Implications of Flat Optically Thick Microwave Spectra in Solar Flares for Source Size and Morphology

The low-frequency optically thick spectral index $\alpha_{l}$, as discussed in Section 1, is a sensitive parameter of the MW burst spectrum that indicates the source spatial characteristics as a function of frequency. This slope and the shape of the non-thermal gyrosynchrotron radiation spectrum are conveniently defined by numerical approximations to the theory (Dulk & Marsh, 1982; Dulk, 1985) for an isolated homogeneous source. The relationship between the brightness temperature $T_{b}$, the effective temperature of the radiating electrons $T_{eff}$, and the emitted flux density $S$ of the radio source are given in the following equations for the optically thick regime:

where $\nu$ and $\nu_{B}$ are the observed frequency and gyrofrequency respectively, $\theta$ is the viewing angle and $\delta$ is the electron spectral index. In addition, the total power flux density is the brightness temperature integrated over the source,

where $k$ and $c$ are the Boltzmann constant and velocity of light, $d\Omega$ is the differential solid angle in steradians and $T_{b}$ is the effective temperature in Kelvin for the optically-thick emission as in equation (5).

For the typically observed values of $\delta$ in the range of $2\leq\delta\leq 7$ and from Equations (5) to (7),

with $0.75\leq x\leq 1.095$.

Therefore, for a single homogeneous burst source, one would expect the observed flux density spectrum to have an optically thick slope $\alpha_{l}\approx 2.75$–3.1 for $\delta\approx 2$–7. The average $\alpha_{l}$ of this relatively small range can thus be taken as typical value for a homogeneous source,

Any value of the slope far from this range indicates some peculiarity in the characteristics of the source. Steeper spectra, $\alpha_{l}\geq 3.1$ can only be due to Razin suppression or absorption by a different source of cooler, intervening plasma (Ramaty, 1969; Bastian et al., 2007). Shallow/flat spectra with $\alpha_{l}\leq 2.75$ indicate spatial inhomogeneity of the source emission. Therefore, examining the low-frequency spectral index of the gyrosynchrotron spectrum provides a sensitive means to reveal complexity in the source morphology.

It should be noted that $\alpha_{l}\approx 2.9$ is for a non-thermal distribution of electrons. Emission from a hot ($T\sim 10$ MK) thermal distribution can produce a low-frequency slope of 2 but, for a homogeneous source, it would also produce an extremely steep high-frequency slope (Gary & Hurford, 1989). Since our bursts do not show such steep high-frequency slopes and have low-frequency slopes shallower than 2, we interpret the emission as non-thermal and inhomogeneous.

Initially, all the events in Table 1 are analyzed for the low-frequency index $\alpha_{l}$ of their spectra. The procedure is illustrated in Figure 1, which gives the overview of the first event, 2015 March 10 in Table 1. Figure 1a shows the total power dynamic spectrum over a 2 minute period, while Figure 1b shows the flux density time profile at the peak frequency, 11.3 GHz. The corresponding flux density spectral evolution is shown in Figure 1c–g. The vertical dashed lines in Figure 1b indicate five times in the burst—2 times during the rise phase, the peak time, and two times in the decay phase—selected for the spectra shown in Figure 1c–g and designated as T1 to T5. For each time, the spectral fitting is applied using the Stähli equation as discussed in Section 2. The fitting parameters, low-frequency index $\alpha_{l}$, high-frequency index $\alpha_{h}$, and peak frequency $\nu_{p}$ are marked at the bottom of each spectral plot.

In Figure 1c–g, the main point of interest of this study, the low-frequency spectral index always remains well below the theoretical value of around 2.9 predicted for a homogeneous source by Equation (8). This low index value is observed more pronounced in the decay phase of the burst. Additionally, in agreement with the previous studies (Lee et al., 1994; Melnikov et al., 2008), the peak frequency $\nu_{p}$ clearly increases in the rise phase (T2 to T3) and decreases during the decay phase, which is an indicator (Melnikov et al., 2008) that the peak is controlled by optically thick gyrosynchrotron emission.

To statistically examine $\alpha_{l}$ and its evolution over the duration of the bursts, the time profiles of $\alpha_{l}$ for all 12 events are determined in the same manner. Figure 2 shows $\alpha_{l}$ over the duration of six of the bursts selected for their representative trends. In each panel, the spectral index is plotted in red (scale on the right side of each plot) and the flux density at the peak frequency in blue (scale on the left). The error bars in the spectral index curves are calculated from the standard deviation over the neighboring 5 s (5 data points).

