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Appendix A: Electron beam instability

We initially assume a one dimensional electron beam injected at height r = 0 of the form (see Reid et al. 2011) (A.1)where g₀(v) ∝ v^{− α} and α is the spectral index of the electron beam (in velocity space). d is the characteristic size of the electron beam (and consequently the size of the acceleration region). At t> 0 the electron beam propagates through space (reaching distance vt₁ at time t₁), creating a bump-in-tail distribution that causes resonant Langmuir wave growth as ∂f/∂v> 0 (Drummond & Pines 1962; Vedenov et al. 1962). The Langmuir wave quasilinear growth rate γ(v,r) and the collisional absorption of Langmuir waves γ_c are given by (A.2)where lnΛ is the Coulomb logarithm, taken as 20 for the parameters in the corona. ω_pe(r), n_e(r) and v_Te are the background plasma frequency, density and thermal velocity respectively. When the growth of Langmuir waves becomes larger than the collisional absorption of Langmuir waves from the background plasma we can obtain Langmuir waves orders of magnitude above thermal levels. Radio waves can then be produced through wave-wave interactions at the local plasma frequency and the harmonics, which we observe as type III radio bursts (e.g. Kontar & Pécseli 2002; Li et al. 2008).

At time t> 0 we can describe the distribution function assuming no energy loss using (A.3)and the growth rate for Langmuir waves becomes (A.4)Langmuir waves will occur at a height h_typeIII which can be found at a distance Δr = h_typeIII − h_acc from the acceleration site at h_acc. Using the condition that Langmuir waves will be prolific when the growth rate exceeds the collisional absorption rate we can find the distance Δr via (A.5)The second term in the brackets can be approximated using coronal parameters. We can use vg₀(v) = n_b were n_b is the electron beam density. We assume n_e(r) = 10⁹ cm^-3, T_e = 2 MK, n_b = 10⁴ cm and we find that this term is around 10^-3 ≪ α. Thus we find the simple relation (A.6)that equates the known quantities h_typeIII and α we can deduce from observations to the unknown quantities of d,

the vertical extent of the acceleration region and h_acc the height of the acceleration region.

Appendix B: Isotropic electron beam instability

We now assume an initial electron beam that can vary with pitch angle such that at t = 0 we inject (B.1)where ψ(μ) is the pitch angle distribution and . We consider an initial isotropic distribution function where ψ(μ) = const at t = 0. Again g₀(v) ∝ v^{− α} and α is then spectral index of the electron beam (in velocity space). d is the characteristic size of the electron beam (and consequently the size of the acceleration region). The distribution function at t> 0 becomes (B.2)where v is the speed of the electrons. We can find the reduced distribution along B using (B.3)where we integrated for r − vμt> 0 as we are interested in the growing part of the electron beam where ∂f/∂v> 0. Assuming that r ≫ d and d/vt ≪ 1, as predicted by Eq. (2), we can approximate Eq. (B.3) as (B.4)By finding ∂f_∥/∂v we obtain the equation for the growth rate of Langmuir waves (B.5)Using the same analysis demonstrated between Eqs. (A.4) and (A.6) we find the simple relation (B.6)that equates the known quantities h_typeIII and α we can deduce from observations to the unknown quantities of d, the vertical extent of the acceleration region and h_acc the height of the acceleration region.

© ESO, 2014