Evolution of Molecular Clumps in Quasar Outflows

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Abstract

Recent observations of active galaxies show that a large amount of outflowing gas is in molecular form up to a radius of about 1 kpc. Molecular gas is expected to form in outflows beacuse of hydrodynamical instabilities at the interface between the black hole wind and the interstellar medium, presumably in the form of clumps. In this work, we study how the structure of an hydrogen clump changes while it is carried with the outflow and exposed to quasar radiation, assuming spherical geometry and a Bonnor-Ebert density profile (with central density ∼ 1.8 × 105cm−2). We expect the clump to present an H2 core, surrounded by an atomic shell and an ionized shell.

Thus, we investigate the propagation and interaction of shock and rarefaction waves which originate at the discontinuities between different layers. We show that the molecular core contracts to high densities (∼ 4.5 × 108_cm−2_{) and after that it starts}

an expansion phase, which allows radiation to penetrate further inside, until the clump is completely evaporated. We estimate clump lifetimes of the order of 1 Myr, implying that after its formation molecular gas can travel 1 ÷ 2 kpc in SMBHs of 108_M

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Introduction

The field of Active Galactic Nuclei (AGNs) has been one of the richest area of research since their discovery in the early sixties. All these objects appeared as very bright galaxies, but with different spectra, so that at first astronomers classified them in distinct categories of sources (Seyfert Galaxies, Quasars, Radio Galaxies,...). Nevertheless, today there is a wide agreement that most differences depend on the viewing angle and on obscuration from intervening material. A unified model has therefore been developed: an AGN (or a quasar) consists in a supermassive black hole (SMBH) which is accreting material via an accretion disk, surrounded by clouds of gas (broad and narrow line regions), a dusty torus and the host galaxy.

Observations show that the properties of the SMBH are tightly related to the properties of the host galaxy, as for the black hole mass - bulge mass relation or the M − σ relation. These evidences motivate the study of possible mechanisms which could couple SMBHs with their surroundings, a key factor to understand how they co-evolve and what brings them to be eventually inactive.

In this direction, it is promising to investigate ultra fast kpc-scale outflows, which have been observed in multiple sources in the last twenty years. The huge intensity of the radiation emitted by the quasar drives a wind which impacts on the interstellar medium (ISM), thus generating outflows of 108 _{÷ 10}9_M

. This phenomenon could

explain the regulation of SMBH growth and the quenching of star formation in the galaxy.

A feature of quasar outflows, recently revealed by observations in the millimetre band, is that a large part of the outflowing gas is actually in molecular form up to a radius around 1 kpc. This gas is presumably structured in clumps, which originates

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at the wind - ISM interface beacuse of hydrodynamical instabilities.

The aim of this thesis is to understand clump dynamical and chemical evolution while they are carried with the outflow. A typical clump is modelled as a Bonnor-Ebert sphere exposed to quasar radiation, thus developing an H2 core, a neutral HI

shell, and an outer ionized shell. Shocks are driven inward from the two external layers, with the global effect of a contraction of the clump. After the contraction phase the cloud presents high density and pressure, while the external shells are almost dispersed. Thus, an expansion of the clump follows, progressively allowing further penetration by radiation, until its complete photoevaporation.

In chapter 1, we describe the physical and observational properties of accreting black holes and quasars. In particular, we concentrate on the accretion mechanism responsible for the high luminosities of these objects. Indeed, its is the radiation pressure that drives ultra fast outflows, whose structure is described in detail in the last part of the chapter, together with observational evidences.

In chapter 2, basic properties of ionized regions and photodissociation regions are introduced, as their thickness and temperature. Then we describe the propaga-tion of shock and rarefacpropaga-tion waves in media with density gradient and spherical geometry, and the interaction of two waves both with a numerical approach and an analytic argument.

In chapter 3, we give some clues of how clumps form, in order to estimate their basic properties, as its mass, size and density profile. The initial structure of a typical clump is described, investigating the effect of radiation coming from the quasar.

Finally, in chapter 4 we study the evolution of clump structure, taking into account the propagation of shock fronts and their interaction, together with the effect of photoevaporation. This analysis allows to estimate the evaporation time of a clump and the radial extension of a molecular outflow.

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Chapter 1 Quasar outflows

1.1 Active Galactic Nuclei and Quasars

Nowadays it is widely accepted that every galaxy has got a central supermassive black hole (SMBH). Active galaxies are a particular subclass of galaxies that present a SMBH accreting material: the infall of gas in the central black hole happens via an accretion disk, a mechanism which explains the high luminosities of these objects. The central structure of an active galaxy is called an Active Galactic Nucleus (AGN), and these are indeed one of the most powerful emitting sources of the universe. Observed bolometric luminosities for this objects range from 1045erg/s to 1048erg/s, which means up to 5 orders of magnitude more luminous of ordinary galaxies as our Milky Way (L_mw∼ 5 × 1043_erg/s).

Another observational signature of AGNs is the broad spectrum, covering up to 15 orders of magnitude in frequency, which presents both thermal (black body shape) and non-thermal emission (power law shape). A typical spectral energy dis-tribution (SED) is shown in figure 1.1.1. The slope at low frequencies is due to either synchrotron self-absorbed emission or the Rayleigh-Jeans tail of dust grains thermal emissions (T ' 2000 K), the latter being responsible for the IR bump. On the other hand, the big blue bump is due to thermal emission by the gas in the accretion disk, since it can reach temperatures around 106K. The extension of the spectrum to the X rays and γ rays is justified with the strong inverse Compton scattering of photons

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CHAPTER 1. QUASAR OUTFLOWS

Figure 1.1.1: Typical broadband spectrum of an AGN, showing the most important features, as the radio continuum, the IR bump and the big blue bump. Figure from Carroll and Ostlie (2006).

coming from the gas corona lying above the disk by the electrons in the accretion disk .

Despite usually showing these common features, spectra of AGNs tend to present some differences, which historically led to a complicate classification. The most important classes of AGNs are

• Seyfert galaxies: weak radio emission, usually spiral galaxies, not variable; • Quasars: strong or weak radio emission, variable, at redshift z > 0.2; • Radio galaxies: strong radio emission, elliptical galaxies, variable or not; • Blazars: strong radio, X ray and γ ray emission, strong polarization, strong

variability;

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1.1. ACTIVE GALACTIC NUCLEI AND QUASARS

Figure 1.1.2: AGN structure according to the unified model. Figure from Urry and Padovani (1995).

Anyway, it now seems likely that all AGNs present the same structure with the SMBH and an accretion disk, with the differences arising from the different orientation of the object with respect to the Earth and the presence of obscuring material. Thus, the classification remains useful for observational purposes, but often loses significance from a theoretical point of view. In this work we will address to these objects as AGNs or quasars without any distinction. The unified model for AGNs is nowadays widely accepted (Antonucci (1993), Kauffmann and Haehnelt (2000)), and the basic structure is represented in figure 1.1.2. The main features are

• the central supermassive black hole, of mass 106÷ 109M;

• the accretion disk, which usually extends to a radius of the order of 1 µpc; • the torus, a thick dust structure at a distance of few parsecs from the SMBH,

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• the broad line region (BLR), made up of cloud enclosed by the torus, whose

emission lines are broadened because of the high velocity of rotation;

• the narrow line region (NLR), made up of clouds out of the torus, up to few

kpc;

• the jets, highly relativistic flows of material collimated in two opposite

direc-tions.

In the remainder of the chapter, we will focus mainly on the accretion mechanism which explains AGN luminosities, and on the structure of outflows which form as a consequence of the huge radiation pressure exerted on the gas surrounding the central black hole.

