MOKA3D: 3D Kinematic Modeling of AGN Outflows

• MOKA3D is a 3D kinematic modeling framework that reconstructs the true distribution of ionized gas clouds in AGN outflows.
• It integrates projection effects, beam smearing, and data-driven emissivity to overcome limitations of smooth analytic models.
• The framework enables spatially resolved estimation of outflow energetics and dynamics, crucial for advancing AGN feedback studies.

Searching arXiv for the MOKA3D paper and closely related context papers. MOKA3D, short for “Modelling Outflows and Kinematics of AGN in 3D,” is a 3D kinematic modelling framework for integral-field spectroscopy (IFS) observations of AGN-driven ionized outflows. It was introduced to recover the true 3D distribution of emitting gas clouds and to infer spatially resolved outflow physical properties while explicitly accounting for projection effects, beam smearing / PSF blurring, and the observed clumpy and irregular surface brightness of nearby AGN outflows. In contrast to simplified cone models that impose smooth analytic emissivity laws, MOKA3D derives the final emissivity distribution from the observed cube itself and uses a cloud-based tomographic reconstruction to connect geometry, line profiles, and energetics (Marconcini et al., 2023).

1. Problem setting and motivation

MOKA3D was developed in the context of AGN feedback studies, where one seeks to characterize multiple kinematical components, including rotating gas and stellar disks, outflows, inflows, and jets. The immediate motivation is methodological: existing kinematic models for AGN outflows are often highly idealized, typically assuming smooth analytical emissivity laws, simplifying or ignoring projection effects, neglecting beam smearing, and failing to represent the observed clumpy emission seen in nearby systems. These limitations matter because reliable comparison with feedback and galaxy-evolution models requires robust estimates of geometry, velocity, mass flow, kinetic power, and momentum rate (Marconcini et al., 2023).

A central premise of MOKA3D is that apparently irregular or “messy” velocity fields need not imply intrinsically complex velocity laws. The framework is motivated by the idea that the observed kinematics of ionized cones may instead arise from a simple velocity field plus a clumpy cloud distribution and projection effects. This shifts the modelling emphasis away from increasingly elaborate analytic prescriptions and toward a more explicit reconstruction of the emitting medium in three dimensions.

The framework is presented in the specific case of AGN ionised outflows, where clumpiness and strong line-profile asymmetries are especially prominent. This focus is astrophysically consequential because AGN outflows are invoked as a mechanism capable of removing gas, suppressing star formation, and regulating black hole–galaxy coevolution.

2. Cloud-based formalism and source-frame parameterization

MOKA3D represents the outflow as a large ensemble of point-like emitting clouds distributed in 3D. In the source frame, each cloud is assigned a position and velocity, after which the system is rotated into the observer frame, convolved with observational effects, binned into the same cube as the data, and iteratively fit to the observed spectral line profiles. The end product is a tomographic reconstruction of the emitting gas distribution (Marconcini et al., 2023).

The geometry is formulated in a spherical coordinate system, with clouds described by semi-polar angle $\theta$, azimuthal angle $\phi$, and radial distance $r$. The source-frame Cartesian coordinates are

$$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$

The model typically adopts a conical or biconical outflow geometry. When hollow cones are considered, the geometry is specified by an inner semi-aperture $\theta_{\rm IN}$ and an outer semi-aperture $\theta_{\rm OUT}$; a filled cone corresponds to $\theta_{\rm IN}=0$. The implementation uses a large number of synthetic clouds, typically $10^7$, so that each observed voxel is sampled by multiple model clouds.

An initial radial flux law is introduced as

$f(r)=f_0\,e^{-r/r_0},$

where $f_0$ is the flux at the cone apex and $\phi$0 is a scale radius. In the framework, however, this is only an initial bookkeeping choice. The crucial point is that the final cloud weights are imposed by the observations rather than fixed by this analytic profile.

