Electron densities and filling factors of extragalactic HII regions: NGC 2403 and NGC 628

Imagine the universe as a giant, bustling city. In this city, the "Interstellar Medium" (the space between stars) isn't empty air; it's more like a foggy, chaotic neighborhood filled with clouds of gas and dust. Within this fog, massive stars are born, acting like streetlights that blast intense ultraviolet radiation, carving out glowing bubbles of ionized gas called H II regions.

This paper is like a detailed city survey of two specific neighborhoods: NGC 2403 and NGC 628. The astronomers (Zurita and her team) wanted to answer a very specific question: How "porous" or "leaky" are these gas bubbles?

Why does this matter? Because if these bubbles are full of holes (porous), the intense radiation from the stars can escape the galaxy and travel into deep space. This escaping radiation is the key to understanding how the early universe became transparent billions of years ago.

Here is a breakdown of their findings using simple analogies:

1. The Two Types of "Crowd Density"

To understand the bubbles, the team had to measure the "density" of the gas inside them. But there are two ways to measure this, and they tell very different stories:

The "In-Situ" Density ( $n_e$ ): Imagine standing inside a dense crowd at a concert. You can only see the people right next to you. This is the local density. The team found that inside the brightest, densest clumps of gas, the crowd is quite thick (up to 300 people per cubic meter).
The "RMS" Density ( $\langle n_e \rangle_{rms}$ ): Now, imagine looking at the entire concert hall from a drone. You see the dense crowd in the center, but you also see huge empty aisles and empty seats in the back. The average density of the whole room is much, much lower than the density of the crowd in the center.

The Discovery: The team found that the "average" density of these gas bubbles is 10 to 100 times lower than the "local" density.

The Analogy: Think of a sponge. If you squeeze a tiny piece of the sponge, it feels hard and dense (high local density). But if you look at the whole sponge, it's mostly air holes (low average density).
The Result: These gas bubbles are like giant, porous sponges. They are mostly empty space with tiny, dense islands of gas floating inside.

2. The "Filling Factor" (How much of the sponge is actually sponge?)

Because the gas is so clumpy, the team calculated a "filling factor." This is the percentage of the bubble's volume that is actually filled with gas, versus how much is just empty space.

The Finding: The gas only fills about 0.01% to 10% of the total volume.
The Metaphor: If an H II region were a football stadium, the actual gas would only be enough to fill a few rows of seats. The rest is empty air. This explains why radiation can easily leak out—there are plenty of "holes" in the fence.

3. The Size vs. Density Rule

The team looked at how the size of these bubbles relates to their density.

The Old Theory: Scientists used to think that as bubbles get bigger, they get proportionally less dense (like a balloon expanding).
The New Discovery: The team found this rule only works for small bubbles (under 50 parsecs, or roughly 163 light-years, across).
- Small Bubbles: As they get bigger, they get less dense (just like the old theory).
- Big Bubbles: Once they get larger than 50 parsecs (163 light-years), the rule breaks. They don't get less dense; they actually stay roughly the same density or even get slightly denser.
The Analogy: Imagine inflating a balloon. At first, it gets bigger and the rubber gets thinner. But if you keep inflating it past a certain point, the rubber stops stretching and just gets tighter. The big bubbles seem to be hitting a "pressure limit" set by the surrounding galaxy.

4. The "Star Formation" Connection

The team combined their data with other galaxies to see if there was a pattern. They found a fascinating link:

Busy Galaxies = Denser Gas: Galaxies that are forming stars very rapidly (high "Star Formation Rate") tend to have gas bubbles that are, on average, denser.
The Analogy: In a city with a lot of construction (star formation), the air is thicker with dust and debris. In a quiet city, the air is thinner.
Why it matters: This suggests that the "pressure" of the surrounding gas in a galaxy controls how big these bubbles can get. High pressure keeps the bubbles smaller and denser; low pressure lets them expand and become more porous.

5. Why This is a Big Deal

The paper introduces a new, uniform way of mapping these gas bubbles (using a "segmentation" method, like cutting a cake into perfect slices rather than guessing where the crumbs are). This is crucial because previous studies used different methods, making it hard to compare galaxies.

The Bottom Line:
The universe is not a smooth fog; it is a Swiss cheese of gas. The "holes" in the cheese are what allow the light from the first stars to escape and light up the universe. By studying these two galaxies, the team has provided a new map of how "leaky" the universe is, which helps us understand how the cosmos evolved from a dark, foggy place into the bright, clear universe we see today.

