Cyclotron Resonant Interactions in Cosmic Particle Accelerators

Abstract

A review is given for cyclotron resonant interactions in space plasmas. After giving a simple formulation for the test particle approach, illustrative examples for resonant interactions are given. It is shown that for obliquely propagating whistler waves, not only fundamental cyclotron resonance, but also other resonances, such as transit-time resonance, anomalous cyclotron resonance, higher-harmonic cyclotron resonance, and even subharmonic resonance can come into play. A few recent topics of cyclotron resonant interactions, such as electron injection in shocks, cyclotron resonant heating of solar wind heavy ions, and relativistic modifications, are also reviewed.

Appendices

Appendix A

For the j species (j=i or e), the plasma frequency is defined as $\omega_{pj} = \sqrt{4\pi n_{0} e^{2}/m_{j}}$, and the cyclotron frequency as Ω _cj =eB ₀/m _j c. The Alfvén velocity V _A is defined as $B_{0}/\sqrt{4\pi n_{0} (m_{i}+m_{e})}$, which is chosen to be 10⁻⁴ c in the test particle calculation. The ion mass m _i is taken 1836 times m _e .

The wave electric field E _w ∝expi(kr−ωt) should satisfy the following matrix relation (see, e.g., Chap. 2 of Stix 1992),

$${\tilde{M}} {\mathbf{E}_w}=0, $$

(17)

with a matrix ${\tilde{M}}$ given as

$${\tilde{M}} \equiv \left \{ \begin{array}{c@{\quad}c@{\quad}c}P-N^2 \sin^2 \theta& 0 & N^2 \sin\theta\cos\theta\\0 & S-N^2 & -iD \\N^2 \sin\theta\cos\theta& iD & S-N^2 \cos^2 \theta \end{array} \right \},$$

(18)

where N≡kc/ω is a refractive index, and P, S, and D are defined as

(19)

Note that the condition $\det {\tilde{M}} =0$ gives the wave dispersion relation.

From the first and second column of the matrix equation (17), we have

(20)

Now switching from the complex formulation to the real formulation, we write the xyz components of the electric field, (E _w,x , E _w,y , E _w,z ), as

$$(E_{w,x}, E_{w,y}, E_{w,z})= (g_x \cos \varphi, g_y \sin\varphi, g_z \cos\varphi),$$

(21)

where the relative amplitudes of (g _x , g _y , g _z ) are determined by (20), and φ is the phase angle,

$$\varphi\equiv{\mathbf{k}} \cdot{\mathbf{r}} - \omega t + \varphi_0,$$

(22)

where φ ₀ is the initial phase angle, respectively. From the Faraday’s law, we have

$${\mathbf{B}_w} = N (\cos\theta, 0, \sin\theta) \times{\mathbf{E}_w},$$

(23)

from which the xyz components of the magnetic field, (B _w,x , B _w,y , B _w,z ), are obtained as

(24)

Appendix B

In a finite amplitude monochromatic electromagnetic wave propagating parallel to the background magnetic field B ₀ charged particles can be phase-space trapped around the resonant velocity (see e.g., Palmadesso and Schmidt 1971; Helliwell 1974; Schmitt 1976; Karpman 1974; Matsumoto 1979; Hoshino and Terasawa 1985; Kuramitsu and Krasnoselskikh 2005b). Here we follow the description in a recent article by Kuramitsu and Krasnoselskikh (2005b). Firstly, the gyrophase angle for an ion is defined as ϕ _p ≡tan⁻¹(v _z /v _y ), the wave phase angle as ϕ _w ≡kx−ωt+α _w (α _w : the initial value), and their difference as ψ≡ϕ _w −ϕ _p (Fig. 11). Next, with the velocity of ions in the wave rest frame (u _x ,u _⊥)=(v _x −ω/k,v _⊥) the cosine of the pitch angle is defined as μ≡u _x /|u|, where $|u| = (u_{x}^{2} + u_{\perp}^{2})^{1/2}$ is the constant of motion (namely, the ion energy is conserved in the wave rest frame). The second integral of the ion motion χ can be written as,

$$\chi= \frac{\kappa}{2} \biggl( \mu+ \frac{1}{\kappa} \biggr)^2 +b \bigl(1-\mu^2\bigr)^{1/2} \cos\psi,$$

(25)

where κ=ku/Ω _ci , and b the wave amplitude normalized by |B ₀|. With χ, the equation of motion is simplified as

$${\dot{\mu}} = -\frac{\partial\chi}{\partial\psi},\qquad {\dot{\psi}} = \frac{\partial\chi}{\partial\mu}.$$

(26)

Figure 12 shows the ion trajectories in the μ−ψ phase space with κ=3 and b=0.01, 0.1, 1, 10. When the wave amplitude is small ((a): b=0.01), ions are trapped around (μ,ψ)=(−1/κ,π), which is due to the linear cyclotron resonant interaction. As the wave amplitude b becomes larger ((c) and (d)), the trapping region also becomes larger and a new trapping region is born in a location far from the original point (−1/κ,π).

Fig. 11

Figure from Kuramitsu and Krasnoselskikh (2005b) showing the definition of angles, ϕ _p , ϕ _w , and ψ

Fig. 12

Figure from Kuramitsu and Krasnoselskikh (2005b) showing particle trajectories in the μ–ψ phase space with κ=3 when (a) b=0.01, (b) b=0.1, (c) b=1, and (d) b=10. One curve corresponds to one constant of motion χ