AGB Density Maps: Methods and Applications

• AGB Density Maps are spatial representations of asymptotic giant branch stars and their circumstellar material, crucial for quantifying stellar populations and mass-loss processes.
• Catalog-based maps leverage photometric and astrometric surveys with HEALPix binning to produce 2D/3D density distributions for both Galactic and extragalactic contexts.
• High-resolution imaging and hydrodynamic simulations reveal detailed gas and dust structures, enabling analysis of spiral patterns, equatorial enhancements, and binary interactions.

An AGB density map is a spatial representation—2D or 3D—of the number density, surface density, or volume density of asymptotic giant branch (AGB) stars, or of gas/dust constituents in their circumstellar environments. Such maps are essential for quantifying stellar populations in galaxies, diagnosing the mass-loss structure from single or binary AGB stars, and tracing the contribution of AGB material to the interstellar medium. Construction and interpretation of AGB density maps span techniques from catalog-level star counts in Galactic/extra-galactic surveys, through angular-resolved observations of gaseous and dusty envelopes, to hydrodynamic and radiative-transfer modeling of post-AGB structures.

1. Catalog-based AGB Density Maps: All-sky and Galaxy-scale Applications

Catalog-level density maps count and spatially bin AGB-classified stellar sources extracted from large photometric or astrometric datasets. The canonical workflow leverages archives such as Gaia DR3, accessed via TAP services (e.g., GACS, ARI-Gaia) supporting the on-the-fly HEALPix indexing and aggregation of sources:

• AGB candidates are selected with photometric and astrometric cuts, e.g. $(G_{BP} - G_{RP}) > 1.5$, $G < 17$, $phot\_variable\_flag = 'VARIABLE'$, parallax $< 1$ mas, $parallax\_over\_error < 5$.
• Binning on the sky is accomplished using NESTED HEALPix pixelization, with $n_{side}=32$–$256$ balancing spatial resolution and output row count ($\sim$0.23 deg$^2$ for $n_{side}=128$). The density per pixel is computed as $N_i = \sum_{j \in \text{pixel }i} 1$ or a weighted sum if a weight function is relevant.
• Data queries return pixelated counts, which are then visualized using tools such as TOPCAT, supporting FITS/VOTable export and interactive all-sky mapping (Mollweide projection, log-scaling, overlays).
• These maps efficiently render the Galactic distribution of AGB stars and their structural subcomponents, enabling direct comparison to models of the Milky Way or nearby galaxies (Taylor et al., 2016).

In extragalactic contexts (e.g., Local Group galaxies), 2D surface-density maps are built by binning spatial positions of AGB/RSG candidates identified from near-IR/optical time-domain imaging. Deprojection into galactocentric coordinates allows analysis of disks, spiral arms, bars, and halos. Kernel Density Estimation or adaptive binning are used to achieve Poisson-limited smoothing, with completeness corrections and foreground subtraction applied for unbiased densities. Resulting maps provide exponentially declining disks, Sèrsic bulges, and can be dissected by stellar mass or age to probe population gradients and dynamical mixing (Javadi et al., 2018).

2. Imaging and Mapping Density in Circumstellar Envelopes

At sub-arcsecond scales, resolved mapping of gas and dust density in the circumstellar environments of AGB and post-AGB stars utilizes high-angular-resolution observations (e.g., ALMA, IRAM/NOEMA, optical polarimetry):

