Cold clouds as cosmic-ray detectors

The ionization fraction of atomic and molecular clouds is a key factor in interstellar gas evolution: it controls heating and cooling efficiencies, drives chemistry and molecule formation, and enables coupling to Galactic magnetic fields. While ultraviolet radiation from starlight can ionize gas, it is effective only in regions near massive stars and does not dominate in the bulk of Galactic gas. Instead, low-energy cosmic rays (energies up to about a GeV) govern ionization in the interstellar medium, yet direct observations from Earth probe mainly higher-energy CRs. The cosmic-ray ionization rate (CRIR), often inferred from secondary molecular species combined with chemical models, spans roughly 10^-17 to 10^-15 s^-1 in dense and diffuse Galactic clouds and up to about 10^-14 s^-1 in Galactic centers and some extragalactic environments. However, such determinations depend on model assumptions (e.g., densities, reaction rates, abundances, and line-of-sight structure). Other indirect approaches invoke thermal balance, deuterium fractionation, radio recombination lines, and synchrotron emission. Given that H2 dominates the mass in molecular clouds, the study proposes a more direct probe: using H2 rovibrational line emission excited by CR-produced secondary electrons in cold (T ≤ 100 K), quiescent (interstellar UV field strength ≤ 1) clouds where CR excitation should dominate. The goal is to derive conditions under which CR pumping governs line emission and to assess detectability with current instruments to constrain CRIR inside clouds.

Prior CRIR estimates have relied on abundances of species such as OH, OH+, H2O+, H3+, ArH+, combined with chemical modeling, yielding CRIRs from ~10^-17 to 10^-15 s^-1 in Galactic clouds and higher in centers and some galaxies. Additional indirect methods include analyses of dust and gas thermal balance, effects on deuterium fractionation, radio recombination lines, and synchrotron radiation. Regarding H2 excitation mechanisms, past work modeled excitation by energetic electrons and X-rays, UV fluorescence pumping via Lyman–Werner absorption and cascades, and formation pumping during H2 creation on dust grains, with several formation excitation distributions explored. Observationally, H2 rovibrational lines have been seen in shocked warm gas and bright photon-dominated regions where UV pumping is intense, but typical cold, quiescent molecular gas has been less explored for CR-driven excitation. This study builds on those theoretical frameworks to isolate CR excitation and compare it against UV and formation pumping under realistic ISM conditions.

An analytic framework is developed for H2 rovibrational line emission in cold molecular clouds where radiative decay is rapid compared to excitation, allowing separation of contributions from CR excitation, UV pumping, and formation pumping. For CR pumping, the surface brightness of a transition u→l is proportional to the product of the CR excitation rate (which scales with CRIR), the probability of populating level u by CR-produced electrons, the branching ratio to level l, the transition energy, and a dust attenuation factor that saturates in the optically thick limit. The total CR-driven line power sums to an average transition energy of approximately 0.486 eV times the total number of excitations per ionization (~5.8), yielding normalized line fractions f_ul^(cr) that identify the brightest lines. For UV pumping, assuming H2 formation–photodissociation steady state in cloud envelopes, the total line brightness is linked to the H2 formation rate coefficient, gas density, free-space photodissociation rate, pumping rate, and an effective transition energy ~1.82 eV; in the regime of x/n ≤ 0.05 cm^3 the brightness scales linearly with the UV field strength and is independent of density. Relative line fractions f_ul^(uv) depend on radiative cascades and are typically ~1%, lower than for CR pumping. For formation pumping, emission is separated into molecular cores (where CR destruction dominates) and outer envelopes (where UV photodissociation dominates), producing two terms for total brightness that scale with CRIR or UV field, respectively; three different formation excitation models bracket uncertainties in the excitation distribution. Critical conditions are derived for when CR pumping dominates over UV pumping, yielding thresholds in zeta (expressed as zeta_16 = zeta/10^-16 s^-1) that depend on column density and dust attenuation, and for when core vs envelope formation pumping dominates. The study also models the near-IR continuum background from dust-scattered starlight and small-grain/PAH emission, integrating over a spectral bin with resolving power R and applying dust attenuation, finding scattered light dominant at 2–3 microns. Detectability is assessed for ground-based observations where sky thermal emission dominates noise, using the VLT X-shooter instrument as a case study. Predicted line brightnesses for S(0) and Q(2) at zeta_16=1 and N_H2≈10^22 cm^-2 are combined with instrument parameters (slit, resolving power, exposure time, spatial sampling) to estimate signal-to-noise ratios per pixel and after integrating along the slit. The methods section further details: derivation of excitation probabilities P_u^(cr) and relative line fractions from published cross sections; temperature dependence of line fractions (T=30–300 K), showing para-H2 dominance at T<60 K; treatment of CRIR attenuation with depth using power-law zeta–column relations parameterized by spectral index alpha, showing interior zeta controls brightness; conditions where linearity in zeta breaks due to rising electron fraction and Coulomb losses; collisional de-excitation thresholds demonstrating radiative decay dominance for typical cold cloud densities; and negligible thermal excitation at T≤100 K. Exposure time calculator settings for X-shooter are documented.

