Self-consistent model of cosmic ray penetration into molecular clouds: Effect of energy losses

TL;DR

The paper extends the self-consistent CR penetration model to non-relativistic energies, showing that ionization losses in the envelope and wave excitation by both protons and electrons significantly alter the escaping CR flux into molecular clouds. Using a loss-influenced transport equation and a transformed-variable solution, it derives outer and inner diffusion-zone boundaries that depend on the loss scale $\lambda$ and the density inhomogeneity scale $\Lambda$, revealing that losses can dramatically shift the outer boundary for soft spectra while leaving hard spectra largely flux-conserving. It demonstrates that electron-driven turbulence can coexist with proton-driven turbulence, leading to complex, energy-dependent modulation, including a transition from Alfvenic to whistler turbulence at small $p$ and a secondary-particle contribution to the electron flux. The results imply strong suppression of low-energy CRs inside clouds, affecting ionization rates and line emissions, and suggest a unified modulation of disparate interstellar CR spectra inside molecular clouds.

Abstract

The theory of cosmic-ray (CR) penetration into dense molecular clouds developed recently for relativistic particles by Chernyshov et al. (2024) is extended to non-relativistic CRs. Interstellar CRs streaming into the clouds are able to resonantly excite MHD waves in diffuse cloud envelopes. This leads to the self-modulation, such that streaming particles are scattered at the self-generated waves. In contrast to relativistic CRs, transport of lower-energy particles in the envelopes is generally heavily affected by ionization losses; furthermore, both CR protons and electrons contribute to wave excitation. We show that these effects have profound impact on the self-modulation, and can dramatically reduce CR spectra even for clouds with moderate column densities of a few times $10^{21}$ cm$^{-2}$.

Figures (10)

• Figure 1: A sketch illustrating self-modulation of CRs penetrating a dense molecular cloud. Interstellar CRs, being isotropic in the absence of the cloud, stream along the local magnetic field lines in both directions. The net flux of CRs into the cloud is formed due to their attenuation in the central dense clump. This triggers the resonant streaming instability in the diffuse envelope surrounding the clump, generating the turbulent zones in outer envelope regions. The gas density $n$ in the envelope increases monotonically toward the dense clump; the inset highlights the fact that the outer and inner boundaries of the turbulent zone, $n_1(p)$ and $n_2(p)$, are functions of the particle momentum $p$.
• Figure 2: Characteristic spectra of interstellar protons adopted in the present paper: "model $\mathscr{L}$" given by Eq. (\ref{['spectrum_L']}) with $\alpha_0=-0.2$ (black solid line), and a power-law spectrum given by Eq. (\ref{['spectrum_PL']}) with $\alpha_0=2$ ("PL2", black dashed line). The gray solid line with hatching shows where inequality (\ref{['eq:n1_limits']}) is violated.
• Figure 3: Diffusion zone of CR protons, computed for interstellar spectrum $\mathscr{L}$ (top panel) and PL2 (bottom panel). For clarity, the results are plotted in the plane of proton kinetic energy $E$ (instead of momentum) and gas density $n$. The solid lines show the outer, $n_1(E)$, and inner, $n_2(E)$, boundaries merging at the excitation threshold $E_{\rm ex}$. The dashed lines indicate the (outer boundary) critical density $n_{\rm cr}(E)$ in the loss-free case, given by Eq. (\ref{['eq:n1_noloss']}). The results are for the cloud column density of $\mathcal{N} = 10^{22}$ cm$^{-2}$.
• Figure 4: Net flux $S$ (top panels) and spectrum $f$ (bottom panels) of protons at the outer boundary $n_1$ (where $f_1=f_0$, dashed lines) and inner boundary $n_2$ (solid lines) of the diffusion zone. The results are for the interstellar spectrum $\mathscr{L}$ (left) and PL2 (right), assuming the cloud column density of $\mathcal{N} = 10^{22}$ cm$^{-2}$.
• Figure 5: Modulation of CR electrons in the proton diffusion zone, computed from Eq. (\ref{['eq:f_integral_form']}). The interstellar electron spectrum, Eq. (\ref{['spectrum_e']}), is plotted by the gray line, the modulated spectra at the inner boundary of the diffusion zone are depicted by the black lines, representing model $\mathscr{L}$ (solid line) and PL2 (dashed line) of interstellar protons. The results are for the cloud column density of $\mathcal{N} = 10^{22}$ cm$^{-2}$.