Dust Dynamics in AGN Winds A New Mechanism For Multiwavelength AGN Variability

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Dust Dynamics in AGN Winds: A New Mechanism For Multiwavelength

AGN Variability

Nadine H. Soliman1, Philip F. Hopkins1

1TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

Partial dust obscuration in active galactic nuclei (AGN) has been proposed as a potential explanation for some cases of AGN

variability. The dust-gas mixture present in AGN tori is accelerated by radiation pressure, leading to the launching of an AGN

wind. Dust under these conditions has been shown to be unstable to a generic class of fast-growing resonant drag instabilities

(RDIs). In this work, we present the ﬁrst numerical simulations of radiation-driven outﬂows that explicitly include dust dynamics

in conditions resembling AGN winds. We investigate the implications of RDIs on the torus morphology, AGN variability, and the

ability of radiation to eﬀectively launch a wind. We ﬁnd that the RDIs rapidly develop, reaching saturation at times much shorter

than the global timescales of the outﬂows, resulting in the formation of ﬁlamentary structure on box-size scales with strong dust

clumping and super-Alfvénic velocity dispersions. The instabilities lead to ﬂuctuations in dust opacity and gas column density of

10-20% when integrated along mock observed lines-of-sight to the quasar accretion disk. These ﬂuctuations occur over year to

decade timescales and exhibit a red-noise power spectrum commonly observed for AGN. Additionally, we ﬁnd that the radiation

eﬀectively couples with the dust-gas mixture, launching highly supersonic winds that entrain 70-90% of the gas, with a factor

of ≲3photon momentum loss relative to the predicted multiple-scattering momentum loading rate. Therefore, our ﬁndings

suggest that RDIs play an important role in driving the clumpy nature of AGN tori and generating AGN variability consistent

with observations.

Key words: instabilities — turbulence — ISM: kinematics and dynamics — star formation: general — galaxies: formation —

dust, extinction

1 INTRODUCTION

Dust plays a critical role in how a wide range of astrophysical systems

form, evolve, and are observed. It is involved in processes such as

planetary formation and evolution (Lissauer 1993;Liu & Ji 2020;

Apai & Lauretta 2010); chemical evolution (Watanabe & Kouchi

2008;Whittet et al. 1993;Weingartner & Draine 2001b;Minissale

et al. 2016), heating, and cooling within the interstellar medium

(ISM) and star formation (Dorschner 2003;Weingartner & Draine

2001a;Draine 2003;Salpeter 1977;Spitzer Jr 2008); as well as feed-

back and outﬂow launching in star-forming regions, cool stars and

active galactic nuclei (AGN) (King & Pounds 2015;Murray et al.

2005;Höfner & Olofsson 2018). Moreover, dust imprints ubiquitous

observable signatures, such as the attenuation and extinction of ob-

served light (Savage & Mathis 1979;Draine & Lee 1984;Mathis

1990).

One particular regime where dust is believed to play a central

role in both dynamics and observations is the “dusty torus” region

around AGN (Antonucci 1982;Lawrence & Elvis 1982;Urry &

Padovani 1995;Choi et al. 2022). It is well established that outside

of the dust sublimation radius, AGN and quasars are surrounded by

a dust-laden region with extinction and column densities ranging

from ∼1022 cm−2in the polar direction to ∼1026 cm−2in the mid-

plane (on average), exhibiting “clumpy” sub-structure in both dust

and gas, ubiquitous time variability on ≳yr timescales, and a diverse

array of detailed geometric and reddening properties (see Krolik &

Begelman 1988;Elitzur & Shlosman 2006;Tristram et al. 2007;

Nenkova et al. 2008b,a;Stalevski et al. 2012;Leighly et al. 2015, or

for recent reviews see Netzer 2015;Padovani et al. 2017;Hickox &

Alexander 2018;Baloković et al. 2018), as well as a broad variety of

diﬀerent extinction curve shapes (Laor & Draine 1993;Hopkins et al.

2004;Maiolino et al. 2004;Hatziminaoglou et al. 2009;Gallerani

et al. 2010;Hönig & Kishimoto 2010). It has been recognized for

decades that the torus represents one (of several) natural locations

where bright AGN should drive outﬂows, and indeed many have

gone so far as to propose the “torus” is, itself, an outﬂow (see e.g.

