AA YYYY

# Photodissociation and X-Ray Dominated Regions

Mark G. Wolfire  1 Livia Vallini  2 and Mélanie Chevance3 1Department of
Astronomy, University of Maryland, College Park, MD, 20742, USA; email:
<EMAIL_ADDRESS>2Scuola Normale Superiore, Piazza dei Cavalieri 7,
I-56126, Pisa, Italy  3Astronomisches Rechen-Institut, Zentrum für Astronomie
der Universität Heidelberg, Mönchhofstrasse 12-14, D-69120 Heidelberg, Germany

###### Abstract

The radiation from stars and active galactic nuclei (AGN) creates
photodissociation regions (PDRs) and X-ray dominated regions (XDRs), where the
chemistry or heating are dominated by far-ultraviolet (FUV) radiation or X-ray
radiation, respectively. PDRs include a wide range of environments from the
diffuse interstellar medium to dense star-forming regions. XDRs are found in
the center of galaxies hosting AGN, in protostellar disks, and in the vicinity
of X-ray binaries. In this review, we describe the dominant thermal, chemical,
and radiation transfer processes in PDRs and XDRs, as well as a brief
description of models and their use to analyze observations. We then present
recent results from Milky Way, nearby extragalactic, and high-redshift
observations.

Several important results are:

  * •

Velocity resolved PDR lines reveal the kinematics of the neutral
atomic gas and provide constraints on the stellar feedback process.
Their interpretation is, however, in dispute as observations suggest a
prominent role for stellar winds while they are much less important
in theoretical models.

  * •

A significant fraction of molecular mass resides in CO-dark gas
especially in low-metallicity/highly irradiated environments.

  * •

The CO ladder and [C I]/[C II] ratios can determine if FUV or X-rays
dominate the ISM heating of extragalactic sources.

  * •

With ALMA, PDR and XDR tracers are now routinely detected
on galactic scales over cosmic time. This makes it possible to link
the star formation history of the Universe to the evolution of the
physical and chemical properties of the gas.

###### doi:

10.1146/((please add article doi))

###### keywords:

photodissociation regions, x-ray dominated regions, interstellar medium,
active galactic nuclei, infrared emission, star forming regions

††journal: Xxxx. Xxx. Xxx. Xxx.

###### Contents

  1. 1 INTRODUCTION
  2. 2 Fundamental Processes in PDRs and XDRs
    1. 2.1 Thermal Balance
    2. 2.2 Radiation Transfer
    3. 2.3 Chemical Balance
  3. 3 The Physics of PDRs
    1. 3.1 1-D Structure
    2. 3.2 Chemistry
      1. 3.2.1 Photoreactions
      2. 3.2.2 Photodissociation of H2
      3. 3.2.3 Formation of H2
      4. 3.2.4 Reactions with ${\rm H_{2}^{*}}$
      5. 3.2.5 Cosmic-ray reactions
      6. 3.2.6 Grain assisted recombination
      7. 3.2.7 Grain surface reactions
    3. 3.3 Gas Heating
    4. 3.4 Gas Cooling
  4. 4 The Physics of XDRs
    1. 4.1 1-D Structure
    2. 4.2 Gas Heating
    3. 4.3 Gas Cooling
    4. 4.4 Chemistry
  5. 5 Using Models to Analyze Observations
    1. 5.1 Overview
    2. 5.2 Main input parameters
    3. 5.3 Diagnostic plots
      1. 5.3.1 Diagnostic to determine the gas density $n$, and radiation field strength, $G_{0}$
      2. 5.3.2 Diagnostics to distinguish X-rays from other heating mechanisms
    4. 5.4 Origin of [C II]
    5. 5.5 Caveats in using models to analyze observations
  6. 6 Time Dependent and Multi-Dimensional Models
    1. 6.1 One-dimensional time dependent models
    2. 6.2 Multidimensional codes with steady state chemistry
    3. 6.3 3-D Magnetohydrodynamic codes with time dependent chemistry
  7. 7 Galactic Observations
    1. 7.1 High FUV field PDRs
    2. 7.2 Intermediate to Low FUV fields
    3. 7.3 Diffuse Gas
  8. 8 Extragalactic Observations
    1. 8.1 Observations on galaxy scales
    2. 8.2 Mapping and velocity resolved observations applied to galaxies
    3. 8.3 Galactic centers
    4. 8.4 Low metallicity environments
  9. 9 PDRs and XDRs in the high-redshift Universe

##  1 INTRODUCTION

Massive stars produce a prodigious amount of radiative energy that interacts
with the interstellar medium (ISM). The radiation drives chemical processing
and heats the gas and dust which cool in bright line and continuum emission.
Regions where the heating or chemistry are dominated by far-ultraviolet (FUV;
$6\,{\rm eV}<h\nu<13.6\,{\rm eV}$) radiation are called photodissociation
regions (or PDRs; Tielens & Hollenbach, 1985a, hereafter TH85).111 Sometime
referred to as Photon-Dominated Regions (also PDRs). This term was first
coined by Alex Dalgarno and Amiel Sternberg to contrast PDRs with optically
thick gas in which the chemistry is driven by cosmic-ray ionization, rather
than by FUV photons that stimulate molecule production via ionization and
heating in addition to destroying molecules via photodissociation (A.
Sternberg, private communication). In this review we will adopt the original
and more widely used term Photodissociation Regions. PDRs can have a range of
incident FUV flux, densities, and column densities, depending on the distance
of the stars that produce the FUV radiation and the gas environment where the
FUV is absorbed. The “classic” PDR discussed in TH85 is one in which FUV
radiation is emitted by an O or early B star and passes through an H II region
to be absorbed in an adjoining molecular cloud. In general, these are high
density and high FUV field PDRs (e.g., Tielens & Hollenbach, 1985b, Meixner et
al., 1992, Tielens et al., 1993) and is where feedback from radiative and
mechanical energy of massive stars acts on the neutral gas. The FUV radiation
from the OB association may also illuminate more distant parts of the
molecular cloud and produce PDRs with low radiation fields but greater area
than the high intensity PDRs (Stacey et al., 1993). In addition, the OB stars
contribute FUV radiation to the interstellar radiation field that illuminates
molecular clouds but also illuminates the diffuse ISM where the field
strength, density and column densities are lower than the classic PDRs
(Wolfire et al., 1995a, 2003). PDRs can also be found in reflection nebula
(Chokshi et al., 1988, Steiman-Cameron et al., 1997), in planetary nebula
(Graham et al., 1993, Latter et al., 2000), on the surfaces of pillars and
globules (Mookerjea et al., 2019, Goicoechea et al., 2020, Schneider et al.,
2021), in embedded protostars (van Kempen et al., 2010, Visser et al., 2012),
and protostellar and protoplanetary disks (Aikawa et al., 2002, Gorti &
Hollenbach, 2004, Woitke et al., 2010, Kamp et al., 2010, Öberg & Bergin,
2021).

PDRs include all regions of the neutral ISM where FUV radiation plays a role
in the physics and/or chemistry. This includes the atomic gas, but also the
deeper molecular layers where FUV radiation plays a role in the organic
inventory. Most of the molecular gas in giant molecular clouds resides in PDRs
and thus almost all of the atomic gas and molecular gas in the Galaxy is in
PDRs. This conclusion holds for other galaxies as well so that most of the
non-stellar baryons within galaxies are in PDRs.

