Identification of Multiple Hard X-Ray Sources in Solar Flares: A Bayesian Analysis of the 2002 February 20 Event

Abstract

The hard X-ray emission in a solar flare is typically characterized by a number of discrete sources, each with its own spectral, temporal, and spatial variability. Establishing the relationship among these sources is critical to determining the role of each in the energy release and transport processes that occur within the flare. In this paper we present a novel method to identify and characterize each source of hard X-ray emission. The method permits a quantitative determination of the most likely number of subsources present, and of the relative probabilities that the hard X-ray emission in a given subregion of the flare is represented by a complicated multiple source structure or by a simpler single source. We apply the method to a well-studied flare on 2002 February 20 in order to assess competing claims as to the number of chromospheric footpoint sources present, and hence to the complexity of the underlying magnetic geometry/topology. Contrary to previous claims of the need for multiple sources to account for the chromospheric hard X-ray emission at different locations and times, we find that a simple two-footpoint-plus-coronal-source model is the most probable explanation for the data. We also find that one of the footpoint sources moves quite rapidly throughout the event, a factor that presumably complicated previous analyses. The inferred velocity of the footpoint corresponds to a very high induced electric field, compatible with the fields in thin reconnecting current sheets.

Keywords: Sun: X-rays, gamma rays; Sun: flares; methods: data analysis; methods: observational; methods: statistical.

Figure 1.

The three source types: circle, ellipse and loop.

Figure 2.

Posterior probability for the number of sources, color-coded: in blue the probability of a single source, in green that of two sources, in yellow that of three sources, and so on. The parameter λ has been set as follows: λ = 1 for the energy band 6–10 keV, λ = 2 for 10–14 keV and 50–70 keV, and λ = 3 for 14–50 keV.

Figure 3.

Distribution of the particles in the last iteration and (black lines) Clean reconstruction. The parameter λ has been set as in Figure 2. Clean contour levels are 0.08, 0.2, 0.3, and 0.7.

Figure 4.

Distribution of the particles in the last iteration and (black lines) MEM-NJIT reconstruction. The parameter λ has been set as in Figure 2. MEM-NJIT contour levels are 0.08, 0.2, 0.3, and 0.7.

Figure 5.

Images reconstructed in the 6–10 keV interval by Clean (left), forward-fit (middle), and the Bayesian approach with λ = 2 and p_C = 1/2, p_E = p_L = 1/4 (right).

Figure 6.

Histograms of the Monte Carlo samples for the three sources in the energy band 20–30 keV; these histograms can be interpreted as (unnormalized) posterior distributions for the respective parameters.

Figure 7.

Distribution of the particles (first row) for the February 20 flare at 11:06:08 UT, 11:06:15 UT, 11:06:24 UT (30–80 keV, 7″ resolution). The time intervals are ∼8 s, ∼8 s, and ∼16 s, respectively. Histograms of the positions of the northern and southern sources are shown in the second and third rows, respectively.

Figure 8.

Distribution of the particles obtained by applying the Bayesian approach to visibilities taken every 2 s, with 4 s integration time. Energy interval: 30–80 keV, 7″ resolution. All times in UT. The parameter λ has been set to 2 for all the time intervals.

Figure 9.

Temporal evolution of the x-coordinate, the y-coordinate, and the flux of the southern source. Error bars represent one standard deviation of the posterior distribution. The x-axis represents seconds after 11:06 UT.

Figure 10.

Posterior probabilities for the number of sources, for different values of the prior parameter λ.

Figure 11.

Posterior probabilities for the source types, using different a priori probability sets. Left: p_{C $=\frac{1}{2},p_E\frac{1}{4},p_L=\frac{1}{4}$};middle: P_{C $=\frac{1}{4},p_E=\frac{1}{2},p_L=\frac{1}{4}$}; right: p_{C $=\frac{1}{4},p_E=\frac{1}{4},p_L\frac{1}{2}$}.