Sanders et al. 1988;Pier & Krolik 1992;Konigl & Kartje 1994;

Elvis 2000;Elitzur & Shlosman 2006). Put simply, because the dust

cross-section to radiation scattering and absorption is generally much

larger than the Thompson cross section, which deﬁnes the Eddington

limit, any AGN accreting at even modest fractions of Eddington

should be able to unbind material via radiation pressure on dust,

launching strong outﬂows. This concept has led to an enormous body

of detailed observational followup (Hönig & Kishimoto 2010;Horst

et al. 2008;Tristram et al. 2009a;Bianchi et al. 2009;Alonso-Herrero

et al. 2011;Kishimoto et al. 2011a;Ricci et al. 2017;Hönig 2019)

arXiv:2210.13517v2 [astro-ph.GA] 6 Oct 2024

2Soliman et al.

and detailed theoretical simulations and models of dust-radiation

pressure-driven outﬂows from AGN in the torus region (Thompson

et al. 2005;Debuhr et al. 2010;Wada et al. 2009;Wada 2012;Roth

et al. 2012;Costa et al. 2018;Thompson et al. 2015;Ishibashi &

Fabian 2015;Chan & Krolik 2016;Baskin & Laor 2018;Ishibashi

et al. 2018;Kawakatu et al. 2020;Venanzi et al. 2020).

Yet despite this extensive literature, almost all the theoretical work

discussed above has assumed that the dust dynamics are perfectly

coupled to the dynamics of the surrounding gas – eﬀectively that

the two “move together” and the dust (even as it is created or de-

stroyed) can simply be treated as some “additional opacity” of the

gas. But in reality, radiation absorbed/scattered by grains acceler-

ates those grains, which then interact with gas via a combination of

electromagnetic (Lorentz, Coulomb) and collisional (drag) forces,

re-distributing that momentum.

Accurately accounting for these interactions is crucial for un-

derstanding any radiation-dust-driven outﬂows. If the dust “free-

streaming length” is very large, grains could simply be expelled

before sharing their momentum with gas (Elvis et al. 2002). If dust

can be pushed into channels, creating low-opacity sight-lines through

which radiation can leak out eﬃciently, some authors have argued

that the coupled photon momentum might be far smaller than the

standard expectation ∼𝜏IR 𝐿/𝑐(where 𝜏IR is the infrared optical

depth; see Krumholz & Thompson 2012 but also Kuiper et al. 2012;

Wise et al. 2012;Tsang & Milosavljević 2015).

Perhaps most importantly, Squire & Hopkins (2018b) showed that

radiation-dust-driven outﬂows are generically unstable to a class of

“resonant drag instabilities” (RDIs). RDIs occur due to diﬀerences in

the forces acting on the dust versus the gas and are inherently unstable

across a broad range of wavelengths. However, the fastest growing

modes, “resonant modes”, arise when the natural frequency of a dust

mode matches that of a gas mode. Each pair of resonant modes leads

to a unique instability with a characteristic growth rate, resonance

and mode structure. In subsequent work (Hopkins & Squire 2018b;

Squire & Hopkins 2018a;Hopkins & Squire 2018a), the authors

showed that systems like radiation-dust-driven outﬂows are unstable

to the RDIs on all wavelengths – even scales much larger than the dust

free-streaming length or mean free path. Subsequent idealized sim-

ulations of these instabilities (Moseley et al. 2019;Seligman et al.

2019a;Hopkins & Squire 2018a) have shown that they can grow

rapidly, reaching signiﬁcant non-linear amplitudes on large scales.

Furthermore, the simulations demonstrated time-dependent cluster-

ing in both dust and gas, and a separation of dust and gas that is

dependent on grain size. Additionally, the RDIs could drive ﬂuctua-

tions in the local dust-to-gas ratios which would aﬀect the absorption

and re-emission of radiation at diﬀerent wavelengths. Speciﬁcally,

as dust dominates the variability in the optical-UV bands but has a

weaker eﬀect on the IR and X-ray bands, dust-to-gas ﬂuctuations can

result in diﬀerences in the observed variability of the AGN emission

across the electromagnetic spectrum.

The insights gained from these simulations are crucial not only

for determining the initiation of an outﬂow but also for explaining

various related phenomena. These include clumping in the torus,

variations in AGN extinction curves, and speciﬁc forms of temporal

variability. AGN sources are known to exhibit variability at essen-

tially all wavelengths and timescales, ranging from hours to billions

of years (Uttley & McHardy 2004;Paolillo et al. 2004,2017;Assef

et al. 2018;Caplar et al. 2017). However, there have been observa-

tions of sources where the X-ray ﬂux varies by approximately 20%

to 80% over a few years, with no apparent variation in the optical

component (Risaliti et al. 2002,2005;Markowitz et al. 2014;Laha

et al. 2020;De Rosa et al. 2007;Smith & Vaughan 2007). In some

cases, ’changing-look’ AGN have shown order of magnitude vari-

ability on timescales as short as a few days to a couple hours (e.g.,

LaMassa et al. 2015;Runnoe et al. 2016;Ruan et al. 2016;McElroy

et al. 2016;Yang et al. 2018;Mathur et al. 2018;Wang et al. 2018;

Stern et al. 2018;Ross et al. 2020;Trakhtenbrot et al. 2019;Hon

et al. 2020). However, the processes driving such variability and the

clumpy nature of the torus remain unexplained.

