Acceleration and Storage of Energetic Electrons in Magnetic Loops in the Course of Electric Current Oscillations by V.V. Zaitsev and A.V. Stepanov

There are long-lived radio events on the Sun and stars like in type IV solar radio bursts with sudden reductions and pulsating type III bursts (Slottje, 1972; Huang et al. 2016) as well as intriguing intense radio emission from ultracool stars that lasts for several rotation periods (Hallinan et al. 2007). This can be the result of the multiple injections of accelerated electrons into the coronal magnetic loops. The idea of the acceleration and storage of energetic electrons in the magnetic loops in the course of electric current oscillations was suggested by Zlotnik et al. (2003). This idea is developed here on the base of the analogy of the coronal loop with a RLC-circuit and on the modern observations.

Excitation of Induced Electric Field

Convective motions of the photosphere matter interacting with the magnetic field near the loop legs generate the electric current in the loop. The current-carrying coronal loop can be presented as an electric (RLC) circuit whose eigen frequency depends on the magnitude of the electric current $I_0$, the electron density $n$, the loop radius $r_0$, and the length $l$ of the loop (Zaitsev et al. 1998):
$\nu_{\rm{RLC}}=\frac{c}{2\pi \sqrt{LC(I_0)}} \approx \frac{1}{2\pi \sqrt{2\pi \zeta}} \frac{I_0}{c r_0^2 \sqrt{nm_i}}, \zeta=\ln (4l/\pi r_0) – (7/4).$
Electric current oscillations are connected with the oscillations of the azimuthal component of the magnetic field $B_{\varphi}(r,t)=2rI_z (t)/cr^2_0$. These oscillations in accordance with the equation $\rm{curl}\mathbf{E}=-(1/c)\partial \mathbf{B}_{\varphi}/\partial t$ lead to generation of an electric field directed along the component of the magnetic field $B_z$, and therefore it should efficiently accelerate charged particles. Assuming $I_z(t)=I_0+\Delta I \sin (2\pi \nu _{\rm{RLC}} t)$ one can obtain the mean value of the electric field along the loop radius:
$\overline{E}=\frac{4}{3}\frac{\nu_{\rm{RLC}}I_0}{c^2} \frac{\Delta I}{I_0} \propto I^2_0 (\Delta I/I_0).$
The self-consistent equation for the electric current oscillations in an equivalent electric circuit can be written in the form (Khodachenko et al. 2009):
$\frac{d^2 y}{d\tau ^2} – \epsilon \left( \delta -2y-y^2 \right) \frac{dy}{d\tau} + \left( 1+\frac{3}{2} y+ \frac{1}{2} y^2 \right) y=0, y=( I – I_0) / I_0.$
Here $\delta = \big[ (\left| V_r \right| l_1) / c^2 r_1 R(I_0) \big] -1$, $V_r$ is the radial component of the convergent flow of the photosphere matter in the loop legs, $l_1$ and $r_1$ are the length and radius of the flux tube near loop footpoints. Because Q-factor of RLC-oscillation $G \gg 1$ the small parameter $\epsilon =1/Q \ll 1$ in the last equation makes it possible to apply the Van der Pol method and the solution has the form
$y=2\sqrt{\delta}\cos \big[ 2\pi \nu_{\rm{RLC}}(1+\frac{3}{4} \delta) t \big].$
Thus, the nonlinearity leads to the establishment of a finite amplitude of oscillations, as well as to a small shift in the oscillation frequency. Note that our lumped circuit approach suggests that electric field oscillations must be in-phase at all points of the loop. On the other hand, the electric current variations propagate along the loop with the Alfvén speed. Therefore, for the in-phase condition, the Alfvén time $\tau_A = l/V_A \approx 100$ s must be less than the period of RLC oscillations $(\nu_{\rm{RLC}})^{-1}$.