The overall behavior of the plots illustrates three types of $\alpha_{l}$ evolutionary trend. In the first type, as shown in Figure 2a and b, $\alpha_{l}$ starts to decrease before the peak and continues to drop into the decay phase except for a slight, short-duration increase near the peak. During the late decay phase, $\alpha_{l}$ shifts back to higher values. For example, in the 2015 March 10 event shown in Figure 2a the index starts with a value of $\sim 3$ in the rise phase then smoothly reduces to a value of $1$ at peak and eventually decreases almost to $0$ in the decay phase. In the second type (Figure 2c and d), $\alpha_{l}$ shows a sharp decrease at the peak and increases in the early decay phase. In the third type of $\alpha_{l}$ evolution, shown in Figure 2e and f, the spectral index follows the same trend as the flux density, growing steeper during the rise phase and dropping rapidly after the main peak and in the decay phase.

To better compare the behavior of the spectral index over all the events, Figure 3 presents a bar chart for the trend of $\alpha_{l}$ versus each event number listed in Table 1. The chart reports the indices averaged over a 10 s period in the rise phase, peak, and decay phase of each event (e.g., at times T1, T3, and T5 in Figure 1 corresponding to each phase). The rise phase for a given event is defined as the time when the flux density first reaches 10% of the maximum, while the decay phase is at the same flux density during the decay. The theoretical homogeneous source spectral index of $\sim$2.9 is marked by the horizontal dashed line. Note that the $\alpha_{l}$ value in each phase of the burst differs slightly from the value shown in Figure 1 for the 2015 March 10 event due to the averaging of the ten values in each corresponding phase in Figure 3. The error bars are the standard deviation of fluctuations in $\alpha_{l}$ during each 10 s period.

The observations of the chart are summarized as follows.

1. 1.

   Index evolution: Over the evolution of the burst, the spectral index decreases from peak to decay phase for nine out of the 12 events (Event numbers 1–3, 7–12).

2. 2.

   Index value: Compared to the theoretical value, most of the events have $\alpha_{l}$ below 2.9. Nine of the events have spectral indices less than 2.9 in at least one of the three phases or all the phases of the burst (Event numbers 1, 2, 5, 6, 8–12). In particular, five of them (1, 2, 5, 8, and 12) show extreme spectral index values $<$ 1 in at least one phase of the burst, which we henceforth define as flat events.

3. 3.

   Contrary to that suggested in earlier studies, flat low-frequency spectral indices are not limited to large, high-flux-density (X-class) flares (Hachenberg & Valais, 1961; Ramaty & Petrosian, 1972; Lee et al., 1994) but can also appear in a relatively weak C9.3 flare (event 12) with a low peak flux $\sim 125$ sfu.

4. 4.

   All the events but one (event 5, the sole X-class flare in our list) that show flat indices originate from active regions with a complex magnetic configuration of $\beta\gamma\delta$ as seen in Table 1.

5. 5.

   The variation of the spectral index within an event is generally smaller than between the events. An event with either a large or small index maintains similar values during its evolution.

Interpreting these points in terms of the homogeneity of the emitting source, a very low value of $\alpha_{l}$ signifies that the emission is from a highly inhomogeneous source. The area/emitting volume of this source grows with decreasing frequency due to the non-uniform physical parameters within. The declining value of $\alpha_{l}$ with time in some events suggests that the inhomogeneity and the complexity of the burst source grow as the flare evolves with time.

This spectral index analysis leads to a representative set of 5 events (42$\%$ of our sample) that have a flat ($\alpha_{l}<1$) spectrum, which is further investigated for additional evidence of source inhomogeneity in the next section.

The MW flux density as a function of frequency $S({\nu})$ for a simple homogeneous source as mentioned in Equation (7) can be written as

At the observed frequency $\nu$, for a constant brightness temperature $T_{b}$, the emitted flux density $S({\nu})$ is directly proportional to the solid angle area of the source $\Omega(\nu)$ ($\Omega$ as a function of $\nu$ emphasizes that the source area indeed depends and changes with frequency). We expect that most of the flares observed in MW emission exhibit some level of inhomogeneity, leading to an increase in source area with decreasing frequency. The flat events that we have identified require an extreme rise in size and hence inhomogeneity.

