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2.2 Geometry of hard X-ray emissions

When an electron hits another particle in a Coulomb collision, it emits bremsstrahlung. In fully ionized plasma, this is the physical process also known as free-free radiation of thermal electrons. The change in direction and momentum causes a particle with some keV energy or greater to emit X-ray photons. Bremsstrahlung (X-ray emissions) of thermal and non-thermal electron populations are shown in Figure 7View Image.
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Figure 7: Typical X-ray spectrum of flare observed by the RHESSI satellite. The soft part is fitted with a thermal component (green) having a temperature of 16.7 MK, and the high-energy part with a power law having two breaks at 12 keV (possibly due to the acceleration process if real) and at 50 keV, of which the origin is unknown (Grigis and Benz, 2004Jump To The Next Citation Point).

The cross-section of a single electron for producing an X-ray photon of a certain energy is given by the quantum mechanical Bethe–Heitler formalism. The instantaneous emissivity of an electron beam propagating in a plasma is named thin target solution. It does not account for the resulting particle energy change to be considered in further collisions. If the target is deep enough, the incident particle slows down to thermal speed. The total radiation is therefore the integral in time over all emissions starting at the entry into the target until the particle energy is thermalized. If other deceleration processes than Coulomb collisions can be excluded, this is the thick target situation (Brown, 1971Jump To The Next Citation Point). The thick target photon spectrum of an electron beam is flatter (usually called harder) than any thin target it may transverse before, as it includes emission of the decelerating electrons. In an ideal situation including a power-law distribution of electron energies, the power-law index of thin target emission is smaller by 2 than the thick target spectrum.

The emission of hard X-rays with a non-thermal energy distribution was first located by Hoyng et al. (1981) at footpoints of coronal loops. Based on stereoscopic observations, Kane (1983Jump To The Next Citation Point) reported that 95% of the > ∼ 150 keV X-ray emission in impulsive flares originates at altitudes < ∼ 2500 km, that is at the level of the chromosphere. Brown et al. (1983) concluded that this upper limit in altitude satisfies the collisional thick-target model in which precipitating electrons loose their energy in the dense, cold target. The emission presumably originates from flare-accelerated electrons precipitating into the chromosphere. They follow the field line until they reach a density high enough for collisions and emit bremsstrahlung by scattering and slowing down. Footpoint sources are also occasionally found on Hα flare ribbons some 30,000 km apart, tracing the base of a magnetic arcade (Masuda et al., 2001). The connecting loop are observed in thermal soft X-ray emission, indicating that the density has greatly increased. Densities of more than 1011 cm–3 are frequently reported (e.g., Tsuneta et al., 1997Jump To The Next Citation Point) and agree with the concept of chromospheric evaporation. Up-streaming plasma, presumably heated by precipitating particles and filling up a coronal loop, has been observed in blue shifted lines (Antonucci et al., 1982Jump To The Next Citation PointMariska et al., 1993).

Flare footpoints are usually observed to move, indicating propagation of the energy release site (Krucker et al., 2003). Reconnection in a cusp above the flare ribbons predicts that footpoints move apart. Contrary to expectation, parallel motion along the flare arcade has also been reported by Grigis and Benz (2005aJump To The Next Citation Point) and Bogachev et al. (2005) (see movie in Figure 8Watch/download Movie). Stochastic jumps in the time evolution are frequent (Fletcher et al., 2004Krucker et al., 2005a). These hard X-ray observations suggests that most of the flare energy at a given site is released in an initial burst. The separation of the footpoints perpendicular to the flare ribbon, well observed in Hα, appears to be the result of a more gentle energy release in a later flare phase (Švestka, 1976Martin, 1989).

