Effects of opacity temperature dependence on radiatively accelerated clouds

ABSTRACT

We study how different opacity–temperature scalings affect the dynamical evolution of irradiated gas clouds using time-dependent radiation-hydrodynamics simulations. When clouds are optically thick, the bright side heats up and expands, accelerating the cloud via the rocket effect. Clouds that become more optically thick as they heat accelerate |$\sim\! 35{{\ \rm per\ cent}}$| faster than clouds that become optically thin. An enhancement of |$\sim\! 85{{\ \rm per\ cent}}$| in the acceleration can be achieved by having a broken power-law opacity profile, which allows the evaporating gas driving the cloud to become optically thin and not attenuate the driving radiation flux. We find that up to |$\sim\! 2{{\ \rm per\ cent}}$| of incident radiation is re-emitted by accelerating clouds, which we estimate as the contribution of a single accelerating cloud to an emission or absorption line. Re-emission is suppressed by ‘bumps’ in the opacity–temperature relation since these decrease the opacity of the hot, evaporating gas, primarily responsible for the reradiation. If clouds are optically thin, they heat nearly uniformly, expand and form shocks. This triggers the Richtmyer–Meshkov instability, leading to cloud disruption and dissipation on thermal time-scales. Our work shows that, for some parameters, the rocket effect due to radiation-ablated matter leaving the back of the cloud is important for cloud acceleration. We suggest that this rocket effect can be at work in active galactic nuclei outflows.

hydrodynamics, radiation: dynamics, stars: massive, stars: winds, outflows, quasars: general, X-rays: galaxies

1 INTRODUCTION

Gas clouds appear in many astrophysical systems such as the interstellar medium (ISM), intergalactic medium (IGM), and active galactic nuclei (AGN). In AGN, X-ray studies show that observed column densities are variable, suggesting that the torus and broad-line region is clumpy and very dynamic (e.g. Krumpe, Markowitz & Nikutta 2014; Ramos Almeida & Ricci 2017, and references therein). Likewise multiwavelength observations of galactic winds from radio to X-ray find multicomponent, multitemperature outflows of gas and dust (e.g. Veilleux, Cecil & Bland-Hawthorn 2005, and references therein). Two important avenues of study are what physical processes are responsible for accelerating the clouds and how efficiently is energy deposited from the radiation field to the gas (Tombesi et al. 2015)?

In one picture, clouds are accelerated indirectly by advecting them into an accelerating hot wind (Murray et al. 2007). However, they are susceptible to being destroyed via hydrodynamic instabilities before being accelerated to significant velocities (Poludnenko, Frank & Blackman 2002; Scannapieco & Brüggen 2015; Brüggen & Scannapieco 2016), though cloud magnetization may suppress such fragmentation (Cooper et al. 2009; McCourt et al. 2015, 2018). Thompson & Krumholz (2016) proposed that though clouds may be destroyed, the entraining hot wind may radiatively cool on larger scales and clouds may form further downstream in the flow via the classical thermal instability (Field 1965).

An alternative to the entrainment picture is one where clouds are accelerated directly via radiation pressure on dust (Murray, Quataert & Thompson 2005). Assuming a central source of luminosity L, a momentum flux |$\dot{P} \sim L/c$| is imparted on the gas (Krumholz & Thompson 2012, 2013). Observations in AGN find that clouds have momenta ≳10L/c (Rupke & Veilleux 2011; Faucher-Giguère, Quataert & Murray 2012), suggesting a direct transfer of momentum from the radiation field to the gas is insufficient to accelerate the clouds to sufficiently high velocities.

Finally, energy may be deposited directly into clouds when the wind shocks the ISM (Chevalier & Imamura 1982). Though the shock conserves energy, inverse Compton cooling (King 2003) may subsequently allow some energy to radiatively escape. The deposited energy can then accelerate clouds via the rocket effect (Oort & Spitzer 1955, hereafter OS55).

The key to modelling these systems is understanding the radiation and gas coupling. When the gas is mostly neutral, the coupling is dominated by bound–free interactions, whereas when it is ionized the interactions are dominated by free–free processes. Photoionization calculations (see e.g. Iglesias & Rogers 1996) show that opacity is a complex function of gas density and temperature. Before we introduce sophisticated microphysics into our models, such as Compton heating/cooling, radiation pressure due to spectral lines or dust, we aim to understand how different opacity scalings as a function of temperature affect cloud dynamics.

We consider an opacity parametrized by a power law:

$$\begin{eqnarray*} \kappa = \kappa _0 \left(\frac{T}{T_0}\right)^{-s} \left(\frac{\rho }{\rho _0}\right)^{n-1}, \end{eqnarray*}$$

(1)

where T and ρ are the temperature and density, respectively, and the 0 subscript denotes some fiducial value. The power-law indices s and n encapsulate the microphysics responsible for the gas opacity.

