BELMAC: Broad Emission Line Mapping Code

• BELMAC is a forward-modeling framework for AGN BLR studies, simulating the velocity-resolved response of emission lines using 3D cloud ensembles.
• It integrates geometry, kinematics, and photoionization physics to link observable light curves with underlying cloud dynamics.
• BELMAC improves black hole mass calibration by combining physical emissivity models with measured reverberation lags and line profiles.

Broad Emission Line MApping Code (BELMAC) is a forward-modeling framework for broad-line region (BLR) reverberation studies in active galactic nuclei, designed to simulate the velocity-resolved response of broad emission-line gas to a driving continuum by constructing a three-dimensional ensemble of BLR clouds with specified geometry, kinematics, and cloud properties. In the BELMAC literature, the code is presented both as a generic reverberation-response simulator and, in a later version, as a photoionization-based model that uses Cloudy grids to compute cloud luminosities for specific broad emission lines rather than relying on a simple hydrogen recombination approximation (Rosborough et al., 2023, Rosborough et al., 18 Jul 2025).

1. Origins, scope, and scientific role

BELMAC was introduced as a tool for modeling the reverberation response of the BLR to continuum variability, with the explicit aim of connecting assumed BLR structure and dynamics to observable quantities such as velocity–delay maps, integrated line light curves, and time-dependent line profiles (Rosborough et al., 2023). In the NGC 3227 reverberation study, BELMAC was described as a photoionization-based BLR modeling tool distinct from phenomenological codes such as CARAMEL, because it couples a physically motivated cloud distribution, geometry, and kinematics to radiative transfer and emissivity computed from photoionization grids (Bentz et al., 2023).

The code’s scientific role is therefore narrower and more physical than a purely descriptive transfer-function fit. Its operating assumption is that the emitting gas is represented by clouds distributed in three dimensions, with cloud emissivity depending on radius, density, optical depth, ionization state, and anisotropy rather than merely on geometric weighting. This makes BELMAC particularly relevant where phenomenological reverberation mapping must be supplemented by explicit line-formation physics.

A second strand in the literature situates BELMAC in relation to adjacent BLR-inference frameworks. In a single-epoch, multi-line context, the DyBEL package was presented as functionally what BELMAC aims to be: a framework that infers shared BLR geometry and kinematics together with line-dependent radial emissivity distributions from broad-line profiles, returning quantities such as inclination, virial factor, characteristic radii, and relative lag ratios (Kuhn et al., 2024). This suggests that “BELMAC” denotes not only a specific reverberation code but also a broader modeling program linking BLR dynamics, emissivity stratification, and observables across time-domain and single-epoch regimes.

2. Reverberation formalism and forward modeling

BELMAC adopts the standard reverberation-mapping convolution between a driving continuum and a velocity-dependent transfer function. In the NGC 3227 study the forward model was written as

$L(v,t)=\int \Psi(v,\tau)\,C(t-\tau)\,d\tau,$

where each cloud contributes according to its line-of-sight velocity $v$ and light-travel delay $\tau$, with weights set by photoionization emissivity and geometric factors (Bentz et al., 2023). In the later photoionization version, the corresponding formulation for a line $\lambda$ was

$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$

with BELMAC constructing $\Psi_\lambda$ by summing delayed, Doppler-shifted, and attenuated line emission from a 3D cloud ensemble at each observer time step (Rosborough et al., 18 Jul 2025).

The core computational object is therefore the transfer function $\Psi(v,\tau)$ or $\Psi_\lambda(t',v^{||})$. In BELMAC, this function simultaneously encodes geometric mapping, kinematic mapping, and photoionization weighting. Geometry determines where clouds lie in three dimensions; kinematics determine their projected velocities and light-travel delays; photoionization weighting determines which clouds are luminous in a given line under a specified spectral energy distribution (SED), luminosity, density law, and column-density structure. The output is not only an integrated lag estimate but a synthetic, time-dependent emission-line signal.

The 2025 version introduced additional diagnostics. It defined a 1D normalized response function,

$$L_{\lambda,\,\mathrm{norm}}(\tau)=\frac{L_\lambda(\tau)-L_\lambda(0)}{L_{\lambda,\mathrm{max}}-L_\lambda(0)},$$

and a velocity-resolved RMS profile,

$$L_{\lambda,\,\mathrm{RMS}}(v^{||})=\frac{1}{\mathcal{N}}\sqrt{\sum_{j=1}^{\mathcal{N}}L_{\lambda,\,\mathrm{norm}}(\tau_j,\,v^{||})^2},$$

which were used to characterize the variable component of the response. BELMAC also measured a response-weighted delay (RWD) and compared it to a luminosity-weighted radius (LWR), emphasizing that these need not coincide under general emission physics (Rosborough et al., 18 Jul 2025).