Much of the infrared line and continuum radiation from galaxies arises in PDRs
(Crawford et al., 1985, Stacey et al., 2010). The FUV radiation is mainly
absorbed by grains and is radiated away as infrared continuum, but a fraction
of the FUV radiation heats the gas via the photoelectric effect on small
grains and large molecules. The PDR line emission is the dominant coolant in
the neutral gas and can be as bright as $0.1-1$% of the infrared continuum.
The line and continuum emission can be used to determine the gas physical
conditions in the PDR gas, and thus in most of the mass of the ISM.

The observation and modeling of PDRs is also important for star formation. The
line emission reveals the local environment that gives rise to star formation
and the feedback processes that might inhibit it. PDRs measure the star
formation rate by probing the radiative energy produced by embedded massive
stars. In addition, PDRs play an important role in the global processing of
material between thermal phases, which results in cold dense gas from which
molecules and stars can form (Ostriker et al., 2010). The ionization produced
by FUV radiation in molecular clouds provides coupling to magnetic fields,
helping to regulate star formation (McKee, 1989).

In addition to FUV radiation, X-ray radiation can dominate the heating,
ionization and chemical composition in X-ray dominated regions (XDRs; Maloney
et al., 1996)222The term X-ray Dissociation Regions - in analogy to the
Photodissociation Regions - was first coined by Maloney et al. (1996) to refer
to the portion of a gas cloud illuminated by an X-ray source where the
molecule dissociation, but also the ionization, heating, and chemical
composition are driven by the X-rays. For this reason the term XDR is now
widely used with the more general meaning of X-ray Dominated Region. In this
review we will adopt this definition.. The X-rays can be produced by a variety
of sources and processes such as young stellar objects, gas accretion onto a
super massive black hole, and X-ray binaries. Early seminal works by Krolik &
Kallman (1983) and Lepp & Shull (1983) concentrated on the effects of X-rays
from embedded stellar sources within molecular clouds, and on the impact of
X-rays on the molecular gas within the obscuring torus of Active Galactic
Nuclei (AGN; e.g., Krolik & Lepp, 1989). The first comprehensive analysis of
the chemistry, ionization, and thermal balance of XDRs was that of Maloney et
al. (1996). Since then, the physics and chemistry of XDRs has been examined
from several different perspectives concentrating on protoplanetary disks
(Glassgold et al., 1997, Igea & Glassgold, 1999, Stäuber et al., 2005,
Ercolano et al., 2008, Owen et al., 2011, Meijerink et al., 2012, Aresu et
al., 2012), galactic centers of AGN host sources (e.g., Meijerink et al.,
2007, 2011, van der Werf et al., 2010, García-Burillo et al., 2010, Harada et
al., 2013, Mingozzi et al., 2018), and X-ray binaries (Moser et al., 2017,
Lebouteiller et al., 2017).

Several previous reviews have been published on both PDRs and XDRs (e.g.,
Hollenbach & Tielens, 1997, 1999, Sternberg, 2005, Snow & McCall, 2006,
Bolatto et al., 2013). In this review we provide the basic micro-physics
required to understand the dominant thermal, chemical, and radiative processes
in PDRs and XDRs while striving to summarize results from theoretical,
observational, and laboratory work since the previous reviews. In order to
limit the scope of such a vast field of research we will only lightly touch
upon topics that are covered in recent reviews such as the emission from
Polycyclic Aromatic Hydrocarbons (PAHs; Tielens, 2008), chemistry in diffuse
gas (Gerin et al., 2016), and protoplanetary disks (e.g., Bergin et al.,
2007).

##  2 Fundamental Processes in PDRs and XDRs

All PDR and XDR models must self-consistently solve for the gas temperature,
the radiation field, and the abundances of atomic and molecular species, as a
function of depth into the cloud. These components are coupled and thus
require an iterative scheme to solve (see Figure 1). Some simplification can
be made by assuming a plane-parallel geometry, thermal balance in which the
heating rate equals the cooling rate, chemical balance in which the formation
rate of each species is balanced by the destruction rate, and parameterized
fits to the depth dependence of photo processes. In this section we briefly
discuss the three main components of a model under these assumptions and
discuss models that lift them in later sections. Note that while we will
mostly refer to a prototypical PDR model, the same considerations apply to XDR
models, modulo their peculiar heating mechanisms that are addressed in Section
4.2.

Figure 1: Schematics of the iterative calculation required between chemical
balance, thermal balance and radiative transfer (blue boxes) in PDR models.
Numbers indicate subsection in text. Some relevant processes are indicated in
green. The main outputs of the calculation are indicated in yellow.

A common iterative scheme is to start at the outer edge of a cloud, with an
initial guess of temperature, abundances, and radiation field and solve for
abundances in chemical balance. Using these abundances, the temperature is
adjusted lower (higher) if the cooling rate is higher (lower) than the heating
rate and the chemistry is recalculated. Both the chemistry and temperature are
iterated until both chemical balance and thermal equilibrium are achieved.
This cycle is repeated at the next step into the cloud where fitted functions
are used for the dependence of radiative processes with depth.

###  2.1 Thermal Balance

The heating rates are generally of the form $n\Gamma$ erg ${\rm cm^{-3}}$
${\rm s^{-1}}$ where $n$ is the hydrogen nucleus density and $\Gamma$ is the
heating rate per hydrogen in units erg ${\rm s^{-1}}$. For example, an
important heating process in PDRs is the photoelectric effect on small grains
and PAHs which is given by

$n\Gamma=10^{-24}nG\epsilon\,\,{\rm erg}\,{\rm cm^{-3}}\,{\rm s^{-1}}\,,$ (1)

where $n$ is the hydrogen nucleus density, $G$ is a measure of the integrated
FUV radiation field, and $\epsilon$ is the photoelectric heating efficiency.
TH85 defined $G_{0}$ to be the incident field strength in units of the Habing
(1968) interstellar radiation field, while $G$ is the local (attenuated) field
strength within the cloud. The Habing field has an integrated FUV flux of
$1.6\times 10^{-3}$ ${\rm erg}\,{\rm cm^{-2}}\,{\rm s^{-1}}$. Since $\epsilon$
depends on $G$, the gas temperature $T$, and the electron abundance $n_{e}$,
the photoelectric heating rate is not fixed until $G$, $T$, and $n_{e}$ are
converged.

