Two-Region Ionisation Model

• Two-Region Ionisation Model is an analytical framework that divides astrophysical plasmas into distinct regimes based on dominant ionisation processes.
• It employs observational proxies like CO depletion and deuteration fractions to map spatial variations in cosmic-ray ionisation rates.
• The model enhances diagnostic precision by linking microphysical processes with high-resolution data, guiding improvements in astrochemical simulations.

A Two-Region Ionisation Model identifies and exploits the existence of distinct physical zones or regimes within a plasma or astrophysical environment, each characterized by different dominant ionisation processes, microphysics, or response to external stimuli. Such models combine analytically motivated treatments or simulation frameworks that interpolate between local behaviors—grounded in atomic or subatomic processes—and macroscopic spatially differentiated effects, thereby refining both ionisation rate estimates and emergent diagnostics compared to single-regime or equilibrium approaches.

1. Fundamental Concepts of Two-Region Ionisation

A two-region ionisation framework divides the system, either spatially or by physical regime, into two principal regions:

• A region where a specific set of ionisation or recombination mechanisms dominates (e.g., low-density coronal equilibrium, low column-density cosmic-ray penetration, direct electron impact from ground states), and
• A contrasting region where other processes become significant (e.g., high-density plasmas with metastable populations altering ionisation/recombination rates, thick columns where the cosmic-ray flux is attenuated, strongly temperature- or radiation-modified outer shells).

This division is motivated by the observation that key microphysical parameters—such as metastable level populations, radiative transfer, collisional (de-)excitation, and cosmic ray flux—change nonlinearly with density, temperature, or location, thereby producing stepwise or layered ionisation structures not captured by a spatial or temporal average. The two-region approach seeks to explicitly represent or link models tailored for these regimes—yielding superior quantitative or diagnostic performance.

2. Application in Astrochemical and Molecular Cloud Ionisation

In the context of molecular clouds and star-forming regions, as investigated in Orion’s OMC-2 and OMC-3, the two-region model is crystallized via differentiation of zones according to column density and associated tracers such as CO depletion and deuterium fractionation (Socci et al., 24 Apr 2024). The methodology leverages:

• The CO depletion factor ($f_D$), measuring the freeze-out of CO onto dust grains, directly linked to ambient conditions favoring enhanced deuterium chemistry and lowered ionisation rates.
• The deuteration fraction ($R_D = N(\mathrm{DCO}^+)/N(\mathrm{HCO}^+)$), which provides a sensitive diagnostic of cold, dense, and chemically evolved environments.
• The empirical relationship between $f_D$ and o-H$_2$D$^+$ abundance:

$\log_{10}[X(\mathrm{o}\mbox{-}\mathrm{H}_2\mathrm{D}^+)] = 0.05\,f_D - 10.46,$

effectively providing a handle on unobservable ions controlling low-temperature, cosmic ray driven chemistry.

The core quantitative estimate for the cosmic-ray ionisation rate is:

$\zeta^\mathrm{ion}_{\mathrm{H}_2} = \frac{k_\mathrm{CO}^{\mathrm{o\mbox{-}H}_3^+}\,N(\mathrm{o}\mbox{-}\mathrm{H}_2\mathrm{D}^+)\,N(\mathrm{CO})}{3\,R_D\,N(\mathrm{H}_2)\,l},$

with $k_\mathrm{CO}^{\mathrm{o\mbox{-}H}_3^+}$ the destruction rate coefficient, $l$ the effective path length, and other quantities as above.

Empirical findings demonstrate a monotonic decrease in $\zeta^\mathrm{ion}_{\mathrm{H}_2}$ with increasing $N(\mathrm{H}_2)$, directly reflecting the attenuation of cosmic rays as the environment transitions from a lower-density, higher-ionisation-rate outer region to a denser, more shielded domain with lower ionisation rates. The two-region picture thus captures the spatial stratification (low column/high $\zeta$; high column/attenuated $\zeta$) in cosmic-ray driven molecular cloud ionisation.

