tokens: light-to-Alfvén speed ratio is fixed at c / v Ai = 10 4 . The top panel shows the normalized real part of the wave frequency, and the bottom panel shows the normalized imaginary part of the wave frequency as functions of the normalized wavenumber k . The vertical lines in the bottom panel mark the positions of kd e = 1 for the given mass ratios. In Fig. 3 , we show the dispersion relations for the FM/W wave in quasi-parallel direction of propagation. This mode shows the typical quadratic whistler-mode behaviour at . The FM/W wave is dominated by electron dynamics unless kd i ≲ 1. With decreasing m i / m e , ω r /Ω i decreases while | γ |/Ω i increases. We attribute this behaviour to increasing cyclotron damping by electrons, which is, for the quasi-parallel FM/W wave, most efficient when n = −1 in equation ( 17 ). Thermal electrons with v ∥ ≈ − w e thus efficiently resonate with the quasi-parallel FM/W mode when in our normalization. In the electron–positron case with m i / m e = 1, a significant number of electrons resonate with the FM/W mode at ω r /Ω i = ω r /|Ω e | ≲ 1 and at kd i = kd e ≲ 1 (see fig. 2 by Gary Karimabadi 2009 ). Like in the case of the quasi-parallel A/IC wave, the resonant cut-off of the dispersion relation for ω r → |Ω e | is also present in the cold-plasma limit (Stix 1992 ). The quasi-parallel FM/W mode reaches the wavenumber range kd e ≥ 1 with | γ | ω r for all mass ratios shown in Fig. 3 . Figure 3. Dispersion relations of the FM/W wave in quasi-parallel propagation for different values of m i / m e . The light-to-Alfvén speed ratio is fixed at c / v Ai = 10 4 . The plot follows the same format as Fig. 2 . In Fig. 4 , we show the dispersion relations for the FM/W wave in quasi-perpendicular direction of propagation. The dispersion relation does not depend significantly on m i / m e for m i / m e ≳ 10. The quasi-perpendicular FM/W wave reaches the ion-Bernstein regime at ω r /Ω i ≈ 1. In the perpendicular limit ( θ → 90°), the susceptibilities for both ions and electrons include sums over all n of contributions that are each proportional to 1/( ω r − n Ω j ), leading to this resonant behaviour. As long as m i / m e ≫ 1, electron harmonics do not interfere with the ion harmonics since all contributions to decrease with increasing n . For m i / m e = 2, however, the second-order ion resonance makes a contribution to that is comparable with the first-order electron resonance. The quasi-perpendicular FM/W mode reaches the wavenumber range kd e ≥ 1 with | γ | ω r for all shown parameter combinations with m i / m e ≤ 125. Figure 4. Dispersion relations of the FM/W wave in quasi-perpendicular propagation for different values of m i / m e . The light-to-Alfvén speed ratio is fixed at c / v Ai = 10 4 . The plot follows the same format as Fig. 2 . 2.2 Dependence on light-to-Alfvén speed ratio We now investigate the dependence of the dispersion relation on c / v Ai . For this study, we fix the mass ratio at m i / m e = 100
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