• In dust-rich AGB envelopes (e.g., IRC+10216), dual-beam optical polarimetric imaging reveals equatorially concentrated "dark lanes"—regions of enhanced dust density which are fit using radiative-transfer models (e.g., MCMax) with parameterized 2D density structures: a torus-like equatorial enhancement or latitude-localized rings. Fits are constrained by the polarized intensity image, SED, and reproduce high equator-to-pole density contrasts ($A\simeq 31$ at $R_0=15$ AU, density $\rho_0 \sim 5.6 \times 10^{-19}$ g cm$^{-3}$) (Jeffers et al., 2014).
• Gas density maps for inner envelopes (as in R Doradus) are derived from multi-frequency ALMA CO line cubes. Non-LTE 3D radiative-transfer (e.g., LIME) fits both emission/absorption profiles and spatial morphology. Density is found to decline steeply with radius in "extended atmospheres" ($n_\mathrm{H_2} \propto r^{-7.0}$ for $r<1.6\,R_*$), transitioning to a wind-like $n \propto r^{-2}$ profile at larger radii. Asymmetries and high-density "blobs" imply episodic, localized mass-ejection or convection-driven plumes, while the overall radial profile provides key input for dust-nucleation and wind-acceleration models (Khouri et al., 21 Feb 2024).
• In binary and post-AGB systems, interferometric mapping of CO and radiative transfer modeling yield full 3D (cylindrical/polar) density distributions for disks and outflows, distinguishing disk-dominated from outflow-dominated sources based on the partitioning of nebular mass and density (Cava, 2023).

3. Theoretical Modeling: Hydrodynamics and Morphological Classification

Sophisticated 3D hydrodynamic simulations elucidate the density structures resulting from wind-binary interactions:

• Smoothed Particle Hydrodynamics (e.g., PHANTOM) and grid-based radiation-hydrodynamics (AstroBEAR) model the gravitational and radiative forcing that generate spirals, arcs, vortices, circumbinary disks, and equatorial density enhancements (EDEs).
• The morphology and density contrasts are governed by the ratio $\xi = |U_\text{grav}| / E_\text{kin}$, i.e., the companion’s gravitational potential energy density relative to the wind’s kinetic energy density. Three morphological classes are distinguished:
  • $\xi < 0.1$: Class I, quasi-spherical, weak spiral perturbations,
  • $0.1 \leq \xi \leq 1$: Class II, regular spiral+arcs,
  • $\xi>1$: Class III, complex/broken spirals, significant EDE, global outflow flattening.
• Quantitative density enhancements observed in simulations (spiral/inter-arm contrasts up to 5 for $\xi\sim1$, up to 10 or more in disks) are consistent with values derived from interferometric maps and forward-modeling of observed spirals/arcs (Maes et al., 2021, Chen et al., 2019, Kim et al., 2011).

The transition from wind-Roche-lobe overflow (WRLOF) to Bondi-Hoyle-Lyttleton (BHL) accretion regimes in binaries is associated with clear transitions in the density morphology (formation of disks vs. persistent spirals), and density maps extracted from simulations exhibit the expected power-law behavior (e.g., broken power-laws in equatorial disks, $p=1.5$--2.5) and azimuthal density enhancements (Chen et al., 2019).

4. Methodologies for Constructing 3D and Projected Density Maps

Methodological frameworks for deriving AGB density maps include:

• Catalog binning (HEALPix, Cartesian, or galactocentric coordinates) with photometric/variability-based selection, and aggregation of counts or weighted sums (Taylor et al., 2016, Javadi et al., 2018).
• Hierarchical Bayesian inversion of line-of-sight integrals (DIBs, extinction) sampled along thousands of stellar lines of sight, discretizing the volume into voxels at multiple resolutions (down to 50 pc). The local densities $\rho_\text{DIB}$, $\rho_\text{dust}$ are derived by inversion of $W_i = \int_0^{d_i} \rho_\text{DIB}(\ell) d\ell$, with spatial regularization. The resulting maps permit calculation of DIB-to-dust density ratios and comparison with AGB ejecta mass-flux distributions (Cox et al., 2 Jun 2024).
• Grid- and ray-tracing radiative-transfer methods (e.g., LIME, MCMax) for reconstructing spatially resolved dust/gas density from resolved emission or polarimetric images, assuming parameterized radial and/or latitudinal profiles and empirically constrained by matched synthetic imaging (Jeffers et al., 2014, Khouri et al., 21 Feb 2024).
• Hydrodynamical snapshots from simulations provide 3D fields of $\rho(\mathbf{r})$; isosurface rendering and slice maps (e.g., equatorial, meridional planes) visualize spatial and angular variations, and enable comparison with high-resolution observations.