Previous numerical simulations have studied clouds in different physical regimes. For example, Proga et al. (2014, hereafter P14) used the Kramers form of opacity, s = 3.5 and n = 2, to study cloud evolution in the broad-line region of AGN. They found the clouds, which are optically thick to absorption, disperse before they can move more than a few cloud radii. Zhang et al. (2018, hereafter Z18) studied the evolution of dusty clouds in rapidly star-forming galaxies where radiation flux is dominated by the infrared (IR). They used an opacity scaling with s = −2 and n = 1 and found that clouds can be significantly accelerated without being dispersed. The scaling of opacity with temperature in these two studies is different, leading to qualitatively different responses from the irradiated cloud. In P14, as the cloud absorbs radiation, it heats up and because s > 0 becomes less optically thick, thereby slowing down the heating. In contrast, the Z18 case has s < 0, so cloud heating is a runaway process as opacity increases with temperature.

These earlier cloud models used a simplified, power-law expression for the opacity (equation 1). Photoionization calculations have shown that the Rosseland mean opacity is not monotonic, with features due to H, He, and Fe. The iron opacity peak has been shown to be important in the structure and stability of massive star envelopes (Jiang et al. 2015) and AGN discs (Jiang, Davis & Stone 2016). On either side of these features the opacity scaling changes sign, potentially affecting the cloud dynamics. To build an intuition for models where opacity is computed self-consistently with photoionization codes, we study cloud acceleration models where opacity scales like equation (1) for different temperature power-law scalings s.

Our models consist of overdense, cold, spherical clouds in pressure equilibrium with a dilute, hot, ambient gas irradiated from one side. We consider two sets of simulations exploring the effects of the temperature scaling of cloud opacity. In one set of models, we keep the optical depth of the cloud constant but vary κ₀. In another set of models we keep the opacity of the ambient gas fixed, and vary the optical depth of the cloud. In all our models the cloud is initially optically thick and the ambient gas is optically thin.

We find two types of behaviour: clouds can balloon outward or they may accelerate away from the radiation source. The former occurs if the cloud becomes optically thin and thus heats nearly uniformly, as in the P14 models. The later occurs if the cloud remains optically thick and heats non-uniformly, accelerating the cloud via the rocket effect (OS55) as hot gas evaporates away, as in the Z18 models (see also Mellema 1998 and references therein). We then consider models where the opacity scaling with temperature changes sign at a critical temperature, to model the effect of a ‘bump’ in the opacity (see e.g. fig. 5.2 in Hansen, Kawaler & Trimble 2004). We find that this can change the heating rate or acceleration efficiency, but it does not qualitatively change the dynamics.

The outline of our paper is as follows. In Section 2, we describe our numerical set-up for modelling the clouds. In Section 3.1, we describe our main results for clouds with monotonic dependence of opacity on temperature, and in Section 3.2 describe results for models with broken power-law opacity, simulating a feature in the opacity profile. In Section 4, we discuss applications of this work, in particular to modelling clouds in the broad-line region of AGN and for heating gas in the IGM. We conclude in Section 5 where we discuss the physical processes we would like to include in future simulations of clouds and the prospects for studying multicloud systems.

2 NUMERICAL METHODS

We performed all numerical simulations with the developmental version of the radiation magnetohydrodynamics (rad-MHD) code athena++  (Stone et al. in preparation), a rewrite of the MHD code athena (Gardiner & Stone 2005, 2008), optimized for adaptive mesh refinement and various modules incorporating new physics including, crucially for this work, radiation transport (Jiang, Stone & Davis 2012, 2014). The basic physical set-up is a 2D box with initially constant gas pressure, centred on an overdense spherical cloud. Radiation flux enters the box along a fixed direction, which is assumed to be emitted from a faraway blackbody, hotter than the gas. The radiation causes the cloud to heat, accelerate, and shear, depending on the strength of the opacity. We describe our set-up in more detail below.

2.1 Basic equations

The equations for fluid hydrodynamics coupled to a radiation field are

$$\begin{eqnarray*} \frac{\partial \rho }{\partial t} + \nabla \cdot \left(\rho \boldsymbol{v} \right) = 0, \end{eqnarray*}$$

(2a)

$$\begin{eqnarray*} \frac{\partial (\rho \boldsymbol{v})}{\partial t} + \nabla \cdot \left(\rho \boldsymbol{vv} + {\sf {P}} \right) = -\mathbb {P} \boldsymbol{S}_\mathrm{ r}(\mathbf {\boldsymbol{P}}), \end{eqnarray*}$$

(2b)

$$\begin{eqnarray*} \frac{\partial E}{\partial t} + \nabla \cdot \left((E + P)\boldsymbol{v} \right) = - \mathbb {PC} S_\mathrm{ r}(E), \end{eqnarray*}$$