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Figure 8: mpg-Movie (1895 KB) Hard X-ray footpoints (with error bars) observed with the RHESSI satellite during the flare of November 9, 2002. Simultaneous footpoints are connected by a line colored sequentially with time. The footpoint information is overlaid on a SOHO/EIT image at 195 Å, indicating enhanced density in the corona. The movie shows two simultaneous footpoints connected by a vertical half-circle. The flux at each footpoint is indicated by the size of the purple circle at logarithmic scale (from Grigis and Benz, 2005a), courtesy of Paolo Grigis.

Hard X-rays have been noted to correlate with Hα kernels (Vorpahl, 1972Wuelser and Marti, 1989), indicating that the flare energy reaches the dense part of the chromosphere within less than 10 seconds. Hard X-ray footpoints coincide spatially with Hα kernels (Radziszewski et al., 2007, see Figure 9Watch/download Movie).

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Figure 9: mpg-Movie (9784 KB) Left: RHESSI flare observations of soft X-rays (red, 8 – 12 keV) and hard X-rays (blue, 20 – 50 keV) overlaid on an Hα background. Note the high-energy footpoints moving on the Hα flare ribbons, which moves apart in the very late phase. Visualization by RHESSI scientists.

Hard X-ray footpoints and soft X-ray flare loops are consistent with the standard flare scenario of energy release in the corona, energy transport to the chromosphere and chromospheric evaporation, but where is the energy released? First hints came from a third hard X-ray source above the soft X-ray loop (Masuda et al., 1994Jump To The Next Citation Point). The coronal X-ray emission contains a thermal part, dominating a low energies, and a weak non-thermal part above about 8 – 10 keV. The non-thermal emission in the corona is usually soft (Mariska and McTiernan, 1999Petrosian et al., 2002Jump To The Next Citation Point), consistent with the idea of a thin target (Datlowe and Lin, 1973). Thus, the accelerated electrons lose only a small fraction of their energy and continue to propagate towards the chromosphere. In the chromosphere they meet a thick target, yielding a harder spectrum (Brown, 1971Hudson, 1972). An example showing two footpoints and a coronal source is shown in Figure 10View Image. In exceptional cases, only the fastest electrons reach the chromosphere (Veronig and Brown, 2004Jump To The Next Citation Point). The coronal source often appears before the main flare hard X-ray increase, but is well correlated in time and spectrum with the footpoints (Emslie et al., 2003Jump To The Next Citation Point). These observations suggest strong coupling between corona and chromosphere during flares.

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Figure 10: Image of a solar flare in hard X-rays observed by the RHESSI satellite. The curved line indicates the limb of the photosphere. The displayed energy range 12 – 50 keV is dominated by the low energies, where the coronal source (right) prevails. Two footpoints (left) are clearly visible on the disk. The spider like structures are mostly artifacts of image reconstruction (from Battaglia and Benz, 2007Jump To The Next Citation Point).

The altitude of coronal hard X-ray sources, some 6000 – 25,000 km, is compatible with observed time delays of hard X-ray peaks emitted at the footpoints (Aschwanden et al., 1995). Low energy photons are emitted later compared to higher energy photons. Delay-time observations scale with the observed lengths of the soft X-ray loop (Aschwanden et al., 1996), consistent with the interpretation of longer propagation times of lower energy electrons. If only time-of-flight effects are assumed, the propagation path would put the acceleration above the loop-top hard X-ray source. The assumption of simultaneous injection puts strong constraints on the timescales involved on the acceleration process (Brown et al., 1998).

The difference in the spectrum of coronal source and footpoints can be taken as a test for the geometry. Measurements by Petrosian et al. (2002) show that the power-law indices differ by about 1 on the average, contrary to expectations. A possible interpretation for a value less than 2 may be a mixture of thick and thin target emissions, overlapping in the observed resolution element. On the other hand, an analysis based on RHESSI data demonstrates the existence of index differences larger than two in 3 out of 9 events (Battaglia and Benz, 2007Jump To The Next Citation Point). This discrepancy requires a more complete physical scenario than ballistic particle motion and will be discussed in the following section.


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