The cooling rate is generally of the form $n^{2}\Lambda$ erg ${\rm cm^{-3}}$
${\rm s^{-1}}$ where $\Lambda$ is the cooling rate coefficient in erg ${\rm
cm^{3}}$ ${\rm s^{-1}}$. To illustrate the cooling process, we consider
collisional excitation followed by radiative de-excitation of a two-level
system of species $i$, with a density $n_{\rm i}$, and collisions with atomic
hydrogen of density $n$. In the absence of background radiation the cooling
rate is

$n^{2}\Lambda=nx_{\rm i}\frac{g_{\rm u}/g_{\rm l}{\rm e}^{-E_{\rm
ul}/kT}A_{\rm ul}E_{\rm ul}\beta_{\rm esc}(\tau_{\rm lu})}{(g_{\rm u}/g_{\rm
l}{\rm e}^{-E_{\rm ul}/kT}+1)+A_{\rm ul}\beta_{\rm esc}(\tau_{\rm
lu})/n\gamma_{\rm ul}}\,,$ (2)

where $x_{\rm i}=n_{\rm i}/n$ is the fractional abundance of species $i$,
$g_{\rm u}$ and $g_{\rm l}$ are the statistical weights of the upper and lower
states, $E_{\rm ul}$ is the energy of the transition, $k$ is the Boltzmann
constant, $T$ is the gas temperature, $A_{\rm ul}$ is the Einstein A
coefficient, $\beta_{\rm esc}(\tau_{\rm lu})$ is the escape probability,
$\tau_{\rm lu}$ is the optical depth in the line, and $\gamma_{\rm ul}$ is the
rate coefficient for collisional de-excitation. $\gamma_{ul}$ is often fit
with a form $\gamma_{\rm ul}=CT^{\alpha}$. The derivation of equation 2 can be
found, for example, in Tielens (2005).

$\beta_{\rm esc}(\tau_{lu})$The probability that a photon escapes the cloud
given a line optical depth $\tau_{\rm lu}$ to the surface.

A useful characterization of a line transition is given by the critical
density, $n_{\rm cr}$, which is the density at which the rate of collisional
de-excitations is equal to the rate of spontaneous radiative de-excitations,
$n_{\rm cr}=A_{\rm ul}/\gamma_{\rm ul}$333For a multi-level system, $n_{\rm
cr}$ includes the sum of all radiative and collisional rates out of level u to
lower levels.. Through $\gamma_{\rm ul}$, $n_{\rm cr}$ has a weak dependence
on temperature. For densities, $n\ll n_{\rm cr}$ the cooling rate per volume
is

$n^{2}\Lambda=n^{2}x_{i}\gamma_{\rm ul}g_{\rm u}/g_{\rm l}{\rm e}^{-E_{\rm
ul}/kT}E_{\rm ul}\,,$ (3)

which is proportional to $n^{2}$ and each collisional excitation results in a
photon. For densities $n\gg n_{\rm cr}$ the levels are thermalized (they
depend only on gas temperature and are populated according to the Boltzmann
distribution) and the cooling rate is proportional to $n$ (see equation 2).
Note that even though the rate of collisional de-excitation exceeds the rate
of radiative de-excitation, the upper level is still radiatively de-excited at
the Einstein $A$ rate.

[] $n_{\rm cr}$Critical density at which the rate of collisional de-
excitations is equal to the rate of spontaneous radiative de-excitations..

The emergent line intensity is given by the integral of the cooling rate from
the cloud surface into the cloud,

$I=\frac{1}{2\pi}\int_{0}^{z}dz\,n^{2}\Lambda\,,$ (4)

where the factor $1/2\pi$ is appropriate for a semi-infinite slab where
photons escape through only the front surface, and $n^{2}\Lambda$ is given by
equation 2 and accounts for optical depth effects in the line. The line
optical depth, $\tau_{\rm lu}$, and hence $\beta_{\rm esc}$ depends on the
densities of the species in the upper and lower levels and the Doppler line
width, $\delta v_{\rm D}$, which includes both thermal and turbulent
broadening444The Doppler width is related to the full width at half maximum
width by $\delta v_{\rm FWHM}=1.665\delta v_{\rm D}$. (see Section 3.4). The
effects of background radiation, multiple collision partners, and multilevel
systems can be included in the excitation, cooling, and line emission
following Tielens (2005) and Draine (2011). Thermal balance is achieved when
$n^{2}\Lambda=n\Gamma$. Heating rates for PDRs are proportional to
$G_{0}n\epsilon$ or $\propto n^{1.7}G_{0}^{0.3}$ for high charge parameter
(see equation 8) ($\propto nH_{\rm X}$ for XDRs; see Section 4.2) and cooling
rates are proportional to $n^{2}$, and thus the thermal structure is a
function of $(G_{0}/n)^{0.3}$ (or $H_{\rm X}/n$ for XDRs, see Section 4). The
$[\mbox{\rm C\,{\small\footnotesize II}}]$ (${}^{2}P_{3/2}-\,^{2}\\!P_{1/2}$)
158 $\mu$m line is an example of a two-level, fine-structure transition, with
$E_{\rm ul}/k=92$ K, $A_{\rm ul}=2.4\times 10^{-6}$ ${\rm s^{-1}}$, and at
$T=100$ K, critical densities of 9, 3000, and 6100 ${\rm cm^{-3}}$ for
collisions with $e^{-}$, H, and ${\rm H_{2}}$ respectively. The statistical
weights are given by $2J+1$ so that $g_{\rm u}=4$ and $g_{\rm l}=2$. Goldsmith
et al. (2012) give analytic solutions for the excitation of ${\rm C^{+}}$ and
the $[\mbox{\rm C\,{\small\footnotesize II}}]$ line intensity in various
limits of the critical density, optical depth, and background radiation
fields. A similar analysis for the $[\mbox{\rm O\,{\small\footnotesize I}}]$
lines is given in Goldsmith (2019).

###  2.2 Radiation Transfer

The equation of transfer for the FUV continuum radiation field (or X-ray
continuum, see Section 5.2), including non-isotropic grain scattering, has
been solved using a number of methods including spherical harmonics (Flannery
et al., 1980, Le Petit et al., 2006, Goicoechea & Le Bourlot, 2007), and ray
tracing (Röllig et al., 2013, Yorke, 1980). Photo rates at each depth into the
cloud are given by $4\pi\int J(\lambda)(\lambda/hc)\sigma(\lambda)d\lambda$
where $\sigma(\lambda)$ is the cross section of the process, and $J(\lambda)$
is the local mean intensity defined as the angle average over the specific
intensity $I(\lambda)$, $J(\lambda)=1/(4\pi)\int_{\rm 4\pi}I(\lambda)d\Omega$,
where $d\Omega$ is the differential element of solid angle. Instead of
explicitly solving for $J(\lambda)$, a simplification can be made by using
pre-calculated fits to the photo rates which take into account the depth
dependence of the radiation field. These are calculated for a specific angle
of incidence and spectral energy distribution, grain type, and geometry (e.g.,
Heays et al., 2017). For a radiation field that is normally incident on a
plane parallel layer, the photo rates generally take the form

$k_{i}=\alpha G_{0}{\rm e}^{-\gamma_{\rm exp}A_{\rm V}}\,\,{\rm s^{-1}}$ (5)

where $\alpha$ is the unattenuated rate in the local interstellar radiation
field, $G_{0}$ is the incident field and and $\gamma_{\rm exp}$ gives the
(exponential) depth dependence of the dust opacity relative to the visual
extinction, $A_{\rm V}$, into the cloud. The effects of scattering are usually
included in $\gamma_{\rm exp}$ for a specific grain model. The hydrogen
nucleus column density, $N=N_{\rm H}+2N_{\rm H_{2}}$, is related to $A_{\rm
V}$ by $N=1.9\times 10^{21}A_{\rm V}$ $\rm cm^{-2}$ for the local Galaxy.
Several different radiation fields are in use besides the Habing field,
including the Draine (1978) field, $\chi$, and the Mathis et al. (1983) field,
$U$555The symbol $U$ is also commonly used for the ionization parameter in H
II regions.. The integrated field strengths are related to $G_{0}$ by
$\chi{\sim}G_{0}/1.7$ and $U{\sim}G_{0}/1.1$.666Note that $\alpha$ calculated
for a Draine field will be approximately 1.7 times higher than that for a
Habing field, but this scaling does not account for differences in the
spectral energy distribution. See also Section 5.2 for conversion between
different radiation fields.

