height of a coronal source before the peak in X-ray emission in terms of Taylor relaxation. They also suggested that this type of event can only occur if the coronal loop is already filled with hot, dense plasma before the onset of new event, that is from a previous event in that region. The fact that we do not see EUV emission during the rise phase of Flare 1 seems to correlate with the observations and suggestions of Liu et al. (2009). Figure 9 shows SOHO (Solar and Heliospheric Observatory) EIT (Extreme Ultraviolet Imaging Telescope) 195 Å images at the times of 14:21:12 and 14:34:51 for Flare 1. X-ray emission contours at 10–12 keV for 14:22:00–14:26:00 and 14:34:00–14:36:00 are also over plotted. From Figure 9, we can see that during the rise phase there is no bright 195 Å EUV emission emanating from the loop, only 10–20 keV X-ray emission. After the peak in X-ray emission, we do see EUV emission from the loop. We see an overall increase in the number of particles, N, within the loop region throughout the duration of all three events. Due to low coronal densities, chromospheric evaporation probably accounts for this increasing N, initially driven by thermal conduction and possibly at later times by electrons reaching the chromosphere, when we see weak footpoint emission and EUV emission from the loop.

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5.1. Three Temporal Phases and Suggested Explanations for the Observations

For each flare, Figure 10 replots the plasma temperatures, X-ray emissions, emitting loop widths and thermal pressures but now at a single energy band of 10–20 keV (14–25 keV for Flare 3). From Figures 5 and 7 and now more clearly in Figure 10, we can see that collectively, the observations of each flare display three distinct phases and each of these phases will form the basis of our suggested explanation. Each phase is represented by a shaded orange bar. During Phase 1, there is a peak in plasma temperature and during Phase 2, a peak in X-ray emission. At Phase 2 we see the smallest emitting loop width, length and hence volume. Finally, during Phase 3, the thermal pressure of the region peaks. For Flare 1 (left column), each phase is well separated and can be clearly seen. The pattern can also be seen for Flares 2 (middle column) and 3 (right column) but each phase is not as clearly defined as for Flare 1. During Flare 3, each phase occurs over much shorter time intervals and therefore each of the phases overlap slightly. During Flare 2, there are multiple peaks, which along with the shorter timescales for each process, makes each individual phase harder to see. However, the overall pattern is observed for all three flares. We can see from Figure 10 that each phase can only be easily seen for slower events. The quicker the event, the harder it is to distinguish between each of the three phases as each phase overlaps in time. Many solar eruptive events may exhibit a similar pattern but each phase may occur at timescales too fast to be currently observed.

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In order to understand our observations, we believe it is crucial to understand the decreasing X-ray widths before the X-ray peak. Length variations along the field can always be explained in terms of changing number density, but in order to explain the corpulence variations, where electrons are tied to the magnetic field, the most plausible explanation is the movement or the diffusion of magnetic field lines. We have already noted the energetic loop width increases at a given time for each flare, suggesting the presence of turbulence in the region, but overall in time the width at a given energy band shrinks until the X-ray peak is reached, implying the presence of both turbulence and the shrinking of the cross-sectional area of the field at this stage of the flare. Therefore, we suggest the plasma within our emitting X-ray loop region is tied to the magnetic field lines and hence the contraction of the emitting loop cross-section or corpulence during the rise phase is ultimately due to the contraction of the cross-sectional area of field lines that thread the region or possibly the expansion of field lines above the region. It is also sensible to assume that the region we call the loop in X-rays actually consists of multiple coronal loops that cannot be resolved in X-rays using RHESSI. Though we can only see what is happening to the region of the loop that is emitting X-rays, we will assume the entire loop region is contracting since the plasma is tied to the field lines. Hopefully, from future observations we will find similar, newer coronal events with complementary AIA/SDO data where it may be possible to see if the region consists of multiple coronal loops. From our observations, the reason for this cross-sectional width shrinking is unknown and beyond the scope of this paper, but we can speculate for further study that it may be due to a reduction in magnetic pressure within the X-ray loop region, as often suggested to describe height decreases (coronal implosion or loop contraction) discussed earlier (Li & Gan 2005; Liu et al. 2009; Joshi et al. 2009; Reznikova et al. 2010; Gosain 2012). The magnetic pressure decreases as the magnetic field relaxes from a non-potential state. Although it is reasonable to assume that during a solar eruptive event such as a flare, the field will reside in a non-potential state, X-ray observations of these events do provide evidence for the non-potentiality, again through the inference of magnetic turbulence and hence a non-parallel field component within the loop region. The non-potential state and decreasing magnetic pressure could be due to reconnection above the region or maybe even along the loop itself (Vlahos et al. 2004; Gordovskyy & Browning 2011, 2012; Gordovskyy et al. 2012). One such model is caused by a kink instability as shown in recent simulations by Gordovskyy & Browning (2011, 2012) and Gordovskyy et al. (2012). In this model, the energy of twisted loops is released by reconnection inside the loop and transferred to plasma heating and particle accelerations.

