Locally Optimally Emitting Clouds (LOC)

• LOC is a photoionization framework for AGN emission-line regions that integrates over varied cloud properties to achieve optimal local emissivity.
• The model uses distribution functions in radius and density to reproduce observed BLR and NLR line ratios, variability, and spectral diagnostics.
• LOC links cloud dynamics, radiative selection, and turbulence, providing insights into changing-look AGNs and low-ionization emission features.

to=arxiv_search 天天中彩票不ి 手机版天天中彩票? {"query":"Locally Optimally Emitting Clouds AGN (Guo et al., 2019, Waters et al., 2019, Moloney et al., 2014, Richardson et al., 2013, Matthews et al., 2020, 0911.1173)","max_results":10} Locally Optimally Emitting Clouds (LOC) is a photoionization framework for active galactic nucleus (AGN) emission-line regions in which the broad-line region (BLR) or narrow-line region (NLR) is not treated as a single-zone cloud population, but as a broad distribution of clouds spanning radius, density, and, equivalently, ionizing flux. Its defining principle is local optimal emission: each spectral line is produced most efficiently by the subset of clouds whose local physical conditions maximize that line’s emissivity, so the observed spectrum is the integrated output of an ensemble rather than the signature of a uniquely tuned cloud. In the literature considered here, LOC is used to model BLR line ratios, EUV diagnostics, broad-line variability, changing-look behavior, Fe II anisotropy, and NLR ionization sequences, and it is also reinterpreted physically in terms of clumpy turbulent outflows and stratified disc winds (Guo et al., 2019, Moloney et al., 2014, Waters et al., 2019, Matthews et al., 2020, 0911.1173, Richardson et al., 2013).

1. Formal definition and mathematical structure

In LOC modeling, the observable line luminosity is obtained by integrating single-cloud emissivities over a cloud ensemble weighted by distribution functions in radius and density. One implementation writes

$$L_{\rm line} \propto \int\!\!\int F(r)\, f(r)\, g(n)\, dn\, dr,$$

with empirical choices $f(r)\propto r^\Gamma$ and $g(n)\propto n^\beta$ (Guo et al., 2019). An equivalent BLR formulation integrates over hydrogen density and hydrogen-ionizing photon flux,

$$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$

where $F(n_H,\Phi_H)$ is the photoionization-predicted line flux and the exponents describe the weighting of clouds across parameter space (Moloney et al., 2014). For NLR applications, the same idea is expressed as

$$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$

with $f(r)\propto r^\gamma$ and $g(n_H)\propto n_H^\beta$ (Richardson et al., 2013).

The conceptual content of these expressions is the same across implementations. LOC assumes that different lines are “optimally emitted” in different restricted regions of the density–radius or density–flux plane, so the integrated spectrum is controlled by the cloud distribution functions and by the emissivity contours of the lines themselves. This is why LOC is a selection-effect model rather than a one-zone prescription: the relevant observable is the sum over the ensemble, not the spectrum of any individual cloud (Richardson et al., 2013).

Several quantities recur in LOC parameterizations. The hydrogen-ionizing photon flux $\Phi_H$ is used as a proxy for distance from the ionizing source because $\Phi_H \propto r^{-2}$ in the optically thin limit (Moloney et al., 2014). The ionization parameter is commonly written as

$f(r)\propto r^\Gamma$0

and in one turbulent-outflow treatment the X-ray ionization parameter is

$f(r)\propto r^\Gamma$1

with $f(r)\propto r^\Gamma$2 for the adopted NGC 5548 spectral energy distribution (Waters et al., 2019).

2. Parameter-space coverage and spectroscopic applications

A central strength of LOC is its ability to reproduce multiple lines that peak at different physical conditions. In EUV BLR spectroscopy, CLOUDY grids were computed over $f(r)\propto r^\Gamma$3 and $f(r)\propto r^\Gamma$4, assuming a one-sided 1D slab, constant density, $f(r)\propto r^\Gamma$5, solar metallicity and solar abundance ratios, no cloud shielding of the continuum source by other clouds, purely thermal motions, and a BLR covering factor of 40% (Moloney et al., 2014). In that application, the LOC models provide good fits to the measured fluxes, while the single-component models do not. By contrast, the single-component models could fit only two of the six individual AGN spectra, and only after excluding the problematic N $f(r)\propto r^\Gamma$6991 line; the LOC models gave good fits to four of the six individual AGN and to the EUV subset of the 159-object composite spectrum (Moloney et al., 2014).