In this study, we investigate the behaviour of radiation-dust-driven

outﬂows for AGN tori, including explicit dust-gas radiation dynam-

ics for the ﬁrst time. We introduce our numerical methods and ini-

tial conditions in §2, followed by an analysis of our results in §4.

We analyze the morphology, dynamics, and non-linear evolution

of the dusty gas in the simulations, and in §4.2 we compare our

standard simulations results to simulations with full radiation-dust-

magnetohydrodynamics. Additionally, we investigate the feasibility

of launching radiation-driven outﬂows and measure the momentum

coupling eﬃciency within the wind in §4.3. In §5.1, we examine

how the presence of RDIs aﬀects observable AGN properties, such

as time variability. Finally, we provide a summary of our ﬁndings in

§6.

2 METHODS & PARAMETERS

We consider an initially vertically-stratiﬁed mixture of magnetized

gas (obeying the ideal MHD equations) and an observationally-

motivated spectrum of dust grains with varying size, mass, and

charge. The dust and gas are coupled to one another via a com-

bination of electromagnetic and collisional/drag forces. The system

is subject to an external gravitational ﬁeld, and the dust absorbs and

scatters radiation from an external source. In Figure 1, we show a car-

toon illustrating the geometry of our idealized setup and its relation

to an AGN torus.

2.1 Numerical Methods

The numerical methods for our simulations are identical to those in

Hopkins et al. (2022), to which we refer for more details (see also

Hopkins & Lee 2016;Lee et al. 2017;Moseley et al. 2019;Selig-

man et al. 2019b;Hopkins et al. 2020b;Steinwandel et al. 2021;

Ji et al. 2021;Squire et al. 2022 for additional details and appli-

cations of these methods). Brieﬂy, we run our simulations with the

code GIZMO1(Hopkins 2015), utilizing the Lagrangian “meshless

ﬁnite mass method” (MFM) to solve the equations of ideal magne-

tohydrodynamics (MHD; Hopkins & Raives 2016;Hopkins 2016,

2017;Su et al. 2017). Dust grains are modelled as “super-particles”

(Carballido et al. 2008;Johansen et al. 2009;Bai & Stone 2010;

Pan et al. 2011;McKinnon et al. 2018) where each simulated “dust

particle” represents an ensemble of dust grains with a similar grain

size (𝜖grain), charge (𝑞grain), and mass (𝑚grain).

We simulate a 3D box with a base of length 𝐻gas =𝐿xy in the

𝑥𝑦 plane and periodic ˆ𝑥,ˆ𝑦boundaries, and height 𝐿box =𝐿z=

20 𝐿xy in the ˆ𝑧direction with a reﬂecting lower (𝑧=0) and outﬂow

upper (𝑧=+𝐿z) boundary. Dust and gas feel a uniform external

gravitational ﬁeld g=−𝑔ˆ𝑧. The gas has initial uniform velocity

𝑔=0, initial magnetic ﬁeld B0≡𝐵0ˆ

B0in the 𝑥𝑧 plane ( ˆ

B0=

sin(𝜃0

𝐵)ˆ𝑥+cos(𝜃0

𝐵)ˆ𝑧), obeys a strictly isothermal equation of state

1A public version of the code is available at http://www.tapir.caltech.

edu/~phopkins/Site/GIZMO.html

MNRAS 000,1–23 (2021)

Dust Dynamics in AGN Winds 3

(𝑃=𝜌𝑔𝑐2

𝑠), and the initial gas density is stratiﬁed with 𝜌0

𝑔≡𝜌𝑔(𝑡=

0)=𝜌base exp (−𝑧/𝐻gas)(with 𝜌base ≈𝑀gas,box/𝐻3

gas).