3. Diagnostic Proxies and Observational Correlations

The methodology employs species- and environment-specific molecular proxies to map ionisation stratification:

• C$^{18}$O depletion traces the progressive chemical isolation and shielding in dense regions;
• N$_2$H$^+$ emission and the growth in deuterium fractionation index the degree of chemical evolution and cosmic-ray penetration;
• Drop in $\zeta^\mathrm{ion}_{\mathrm{H}_2}$ with $N(\mathrm{H}_2)$ corresponds to the established theoretical expectation from particle transport models of cosmic rays.

The two-region model is thus constrained by the spatial correlation of these proxies across parsec scales (Socci et al., 24 Apr 2024). The division between regions is not a priori but is emergent from continuous mapping of diagnostics and can be empirically located where, e.g., the transition in CO depletion or a marked inflection in deuteration fraction is observed.

4. Analytical and Modelling Frameworks

The analytical approach uses empirical relations and observationally determined quantities to bypass the lack of direct measurements for short-lived or spatially unresolved ions. This is achieved via:

• Proxy-derived abundances (notably o-H$_2$D$^+$ via CO depletion),
• Population ratios from rare isotope lines,
• Path length estimation from spatially resolved maps.

Limitations of the model are acknowledged, including sensitivity to the choice of $l$, dependence on the calibration of empirical relations (which are themselves scale and environment-dependent), and the breakdown of proxies at elevated temperatures ($T_K\gtrsim 20$–25 K). These issues delimit the regime of validity for a two-region approach and are central in prioritizing future observational campaigns (notably, the need for high-spatial-resolution interferometry to better constrain $l$).

5. Interpretive Implications and Theoretical Significance

The overall two-region ionisation paradigm enables:

• The reconciliation of spatially (or temporally) heterogeneous ionisation rates within a single analytic or simulation framework;
• Empirical validation for core predictions of cosmic-ray propagation and the resultant chemistry in clouds, where $\zeta_{\rm H_2}^{\rm ion}$ is highest at low columns and falls, sometimes by orders of magnitude, in dense, shielded inner regions;
• A direct mapping of chemical and physical transitions that underlie large-scale star formation and ISM evolution, demonstrating the general necessity of multi-regime treatments to recover observed molecular, atomic, and ionic stratification;
• An opportunity to assess the robustness and portability of analytic proxies for ionisation rates by extending them across environments and scales.

6. Practical Limitations, Improvements, and Future Directions

Critical limitations, as detailed in (Socci et al., 24 Apr 2024), include:

• Uncertainties associated with the assumed or empirically determined path length $l$, which can vary substantially across spatial resolution and can induce order-of-magnitude changes in inferred $\zeta_{\rm H_2}^{\rm ion}$;
• The dependence on empirical f_D–o-H$_2$D$^+$ relations, which may not be universal and could be affected by environmental parameters (temperature, turbulence, time since collapse);
• The need to improve temperature mapping, as deuteration-based diagnostics lose reliability above $T_K\sim25$ K.

Future research is likely to focus on:

• Maximizing spatial and spectral resolution (with facilities such as ALMA) to refine local $l$ estimation and resolve chemical inhomogeneities;
• Direct mapping of o-H$_2$D$^+$ emission wherever possible to calibrate and potentially localize empirical proxy relations;
• Integrating these empirical findings into more comprehensive, perhaps three-dimensional, simulation frameworks that explicitly distinguish and evolve the two (or multiple) ionisation zones over dynamical timescales.

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The two-region ionisation model in the context of cosmic-ray driven ISM chemistry, as exemplified in Orion’s OMC-2 and OMC-3, operationalizes the interplay between density-dependent molecular depletion, deuteration chemistry, and cosmic-ray penetration into an analytically tractable and observationally testable framework. By empirically correlating proxies and spatial structure with theoretical expectations for $\zeta_{\rm H_2}^{\rm ion}$ stratification, the approach simultaneously validates fundamental ISM ionisation models and reveals the necessity of spatially resolved, regime-specific treatments for accurate modelling of astrophysical plasmas (Socci et al., 24 Apr 2024).