• CR-excited H2 rovibrational lines provide a direct probe of the cosmic-ray ionization rate inside cold molecular clouds; line brightness is proportional to CRIR in the relevant regime.
• Brightest CR-pumped lines are the (1–0) O(2) at 2.63 µm, Q(2) at 2.41 µm, S(0) at 2.22 µm, and O(4) at 3.00 µm (para-H2), with large relative brightness fractions f_ul^(cr) ~ 13–45%, far exceeding the ~1% typical for UV-pumped cascades.
• Under typical conditions (N_H2 ≈ 10^22 cm^-2; dust optical depth at 2–3 µm ~0.9; mean interstellar UV field), CR pumping dominates line emission when zeta_16 exceeds critical values: for N_22=1 (attenuation factor g≈0.66), zeta_16,c ≈ 0.08 for O(2), and ≈ 0.2 for Q(2), S(0), and O(4). The general dominance condition scales as (I_cr/I_uv) ∝ (f_ul^(cr)/f_ul^(uv)) × zeta_16 × g/N_22.
• H2 formation pumping never dominates over CR or UV pumping in the explored regimes. In high-UV conditions (chi >> chi_crit), I_cr/I_f ≈ 2.2/(gamma Phi) × (f_ul^(cr)/f_ul^(f)) with f_ul^(cr)=15–45% versus f_ul^(f)=0.1–1%, ensuring CR dominance. At very low UV fields, formation pumping can exceed CR pumping but remains subdominant to UV pumping overall unless unlikely excitation distributions occur.
• Predictive detectability: For zeta_16 = 1 and N_22 = 1, predicted surface brightnesses are I ≈ 3.8×10^-8 and 4.2×10^-8 erg cm^-2 s^-1 sr^-1 for the 1–0 S(0) and Q(2) lines, respectively. With VLT X-shooter (R≈11,600, 0.4 arcsec slit), the estimated S/N per pixel in 1 hour is ~0.29 (S(0)) and ~0.14 (Q(2)); over 8 hours and integrating along the slit (55 pixels), S/N ≈ 6.1 and 2.9, respectively.
• Instruments with large fields of view (e.g., integral field units or narrowband imagers) can boost S/N substantially; an example geometry yields a field solid angle ~6800 times larger than an 11″ slit, implying ~82× S/N improvement. Space-based platforms (e.g., JWST) offer lower IR background and access to the O(2) line, which is ~4× brighter than S(0).
• Near-IR continuum at 2–3 µm is dominated by dust-scattered starlight, with emission from small grains and PAHs subdominant; continuum levels scale with column density and spectral resolving power, informing line-to-continuum contrast for detection.
• Temperature effects: At T<60 K, para-H2 lines dominate and remain weakly sensitive to temperature; as T increases, ortho-H2 lines (S(1), Q(1), Q(3), O(3), O(5)) gain strength due to population in (v,J)=(0,1), but total emitted power is conserved.
• CR attenuation with depth modifies the effective CRIR controlling line emission to the interior value; for relatively flat CR spectra the brightness becomes largely independent of spectral shape.
• Line brightness remains linear with CRIR until electron fractions become high (x_e ≳ 10^-4), where Coulomb losses quench excitation; such conditions correspond to extreme CRIR where gas is no longer molecular.

The work addresses the challenge of constraining low-energy cosmic rays—dominant ionizers in the ISM—by proposing H2 rovibrational line emission as a direct and robust probe of CRIR in cold molecular clouds. By deriving the regimes where CR pumping dominates over UV and formation pumping, and quantifying expected brightness and detectability, the study shows that feasible observations can yield CRIR measurements anchored in the main mass component (H2) rather than sensitive chemical tracers. Diagnostic line ratios can distinguish excitation mechanisms: for example, 1–0 S(1)/S(0) ≈ 2 for UV fluorescence, 3.5–5.6 for formation pumping, but ≈ 1 for CR excitation in cold clouds, enabling identification of CR-driven emission. Applying these diagnostics across multiple clouds and Galactic environments would allow mapping CRIR variations, comparing dense-cloud CRIR with diffuse-cloud estimates from ions like ArH+, OH+, and H2O+, and thereby constraining CR attenuation with depth and the low-energy CR spectrum. Such constraints would inform models of CR origins, propagation, and their coupling to interstellar gas and magnetic fields.

This study provides an analytic framework demonstrating that H2 rovibrational line emission from cold, quiescent molecular clouds can be dominated by cosmic-ray pumping and thus used to measure the cosmic-ray ionization rate. It identifies the brightest diagnostic lines, establishes critical conditions under which CR excitation exceeds UV and formation pumping, quantifies line-to-continuum contrasts, and shows that detections are feasible with current and near-future facilities, especially with instruments offering large fields of view or space-based IR capabilities. Future work should conduct targeted surveys of multiple clouds across Galactic environments to map CRIR variations, combine H2 line observations with diffuse-cloud chemical tracers to derive CR attenuation profiles and constrain the low-energy CR spectrum, and exploit facilities like JWST and integral field spectrographs for improved sensitivity and access to key lines such as 1–0 O(2).

The analysis is primarily analytic and focused on cold (T ≤ 50–60 K), quiescent molecular cloud interiors, assuming rapid radiative decay relative to excitation and negligible thermal collisional excitation. Line brightness linearity with CRIR holds only up to electron fractions below ~10^-4; at higher CRIR, Coulomb losses quench excitation and the gas may become largely atomic. While collisional de-excitation is shown to be negligible for typical cold-cloud densities (10^4–10^6 cm^-3), at much higher densities it may reduce line emission. The treatment of H2 formation pumping carries uncertainties in the excitation distribution, addressed via multiple models but still model-dependent. CRIR attenuation with depth is approximated via power-law relations; interior CRIR governs emitted brightness but detailed spectral shapes and geometries can introduce additional variability. Observationally, atmospheric transmission blocks some lines (e.g., O(2) from the ground), sky background dominates noise, and detectability depends on instrument field of view, calibration (e.g., flat-fielding), cloud extent and homogeneity, dust extinction, and line-of-sight structure.