This formalism makes BELMAC a forward simulator rather than an inversion code in its currently documented reverberation applications. In the NGC 3227 deployment, BELMAC did not yet have automated parameter optimization with priors, likelihood, or MCMC; instead, it was used in confirmatory and exploratory mode, with assessment based on visual and qualitative comparison of modeled and observed H$v$0 light curves and mean/RMS profiles (Bentz et al., 2023).

3. Physical ingredients: geometry, kinematics, and photoionization

BELMAC populates the BLR with clouds drawn from parameterized spatial and density distributions. In the NGC 3227 analysis, the number density of clouds with radius followed $v$1, while the gas density followed $v$2; cloud covering fraction set the cross-sectional area intercepting ionizing radiation (Bentz et al., 2023). In the 2025 photoionization implementation, cloud ensembles were further specified by total number of clouds, covering fraction, radial number-density law, gas-density law, and constant cloud mass with size scaling $v$3 (Rosborough et al., 18 Jul 2025).

BELMAC’s geometric options include flared disks, spheres, and bicones. The later paper explicitly added a hollow biconical zone defined by a cone-wall thickness $v$4 and half-opening angle $v$5, while retaining disk realizations parameterized by half-angular width and inclination (Rosborough et al., 18 Jul 2025). In the NGC 3227 study, the authors tried both a biconical geometry guided by CARAMEL and a thick disk geometry, with BELMAC ultimately preferring the thick disk (Bentz et al., 2023).

Kinematically, BELMAC includes rotational and radial motions. In the NGC 3227 work, radial motion was stated to be influenced by radiation pressure and gravity, and clouds could occupy bound elliptical orbits or unbound hyperbolic orbits, allowing inflow or outflow (Bentz et al., 2023). The 2025 version made this more explicit through a generalized radial velocity law and by distinguishing Keplerian rotation, radial flows, and optional turbulence. The sign of radial acceleration was set by

$v$6

and a generalized integrated radial velocity law was used to describe inflow, outflow, and failed-wind behavior (Rosborough et al., 18 Jul 2025).

A defining BELMAC feature is its use of photoionization calculations. In NGC 3227, emissivities were weighted using CLOUDY v17.03 grids computed for $v$7, with $v$8 and an AGN SED based on the average from 17 local unobscured AGNs with Eddington ratios $v$9 (Bentz et al., 2023). In the 2025 version, BELMAC used Cloudy v23 grids spanning ionization parameter, column density, and gas density, and included selected lines such as H$\tau$0, H$\tau$1, MgII $\tau$2, and CIV $\tau$3 (Rosborough et al., 18 Jul 2025). The ionization parameter was defined as

$\tau$4

and cloud line emission was modulated by anisotropic cloud emission and attenuation by Thomson scattering in a hot inter-cloud medium (Rosborough et al., 18 Jul 2025).

This architecture produces line-dependent reverberation behavior. In the photoionization version, CIV generally responded at shorter delays than low-ionization lines in disk models, while two-zone models could reproduce observed differences between high- and low-ionization lines that single-zone models could not (Rosborough et al., 18 Jul 2025). A plausible implication is that BELMAC was built to treat BLR “structure” and line emissivity as inseparable components of the same inference problem.

4. The NGC 3227 application

The first detailed observational deployment of BELMAC was in the reverberation mapping campaign on NGC 3227, where photometric and spectroscopic monitoring from 2022 December to 2023 June measured several optical broad-line delays, including $\tau$5 days for H$\tau$6 and $\tau$7 days for He II (Bentz et al., 2023). BELMAC was applied only to H$\tau$8 in this work. Its targets were the observed integrated H$\tau$9 light curve and the mean and RMS H$\lambda$0 profiles; the paper did not report BELMAC modeling for He II, and the separate five-bin velocity-resolved delays were not used as BELMAC inputs (Bentz et al., 2023).

The continuum driver was the V-band light curve constructed from image subtraction on the Las Cumbres Observatory network. The light curve used for reverberation analysis had site offsets corrected, uncertainties inflated by a multiplicative factor of approximately 9, and time bins of 0.5 days (Bentz et al., 2023). BELMAC therefore operated on preprocessed observational inputs rather than raw spectroscopy.