Each dust grain obeys an equation of motion

dv𝑑

dt =agas,dust +agrav +arad (1)

=−w𝑠

𝑡𝑠−w𝑠×ˆ

𝑡𝐿+g+

𝜋 𝜖2

grain

𝑚grain 𝑐⟨𝑄⟩ext Grad

where v𝑑is the grain velocity; w𝑠≡v𝑑−u𝑔is the drift velocity for

a dust grain with velocity v𝑑and gas velocity u𝑔at the same position

x;Bis the local magnetic ﬁeld; agas,dust =−w𝑠/𝑡𝑠−w𝑠×ˆ

B/𝑡𝐿

includes the forces from gas on dust including drag (in terms

of the “stopping time” 𝑡𝑠) and Lorentz forces (with gyro/Larmor

time 𝑡𝐿); agrav =gis the external gravitational force; and arad

is the force from radiation in terms of the grain size 𝜖grain, mass

𝑚grain ≡ (4𝜋/3)¯𝜌𝑖

grain 𝜖3

grain (in terms of the internal grain density

¯𝜌𝑖

grain), dimensionless absorption+scattering eﬃciency ⟨𝑄⟩ext, speed

of light 𝑐, and radiation ﬁeld Grad ≡Frad −v𝑑·(𝑒rad I+Prad)in terms

of the radiation ﬂux/energy density/pressure density Frad,𝑒rad,Prad.

The dust is initialized with the local homogeneous steady-state equi-

librium drift and a spatially-uniform dust-to-gas ratio 𝜌0

𝑑=𝜇dg 𝜌0

𝑔.

For all forces “from gas on dust” 𝑎gas,dust the gas feels an equal-and-

opposite force (“back-reaction”). The dust gyro time is given in terms

of the grain charge 𝑞grain =𝑍grain 𝑒as 𝑡𝐿≡𝑚grain 𝑐/|𝑞grain B|, and

for the parameter space of our study the drag is given by Epstein drag

(as opposed to Coulomb or Stokes drag) with

𝑡𝑠≡𝜋𝛾

¯𝜌𝑖

grain 𝜖grain

𝜌𝑔𝑐𝑠1+9𝜋𝛾

128 |w𝑠|2

𝑐2

𝑠−1/2

,(2)

We adopt a standard empirical Mathis et al. (1977) power-law grain

size spectrum with diﬀerential number 𝑑𝑁d/𝑑𝜖grain ∝𝜖−3.5

grain with a

range of a factor of 100 in grain size (𝜖max

grain =100 𝜖min

grain). We as-

sume the grain internal density/composition is independent of grain

size, and assume the charge-to-mass ratio scales as |𝑞grain|/𝑚grain ∝

𝜖−2

grain, consistent with grains charged by a range of processes rel-

evant in this regime such as collisions, Coulomb, photo-electric,

or electrostatically-limited processes (Draine & Sutin 1987;Tielens

2005).

As in Hopkins et al. (2022), we consider two diﬀerent treat-

ments of the radiation ﬁelds. Given the range of column densi-

ties we will explore, we are interested in the multiple-scattering

regime, or equivalently Rayleigh scattering. In this regime, the ra-

diation should be in the long-wavelength limit (spectrum peaked at

wavelengths 𝜆rad ≫𝜖grain), so we expect and assume the spectrally-

averaged ⟨𝑄⟩ext ∝𝜖grain, and we approximate the radiation with a

single band (spectrally-integrated), so eﬀectively treat the grains as

introducing a grain size-dependent but otherwise “grey” isotropic

scattering opacity. In our ﬁrst simpliﬁed treatment (our “constant

ﬂux” simulations), we assume the radiation ﬁelds obey their homo-

geneous equilibrium solution, giving Grad ≈Frad ≈F0=𝐹0ˆ𝑧.

This is a reasonable approximation so long as the radiation is not

“trapped” in highly-inhomogeneous dust clumps. But we also run

a subset of “full radiation-dust-magnetohydrodynamic” (RDMHD)

simulations where the radiation ﬁeld is explicitly evolved using to

the full M1 radiation-hydrodynamics treatment in GIZMO (Lupi

et al. 2017,2018;Hopkins & Grudić 2019;Hopkins et al. 2020a;

Grudić et al. 2021), including terms to O(𝑣2/𝑐2):𝜕𝑡𝑒rad +∇·Frad =

−𝑅dust v𝑑·Grad/𝑐2,𝜕𝑡Frad +𝑐2∇ · Prad =−𝑅dust Grad, where the

absorption/scattering coeﬃcients 𝑅dust are calculated directly from

the explicitly-resolved dust grain populations (consistent exactly with

the radiation ﬂux they see in arad).