For the application to AGN outflows, the simplest adopted velocity field is a constant radial velocity,

$\phi$1

with 3D cloud velocity vector

$\phi$2

The method can also allow random velocity dispersion terms, although these are not exploited in the main analysis.

3. Projection, beam smearing, and data-driven emissivity reconstruction

Projection into the observer frame is handled explicitly through Euler rotations,

$\phi$3

where $\phi$4 is the rotation relative to the line of nodes, $\phi$5 is the inclination of the outflow axis with respect to the plane of the sky, and $\phi$6 is the projected position angle on the sky. The corresponding LOS velocity is

$\phi$7

with $\phi$8 the systemic velocity of the outflow relative to the host galaxy (Marconcini et al., 2023).

MOKA3D incorporates observational effects directly. PSF convolution is implemented by randomly shifting each cloud’s projected $\phi$9 position according to the measured 2D PSF, and LSF convolution can be applied to LOS velocities if needed, although it is not used in the main applications. This ensures that the model cube has the same effective spatial and spectral resolution as the MUSE observations.

After projection and binning into the observer cube $r$0, the framework first produces an unweighted model, which identifies the occupied regions of phase space implied by the adopted geometry and velocity field. The distinctive step is the subsequent weighted model, in which each cloud is assigned a weight from the observed flux in the corresponding voxel,

$r$1

where $r$2 is the observed flux in that voxel and $r$3 is the number of model clouds falling into it. In this way the data determine the emissivity distribution. The final model therefore does not require the surface-brightness distribution to be imposed analytically, and clumpy, irregular emission maps can be reproduced naturally.

4. Inference strategy, degeneracies, and validation

MOKA3D is fit to the full observed spectral cube, not only to moment maps. The comparison between observed and modelled line profiles is performed spaxel by spaxel over the velocity interval defined by the 1st and 99th percentiles of the LOS velocity distribution. The goodness-of-fit statistic is

$r$4

where $r$5 is the observed flux in spaxel $r$6 and channel $r$7, $r$8 is the model flux, and $r$9 is the uncertainty in spaxel $$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$0. The model seeks the parameters that minimize $$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$1 (Marconcini et al., 2023).

The observables used include the full line profiles in each spaxel, integrated flux maps, LOS velocity maps, velocity dispersion maps, and the velocity percentiles of the integrated spectrum. This is stricter than fitting only moment maps or only long-slit position–velocity diagrams.

A major issue is degeneracy among intrinsic outflow velocity, inclination, opening angle, and systemic velocity. Very high intrinsic velocities can formally reproduce a wide range of observed profiles, because the model can populate all velocity channels and then assign zero weight where the data contain no emission. To constrain this, MOKA3D requires the model percentile velocities $$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$2 to match the observed values within a tolerance $$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$3, progressively refined down to $$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$4:

$$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$5

$$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$6

This enforces a physically plausible combination of velocity, inclination, and opening angle.

The method was validated on simulated MUSE-like cubes with different inclinations, cone opening angles, velocity fields, PSFs, and clumpy emission. In these tests it successfully recovered the input geometry and kinematics, especially outflow inclination, radial velocity, and outer opening angle. The reported behavior indicates that the method performs best when fitting up to about four free parameters, whereas additional freedom increases degeneracy. A further mock experiment combined a rotating disk with an outflow; even without explicitly modelling the disk, the outflow properties were still recovered well when the outflow dominated the emission.

5. Application to MAGNUM Seyfert-II galaxies

The first astrophysical application of MOKA3D was carried out on three nearby Seyfert-II galaxies observed with MUSE at the VLT and selected from the MAGNUM survey: NGC 4945, Circinus, and NGC 7582. The analysis uses [OIII]$$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$7 for Circinus and NGC 7582, and [NII]$$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$8 for NGC 4945 because [OIII] is heavily obscured there. The data are continuum-subtracted, the emission lines are fit with multiple Gaussians, and spaxels with S/N $$(X,Y,Z)=\left(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta\right).$$9 are masked (Marconcini et al., 2023).

