5. Discussion and conclusions
We have tested several statistical representations of a large data set of quiet Sun measurements in chromospheric, transition region and coronal lines. The observations have been performed by the EUV spectrometers CDS and SUMER over a period of more than two years including the end of solar activity cycle 22 and the beginning of cycle 23.
The radiance distributions of the quiet Sun observations of both instruments can be almost optimally modelled by a lognormal distribution or with even slightly better accuracy by a combination of a lognormal and a Gaussian distribution. A double Gauss model, although having the advantage of separating an observed intensity distribution into cell and network parts, fails to give an adequate representation of the long tail of particularly bright pixels. In the few cases in which a double Gaussian distribution provides a better representation of the data (CDS observations of the MgX lines) we could show that this is the result of the lower spatial resolution of the instrument.
Is a lognormal distribution alone sufficient to describe the intensity distributions of all lines, or is another component, e.g. a Gaussian, needed? The improved fits to some lines by a lognormal+Gaussian combination speak for the latter. However, the highly variable relative strengths and locations of the two components are a major drawback. Also, a further improvement in the spatial resolution may lead to distributions closer to a pure lognormal, as an extrapolation from our result suggests. Note that at least at coronal temperatures the intensity distributions are not self-similar in the sense that their shapes depend on the spatial resolution of the observations. This confirms that the relevant scales of coronal heating lie at or below the spatial resolution of SUMER.
The fact that in most cases the lognormal distribution is to be preferred to a double Gaussian provides support for the idea that basically the same heating processes are acting in the network and cell interiors. Whether it also supports the conclusion of Griffiths et al. (1999) that small-scale magnetic fragmentation processes could be responsible for the observed radiances also at chromospheric and transition region temperatures is far less clear. They base their conclusion on the work of Bogdan et al. (1988), who studied the size distribution of sunspot umbrae and found it to be lognormally distributed. They (Bogdan et al. 1988) pointed out that the lognormal distribution is an indicator of fragmentation processes (Kolmogorov 1941), in the sense that, starting from an initial quantity , after n fragmentations into fractions and the stochastic quantity is given as a product of independent variables, . The Central Limit Theorem then applies for the logarithm of if the are independent, have the same probability distribution, and their logarithm has finite variance. Thus, this logarithm is normally (Gaussian) distributed.
Lognormal distributions are not unknown in solar and astrophysics. Early studies showed that the distribution of galaxies could be described with a lognormal function (Hubble 1934; Szalay 1987). While Bogdan et al. (1988) used the lognormal distribution to model the distribution of sunspot umbral areas, Martínez Pillet et al. (1993) found sunspot decay rates to be lognormally distributed.
In the case of sunspot umbrae the connection with magnetic fragmentation processes is straightforward. Sunspots represent cross-sections through magnetic flux-tubes. A lognormal distribution of umbral areas thus implies that the flux tubes crossing the solar surface have undergone (or are undergoing) fragmentation. Since the field strength averaged over the cross-section is practically independent of its area (Solanki & Schmidt 1993; Solanki et al. 1999) sunspot umbral area is also rather tightly related to the magnetic flux carried by the flux tube. According to current understanding sunspots and the smaller flux tubes forming pores or underlying plages were part of a single large tube at the bottom of the convection zone. While rising through the convection zone this tube, which carried the flux of the entire active region, was shredded near its apex (Zwaan 1985), thus giving rise to the expectation of a lognormal distribution of the resulting flux tube cross-sections.
The EUV brightenings visible from the chromosphere to the corona are also related to the magnetic field, but far less directly. E.g. Schrijver et al. (1989) found that the relationship between chromospheric brightness and photospheric magnetic field exhibits a huge scatter and is thus far from being one-to-one. Theoretically, the mere presence of a magnetic field is not sufficient to result in an EUV brightening. Rather, the brightness seen in EUV lines is produced by the cooling of gas previously heated up through some dynamic process involving the magnetic field. This may be the dissipation of MHD waves or the release of magnetic tension energy, built up through footpoint motions, by magnetic reconnection. It is thus unclear to what extent the lognormal distribution of EUV brightnesses reflects the lognormal distribution of the magnetic fluxes in the tubes and to what extent it is produced by the statistics of the physical processes generating the brightness. Thus it would be interesting to work out the brightness distribution predicted by the sandpile model of Lu & Hamilton (1991), which predicts a power law distribution of the brightenings. Also of considerable interest would be to follow the change in the shape of the distribution from coronal holes, via the quiet Sun to active regions. We plan to follow up both these paths in future investigations.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000