Our default simulation parameter survey adopts 106gas cells and

4×106dust super-particles. And unless otherwise speciﬁed, our

analysis uses the “full RDMHD” simulations. Readers interested in

details should see Hopkins et al. (2022). In that paper, we applied

these numerical methods to simulations of radiation-dust-driven out-

ﬂows in molecular clouds and HII regions. The key diﬀerences are

(1) we consider a very diﬀerent parameter space (much higher den-

sities and stronger radiation ﬁelds), which lead to qualitatively dif-

ferent instabilities and behaviours, and (2) we speciﬁcally model the

multiple-scattering regime, while Hopkins et al. (2022) focused only

on the single-scattering limit.

2.2 Parameter Choices

Our simulations are then speciﬁed by a set of constants (size and

charge of the largest grains, dust-to-gas ratio, radiation ﬂux, etc.). To

motivate these, we consider a ﬁducial case of dust around a bright

quasar. We expect the most dramatic eﬀects of radiation on dust at the

distances closest to the black hole where grains can survive, i.e. just

outside the dust sublimation radius 𝑟sub ∼ (𝐿QSO/4𝜋 𝜎SB 𝑇4

sub)1/2

where 𝑇sub ∼2000 K is the dust sublimation temperature and we will

consider a typical quasar with 𝐿QSO ∼1046 erg s−1(i.e. 𝑀B∼ −24,

a typical ∼𝐿∗or modestly sub-𝐿∗QSO at redshifts 𝑧∼1−6; see

Shen et al. 2020), so 𝑟sub ∼0.3pc and this corresponds to a BH of

mass 𝑀BH ∼108𝑀⊙accreting near its Eddington limit.

We then take 𝐻gas ∼𝑟sub,𝐹0∼𝐿QSO/(4𝜋 𝑟2

sub),𝑔∼

𝐺 𝑀BH/𝑟2

sub, typical ¯𝜌𝑖

grain ∼1.5 g cm−3and absorption eﬃciency

for the largest grains ⟨𝑄⟩ext (𝜖grain =𝜖max

grain) ∼ 0.2(Draine &

Lee 1984), and initial magnetic ﬁeld strength given by a plasma

𝛽0≡ (𝑐𝑠/𝑣𝐴[𝑧=0])2=4𝜋 𝜌base (𝑐𝑠/𝐵0)2∼1with an arbitrary

angle 𝜃0

𝐵=𝜋/4(though this is essentially a nuisance parameter

here). Observational constraints suggest the dust-to-gas ratios inte-

grated along AGN lines of site range from 0.01-1 times the galac-

tic values (Burtscher et al. 2016;Maiolino et al. 2001;Esparza-

Arredondo et al. 2021). However, these measurements include re-

gions within the dust sublimation radius and therefore should be

interpreted as lower limits. Several studies suggest that the Broad-

Line region (BLR) has super-solar dust-to-gas ratios (Nenkova et al.

2008c;Kishimoto et al. 2009;Sturm et al. 2006). Therefore, given

these uncertainties, we assume a standard (galactic) dust-to-gas ra-

tio 𝜇dg =0.01. Further, we consider various values of 𝜖max

grain from

0.01 𝜇m(smaller grains than typical in the diﬀuse ISM) through

1𝜇m(larger), and also explore variations in the gas density parame-

terized via the gas column density integrated through the box to inﬁn-

ity, 𝑁H,gas ≡𝑚−1

𝑝𝜌0

𝑔𝑑𝑧 =𝜌base 𝐻gas/𝑚𝑝∼1022 −1026 cm−2,

representative of observed values through diﬀerent lines-of-sight of

angles through the AGN torus.

The one remaining parameter is the dust charge. We have consid-

ered both (a) cases where the grains are strongly shielded and the gas

neutral/cold, so collisional charging dominates, and (b) cases where

some photo-electric (non-ionizing UV) ﬂux can reach the grains.

Given the scalings for grain charge in both regimes (Tielens 2005;

Draine & Sutin 1987), if even a small fraction of the QSO photoelec-

tric ﬂux reaches the grains, they will generally reach the electrostatic

photoelectric charging limit such that the equilibrium grain charge

⟨𝑍grain⟩ ∼ 5000 (𝜖grain/𝜇m)(Tielens 2005). For simplicity, we adopt

this by default. However, we note that using the collisional charge

expression from Draine & Sutin (1987), which results in a signiﬁcant

decrease in |𝑍grain |, has little eﬀect. This is because we ﬁnd that in

MNRAS 000,1–23 (2021)

4Soliman et al.