As an illustration, Figure 4a shows the source area spectrum of the 2015 March 10 burst, whose $\alpha_{l}$ varies from 1.3 to 0.5 over the burst duration. For the observed EOVSA flux density, we assume a constant and high brightness temperature of $10^{8}$ K and calculate the source areas as shown in the figure. The three curves are the areas measured for the times selected in Figure 1 (T1, T3, and T5) at the rise phase, peak, and decay phase, respectively. These source size estimates are valid for the optically thick emission well below the spectral peak, shown by the vertical dashed line. In the peak (green) and decay phase (blue) curves, at the low frequencies $\sim 2.9$ GHz, the source areas are $\sim 6200$ and $\sim 5500$ arcsec², respectively. The area spectrum for both the peak and decay phase is relatively flat until $\sim 4$ GHz and then decreases more steeply in the decay phase from frequencies above $\sim 4$ GHz. For the rise phase (red) curve, the source area starts smaller than the other two phases, at $\sim 1500$ arcsec². However, this size would be larger if we make the reasonable assumption of a variable effective temperature, starting from a lower value and increasing with time towards the peak of the burst. Following the source area at one frequency, say $3$ GHz, the rise phase area starts with a large value and grows still larger in the peak and decay phase.

According to Equation (10), for the case of a flat spectrum with $\alpha_{l}\approx 0.5$, the source area goes as $A\propto\nu^{-1.5}_{\rm GHz}$. For example, in the decay phase, a moderate-sized source with area $\sim 900$ arcsec² at $\nu_{p}\sim 10$ GHz must grow to an area of $\sim 5000$ arcsec² at 3 GHz. For the range of spectral index values $\alpha_{l}$ observed in this event, the power-law index $n$ in $A\propto\nu^{-n}_{\rm GHz}$ ranges from $0.7$ to $1.5$.

At $2.9$ GHz, the area for the peak time gives a diameter of $\sim 89\arcsec$ assuming a circular shape of the source (light gray circle in Figure 4b). For comparison, the high-frequency NoRH MW 17 GHz images and the RHESSI Hard X-ray (12–25, 25–50, and 50–100 KeV) sources at the burst peak time are overlaid on a HMI magnetogram as shown in the figure. So, even with our assumption of a high brightness temperature, the estimate of the low-frequency radio source size is many times greater than the 17 GHz and the hard X-ray sources (which themselves may appear larger than they are due to finite resolution). The actual brightness temperature in the flaring site, if not as high as the assumed $T_{b}$, will only lead to a much larger source.

As the emission is optically thick over its volume, these source area measurements characterize the actual area of the source magnetic structure. The changes observed in the area spectrum can be caused by gradients in the magnetic field strength and density (Klein & Trottet, 1984; Kucera et al., 1994; Bastian, 1999; Fleishman et al., 2018) that result in spatially-dependent changes in opacity. The fact that the flux density spectrum becoming flatter with time and the source area spectrum becoming steeper indicates that the sources grow significantly large with decreasing frequencies. Such large sources cannot be homogeneous but have to be non-uniform and inhomogeneous in the flare site. This line of reasoning with inhomogeneity is further discussed in Section 4.3.

We have performed a similar analysis on all the other events and verified that the flat events exhibit the same trend of large areas. We now seek confirmation of these estimates by indirect interferometric measurements via the relative visibility technique described earlier. Before doing that, however, in the next section, we examine the RV technique by comparing its results with the direct EOVSA imaging of source sizes available for the 2017 September 10 event (Gary et al., 2018).

As discussed earlier, relative visibility is a sensitive measure of source size and complexity for a flare MW emission using the observed visibility amplitudes. For a Gaussian source, RV amplitude vs. baseline length shows a smoothly decreasing shape depending only on the source size (Gary & Hurford, 1989). For an extended source, RV is unity at short baselines and decreases with increasing baseline length. Generally, short baselines cannot resolve a single Gaussian source; therefore, cross and auto-correlated data will have almost the same flux density leading their ratio to be unity. When sources become resolved at longer baselines, the cross-correlated data has less power leading to the RV ratio gradually decreasing from unity. Any deviation of the source from a Gaussian shape will modify the manner in which the RV ratio decreases, but the initial drop at short baselines is expected to measure the size of an equivalent Gaussian shape.