(2c)

where ρ and |$\boldsymbol {v}$| are the fluid density and velocity, respectively, and |${\sf {P}}$| is a diagonal tensor with components P the gas pressure. The total gas energy density is |$E = 1/2 \rho |\boldsymbol{v}|^2 + \mathcal {E}$|⁠, where |$\mathcal {E} = P/(\gamma -1)$| is the internal energy density. The isothermal sound speed is a² = P/ρ and the adiabatic sound speed |$c_\mathrm{ s}^2 = \gamma a^2$|⁠. The temperature is |$T = (\gamma -1)\mathcal {E}\mu m_{\rm {p}}/\rho k_{\rm {B}}$|⁠, where μ = 1.0 is the mean molecular weight and other symbols have their standard meaning. The radiation source terms |$\boldsymbol{S}_\mathrm{ r}$| and S_r(E) are calculated from the difference between the angular quadratures of the specific intensity |$I(\boldsymbol{n})$| along unit vectors |$\boldsymbol{n}$| in the lab frame before and after adding the source terms (see Jiang, Stone & Davis 2019). To provide some physical intuition, the radiation source terms are at lowest order in v/c,

$$\begin{eqnarray*} \boldsymbol{S}_\mathrm{ r}(\boldsymbol{P}) = - \left(\sigma _\mathrm{ s} + \sigma _\mathrm{ a} \right) \mathbf {F}_\mathrm{ r} + \mathcal {O}\left(v/\mathbb {C}\right), \end{eqnarray*}$$

(3a)

$$\begin{eqnarray*} S_\mathrm{ r}(E) = \sigma _\mathrm{ a} \left(T^4 - E_\mathrm{ r}\right) + \mathcal {O}\left(v/\mathbb {C}\right), \end{eqnarray*}$$

(3b)

where the radiation energy density and flux are

$$\begin{eqnarray*} E_\mathrm{ r} = 4 \pi \int I \, \mathrm{ d}\Omega, \end{eqnarray*}$$

(4a)

$$\begin{eqnarray*} \boldsymbol{F}_\mathrm{ r} = 4 \pi c \int \boldsymbol{n} I \,\mathrm{ d}\Omega, \end{eqnarray*}$$

(4b)

and the absorption and scattering cross-sections are σ_a and σ_s, respectively, and the integrals are over all solid angles Ω.

2.2 Initial conditions

Initially the box is in hydrostatic equilibrium with gas pressure P₀ and a circular cloud at the centre of the grid with density profile

$$\begin{eqnarray*} \rho = \rho _0 + \frac{\rho _1 - \rho _0}{1 + \exp (10(r-1))}, \end{eqnarray*}$$

(5)

where ρ₁ = 200ρ₀ is the maximum cloud density and r² = (x/x₀)² + (y/y₀)². Here x₀ = y₀ = 0.05 is the radius of the cloud. Because the higher density cloud is in pressure equilibrium with the ambient gas, its temperature is less than T₀.

Our set-up is meant to simulate a cloud far from the radiation source. athena++  assumes light is emitted isotropically for point sources, for rays along angles computed from the algorithm described in Lowrie, Morel & Hittinger (1999). To simulate plane-parallel radiation we use four uniformly distributed rays in the 2D plane-parallel to the diagonals of the box (i.e. we set the code parameter n_ang = 4). This set-up allows us to resolve the rectangularly shaped ‘shadow’ behind optically thick clouds irradiated by a plane-parallel source. We study the cloud dynamics in the rotated coordinate system with x-axis parallel to the incident flux.

Radiation is coupled to the gas via an absorption coefficients σ_a = ρκ. Substituting our parametrization for the opacity (1), the absorption coefficient is

$$\begin{eqnarray*} \sigma _\mathrm{ a} = \sigma _{\mathrm{ a},0} \left(\frac{T}{T_0}\right)^{-s} \left(\frac{\rho }{\rho _0}\right)^n. \end{eqnarray*}$$

(6)

For simplicity, we set the scattering coefficient σ_s = 0. In our fiducial model the absorption coefficient σ_{a, 0} = 0.1 and the optical depth of the cloud is initially τ_a = 2x_oσ_a = 2. In one set of simulations we vary the power-law coefficient s but keep |$\sigma _{\mathrm{ a},0} = \rm {const}$|⁠, thus effectively varying the cloud optical depth τ_a. In the second set of models, as we vary s we keep the optical depth |$\tau _\mathrm{ a} = \rm {const}$| and thus vary σ_{a, 0}. In all cases the absorption optical depth of the ambient gas τ_{a, g} ≤ 0.1, so we refer to it as optically thin.

On the top and right-hand sides of the box we impose outflow conditions on the gas variables and vacuum conditions on the radiation. Along the bottom and left-hand side of the box we keep density and pressure fixed at ρ₀ and P₀, respectively, while ensuring velocity is conserved when we perform this update.