In addition to the dust opacity, gas opacity can significantly modify the
radiation field, mainly due to absorption by H, ${\rm H_{2}}$, C, and CO, and
in protostellar disks OH and ${\rm H_{2}O}$. The gas opacity decreases the
photorates faster with depth than by dust opacity alone (van Dishoeck & Black,
1988, Visser et al., 2009).

###  2.3 Chemical Balance

In chemical balance a kinetic approach is used to find the abundances. For
each species the destruction and formation rates are calculated and the
abundances found for which the formation rates and destruction rates are
equal.

Two-body reaction rates between species $i$ and $j$ are given by
$n_{i}n_{j}k_{ij}$ ${\rm cm^{-3}}$ ${\rm s^{-1}}$ where $n_{i}$ and $n_{j}$
are the volume densities of $i$ and $j$, and $k_{ij}$ is the reaction rate
coefficient in ${\rm cm^{3}}$ ${\rm s^{-1}}$. The densities of PDRs considered
in this review are sufficiently low so that three-body reactions are not
important but they can be important at higher densities or lower grain
abundances found, for example, in protostellar disks.

Fits to $k_{ij}$ are often given as

$k_{\rm ij}=a\left(\frac{T}{300\,{\rm K}}\right)^{b}{\rm e}^{-c/T}$ (6)

where $c$ is the activation energy for the reaction to proceed in temperature
units. For example, the reaction ${\rm C^{+}+H_{2}\rightarrow CH^{+}+H}$ has a
rate coefficient $k=1.5\times 10^{-10}\exp(-4640/T)$ ${\rm cm^{3}}$ ${\rm
s^{-1}}$ (with $b=0$). The photodestruction rates and cosmic-ray destruction
rates of a species $i$ are proportional to $G_{0}n_{i}$ and $\zeta n_{i}$
respectively where $\zeta$ is the cosmic-ray ionization rate. For two-body
formation rates $(\propto n^{2})$ balanced by photo- or cosmic-ray destruction
($\propto nG_{0}$, $\propto n\zeta$) the chemical structure is a function of
$G_{0}/n$ and $\zeta/n$.

The previous subsections serve as illustrations for the basic processes in
PDRs and the interdependence of chemistry, thermal processes, and radiation
transfer. The same apply to XDRs, whose peculiar physics and chemistry are
discussed in Section 4. In the next two sections we discuss PDRs and XDRs in
more detail.

##  3 The Physics of PDRs

###  3.1 1-D Structure

Figure 2: Schematic of a PDR as a function of visual extinction, $A_{\rm V}$,
into the cloud where $A_{\rm V}$ is related to the hydrogen nucleus column
density by $N=1.9\times 10^{21}$ ${\rm cm^{-2}}A_{\rm V}$. Typically, in the
outer layers, the gas consists of neutral H, ${\rm He}$, O, and singly ionized
states of metals (e.g., ${\rm C^{+}}$, ${\rm Si^{+}}$, ${\rm Fe^{+}}$). The
electron fraction is $x_{e}{\sim}10^{-4}$ and is provided by the
photoionization of C to ${\rm C^{+}}$. The gas temperature can be
$T{\sim}100-1000$ K, heated by the photoelectric effect (P.E.) on small grains
and PAHs, and cooled by fine-structure lines of [C II] and [O I]. At $A_{\rm
V}{\sim}2$, H is converted to H2, and at $A_{\rm V}{\sim}2-4$, ${\rm C^{+}}$
recombines with electrons, and C and CO are formed. CO-dark molecular gas lies
between the H2 and CO layers where [C II] and [C I] dominate the cooling.
Atoms and molecules freeze out on grains beyond the snow line. At greater
depths the gas temperatures are $T{\sim}10-100$ K, dominated by cosmic-ray
(C.R.) and gas-grain heating, and CO rotational line cooling. The electron
fraction drops to $x_{e}{\sim}10^{-7}$. Lower $G_{0}/n$ moves the transitions
to lower $A_{\rm V}$. Figure adapted with permission from TH85, ©AAS.

We first discuss the basic structure of PDRs in terms of a one-dimensional
(1-D) layer. Figure 2 shows the 1-D structure of a classic PDR adjacent to an
H II region as a function of $A_{\rm V}$ into the cloud (the exact value of
$A_{\rm V}$ depends on $G_{0}/n$). The stellar extreme-ultraviolet (EUV;
$h\nu>13.6$ eV) photons, emitted by the OB stars, ionize the surrounding gas
and produce an H II region. The FUV radiation field that emerges from the H II
region illuminates the (hydrogen) neutral gas beyond and produces the PDR.
Molecules are photodissociated and metals with ionization potentials less than
13.6 eV are singly ionized (e.g., ${\rm C^{+}}$, ${\rm Si^{+}}$, ${\rm
S^{+}}$). Atomic hydrogen, helium, and oxygen are neutral since these have
ionization potentials greater than 13.6 eV. Here, the gas is heated mainly by
ejection of electrons from small grains and PAHs and cooled by fine-structure
line emission and attains a temperature typically $T{\sim}100-1000$ K. The gas
temperature exceeds the grain temperature since the gas cools by line
radiation while grains cool by continuum radiation. The absorption of FUV
radiation excites emission from PAHs (e.g., Allamandola et al. 1989, Peeters
et al. 2002, 2017, see also reviews by Puget & Leger 1989, Tielens 2008, 2021)
while the FUV and optical radiation heats larger grains that emit an IR
continuum. Deeper into the cloud the FUV radiation diminishes due to dust
opacity and at $A_{\rm V}{\sim}2$, H is converted to ${\rm H_{2}}$. Deeper
into the cloud ($A_{\rm V}{\sim}2-4$), ${\rm C^{+}}$ recombines with electrons
to produce atomic C, and CO can form through a series of ion-neutral reactions
(van Dishoeck & Black, 1988, Sternberg & Dalgarno, 1995). The region between
${\rm H_{2}}$ formation and CO formation is known as the CO-poor (Lada &
Blitz, 1988, van Dishoeck, 1990) or CO-dark molecular gas (Grenier et al.,
2005, Wolfire et al., 2010) and is H2 molecular gas that is not associated
with CO millimeter line emission. The H2 and CO photodissociate via line
absorption and can therefore self-shield and thus appear closer to the surface
than ${\rm H_{2}O}$ or ${\rm O_{2}}$, or other molecules which dissociate via
FUV continuum. In the deepest layers the gas attains temperatures
$T{\sim}10-30$ K and is heated mainly by cosmic-ray ionization and, in dense
regions, from gas collisions with grains heated by the IR radiation from the
surface. The gas is cooled by rotational transitions of CO. Depending on the
physical conditions, atoms and molecules may freeze out on grain surfaces at
$A_{\rm V}\gtrsim 3$, greatly affecting the gas-phase abundances and altering
the gas cooling and chemistry. Although we have presented the PDR as a static
layer sandwiched between the H II region and the molecular cloud, a complete
picture involves the evolution of the H II region, the PDR internal dynamics,
and disruption of the cloud. We will touch upon these topics in Section 6. The
PDR structure changes depending on the type and evolution of the illuminating
source. For example, while O and early B stars produce H II regions, later B
stars produce reflection nebulae with FUV radiation but little EUV radiation
and hence no H II region. Lower $G_{0}/n$, or illumination by cooler stars,
moves the transitions between H/H2 and ${\rm C^{+}}$/C/CO to lower $A_{\rm
V}$.