AGN

rsub

Frad

Hgas

20 Hgas

Dusty Torus

Simulation Box

arad, dust

Dust:

NH ~ 1026 cm-2

NH ~ 1024 cm-2

NH ~ 1022 cm-2

Figure 1. Cartoon illustrating our simulation setup. We simulate 3D boxes of size 𝐻gas ×𝐻gas ×20 𝐻gas along the ˆ𝑥,ˆ𝑦and ˆ𝑧directions respectively with

∼106resolution elements. We enforce outﬂow upper and reﬂecting lower boundary conditions with periodic sides. The gas and dust are initially stratiﬁed such

that 𝜌gas ∝𝑒−𝑧/𝐻gas , and 𝜌d=𝜇dg

0𝜌gas where 𝜇dg

0=0.01 corresponding to a uniform dust-to-gas ratio. The gas follows an isothermal (𝛾=1) EOS with sound

speed 𝑐𝑠, an initial magnetic ﬁeld B0=|B|(sin𝜃0

Bˆx +cos𝜃0

Bˆz)in the ˆ𝑥−ˆ𝑧plane and gravitational acceleration g=−𝑔ˆ𝑧. The dust grains are modelled as

super-particles each representing a population of grains of a given size sampled from a standard MRN spectrum with a factor = 100 range of sizes. The grains

are photo-electrically charged, with the charge appropriately scaled according to grain size. They experience an upward acceleration arad,dust due to absorption

of an initial upward radiation ﬂux 𝐹0=+𝐹0ˆ𝑧corresponding to radiation from an AGN located a sublimation radius rsub distance away, and are coupled to the

gas through drag and Lorentz forces. We consider a range of 1022 −1026 cm−2in column densities representing diﬀerent lines-of-sight angles through the dusty

torus.

the parameter space of interest, the magnetic grain-gas interactions

(grain charge eﬀects) are sub-dominant, even with the larger |𝑍grain |.

In Appendix A, we provide a table that lists the speciﬁc parameters

for each simulation.

3 ANALYTIC EXPECTATIONS & BACKGROUND

Hopkins & Squire (2018b) analyzed the equations of mass and mo-

mentum conservation using a linear stability approach to investigate

the behaviour of an unstable RDI mode in a dust-gas mixture similar

to those simulated in our study. They found that the behaviour of an

unstable mode with wave-vector kis characterized by the dimension-

less parameter k·ws⟨𝑡𝑠⟩, where ⟨𝑡𝑠⟩=𝑡𝑠(⟨𝜌𝑔⟩,⟨ws⟩) corresponds to

the stopping time at the equilibrium gas density ⟨𝜌𝑔⟩and equilibrium

drift velocity ⟨ws⟩of the dust particles. This parameter represents

the ratio of the dust stopping length to the wavelength of the mode,

and deﬁnes three regimes of the instabilities,











k·ws⟨𝑡𝑠⟩≲𝜇dg (Low-k, long-wavelength)

𝜇dg ≲k·ws⟨𝑡𝑠⟩≲(𝜇dg)−1(Mid-k, intermediate wavelength)

k·ws⟨𝑡𝑠⟩≳(𝜇dg)−1(High-k, short-wavelength).

(3)

separated by their linear growth rate scaling and mode structure.

The diﬀerent regimes can be further understood by considering the

parameter 𝜇dg k·ws⟨𝑡𝑠⟩, which can be interpreted as the ratio of the

force exerted by the dust on the gas to the gas pressure forces for a

given scale |k|(Moseley et al. 2019). The mid-k and high-k regimes

exhibit similar behaviour and occur when the gas pressure dominates

the dynamics on the scales being considered. Therefore, the resonant

mode occurs when the drift velocity aligns with the propagation

direction of the gas mode, as given by ˆ

k·ws=±𝑐𝑠. On the other

hand, the low-k regime arises when the bulk force exerted by the

dust on the gas becomes stronger than the gas pressure forces, and

the dust dominates the ﬂow. Resonant modes in this regime typically

align with ws.

As shown in Equation 3, the dust-to-gas ratio plays an important

role in distinguishing the diﬀerent RDI regimes. However, for most

of our simulations, transitioning into a diﬀerent regime would re-

quire a signiﬁcant adjustment of 𝜇dg by several orders of magnitude.

Given the speciﬁc environmental conditions we aim to model and

the likelihood of accurately representing the intended scenario while

having such drastic variations in 𝜇dg, we choose to use our ﬁducial

value for 𝜇dg in all simulations. For a study of the eﬀect of varying

𝜇dg on the behaviour of the RDIs, we refer readers to Moseley et al.

(2019).

