5.8 Coronal flares in time

5.8.1 Flare energy distributions and coronal heating

Magnetic flares in the solar and stellar atmospheres heat plasma explosively to > ∼ 10 MK on time scales of minutes. Large amounts of emission measure are produced, presumably by bringing up heated material from the chromospheric layers, thus increasing the density in coronal regions of magnetic loops. It has been suggested that much, if not all, of the hot coronal plasma has been heated (and evaporated) by flares that may be too small and too frequent to be detected individually. This hypothesis is known as the “microflare” or “nanoflare” hypothesis in solar physics (Parker, 1988). There is observational evidence that numerous small-scale flare events occur in the solar corona at any time (e.g., Lin et al., 1984). Hard-X-ray studies have shown that they are distributed in energy according to a power law,

dN--= kE −α , (20 ) dE

where dN is the number of flares per unit time with a total energy in the interval [E,E+dE], and k is a constant. If α ≥ 2, then the energy integration (for a given time interval, ∫Emax Emin E[dN/dE]dE) diverges for Emin → 0, that is, if the power law is extrapolated to small flare energies, a lower cut-off is required for the power-law distribution; arbitrary energy release powers are possible depending on the value of Emin. From solar studies, α ≈ 1.6 – 1.8 for ordinary solar flares (Crosby et al., 1993), but some recent studies of low-level flaring suggest α = 2.0 – 2.6 (Krucker and Benz, 1998Parnell and Jupp, 2000).

5.8.2 Phenomenological evidence

There is abundant evidence that flare-like processes are important in magnetically active stars:

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Figure 25: X-ray light curve of the young (ZAMS) solar analog EK Dra, obtained with XMM-Newton. The curves refer to the total 0.2 – 10 keV photon energy range (black), the soft band (0.2 – 1 keV; green), the hard band (>1 keV; red), and the ratio of hard/soft (blue; from Telleschi et al. 2005Jump To The Next Citation Point, reproduced by permission of AAS).

5.8.3 Stellar flare energy distributions

Motivated by the above phenomenology, several studies have addressed flare distributions in stellar coronae. Earlier work by Collura et al. (1988) and Pallavicini et al. (1990) indicated power laws with α < 2 although the sensitivities used in those observations were quite limited, and flares from different stars at different distances were lumped together. This also holds for a study by Osten and Brown (1999) for RS CVn-type binaries.

To avoid bias with regard to detection limits at lower energies, new methods have been devised by Audard et al. (1999Jump To The Next Citation Point), Audard et al. (2000Jump To The Next Citation Point), Kashyap et al. (2002), Güdel et al. (2003bJump To The Next Citation Point), and Arzner and Güdel (2004), the latter three using Monte Carlo forward methods and analytical inversion. Work by Wolk et al. (2005Jump To The Next Citation Point), Arzner et al. (2007), and Stelzer et al. (2007Jump To The Next Citation Point) has extended flare statistics into the PMS domain.

Most of these recent studies converge to α ≈ 2.0 – 2.5 (Figure 26View Image), indicating a potentially crucial role of flares in coronal heating if the power-law flare energy distribution extends by about 1 – 2 orders of magnitude below the actual detection limit in the light curves. The X-ray coronae of active stars would be an entirely hydrodynamic phenomenon rather than an ensemble of hydrostatic loops.

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Figure 26: Cumulative flare rate distributions in energy for 47 Cas, EK Dra, and both combined. The indices of power-law fits to the cumulative distributions are smaller than those of the differential distributions by one, α –1. The distributions plotted as solid histograms have been corrected for flare overlap. Also shown are power-law fits to the corrected distributions (solid lines; see Audard et al. 1999Jump To The Next Citation Point for more details; reproduced by permission of AAS).

5.8.4 Stochastic flares and coronal observations

This mechanism has been studied specifically for young solar analogs by Güdel et al. (1997bJump To The Next Citation Point), Güdel (1997Jump To The Next Citation Point), Audard et al. (1999Jump To The Next Citation Point), and Telleschi et al. (2005Jump To The Next Citation Point), in particular with regard to synthetic light curves and emission measure distributions.

Güdel et al. (1997b) found that the time-averaged emission measure distribution of solar flares adds a characteristic, separate emission measure component at high temperatures (> 10 MK); numerical simulations support this picture (Güdel, 1997), and corresponding hot emission measure components are indeed also identified in active solar analogs from X-ray spectroscopic observations. If a full distribution of flares contributes, including small flares with lower temperature, then the entire observed X-ray emission measure could be formed by the continually heating and cooling plasma in flares. The predicted steep low-T slope (up to ≈ 4) of an emission measure distribution induced by stochastic flaring compares very favorably with observations of active stars (Güdel et al., 2003b). This holds true specifically for solar analogs for which α = 2.2 – 2.8 is implied from the emission measure slopes (Telleschi et al., 2005Jump To The Next Citation Point).

Audard et al. (1999) and Audard et al. (2000Jump To The Next Citation Point) derived the flare energy distribution of young solar analogs (Figure 26View Image, where the cumulative distributions are plotted, with a power-law index that is shallower by one unit) and studied consequences for the thermal structure of the active coronae. They speculated that the higher temperatures in active coronae are due to a larger rate of (larger) flares, although they left undecidedwhether there is a larger rate of reheating events of the same coronal structures or whether a higher number of active regions produce the higher flare rate.

5.8.5 Summary: The importance of stochastic flares

The above overview suggests that the young solar corona may have been strongly driven by flares erupting as a consequence of magnetic instabilities. There is no conclusive proof that stochastic flares dominate coronal heating although there is clear evidence that active stellar coronae are continuously variable on time scales similar to flare events, and there is ample circumstantial albeit indirect evidence supporting this model, as described above. There are important implications from flare-heating coronal models:

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