1.2 Accretion mechanism and Eddington

lumin-osity

The main source of quasar radiation is the accretion of surrounding gas. From general relativity, we know that an innermost stable orbit exists for material around a black hole, whose radius depends only on the mass of the black hole and its spin. The last stable circular orbit for a particle rotating in the same direction as the black hole has a radius given by (Bardeen et al. (1972))

rms = Mbh{3 + Z2− [(3 − Z1)(3 + Z1+ 2Z2)]1/2} (1.2.1)

where Z1and Z2are constants depending only on the ratio abh/Mbh, with abh angular

momentum per unit mass (geometrical units with G = c = 1 are used). For radii greater then the innermost stable radius, gas particles arrange in an almost flat disk, whose structure depends on the viscosity coefficient ν of the gas and the accretion rate ˙M_bh of the black hole. Assuming a flat disk where particles have a Keplerian tangential velocity, Pringle (1981) shows that the radial surface density profile of the

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1.2. ACCRETION MECHANISM AND EDDINGTON LUMINOSITY

Figure 1.2.1: In a rotating black hole, the event horizon radius (r+) is spin dependent

and it is smaller if the spin value is higher. r0(θ) is the radius of the the ergosphere,

inside which a particle can stay at a fixed distance from the singularity only if it has a non-zero angular velocity, where θ = π/2 on the equatorial plane. rms is the radius

of the last stable orbit, which has different values whether the particle is rotating in the same direction of the black hole or in the opposite one. The axes of the plot are expressed in the geometrical unit system (G = 1, c = 1). Figure from Bardeen et al. (1972).

gas in the disk is

Σ(r) = 1 ν ˙ M_bh 3π 1 −r rms r (1.2.2) The friction between adjacent rings of the disk rotating at different speed produces a dissipation of energy, with an effective temperature with a radial profile given by

Te(r) = " 3GM_bhM˙_bh 8πσ_sbr3 1 −r rms r #1/4 (1.2.3)

where σ_sb is the Stefan-Boltzmann constant. This thermal emission extends to the UV, but it doesn’t justify the radiation emitted by a quasar in the X rays. As we said earlier, the mechanism responsible for this high frequency emission is the

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inverse Compton effect: photons coming from gas above the disk scatter with ultra relativistic thermalized electrons, resulting in an increase of photon energy.

Accretion of material, falling into the black hole via a disk, is the only possible mechanism to explain the high bolometric luminosity of quasars. If we assume that all the potential energy lost by an infalling particle is radiated, then the total luminosity is

L = η ˙M_bhc2 (1.2.4)

where η is the efficiency of the process, corresponding to the difference of potential energy from the infinite to the last stable orbit. Using the expression (1.2.1) it is possible to find η ' 5.7% ÷ 42.2%, where the lowest value is for a non rotating black hole, while the upper value is for a maximally rotating black hole (usually a value η ' 0.1 is used). This efficiency is far higher then the one we can obtain from other processes: for example, from nuclear reactions we have a 0.13% efficiency at best (taking the proton-proton fusion), which would require unreasonable accretion rates to give the observed luminosities.

It is possible to compute the maximum luminosity of an object which is accreting material from a simple reasoning which doesn’t deal with the detail of the accretion mechanism. Indeed, for a particle to not be swiped away and to eventually fall to the centre, the force exerted by the radiation pressure has to be smaller then the gravitational pull. The force due to radiation over an hydrogen nucleus at radius r is

Frad(r) =

L

4πr2_cσt (1.2.5)

where L is the luminosity, c is the speed of light and σ_t is the Thomson cross section. On the other hand, the gravitational force is

Fgrav(r) =

GM_bhmp

r2 (1.2.6)

Hence the condition to have accretion is Frad(r) ≤ Fgrav(r), which leads to

L ≤ 4πGMbhc

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1.2. ACCRETION MECHANISM AND EDDINGTON LUMINOSITY

defining κ = σ_t/mp. The maximum luminosity is usually called Eddington

luminos-ity L_edd and, putting the constants in, we get L_edd = 1048 M_bh 108_M erg/s (1.2.8)

For a measured luminosity of 1045÷ 1048_{erg/s, this gives a lower limit on the mass}

of the central source of these kind of objects:

M_bh ≥ 105_{÷ 10}8_M

(1.2.9)

Since in many sources we observe a variability with very short timescales (the smaller value is about 1 hour, according to MacLeod et al. (2012)), this implies a small size of the source. This is an evidence of the fact the central engine has to be a supermassive black hole. The Eddington luminosity provides us an estimate of the accretion rate, as well. Considering the relation (1.2.4) between the luminosity and the accretion rate, we get the upper bound

˙ M_bh ≤ Ledd ηc2 ' 1.7 × 10 2 M_bh 108_M 0.1 η M/yr (1.2.10)

Nevertheless, super-Eddington accretion is often observed in accreting binary systems (for example see King et al. (2000)), implying that a way exists for photons to escape the accretion flow. This seems to be possible thanks to density inhomogeneities in actual disks, as Begelman (2002) suggests. As a result, super-Eddington luminosities are possible and they can drive powerful gas outflows via radiation pressure.

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1.3 Interaction with the host galaxy

It is clear from observations in the last years that SMBHs at the centre of galaxies play an important role in the evolution of the galaxy. Various galaxy properties can be related to the BH mass even in galaxies which are no longer active. The total mass of the bulge of the galaxy appears to be proportional to M_bh, and H¨aring and Rix (2004) find

Mb ' 103Mbh (1.3.1)

Furthermore, there is a relation with the velocity dispersion in the bulge (Gebhardt et al. (2000)):

M_bh ' 3 × 108_M

σ₂₀₀α (1.3.2)

where σ200 is the velocity dispersion divided by 200 Km/s and α = 4.4 ± 0.3. The

observational evidence that relations like these hold rises the problem of how the black hole interacts with the host galaxy. In fact, comparing the gravitational pull with the mean kinetic energy of the gas in the galaxy, we get a sphere of influence with radius

Rinf '

GM_bh

σ2 (1.3.3)

which means a radius of few parsecs. This means that the black hole cannot have an influence on the galaxy via the gravitational force. Nevertheless, the great amount of radiation released by an AGN can be a satisfying mechanism for the black hole to affect its surroundings. The direct interaction of this radiation with the galactic gas is not thought to be so efficient, inasmuch as observations tell us that most light seems to escape freely from the AGN (King and Pounds (2015)). On the other hand, mechanical coupling seems more plausible: radiation pressure drives a flow of material from within the nearest parsecs, which go collide with the gas of the host galaxy. Flows of this kind are either highly collimated jets or isotropic winds, and in this work we will focus on the latter.

As mentioned in the previous section, a super-Eddington accretion rate is required to support a black hole wind. Nevertheless, an upper limit over this rate is obtained considering the dynamical timescale that gas require to fall freely. Assuming an

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1.3. INTERACTION WITH THE HOST GALAXY

isothermal density profile of dispersion σ (see Keeton (2001))

ρ(R) = fg

σ2

2πGR2 (1.3.4)

for a protogalaxy, with fg ' 0.16 gas fraction (Spergel et al.), the gas mass within a

radius R is Mg(R) = 4π ˆ R 0 ρ(r)r2dr = 2fgσ 2_R G (1.3.5)

After a destabilization, the mass requires a time tdyn∼ R/σ to fall inwards, so that

we get an accretion rate ˙ Mdyn ∼ fgσ3 G ' 280 σ 3 200M/yr (1.3.6)

We can write the Eddington accretion rate as

˙ M_edd= Ledd ηc2 = 4πGM_bh κηc ' 4.4 σ 4 200M/yr (1.3.7)

using the M − σ relation (1.3.2) with α ' 4. Thus, a super-Eddington accreting black hole has an accretion ratio with respect to Eddington

˙ m < M˙dyn ˙ M_edd ' 64 σ −1 200 (1.3.8)

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1.4 Structure of an outflow

As shown by (1.3.8), super-Eddington accreting black holes have actually a modest accretion ratio with respect to Eddington accretion rate. Then, we will consider a quasar accreting at dynamical accretion rate ˙Macc' ˙Medd and luminosity L ' Ledd.

