Astronomy & Astrophysics

All Figures

$\begin{figure} \par\includegraphics[width=8cm,clip]{2283f1.eps}\end{figure}$ Figure 1: Resonant particle (solid lines) trajectories are shown a) in x-v_x space with electrostatic potential (dotted line); b) in the perpendicular plane; and c) the time evolution of $v_{\perp }$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{2283f2.eps}\end{figure}$ Figure 2: a) Time evolution of $\psi$ . Regardless of the initial values, $\psi$ tends to $\pi /2$ (dotted line) as time passes. b) Time evolution of v_y (solid lines) and E_y (dotted line) are shown.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2283f3.eps}\end{figure}$ Figure 3: Same as Fig. 1, except k=0.5, v=1, and $v_x=v_{\rm r}=-1$ . The electrostatic field "pushes'' the particle in the parallel direction along the trajectory.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{2283f4.eps} \end{figure}$ Figure 4: Definition of the phase relation in the perpendicular plane is shown.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{2283f5.eps} \end{figure}$ Figure 5: Particle trajectories are shown in the $\mu -\psi$ phase space with $\kappa =3$ when a) b=0.01; b) b=0.1; c) b=1; and d) b=10. One line corresponds to one constant of motion H (Eq. (10)).
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In the text

$\begin{figure} \par\includegraphics[width=7.9cm,clip]{2283f6.eps} % \end{figure}$ Figure 6: Same as Fig. 5 except the $\kappa =0.3$ . Particles cannot satisfy the linear cyclotron resonance condition Eq. (12).
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{2283f7.eps} % \end{figure}$ Figure 7: Initial pitch angle and phase dependence of gyroresonant surfing acceleration is shown in the presence of a) R^-; b) L⁺; c) L^-; and d) R⁺, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=7.9cm,clip]{2283f8.eps}\end{figure}$ Figure 8: Trajectories of upstream particles in the same phase space shown using the same parameters as Fig. 7. Note that these are projections of trajectories from higher dimensional phase space.
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In the text

$\begin{figure} \par\includegraphics[width=11.1cm,clip]{2283f9.eps}\end{figure}$ Figure 9: Dependence of acceleration efficiency on wave number. a) Ensemble averages and b) maximum values of perpendicular velocity after interaction with the profile against the absolute value of wave number. In c) and d) the same values as a) and b) are plotted against the resonant velocity and absolute value in the S-system. Except for the abscissa of c), all panels are plotted in the logarithmic scale. In all panels circles, squares, triangles and inverse triangles represent the presence of R^-, L⁺, L^- and R⁺, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{2283f10.eps}\end{figure}$ Figure 10: Particle trajectories in the presence of L- (solid lines) and L⁺ (dashed lines) using the same $v_{\rm r}$ and the same |k| but different signs a) in $v_{\perp }-x$ space; b) in v_x-x space; and c) v_y-v_z space.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2283f11.eps}\end{figure}$ Figure 11: a) Difference of perpendicular velocity; and b) parallel position after the interaction of particles in the same phase space as Fig. 7. c) The distribution of particles in velocity space are shown with the resonant velocity (dotted line).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{2283f12.eps}\end{figure}$ Figure 12: Particle trajectories are shown a) in $x-v_{\perp }$ space; b) in $x-\psi$ space; and c) in velocity space. Parameters are the same as Fig. 11.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{2283f13.eps}\end{figure}$ Figure 13: a) Dependence of $\langle\Delta v_{\perp}\rangle$ ; and b) the maximum $\Delta v_{\perp }$ on the potential width and the wave amplitude is shown in logarithmic scale.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{2283f1.eps}\end{figure}$	Figure 1: Resonant particle (solid lines) trajectories are shown a) in x-v_x space with electrostatic potential (dotted line); b) in the perpendicular plane; and c) the time evolution of $v_{\perp }$ .
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$\begin{figure} \par\includegraphics[width=8.2cm,clip]{2283f2.eps}\end{figure}$	Figure 2: a) Time evolution of $\psi$ . Regardless of the initial values, $\psi$ tends to $\pi /2$ (dotted line) as time passes. b) Time evolution of v_y (solid lines) and E_y (dotted line) are shown.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2283f3.eps}\end{figure}$	Figure 3: Same as Fig. 1, except k=0.5, v=1, and $v_x=v_{\rm r}=-1$ . The electrostatic field "pushes'' the particle in the parallel direction along the trajectory.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{2283f4.eps} \end{figure}$	Figure 4: Definition of the phase relation in the perpendicular plane is shown.
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$\begin{figure} \par\includegraphics[width=8cm,clip]{2283f5.eps} \end{figure}$	Figure 5: Particle trajectories are shown in the $\mu -\psi$ phase space with $\kappa =3$ when a) b=0.01; b) b=0.1; c) b=1; and d) b=10. One line corresponds to one constant of motion H (Eq. (10)).
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$\begin{figure} \par\includegraphics[width=7.9cm,clip]{2283f6.eps} % \end{figure}$	Figure 6: Same as Fig. 5 except the $\kappa =0.3$ . Particles cannot satisfy the linear cyclotron resonance condition Eq. (12).
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{2283f7.eps} % \end{figure}$	Figure 7: Initial pitch angle and phase dependence of gyroresonant surfing acceleration is shown in the presence of a) R^-; b) L⁺; c) L^-; and d) R⁺, respectively.
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$\begin{figure} \par\includegraphics[width=7.9cm,clip]{2283f8.eps}\end{figure}$	Figure 8: Trajectories of upstream particles in the same phase space shown using the same parameters as Fig. 7. Note that these are projections of trajectories from higher dimensional phase space.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{2283f10.eps}\end{figure}$	Figure 10: Particle trajectories in the presence of L- (solid lines) and L⁺ (dashed lines) using the same $v_{\rm r}$ and the same \|k\| but different signs a) in $v_{\perp }-x$ space; b) in v_x-x space; and c) v_y-v_z space.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2283f11.eps}\end{figure}$	Figure 11: a) Difference of perpendicular velocity; and b) parallel position after the interaction of particles in the same phase space as Fig. 7. c) The distribution of particles in velocity space are shown with the resonant velocity (dotted line).
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$\begin{figure} \par\includegraphics[width=8cm,clip]{2283f12.eps}\end{figure}$	Figure 12: Particle trajectories are shown a) in $x-v_{\perp }$ space; b) in $x-\psi$ space; and c) in velocity space. Parameters are the same as Fig. 11.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{2283f13.eps}\end{figure}$	Figure 13: a) Dependence of $\langle\Delta v_{\perp}\rangle$ ; and b) the maximum $\Delta v_{\perp }$ on the potential width and the wave amplitude is shown in logarithmic scale.
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