###  3.2 Chemistry

The basic chemical pathways (e.g., van Dishoeck & Black, 1986, Sternberg &
Dalgarno, 1995) have not changed significantly since the last reviews although
many of the rates have been updated through laboratory and theoretical work
and several processes are now found to be more significant. For example, the
dissociative recombination of ${\rm H_{3}^{+}}$ measured by McCall et al.
(2004) was determined to be higher than previous estimates and led to an order
of magnitude increase in the cosmic-ray ionization rate required to match
observations of molecular ions (Indriolo et al., 2007, Hollenbach et al.,
2012, Neufeld & Wolfire, 2017). Freeze-out of O and CO onto grains, and
reactions occurring on grain surfaces were not accounted for but are now found
to dramatically change gas phase abundances, mainly for $A_{\rm V}\gtrsim 3$
(e.g., Hollenbach et al., 2009), and are essential for the production of
complex molecules.

####  3.2.1 Photoreactions

The photo rates have been reviewed extensively by Heays et al. (2017) for
ionization and dissociation of a number of atomic and molecular species. Rates
are provided in free space for a Draine (1978) interstellar radiation field.
The depth dependence of the dust opacity, including scattering, for an
isotropically illuminated layer is fit both as $\exp(-\gamma_{\rm exp}A_{\rm
V})$, and as $E_{2}(\gamma_{\rm E2}A_{\rm V})$ where $E_{2}$ is the 2nd
exponential integral function and results from the formal solution to the
transfer equation. Isotropic radiation falls off faster than normally incident
radiation at the cloud edge. The dependence for a normally incident field can
be taken from Heays et al. to be $\propto\exp(-\gamma_{\rm E2}A_{\rm V})$.
Only the normally incident rays penetrate to large depth and the depth
dependence of both isotropic and normally incident fields become the same. A
source of FUV radiation can be produced in cloud interiors when cosmic rays
excite H2 to electronic levels that are then radiatively de-excited (Prasad &
Tarafdar, 1983). The rates for these photon interactions are also found in
Heays et al. (2017).

####  3.2.2 Photodissociation of H2

The photodissociation of H2 proceeds by line absorption of photons at
$912<\lambda<1108$ Å within the Lyman and Werner bands (Black & van Dishoeck,
1987, Sternberg & Dalgarno, 1989, Abgrall et al., 1992, Sternberg et al.,
2014). The molecule is pumped from the ground electronic state
$X^{1}\Sigma_{g}^{+}$ to an excited $B^{1}\Sigma_{u}$ or $C^{1}\Pi_{u}$
electronic state. For ${\sim}90$% of the pumps a radiative de-excitation
(emitting a UV photon) lands in a bound vibrational state of the electronic
ground state $X$, where a cascade through vibration and rotation levels ensues
producing an IR line spectrum (see e.g., Shaw et al., 2005, Roueff et al.,
2019, Zhang et al., 2021, for calculations of FUV pumped and collisional H2
line spectra). However, at high gas density, the FUV pumped $v,J$ levels are
collisionally de-excited and the energy goes into heat rather than a radiative
cascade. The remaining ${\sim}10$% of the pumps land in the vibrational
continuum thereby dissociating the H2. When the FUV absorption lines become
optically thick, the FUV pumping and dissociation rates rapidly drop with
increasing H2 column. This optical depth effect is known as H2 self-shielding
and is an essential part of H2 chemistry. Fits for H2 self-shielding as a
function of the H2 column density $f_{\rm H_{2},shield}(N_{\rm H_{2}})$ are
given in Draine & Bertoldi (1996), yielding a photodissociation rate per H2 of
$D=5.8\times 10^{-11}\chi f_{\rm H_{2},shield}\exp(-3.1A_{\rm V})$ ${\rm
s^{-1}}$ (see Sternberg et al., 2014, for a comprehensive analysis of H2
photodissociation)777Here we have written the photo rate using $\chi$, the
Draine (1978) field, since it has been calculated explicitly for that energy
distribution.. Similar to H2, CO also photodissociates in lines but the lines
are nearly 100% predissociated, i.e., absorption leads directly to
dissociation. The absorption from H2 can also shield CO from photodissociation
deeper into the cloud. Shielding functions for CO including H2 and CO self-
shielding are given in Visser et al. (2009). Atomic C is photoionized by
continuum radiation at $\lambda<1102$ Å and can also shield molecules at
larger depth (Rollins & Rawlings, 2012).

[] AdsorptionAdsorption is when an atom or molecule attaches to a grain
surface upon collision.

[] Physisorb/chemisorb Physisorb is attachment by weak long-range van der
Waals forces. Chemisorb is attachment by strong chemical bonds.

[] DesorptionDesorption is when an atom or molecule becomes detached from a
grain surface.

####  3.2.3 Formation of H2

The formation of H2 in the ISM occurs on grain surfaces (e.g., Cazaux &
Tielens 2004, 2010, see also review by Wakelam et al. 2017). In diffuse gas,
H2 is formed by the Langmuir-Hinshelwood (LH) mechanism where H atoms are
physisorbed to the surface by weak long-range van der Waals forces. The H
atoms migrate around the surface until they meet and react to form H2. A rate
coefficient of $k_{\rm H_{2}}=3\times 10^{-17}$ ${\rm cm^{3}}$ ${\rm s^{-1}}$
was fit to observations of H2 columns in diffuse gas and has been known as the
“standard” or diffuse gas value (Jura, 1974). The rate per volume is given by
$R_{\rm H_{2}}=nn_{\rm H}k_{\rm H_{2}}$ ${\rm cm^{-3}}$ ${\rm s^{-1}}$ where
$n$ is the hydrogen nucleus density and $n_{\rm H}$ is the density of atomic
hydrogen. Theoretical modeling (Hollenbach & McKee, 1979) using
observationally constrained parameters reproduce this rate and provide a
functional form for the rate coefficient