The momentum transferred to the surrounding gas via radiation pressure is ˙

Mwv '

L_edd

c (1.4.1)

where ˙Mwis the wind mass flux and v is the terminal wind velocity, assuming that the

scattering optical depth is τ ∼ 1 (King (2003)), i.e. each photon scatters once before escaping. As observations of bright quasars suggest (Pounds et al. (2003)), winds have mass flows comparable to the accretion rate, then we will assume ˙Mw ' ˙Macc

from now on. Thus, from expressions (1.3.7) and (1.4.1) we get a velocity

v ' ηc (1.4.2)

This wind is going to impact with the interstellar medium (ISM) of the galaxy, which we assume to be uniform and not in motion. The collision generates a shock propagating into the ISM, speeding it up, and a reverse shock into the wind itself, while the contact discontinuity between the wind and the ISM continues to move forward. Depending on the efficiency of cooling mechanisms in the gas, the reverse shock can be considered isothermal or adiabatic. Because of the very high temper-ature of the reverse-shocked wind and the presence of an intense radiation field, we expect inverse Compton cooling to be very a efficient way of cooling the gas. King (2003) computes the shocked wind temperature (1010_{÷ 10}11_{K) and the timescale for}

Compton cooling of the electrons in the wind:

t_c ' 105R2_kpcc v

2

M₈−1yr (1.4.3)

where M8 = Mbh/108M and Rkpc is the distance from the black hole expressed in

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1.4. STRUCTURE OF AN OUTFLOW

t_f = 8 × 106Rkpcσ200M −1/2

8 yr (1.4.4)

cooling is efficient only when t_c < t_f, that means within a radius

R_c ' 500 M₈1/2σ200pc (1.4.5)

We will then study two different regimes of the outflow: an isothermal momentum-driven outflow when R < R_c and an adiabatic energy-driven outflow when R > R_c. Since an outflow can originate close to the black hole, we expect it to be isothermal and then to switch to being adiabatic when it reaches the critical radius.

Isothermal outflows

The expression for the ISM mass within a radius R is given by (1.3.5), without the factor fg if we are considering the whole mass M (R) and not only the gas component.

The wind reaching radius R means that all the ISM gas mass up to R has been swiped by the wind, and then it is compressed in a shell ahead of the wind. The pressure in the post-shock wind gas for a strong isothermal shock is (see Dyson and Williams (1997))

P = ρwvw2 (1.4.6)

so that the force on the shell is given by

F = 4πR2P = (4πR2ρwvw)vw = ˙Mwvw =

L_edd

c (1.4.7)

Thus, the equation of motion of the gas shell is d dt[Mg(R) ˙R] + GMg(R)[Mbh+ M (R)] R2 = L_edd c (1.4.8) which simplifies to d dt(R ˙R) + GM_bh R = −2σ 1 −Mbh Mσ (1.4.9)

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Figure 1.4.1: Representation of an isothermal (top figure) and an adiabatic (bottom figure) outflow. In the isothermal outflow, the reverse-shocked wind cools rapidly and doesn’t expand, so that the thin layer just communicates the ram pressure of the wind to the ISM. In the adiabatic case energy is not lost, resulting in a powerful outflow with higher momentum. Figure from King and Pounds (2015).

defining Mσ = fgκ πG2σ 4 _{' 3.2 × 10}8 Mσ4200 (1.4.10)

For large R the equation gives ˙ R2 ' −2σ2 1 − Mbh Mσ (1.4.11)

Notice that this is impossible if M_bh < Mσ, which means that the outflow doesn’t

make it to large R at all. Nevertheless the black hole is accreting mass, then it will be able to swipe the surrounding gas away when it is grown to a mass Mσ. This also

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1.4. STRUCTURE OF AN OUTFLOW

means that, once the black hole reaches Mσ, then it stops accreting because all the

gas is swiped away. The relation (1.4.10) is very close to the observational M − σ relation (1.3.2), suggesting that outflows are a good mechanism to regulate SMBH growth.

Adiabatic outflows

When the mass of the SMBH is larger than the critical mass Mσ, outflows are able

to reach an arbitrary large radius. This means that for R > R_c the gas wind is no longer efficiently cooled, and the outflow can be approximated to be adiabatic. The equation of motion is d dt[Mg(R) ˙R] + GMg(R)M (R) R2 = 4πR 2 P (1.4.12)

where this time we have neglected the black hole mass. P is determined by the energy conservation equation:

d dtUth = ˙Ew,kin− P dV dt − Fgrav ˙ R (1.4.13) where Uth = 4 3πR 3_· 3 2P (1.4.14) ˙ Ew,kin= 1 2 ˙ Mwv2 = η 2Ledd (1.4.15) Fgrav = GMg(R)M (R) R2 = 4fg σ4 G ˙ R (1.4.16)

Substituting P from (1.4.12) into (1.4.13), after some algebra we get to the equation η 2Ledd= 2fgσ2 G 1 2R 2... R + 3R ˙R ¨R + 3 2 ˙ R3 + 10fg σ4 G ˙ R (1.4.17)

The equation has a solution of the form R = vet with ve satisfying

2ηc = 3v

3 e

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The assumption ve σ leads to the contradiction ve ' 0.01c σ, then assuming

ve σ we get the following expression for ve:

ve ' 2 3ησ 2 c 1/3 ' 925 σ2/3₂₀₀km/s (1.4.19)

From physics of shocks, which will be reviewed in section 2.5, Zubovas and King (2014) compute post-shock ISM temperature and density:

T_ism' 2.2 × 107_σ4/3 200f −2/3 K (1.4.20) n_ism' 60 σ2 200f R −2 kpccm −3 (1.4.21) where f is the ratio of fg to the cosmological mean value 0.16 (Spergel et al.).

1.5 Outflows observations

The detection of ultra fast outflows (UFOs) in AGNs has become possible thanks to the modern high-resolution X-ray observatories, such as Suzaku, Chandra and XMM-Newton. The first UFO was found by XMM-Newton analysing radiation with energy above 1 keV of PG1211+143. The main features of the spectrum were blue-shifted absorption lines of highly ionized metals, providing evidence of fast ionized outflows: in particular, the Doppler shifted Lyα line of iron indicated a speed of about 0.09 c (Pounds et al. (2003)). Modelling the absorption in a photoionized gas with the XSTAR code of Kallman et al. (1996), by comparing with the data it was also possible to estimate the angular covering factor b = Ω/2π of the outflow, which resulted 75% (Pounds and Reeves (2009)). It follows that the mass rate of the ionized component of the outflow is

˙

M_out(ion)' 4πbρr2_{v ' 2.5 M}

/yr (1.5.1)

After this first example many other observations reported evidences for the existence of UFOs, and lists of the related objects are for example included in Cappi et al.

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1.5. OUTFLOWS OBSERVATIONS

Table 1.1: Information inferred by observation of a set of active galaxies. Columns: (1) name, (2) mass of the molecular gas in the outflow, (3) molecular outflow mass rate, (4) radial extension of the outflow, (5) average speed of the outflow, (6) max-imum speed of the outflow, (10) references: 1. Cicone et al. (2014) 2. Cicone et al. (2012), 3. Feruglio et al. (2013), 4. Krips et al. (2011). Table from Cicone et al. (2014).

(2006) and Tombesi et al. (2010).

More recent observations have used the technique of interferometric mapping of the CO millimetre emission, showing that a massive component of the outflowing gas is actually in molecular form (Cicone et al. (2014)). A molecular phase in the wind has been detected for example in Mkr 231 (Feruglio et al. (2010), Cicone et al. (2012), Feruglio et al. (2015)) with the IRAM/PdBI, measuring the CO(2-1) and CO(3-2) rotational molecular transition lines. The high-resolution spectroscopy allows to map the emission in space, thus giving information on the mass and the radial extension of the molecular outflow. As we can see from table 1.1, molecular gas is present up to radius of about 1 kpc, implying that molecules are dissociated while carried by the outflow. We can also notice that the mass rate is many hundreds of solar masses per year, which is much grater than a typical ionized outflow, as we can see from (1.5.1). This is an observational evidence that most of the gas in an outflow is in molecular form, and thus a theoretical model to explain how molecules might form and how long they survive is needed. As we will see in detail in the next chapters, a

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possible mechanism to put the gas in the molecular form are thermal instabilities at the boundary between the shocked ISM and the reverse-shocked wind. This means that the molecular gas can form starting from the cooling radius R_c in SMBH with M ≥ Mσ, since otherwise the momentum-driven outflow would fall back toward the

black hole, and the molecules dissociate because of the radiation from the quasar within few hundred parsecs. The fact that molecular gas survives for a good amount of time into the flow means that it has to be structured in clumps, so that self-shielding by a shell of photodissociated molecules can extend the lifespan of the clump.

