We present a simple numerical model that simulates the expected radio emission from Nanoflares in an active region, which we use to test and calibrate the technique.

Model & Results

We consider a simple model with symmetric loops in static equilibrium with uniform heating. The loops models are derived from solutions to the 1-D fluid equations from Martens (2010). We consider a distribution of loop lengths and a heating rate that scales with the length as: \(Q=cL^{-3}\) (Mandrini et. al. 2000), giving different density and temperature profiles for each loop. The direct dependence of plasma frequency on ambient density, i.e., \(\nu_p=8980\sqrt{n_e}\) then allows us to obtain plasma frequency as a function of position. 

The time-lag technique correlates light curves (intensity vs. time) in two observing channels/frequencies to find the temporal offset that maximizes the correlation. Figure 1 (right) shows a sample loop in which a nanoflare occurs at a position marked by the star. The accelerated electron beam travels along the loop.

The two time-lags Δt₁ and Δt₂ due to the distance between the frequency-positions are shown by the green lines. The negative pair of lags is simply a resultant of cross-correlation between frequency pairs \(\nu_1-\nu_2\) and \(\nu_2-\nu_1\). On the left is the Cross-COrelation Power Spectrum (CCOPS) showing peaks at the expected lags.

Figure 1. Left: Cross-COrelation Power Spectrum (CCOPS) for ν1−ν2 from light curves obtained where a nanoflare produces electron beams moving in both directions along a loop. The right side demonstrates the multiple time lags that peak in the CCOPS.

The simulations are repeated for hundreds of loops, each with their own set of time-lags dependent on their density profiles. Some other parameters considered in the model are the duration of type IIIs, the occurrence rate of the bursts (bursts per second), and the addition of realistic noise. As shown in Figure 2, the technique does show signatures of these bursts at the expected positions (peaks overlaid with orange and green dashed lines). An important result is that, because of the density structure in closed loops (shallow gradient in most of the corona and steep gradient in the transition region footpoints), the emission is a very strong function of frequency and is brightest at the loop-top.  The CCOP is the maximum for closely-separated frequency pairs. 

Figure 2. CCOPS for 30 bursts s⁻¹ with expected time lags overlaid. The orange dashed lines mark the expected time lags Δt₁, and the green dashed lines mark the expected time lags Δt₂.

Conclusions

We find that in the case of closed loops, the frequency spectrum of type III bursts is expected to be extremely steep such that significant emission is produced at a given frequency only for a rather narrow range of loop lengths. We also find that the signature of bursts in the time-lag signal diminishes as: (1) the variety of participating loops within that range increases; (2) the occurrence rate of bursts increases; (3) the duration of bursts increases; and (4) the brightness of bursts decreases relative to noise. In addition, our model suggests a possible origin of type I bursts as a natural consequence of type III emission in a closed-loop geometry.

The paper can be accessed here:

Chhabra, S., Klimchuk, J. A. & Gary, D. E., 2021, ApJ, 922, 128  doi:  10.3847/1538-4357/ac2364

References

Dahlin, J. T., Drake, J. F., & Swisdak, M. 2015, PhPl, 22, 100704

Mandrini, C. H., D moulin, P., & Klimchuk, J. A. 2000, ApJ, 530, 999

Martens, P. C. H. 2010, ApJ, 714, 1290

Viall, N. M., & Klimchuk, J. A. 2012, ApJ, 753, 35

Vievering, J. T., Glesener, L., Athiray, P. S., et al. 2021, ApJ, 913, 15

Full list of authors: Sherry Chhabra, James A. Klimchuk, and Dale E. Gary