Effect of Magnetised Discontinuity on Diffusive Shock Acceleration

Abstract

We investigate the impact of magnetic fields and diffusion mechanisms on the energy spectra of particles accelerated via diffusive shock acceleration. We analyse magnetised shock jump conditions and demonstrate how magnetisation and angular dependence modify upstream and downstream velocities, which enter the transport equation within a Monte Carlo simulation framework. We consider constant, momentum-dependent, and pitch-angle-dependent diffusion coefficients to assess their influence on particle acceleration. Our results show that magnetic fields enhance particle confinement and facilitate more efficient energy gain. In the absence of magnetisation, particle spectra tend to be steeper due to rapid escape and weaker scattering effects, whereas magnetised shocks systematically produce flatter spectra across all diffusion models. Among them, pitch-angle-dependent diffusion leads to the strongest spectral flattening, underscoring its role in sustaining extended acceleration. It is also seen that an increased upstream pressure, associated with enhanced magnetic turbulence, broadens the spectral range by improving particle scattering efficiency and enabling multiple shock crossings. As the shock inclination angle increases, the velocity contrast between upstream and downstream regions diminishes, modulating the spatial extent of the acceleration zone. Notably, pitch-angle-dependent diffusion remains robust under varying shock conditions, ensuring sustained acceleration.

Figures (7)

• Figure 1: Schematic diagram illustrating a moving shock discontinuity in both the NI-frame (primed) and the HT-frame (unprimed). In both reference frames, the shock front propagates from left to right. The matter velocities on either side of the shock, denoted as $v_a$ and $v_b$ (with primed/unprimed variations), are oriented at angles $\theta_a$ (incident angle, primed/unprimed) and $\theta_b$ (reflected angle, primed/unprimed) relative to the shock normal. In the HT-frame, the velocity and magnetic field vectors become collinear, simplifying the analysis of the problem.
• Figure 2: The figure illustrates the dependence of upstream and downstream velocities on the energy density ($\epsilon$) for an unmagnetized case, with the same analysis in the presence of a magnetic field for comparison. Panel (a) represents the case of an unmagnetized shock, while panel (b) corresponds to a magnetised shock with a zero-incident angle ($\theta_a \approx 0$) in both cases. The solid line denotes the upstream velocity, whereas the line with crosses indicates the downstream velocity. The overall trends exhibit similar behaviour across both cases; however, in the unmagnetized scenario, both upstream and downstream velocities are slightly lower for upstream pressures of $p_{a1} = 0.15$ MeV/fm$^3$ and $p_{a2} = 0.25$ MeV/fm$^3$ compared to the magnetized case. The presence of a magnetic field enhances particle scattering and modifies the shock dynamics, leading to marginally higher post-shock velocities. As the upstream pressure increases significantly, the upstream and downstream velocities progressively converge, approaching comparable magnitudes. This trend suggests that the shock undergoes a transition at sufficiently high pressures where magnetic effects become increasingly prominent, reducing the velocity contrast across the discontinuity.
• Figure 3: The figure illustrates the variation of upstream and downstream velocities as a function of energy density ($\epsilon$) for a magnetised shock. The solid line represents the upstream velocity, while the line marked with crosses depicts the downstream velocity. Figure (a) presents the velocity variations for different incident angles with an upstream pressure of $p_{a1} = 0.15\, MeV/fm^3$, whereas Figure (b) corresponds to an upstream pressure of $p_{a2} = 0.25\, MeV/fm^3$. The downstream pressure can be selected arbitrarily, provided it remains greater than the upstream pressure to ensure the occurrence of a strong shock process. The nature of the plots indicates that as the incident angle increases from lower to higher values, the upstream and downstream velocities progressively converge. This behaviour is consistent with the theoretical expectation that these velocities converge as the incident angle increases.
• Figure 4: Energy spectrum with no magnetic fields under three different diffusion coefficients with two different upstream pressure values: (a) $p_{a1} = 0.15 Mev/fm^3$; (b) $p_{a2} = 0.25 Mev/fm^3$.
• Figure 5: The energy spectrum in the presence of a magnetic field at zero incident angle is analyzed for three distinct diffusion coefficients under two different upstream pressure conditions: (a) $p_{a1} = 0.15$ MeV/fm$^3$ and (b) $p_{a2} = 0.25$ MeV/fm$^3$.