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Chapter 2 Structure and dynamics of gas

clouds

2.1 Ionization of an H nebula

If a cloud of atomic hydrogen is exposed to ionizing radiation (energy greater than hνo= 13.6 eV), it develops a layer where concentration of ionized hydrogen increases

towards the face of the cloud, where the gas is expected to be completely ionized. We want to describe the structure of this layer, computing the HII fraction and its temperature. This analysis applies to many astrophysical situations, such as interstellar gas subject to radiation coming from a star (HII regions), or intergalactic gas exposed to quasar intense ionizing radiation field. We are interested in the latter, as we want to study the structure of galactic outflows driven by the quasar luminosity. When the gas exposed to radiation is in ionization equilibrium, at each point in the layer the concentration of HII is constant. Two processes can modify the ionic concentration in the gas: photoionization (H0+ γ −→ H++ e−) and recombination, acting in the opposite way (H++ e− −→ H0 _{+ γ). If ionization equilibrium holds,}

their rates must be equal at each point r:

n_h(r) ˆ ∞

ν0

4πJν(r)

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CHAPTER 2. STRUCTURE AND DYNAMICS OF GAS CLOUDS

The left hand side is the number of photoionization per unit time per unit volume, since Jν is the mean spectral intensity (erg/cm2s sr) and aν the cross section of the

process, while on the right α_a is a recombination coefficient that we will describe below. The mean lifetime of a neutral atom because of photoionization is the inverse of the integral´_ν∞

0

4πJν

hν aν(H) dν , and its typical values are of the order of τph∼ 10

8_s.

Comparing with mean lifetimes of hydrogen in an excited state (∼ 10−8s, for example Ankudinov et al. (1965)), we notice that the decay to ground state is a much faster process than photoionization. This allows us to work in the following approximation:

1. all neutral hydrogen is in the ground state 1s;

2. all photoionization events involve H in the ground state, but are balanced by recombination to all hydrogen energy levels;

3. all recombination events to excited levels are immediately followed by a decay to the ground state.

For an hydrogen atom in the ground state, the photoionization cross section is given by (Osterbrock and Ferland (2006))

aν(H) = Ao ν_o ν 4 exp[4 − 4(tan−1ε)/ε] 1 − exp(−2π/ε) ν ≥ νo (2.1.2) where Ao ' 6.30 × 10−18cm−2 ε = p ν/νo− 1 (2.1.3)

Electrons produced via photoionization have a characteristic initial energy distribu-tion, which depends on the intensity of the ionizing flux. Nevertheless, elastic scat-tering between electrons has an higher rate than photoionization, so that a Maxwell-Boltzmann thermal distribution is set up. As a result, a gas temperature is well defined, and it is the only parameter describing the kinetic aspects of the gas.

The right hand side of the equation (2.1.1) represents the number of recombina-tions per unit time per unit volume. α_a is a recombination coefficient, which is the sum of rates for recombination of ions with the electron placing in different energy

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2.1. IONIZATION OF AN H NEBULA levels (n, `): α_a(H, T ) = ∞ X n=1 αn(H, T ) = ∞ X n=0 n−1 X `=0 αn`(H, T ) (2.1.4) with αn`(H, T ) = ˆ ∞ 0

uσn`(H, u)f (u) du (2.1.5)

where u is the electron velocity, f (u) is the Maxwell-Boltzmann distribution and σn`(H, u) is the recombination cross section for an electron at velocity u to the level

(n, `). It will also be used the coefficient

α_b(H, T ) =

∞

X

n=2

αn(H, T ) = αa− α1 (2.1.6)

The values of recombination coefficients are listed for instance in Verner and Ferland (1996), but the following fit by Cen (1992) comes in useful:

α_b' 8.4 × 10−11T−0.5 T 103_K −0.2" 1 + T 106_K 0.7#−1 (2.1.7)

To solve equation 2.1.1 we need to explicit the intensity Jν. It is not only due to the

central source, but also the gas in the nebula itself emits radiation:

Jν(r) = Jν,s(r) + Jν,d(r) (2.1.8)

where Jν,d(r) is the “diffuse” contribution. The mean intensity from the central source

is given by Jν,s(r) = R r 2 Jν,s(R)e−τν(r) r > R (2.1.9)

where R is a distance at which we know the flux (for example at the surface of a star, where still there is no absorption) and τν(r) is the optical depth defined as

τν(r) =

ˆ r R

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The diffuse radiation satisfies the transfer equation d

drIν,d(r) = −nh(r)aν(H)Iν,d(r) + jν,d(r) (2.1.11) where Iν,d is the specific radiant intensity (erg/cm2s sr). Re-emitted radiation is

quantified by the emission coefficient jν,d (erg/cm3s sr), and these photons are

pro-duced by recombination events from the ground state:

4π ˆ ∞

ν0

jν,d(r)

hν dν = np(r)ne(r)αa(H, T ) (2.1.12) In an optically thick cloud, assuming that every photon is absorbed very near to where it was emitted, the relation

jν,d(r) = nh(r)aν(H)jν,d(r) (2.1.13)

holds (the on-the-spot approximation). We can then use (2.1.12) and (2.1.13) to rewrite the ionization equilibrium equation (2.1.1). We also introduce the variable x = np/n = ne/n, which represents the ionized fraction of hydrogen (n = np+ nh).

For a central star with radius R we have Jν,s = 1₄Fν,s, where Fν,sis the specific energy

flux from the star, and so 1 − x(r) x2_(r) R r 2ˆ ∞ ν0 πFν,s(R) hν aν(H)e −τν(r)_{dν = nα} b(H, T ) (2.1.14)

When dealing with a clump developing an external HII shell, in the context of quasar outflows, we will assume that the flux is known at a radius R much greater than the radius of the clump and the emitting surface of the quasar (i.e. the accretion disk). Then the r−2 attenuation of flux is not an important factor and can be omitted, and in addition Jν,s= _4π1 Fν,s. The spatial coordinate is the depth into the clump, which

we denote z, and the ionization equilibrium equation becomes 1 − x(z) x2_(z) ˆ ∞ ν0 Fν,s(R) hν aν(H)e −τν(z)_{dν = nα} b(H, T ) (2.1.15)

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2.1. IONIZATION OF AN H NEBULA

The transition between x ' 0 and x ' 1 is very steep, then the HII shell of the clump have a thickness δ_hii, which we can obtain integrating from 0 to δ_hii and substituting x = 1: δ_hii = 1 n2_α b(H, T ) ˆ ∞ ν0 Fν,s(R) hν dν (2.1.16)

Finally, let’s estimate the the characteristic ionization time: an ionization front propagates into the clump, at a speed regulated by the photon flux. We expect this speed to decade exponentially, when the flux is absorbed in the already ionized layer, to balance recombination. Each arriving photon is absorbed before it reaches the ionization front or it ionizes an atom, then we have

4πR2 ˆ ∞ ν0 Fν,s(R) hν dν = 4πR 2_n h˙δhii(t) + 4 3π(R + δhii(t)) 3_{− R}3 (2.1.17) where (R + δ_hii(t))3− R3 ' 3R2δ_hii(t) (2.1.18) Solving the differential equation, we get

δ_hii(t) = δ_hii1 − e−nhαbt _(2.1.19)

˙δhii(t) = δhiinhαbe

−n_hαbt _(2.1.20)

If many dense clouds, a term due to collisional ionization of hydrogen may introduce a significant contribution. It has the form

γ(H, T )nenh (2.1.21) where (Cen (1992)) γ(H, T ) = 5.85 × 10−11 T 0.5 1 + (T /105_K)0.5 e −157809.1/T (2.1.22)

The ionization profile with respect to temperature tends to be very steep for ISM around stars or quasars, and neutral hydrogen concentration drops rapidly to values of the order of 10−3 at temperatures of around 103_K.