The same study showed why the distributed-cloud approach matters physically. EUV lines such as N IV $f(r)\propto r^\Gamma$7, O II $f(r)\propto r^\Gamma$8, and O III $f(r)\propto r^\Gamma$9 originate primarily from gas with electron temperatures between 37000 K and 55000 K, in BLR clouds with high hydrogen densities ($g(n)\propto n^\beta$0) and hydrogen ionizing photon fluxes ($g(n)\propto n^\beta$1) (Moloney et al., 2014). Different lines therefore require access to different regions of the grid. This was one reason single-zone models failed for most objects: even when lines align roughly along constant $g(n)\propto n^\beta$2, they do not all peak at the same $g(n)\propto n^\beta$3, and collisionally excited emissivities remain strongly sensitive to temperature, density, and flux (Moloney et al., 2014).

In the NLR, LOC has been used to fit composite spectra along an AGN ionization sequence. Cloudy v10.0 grids with 7171 cloud models sampled $g(n)\propto n^\beta$4 and $g(n)\propto n^\beta$5, excluding gas satisfying $g(n)\propto n^\beta$6, corresponding roughly to $g(n)\propto n^\beta$7 (Richardson et al., 2013). The resulting line-emissivity maps show that different optical diagnostics peak at different radii and densities; for example, $g(n)\propto n^\beta$8 peaks at smaller radii than $g(n)\propto n^\beta$9, and $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$0 peaks at smaller radii than $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$1 (Richardson et al., 2013). This is the NLR analogue of the BLR selection effect.

3. Physical interpretations: from empirical ensemble to dynamical medium

LOC has often been implemented as a large grid of photoionization calculations without a specific cloud-formation mechanism. A major development is the proposal that the BLR cloud population is the observational imprint of a clumpy turbulent outflow. In that picture, condensations form in thermally unstable zones of an AGN outflow, and the relevant cloud sizes are many orders of magnitude smaller than the global outflow scale $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$2, so cloud dynamics can be modeled locally (Waters et al., 2019). The characteristic cooling length is written as

$$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$3

with the estimate

$$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$4

and the essential BLR result is that $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$5, which justifies local multiphase turbulence simulations (Waters et al., 2019).

Using Athena, non-adiabatic hydrodynamics, thermal conduction, heating/cooling, and purely solenoidal forcing, the first 3D local clumpy turbulent outflow simulations for this problem were presented with subsonic turbulent Mach numbers $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$6 and fiducial box size $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$7 (Waters et al., 2019). Those simulations show that condensations form only in a restricted wavenumber interval inside the inertial range,

$$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$8

where $$\int\!\!\int A\,F(n_H,\Phi_H)\,n_H^{\beta_n}\,\Phi_H^{\beta_\Phi}\, d(\log n_H)\,d(\log \Phi_H),$$9 corresponds to the Field length (Waters et al., 2019). The authors argued that this reproduces the same selection effects characteristic of LOC. A plausible implication is that the broad grid of ionization parameters and densities used in empirical LOC work may arise dynamically from thermal instability plus turbulence rather than being imposed ad hoc.

A related reinterpretation replaces discrete clouds with a continuous, self-shielded, biconical disc wind. In that framework, the wind spans a large portion of the ionizing flux-density $F(n_H,\Phi_H)$0 plane because density varies along streamlines through mass conservation, while ionizing flux is attenuated and reprocessed by the flow itself (Matthews et al., 2020). The hydrogen-ionizing photon flux density is

$F(n_H,\Phi_H)$1

with $F(n_H,\Phi_H)$2, and the authors explicitly stated that the behavior of a line-emitting wind is similar to that of LOC except that the gradients in ionization state and temperature are large-scale and continuous rather than within or between distinct clouds (Matthews et al., 2020). They also found that clumpy biconical disc winds can produce BLR-like spectra provided that the wind has a volume filling factor of $F(n_H,\Phi_H)$3, with the most successful models typically using $F(n_H,\Phi_H)$4, and that line emission arises almost exclusively from plasma travelling below the escape velocity, implying that “failed winds” are important BLR candidates (Matthews et al., 2020).

