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Excitation of molecular hydrogen by cosmic-ray protons

Marco Padovani, Daniele Galli, Corey T. Plowman, Liam H. Scarlett, Mark C. Zammit, Igor Bray, Dmitry V. Fursa

TL;DR

This work demonstrates that primary cosmic-ray protons can drive H$_2$ excitation as effectively as, or more than, secondary electrons in molecular clouds. By applying a semi-classical molecular convergent close-coupling framework to proton-H$_2$ collisions, the authors generate rovibrationally resolved cross sections for electronic excitations to the $B$ and $C$ states and show protons can dominate excitation rates under typical interstellar conditions. The study also develops analytic approximations for the secondary-electron spectrum and shows that proton-induced excitation increases the total H$_2$ excitation relative to ionisation, with implications for interpreting near-infrared and ultraviolet H$_2$ emission and for constraining the cosmic-ray ionisation rate. Overall, the results provide a robust cross-section database and a practical secondary-electron parameterisation that can be readily implemented in astrochemical and radiative-transfer models, refining our understanding of energy balance in the interstellar medium.

Abstract

Low-energy cosmic rays ($E\lesssim 1$ GeV) are responsible for the ionisation and heating of molecular clouds. While the role of supra-thermal electrons produced in the ionisation process in inducing excitation of the ambient gas (mostly molecular hydrogen) has been studied in detail, the role of primary cosmic-ray nuclei (protons and heavier nuclei) has been generally neglected. Here, we introduce, for the first time, cross sections for proton impact on H$_2$, calculated using the semi-classical implementation of the molecular convergent close-coupling method. Our findings show that proton-induced H$_2$ excitation is comparable in magnitude to that caused by electrons. We discuss the possible implications on the estimate of the cosmic-ray ionisation rate from observations in the near-infrared domain and on the cosmic-ray-induced H$_2$ ultraviolet luminescence. We also derive a new approximated analytical parameterisation of the spectrum of secondary electrons that can be easily incorporated in numerical codes.

Excitation of molecular hydrogen by cosmic-ray protons

TL;DR

This work demonstrates that primary cosmic-ray protons can drive H excitation as effectively as, or more than, secondary electrons in molecular clouds. By applying a semi-classical molecular convergent close-coupling framework to proton-H collisions, the authors generate rovibrationally resolved cross sections for electronic excitations to the and states and show protons can dominate excitation rates under typical interstellar conditions. The study also develops analytic approximations for the secondary-electron spectrum and shows that proton-induced excitation increases the total H excitation relative to ionisation, with implications for interpreting near-infrared and ultraviolet H emission and for constraining the cosmic-ray ionisation rate. Overall, the results provide a robust cross-section database and a practical secondary-electron parameterisation that can be readily implemented in astrochemical and radiative-transfer models, refining our understanding of energy balance in the interstellar medium.

Abstract

Low-energy cosmic rays ( GeV) are responsible for the ionisation and heating of molecular clouds. While the role of supra-thermal electrons produced in the ionisation process in inducing excitation of the ambient gas (mostly molecular hydrogen) has been studied in detail, the role of primary cosmic-ray nuclei (protons and heavier nuclei) has been generally neglected. Here, we introduce, for the first time, cross sections for proton impact on H, calculated using the semi-classical implementation of the molecular convergent close-coupling method. Our findings show that proton-induced H excitation is comparable in magnitude to that caused by electrons. We discuss the possible implications on the estimate of the cosmic-ray ionisation rate from observations in the near-infrared domain and on the cosmic-ray-induced H ultraviolet luminescence. We also derive a new approximated analytical parameterisation of the spectrum of secondary electrons that can be easily incorporated in numerical codes.

Paper Structure

This paper contains 10 sections, 21 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. The semi-classical MCCC method
  4. Proton-impact excitation of H2
  5. Results
  6. Spectrum of secondary electrons
  7. On-the-spot approximation
  8. Steady-state approximation
  9. Proton-impact ionisation of atomic hydrogen
  10. Discussion and Conclusions

Figures (5)

  • Figure 1: Ratio between proton-impact H2 excitation rates ($\zeta^{\rm exc}_{p,{\rm H}_2}$) summed over the initial and final rotational states and the cosmic-ray ionisation rate ($\zeta^{\rm ion}_{{\rm H}_2}$) as a function of the upper vibrational level, $v_u$. Results are shown for transitions from the lowest vibrational level of the ground state $X(v_l=0)$ to the excited electronic states $B$ (solid blue circles) and $C$ (solid orange triangles). Empty black symbols show a subset of previous estimatesCecchi-PestelliniAiello1992.
  • Figure 2: Cumulative integral of the excitation rates of H2 by cosmic-ray protons and secondary electrons, $\zeta^{\rm exc}_{p,\ce{H2}}(E)$ and $\zeta^{\rm exc}_{{\rm sec},\ce{H2}}(\varepsilon)$, and the corresponding excitation cross sections ($\sigma^{\rm exc}_{p,\ce{H2}}$ and $\sigma^{\rm exc}_{e,\ce{H2}}$, purple short-dashed lines) for model $\mathscr{L}$ and $\mathscr{H}$ (blue and green lines, respectively) computed at $N(\ce{H2})=10^{22}$ cm$^{-2}$.
  • Figure 3: Auxiliary functions for the secondary electron spectrum: $\varphi_{\ce{H2}}(\varepsilon)$ (Eq. \ref{['eq:phi']}), $\varepsilon\varphi_{\ce{H2}}(\varepsilon)$ (Eq. \ref{['eq:jsec_osa']}), and $\Phi_{\ce{H2}}(\varepsilon)$ (Eqs. \ref{['eq:jsec_essa']}, \ref{['eq:Phi']}). The dashed curves show the same quantities according to the BEA (Eqs. \ref{['eq:phibeb']}, \ref{['eq:PHIbeb']}).
  • Figure 4: Flux of secondary electrons computed exactly by solving the balance equationIvlev+2021 (solid blue line), using the on-the-spot (OS) approximation (dotted red line, Eq. \ref{['eq:jsec_osa']}), and the steady-state (SS) approximation (dashed red line, Eq. \ref{['eq:jsec_essa']}). The input cosmic-ray proton spectrum is the model $\mathscr{L}$ at $N(\ce{H2})=10^{22}$ cm$^{-2}$.
  • Figure 5: Ratio between the cosmic-ray ionisation rate of H2 (primary plus secondary contributions, $\zeta_2$) and of H (primary plus secondary contributions, $\zeta_1$) as a function of the total hydrogen column density, $N_{\ce{H}}$, obtained from the exact solution for the model $\mathscr{L}$ and $\mathscr{H}$ (solid blue and long-dashed green line, respectively) and from the SS approximation (short-dashed black line, Eq. \ref{['eq:jsec_essa']}) compared to the canonical valueGlassgoldLanger1974 (dotted orange line).