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2.2 Temperature of ionized hydrogen

The temperature of an ionized region in a cloud can be found by imposing the equi-librium among the various energy exchange mechanisms. Here we consider heating by photoionization, cooling by recombinations and radiative cooling (electron colli-sional excitation in ions and bremsstrahlung). Photoionization heats the gas since ionized electrons have an energy higher than the mean electron energy in the gas, depending on the incident photon, then they transfer energy to the gas when they thermalize. For a pure H cloud, the heating rate per unit volume is given by

G(H) = n_h ˆ ∞

ν0

4πJν

hν h(ν − ν0)aν(H) dν (2.2.1)

Since the gas in ionization equilibrium, we can put in n_h from equation (2.1.1) to get

G(H) = nenpαa(H, T ) ´∞ ν0 4πJν hν h(ν − ν0)aν(H) dν ´∞ ν0 4πJν hν aν(H) dν (2.2.2)

Looking at recombination, the energy subtracted from the gas is

LR(H, T ) = nenpkbT βa(H, T ) (2.2.3) where β_a(H, T ) = ∞ X n=1 βn(H, T ) = ∞ X n=1 n−1 X `=0 βn`(H, T ) (2.2.4) βn`(H, T ) = 1 k_bT ˆ ∞ 0 uσm`(H, T ) 1 2meu 2_{f (u) du} _(2.2.5)

in complete analogy with coefficients α_A, αn, αn`. Proceeding in the same way as in

the previous section, the on-the-spot approximation corresponds to not including in the energy balance equation photon production by diffuse radiation and recombina-tion to the ground level (as in equarecombina-tion (2.1.14)):

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2.2. TEMPERATURE OF IONIZED HYDROGEN G(H, T ) ' nenpαb(H, T ) ´∞ ν0 4πJν,s hν h(ν − ν0)aν(H) dν ´∞ ν0 4πJν,s hν aν(H) dν (2.2.6) LR(H, T ) ' nenpkbT βb(H, T ) (2.2.7)

with β_b= β_a− β1. We can include helium with the same formalism:

G(He, T ) ' nenheiiαb(He, T )

´∞ ν0 4πJν,s hν h(ν − ν0)aν(He) dν ´∞ ν0 4πJν,s hν aν(He) dν (2.2.8)

LR(He, T ) ' nenheiikbT βb(He, T ) (2.2.9)

The contribution of metals can be safely neglected since G and LR are proportional

to ion number density.

Bremsstrahlung (or free-free) contribution to gas cooling, for an ion with global charge Ze, is given by the expression (Rybicki and Lightman (2008))

LF F(Z) ' 1.42 × 10−27Z2T1/2gf fneni(z) (2.2.10)

where n_i(z) is the number density of the ion and gf f is the Gaunt factor, which for

nebula usually is ∼ 1.3.

A collision between an electron and an ion may result in the transition of the ion into a low lying excited level. The collisional excitation rate is

nen1q12 = nen1 8.629 × 10−6 T1/2 Υ12 g1 e−χ/kbT _(2.2.11)

where Υ12 is a tabulated value for the particular transition, and n1 is the number

density of ions in the lower state. In the limit of very low electron density, every collisional excitation is followed by an emission of a photon, so that the cooling rate per unit volume is

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Figure 2.2.1: Cooling function (solid line) for an ionized plasma, considering bremsstrahlung and line cooling, appropriated for an optically thin gas of cosmic abundances. Figure from Raymond et al. (1976).

In general excited electrons cannot emit a photon if the deexcitation happens before via another collision. In that case the equilibrium between excitation and deexcitation is

nen1q12 = nen2q21+ n2A21 (2.2.13)

where A21 is the spontaneous decay rate, so that

n2

n1

= neq12/A21 1 + neq21/A21

(2.2.14)

Thus the cooling rate is

LC = n2A21hν21=

n1neq12

1 + neq21/A21

hν21 (2.2.15)

Other mechanisms that contribute to heating or cooling the gas are collisional pho-toionization of H and He and Compton heating. A cooling function which accounts for all the cooling effects can be computed for a gas when its composition and the

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2.3. DENSITY PROFILE OF AN ISOTHERMAL CLOUD

degree of ionization of each species are specified. Regarding two-body processes, as bremsstrahlung and line cooling, a cooling function has been provided by Raymond et al. (1976) and then fitted by Mathews and Bregman (1978):

Λ(T ) =      2.18 × 10−18T erg cm3s−1 103K ≤ T ≤ 1.3 × 105K 2.18 × 10−18T−0.6826+ 2.706 × 10−47T2.976 _{erg cm}3_s−1 _{1.3 × 10}5_{K ≤ T ≤ 10}8_K (2.2.16) Such cooling function has been calculated for an optically thin ionized gas of cosmic abundances in the range 105− 108_{K, and it is appropriate for studying gas in quasar}

outflows.

2.3 Density profile of an isothermal cloud

Consider a spherical gas cloud of mass M surrounded by a medium with constant pressure Po. We want to describe the density as a function of radius of the cloud

con-sidering its self-gravity, usually called Bonnor-Ebert density profile (Bonnor (1956) and Ebert (1955)). The equation of hydrostatic equilibrium is

d drP (r) = − GM (r) r2 ρ(r) = − dΦ drρ(r) (2.3.1)

where M (r) is the mass enclosed within radius r, and Φ(r) is the gravitational potential. The equation of state of an ideal gas is

P (r) = ρ(r)c2_s (2.3.2)

where cs is the isothermal sound speed, constant throughout the sphere. Then we

can write (2.3.1) as d drln ρ(r) = − d dr Φ c2 s (2.3.3) Denoting ρcthe central density (that we don’t know in our problem) and noting that

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Figure 2.3.1: Solution of the dimensionless Lane-Emden equation (2.3.7) for a Bonnor-Ebert sphere.

ρ(r) = ρcexp[−Φ(r)/cs] (2.3.4)

The potential is given by the Poisson equation

∇2_{Φ =} 1 r2 d dr r2 d drΦ(r) = 4πGρ(r) = 4πGρcexp[−Φ(r)/c2s] (2.3.5)

In the dimensionless variables

ψ = Φ(r)/c2_s ξ = 4πGρc c2

s

1/2

r (2.3.6)

the equation becomes

1 ξ2 d dξ ξ2dψ dξ = exp(−ψ) (2.3.7)

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Figure 2.3.2: Dimensionless mass for a Bonnor-Ebert sphere, as a function of density contrast between the centre and the edge of the cloud. At a density contrast ρc/ρo'

14.1 the mass rises to a maximum value, above which the cloud is unstable for collapse. Actually, it can be shown that only cloud with ρc/ρo . 14.1 are stable,

otherwise undergoing a run-away collapse or run-away growth.

ψ(0) = 0 dψ dr ξ=0 = 0 (2.3.8)

The solution of equation (2.3.7) can be found numerically: the function ψ and ρ/ρc=

e−ψ are plotted in figure 2.3.1.

The radius R of the sphere is given by the value ξo at which ψ is zero, and the

total mass can be expressed as

M = ˆ R 0 4πr2ρ(r) dr = 4πρc c2_s 4πGρc 3/2ˆ ξo 0 ξ2e−ψdξ = 4πρc c2_s 4πGρc 3/2 ξ2dψ dξ ξo (2.3.9) using the equation (2.3.7) in the last passage. The mass M is given, but it’s more

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convenient to use the dimensionless mass

m(ξo) = Po1/2G3/2M c4 s = 4πρc ρo −1/2 ξ2dψ dξ ξo (2.3.10)

where ρo is the density at the edge of the cloud (easily calculated by Po = ρoc2s).

Plotting m(ξ) = 4πeψ(ξ)−1/2ξ2dψ

dξ , is possible to find the the value of ξo knowing m(ξo), which we have computed from M . Then we get ρc and the the density profile

from (2.3.4). Finally, the radius of the cloud is given by

R = 4πGρc c2

s

−1/2

ξo (2.3.11)

We apply this prescription to write the density profile for Bonnor-Ebert spheres of atomic hydrogen with T = 104K, Po = 1.8 × 10−7erg/cm3 and total mass ranging

between 102M and 104M, as they will be in our interest when studying molecular

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Figure 2.3.3: Density profiles of Bonnor-Ebert spheres with T = 104K, Po = 1.8 ×

10−7erg/cm3 _{and total mass ranging between 10}2_M

and 104M. We notice that

number densities are of the order of 105_cm−3 _{with density contrasts between the}

centre and the edge not grater then 1.4. The cloud radii are around some tenths of parsec.