Our simulation uses dimensionless parameters, but for AGN clouds reasonable parameters might be cloud temperature |$T_0 = 2.44 \times 10^{6}\, \rm {K}$| and the cloud density is |$\rho _0 = 1 \, \rm {g \, cm^{-3}}$|⁠. The pressure is then |$P_0 = \rho _0 T_0 k_{\rm {B}}/\mu m_\mathrm{ p} = 2.02 \times 10^{14}\, \rm {erg \, cm^{-3}}$|⁠, where we assumed μ = 1. The isothermal sound speed |$a_0 = \sqrt{P_0/\rho _0} = 1.42 \times 10^{7}\, \rm {cm \, s^{-1}}$| and the adiabatic sound speed |$c_\mathrm{ s} = \sqrt{\gamma }a_0 = 1.83 \times 10^{7}\, \rm {cm \, s^{-1}}$|⁠. The dimensionless speed of light |$\mathbb {C} = c/a_0 = 2.1 \times 10^{3}$| and the ratio of radiation pressure to gas pressure |$\mathbb {P} = a_\mathrm{ r} T_0^4/P_0 = 10^{-3}$|⁠.

3 RESULTS

We investigate models with a variety of opacity coefficients of the form (6), with power-law scalings ranging −1 ≤ s ≤ 1 and n = 2. We describe the basic physics governing the dynamics of these monotonic power-law models in Section 3.1.

More realistic modelling using photoionization codes finds that gas opacity is not a simple power law but has features due to specific chemical elements, notably H, He, and Fe. In Section 3.2, we investigate the effect of such features by studying models where the opacity is a broken power law, turning over from s = −1 to s = 1 at a critical temperature T_c.

3.1 Monotonic power-law opacity

A summary of our models with power-law scalings of the form (6) is shown in Table 1, where we indicate the relevant parameters such as cloud optical depth τ_a, absorption cross-section σ_{a, 0}, power-law scaling s, and the sign of the opacity slope |$\partial$|σ_a/|$\partial$|T. We also list the various time-scales of the model, the diffusion time |$t_{\rm {dif}} = 4 x_0^2 \sigma _\mathrm{ t}/\mathbb {C}$| and the thermal time |$t_{\rm {th}}= P/(\mathbb {P}\mathbb {C}E_\mathrm{ r}\sigma _\mathrm{ a})$| in units of the sound-crossing time t_sc = 2x₀/c_s. Finally, we list the qualitative behaviour of the cloud due to irradiation. For clouds undergoing significant acceleration we list the velocity of the core v and centre of mass v_cm at representative time t = 0.6 and when the cloud has mass m = 2/3m₀ remaining.

Table 1.

Summary of monotonic opacity cloud models. We indicate the cloud optical depth τ_a, the absorption cross-section σ_{a, 0}, and the opacity power-law scalings on temperature s (see equation 6). We list the corresponding diffusion t_dif and thermal t_th time-scales in units of the sound-crossing time t_sc and a qualitative description of the dynamics. For clouds undergoing rocket acceleration we indicate the cloud core velocity v and centre of mass velocity v_cm of the cold gas at representative time |$t = 0.6 \,\rm {s}$| and when the cloud has evaporated to m = 2/3m₀ of its initial mass.

	Opacity properties				Time-scales (tsc)			t = 0.6		m = 2/3m0
Model	τa	σa, 0	s	|$\partial$|σa/|$\partial$|T	tdif	tth	Summary	v	vcm	v	vcm
R0	2.0	1.0 × 10−1	−1	>0	8.6 × 10−5	1.4 × 10−3	Rocket	0.79	0.38	0.64	0.30
B1	2.0	5.0 × 10−4	0	=0	8.6 × 10−5	1.4 × 10−3	Balloon	–	–	–	–
B2	2.0	2.5 × 10−6	1	<0	8.6 × 10−5	1.4 × 10−3	Balloon	–	–	–	–
R1	4.0 × 102	1.0 × 10−1	0	=0	1.7 × 10−2	6.7 × 10−6	Rocket	0.58	0.54	0.6	0.41
R2	8.0 × 104	1.0 × 10−1	1	<0	3.4 × 100	3.4 × 10−8	Rocket	0.59	0.60	0.57	0.50

We summarize our results with snapshots of the density (Fig. 1) and temperature (Fig. 2) for models with −1 ≤ s ≤ 1 and either |$\tau = \rm {const}$| (three leftmost columns) or |$\sigma _{\mathrm{ a},0} = \rm {const}$| (three rightmost columns). Our fiducial model R0 (centre column) has s = −1, an initial cloud optical depth τ = 2, and σ_{a, 0} = 0.1. In this regime the cloud is optically thick for the duration of the simulation, since it is initially optically thick and s < 0 ensures that opacity increases as it heats. Clouds exhibit different qualitative behaviour across the parameter space of models. The clouds can accelerate via the rocket effect or diffuse away like a balloon (see also P14, models A40 and A10, respectively).

Figure 1.

