Cyclotron Maser Emission in Solar and Stellar Flares 1

32nd URSI GASS, Montreal, 19–26 August 2017

CYCLOTRON MASER EMISSION IN SOLAR AND STELLAR FLARES

Donald B. Melrose SIfA, School of Physics, University of Sydney, NSW 2006, Australia, http://www.physics.usyd.edu.au/ melrose

1 Introduction

The problem discussed in this paper is whether the now accepted form of electron cyclotron maser emission (ECME) for the Earth’s auroral kilometric radiation (AKR) also applies to other suggested applications of ECME, which would have radical implications for solar and stellar applications, or whether AKR is an exception and a different form of ECME applies to other sources.

Since circa 1980, ECME has been the accepted emission mechanism for AKR, for Jupiter’s decametric radiation (DAM), for solar spike bursts and for bright radio emission from flare stars. Originally, a loss-cone driven form of ECME [1] was widely accepted for all these application, but this changed for AKR in the 1990s when it was recognized that the driving electrons have a horseshoe distribution, and a horseshoe-driven form of ECME [2, 3] became the favored mechanism [13]. This form of ECME requires an extremely low electron density in the source region, in order for the radio emission to escape. This extreme condition appears to be satisfied in the “auroral cavity” that develops in association with the precipitating electrons [4]. The question discussed in this paper is whether the horseshoe-driven form of ECME, known to apply to AKR, applies to other sources of ECME, specifically to DAM, solar spike bursts and bright emission from flare stars.

2 Forms of ECME

ECME corresponds to negative gyromagnetic absorption. The gyroresonance condition must be satisﬁed,

ω − sΩ − kkvk = 0, (1) where ω is the frequency, s = 1 here is the harmonic number, Ω = Ωe/γ is the relativistic gyrofrequency, with Ωe = eB/m the electron cyclotron frequency and γ the Lorentz factor, and kk and vk are the components of the wave vector and the electron velocity parallel to the magnetic field. The sign of the absorption coefficient depends on two terms that involve derivatives of the distribution function, f , with respect to momentum components: ∂ f /∂ p⊥ and ∂ f /∂ pk with p⊥ = γmv⊥ and pk = γmvk. These derivatives and be rewritten in term of momentum p and pitch angle α, with p⊥ = psinα and pk = pcosα. For absorption to be negative, one or other of these terms must be positive. Thus ECME can be identified as perpendicular-driven (∂ f /∂ p⊥ > 0 ) or parallel-driven (kk ∂ f /∂ pk > 0) [5], or it can be identified as ring-driven (∂ f /∂ p > 0) or loss-cone-driven (cosα ∂ f /∂α > 0). A caveat is that if one makes the nonrelativistic approximation, γ → 1, one can partially integrate with respect to p⊥ and prove that the ∂ f /∂ p⊥ term contributes only to positive absorption; a corollary is that for an isotropic distribution (∂ f /∂α = 0), ring-driven ECME is possible only if the relativistic term, γ 6= 1, is retained in the resonance condition.

The relation between the driver on the ECME and the growing waves can be understood by plotting equation (1) in v⊥–vk space [5]: for given ω and kk, the curve is an ellipse, which reduces to a circle centered on the origin for kk = 0. The electrons that can resonate with the wave lie on the ellipse, and the absorption coefﬁcient can be written as an integral around the ellipse. The maximum growth rate is determined by the ellipse that correspondsto the largest positive value of this integral, thereby determining the values of ω and kk for the fastest growing wave.

The earliest versions of ECME [6, 7] were ring-driven, and assumed emission in vacuo. An idealized ring distribution is confined to a narrow range of speed, with a peak at v = v0 say, with ∂ f /∂ p > 0 at v < v0. The maximum growth rate corresponds to a circle centered on the origin with radius v slightly less than v0. The resulting ECME is at ω = Ωe/γ0, where γ0 is the Lorentz factor corresponding to v0. These earliest versions of ECME were not motivated by any specific application. The first suggested application of ECME was to Jupiter’s DAM [8]. Non-maser models for cyclotron emission in DAM already existed, and included the requirement that the emission be in the x mode of magneto-ionic theory above its cutoff [9]. The x mode cutoff is at 1 ω = Ω + (Ω2 + 4ω2)1/2 ≈ Ω + ω2/Ω , (2) x 2 h e e p i e p e where the approximation applies for ωp ≪ Ωe. This requires that the Doppler term kkvk in equation (1) be positive to give ω > ωx, so that x mode waves can propagate and escape. A parallel-driven form of ECME that satisfies this requirement was proposed [10], but it requires a distribution function for which there is no evidence. A distribution for which there is observational evidence is an upward directed loss-cone distribution which causes upward propagating waves to grow [1]. In this case, the most favorable resonance ellipse has its center displaced from the center of velocity space, implying upward emission in a small range of angles, θ, on the surface of a cone at a large angle to the magnetic field [11], providing a natural explanation for the seemingly bizarre angular distribution identified observationally [12].

3 Horseshoe-driven ECME

A horseshoe distribution may be defined as a ring distributionwith a one-sidedloss cone [13]. A 1D analytic model for the formation of a horseshoe distribution involves isotropic, low-energy electrons escaping from a scattering region at the top of a flux loop and accelerated towards a footpoint by a parallel electric field [14]. ECME could be driven either by the ring feature or by the loss-cone feature. A much large fraction of the electrons contribute to the ring-driven growth than to the loss-cone driven growth, so that ring-driven ECME should dominate, and this is assumed in horseshoe-driven ECME models applied to AKR. The growing waves are then at ω < Ωe < ωx, and so cannot escape in the presence of ambient plasma. Waves at ω < Ωe can propagate if the plasma frequency is so low that vacuum dispersion applies, and this is assumed to be the case in the auroral cavity. The emission escapes from the cavity after reflections off the walls of the cavity leads to upward ducting to a height where the density in the surrounding plasma has decreased sufficiently to satisfy ω > ωx [13].

Loss-cone driven ECME is also possible for a horseshoe distribution. It is usually neglected both because the growth rate is smaller than for the ring-driven ECME, and because there is no signature of it in AKR. However, there are two arguments that favor loss-cone driven ECME in DAM. One argument is that loss-cone driven ECME accounts naturally for the bizarre angular pattern of DAM. The other is that the intrinsically elliptical polarization of DAM [15] is consistent with the expected polarization of loss-cone driven ECME, but not with ring-driven ECME. This suggests that either DAM is not driven by a horseshoe distribution or, perhaps more likely, in this case the ring-driven growth is either suppressed or not observed.

4 Applications of ECME to the Sun and ﬂare stars

Solar spike bursts [16] are plausibly generated by precipitating electrons, and there is analogous bright emission from ﬂare stars, leading to the suggestion that the radiation is dueto ECME [17, 18]. It has also been suggested that ECME operates in several other astrophysical situations [19, 20]. The question arises as to what form of ECME is appropriate in these cases: ring-driven or loss-cone driven. It is plausible, but not necessarily widely accepted, that the acceleration in these cases is due to a parallel electric ﬁeld, and hence that a horseshoe distribution should result. Assuming this to be the case, loss-cone driven ECME could operate, but one would expect ring-driven ECME to dominate. The requirement of vacuum-like dispersion to allow ring-driven ECME to escape may seem implausible in a stellar environment, but it cannot easily be ruled out [14]. Another alternative is that ring-driven ECME produces z mode radiation which cannot escape directly but which can produce escaping radiation through wave-wave coalescence or mode coupling due to plasma inhomogeneity.

A critical discussion of these possibilities and their implications will be presented.

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