Two BELMAC tests were performed. The first adopted a model guided by CARAMEL’s median parameters to assess consistency with the observed H$\lambda$1 light curve and mean/RMS profiles. In that run, the authors used $\lambda$2, $\lambda$3, outer-radius gas density $\lambda$4, covering fraction $\lambda$5, and rotational plus purely inflowing radial motions within a biconical geometry (Bentz et al., 2023). The second test manually adjusted BELMAC parameters to find a better qualitative match. The resulting preferred model was a thick disk with half-angular width $\lambda$6, inclination $\lambda$7, $\lambda$8, $\lambda$9, outer-radius density $$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$0, and covering fraction $$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$1; its kinematic mix comprised 53% bound elliptical orbits and 47% unbound hyperbolic orbits, of which 98% were inflowing (Bentz et al., 2023).

Aspect	CARAMEL-guided BELMAC run	Preferred BELMAC model
Geometry	Biconical geometry	Thick disk
Inclination	Guided by CARAMEL median values	$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$2
Cloud laws	$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$3, $$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$4	$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$5, $$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$6
Outer density	$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$7	$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$8
Covering fraction	$$L_\lambda(t,\,v^{||})=\int_{-\infty}^{\infty}\Psi_\lambda(t^\prime,v^{||})\,L_c(t-t^\prime)\,dt^\prime,$$9	$\Psi_\lambda$0
Kinematics	Rotation + pure inflow	Roughly equal rotation and inflow

BELMAC and CARAMEL converged on several qualitative conclusions: a radially extended and flattened BLR, similar overall size, low-to-intermediate inclination, and an absence of outflow-dominated kinematics. They diverged in detail. CARAMEL preferred a biconical or flared-disk “skin” and was more rotation-dominated, while BELMAC preferred a thick disk and required more inflow to improve the match to the wings and blue side of the RMS profile (Bentz et al., 2023). BELMAC did not independently fit the black hole mass in this study; the quoted reverberation estimate,

$\Psi_\lambda$1

with $\Psi_\lambda$2, yielded $\Psi_\lambda$3 from the H$\Psi_\lambda$4 lag and $\Psi_\lambda$5, while CARAMEL found $\Psi_\lambda$6 (Bentz et al., 2023).

5. Relation to phenomenological, multi-component, and single-epoch BLR modeling

BELMAC is most clearly distinguished from CARAMEL by its treatment of emissivity. CARAMEL fits parametric geometric and kinematic distributions directly to reverberation data, whereas BELMAC weights cloud emission through photoionization physics, including ionization state, optical depth, density dependence, and anisotropy (Bentz et al., 2023). This difference was explicitly invoked to explain why the two codes could agree on a flattened, radially extended BLR while preferring different detailed geometries and different balances between rotation and inflow.

The literature also links BELMAC to multi-component BLR modeling. In NGC 5548, the observed combination of a single-peaked Ly$\Psi_\lambda$7 line profile and a distinctly double-peaked velocity-resolved delay profile was argued to be incompatible with simple single-component cloud or thin-disk models. The same study formulated direct guidance for BELMAC: it proposed support for additive multi-component transfer functions,

$\Psi_\lambda$8

together with anisotropic emissivity, obscuration, optical-depth effects, non-linear response, and regularized inversion that fits line profiles, delay profiles, and light curves simultaneously (Long et al., 10 Jul 2025). Those proposals were not presented as existing BELMAC capabilities in that paper; they were design requirements motivated by the NGC 5548 data.

A related comparison arises from the single-epoch code DyBEL. DyBEL adapts a Pancoast-style BLR cloud/dynamical framework to fit multiple broad lines simultaneously from one spectrum, tying global geometry and kinematics across lines while allowing line-specific radial emissivity distributions. Its parameterization includes inclination $\Psi_\lambda$9, angular thickness $\Psi(v,\tau)$0, mid-plane transparency $\Psi(v,\tau)$1, anisotropy $\Psi(v,\tau)$2, the fraction on bound elliptical orbits $\Psi(v,\tau)$3, and shifted-Gamma radial distributions for individual lines (Kuhn et al., 2024). The same paper explicitly described DyBEL as functionally what BELMAC aims to be in a single-epoch setting, and as a practical realization of BELMAC’s architecture for profile-based inference.

