Escape dynamics through a continuously growing leak

We formulate a model that describes the escape dynamics in a leaky chaotic system in which the size of the leak depends on the number of the in-falling particles. The basic motivation of this work is the astrophysical process, which describes the planetary accretion. In order to study the dynamics generally, the standard map is investigated in two cases when the dynamics is fully hyperbolic and in the presence of Kolmogorov–Arnold–Moser islands. In addition to the numerical calculations, an analytic solution to the temporal behavior of the model is also derived. We show that in the early phase of the leak expansion, as long as there are enough particles in the system, the number of survivors deviates from the well-known exponential decay. Furthermore, the analytic solution returns the classical result in the limiting case when the number of particles does not affect the leak size.

Figure 1

(a) An example of a growing leak in dynamical astronomy. The plot shows initial conditions from the annulus around the planet's orbit. The end states of the particles are color coded as described in the text. Particles have been started on circular Keplerian orbit. The size of the planet and star are enlarged for better visualization. The triangular Lagrangian points ($L_4\text{and}{L}_5$) are also marked. (b)–(d) Examples for individual orbits corresponding to certain initial conditions in (a). Note that the end point of the light gray (blue) trajectory is outside the plotted region.

Figure 2

Visualization of the numerical setup. The invariant curves (blue), plotted for completeness, are related to different initial conditions than those show by dots representing the chaotic trajectories.

Figure 3

Escape dynamics in SM. Parameters of the simulation: $K=2.7, \gamma =1, m=1, C_p={10}^{-7}/{\left(2\pi \right)}^2, N_0={10}^6$, and $M_0=1000$. The leak reaches its final mass at $t\approx 6000.$ For more details see the text.

Figure 4

The mass growth of the leak $x\left(t\right)$ for different $\gamma \mathrm{s}$. For better visibility, the constants of integration ($\tau$) are chosen with $x_0=x_{\mathrm{PoI}}$ taken at 0 see Eq. (10). Parameter ${\kappa }_{\infty }$ is taken equal to $1/{\left(2\pi \right)}^2\approx 0.0253$.

Figure 5

(a) Survival probabilities for different parameters $K=5.19$ and 2.7 in SM. Parameters of the simulation: $\gamma =1, m=1, C_p={10}^{-7}/{\left(2\pi \right)}^2, N_0={10}^6$, and $M_0=1000$. The gray dashed ($K=5.19$) and dashed-dotted ($K=2.7$) lines represent the analytic formula (B4) with ${\kappa }_{\infty }=0.1/{\left(2\pi \right)}^2\approx 0.00253, x_0={10}^{-3}$ and ${\kappa }_{\infty }=0.00285, x_0={10}^{-3}$, respectively. (b) The difference between the numerical simulation and the analytic formula for $K=5.19$.

Figure 6

(a) The growth of the leak's mass and the decay of particles for $K=2.7$. The other parameters are $\gamma =4/3, m=1, C_p=0.2845, N_0={10}^6$, and $M_0=5631$. The dashed lines illustrate the analytic solutions with ${\kappa }_{\infty }=0.1/{\left(2\pi \right)}^2\approx 0.002533, x_0=0.0055$. (b) $S-C$ curve shows the difference in leak mass.

Figure 7

(a) The leak mass vs time for different mass distributions of particles each of them with $\text{mean}=1$ and $\text{std}=0.667$. Squares represent equal masses $m=1.$ (b) The differences between the leak's masses are significant only for the first 200 iterations. $K=2.7.$

Figure 8

(a) Number of nonescaped particles and leak size and growth vs time for $K=2.7,\gamma =1/2,m_i=1.$ (b) Magnification during growing process.

Figure 9

The relative error of the approximation Eq. (C1) after a single iteration as a function of the mass ratio $x$. The inset helps to understand the formula of the relative error. The black dots represent the two successive terms. The black solid line and the (blue) dashed line display the continuous solution $t\left(x\right)$ and its tangent curve, respectively. The parameters are the same as in Fig. 11.

Figure 10

The first (red), the fifth (green), and the ninth (blue) deciles of the series of distributions in the case of $\gamma =4/3.$ To make the distinctions of the three curves easier, we cut off their first parts, $t<900$. The black curve shows the particle number $y\left(t\right).$ The insets (a) and (b) show two distributions corresponding to the iterations indicated by the two vertical black lines.

Figure 11

(a) The first (red), the fifth (green), and the ninth (blue) deciles of the three calculated series of distributions The black curves show the particle number ratio $\left[\right(1-x\left)\right]$ calculated in the Sec. 3a. (b) Standard deviations of the distributions for different $\gamma \mathrm{s}$.