The Stochastic-Dissipative Störmer Problem-Trajectories and Radiation Patterns

TL;DR

The paper extends the classical Störmer problem by embedding a frictional (dissipative) and a stochastic force into the motion of a nonrelativistic charged particle in a pure dipole magnetic field, formulating the dynamics as a Lorentz-Langevin equation. It provides a dimensionless framework, analyzes energy loss and radiation emission, and implements a Milstein-based numerical scheme to study CSP, CDSP, and SDSP trajectories, along with their power spectra. Key findings include how dissipation suppresses radiation and alters PSD structure, how stochastic forcing reintroduces spectral complexity and can sustain motion against damping, and how escape rates depend on noise and friction. The results offer insights into magnetospheric dynamics and radiation patterns, suggesting potential connections to geomagnetic storms and observational diagnostics via PSD and energy-balance analyses, with broader applicability to stochastic-dissipative charged-particle dynamics in dipole fields.

Abstract

We consider a generalization of the classical nonrelativistic Störmer problem, describing the motion of charged particles in a purely magnetic dipole field, by taking into account the effects of the dissipation, assumed to be of friction type, proportional to the velocity of the particle, and of the presence of stochastic forces. In the presence of dissipative/stochastic effects, the motion of the particle in the magnetic dipole field can be described by a generalized Langevin type equation, which generalizes the standard Lorentz force equation. We perform a detailed numerical analysis of the dynamical behavior of the particles in a magnetic dipolar field in the presence of dissipative and stochastic forces, as well as of the electromagnetic radiation patterns emitted during the motion. The effects of the dissipation coefficient and of the stochastic force on the particle motion and on the emitted electromagnetic power are investigated, and thus a full description of the spectrum of the magnetic dipole type electromagnetic radiation and of the physical properties of the motion is also obtained. The power spectral density of the emitted electromagnetic power is also obtained for each case, and, for all considered Störmer type models, it shows the presence of peaks in the radiation spectrum, corresponding to certain intervals of the frequency.

Figures (12)

• Figure 1: Periodic motion, radiation power, and PSD in the Classical Störmer Problem (CSP) for $\vec{R}_0=\left(0.7,0.8,0\right)$, $\left|\vec{R}_0\right|=1.063$, and $\vec{V}_0=\left(0.10,0,0\right)$. For the numerical simulations the values $h=0.001$ and $L=350000$ have been adopted.
• Figure 2: Chaotic motion, radiation power and PSD in the Classical Störmer Problem (CSP) for $\vec{R}_0=\left(0.7,0.8,0\right)$, $\left|\vec{R}_0\right|=1.063$, and $\vec{V}_0=\left(0.01,0.10,0.10\right)$. For the numerical simulations the values $h=0.001$ and $L=180000$ have been adopted.
• Figure 3: Periodic motion in the Classical Dissipative Störmer Problem (CDSP): trajectory, radiation and PSD for $\vec{R}_0=\left(0.7,0.8,0\right)$, $\left|\vec{R}_0\right|=1.063$, and $\vec{V}_0=\left(0.10,0,0\right)$, $h=0.001$, $L=350000$ and $\Gamma=10^{-2}$, respectively. Due to increasing friction, the particle no longer covers the $xOy$ plane as in the CSP.
• Figure 4: Chaotic motion in the Classical Dissipative Störmer Problem (CDSP): trajectory, radiation and PSD for $\vec{R}_0=\left(0.7,0.8,0\right)$, $\left|\vec{R}_0\right|=1.063$, and $\vec{V}_0=\left(0.01,0.10,0.10\right)$, $h=0.001$, $L=180000$ and $\Gamma=10^{-2}$.
• Figure 5: Periodic motion in the Stochastic-Dissipative Störmer Problem (SDSP): trajectory, radiation and PSD for $\vec{R}_0=\left(0.7,0.8,0\right)$, $\left|\vec{R}_0\right|=1.063$, and $\vec{V}_0=\left(0.10,0,0\right)$, $h=0.001$, $L=150000$, for $\sigma _S= 10^{-6}$ and $\Gamma = 10^{-4}$.