4. Responsivity, variability, and broad-line reverberation

LOC has been used to explain why broad Mg II behaves differently from broad Balmer lines in quasars. In one fiducial implementation, the cloud distribution adopted $F(n_H,\Phi_H)$5, $F(n_H,\Phi_H)$6, solar abundance, $F(n_H,\Phi_H)$7, and a global covering factor of 50%, with a fiducial quasar of $F(n_H,\Phi_H)$8, $F(n_H,\Phi_H)$9, inner boundary $$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$0 cm, and outer boundary $$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$1 cm, or about 0.3 pc (Guo et al., 2019). In that model, Mg II-emitting gas is on average more distant from the ionizing source than the H$$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$2/H$$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$3 gas, and responds with a lower amplitude to continuum variations (Guo et al., 2019).

The physical distinction between Mg II and the Balmer lines is central. H$$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$4 and H$$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$5 are recombination lines, while Mg II is dominated by collisional excitation and has a low excitation energy of 4.4 eV (Guo et al., 2019). For typical BLR conditions, $$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$6 K and $$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$7, the Balmer recombination timescale is

$$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$8

while the Mg II collisional timescale is $$L_{\rm line}\propto \iint F(r,n_H)\, f(r)\, g(n_H)\, dr\, dn_H,$$9 s (Guo et al., 2019). The line response is therefore not limited by atomic timescales; it is set by where in the BLR the line forms. One-cloud calculations showed that Mg II emission becomes significant only at large enough column density, peaking around $f(r)\propto r^\gamma$0, and dropping again at very high columns where clouds become too optically thick (Guo et al., 2019).

Responsivity was formalized as

$f(r)\propto r^\gamma$1

so that if $f(r)\propto r^\gamma$2, then $f(r)\propto r^\gamma$3 (Guo et al., 2019). In the fiducial model, when the continuum drops by 1 dex, Mg II luminosity falls by only $f(r)\propto r^\gamma$4 dex, compared with about 0.6 dex for H$f(r)\propto r^\gamma$5 and 0.7 dex for H$f(r)\propto r^\gamma$6 (Guo et al., 2019). This lower responsivity, combined with a larger average formation radius, naturally dilutes and slows Mg II variability. If the BLR is truncated at $f(r)\propto r^\gamma$7 pc, most of the Mg II flux is emitted near that outer boundary, so the line does not display strong breathing; depending on $f(r)\propto r^\gamma$8, the same LOC framework can produce fully breathing, partially breathing, or no-breathing behavior (Guo et al., 2019).

These results have direct implications for reverberation mapping. Broad Mg II lags are intrinsically harder to measure because the line varies less strongly, and because the Mg II-emitting region is somewhat farther out, longer monitoring baselines are needed (Guo et al., 2019). Even so, a measured Mg II lag can still be used to infer a BLR size and black hole mass through

$f(r)\propto r^\gamma$9

The same model suggests, however, that Mg II may not have a strong intrinsic size-luminosity relation for an individual quasar, because its emission can remain tied to the outer truncation radius rather than tracking luminosity in the same way as H$g(n_H)\propto n_H^\beta$0 (Guo et al., 2019).

Changing-look behavior is interpreted similarly. Simulations of a continuum decline from $g(n_H)\propto n_H^\beta$1 to $g(n_H)\propto n_H^\beta$2 showed Balmer lines fading first while Mg II remains detectable over a wider luminosity range (Guo et al., 2019). This provides a natural LOC explanation for the persistence of broad Mg II in changing-look quasars defined on H$g(n_H)\propto n_H^\beta$3/H$g(n_H)\propto n_H^\beta$4, and for the rare population of broad Mg II emitters in otherwise normal galaxy spectra (Guo et al., 2019).

5. Low-ionization clouds, column density, and anisotropic emission

LOC has also been extended by adding dynamical filtering to the low-ionization part of parameter space. For Fe II-emitting gas in quasars, the relevant clouds occupy the low-ionization region of the ionizing flux–density plane, but the decisive parameter is column density because radiation pressure can expel low-column clouds (0911.1173). The force multiplier, defined as the ratio of total gas opacity to electron scattering opacity, is $g(n_H)\propto n_H^\beta$5 in Fe II-emitting gas (0911.1173). Since observed systems typically have $g(n_H)\propto n_H^\beta$6, whereas optically thin low-ionization gas would require $g(n_H)\propto n_H^\beta$7 for infall, the conclusion is that infalling Fe II clouds must be sufficiently thick that radiation acts primarily on an illuminated surface layer rather than the entire cloud (0911.1173).

