Investigating the signs of evolutionary characteristics in the energy spectrum of shock wave acceleration

TL;DR

This work confronts the expectation of a universal spectral index in shock-accelerated cosmic rays with high-precision element-dependent spectra from AMS-02 and DAMPE. Using a Spatially Dependent Propagation (SDP) model solved with DRAGON, they constrain the injection index $\gamma_2$ and find a robust positive correlation with both $A$ and $Z$ for $A/Z \approx 2$, consistent with fragmentation effects during acceleration. They further predict that Ni and Zn spectra should align with Fe, while their injection indices are slightly softer than Fe, and they highlight the need for new data from AMS-02, DAMPE, and HERD plus theoretical developments to explain the trend. The results have significant implications for understanding acceleration physics and the role of fragmentation in shaping cosmic-ray spectra.

Abstract

Under ideal conditions, the theory of shock acceleration for cosmic rays predicts that different elements should exhibit strictly identical spectral indices when accelerated to the same rigidity (R). However, recent high-precision measurements of elemental energy spectra have definitively established the existence of variations in spectral indices across different elements. This study constrains the spectral indices of cosmic-ray elements using AMS-02 and DAMPE observations within the Spatially Dependent Propagation (SDP) model. For elements with A/Z = 2, spectral indices shows significant positive correlations with both atomic number Z and mass number A, likely due to A or Z-dependent fragmentation cross-sections. Predictions indicate that the observed spectra of Ni and Zn will align with the Fe spectrum, while their injection spectra will exhibit slightly softer spectral indices compared to Fe. Future observations from AMS-02, DAMPE and HERD are expected to verify these findings, while theoretical models are needed to systematically explain this phenomenon.

Figures (3)

• Figure 1: Correlation between the injection spectral index $\gamma_2$ and the nucleon number ${\mathrm{A}}$ or proton number ${\mathrm{Z}}$ for various elements. The red and blue data points represent the fitting results derived from the AMS-02 and DAMPE data, respectively. The uncertainties in $\gamma_2$ were determined by constraining $\chi^2/\mathrm{d.o.f.} < 2$. The solid red and blue lines correspond to the linear fits to the AMS-02 and DAMPE data, respectively, with the shaded areas indicating the associated uncertainties of the fits. Top left: $\gamma_2$ as a function of ${\mathrm{A}}$ from AMS-02; Top right: $\gamma_2$ as a function of ${\mathrm{Z}}$ from AMS-02; Bottom left: $\gamma_2$ as a function of ${\mathrm{A}}$ from DAMPE; Bottom right: $\gamma_2$ as a function of ${\mathrm{Z}}$ from DAMPE.
• Figure 2: The injection energy spectra of various elements derived from the spatially dependent propagation (SDP) model fitting to AMS-02 data (left) and DAMPE data (right).
• Figure 3: The fluxes of various elements calculated by the model are compared with the experimental observations from AMS-02 and DAMPE. In the figure, red points represent AMS-02 observational results, blue points represent DAMPE observational results, the black solid line denotes the total flux calculated by the model, and the shaded black area indicates the error range obtained under the condition of $\chi^2/\mathrm{d.o.f.} < 2$. "Primary" and "Secondary" refer to the energy spectra of the primary and secondary components, respectively.