Lorentzian-Weighted HI Column Density

• Lorentzian-weighted HI column density is defined by emphasizing the damping wings in Voigt profiles to accurately capture high-density neutral hydrogen features.
• This method improves spectral fitting in Damped Lyman-α absorbers by robustly accounting for extended Lorentzian components in the absorption profiles.
• It underpins models of radiative transfer and molecular transition in the ISM, providing actionable insights into galaxy evolution and star formation processes.

The Lorentzian-weighted HI column density refers to the calculation, modeling, and physical interpretation of neutral hydrogen (HI) column density, where the weighting or sensitivity to broad Lorentzian profiles—particularly the damping wings in radiative transfer and spectral fitting—plays a pivotal role. This weighting arises both in the analysis of absorption line profiles at high HI column densities and in the theoretical treatment of atomic-to-molecular hydrogen transition regimes, often linked to self-shielding, pressure effects, and molecular formation, especially in cosmological and ISM environments.

1. Definition and Physical Basis

The HI column density distribution function, customarily denoted as $f(N_{\rm HI}) \equiv d^2n/(dN_{\rm HI} dX)$, encapsulates the number of absorption systems per unit HI column density ($N_{\rm HI}$) and per unit absorption distance ($X$), with $dX/dz = H_0 (1+z)^2 / H(z)$ in cosmological applications (Altay et al., 2010, Erkal et al., 2012). At high column densities ($N_{\rm HI} \gtrsim 10^{20.3}$ cm$^{-2}$), absorption profiles are well-described by Voigt profiles—a convolution of a Gaussian (Doppler broadening) and a Lorentzian (natural line broadening). In this regime, the Lorentzian component dominates (“damping wings”), and the absorption strength—and thus the inferred column density—is strongly affected by the Lorentzian profile. The weighting thus naturally emerges from the physics of radiative transfer, the spectral fitting methodology, and the instrumental response in observations of high-$N_{\rm HI}$ systems.

2. Lorentzian Weighting in Spectral Line Fitting

For Damped Lyman-$\alpha$ Absorbers (DLAs), and in general for systems with $N_{\rm HI} \gtrsim 10^{20.3}$ cm$^{-2}$, the absorption line profile is best modeled as a Voigt function:

$V(\nu) = \frac{H(a, x)}{\sqrt{\pi} \Delta\nu_D}$

Here, $H(a,x)$ is the Voigt function comprising the Gaussian core and Lorentzian wings, $a$ is the damping parameter, and $x$ is the dimensionless frequency offset. In the damping wings, $V(\nu)$ approaches the Lorentzian limit, and the “weighting” of the observed absorption to match a given $N_{\rm HI}$ follows the Lorentzian profile. This influences how $N_{\rm HI}$ is extracted from fitting observed or simulated spectra—an absorber contributing a larger equivalent width not from its Doppler-broadened core, but its extended Lorentzian damping wings (Altay et al., 2010).

The optical depth as a function of frequency for HI is given by:

$$\tau(\nu) = N_{\rm HI}\,f\,\left(\frac{\pi e^2}{m_e c}\right) V(\nu)$$

where the weighting in the damping wings is strictly Lorentzian. This is critical for the identification and quantification of column densities in DLAs and high-$N_{\rm HI}$ systems because it sets the scale at which the real (often saturated) absorber signature is discerned (Altay et al., 2010, Kanekar et al., 2011).

3. Radiative Transfer, Self-Shielding, and High-$N_{\rm HI}$ Regimes

Self-shielding is a decisive process governing the transition from a predominantly ionized to a neutral and then molecular ISM. In cosmological simulations, the accurate modeling of radiative transfer—using algorithms such as reverse ray-tracing—assigns each gas parcel an effective optical depth:

$$\tau_{\rm eff} = -\ln(\Gamma^{\rm shld}/\Gamma^{\rm thin})$$

where $\Gamma^{\rm thin}$ is the optically thin photoionization rate and $\Gamma^{\rm shld}$ accounts for local UV field attenuation (Altay et al., 2010). For $N_{\rm HI} \gtrsim 10^{18}$ cm$^{-2}$, self-shielding becomes efficient, flattening the $f(N_{\rm HI})$ distribution and increasing the neutral fraction. This transition, while a function of total column density, is reflected observationally in the emergence of strong Lorentzian wings, directly linking radiative transfer physics to Lorentzian-weighted line fitting.

At even higher HI column densities ($N_{\rm HI} \gtrsim 10^{20.3}$ cm$^{-2}$), pressure effects and the onset of molecular conversion come into play. Empirical relations such as $$R \equiv \Sigma_{H_2}/\Sigma_{HI} \approx (P/P_0)^{\alpha}$$ are used to convert atomic to molecular mass fractions—removing HI from the $f(N_{\rm HI})$ high-density tail and steepening the distribution (Altay et al., 2010, Sternberg et al., 2014).

4. Column Density Distribution and Turnover Features

Observational and simulated HI column density distributions display characteristic turnovers:

• At $N_{\rm HI} \simeq 10^{21}$ cm$^{-2}$, both $z=0$ and $z\sim3$ samples show a pronounced turnover, argued not to result from the HI–H$_2$ transition but from the maximum HI surface density established by global disk structure and projection effects. Specifically, universal processes such as disk formation and random inclination of sightlines set the observed turnover; the Lorentzian-dominated wings in spectral fitting ensure the correspondence between line profile and true column density (Erkal et al., 2012).
• At $N_{\rm HI} \gtrsim 10^{22}$ cm$^{-2}$, a lack of systems at $z=0$ is attributed to efficient conversion to molecular hydrogen. Here, the weighting by Lorentzian wings is essential to recognizing even rare high-$N_{\rm HI}$ cases.

