Interplanetary Type III Bursts and Electron Density Fluctuations in the Solar Wind

We performed a statistical analysis of 152 type III radio bursts observed by STEREO between 2007 May and 2013 February. The separation angle between the spacecraft in the ecliptic plane ranged between 7° (2007 May) and 180° (2011 February). Only intense, simple, and isolated cases have been included. This data set has already been used to study radio source locations and radio flux variations with frequency from Krupar et al. (2014a, 2014b). As an example from our list of events, we show an analysis of a type III burst from 2010 November 13 that has been linked to a C1.3 solar flare located at S23°W16°. Figure 1 shows the flux density S from STEREO-A and STEREO-B. Both spacecraft detected a simple and isolated type III burst with an onset time at about 11:35 UT. During this event, STEREO-A was at 84° west from a Sun–Earth line at 0.97 au from the Sun, whereas STEREO-B was at 83° east and 1.08 au from the Sun.

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Figure 2 displays fixed-frequency light curves of the same event in four frequency channels (125, 375, 675, and 975 kHz). The exponential decay of the flux density S over several decades can be clearly seen. Through further analysis, we calculated median values of the flux density S for each frequency channel separately to identify background level (red lines in Figure 2). We use only data points above this threshold. We analyze data points between the peak time (t_peak) and the last value above this level (i.e., between dashed blue lines in Figure 2). We assume an exponential decay profile of the flux density S, which can be described by the following equation:

$$\begin{eqnarray}&&S(t)=\displaystyle \frac{I}{\tau }\exp \left(\displaystyle \frac{{t}_{\mathrm{peak}}-t}{\tau }\right),\end{eqnarray} \tag{ 1 }$$

where t is the time and ${t}_{\mathrm{peak}}$ corresponds to the time of the peak flux density. Coefficients I and τ are parameters of a gradient-expansion algorithm to compute a nonlinear least-squares fit. Figure 2 shows the results of this fitting for decay flux density profiles in green. As expected, the calculated decay times τ increase with decreasing frequency f. Results from both spacecraft are about the same.

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We performed the above-described analysis of the exponential decay times τ on 152 type III bursts case by case. Although our data set spans periods of solar minimum to near solar maximum, we do not observe any significant variations of τ in time. Figure 3 displays histograms of the decay times τ for four frequency channels (125, 375, 675, and 975 kHz). As expected, τ increases with decreasing frequency. We note that only 35% (STEREO-A) and 25% (STEREO-B) of events were measured down to 125 kHz (Krupar et al. 2014b). This low-frequency cut-off can be explained as an intrinsic characteristic of the radiation mechanism, an effect of the directivity of the radiation, and propagation effects between the source and the observer (Leblanc et al. 1995). For the statistical analysis, we use median values due to the log-normal character of the τ distributions. Figure 4 shows median values of decay times τ as a function of frequency. We assume that the decay times τ are statistically frequency depend as

$$\begin{eqnarray}&&\tau (f)=\alpha {f}^{\beta }.\end{eqnarray} \tag{ 2 }$$

This model fit the data very well for both spacecraft. We obtained spectral indices β of −1.18 ± 0.2 and −1.25 ± 0.2 for STEREO-A and STEREO-B, respectively. These values were calculated by minimizing the χ² error statistic with the 1σ uncertainty estimates. A similar study of 79 type III bursts observed by the OGO-5 spacecraft was performed by Alvarez & Haddock (1973). They analyzed nine frequency channels between 50 kHz and 3.5 MHz, obtaining a spectral index β of −0.95. Evans et al. (1973) investigated 35 type III bursts observed by the RAE-1 and IMP-6 spacecraft between 67 kHz and 2.8 MHz. They obtained a spectral index β of −1.08, which is close to our findings.

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Thejappa et al. (2007) developed a Monte Carlo simulation code to quantify effects of refraction and scattering on propagation of radio emissions generated at f = 120 kHz, when observed at 1 au. For this purpose, they employed a set of six first-order differential equations derived by Haselgrove (1963) to retrieve the position and direction vectors R and T, respectively. They launched 1000 randomly directed rays from the isotropic point source located at altitudes with plasma levels corresponding to 115 kHz (the fundamental component; $f=1.05{f}_{\mathrm{pe}}$) and 60 kHz (the harmonic component; $f=2{f}_{\mathrm{pe}}$). For the refraction, the spherically symmetric solar wind electron density model of Bougeret et al. (1984) was employed ($n\sim {r}^{-2.10}$). For the scattering, Thejappa et al. (2007) assumed the power spectrum of electron density fluctuations in the solar wind P_n in the inertial range to be proportional to the spatial wavenumber q as ${P}_{n}(q)\sim {q}^{-11/3}$ (i.e., the Kolmogorov spectrum). They used an empirical formula for the outer scale of the electron density fluctuations l_o by Wohlmuth et al. (2001), while the inner scale l_i was assumed to be 100 km (Manoharan et al. 1987; Coles & Harmon 1989). The relative electron density fluctuations $\epsilon =\langle \delta {n}_{{\rm{e}}}\rangle /{n}_{{\rm{e}}}$ was set to be 0.07, which corresponds to results by Bavassano & Bruno (1995). They investigated 4693 intervals with a length of 45 minutes recorded by the two Helios spacecraft covering heliospheric distances between 0.3 au and 1 au to retrieve the relative density fluctuations . Bavassano & Bruno (1995) concluded that in about 86% of cases, has a value below 0.1. The scattering is included by adding a perturbation vector $\langle {\boldsymbol{q}}\rangle$ to T after each step when a ray suffers a regular refraction in a layer of thickness ΔS corresponding to the ray path length. Thejappa et al. (2007) used ΔS = 10l in the simulation code, where l is the effective scale of electron density fluctuations defined as $l={l}_{i}^{1/3}{l}_{o}^{2/3}$. The vector $\langle {\boldsymbol{q}}\rangle$ is calculated from a Gaussian distribution of random numbers with a zero mean and a standard deviation of