Rewriting the regimes above in terms of wavelength, we can see

that 𝜆crit ∼ ( ¯𝜌𝑖

grain 𝜖grain)/(𝜇dg 𝜌𝑔) ∼ ˜𝛼Hgas/𝜇dg deﬁnes the criti-

cal wavelength above which modes are in the low-k regime, where

˜𝛼≡ ( ¯𝜌𝑖

grain 𝜖grain)/(𝜌base 𝐻gas)is the dimensionless grain size pa-

rameter which characterizes the coupling strength between the dust

and gas. For the parameter set explored here, ˜𝛼≪𝜇dg, we ﬁnd that

largest-wavelength interesting modes (𝜆∼Hgas ≫𝜆crit) always lie

in the "long-wavelength" regime. Within the linear theory frame-

work, this mode behaves as a "compressible wave", with similar dust

and gas velocity perturbations that are nearly in phase and parallel to

the wave-vector ˆ

k. This will therefore drive relatively weak dust-gas

separation with respect to other regimes previously studied in Hop-

kins et al. (2022). The linear growth timescale 𝑡grow of the fastest

growing modes in this regime scales approximately as:

𝑡grow(𝑘) ∼ 1

𝔉(𝑘)∼𝜇dg⟨w2

𝑠⟩𝑘2

⟨𝑡𝑠⟩−1/3,(4)

MNRAS 000,1–23 (2021)

Dust Dynamics in AGN Winds 5

where 𝔉(𝑘)is the linear growth rate for a mode with wave-number 𝑘

(Hopkins & Squire 2018b). Importantly, as shown therein, the fastest

growing mode in the linear long-wavelength regime is the “pressure-

free” mode, which is weakly dependent on the magnetization and

thermal physics of the gas. We discuss this further below.

We deﬁne the geometrical optical depth 𝜏geo instead of the

“observed” optical depth 𝜏𝜆since the latter depends on the

observed wavelength (the same integral replacing 𝜋𝜖2

grain →

𝑄𝜆(𝜖grain, 𝜆)𝜋𝜖2

grain), integrated from the base of the box to inﬁnity.

Assuming a vertically stratiﬁed environment and dust grains with a

power-law grain size spectrum, we can express 𝜏geo strictly in terms

of our simulation parameters,

𝜏geo ≡∞

0𝜋𝜖2𝑛grain 𝑑𝑧

=𝐶 𝜇dg 𝜌𝑔𝐻gas

𝜌𝑑𝜖max

grain

=𝐶𝜇dg

˜𝛼m,(5)

where 𝑛grain is the number density of dust grains, ˜𝛼mis the di-

mensionless maximum grain size parameter ( ˜𝛼evaluated at 𝜖grain =

𝜖max

grain), and 𝐶is a constant of order 20.

Another useful parameter is the “free streaming length” of the dust

(relative to the gas),

ℓstream,dust

𝐻gas ∼10−4𝜖grain

𝜇m 1024 cm−2

𝑁H,gas ∝𝜏−1

geo.(6)

Therefore, for all our simulations, the grains are “well-coupled”

to the gas in the sense that ℓstream,dust ≪𝐻gas, so we do not expect

them to simply “eject” from the gas without interacting and sharing

momentum.

3.1 Parameters & Physics with Weak Eﬀects

We now discuss physical parameters that we tested, but found to have

weak to no eﬀect on the behaviour of the instabilities within this

regime including magnetic ﬁeld strength, magnetic ﬁeld direction,

AGN luminosity, grain charge, and strength of gravity.

3.1.1 Charging Physics & Magnetic Field Strength

We ran tests varying the magnetic ﬁeld strength 𝐵0, or equivalently

the plasma 𝛽, and magnetic ﬁeld orientation 𝜃𝐵within the box. Simi-

larly, as the grain charge is unconstrained, we consider diﬀerent grain

charging mechanisms (collisional vs. photoelectric) and found these

parameters to have a negligible eﬀect on the long-term behaviour

of the instabilities. This is due to two reasons. Firstly, this arises

naturally within AGN-like environments where Lorentz forces are

weak relative to the drag force, i.e., 𝑡𝑠/𝑡𝐿∼˜

𝜙m/˜𝑎1/2

d≪1where

𝜙≡3𝑍0

grain [𝜖max

grain]𝑒/(4𝜋 𝑐 (𝜖max

grain)2𝜌1/2

base)is the dimensionless

grain charge parameter, and ˜𝑎d≡ (3/4) (𝐹0⟨𝑄⟩ext /𝑐)/(𝜌base 𝑐2

𝑠)

is the dimensionless dust acceleration parameter. Secondly, the dom-

inant modes in our simulations are in the “long-wavelength regime”,

and hence, are only weakly sensitive to magnetic eﬀects as the mag-

netic pressure and tension provide only second-order corrections to

what is to leading order a “collisionless” or “pressure-free” mode

(Hopkins & Squire 2018a). Therefore, we observe that at early stages

of the RDIs’ development, ampliﬁed magnetic ﬁelds, or higher grain

charge-to-mass ratios merely result in density perturbations propa-

gating at slightly diﬀerent angles ∼𝜃𝐵, but the ﬂuid ﬂow retains its

general properties. Further, as the instabilities reach the non-linear

stage of their evolution, this propagation angle decreases till the ﬂuid

is moving roughly parallel to the vertical acceleration, and we see

essentially no eﬀect on the medium.