Summary of models with monotonic opacity law showing density (green scale) and velocity vectors (grey scale) in units of the sound speed at representative moments in time in seconds (top right of each panel). Radiation flux enters from the left-hand boundary and interacts with the cloud. The fiducial model with σ_{a, 0} = 0.1 and τ_a = 2 is in the centre column. Models to the left have constant initial cloud optical depth τ_a and models to the right have constant ambient gas opacity σ_{a, 0}. When clouds heat non-uniformly (models to the left) they accelerate via the rocket effect, whereas models that heat nearly uniformly (models to the right) cause the cloud to balloon and dissipate.

Figure 2.

Same as Fig. 1 but showing logarithmic temperature contours (colour). When clouds heat non-uniformly (models to the left) they accelerate via the rocket effect, whereas models that heat nearly uniformly (models to the right) cause the cloud to balloon and dissipate.

A useful time-scale for this problem is the sound-crossing time, t_sc. Since it is fixed across our models, it is natural to use it to compare the various other time-scales to. We see two qualitatively different behaviours, depending on whether the thermal time is smaller or larger than the diffusion time. When the cloud heats non-uniformly it accelerates via the rocket effect. Diffusion through the cloud is slow compared to the rate of heating, t_dif ≳ t_th. The irradiated side heats up and the back-reaction of this evaporating gas accelerates the cloud. When the cloud heats uniformly the gas expands outwards like a balloon. Physically this regime requires photons to diffuse through the cloud faster than they can heat it, that is to say t_dif ≲ t_th. Since |$t_{\rm {th}}/ t_{\rm {dif}} \sim \tau _\mathrm{ a}^{-2}$|⁠, this occurs when the cloud is optically thin.

Models denoted by R (three leftmost columns of Fig. 1) are initially optically thick and remain so throughout the simulation, irrespective of the sign of s. To become optically thin the cloud density and temperature would have to, respectively, decrease and increase to the ambient backgrounds values. This is not possible, and therefore these models remain optically thick throughout the entire simulation. Clouds undergo acceleration via the rocket effect and exit the simulation in approximately the sound-crossing time.

We consider two possible metrics for quantifying the cloud acceleration. The cloud velocity v is defined at the density maxima, whereas the centre of mass velocity v_cm is the density weighted velocity over all cold gas, that is to say with T < T₀. We see a slight dependence on the temperature scaling and v, with the highest velocity achieved for R0. When s > 0 the evaporating cold gas is more optically thick than for s ≤ 0 and shields the cloud core from radiation and reduces v. However, this evaporating gas is more quickly heated and makes a smaller negative contribution to the centre of mass velocity resulting in a 35 per cent higher v_cm than when s ≤ 0. Models with s ≥ 0 thus have a similar v and v_cm, whereas models with s ≤ 0 efficiently accelerate their core increasing v but conservation of momentum dictates that evaporating cold gas efficiently acquires a large fraction of this velocity thereby reducing v_cm.

To further quantify the efficiency of the rocket effect for accelerating clouds, we compare velocities after the cloud has evaporated to m = 2/3m₀ of its initial mass. By this metric the cloud opacity plays little role in determining the acceleration of the core. The s = −1 model is only |$\sim\! 10{{\ \rm per\ cent}}$| higher velocity than model with s ≥ 0, which is expected from our intuition from the rocket equation as proposed by OS55. The authors modified the standard derivation of the rocket equation as applied to an irradiated cloud and found

$$\begin{eqnarray*} v = v_\mathrm{ e} \ln \left(\frac{m}{m_0}\right), \end{eqnarray*}$$

(7)

where m₀ is the initial cloud mass and v_e the velocity of the evaporating gas. For a fixed mass loss, the velocity is set by v_e. Approximating v_e ≈ c_s for the hot gas, when m = 2/3m₀ we expect v ∼ 0.4c_s = 0.52a₀, which agrees with our result to within |$\sim\! 20{{\ \rm per\ cent}}$|⁠. The energy density in the radiation field is much larger than the kinetic and thermal energy, which are nearly in equipartition. This equipartition is what we would expect from gas evaporated from other objects such as a disc or a star with a thermal wind.

Models B1 and B2 (two rightmost columns of Fig. 1) are initially optically thick but this changes as the clouds heat since s ≥ 0. The radiation diffuses through the cloud faster than it causes the cloud to evaporate, so the radiation energy density in the cloud is nearly uniform and the whole cloud heats approximately at the same rate. They have nearly all their flux passing through the cloud a short time after the start of the simulation, whereas R0 needs about half the simulation time for the flux to exit. The qualitative behaviour of the cloud thus depends on the optical depth of the cloud.