In short: They measured the holes in the cosmic sponge, found that the sponge is much emptier than we thought, and discovered that the "pressure" of the galaxy determines how big the holes get.

Here is a detailed technical summary of the paper "Electron densities and filling factors of extragalactic H ii regions: NGC 2403 and NGC 628" by Zurita et al. (2026).

1. Problem Statement and Motivation

The paper addresses a critical gap in understanding the physical state of the interstellar medium (ISM) in star-forming galaxies. Specifically, it focuses on the porosity of H ii regions, which governs the escape of ionizing Lyman continuum (LyC) photons—a key factor in cosmic reionization and the ionization of the diffuse ionized gas (DIG).

While electron densities derived from emission line ratios ( $n_e$ ) are common, the root-mean-square (rms) electron density ( $\langle n_e \rangle_{\text{rms}}$ ), which reflects the volume-averaged density, has been measured for very few extragalactic H ii regions. The discrepancy between $n_e$ (measured in high-density clumps) and $\langle n_e \rangle_{\text{rms}}$ (derived from total luminosity) allows for the calculation of the volume filling factor ( $\phi$ ). Previous studies have been limited by small sample sizes and, crucially, by a lack of homogeneous methodologies for identifying and characterizing H ii regions across different galaxies. The authors aim to establish a consistent framework to analyze these parameters in nearby galaxies to provide constraints for massive cluster formation models and high-redshift interpretations.

2. Methodology

The study focuses on two pilot galaxies: NGC 2403 (distance $\sim$ 3.16 Mpc) and NGC 628 (distance $\sim$ 9.84 Mpc). The authors developed a novel, homogeneous image-segmentation methodology to construct H ii region catalogues.

A. Observational Data

Imaging: Narrow-band H $\alpha$ and off-band continuum images were obtained using the William Herschel Telescope (WHT).
Spectroscopy: New long-slit spectroscopic observations were obtained with CAFOS (Calar Alto) for NGC 2403. These were combined with a comprehensive compilation of existing spectroscopic data for both galaxies (totaling 137 observations for NGC 2403 and 210 for NGC 628).
Resolution Degradation: To ensure uniformity across the future larger sample (distances 2–28 Mpc), the authors degraded the angular resolution of the images to mimic a galaxy at 12 Mpc. This reduces the bias introduced by differential physical resolution, setting a minimum detectable radius of $\sim$ 40 pc.

B. H ii Region Identification (Segmentation)

The authors employed a Python-based workflow (photutils and Astropy) involving:

Background Subtraction: Creation of a DIG (Diffuse Ionized Gas) background map based on the spatial gradient of H $\alpha$ surface brightness. Regions with gradients below a specific cutoff were masked to isolate discrete H ii regions.
Segmentation: Detection of sources in the background-subtracted image using a flux threshold (3 $\sigma$ above noise) and a minimum pixel count (corresponding to the degraded resolution).
Deblending: Application of multi-thresholding and watershed segmentation to separate overlapping regions, validated against FUV (GALEX) and continuum images to ensure ionizing clusters were correctly associated with H ii regions.
Catalog Construction: Derivation of equivalent radii ( $R_{eq}$ ), H $\alpha$ luminosities ( $L_{H\alpha}$ ), and morphological parameters. Both DIG-corrected and uncorrected luminosities were retained.

C. Density Derivation

In-situ Density ( $n_e$ ): Calculated from the [S ii] $\lambda\lambda$ 6717, 6731 doublet ratios using Pyneb.
RMS Density ( $\langle n_e \rangle_{\text{rms}}$ ): Derived from the H $\alpha$ luminosity and equivalent radius assuming spherical geometry and Case B recombination:
$\langle n_e \rangle_{\text{rms}} \propto \sqrt{\frac{L_{H\alpha}}{R_{eq}^3}}$
Filling Factor ( $\phi$ ): Calculated as $\phi = (\langle n_e \rangle_{\text{rms}} / n_e)^2$ .

