Relativistic chaotic scattering: Unveiling scaling laws for trapped trajectories

In this paper we study different types of phase space structures which appear in the context of relativistic chaotic scattering. By using the relativistic version of the Hénon-Heiles Hamiltonian, we numerically study the topology of different kind of exit basins and compare it with the case of low velocities in which the Newtonian version of the system is valid. Specifically, we numerically study the escapes in the phase space, in the energy plane, and in the $\beta$ plane, which richly characterize the dynamics of the system. In all cases, fractal structures are present, and the escaping dynamics is characterized. In every case a scaling law is numerically obtained in which the percentage of the trapped trajectories as a function of the relativistic parameter $\beta$ and the energy is obtained. Our work could be useful in the context of charged particles which eventually can be trapped in the magnetosphere, where the analysis of these structures can be relevant.

Figure 1

Isopotential curves for the Hénon-Heiles potential: they are closed for energies below the Newtonian threshold energy escape $E_e=1/6$. Three different exits for energy values above $E_e=1/6$ are shown. The dotted lines represent the unstable Lyapunov orbits $L_i$.

Figure 2

Comparison between two trajectories: in black, we show the relativistic trajectory, and, in pale gray, the Newtonian one. Trajectories correspond to particles shot from the same initial condition $(x_0,y_0,\dot{x_0},\dot{y_0})=(0,0,vcos\left(0.3\pi \right),vsin\left(0.3\pi \right))$ with velocity $v=0.583$. The effect of $\gamma$ in Eq. (6), even for a very low velocity ($\beta =0.05$), yields a significantly different result. Despite the consideration of the Galilean corrections resulting in a trajectory going out of the scattering region by Exit 1 (see Fig. 1), the same initial condition contemplating the relativistic corrections yields a trajectory that leaves the scattering region after a finite time by Exit 2.

Figure 3

Exit basins of the Hénon-Heiles Hamiltonian in the coordinates plane $(x,y)$ and the phase space $(y,p_y)$ for different values of $\beta$. The exit basins show the regions of initial conditions that lead to different escape outcomes or bounded trajectories in the Hénon-Heiles system. The colors indicate the exit number [red (dark gray) for 1, yellow (white) for 2, blue (black) for 3] or bounded motion [green (light gray)]. Initial conditions in the coordinates plane are chosen to maintain the symmetry of the problem, while in the phase space they satisfy $x\left(0\right)=0$ and $p_x$ is obtained from the energy $E_N=0.17$. The top row corresponds to the Newtonian case, while the central and bottom rows correspond to the relativistic cases ($\beta =0.2$ and $\beta =0.5$, respectively).

Figure 4

Exit basins of the Hénon-Heiles Hamiltonian in the coordinate plane $(x,y)$ and the phase space $(y,p_y)$ for different values of $\beta$. The exit basins show the regions of initial conditions leading to different escape outcomes or bounded trajectories in the Hénon-Heiles system. The colors indicate the exit number [red (dark gray) for 1, yellow (white) for 2, blue (black) for 3] or bounded motion [green (light gray)]. Initial conditions in the coordinate plane are chosen to maintain the symmetry of the problem, while in the phase space they satisfy $x\left(0\right)=0$, where $p_x$ is obtained from the energy $E_N=0.25$. The top row corresponds to the Newtonian case, while the central and bottom rows correspond to the relativistic cases ($\beta =0.2$ and $\beta =0.5$, respectively).

Figure 5

Left panel: $E_N=0.17$. Plot of bounded region size $F$ vs $\beta$ [in blue (black)]. The linear fitting is denoted in red (dark gray), $F=0.1-0.02\beta$, while the quadratic fit, $F=0.03+0.25\beta -0.24{\beta }^2$, has been plotted in light blue (light gray). Bounded region for the Newtonian Hénon-Heiles is represented in green (light gray), where the particles are permanently trapped. Right panel: $(y,H)$ planes. Right above: $\beta =0.05$. Right below: $\beta =0.95$. In both cases, we can clearly see the effects of the high velocity for which the particles escape faster insofar $\beta$ is increasing.

Figure 6

Plot of the bounded region $F$ (assuming a total size of 1) vs the relativistic energy $H$ for $\beta =0.2$ [in blue (black)]. The linear fit, $F=0.13\text{–}0.17H$, is plotted in red (dark gray). The picture on the right side (up) represents the situation for the Newtonian Hénon-Heiles system. In green (light gray), we plot the bounded trajectories. The $(E_N,y)$ planes below show different situations of the relativistic regime for $\beta =0.2$ and energies $E_N=0.16$, $E_N=0.2$, and $E_N=0.25$ from left to right, respectively. As in Fig. 5, the horizontal dash line in green (light gray) color denotes the bounded region $F$ for the Newtonian Hénon-Heiles system in which the particles are permanently trapped.

Figure 7

Plot of the $(y,\beta )$ plane for $E_N=0.05,0.16,0.17,0.25$. Left panes: colors indicate the exit number [red (dark gray) 1, yellow (white) 2, or blue (black) 3] or bounded motion [green (light gray)], as in the previous figures. Right panels: colors indicate the value of the chaos indicator $OFLI2$. Regular regions (fast escape) [blue (light gray)], chaotic trajectories [red (dark gray)], periodic trajectories [green (white)], and quasiperiodic [yellowish (pale gray)]. The color code is as described in the captions of the previous figures. Notice that for $E_N=0.25$ there are escapes for all values of $\beta$ which show that KAM islands are destroyed.

Figure 8

Quadratic fits for energy values from $E_N=0$ to the end of KAM region. The escape Newtonian energy, $E_e=1/6$, is denoted by a dashed green (pale gray) vertical line. The quadratic fit $F=0.49+0.63E_N-15.92E_N^2$ changes very suddenly to $F=1.84-17.61E_N+42.14E_N^2$ when the particles have enough energy to escape ($E_N>E_e=1/6$). The jump at $E_N=1/6$ is due to the trajectories start to escape from the scattering region, and, therefore, the system becomes open presenting three different channels for which the particles can escape. The second dashed green (light gray) vertical line (located at $E_N\cong 0.21$) denotes the end of the KAM islands. There all trajectories are escaping since it is the beginning of the hyperbolic regime and, therefore, $F\to 0$.