$k_{\rm H_{2}}=3\times 10^{-17}\left(\frac{T}{100\,{\rm
K}}\right)^{1/2}S(T,T_{\rm gr})\eta(T_{\rm gr})Z^{\prime}_{d}\,{\rm
cm^{3}}\,{\rm s^{-1}}\,,$ (7)

where $S(T,T_{\rm gr})$ is the sticking coefficient for H atoms striking
grains of temperature $T_{\rm gr}$, $\eta(T_{\rm gr})$ is the fraction of
atoms that react to form H2 before being thermally desorbed, and
$Z^{\prime}_{d}$ is a scale factor for the dust abundance with
$Z^{\prime}_{d}=1$ the local Galactic abundance. Usually in PDRs, $T>T_{\rm
gr}$. On grains with regular surfaces and only physisorbed sites, H will
thermally desorb before it can react with another H to form H2 at a critical
grain temperature of $T_{\rm cr}{\sim}20$ K. Surface defects or an amorphous
surface could increase the adsorption energies, and $T_{\rm cr}$ is then the
grain temperature for which a H is desorbed before finding a defect site.
Hollenbach & Salpeter (1971) find $T_{\rm cr}{\sim}65$ K for a binding energy
of ${\sim}600$ K which allows for a sufficiently high grain temperature for H2
formation in diffuse gas.888We note, however, that a more stringent constraint
comes from comparing thermal desorption times with the time for another H to
be adsorbed. For $T_{\rm cr}{\sim}50$ K a binding energy of 1500 K is
required. Whether such sites exist on an amorphous silicate surface is
unclear. For cold gas and grains, $\eta$ and $S$ are ${\sim}1$ and equation 7
with these values are often used in both diffuse and dense PDRs.

However, for the higher $T$ and $T_{\rm gr}$ found in PDRs, both $S$ and
$\eta$ decrease since H will bounce rather than stick and adsorbed atoms will
be rapidly desorbed, thus driving down the ${\rm H_{2}}$ formation rate by the
LH mechanism. But the analysis of the H2 rotational and vibrational line
emission in both moderate and low $G_{0}$ PDRs indicate the required rate
coefficient is 1-5 times higher than that in diffuse gas (Habart et al.,
2004). The higher formation rates draw the ${\rm H/H_{2}}$ transition closer
to the warm surface where collisional excitation enhances the rotational line
emission. What could maintain or even increase H2 formation? The effects of
higher velocity collisions and reduced sticking coefficient at higher $T$ may
approximately cancel (Kaufman et al., 1999) leaving a constant rate, but this
does not account for the thermal desorption that would occur for higher grain
temperatures. Another mechanism that might be at work is the Eley-Rideal (ER)
process (see e.g., Cuppen & Hornekær, 2008, Cuppen et al., 2010, Le Bourlot et
al., 2012, Wakelam et al., 2017). For ER the H atoms are chemisorbed to the
surface by strong chemical bonds that hold the H atom in place. A hot atom
from the gas phase may strike the bound atom and react to form H2. The
resulting formation rate is a function of the energy barrier required to
attach the H atom to the surface, surface coverage of chemisorbed H so that a
second H will strike it, the total surface area of grains per hydrogen
nucleus, and the gas temperature. For a graphitic grain the barrier is
measured to be ${\sim}2000$ K (Sha & Jackson, 2002) for attachment to the
surface (basal plane) and high gas temperatures are required to overcome the
barrier. After the first attached H, the barrier becomes minor for additional
H in neighboring sites and high coverage in clusters can be achieved (Hornekær
et al., 2006). On silicates there is no barrier for a H atom adsorption. Le
Bourlot et al. (2012) and Bron et al. (2014) adopt a barrier of ${\sim 300}$ K
for carbon particles suggesting that defects might lower the barrier, although
direct interaction will be limited by the surface coverage of defect sites.
With a barrier of 300 K, and sufficiently high surface coverage of defects and
grain surface area, the rate coefficient at $T{\sim}2\times 10^{3}$ K is 3-4
times the diffuse rate and provides a good fit to H2 line emission in the
Orion Bar and NGC 7023 (Bron et al., 2014, Joblin et al., 2018). Since the
diffuse rate is essentially a maximum in which every two H collisions result
in an H2, the inferred high H2 formation rate in PDRs likely requires
activation of new surface routes (see also discussion in Tielens, 2021).

Bron et al. (2014) considered the effects of temperature fluctuations in very
small grains and PAHs caused by single photon absorption. Between photon
absorptions, an H2 may form by the LH mechanism before another photon arrives
and desorbs the H. The process has a low efficiency for $G_{0}/n{\sim 1}$ but
can be important for $G_{0}<200$ and $n>10^{3}$ ${\rm cm^{-3}}$. Endothermic
reactionChemical reaction where there is a net energy input between energy
state of reactants and energy state of products Exothermic reactionChemical
reaction where there is a net energy release between energy state of reactants
and energy state of products Activation energyEnergy required for a reaction
to occur. Both endothermic and exothermic reactions may require an activation
energy.

Another possibility, that takes advantage of the increased surface area of
PAHs, are H abstraction reactions on PAHs that have adsorbed a large number of
H (a superhydrogenated PAH; Thrower et al., 2012). An incident H from the gas
phase reacts with H on the PAH to form H2. The barrier for attachment is minor
$T{\sim 100}$ K and easily achieved at PDR temperatures. However, the
additional H is easily photodesorbed (Andrews et al., 2016) and if the
abstraction cross section is as low as that measured for the PAH coronene
(Mennella et al., 2012) then this process will be not be efficient. Another
process is by way of photodesorption (Castellanos et al., 2018). An absorbed
UV photon will excite the PAH, allow H to roam, and the energy is dissipated
in a loss of an H or H2. Whether H or H2 is produced is quite sensitive to PAH
structure, and might work in only a narrow band of PAH types. For optimum
conditions, the H2 formation rate coefficient could be as high as
${\sim}2\times 10^{-17}(T/100\,{\rm K})^{1/2}$ ${\rm cm^{3}}$ ${\rm s^{-1}}$
(Tielens 2021, private communication).

Model fits to observations of the mid-infrared H2 line emission and the CO
ladder can help to constrain the H2 formation rate (e.g., Habart et al. 2004,
Sheffer et al. 2011, Stock et al. 2015, Joblin et al. 2018, Wu et al. 2018,
and future observations of H2 with JWST). As already mentioned, higher H2
formation rates draws the H2 to warmer surface layers and increases the H2
line emission but can also produce a small fraction of warm CO through
reactions with excited H2 (Sections 3.2.4 and 7.1).

Although there has been significant advances in our understanding from
theoretical and laboratory work (see review by Wakelam et al., 2017), the H2
formation process is still uncertain, especially as a function of the gas and
grain temperatures. Observations suggest $k_{\rm H_{2}}{\sim}3\times 10^{-17}$
${\rm cm^{-3}}$ ${\rm s^{-1}}$ or larger, even in regions of fairly high gas
temperature ($T{\sim}1000$ K) and warm ($T_{\rm gr}{\sim}75$ K) grains, as in
the Orion Bar PDR. Further experimental and quantum chemical studies as well
as confirmation by comparison to astronomical observations are warranted.