3. Key Contributions

Homogeneous Cataloguing Method: The paper introduces a robust, reproducible image-segmentation technique that accounts for DIG contamination and resolution effects, addressing the inconsistency found in previous literature (where different methods yield vastly different size-luminosity relations).
Large-Scale Pilot Catalogues: The authors generated the first homogeneous catalogues for NGC 2403 (370 regions) and NGC 628 (1458 regions) using this unified approach.
Systematic Comparison: By combining new spectroscopic data with compiled literature data, the study provides a large, consistent dataset to analyze the relationship between $n_e$ , $\langle n_e \rangle_{\text{rms}}$ , and $\phi$ .

4. Key Results

A. Density and Filling Factors

In-situ Density ( $n_e$ ): Values are generally below 300 cm $^{-3}$ , with a median of $\sim$ 46 cm $^{-3}$ for both galaxies. No clear dependence on region size or galactocentric radius was found.
RMS Density ( $\langle n_e \rangle_{\text{rms}}$ ): Significantly lower than $n_e$ $n_{e}$ (by 1–2 orders of magnitude).
- NGC 2403: Median $\sim$ 1.8 cm $^{-3}$ .
- NGC 628: Median $\sim$ 1.1 cm $^{-3}$ .
Filling Factor ( $\phi$ ): Ranges from $\sim$ 10 $^{-4}$ to 10 $^{-1}$ , with most values between 10 $^{-4}$ and 10 $^{-2}$ . No significant dependence on region size or luminosity was detected within the spectroscopic subsample.

B. Size-Density Relations

Break in Scaling: The study confirms a size-density anti-correlation ( $\langle n_e \rangle_{\text{rms}} \propto R_{eq}^{-\alpha}$ $⟨ n_{e} ⟩_{rms} \propto R_{e q}^{- α}$ ) but identifies a distinct break at $R_{eq} \approx 45\text{--}50$ $R_{e q} \approx 45 - 50$ pc.
- For $R_{eq} \lesssim 50$ pc: $\alpha \approx 0.3$ (shallower than the canonical $\alpha=0.5$ or 1.0).
- For $R_{eq} \gtrsim 50$ pc: The relation breaks, with increased scatter and a slight tendency for density to rise with size.
Deviation from HH09 Sequence: When plotted on the classic Hunt & Hirashita (2009) size-density diagram, the bulk of the H ii regions in NGC 2403 and NGC 628 are offset toward smaller diameters compared to the traditional $\langle n_e \rangle_{\text{rms}} \propto D^{-1}$ relation. This suggests the traditional relation does not accurately describe giant extragalactic H ii regions within a single galaxy.

C. Scaling with Galaxy Properties

Combining their data with literature data for NGC 4449 and M51, the authors propose tentative scaling relations:

Large Region Fraction: The fraction of large H ii regions ( $R_{eq} > 50$ pc) decreases with increasing median $\langle n_e \rangle_{\text{rms}}$ ( $N_{large}/N \propto \langle n_e \rangle_{\text{rms}}^{-1.5}$ ).
Star Formation Rate (SFR): A correlation exists between median $\langle n_e \rangle_{\text{rms}}$ and the SFR surface density ( $\Sigma_{\text{SFR}} \propto \langle n_e \rangle_{\text{rms}}^{0.9}$ ).
Ambient Pressure: The variation in median $\langle n_e \rangle_{\text{rms}}$ among galaxies likely traces differences in ambient ISM pressure, which regulates the equilibrium size of H ii regions.

5. Significance and Implications

ISM Porosity: The low filling factors and the specific size-density relations provide direct constraints on the "clumpiness" of the ISM, which is essential for modeling the leakage of LyC photons.
High-Redshift Analogs: The findings suggest that the volume filling factor of ionized gas may be relatively constant with redshift, while the observed density increase at high redshifts may reflect higher initial gas densities in molecular clouds and higher ambient pressures.
Cluster Formation: The correlation between $\langle n_e \rangle_{\text{rms}}$ , SFR surface density, and the fraction of large H ii regions offers new observational constraints for models of massive star cluster formation, linking the local ISM pressure to the efficiency of forming massive bound clusters.
Methodological Standard: The paper establishes a necessary standard for future large-scale surveys (e.g., extending to the full dozen galaxies in the authors' project), ensuring that comparisons of H ii region properties across different galaxies are not biased by inconsistent identification criteria.

In conclusion, this work demonstrates that the physical properties of H ii regions are more complex than previously assumed, with a distinct break in size-density scaling and a strong dependence on the ambient galactic environment, necessitating a shift from simple power-law descriptions to models incorporating pressure equilibrium and environmental variations.