As radiation saturates the cloud, it heats up, causing the central part to expand outwards (t = 0.5). The accelerating, overdense cloud contacts the stationary, less dense ambient gas producing a shock (t  = 1.0). This triggers the Richtmyer–Meshkov instability (RMI, the impulsive acceleration analogue of the Rayleigh–Taylor instability, see Brouillette 2002 for a review), producing cold, dense fingers around the cloud. The instability induces a reverse shock, causing the cloud to recollapse (t = 1.5) after which the cloud slowly dissipates. We see this from the plot of position where the cloud core x undergoes damped harmonic motion but the cold gas centre of mass essentially remains at x_cm ≈ 0. After the initial expansion phase generated by the radiation, the evolution of the cloud is primarily driven by pure hydrodynamics. This is characteristic of the RMI being a purely hydrodynamic instability that only requires an accelerating dense medium and is agnostic of the particular acceleration mechanism. We tested this by turning off the radiation field after triggering the initial cloud expansion and found that cloud evolution was largely unchanged. We see a clear hierarchy of scales in the energy, with E_rad ≫ E_th ≫ E_K, consistent with what we expect for a nearly stationary cloud that evaporates as energy is transferred from the radiation field to the thermal and kinetic energy of the gas.

3.2 Broken power-law opacity

We consider models where the absorption cross-section is a broken power law,

$$\begin{eqnarray*} \sigma _\mathrm{ a} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\sigma _{\mathrm{ a},0} \left(\frac{T}{T_0}\right)^{-s_1} \left(\frac{\rho }{\rho _0}\right)^{n} & T \lt T_\mathrm{ c}, \\ \sigma _{\mathrm{ a},0} \left(\frac{T_\mathrm{ c}}{T_0}\right)^{s_2-s_1} \left(\frac{T}{T_0}\right)^{-s_2} \left(\frac{\rho }{\rho _0}\right)^{n} & T \ge T_\mathrm{ c}, \end{array}\right. \end{eqnarray*}$$

(8)

where s₁ < 0 < s₂ so that the cloud opacity initially increases with temperature but may avoid runaway heating and dissipating away by turning over at the critical temperature T_c.

We set τ_a = 2 and choose s₁ = −1 and s₂ = 1. The critical temperature is chosen as 1/4T₀ ≤ T_c ≤ 3/4T₀. In addition, we can consider the monotonic model R0 as a limiting case with T_c = ∞. A summary of the broken power-law models is listed in Table 2, as well as the core and centre of mass velocities at representative time t = 0.6 and cloud mass m = 2/3m₀.

Table 2.

Summary of broken power-law opacity cloud models that all accelerate via the rocket effect. We indicate the critical temperature T_c, the power-law scalings of the absorption cross-section s₁ and s₂ (see equation 8). We list the cloud core velocity v and centre of mass velocity v_cm at representative time t = 0.6 and after the cloud mass has decreased to m = 2/3m₀ as a metric for acceleration efficiency.

	Opacity properties			t = 0.6		m = 2m0/3
Model	Tc  (T0)	s1	s2	v	vcm	v	vcm
RT1	1/4	−1	1	1.56	0.40	1.96	0.47
RT2	1/2	−1	1	0.85	0.56	0.98	0.65
RT3	3/4	−1	1	0.84	0.51	0.87	0.54
R0	∞	−1	−	0.79	0.38	0.64	0.30

All clouds in these models accelerate via the rocket effect as described in the previous section. Models with lower critical temperature have lower temperature evaporating gas, with the atmosphere in case T_c = 1/4 most closely resembling the balloon behaviour of optically thin monotonic models. Our goal is to quantify how the acceleration efficiency has been affected by introducing a turnover in the opacity scaling, as would be the case if there is an opacity bump.

In Fig. 3, we plot v (top panels) and v_cm (bottom panels) as a function of elapsed time (left-hand panels) and cloud mass (right-hand panels) for different critical temperatures T_c. We show the velocity scaling predicted by the rocket equation (7) (black dashed line). We find that decreasing the critical temperature leads to an increase in the core velocity. For example, at t = 0.6 the RT1 cloud is |$85 {{\ \rm per\ cent}}$| faster than the other models. Evaporating gas from the cloud, and thus accelerating it, requires heating gas to T ∼ T_c. For lower critical temperature the necessary energy deposition required to heat the gas to T_c is lower. Therefore a fixed radiation flux can achieve a higher acceleration. Unlike with monotonic models, the largest cloud velocity is achieved in the case of the most optically thick cloud. In other words, despite the reduced flux incident on the cloud core because of a more optically thick atmosphere, the acceleration is still higher because the cloud atmosphere tends to heat to T ∼ T_c, and thus have a higher v_e, when evaporating. Any additional heating beyond T ∼ T_c causes the evaporating gas to become more optically thin and tends to stop subsequent heating of the atmosphere as the coupling between radiation and gas is weakened. The lower T_c cases thus have a denser, colder, and hence more optically thick atmosphere (since s = −1 for T < T_c).

Figure 3.