We have demonstrated that emission from a large source area at low frequencies is needed to make the flux density spectrum flat. We have suggested that this is due to a rather extreme source inhomogeneity. As a flat MW spectrum diverges from the spectrum produced by a single uniform source, the general homogeneous source theoretical model cannot produce an acceptable fit. Hence, modeling that includes inhomogeneity of the source is needed to explain the observed flat spectrum (Klein et al., 1986). An intermediate step in complexity is to consider an inhomogeneous model consisting of multiple homogeneous sources that are physically discrete, but when combined, will result in the observed flat spectrum. Thus to demonstrate the idea of inhomogeneity, we adopt a model introduced by Hwangbo et al. (2014), which represents the three-dimensional (3D) magnetic field distribution with multiple sources of homogeneous components.

The magnetic field strength is logarithmically scaled between $B_{0}$ and $B_{6}$, with $B_{6}$ being the component with the smallest magnetic field strength. The gyrosynchrotron emission for the components of Equations 11 and 12 is derived from Dulk’s approximations (Dulk, 1985) using Equations $13$ to $17$ of Dulk & Marsh 1982.

As a specific example, in Figure 8 we show the EOVSA total power spectrum of the 2015 August 24 event for the decay phase at 17:45:26 UT (marked with plus symbols). The dashed lines are the model spectra produced from each of the seven spectral components, and the black line shows the total contribution from all the emission of each component. These discrete components can be visualized as the emission from the source regions where the electrons have access, traveling from the main acceleration site during the flare. Individual peaks from these discrete sources are not expected to be observed in the total power flux density as the spectra are measured integrating over the area from the entire flaring region.

Along with the seven values of magnetic field $B_{i}$ (spaced between $1900$ and $380$ G), the parameters given in Table 3 (except $\alpha$ and $\beta$) are varied according to their dependence on the spectral shape (Gary & Hurford, 1989; Stähli et al., 1989; Gary & Hurford, 2005). The high frequency index $\alpha_{h}$ at the time of the spectrum gives electron power-law index $\delta$ $\approx$ $4.1$. Since $\alpha$ in Equation 11 is the factor that controls the slope of the spectrum in the optically thick part, it is set separately from the other parameters. The remaining parameters, $B_{0}$ and $L_{0}$ are fixed with nominal values to obtain the peak frequency close to the observed one. Then, $A_{0}$ is set to match the total flux density, and $N_{NT}$ value is set arbitrarily, which is the spatial density of the non-thermal electrons. Having these multi-variant parameters, the model spectra that best matches the observed spectrum are distinguished having the least $\chi^{2}$ value generated from

where $S(\nu_{i})$ and $S_{m}(\nu_{i})$ are the observed and model fit flux densities with $\sigma_{i}$ being the uncertainty at each frequency $\nu_{i}$.

The area $A_{6}$ and the thickness $L_{6}$ in the table are for the lowest-frequency component 6, peaking at $\sim 3.2$ GHz. The area and thickness of the components (0 to 6) range from 108 to 284 arcsec² and 50 to 69 arcsec, respectively. To resolve the smallest (11$\arcsec$) of these discrete sources requires a radio array with modest baseline lengths of order 1 km depending on frequency (0.38 km at 18 GHz). However, EOVSA imaging spectroscopy already provides a much higher resolution of 3.3$\arcsec$ at 18 GHz.

As the fluxes in this model are summed to match the flux density spectrum, when each component is combined, give an overall area of $\sim$1290 arcsec² (equivalent circular size of $\sim$40 arcsec). This size is although smaller than the size estimates given earlier but nevertheless serves to show that an inhomogeneous source can account for the shape of the spectrum. In addition, the corresponding emission volumes accounting for the LOS thickness, $V_{j}=A_{j}L_{j}$, for any component j, are quite large and grow larger at low frequencies.

These measurements indicate that for reproducing a flat spectrum, the emission either has to be comprised of multiple emission components simultaneously observed within the flare volume or has to be from a huge volume. The model shows that the flat spectrum can be the consequence of a significantly large source structure that is implausible to be homogeneous for such extended physical space over the active region and can only be inhomogeneous in nature.