During phase 1, the process causing the contraction of loop width/cross-section is also probably responsible for the comparable decrease in loop altitude, or at least the change in loop position observed for Flares 2 and 3 sitting on the solar disk. Plots of dW/dt (width contraction/expansion) and dr/dt = v (centroid velocity) for Flare 1 plotted in Figure 6 show that each parameter can be fitted using a straight line during both the contraction and expansion phases. dL/dt (length contraction/expansion) is also plotted in Figure 6 but its trend cannot be described by a straight line fit, suggesting that the changes in length are caused by a different process, itself a consequence of the process producing the decreasing loop corpulence. It is also sensible to assume that this process is responsible for the increased plasma temperature and the acceleration of electrons. The increasing temperature of the region means that energy will be thermally conducted toward the lower levels of the solar atmosphere causing gentle chromospheric evaporation of the denser coronal and chromospheric layers below. The observations show the plasma temperature peaking relatively early before the peak in X-ray emission and then slowly decreasing, slower than by thermal conduction, implying that energy is still being supplied to the loop plasma. This is most likely via the conversion of magnetic energy. Chromospheric evaporation drives plasma into the region producing the increasing number density and hence thermal pressure, along with the shrinking corpulence at this phase. The increasing number density is responsible for the rapid nonlinear decreasing X-ray loop length, since electrons accelerated within the region will travel shorter distances before interacting.

Phase 2. During Phase 2, after the peak in plasma temperature, the loop corpulence stops shrinking. Number density and thermal pressure within the loop still increases due to chromospheric evaporation and the length of the emitting region also reaches its lowest point. The loop corpulence may stop shrinking because the process causing the shrinking ceases or it may be due to the balancing of forces within the region. For example, if the reduction of loop corpulence is due to the reduction of magnetic pressure in the loop, then at this phase, the growing thermal pressure may finally be high enough to balance the reduction in B pressure.

Phase 3. During the final phase, the thermal pressure continues to rise within the loop due to the increasing number density from chromospheric evaporation. We believe the growing thermal pressure in the region is now responsible for the expanding loop corpulence. This expansion, in turn, eventually halts the increasing number density and thermal pressure at a time after the peak in X-ray emission. After Phase 3, the loop corpulence continues to increase and we see slow decreases in both number density and thermal pressure. The X-ray emission continues to decrease during and after Phase 3 and the emitting loop length during this period remains approximately constant, equal to the minimum loop length in Phase 2, even with a decreasing number density. It is sensible to assume that the acceleration mechanism in the loop is slowing during this time. However, Flare 2 is an exception to this trend as we see multiple events/X-ray peaks in the light curve.

Finally, we would like to note that a study of the 2002 July 23 flare by Caspi & Lin (2010) calculated temporal volume changes using the Clean algorithm and assuming an elliptical geometry. Overall this flare shows a general trend consistent with our results; an overall decrease in volume before the peak in X-ray emission and an overall increase in volume after the peak in X-ray emission. This flare also shows a peak in plasma temperature before the peak in X-ray emission and a high number density (and hence thermal pressure) after the X-ray emission first peaks.

N.L.S.J. is funded by STFC and SUPA scholarships. E.P.K. gratefully acknowledges the financial support by STFC Rolling Grant, and by the European Commission through the HESPE (FP7-SPACE-2010-263086) Network acknowledged.