Broken power-law models for critical temperatures T_c = 1/4T₀ (red), 1/2T₀ (green), and 3/4T₀ (blue). We also include the monotonic model R0 (black). Points are plotted in the range 0 ≤ t ≤ 0.75 in intervals Δt = 0.01. Centre of mass velocity v_cm (top panels) and core velocity v (bottom panels) as a function of time (left-hand panels) and cloud mass (right-hand panels). We show the velocity scaling predicted by the rocket equation (7) (black dashed line). At early times, centre of mass velocities are approximately equal but at late times models with 1/2 ≤ T_c/T₀ ≤ 3/4 are approximately 50 per cent faster.

Similarly, after considering the cloud mass as a proxy for acceleration efficiency we find that decreasing the critical temperature increases the core velocity at fixed mass. This is simply a consequence of gas ceasing to heat above T ∼ T_c. The RT1 cloud effectively evaporates gas to accelerate its core, but this gas remains cool T < T₀. It therefore remains part of the overall cloud mass budget and gives the impression that this case is much more efficient at accelerating clouds. When we consider the centre of mass velocity, we find that the broken power-law models have v_cm|$50\!-\!100 {{\ \rm per\ cent}}$| faster than the monotonic model R0. We thus conclude that the broken power-law models, by both metrics, are more efficient at accelerating cold gas.

4 DISCUSSION

Cloud acceleration can be either momentum or energy driven, depending on which quantity is transferred from the radiation field to the cloud. Both regimes can be achieved in a variety of ways. In this work, acceleration is energy driven, as can be seen by comparing the radiative momentum flux incident on clouds and the cloud mass flux, |$F_\mathrm{ r} A/\mathbb {C} \ll \dot{m} v$|⁠, where we have F_r ∼ 16, A ∼ 0.1, |$\dot{m} \sim 3$|⁠, and v ∼ 0.5. The energy-driven regime requires a non-zero absorption cross-section and a sufficiently massive, optically thick cloud that can absorb hot radiation.

This work considers the case of marginally optically thick clouds, τ ≳ 1, whereas P14 studied the case where τ ≫ 1. In both cases the incident radiation is hot relative to the gas, T_r > T_g, which is key for accelerating the cloud via the rocket effect. When clouds are too diffuse or the absorption too strong, clouds will disperse before any significant acceleration (P14 case A10) or accelerate to roughly the sound speed before dispersing (P14 cases A40 and A80 or the accelerating models in this work). Such energy-driven winds (see Faucher-Giguère & Quataert 2012 and references therein) are supported by observations of AGN that find outflows carry more momentum flux than the radiation field driving them.

Alternatively, acceleration can be achieved by directly transferring momentum from the radiation field to the gas. Both the scattering and absorption cross-section mediate this coupling (see equation 3a) so different models have tried to achieve this momentum transfer in different ways. In P14 (see their models S10 or S200) they use a pure scattering coefficient. In Z18 they use a pure absorption coefficient (see their model T1L) but the rocket effect is largely absent since T_r = T_g ensures there is negligible heat transfer. The momentum transferred to the cloud is a function of the incident flux – the optically thick case differs from the optically thin case only by an attenuation factor (see e.g. equation 25 in Z18). For constant incident flux, as studied in these models, this means that momentum-driven acceleration will be approximately constant.

We find that with the rocket effect it is difficult to accelerate clouds much beyond the sound speed v ≳ c_s. Both momentum- and energy-driven clouds dissipate as a result of their acceleration. In the momentum-driven case, the cloud is shredded by the slower moving, ambient medium. The energy-driven cloud dissipates as it is the very gas that is evaporated and responsible for accelerating the cloud. However, as we have shown features in the opacity profile can effectively make the evaporating gas optically thin, which allows it to stay cool relative to the ambient gas. If this cool gas reforms clouds via thermal instability (see e.g. Wareing et al. 2016), it may be possible to accelerate cold gas beyond the sound speed after multiple cycles of acceleration and condensation.

Likewise, we found that optically thin clouds heat uniformly, which causes them to expand and trigger the RMI and ultimately dissipate. Because of our initial mass distribution, this occurs in a circular geometry. Z18 found that clouds are unstable to a Rayleigh–Taylor-type instability, which they resolved in their highest resolution runs (T0.01L) with reduced gas pressure. Likewise they found clouds to be Kelvin–Helmholtz unstable in the case of a hot ambient gas ‘shredding’ clouds (see also Klein, McKee & Colella 1994; Poludnenko et al. 2002). These findings generally support the conclusion that clouds have a variety of mechanisms by which to dissipate and therefore have a finite lifetime.