All three systems show clumpy ionized emission, steep velocity gradients across the outflow cone, velocity dispersion structures consistent with cone-like outflows, and asymmetric receding cones due to dust obscuration. The best-fit solutions indicate axes close to the plane of the sky, constant-radial-velocity conical outflows, and a preference for a filled cone over a hollow cone.

Galaxy	Diagnostic line	Approximate best-fit values
NGC 4945	[NII]$\theta_{\rm IN}$0	$\theta_{\rm IN}$1 km s$\theta_{\rm IN}$2, $\theta_{\rm IN}$3
Circinus	[OIII]$\theta_{\rm IN}$4	$\theta_{\rm IN}$5 km s$\theta_{\rm IN}$6, $\theta_{\rm IN}$7
NGC 7582	[OIII]$\theta_{\rm IN}$8	$\theta_{\rm IN}$9 km s$\theta_{\rm OUT}$0, $\theta_{\rm OUT}$1

The reported best-fit parameter sets include the radial velocity $\theta_{\rm OUT}$2, outflow scale $\theta_{\rm OUT}$3, inclination $\theta_{\rm OUT}$4, systemic velocity $\theta_{\rm OUT}$5, cone opening angle $\theta_{\rm OUT}$6, and position angle $\theta_{\rm OUT}$7. The paper concludes that a full conical geometry with nearly constant radial speed reproduces the observed line profiles and moment maps better than hollow cones or low-velocity models. This suggests that a substantial part of the apparent complexity in these nearby AGN outflows can be attributed to cloud distribution and viewing geometry rather than to intrinsically elaborate velocity laws.

6. Energetics, interpretation, and role in AGN feedback studies

For comparison with standard practice, the MOKA3D study reviews the usual literature recipe in which the outflow velocity is estimated from line percentiles,

$\theta_{\rm OUT}$8

and the ionized mass and mass outflow rate are then derived from [OIII]-based scalings that require assumptions about electron density, oxygen abundance, and geometry. Kinetic energy, kinetic luminosity, and momentum rate follow from those global estimates (Marconcini et al., 2023).

MOKA3D instead uses the reconstructed 3D cloud distribution to estimate ionized mass and the spatially resolved mass outflow rate. Starting from the continuity equation in spherical geometry,

$\theta_{\rm OUT}$9

and writing

$\theta_{\rm IN}=0$0

the local outflow rate becomes

$\theta_{\rm IN}=0$1

Within each radial shell of width $\theta_{\rm IN}=0$2, the model therefore derives the local outflow rate directly from the inferred ionized mass and velocity. The framework then constructs 2D maps of $\theta_{\rm IN}=0$3, radial profiles of $\theta_{\rm IN}=0$4, total kinetic energy, kinetic luminosity, momentum rate, and dynamical timescales,

$\theta_{\rm IN}=0$5

The derived outflows have mass outflow rates of order $\theta_{\rm IN}=0$6–$\theta_{\rm IN}=0$7, dynamical timescales of order $\theta_{\rm IN}=0$8 yr, and energetics described as consistent with AGN-driven ionized winds on kpc scales. The spatially resolved profiles show nonuniform radial behavior, interpreted as potentially reflecting past AGN variability, changes in ionization history, or shell-like structures in the outflow.

Within the broader methodological landscape, MOKA3D differs from earlier simplified models through seven linked features: 3D cloud-based reconstruction, observed-flux weighting of clouds, explicit treatment of projection and PSF smearing, full-cube fitting, tomographic recovery of the emitting gas distribution, direct spatially resolved energetics, and improved handling of clumpiness. A plausible implication is that the framework narrows the gap between observational kinematic modelling and the quantities required for comparison with AGN feedback simulations, especially in nearby systems where high-quality MUSE data make clumpiness and beam smearing impossible to ignore.