The inclination-averaged distribution function is formulated as:

$$f(N_{\rm HI}) = \frac{1}{N_{\rm HI}^3}\int dN_{\rm HI}^\perp (N_{\rm HI}^\perp)^2 f_\perp(N_{\rm HI}^\perp)$$

which encapsulates the geometrical, or “projection-weighted,” contribution to the observed column density statistics (Erkal et al., 2012).

5. Atomic-to-Molecular Transition and Theoretical Frameworks

Theories of the HI-to-H$_2$ transition develop analytic expressions for total HI column density in photodissociation regions—where line absorption is dominated by Lorentzian-damped wings. For planar slabs exposed to beamed radiation:

$$N_{\rm HI,tot} = \frac{1}{\sigma_g} \ln\left(\frac{\alpha G}{2} + 1\right)$$

where $\sigma_g$ is the dust absorption cross-section, and $\alpha G$ encapsulates photodissociation, H$_2$ formation, dust, and self-shielding effects (Sternberg et al., 2014, Bialy et al., 2017). This analytic relation reflects the characteristic Lorentzian damping via its scaling with the effective dissociation bandwidth ($w$), which is reduced in dust-enriched environments.

Radiative transfer computations, such as those performed with the Meudon PDR code, explicitly resolve both the Doppler core (Gaussian) and the Lorentzian damping wings. The growth of absorption in the Lorentzian part of the profile becomes $N_2^{1/2}$, ensuring the robust connection between high-$N_{\rm HI}$ column densities and the line wings—a form of Lorentzian-weighted column density that governs both the mean profile and its probability distribution function (PDF) in turbulent media (Sternberg et al., 2014, Bialy et al., 2017).

6. Observational Techniques and Weighting Schemes

Observational approaches to measuring HI column density involve spectral line mapping with single-dish and interferometric telescopes. In data reduction, weighting functions may be applied during image gridding to optimize flux recovery. While most surveys utilize Gaussian beam weighting (e.g., $w(r) = \exp[-(r/\sigma)^2/2]$), it is conceptually viable to apply Lorentzian weighting when the beam or signal profile exhibits extended wings, potentially improving the measurement of diffuse emission and the recovery of faint, broad spectral features (Popping et al., 2011, Pisano, 2013).

In the context of profile fitting, especially for DLA absorption, the use of the Voigt function ensures explicit Lorentzian weighting in the damping wings. Similarly, in studies probing the atomic-to-molecular transition via PDFs, the log-normal fits emphasize the core, but the inclusion or modeling of Lorentzian-like weighting could better capture the extended distribution in low-intensity tails (Imara et al., 2016).

7. Applications, Implications, and Theoretical Insights

Lorentzian-weighted HI column density modeling is essential in several domains:

• Cosmological simulations: Accurately reproduces the abundance of Ly-$\alpha$ forest, Lyman Limit Systems, and DLAs over ten orders of magnitude in column density, robustly linking physical processes (radiative transfer, molecular conversion, feedback) to observed distributions (Altay et al., 2010).
• ISM structure and star formation: Determines thresholds and transitions for cold neutral medium (CNM) formation, informed by self-shielding regimes where Lorentzian wings dominate spectral signatures (Kanekar et al., 2011, Bihr et al., 2015).
• Galaxy evolution: Governs the mass budget and molecular gas fraction, influencing star-formation rates and gas accretion, particularly where turbulent mixing and interface phenomena are important (Lin et al., 15 Feb 2025).
• Observational surveys: Guides data reduction, weighting protocols, and spectral fitting—either through direct Lorentzian kernels or via the adoption of analytic or numerical line profile models.

A plausible implication is that, in environments where the instrumental response or astrophysical line profiles are wide or non-Gaussian (e.g., due to turbulent mixing layers or interface flows), the explicit adoption of Lorentzian weighting in column density estimation may better recover the true mass and structure of HI reservoirs (Pisano, 2013, Lin et al., 15 Feb 2025).

8. Summary Table: Key Aspects of Lorentzian-Weighted HI Column Density

Context	Lorentzian Weighting Role	Typical Scale/Regime
Spectral Fitting	Damping wings dominate in DLAs	$N_{\rm HI} \gtrsim 10^{20.3}$ cm$^{-2}$
Radiative Transfer	Absorption in Lorentzian wings	High optical depth, PDRs
ISM Transition	Self-shielding via Lorentzian profile	$N_{\rm HI} > 10^{18}$ cm$^{-2}$
Observational Gridding	Potential kernel for diffuse emission	All-sky/data reduction
PDF/Distribution Modeling	Line-wing absorption shapes PDF	Turbulent and molecular clouds

The core principle underlying Lorentzian-weighted HI column densities is the physical and observational dominance of Lorentzian-damped wings (Voigt profiles) in line formation, profile fitting, and radiative transfer across high-column systems. This weighting fundamentally controls the assignment of column density, the interpretation of absorber populations, the delineation of atomic-to-molecular transition regimes, and consequently the inferred structure, dynamics, and evolution of galaxies and the intergalactic medium.