ECAMP 15

When galactic cosmic ray protons propagate through gas clouds in space they collide with atoms and molecules, transferring energy in the process. In order to calculate the photon flux produced as a result of these collisions knowledge of the rovibrationally resolved cross sections for excitation is required [ 1]. The most prevalent species in these environments is the hydrogen molecule. However, there is currently no data, experimental or theoretical, for electronically resolved excitation in p+H$_2$ collisions, let alone rovibrationally resolved, at keV energies where they are most significant. As a substitute, equivelocity scaling of the available data for electron collisions with H$_2$ [2] is currently used to estimate the p+H$_2$ cross sections [ 1]. Without experimental or theoretical data with which to compare there is no way to assess the accuracy of this simple approach and the effect this approximation has on astrophysical models.

We have developed a semi-classical coupled-channel approach to proton collisions with molecular hydrogen to solve this problem. Since the excitation cross section peaks between 10 to 1000 keV we can model the projectile motion as rectilinear, while still treating the electronic dynamics quantum mechanically. Orientationally averaged results are obtained by analytically integrating over the angular coordinates of the internuclear vector [3]. Using the configuration-interaction expansion method developed for the molecular convergent close-coupling (MCCC) approach to electron collisions [2], we are able to generate very accurate fixed-nuclei states for the hydrogen molecule. This enables us to determine electronically resolved cross sections for transitions into the various excited states of H$_2$. A sample of the results is shown in the figure for the dipole-allowed transition $X^1\Sigma_g \rightarrow B^1\Sigma_u$ (Lyman band). Comparison of the present calculations with the scaled electron cross sections reveals significant differences both around the centre of the peak (about 70 keV) and at lower energies where the scaled electron data falls to zero, demonstrating incorrect threshold behaviour. In contrast, our ab initio results for protons correctly incorporate the fact that at keV energies all bound excitation channels are open. This results in a non-negligible cross section for energies less than about 20 keV, compared to the equivelocity electron data.

The present results represent the first data for state-resolved electronic excitations in p+H$_2$ collisions and show that models relying on equivelocity electron cross sections will be underestimating the photon production rate from proton collisions. This is a significant step toward calculating the rovibrationally resolved cross sections for p+H$_2$ collisions that are required for astrophysical modelling.

Fig. 1. Excitation of the $B^1\Sigma_u$ state from the $X^1\Sigma_g$ ground state of H$_2$.

[ 1] M. Padovani et al. Astron. Astro. 682, A131 (2024)
[2] L. H. Scarlett et. al., Atom. Data Nucl. Data Tables 137, 101361 (2021)
[3] I. B. Abdurakhmanov et al. Phys. Rev. Lett. 111, 173201 (2013)