####  3.2.4 Reactions with ${\rm H_{2}^{*}}$

The internal energy of vibrationally or rotationally excited H2, ${\rm
H_{2}^{*}}$, can be used to increase the rate of endothermic chemical
reactions (TH85, Sternberg & Dalgarno, 1995, Agúndez et al., 2010). The
reactions ${\rm C^{+}}\,+{\rm H_{2}^{*}}\rightarrow{\rm CH^{+}}+{\rm H}$ and
${\rm S^{+}}\,+{\rm H_{2}^{*}}\rightarrow{\rm SH^{+}}+{\rm H}$ are especially
important for the formation of ${\rm CH^{+}}$ and ${\rm SH^{+}}$. The
activation energies of $\Delta E/k=4640$ K and $\Delta E/k=9860$ K
respectively are too high to be achieved by gas temperatures in much of the
PDR. However, these endothermic reactions can be driven near the surface of
PDRs where the ${\rm H_{2}}$ is FUV pumped or where gas temperatures are
sufficiently high to drive the reactions (with rates $\propto\exp[-\Delta
E/kT]$). Such enhanced rates are required to match observations of ${\rm
SH^{+}}$ and ${\rm CH^{+}}$ (Nagy et al., 2013, Zanchet et al., 2019,
Goicoechea et al., 2019), and can lead to the production of ${\rm HCO^{+}}$
and CO.

$\zeta_{\rm p}$ Primary cosmic-ray ionization rate per hydrogen atom with
units ${\rm s^{-1}}$.

####  3.2.5 Cosmic-ray reactions

Since there are no stellar EUV photons beyond the H II region boundary, H and
He can not be ionized by the radiation field but can be ionized by cosmic rays
that penetrate the PDR.999Wolfire et al. (2003, 2010) included a thermal soft
X-ray component produced by local Galactic white dwarfs and super novae
remnants to partially ionize the diffuse WNM and molecular cloud surfaces but
these photons do not penetrate more than $N{\sim}10^{19}$ ${\rm cm^{-2}}$.
Although they are important for low column density diffuse gas, they are not
important for dense PDRs. It is this ionization by cosmic rays that drives
much of the ion-neutral chemistry in PDRs (an exception are reactions with
${{\rm H}_{2}^{*}}$). Typical ion-neutral reaction rate coefficients are
${\sim}10^{-9}$ ${\rm cm^{-3}}$ ${\rm s^{-1}}$ with no temperature dependence
nor activation barrier, and thus ion-neutral chemistry proceeds rapidly
compared to neutral-neutral reactions which have typical rate coefficients of
${\sim}10^{-11}$ ${\rm cm^{-3}}$ ${\rm s^{-1}}$. Typical ionization rates in
diffuse gas are inferred from observations of ${\rm ArH^{+}}$, ${\rm OH^{+}}$,
${\rm H_{2}O^{+}}$, and ${\rm H_{3}^{+}}$ to be $\zeta_{\rm p}\sim 2\times
10^{-16}$ ${\rm s^{-1}}$ per H atom (Hollenbach et al. 2012, Indriolo et al.
2015, Neufeld & Wolfire 2017, see also Shaw & Ferland 2021 for the effects of
variable PAH abundance). Each ionization of H, H2 or He, by a cosmic-ray
electron of energy $\lesssim 10$ MeV, produces a secondary electron of energy
${\sim}35$ eV which can further ionize another H or H2 (the fast cosmic ray
being considered as the primary particle). The total rate including secondary
ionizations in H and H2 gas is $\zeta_{\rm H}{\sim}1.5\zeta_{\rm p}$ and
$\zeta_{\rm H_{2}}{\sim}2.3\zeta_{\rm p}$ respectively, and depends in detail
on the molecular and electron fractions (Cravens & Dalgarno, 1978, Glassgold &
Langer, 1974, Dalgarno et al., 1999, Glassgold et al., 2012). Higher electron
fractions result in more energy going into heat rather than secondary
ionizations. The dependence of the ionization rate with column density into
the PDR is not well known and depends on the incident cosmic-ray spectrum and
interaction with turbulence and the magnetic field in the cloud (Padovani et
al., 2009, Silsbee & Ivlev, 2019). The low energy cosmic rays are absorbed at
increasing column density and decrease the total ionization rate compared to
the rate at the surface. Observations indicate $\zeta_{\rm p}\sim 1.1\times
10^{-17}$ ${\rm s^{-1}}$ in protostellar envelopes (van der Tak & van
Dishoeck, 2000) and marginal evidence for a $1/N$ dependence in clouds
(Padovani et al., 2009, Neufeld & Wolfire, 2017). Higher ionization rates
initially increase the abundance of molecular ions (Hollenbach et al., 2012,
Le Petit et al., 2016), however, for $\zeta_{\rm p}/n_{2}\gtrsim 10^{-14}$
${\rm s^{-1}}$ with $n_{2}=n/(100\,{\rm cm^{-3}})$, the abundance of molecular
ions decreases due to the destruction of H2 by cosmic rays and due to an
increasing electron abundance from ionizations leading to faster ${\rm
H_{3}}^{+}$ dissociative recombination (e.g., Le Petit et al., 2016).
Increased ionization rates destroy CO by reactions with ${\rm He^{+}}$,
requiring higher densities for formation and pushing CO deeper into the cloud
(e.g., Gong et al., 2017). Similarly, internal sources of cosmic rays from
e.g., protostars can change the chemical structure by warming the gas in the
interior and increasing the ${\rm He^{+}}$ density which destroys CO and
increases C (Gaches et al., 2019).

####  3.2.6 Grain assisted recombination

The effects of an increased cosmic-ray ionization rate are partly mitigated by
grain assisted recombination where mainly light ions (${\rm H^{+}}$, ${\rm
He^{+}}$ and ${\rm C^{+}}$) recombine with electrons on ${\rm PAH^{-}}$ or
charge exchange with ${\rm PAH^{0}}$. Since the ${\rm PAH^{-}}$ and ${\rm
PAH^{0}}$ species are the result of electron attachment to ${\rm PAH^{0}}$ and
${\rm PAH^{+}}$, the result is to reduce the abundance of free electrons and
increase the abundance of neutral species. If not for this additional
recombination, the production of CO would be inhibited for rates as low as
$\zeta_{\rm p}=10^{-17}/n_{2}$ ${\rm s^{-1}}$ and most of the carbon would
remain as C. The effects of grain assisted recombination have long been known
from absorption line studies in diffuse clouds (e.g., Lepp et al., 1988,
Weingartner & Draine, 2001a). Many modelers adopt the Draine & Sutin (1987)
formalism to obtain the reaction rates as a function of the number of carbon
atoms in a PAH. The rate must be either integrated over the distribution in
PAH size and abundance or multiplied by a typical size and abundance. Wolfire
et al. (2008, 2003) calibrated the theoretical rates to match the observed
${\rm C^{+}/C}$ ratio in diffuse gas and suggested a correction factor
$\Phi_{\rm PAH}=0.5$ is needed, although Liszt (2011) suggests rates should be
recalibrated with the generally lower C columns from Burgh et al. (2010). Fits
for the Draine & Sutin (1987) rates near $T=300$ K can be found in Hollenbach
et al. (2012). We emphasize that the rates and required integrals are quite
uncertain and urge that further work is needed.

