ECME involves negative absorption near the fundamental or harmonics of the cyclotron frequency, û)s = eBlm. The gyroresonance condition is

where (O and 9, are the wave frequency and wave angle, respectively, ¿u is the refractive index, s = 0, ±1, ±2,... is rhe harmonic number, and v = /3c, a and y = (1 - /32)"1'2 are the speed, pitch angle and Lorentz factor, respectively, of the electron. For a given harmonic and a given wave (given s, co, n, 9), the resonance condition (6) defines a resonance ellipse in velocity (fa a) space, which determines those electrons that can resonate with the wave at i. Three examples of resonance ellipses are illustrated in Figure 3. The resonance curve labeled (a) is a vertical line at /fcosa = (ft) - s©B)//ifflcos0, which corresponds to an ellipse with its center far to the left so that the line is actually an arc of a large ellipse. The resonance curve labeled (b) illustrates an intermediate case where the center is closer to the shaded regions.

Figure 3: Examples of resonance ellipses in the nonrelativistic region of velocity space. Thermal electrons occupy the darkly shaded region, and nonthermal electrons occupy the lightly shaded region outside the loss cone a = oc0. Curves (a), (b) and (c) are discussed in the text.

The resonance curve labeled (c) in Figure 3 is the case of interest here. It corresponds to the 'semirelativistic' approximation y » 1 + \ fa in (6), in which case the ellipse reduces to a circle. It is essential that the relativistic correction, be retained even for electrons that can otherwise be treated nonrelativistically [Twiss 1958; Wu and Lee 1979]. The curve (c) is shown to lie within a loss cone, where the distribution is an increasing function ofp_l, and it is this feature which drives the maser. Also, curve (c) is chosen not to intersect the region where the thermal electrons are located; waves corresponding to resonance ellipses that intersect this region suffer strong thermal gyromagnetic absorption.

Figure 3: Examples of resonance ellipses in the nonrelativistic region of velocity space. Thermal electrons occupy the darkly shaded region, and nonthermal electrons occupy the lightly shaded region outside the loss cone a = oc0. Curves (a), (b) and (c) are discussed in the text.

The resonance curve labeled (c) in Figure 3 is the case of interest here. It corresponds to the 'semirelativistic' approximation y » 1 + \ fa in (6), in which case the ellipse reduces to a circle. It is essential that the relativistic correction, be retained even for electrons that can otherwise be treated nonrelativistically [Twiss 1958; Wu and Lee 1979]. The curve (c) is shown to lie within a loss cone, where the distribution is an increasing function ofp_l, and it is this feature which drives the maser. Also, curve (c) is chosen not to intersect the region where the thermal electrons are located; waves corresponding to resonance ellipses that intersect this region suffer strong thermal gyromagnetic absorption.

ECME occurs when the cyclotron absorption coefficient is negative. The growth rate for ECME is then identified as the magnitude of this absorption coefficient. An explicit expression for the absorption coefficient is [e.g., Hewitt, Melrose and Ronnmark 1982]

1 dpp'l i r(s, co, 6) = - 2c I dpp" | d cos kA(s, co, 6; /?, a)Af (7)

with p = yfimc and where A(s, co, &, ¡3, a) involves Bessel functions and a 5-function expressing the resonance condition (6). For an azimuthally symmetric distribution, the integral over azimuthal angle in (7) is trivial, and on carrying out the other integrals using the 5-function, the remaining integral may be rewritten as an integral around the resonance ellipse. The sign of the absorption coefficient is determined by the sign of S0)B 3 ^ fxp cos 0

where / = J(p, cos a) is the distribution function of the electrons. The two terms in (8) allow the possibility of a perpendicular-driven maser, or a parallel-driven maser, respectively [e.g., Melrose 1986]. The perpendicular-driven maser is favored for ECME.

The most favorable case for ECME is emission at s = 1 in the x-mode above the cutoff frequency at t»x, cf. (1). Notable features of this important case are:

1. Fundamental emission at co = coB can be in the x-mode only in a plasma with lOp « (0B

2. ECME above the cutoff (1) then requires co = cog > ct^/cOr, and hence that the electrons that drive the instability satisfy /? cos a cos 6 > OTp/cOg.

3. A region around Q = n/2 is forbidden for fundamental x-mode ECME.

4. The small range of allowed resonance ellipses, cf. Figure 2.3, for fundamental x-mode ECME imply that the emission is restricted to narrow ranges of frequency and angle. The angular restriction is to a thin surface of a hollow cone with half angle 6 ~ cos-'jS^, where /3cos a = /3q is the center of the resonance ellipse.

A loss-cone distribution has a deficiency of electrons with small pitch angles, as illustrated in Figure 2. This implies dfldpj_ > 0 within the loss cone, allowing perpendicular-driven ECME. Let a = <Xq define the edge of the loss cone. The growth rate is given approximately by

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