5. Scientific Applications and Interpretation

AGB density maps serve multiple scientific purposes across galactic and stellar astrophysics:

• Galactic Structure: Surface density distributions of AGB populations reveal bar/disk/halo structure, dynamical mixing timescales, and the star-formation history in Local Group galaxies. The absence of azimuthal offsets between AGB arms and K-band spiral tracers constrains spiral arm lifetimes (Javadi et al., 2018).
• Circumstellar Physics: Gas and dust density maps inform models of mass loss, wind acceleration, and dust nucleation in late stellar evolutionary stages. Detailed radial profiles (e.g., steep gradients in extended atmospheres, breaks at dust-formation radii) provide empirical bounds for wind models in O-rich and C-rich AGB stars (Khouri et al., 21 Feb 2024, Jeffers et al., 2014).
• Binary Interaction: Patterns in density maps (spirals, arcs, equatorial enhancements, disks) encode properties of binary companions (mass, separation) and can be inverted—using analytic and simulation-calibrated formulae—to infer unseen substellar/stellar companions or to classify envelope morphologies (Kim et al., 2011, Maes et al., 2021, Chen et al., 2019).
• ISM Contributions: Comparisons of 3D volume densities for interstellar DIBs and dust, mapped against AGB ejecta fluxes (C-rich vs O-rich), support hypotheses that carbon-rich AGB stars are dominant sources or growth sites for certain organic carriers in the ISM. Spatial gradients in DIB/dust ratios correlate with the distribution of C-rich AGB ejecta, reinforcing the physical connection (Cox et al., 2 Jun 2024).

6. Key Quantitative Results and Morphological Diagnostics

Tables below summarize representative findings for AGB density map construction and morphological classification:

Method/Model	Quantity Mapped	Typical Density Scale
HEALPix+Gaia	$N_\mathrm{AGB}$ per pixel	$0.2$--$3.7$ deg$^2$, counts
ALMA CO ($R$ Dor)	$n_\mathrm{H_2}(r)$	$10^{11}$--$10^{7}$ cm$^{-3}$
MCMax/IRC+10216	$\rho_\mathrm{dust}(R,\theta)$	$5.6 \times 10^{-19}$ g/cm$^3$ (inner)
SPH (PHANTOM)	$\rho(r,\theta,\phi)$	$10^{-14}$--$10^{-11}$ g/cm$^3$ (disk), spiral contrasts up to $5 \times$
Bayesian inversion	$\rho_\mathrm{DIB}, \rho_\mathrm{dust}$	Varies, 50 pc voxel scale

$\xi$ regime (Maes et al., 2021)	Morphology	Density Contrast
$\xi < 0.1$	Smooth, quasi-spherical, weak spiral	Low ($\lesssim 1.5\times$)
$0.1 < \xi < 1$	Single-armed spiral, arcs	Moderate ($1.5$--$5\times$)
$\xi > 1$	Broken spiral, EDE, global flattening	High ($>5\times$ to $>10\times$)

These quantitative diagnostics enable systematic interpretation of observed maps in terms of underlying dynamical and evolutionary processes.

7. Limitations and Outlook

AGB density map construction and analysis remain subject to observational limitations (resolution, sensitivity, foreground/background contamination, completeness, and instrument systematics) and model uncertainties (degeneracies in radiative transfer, symmetry assumptions, dust/gas opacities). For circumstellar mapping, forward-modeling (with axisymmetric or episodic structures) captures first-order geometry, but small-scale clumps, 3D asymmetries, and time variability necessitate more complex approaches. Population-level density maps depend critically on robust source classification and completeness correction.

The combined application of catalog-based mapping, high-resolution imaging, numerical simulation, and multi-wavelength inversion provides an increasingly comprehensive view of AGB populations and mass-loss phenomena. Ongoing advances in survey capability (e.g., deeper Gaia releases, VISTA, LSST), interferometry (ALMA/NOEMA), and computational modeling are expected to further refine the physical fidelity and interpretive power of AGB density maps.