Energization Rate and Energy of Accelerated Electrons

The induced electric field $\overline{E}$ accelerates some part of electron population to a velocity exceeding $V > (E_D / E_z)^{1/2} V_{Te}$, where $V_{Te}$ is the electron thermal velocity, $E_D=e \Lambda \omega_p^2 / V_{Te}^2$ is the Dricer field, $\Lambda$ is the Coulomb logarithm, and $\omega_p$ is the Langmuir frequency. In the case of sub-Dreicer field, $x=E_D / E_z \gg 1$, the theory yields the number rate of runaway electrons accelerated by a DC-electric field: $\dot{N}_s=0.35 n \nu_{ei} V_a x^{3/8} \exp (-\sqrt{2x} –x/4)$, where $\nu_{ei} = 5.5n\Lambda / T^{3/2}$ is the effective frequency of electron-ion collisions, $T$ is the plasma temperature, and $V_a$ is the volume of the acceleration region.
Estimation for the flare on 19 July, 2012 with the set of type III bursts displaying in-phase oscillations with period 270 s ($\nu_{\rm{RLC}}\approx 3.7 \times 10^{-3}$ Hz) within the range of 0.7-3 GHz (Huang et al. 2016) and with $I_0=4 \times 10^{9}$ A, $\Delta I/I_0=10^{-3}$ gives the electron energization rate $\dot{N}_s \approx 3\times 10^{33}$~s$^{-1}$. This is compatible with the energization rate obtained from the RHESSI data. The energy gain for $\overline{E} \approx 2 \times 10^{-5}V$ cm$^{-1}$ at the distance $\Delta l= 10^9$ cm is $\epsilon \approx 20$ keV. For the TVLM 513-46546, a young radio-active M8.5V dwarf with $M_*=0.07M_{\odot}$, $R_* \approx 0.1R_{\odot}$, and an effective temperature $T_{\rm{eff}} \approx 2200$ K, the frequency of high-Q oscillations was estimated as $\nu_{\rm{RLC}} \approx 8\times 10^{-3}$ Hz (period $\approx 130$ s) and the electric field is $\overline{E}_z \approx 8\times 10^{-4} V$ cm$^{-1}$ (Zaitsev and Stepanov 2017). With this electric field, electrons can be accelerated to an energy $\approx 800$  keV at a distance $\approx 10^9$ cm. Energization rate can be as high as $\dot{N}_s \approx 10^{34}$ s$^{-1}$. With this energization rate one can expect quite high level of HXR emission from ultracool stars.

Conclusions

Several models have been suggested to interpret quasi-periodic electron acceleration in coronal magnetic loops. Quasi-periodic acceleration may be associated with the bursty regime of spontaneous magnetic reconnection. The MHD oscillation of flare loops could be a possible option to explain the features under question. However, the sausage and kink MHD modes are not capable of providing synchronous pulsations in a wide frequency interval and have quite low Q-factor. The mechanism of acceleration and storage of electrons driven by the electric current oscillations in a loop as an equivalent RLC-circuit was first suggested by Zlotnik et al. (2003). Here, we develop this idea, but for periodic groups of type III bursts and for the oscillation period exceed 100 s. By this the individual type III bursts forming the periodic groups can be triggered by the bursty regime of magnetic reconnection in the fine loop structure, the flux tubes with cross-section area of about $10^{14} – 10^{15}$ cm$^{2}$. The model developed here can explain the electron acceleration in long-lived type IV radio continuum with the fine structures such as type III bursts and sudden reductions. Proposed model can explain also the origin of accelerated electrons in ultracool stars. Note that if $\nu_{\rm{RLC}}$ coincides with the frequency of the loop MHD oscillations, the ratio $\Delta I/I_0$ grows and the acceleration and storage processes can be more effective.

Based on a recently published article:  Zaitsev V.V., Stepanov A.V., Acceleration and Storage of Energetic Electrons in Magnetic Loops in the Course of Electric Current Oscillations, Solar Physics,  292, 141-152 (2017). doi: 10.1007/s11207-017-1168-2

References:

Hallinan, G. et al. 2007, ApJL, 663, L25. doi: 10.1086/519790

Huang, J. et al. 2016, ApJ, 831, 119. doi: 10.3847/0004-637X/831/2/119

Khodachenko, M.L. et al. 2009, Space Sci. Rev. 149, 83. doi: 10.1007/s11214-009-9538-1

Slottje, C. 1972 Solar Phys. 25, 210. doi: 10.1007/BF00155758

Zaitsev, V.V. et al. 1998, A&A, 337, 887. Bibl. code:1998A&A…337..887Z

Zaitsev, V.V., Stepanov, A.V. 2017, R&QE, 59, 867. doi: 10.1007/s11141-017-9757-3

Zlotnik, E.Y. et al. 2003, A&A, 410, 1011. doi: 10.1051/0004-6361:20031250