For NGC 3783, the simultaneous DyBEL fit produced $\Psi(v,\tau)$4, $\Psi(v,\tau)$5 with approximately $\Psi(v,\tau)$6 uncertainty, $\Psi(v,\tau)$7, and virial factors around 2–3 across H$\Psi(v,\tau)$8, H$\Psi(v,\tau)$9, H$\Psi_\lambda(t',v^{||})$0, Pa$\Psi_\lambda(t',v^{||})$1, and He I $\Psi_\lambda(t',v^{||})$2 (Kuhn et al., 2024). Because these results belong to DyBEL rather than to the documented reverberation versions of BELMAC, they are best interpreted as evidence for architectural continuity rather than as direct BELMAC outputs. This suggests a broader methodological convergence in which photoionization-aware, cloud-based BLR models are applied both to reverberation datasets and to single-epoch multi-line spectra.

6. Mass calibration, limitations, and prospective development

A recurrent BELMAC result is that reverberation lags are not automatically identical to physically meaningful BLR radii. The introductory BELMAC paper stated that the response-weighted delay is only equivalent to the luminosity-weighted radius when clouds emit isotropically and are radiation-bounded; otherwise, the luminosity-weighted radius can be overestimated by up to a factor of 2 (Rosborough et al., 2023). The later photoionization version sharpened this point by showing that negative response can arise in some models, especially when high ionization suppresses line emission at small radii, causing response-weighted delays to overestimate luminosity-weighted radii (Rosborough et al., 18 Jul 2025).

These effects propagate directly into black hole mass estimation. In the 2025 BELMAC study, virial masses inferred from model line widths and response-weighted radii differed dramatically from the actual supermassive black hole mass, depending mainly on disk inclination and velocity field. For H$\Psi_\lambda(t',v^{||})$3 in disk-like models, the inferred scale factor $\Psi_\lambda(t',v^{||})$4 was typically $\Psi_\lambda(t',v^{||})$5–5 at $\Psi_\lambda(t',v^{||})$6, $\Psi_\lambda(t',v^{||})$7–10 at $\Psi_\lambda(t',v^{||})$8, and $\Psi_\lambda(t',v^{||})$9–300 at $$L_{\lambda,\,\mathrm{norm}}(\tau)=\frac{L_\lambda(\tau)-L_\lambda(0)}{L_{\lambda,\mathrm{max}}-L_\lambda(0)},$$0 for rotating and inflowing disks, whereas outflows gave much wider ranges, approximately $$L_{\lambda,\,\mathrm{norm}}(\tau)=\frac{L_\lambda(\tau)-L_\lambda(0)}{L_{\lambda,\mathrm{max}}-L_\lambda(0)},$$1–7 depending on density law and acceleration or deceleration (Rosborough et al., 18 Jul 2025). BELMAC therefore treats the virial factor as a model-dependent consequence of geometry and dynamics rather than as a fixed calibration constant.

The code’s current limitations are also explicit in the literature. In the NGC 3227 application, BELMAC had no reported automated optimization, no formal Bayesian uncertainties, and no statistical goodness-of-fit metrics; parameter selection was manual and qualitative (Bentz et al., 2023). In the 2025 photoionization version, further caveats included the use of a square-wave pulse for transfer-function exploration, a uniform hot inter-cloud medium, and uncertainty in the effective outer BLR boundary set by dust sublimation (Rosborough et al., 18 Jul 2025). Code availability remains limited in the published descriptions: the NGC 3227 paper cited BELMAC as Rosborough et al. (2023, submitted) and did not provide a public repository link or version (Bentz et al., 2023).

Future development directions are correspondingly well defined. The NGC 3227 study noted that automated parameter optimization capability was under development (Bentz et al., 2023). The NGC 5548 multi-component analysis proposed that BELMAC incorporate multi-component priors, regularized inversion, simultaneous fitting in profile and delay space, and formal model comparison metrics such as $$L_{\lambda,\,\mathrm{norm}}(\tau)=\frac{L_\lambda(\tau)-L_\lambda(0)}{L_{\lambda,\mathrm{max}}-L_\lambda(0)},$$2, AIC, and BIC (Long et al., 10 Jul 2025). Taken together, these developments position BELMAC as an evolving BLR-mapping framework whose central technical premise is that reverberation observables, line-profile morphology, and black hole mass calibration must be interpreted through joint modeling of geometry, kinematics, and line-dependent photoionization physics.