We investigated whether the presence of a ‘bump’ in the opacity could alter cloud lifetime. In order for a cloud to change from the accelerating rocket regime to the expanding balloon regime the optical depth of the cloud must transition from optically thick to thin. Consider a cloud that is marginally optically thick, τ_i ≳ 1, that heats isobarically and has s₁ < 0. If the temperature doubles T_i → T_c = 2T_i, then the opacity changes by a factor |$2^{n+s_1}$|⁠. If the opacity is a power law and s₁ ≥ 0, then we find the cloud evolves in the balloon regime. Suppose that at temperature T_c, the opacity function has a break as in equation (8). A similar doubling of the temperature, T → 2T_c, will decrease opacity by a factor |$2^{n+s_2}$|⁠. For the cloud to re-enter the optically thick regime, we need s₂ ≳ 2n + |s₁|. With our choice of parameters, we would need s₂ = 5, a far steeper power law than say Krammers s = 3.5. The steepness of the power law can be reduced by having a greater temperature change as the cloud thickens, but this range is limited by the initial cloud/medium density contrast. We conclude that except for perhaps some finely chosen area of parameter space it is challenging to change from the optically thin balloon regime to the optically thick rocket regime. We note however than when the transition temperature is low, say T_c = 1/4, we did see ballooning of the expanding atmosphere, though the cloud core still underwent rocket acceleration.

We have assumed clouds are initially in hydrostatic pressure balance. Around AGN, radiation pressure from the central object is expected to be important is accelerating clouds, so an equally compelling initial condition is one where the cloud is in pressure equilibrium between its internal gas pressure and the external radiation field (see e.g. Dopita et al. 2002). A fundamental question is to explore how the dynamics of these clouds may be different from those initially in thermal pressure equilibrium. This is particularly an important question for AGN, where radiation pressure is expected to be an important wind-driving mechanism.

Approximately 10 per cent of the incident flux is reprocessed by the accelerating cloud and re-emitted perpendicular to the incident flux. This emission is primarily from the hot evaporating gas and not the cold cloud core. In all rocket cases the reprocessed emission is ∼few per cent. This is what we expect from purely geometric considerations, since this cloud has a covering fraction of |$10{{\ \rm per\ cent}}$|⁠, so we expect ∼1/4 of this radiation to be reradiated out the top part of the box or |$L \sim 2.5{{\ \rm per\ cent}}\, L_0$|⁠. It peaks at ∼0.2 s, which corresponds to the time when the initial transient phase of gas evaporation from the cloud occurs. It then decreases roughly linearly with time before dropping to zero as the cloud exits the simulation domain. We thus estimate the line emission and absorption from a single cloud to be approximately few per cent. In the balloon cases, the emission is negligible, a factor of ∼10³ smaller. As the cloud heats to T₀ its optical depth decreases by ∼10³ causing a similar drop in the reprocessed radiation. The broken power-law models show nearly a factor of ∼10 drop in re-emitted flux. As seen from the spatial distribution of re-emitted flux, most radiation is coming from the hot, evaporating gas. Introducing a cut-off T_c means that this hot gas is less optically thin and therefore reprocesses less incident radiation.

5 CONCLUSION

We have studied the dynamics of a single cloud absorbing radiation from a distant source. We find the cloud behaves in two qualitatively different ways. If the cloud is optically thin, it heats nearly uniformly and expands like a balloon. If the density of the heated gas is higher than of the ambient gas, this triggers the RMI and leads to cloud dissipation. If the cloud is optically thick, it heats preferentially on the radiated side and gas evaporation accelerates the cloud via the rocket effect. The velocity growth is logarithmic, quantitatively different from the linear growth seen in the regime where the radiation and gas are in thermal equilibrium as studied in Z18.

We could not qualitatively alter the behaviour of clouds using broken power-law opacities – accelerating clouds could not be made to balloon and ballooning clouds could not be made to accelerate. We estimate such a qualitative change in behaviour would require a very steep, s ∼ 5, opacity temperature dependence. However the efficiency of cloud acceleration can be increased if hot gas is more optically thin, as gas evaporating from the cloud no longer absorbs incident radiation. These results suggest that features in the opacity profiles due to certain chemical elements can affect cloud dynamics.

After having modelled the dynamics of a single cloud, we are in a position to simulate multiple clouds in a dynamic environment. Proga & Waters (2015, see also Waters & Proga 2019) have shown how clouds can form via thermal instability from initial perturbations. Using this initial set-up we can form clouds in situ and study their evolution in a periodic box. We expect clouds to dissipate as they accelerate/balloon away and reform again via thermal instability. One possible scenario is finding a quasi-steady, multiphase solution as predicted by Krolik, McKee & Tarter (1981). We may then characterize the covering fraction of such a system, as measured by observations of AGN tori.

ACKNOWLEDGEMENTS

All simulations were performed on the UNLV National Supercomputing Institute’s Cherry Creek cluster and the authors acknowledge Ron Young’s technical expertise. We thank the reviewer, Achim Feldmeier, for his thorough reading and constructive feedback on our manuscript. SD acknowledges support from ERC Advanced Grant 340442. CSR thanks the UK Science and Technology Facilities Council (STFC) for support under the New Applicant grant ST/R000867/1, and the European Research Council (ERC) for support under the European Union’s Horizon 2020 research and innovation programme (grant 834203). Support for Program number HST-AR-14579.001-A was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. This work also was supported by NASA under ATP grant 80NSSC18K1011.