$$\begin{eqnarray}&&\sigma =\sqrt{\displaystyle \frac{\pi {\rm{\Delta }}S}{l}}{\left(\displaystyle \frac{{f}_{\mathrm{pe}}}{f}\right)}^{2}\displaystyle \frac{\epsilon }{\sqrt{1-{({f}_{\mathrm{pe}}/f)}^{2}}}.\end{eqnarray} \tag{ 3 }$$

These rays are traced until they cross the sphere of 1 au radius (i.e., $| {\boldsymbol{R}}| =1$ au). Thejappa et al. (2007) concluded that scattering by random electron density fluctuations may extend the angular range of visibilities of the fundamental and harmonic emissions from 18° to 100°, and from 80° to 150°, respectively.

We have implemented the Monte Carlo technique of Thejappa et al. (2007) to simulate arrival times of radio emissions to 1 au for several sets of parameters assuming a presence of both the fundamental and harmonic components. Contrary to Thejappa et al. (2007), we have used a larger number of rays (100,000 rays instead of 1000 rays), a finer simulation grid (ΔS = l versus ΔS = 10l), and variable values of the inner scale l_i. Specifically, we assumed a linear increase of l_i with distance R as ${l}_{i}=(R/{R}_{{\rm{S}}})$ km for R < 100 R_S, and l_i = 100 km for R ≥ 100 R_S, where R_S represents the solar radius (Manoharan et al. 1987; Coles & Harmon 1989). Figure 5 shows histograms of simulated arrival times t_MC of rays generated at 375 kHz for four levels of the relative electron density fluctuations ( = 0.05, 0.10, 0.15, 0.30). We observe a similar exponential decay as for the STEREO/Waves measurements in Figure 2. We assume that the number of rays can be related to the flux density S. We have applied the same procedure as for STEREO/Waves data (Figure 2) to derive the decay times τ from these histograms. We calculated median values of the number of rays to estimate the background level. We analyze the shape of the histogram between the peak time (t_peak) and the last value above this level. We use Equation (1) to calculate the decay times τ. We have found that this model is in good agreement with the simulated data for the fundamental component. A comparison between observations (Figures 2(a) and (c)) and simulations of the fundamental component (Figures 5(a) and (c)) suggests the relative electron density fluctuations to be between 0.05 and 0.10.

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Although the exponential decay is also partly present for simulations of the harmonic component, we observe significant deviations from the exponential decay model for late phases of the arrival time. Moreover, simulated rise times are significantly shorter when compared to the observations. We performed Monte Carlo simulations for 17 frequency channels between 125 and 975 kHz corresponding to the STEREO/Waves measurements to reproduce radio flux density spectra (Figure 6). For each frequency channel, we calculated the simulated arrival time ${t}_{0.1c}$ assuming a constant beam speed of 0.1c, which is typical for interplanetary type III radio bursts (Krupar et al. 2015):

$$\begin{eqnarray}&&{t}_{0.1c}={t}_{\mathrm{MC}}+\displaystyle \frac{{r}_{i}}{0.1c},\end{eqnarray} \tag{ 4 }$$

where r_i represents heliospheric distance from the model of Bougeret et al. (1984). While the simulated radio emission profiles of the fundamental component are consistent with the observations, results for the harmonic component are very narrow with short onset times. This indicates that this Monte Carlo simulation algorithm can be used to explain decay times of the fundamental component only.

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We performed the Monte Carlo simulations for the same set of frequency channels and six levels of the relative electron density fluctuations ( = 0.05, 0.06, 0.07, 0.08, 0.09, 0.10) for the fundamental component only due to above-mentioned discrepancies in the harmonic component. Figure 7 shows simulated and observed decay times τ versus f. We found that the simulated decay times τ have a power-law dependence on the frequency similar to those observed by STEREO/Waves (Figure 4). The simulated decay times increase with increasing the relative electron density fluctuations . We obtained comparable values of the simulated decay times τ with STEREO observations for relative electron density fluctuations  = 0.06–0.07.

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