3.1.2 Thermal State of Gas

We ﬁnd that the choice of the thermal equation-of-state of the gas 𝛾,

and therefore the speed of sound 𝑐𝑠do not aﬀect our results. As the

grains are accelerated to super-sonic velocities, 𝑐𝑠factors out of the

relevant equations such as the stopping time and the growth rates of

the modes to leading order in the linear theory for these particular

long-wavelength modes of interest.

3.1.3 Gravity

Further, as shown in Table A1, for this environment, the strength of

gravity is much weaker than the acceleration due to radiation, i.e.,

˜𝑔/˜𝑎𝑑∼10−3(𝜖max

grain/𝜇m), where ˜𝑔≡ |g|𝐻gas/𝑐2

𝑠is the dimension-

less gravity parameter and ˜𝑎d≡ (3/4) (𝐹0⟨𝑄⟩ext /𝑐)/(𝜌base 𝑐2

𝑠)is

the dimensionless acceleration parameter. Thus, gravity acts merely

to ensure that the gas that is left behind the wind “falls back”, but

does not have a noticeable eﬀect on the general behaviour of the

RDIs. It is easy to verify that for the conditions and timescales we

emulate here, the self-gravity of the gas should also be unimportant.

3.1.4 AGN Luminosity

Naively, the AGN luminosity should have an important eﬀect here.

However, in the dimensionless units in which we will work, i.e. length

in units of ∼𝐻gas ∼𝑟sub, time in units of the “acceleration time”

deﬁned below, the absolute value of the AGN luminosity factors out

completely. Nonetheless, while the AGN luminosity does not aﬀect

the qualitative behaviour of the RDIs (in the appropriate units), it

eﬀectively deﬁnes the characteristic time and spatial scales of the

problem. For example, the AGN luminosity normalizes the subli-

mation radius, i.e. 𝑟sub ∼0.3 pc 𝐿1/2

46 . This means if we deﬁne the

ﬂux at the base of our box as the ﬂux at 𝑟sub (as we do), the AGN

luminosity factors out (the ﬂux at 𝑟sub is, by deﬁnition, ﬁxed (Ivezić

& Elitzur 1997)), and we ﬁnd that the vertical acceleration of the

column, 𝑎eﬀ ≡𝜇dg𝑎dust −𝑔∼𝑎eﬀ ≡𝜇dg𝑎dust, where 𝑎dust is the

acceleration experienced by the dust, has the following scaling,

𝑎eﬀ ∼0.3cm s−21𝜇m

𝜖max

grain ,(7)

which is independent of the AGN luminosity, and only depends on

the maximum size of the grains.

It is worth noting that our choice of normalization is not arbi-

trary. In the context of dust-driven winds, our focus is on regions

where dust is present, i.e., beyond the sublimation radius. When the

radius is much smaller than the sublimation radius (𝑟≪𝑟sub), the

dust is expected to be sublimated, and the dominant mechanism for

driving the wind would be line-driving rather than dust absorption

(Proga et al. 2000). Conversely, when the radius is much larger than

the sublimation radius (𝑟≫𝑟sub), the radiation ﬂux decreases ac-

cording to the inverse square law. In our simulations, we observe

that the wind originates from the base of the column where the ra-

diation ﬂux is strongest, which aligns with our expectations. The

sublimation radius can be derived analytically by assuming thermal

equilibrium, allowing allows us to express the sublimation radius as

MNRAS 000,1–23 (2021)

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MNRAS000,1–23(2021)Preprint8October2024CompiledusingMNRASLATEXstylefilev3.0DustDynamicsinAGNWinds:ANewMechanismForMultiwavelengthAGNVariabilityNadineH.Soliman1,PhilipF.Hopkins11TAPIR,Mailcode350-17,CaliforniaInstituteofTechnology,Pasadena,CA91125,USAAcceptedXXX.ReceivedYYY;inoriginalformZZZABSTRACTP...

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Dust Dynamics in AGN Winds A New Mechanism For Multiwavelength AGN Variability

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