# Cosmic Ray Spectrum Steepening in Supernova Remnants -- I. Loss-Free Self-Similar Solution

## TL;DR

This paper presents an exact self-similar solution showing how spherical expansion, magnetic field tilting, and shock deceleration cause the observed steepening of cosmic ray spectra in supernova remnants, challenging standard DSA predictions.

## Contribution

It introduces a new self-similar model that explains the steepening of cosmic ray spectra due to shock geometry and dynamics, differing from traditional diffusive shock acceleration theory.

## Key findings

- Steepening of CR spectra explained by shock expansion and magnetic field orientation.
- Exact self-similar solution derived for particle acceleration with arbitrary diffusivity energy dependence.
- Spectral steepening results from accumulation of lower-energy particles in expanding shock regions.

## Abstract

The direct measurements of cosmic rays (CRs), after correction for the propagation effects in the interstellar medium, indicate that their source spectra are likely to be significantly steeper than the canonical $E^{-2}$ spectrum predicted by the standard Diffusive Shock Acceleration (DSA) mechanism. The DSA has long been held responsible for the production of galactic CRs in supernova remnant (SNR) shocks. The $\gamma$-ray 'probes' of the acceleration spectra of CRs on-the-spot, inside of the SNRs, lead to the same conclusion. We show that the steep acceleration spectrum can be attributed to the $combination$ of (i) spherical expansion, (ii) tilting of the magnetic field along the shock surface and (iii) shock deceleration. Because of (i) and (ii), the DSA is efficient only on two ``polar caps'' of a spherical shock where the local magnetic field is within $\simeq45^{\circ}$ to its normal. The shock-produced spectrum observed edge-on steepens with the particle energy because the number of freshly accelerated particles with lower energies continually adds up to a growing acceleration region. We demonstrate the steepening effect by obtaining an exact self-similar solution for the particle acceleration at expanding shock surface with an arbitrary energy dependence of particle diffusivity $\kappa$. We show that its increase toward higher energy steepens the spectrum, which deeply contrasts with the standard DSA spectrum where $\kappa$ cancels out.