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Figure 2.4.1: Structure of a PDR according to the model by Tielens and Hollenbach (1985) with initial parameters G0 = 1.0 × 105 and n0 = 2.3 × 105cm−3. Radiation

is coming from the left, completely ionizing a shell of gas. The PDR starts from the HI/HII interface, where ionizing radiation is completely shielded and the FUV portion of the spectrum begins to determine the morphology of the cloud. Hydrogen is found in molecular form at a depth greater then Av ' 2, which corresponds to an

HI column density N_HI' 2 ÷ 5 × 1021_cm−2_.

2.4 Photodissociation regions

A molecular cloud exposed to radiation will develop a region in atomic form because of photodissociation of molecules. The radiation with energy above 13.6 eV ionizes a superficial layer, and once it has reached its final equilibrium thickness, it completely prevents the ionizing radiation to travel deeper into the cloud. The less energetic portion of the spectrum, between 6 eV and 13.6 eV, crosses the HII layer and it is responsible for the photodissociation of molecules. In literature it is often referred as FUV (far ultra-violet) radiation, while the region whose energetic and chemical com-position is basically dominated by this flux is the PDR (photodissociation region). In a molecular cloud the PDR is the layer between the cloud core in molecular form

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2.4. PHOTODISSOCIATION REGIONS

and the fully ionized shell.

A complete analysis of the structure of a PDR has been performed by Tielens and Hollenbach (1985), where a theoretical model is presented for a given FUV flux G0

(in units of Habing flux 1.6 × 10−3erg/cm2_{s) and a given hydrogen nucleus number}

density n0. To compute the abundances of each species and the temperature of the

gas, equations for chemical balance and energy balance are required. The molecules considered are the ones forming from the elements from H to Si, such as H2, OH,

O2, CH, SiO, CO, HCO, H2O, CH2, etc.., and the possible reactions involving them

and relative ions are taken into account (the vibrationally excited H∗₂ is considered separately, since it takes part independently to some reactions).

Here we briefly describe the structure of a PDR as a function of the depth into the cloud, expressed as visual extinction (Av ' 1 corresponds to a column density

2 × 1021_cm−2_{), for a standard model with G}

0 = 1.0 × 105 and n0 = 2.3 × 105cm−3.

Hydrogen is mainly in atomic form until Av ' 2, where FUV radiation is sufficiently

shielded and photodissociation is balanced by H2 formation. The possible

mechan-isms for H2 synthesis are formation on dust grain surfaces (Hollenbach and Salpeter

(1971)) and radiative association reactions followed by associative detachment reac-tions (de Jong (1972))

H + e− −→ H−+ hν H−+ H −→ H2+ e−

(2.4.1)

H2 formation and FUV pumping to vibrationally excited state followed by

colli-sional de-excitation, are important processes contributing to the heating of the gas, while the dominating heating process down to Av ' 6 is photoelectric effect on

dust grains. On the other hand, gas cooling is dominated by infrared fine-structure lines from atoms (mainly the 63 µm line of OI) and rotational lines from molecules. Both photoelectric effect and line cooling decrease inside the cloud because of the increasing FUV optical depth, but the first tends to drop more slowly considering that the charge of grains decreases with depth, with a consequent gain of photoelec-tric effect efficiency. Thus the temperature in the HI region has a profile peaking at T ' 100 ÷ 3000 K around Av ' 1.2, depending on the initial parameters of the

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to form are found: the most important ones are the CII/CI/CO transition at Av ' 7

and O/O2 transition at Av > 10. Anyway, we will be mainly concerned with the HI

shell, since it is heated to an higher temperature than the deeper zones of the PDR, as shown in figure 2.4.2. When a molecular cloud at 10 ÷ 100 K is suddenly exposed to radiation, the formation of an high pressure HI shell drives a shock into the cloud, which is consequently going to shrink.

The thickness of the HI shell can be approximately computed equating the H2

formation rate and the photodissociation rate, considering self-shielding. As in Hol-lenbach and Tielens (1999), we use a semi-empirical rate for H2 formation on grain

surfaces

Rf orm = γh2nh(nh+ 2nh) γh2 ' 1 ÷ 3 × 10

−17

cm3/s (2.4.2)

and the approximation for the photodissociation rate by Draine and Bertoldi (1996)

Rdiss = fshield(Nh2)e

−τgr,1000_I

diss(0)nh2 (2.4.3)

where Idiss(0) = 4 × 10−11G0s−1 is the unshielded radiation intensity, τgr,1000 is the

optical depth of the dust at 1000 ˚A and fshield is a factor given by

fshield(Nh2) =        1 N_h₂ _{> 10}14_cm−2 N_h₂ 1014_cm−2 −3/4 1014_cm−2 > Nh2 > 10 21_cm−2 (2.4.4)

For H2column density greater than 1021cm−2, dust opacity dampens the dissociation

rate. When equilibrium between H2 formation an destruction holds, in the range

1014_cm−2

> Nh2 > 10

21_cm−2_{, neglecting dust opacity, it’s possible to obtain the}

following expression for the molecular fraction:

x_h2 = n_h₂ n0 = 4 γ_h₂n0 4Idiss(0) 4 N0 1014_cm−2 3 (2.4.5)

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2.5. SHOCK AND RAREFACTION WAVES

Figure 2.4.2: Temperature profile of a PDR with initial parameters G0 = 1.0 × 105

and n0 = 2.3 × 105cm−3. The gas shows an high temperature into the HI region,

where incident FUV radiation is intense and photoelectric heating is efficient. Going deeper into the cloud, line cooling by atoms and molecules decrease the temperature below 100 K, and gas and dust temperature tend to couple. Figure from Tielens and Hollenbach (1985).

column density has a dependence N_pdr ∼ (n0/G0)−4/3 on gas density and FUV flux,

which means δ_pdr ∼ n−7/3₀ G4/3₀ for the PDR thickness.

2.5 Shock and rarefaction waves

A plane shock propagating in a fluid is a discontinuity surface of flow variables ρ, P , T , v, moving at a certain velocity D with respect to the lab frame, where conservation equations of fluids are respected across the discontinuity. We will outline the main properties of a shock wave, as developed for the first time in Rankine (1870). We will work in the frame of the shock front, with the x axis of our Cartesian coordinate system perpendicular to the shock surface, and u is the flow velocity in the x direction this frame. The subscript 0 denotes the variables ahead of the shock, in the unperturbed flow, while the subscript 1 stands for variables in the shocked fluid, as in figure 2.5.1 For a steady flow, the continuity equation immediately gives

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Figure 2.5.1: Schematic representation of a shock front propagating into a fluid, as seen in the lab reference frame, and the shock front reference frame. Obviously it has to be v0, v1 < D.