The minimum column density for infall is written as

$g(n_H)\propto n_H^\beta$8

with $g(n_H)\propto n_H^\beta$9 (0911.1173). In the standard cloud, the ionized surface layer has $\Phi_H$0 before the ionization front is reached, and the cloud becomes neutral and Fe II-producing deeper inside (0911.1173). This produces a physically filtered LOC ensemble: clouds with too small a column are pushed out, while only the high-column subset contributes to the infalling Fe II component.

Anisotropy is then unavoidable. The observed spectrum is dominated by the shielded face of the infalling clouds rather than a symmetric distribution of emitters (0911.1173). In this geometry, optical Fe II emission is nearly isotropic, whereas UV Fe II is predominantly inwardly beamed. Once the column exceeds the ionization-front threshold, around $\Phi_H$1, Fe II emission becomes strongly inwardly beamed; as the cloud column increases further into the $\Phi_H$2 range, the Fe II/H$\Phi_H$3 ratio increases, approaching an asymptotic value of about 3 for sufficiently large $\Phi_H$4 or $\Phi_H$5 (0911.1173). The paper interpreted this as a physical driver for Eigenvector 1: cloud column density acts as the hidden variable coupling $\Phi_H$6, Fe II strength, and spectral differences (0911.1173).

Within LOC language, this is a notable conceptual shift. The model preserves the idea that different lines arise from locally optimal conditions, but it adds a dynamical survival criterion to the cloud ensemble. This suggests that, at least for low-ionization species, the cloud distribution sampled by the spectrum may be shaped jointly by photoionization optimality and radiation-pressure selection.

6. NLR sequences, unresolved issues, and nomenclature

In the NLR, LOC has been used to interpret systematic changes in ionization state across AGN composite spectra. After integrating over a wide range of radii and densities, the models indicate that the radial extent of the NLR is the major parameter determining the position of high- to moderate-ionization AGN along the sequence (Richardson et al., 2013). The best dust-free model kept the density weighting nearly fixed at $\Phi_H$7 while varying the radial weighting $\Phi_H$8 from $\Phi_H$9 at the high-ionization end through $\Phi_H \propto r^{-2}$0, $\Phi_H \propto r^{-2}$1, and $\Phi_H \propto r^{-2}$2 to $\Phi_H \propto r^{-2}$3 at the low-ionization end (Richardson et al., 2013). The physical interpretation was explicit: higher-ionization AGN contain optimally emitting clouds more concentrated toward the central continuum source, while lower-ionization AGN have more extended NLRs (Richardson et al., 2013).

The same analysis inferred that ionizing luminosity is anticorrelated with NLR ionization level for the AGN sequence selected by mean-field independent component analysis, and, for a fiducial covering factor $\Phi_H \propto r^{-2}$4, characteristic sizes up to roughly 7 kpc were obtained for the lowest-ionization subset (Richardson et al., 2013). The authors also considered whether the sequence might instead be a mixing curve of star formation and AGN emission, but argued that while many galaxies do show such composite behavior, the AGN sequence they isolated appears to be a special set of objects with negligible star-formation excitation (Richardson et al., 2013). A possible interpretation deserving further exploration was that the ionization sequence might be an age sequence in which lower-ionization objects are older and have systematically cleared out their central regions by radiation pressure (Richardson et al., 2013).

LOC does not eliminate all difficulties. In the NLR application, the observed increases of $\Phi_H \propto r^{-2}$5 and $\Phi_H \propto r^{-2}$6 toward lower ionization were not reproduced, and the temperature-sensitive auroral-line ratios remained a significant unresolved problem (Richardson et al., 2013). In the BLR EUV application, N $\Phi_H \propto r^{-2}$7991 was systematically underpredicted, with the most likely explanation being a combination of line-fitting and deblending uncertainties rather than a definitive abundance anomaly (Moloney et al., 2014). These shortcomings delimit the explanatory range of current LOC implementations without undermining the core selection-effect picture.

A separate source of confusion is terminological rather than physical. The acronym “LOC” is also used for a line radiative transfer program, explicitly “line transfer with OpenCL,” which is a deterministic non-LTE ray-tracing code for 1D and 3D interstellar-medium line transfer and is not related to the astrophysical concept of Locally Optimally Emitting Clouds (Juvela, 2020). In AGN emission-line theory, by contrast, LOC denotes the cloud-ensemble framework summarized above.