ρ0u0 = ρ1u1 (2.5.1)

(u0 and u1 are taken positive) and similarly, the Euler equation for a steady,

non-viscous fluid udu dx = − 1 ρ dP dx (2.5.2) gives ρ0u20 + P0 = ρ1u21+ P1 (2.5.3)

The energy transport equation (kinetic and internal) for an adiabatic process is

1 2u 2 0+ ε0+ P0 ρ0 = 1 2u 2 1+ ε1 + P1 ρ1 (2.5.4) where ε is the internal energy for unit mass, which for an ideal gas with constant specific heat is

ε = 1

γ − 1 P

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2.5. SHOCK AND RAREFACTION WAVES

To sum up, the Rankine-Hugoniot jump conditions are            ρ0u0 = ρ1u1 ρ0u20+ P0 = ρ1u21+ P1 1 2u 2 0+ γ0 γ0− 1 P0 ρ0 = 1 2u 2 1+ γ1 γ1− 1 P1 ρ1 (2.5.6)

If rest flow variables are known and a parameter describing the strength of the shock is given (for example v1, u1, P1, T1 or the shock speed D), then jump conditions

allow to solve for the state of the post-shock fluid. The following useful relations can be obtained in the case γ = γ1 = γ2, involving the Mach number Mi = ui/ci

(i = 0, 1) with ci speed of sound:

u1 u0 = (γ − 1)M 2 0 + 2 (γ + 1)M2 0 (2.5.7) P1 P0 = 2γM 2 0 − (γ − 1) γ + 1 (2.5.8) T1 T0 = 1 + γ − 1 2 M 2 0 1 + γ − 1 2 M 2 1 (2.5.9) M₁2 = M2 0 + 2 γ − 1 2γ γ − 1M 2 0 − 1 (2.5.10)

Consider a fluid at rest and a piston confining the fluid in the region x > 0. A shock wave may be generated if the piston push into the fluid. But if the piston starts moving backwards, then the fluid occupies the region to the left, where the piston is making room. A front propagates to the right, such that the fluid is progressively put into motion toward the piston (the fluid particles close to it move at same velocity of the piston), while pressure and density decrease to lower values. The formalism of characteristics and Riemann invariants allows to find a complete solution of such phenomenon, called rarefaction wave, for a given velocity of the piston. Zel’dovich and Raizer (2002) go through a full derivation, and here we present the resulting

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motion of a rarefied fluid. Assume that the piston starts to move from x = 0 at t = 0 with velocity v = −U , and assume that the system is adiabatic. The fluid has initially a pressure Po, a density ρo and a speed of sound co. Then a “wave head”

moves at the speed of sound toward the right, followed by a “wave tail” at a velocity

vtail = co−

γ + 1

2 U (2.5.11)

The fluid passed by the wave head starts accelerating according to

u = − 2 γ + 1 co− x t (2.5.12)

i.e. it accelerates linearly so that it has reached the speed of the piston when crossing the wave tail. The pressure and density throughout the rarefaction wave are

ρ = ρo 1 −γ − 1 γ + 1 1 − x/t co 2/(γ−1) (2.5.13) P = Po 1 −γ − 1 γ + 1 1 −x/t co 2γ/(γ−1) (2.5.14) so that the final pressure after the wave has completely passed are

ρf = ρo 1 − γ − 1 2 U co 2/(γ−1) (2.5.15) Pf = Po 1 −γ − 1 2 U co 2γ/(γ−1) (2.5.16)

It is interesting that U has to be less than Umax =

2

γ − 1co for these expressions to make sense. Indeed if U is moving too fast, a vacuum is created between the piston and the fluid. Incidentally, we have thus found the maximum speed of a gas expanding into vacuum.

If a system is not adiabatic, cooling processes modify the shock or post-rarefaction temperature. A common case, and a simplifying approximation in some circumstances, is that after the disturbance has passed, temperature goes back to

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2.6. PROPAGATION OF SHOCKS IN CLOUDS

the original value, i.e. the process is isothermal. For a shock, assuming that the gas suddenly cools downstream, the energy equation in (2.5.6) can be substituted with T1 = T0, or P1/ρ1 = P0/ρ0. The theory of rarefaction waves can also be adapted to

the isothermal case, and the result is that equations (2.5.13), (2.5.13) and (2.5.14) become u = −co− x t ρ = ρ0exp x/t co − 1 P = P0exp x/t co − 1 (2.5.17)

2.6 Propagation of shocks in clouds

Now we want to study how a shock wave propagate inside a spherical cloud de-scribed by the Bonnor-Ebert density profile. In this situation the shock front doesn’t propagate with a constant speed, since the effects of sphericity and the radial density distribution have to be taken into account. For an isentropic flow in one dimension, the three flow equations

           ∂tρ + ∂x(ρu) = 0 ∂tu + u∂xu = − 1 ρ∂xP (∂t+ u∂x)S = 0 (2.6.1)

hold, where u is the Eulerian velocity, ρ is the density and c is the local sound speed. Combining together the equations, it is possible to show that they are equivalent to

             [∂t+ (u + c)∂x]u + 1 ρc[∂t+ (u + c)∂x]u = 0 [∂t+ (u − c)∂x]u − 1 ρc[∂t+ (u − c)∂x]u = 0 DS Dt = 0 (2.6.2)

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CHAPTER 2. STRUCTURE AND DYNAMICS OF GAS CLOUDS            dP + ρc du = 0 along C+ dP − ρc du = 0 along C− dS = 0 along C0 (2.6.3)

defining the curves in the x, t plane (called characteristics)

C± : dx dt = x ± cs (2.6.4) C0 : dx dt = 0 (2.6.5)

This way of rewriting the flow equations is sometimes more convenient, and this is the case when dealing with non-constant equilibrium density or sectional area. This way of tackling the problem has been developed in Whitham (1958), and here we present the crucial steps of the procedure.

A flow with spherical symmetry can be described as a flow in a tube with a sectional area changing according to A(x) = 4πx2, where x is the distance from the centre. The equation of continuity can be written as

∂t(ρA) + ∂x(ρuA) = 0 (2.6.6)

integrating over a truncated cone with height dx and base areas A(x) and A(x + dx), and after few steps it becomes

∂tρ + ∂x(ρu) + ρu

A0(x)

A(x) = 0 (2.6.7)

Consider now a fluid with initial density and pressure ρ0(x) and P0(x). If this

configuration is stable, than a force must exist to maintain the pressure gradient, and its intensity (per unit mass) is

F (x) = 1 ρ0(x)

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2.6. PROPAGATION OF SHOCKS IN CLOUDS

Thus momentum conservation equation becomes

∂tu + u∂xu = −

1

ρ∂xP + F (x) (2.6.9)

With the two extra terms introduced in equations (2.6.7) and (2.6.9), the equation along C+ is dP + ρc du + ρc 2_u u + cA(x) − ρc u + cF (x) = 0 (2.6.10) where A(x) = A 0_(x) A(x) (2.6.11)

If a shock wave is propagating in a fluid with such properties, the above equation can be applied along the shock, approximating the shock front trajectory in the x, t plane with a characteristic C+ (approximation found to be accurate by Whitham

(1958)). Plugging in in the relations u(M ), P (M ) and ρ(M ) for the post shock variables (from (2.5.7) and (2.5.10)), we get and equation of the form

dM dx =

a1(M )F (x) + a2(M )A(x)

a3(M )

(2.6.12)

that can be solved for the Mach number of the shock front M (x) (with respect to the pre-shock sound speed). This formalism may than been applied to the the problem of a shock propagating into an isothermal cloud, taking ρ0(x) and P0(x) from

the Bonnor-Ebert profile, and A(x) = 4πx2 _{for a sphere. The results obtained by}

numerical integration show that, for typical quantities of clumps in quasar outflows, the density profile does not introduce a considerable correction to the shock speed. Thus the solution for M (x) that we find is nearly the same of the exact solution for spherical imploding shock by Guderley (1942), according to which

M (x) = M (x0)

x₀ x

n(γ)

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Figure 2.6.1: A typical clump in a quasar outflow has a mass of 104_M

, a radius

of 0.5 pc and is confined by a pressure ∼ 1.8 × 10−7Ba. If a shock originates near the surface, with initial Mach number M0, then it propagates to the centre with

increasing velocity according to the equations developed in this chapter. The plots show the profile M (r) for different initial Mach numbers, computed considering the density variation in the cloud and the sphericity (solid line) or using the power law (2.6.13) (dashed line). In the plot on the left the solid and dashed line are undistinguishable, while we can notice from the zoom in the plot on the right that there is a more important difference for small Mach numbers.

where n is an exponent dependent on the index γ: n(5/3) ' 0.453 n(7/5) ' 0.394

(2.6.14)

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2.7. ARBITRARY DISCONTINUITIES

2.7 Arbitrary discontinuities

Suppose that in a fluid an initial discontinuity (say at t = 0) of some of the flow variables is present (for example two fluid with different pressures separated by a membrane), with no relation between the variables across the discontinuity. We want to find the resulting motion of the fluid for t > 0. Since the problem is self-similar, only one wave can propagate in each direction, otherwise there would be a characteristic length given by the distance between two waves. In general, we will have two waves (shock or rarefaction) propagating in opposite directions starting from the discontinuity surface. The shocked or rarefied regions have uniform flow variables, and are separated by the discontinuity surface. These two regions must have the same pressure and the same velocity (not necessarily temperature or density), otherwise a flow would generate and the two regions would not be uniform anymore. Let’s analyse in more detail the behaviour of variables in the various regions, with an argument similar to Shugaev and Shtemenko (1998).

Let’s consider a plane discontinuity in a fluid, where pressure and velocities P_l, v_l to the left of the surface and P_r, v_r to the right, while density, temperature and γ are continuous. Assuming P_l > P_r and v_r = 0, we look for the solutions of the problem for different values of v_l. It is convenient to work in a P , v plain, drawing the lines connecting the initial state of the fluid with the possible final states after it has been shocked or rarefied. Rarefaction waves decrease the fluid pressure, while shocks always increase the pressure. Thus, starting from the point (Pi, vi) with i = l, r, the

possible states (P, v) of the fluid processed by the wave are

v =          vi+ 2ci γi− 1 " P Pi γ−1_2γ − 1 # P < Pi vi+ ci r 2 γ P/Pi− 1 p(γ + 1)P/Pi+ (γ − 1) P ≥ Pi (2.7.1)

The intersection of the lines starting from (P_l, v_l) and (P_r, v_r) gives the final state of the central region between the two waves. Depending on v_l, different types of wave have to originate from the discontinuity, in order to have the same final velocity and

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pressure in the regions to the left and to the right. In figure 2.7.1 the graphical solution of the problem is represented, showing the three possible situations that can happen:

1. for v_l = v_l,1 two shocks propagate in opposite directions with respect to the discontinuity, the final pressure in the central region is higher that the two initial pressure.

2. for v_l = v_l,2 a shock propagates to the left, while the fluid to the left of the discontinuity is rarefied, so that the central region has an intermediate pressure between the initial ones.

3. for v_l = v_l,3two rarefaction waves propagate in the fluid, resulting in a pressure lower than both initial pressures.

The discriminant between the three possibilities is given by a comparison between v_l and the two limiting values va and vb. If vl = va, that simply means that

dis-continuity is not really arbitrary, but Rankine-Hugoniot conditions hold, so that the discontinuity is a shock front. Similarly, v_l = vb means that a rarefaction waves is

propagating from the right to the left. Thus in general we have

v_l− v_r > va → two shock waves

vb < vl− vr < va → a shock wave and a rarefaction wave

v_l− v_r < vb → two rarefaction waves

(2.7.2) va= cr r 2 γ_r P_l/P_r− 1 p(γ_r+ 1)P_l/P_r+ (γ_r− 1) vb = 2c_l γ_l− 1 " P_r P_l γl−1_2γl − 1 # (2.7.3) Using what just developed, it’s possible to tackle the problem of interaction between two shocks. The formalism of arbitrary discontinuities comes in handy, since the moment when the two shock fronts interact is a contact discontinuity, and we can use it as initial conditions to understand the following evolution of the sys-tem. It is to be noticed that without an hydrodynamical code, it’s not possible to follow the complete dynamics of the interaction, but we get the resulting asymptotic

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2.7. ARBITRARY DISCONTINUITIES

states. This is what we need to study how shocks determine the evolution of gas clouds.

The problem in the isothermal case is completely equivalent, with

v(isoth)_a = s P_l ρ_r 1 − Pl P_r v_b(isoth) = c_llog(P_r/P_l) (2.7.4)

Figure 2.7.1: Graphic solution of the arbitrary discontinuity problem with initial conditions (P_l, v_l) and (P_r, v_r) to the right and to the left of the discontinuity, with P_l > P_r and v_r = 0. The other flow variables and γ are continuous. The blue-black line connects all the possible final states to the right, and the green-red lines the possible final states to the left, for different values of v_l. The intersection of the lines for the regions to the left and to the right gives the solution, and different types of waves are required to get to the final state, according to the value of v_l.

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2.8 Wave interaction

To solve the problem of the interaction between shock waves or rarefaction waves, it’s in general necessary to solve the flow equations

           ∂u ∂t + ∂ ∂x(ρu) = 0 ∂ ∂t(ρu) + ∂ ∂x(ρu 2_{+ P ) = 0} ∂ε ∂t + ∂ ∂x[(ε + P )u] = 0 (2.8.1)

with ε energy density. The equations can be written in the matricial form ∂U ∂t + ∂E ∂x = 0 (2.8.2) U = [ u, ρu, ε ] (2.8.3) E = [ ρu, ρu2+ P, (ε + P )u ] (2.8.4)

which is an hyperbolic partial differential equation. A simple algorithm to deal with these equations is the MacCormack algorithm (MacCormack (2003)), a time-marching method which gives the flow variables of the system at each point in space. Time and space are discretized in a grid defined by {xn, tm}, where n = 0, . . . , N1

and m = 0, . . . , N2. Assume that U is know at the initial time t = 0, i.e. the values

Un,0 are known for every n. Then a “predictor value” of U at time t1 is constructed

for every n:

U_n,1pred = Un,0− a

t1− t0

xn+1− xn

(Un+1,0− Un,0) (2.8.5)

successively corrected with the final value

Un,1= 1 2(Un,0+ U pred n,1 ) − 1 2a t1− t0 xn+1− xn (U_n,1pred− U_n−1,1pred ) (2.8.6) Recursively, the solution is found for every time:

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2.8. WAVE INTERACTION        U_n,m+1pred = Un,m− a tm+1− tm xn+1− xn (Un+1,m− Un,m) Un,m+1= 1 2(Un,m+ U pred n,m) − 1 2a tm+1− tm xn+1− xn (Upred n,m − U pred n−1,m) (2.8.7)

For time-marching numerical solutions of PDFs, the CFL condition (Courant et al. (1928)) on the time steps has to be satisfied to ensure the convergence of the al-gorithm. The condition is usually expressed as

∆t

∆xV ≤ 1 (2.8.8)

where V is the speed at which “information” travels between two grid points. In the case in analysis, we take V = |u| + cs. Since V is different at each point of the spatial

grid, actually at the m time step we have a set of values {Vn,m}n. Then to compute

the (m + 1) time step, we take as time increment the minimum value of

∆tn,m =

(xn− xn−1)

Vn,m

(2.8.9)

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Chapter 3 Molecular clumps in the outflow

3.1 Gas cooling in the outflow

The presence of molecular gas in quasar outflow is a problem in principle, because the temperature of the shocked gas reaches 107K, a temperature higher than mo-lecule dissociation temperature (the binding energy in H2 is roughly 4.52 eV, which

corresponds to about 50, 000 K). Thus, efficient cooling mechanisms must operate, so that a fraction of gas is put in molecular form, and in the following we will see the conditions for the gas to be stable in two different phases.

Cooling mechanism that should be considered are Compton cooling, bremsstrahlung and line cooling. However, the shocked gas is at a temperature similar to Compton temperature for radiation in the extreme UV

TC(h¯ν ' 103eV) =

h¯ν

4k_b ' 3 × 10

6_K _(3.1.1)

(The extreme UV is a reasonable mean photon energy in a quasar spectrum). Then we do not expect Compton cooling to be relevant to the purpose of decreasing the gas temperature to about 104_{K . The cooling function (2.2.16) is appropriate for a}

fully ionized optically thin plasma, and indeed the optical depth of the shocked ISM is

Evolution of Molecular Clumps in Quasar Outflows

Contents

Abstract

Introduction

Chapter 1

Quasar outflows

1.1

Active Galactic Nuclei and Quasars

1.2

Accretion mechanism and Eddington

lumin-osity

1.3

Interaction with the host galaxy

1.4

Structure of an outflow

1.5

Outflows observations

Chapter 2

Structure and dynamics of gas

clouds

2.1

Ionization of an H nebula

2.2

Temperature of ionized hydrogen

2.3

Density profile of an isothermal cloud

2.4

Photodissociation regions

2.5

Shock and rarefaction waves

2.6

Propagation of shocks in clouds

2.7

Arbitrary discontinuities

2.8

Wave interaction

Chapter 3

Molecular clumps in the outflow

3.1

Gas cooling in the outflow