The photoionization of metals, in particular S, produces free electrons and
affects the ion-neutral chemistry at intermediate depths $A_{\rm V}{\sim}1-5$.
Similar to the effects of cosmic rays, a high electron abundance from metal
ions can suppress the ion-neutral chemistry. The effects are mitigated by
recombination on PAHs or freeze-out of metals (e.g., Hollenbach et al., 2012).

####  3.2.7 Grain surface reactions

Atoms and molecules collide with grains, and if the grains are sufficiently
cold, they will be adsorbed on grain surfaces thereby depleting the species
from the gas phase and freeze out as ice on grains (e.g., Hollenbach et al.,
2009, Esplugues et al., 2016, 2017, Tielens, 2021). Freeze out affects the
gas-phase chemistry and cooling, and induces grain-surface chemistry which can
produce simple molecules (e.g., ${\rm H_{2}O}$) and complex organic molecules
(COMs; e.g., methanol ${\rm CH_{3}OH}$; Garrod et al., 2008), which can then
be desorbed from the grain back into the gas. Reviews of surface chemistry in
PDRs and dark cores can be found in Cuppen et al. (2017) and in pre- and
protostellar environments in Öberg & Bergin (2021).

Similar to H, adsorbed species may diffuse across the surface, although
heavier atoms or molecules diffuse slower, and undergo chemical reactions when
meeting (the LH mechanism) or may be hit by a gas-phase species and chemically
react (the ER mechanism). The atoms and molecules on the grain surface can be
desorbed if they can overcome their binding energy. Potential desorption
processes important near the cloud surface include photodesorption (e.g.,
Öberg et al., 2009), thermal desorption (Tielens & Allamandola, 1987), and
thermal fluctuations (Bron et al., 2014), while deeper into the cloud, cosmic-
ray desorption (e.g., Hasegawa & Herbst, 1993), cosmic-ray induced FUV
photodesorption, and chemical desorption (Dulieu et al., 2013, Garrod et al.,
2007) dominate. Photon and cosmic-ray desorptions are individual photon or
cosmic-ray events, thermal desorption depends on the equilibrium (large) grain
temperature, thermal fluctuations are the temperature spikes of small grains
and PAHs after absorbing a photon, and chemical desorption relies on
exothermic chemical reactions on the grain surface. The cosmic-ray rates, and
chemical desorption in particular, are not well constrained. In addition,
thermal desorption and chemical reaction rates are sensitive to variations in
grain temperature (Esplugues et al., 2019). Additional photoprocessing can
occur on the grain surface and species can be photodissociated during
photodesorption. The formation of molecules is then a competition between the
rates of adsorption and surface reactions versus the desorption rates. See
e.g., Hollenbach et al. (2009), van Dishoeck et al. (2021), and Tielens (2021)
for ${\rm H_{2}O}$ ice formation.

The ice is observed to be mainly composed of CO, ${\rm H_{2}O}$, ${\rm
CO_{2}}$, ${\rm NH_{3}}$ (ammonia), ${\rm H_{2}CO}$ (formaldehyde), ${\rm
CH_{4}}$ (methane), and ${\rm CH_{3}OH}$ (methanol) (e.g., Gibb et al., 2004)
but can vary considerably depending on grain temperature, radiation field
strength, gas density, and depth into the cloud. For example, ${\rm H_{2}O}$
ice can be thermally desorbed at grain temperatures $T_{\rm gr}\gtrsim 100$ K
while CO is desorbed at grain temperatures $T_{\rm gr}\gtrsim 20$ K. The depth
for water ice formation is proportional to $\ln(G_{0}/n)$ and is typically
$A_{\rm V}$ ${\sim}1.6$ for moderate density ($n{\sim}10^{3}\,{\rm cm^{-3}}$)
and low FUV field ($G_{0}\sim 5$) sources (see review and papers therein by
Boogert et al., 2015).101010In previous PDR reviews the ${\rm O/O_{2}}$
transition was depicted at $A_{\rm V}\gtrsim 10$. However in Hollenbach et al.
(2009) it was shown that the abundance of ${\rm O_{2}}$ peaks at $A_{\rm
V}$$\sim 5$. At lower $A_{\rm V}$ the photodesorption of OH and ${\rm H_{2}O}$
followed by the gas-phase reaction ${\rm OH\,+O\rightarrow O_{2}+H}$ increases
${\rm O_{2}}$ while at larger $A_{\rm V}$ freeze-out of O in ${\rm H_{2}O}$
ice diminishes all O bearing gas-phase species. However, grain surface
chemistry can produce minimum ${\rm H_{2}O}$, OH, and NH gas-phase abundances
even in diffuse gas (Crutcher & Watson, 1976, Sonnentrucker et al., 2015)
although the fraction in ice is insignificant. The time scale for several
processes can become longer than cloud lifetimes and these require a time
dependent approach (see Section 6.1).

PDR chemical networks have been extended and updated to include e.g., N
chemistry (e.g., Young Owl et al., 2002, Boger & Sternberg, 2005, Li et al.,
2013, Persson et al., 2014), S chemistry (Sternberg & Dalgarno, 1995,
Goicoechea et al., 2021), and H, C, and N isotopes (Le Petit et al., 2002,
Heays et al., 2014, Szűcs et al., 2014, Roueff et al., 2015, Röllig &
Ossenkopf, 2013, Visser et al., 2018). Photo rates and cross sections can be
found on-line at https://home.strw.leidenuniv.nl/~ewine/photo/ and
dielectronic and radiative recombination rates at
http://amdpp.phys.strath.ac.uk/tamoc. Data bases for gas-phase reactions are
KIDA (Wakelam et al., 2012) and UMIST (McElroy et al., 2013) although the
latter is becoming somewhat dated.

###  3.3 Gas Heating

Several processes may contribute to the gas heating depending on the incident
FUV field strength, cosmic-ray ionization rate, gas density, and depth into
the cloud. At moderate depths ($A_{V}\lesssim 5$), the dominant gas heating
process is from photoelectric ejection of electrons from small grains and PAHs
(Bakes & Tielens, 1994, Weingartner & Draine, 2001c). The fraction of FUV
photon energy that goes into gas heating is the heating efficiency,
$\epsilon$, and is a function of the grain charge. The charge parameter
$\gamma=GT^{1/2}/n_{e}$ ${\rm K^{1/2}}$ ${\rm cm^{3}}$, is proportional to the
ratio of the ionization rate of grains to the electron recombination rate
where $G$ is the local (attenuated) field strength, $T$ is the gas
temperature, and $n_{e}$ is the electron density. For $\gamma\lesssim 10^{2}$
${\rm K^{1/2}}$ ${\rm cm^{3}}$ grains are neutral and $\epsilon$ is at a
maximum of about ${\sim}5$%, while for higher $\gamma$, grains become charged
and $\epsilon$ drops due to the electron kinetic energy loss in escaping the
Coulomb potential and because fewer photons can ionize a more highly charged
species. Since the ionization to recombination rate is proportional to the
grain size, small grains and PAHs have on average lower charge than larger
species. In addition, the larger grains contribute less because the electron
escape length is smaller than the photon absorption depth; i.e, the yield goes
down. Thus, small grains and PAHs dominate the heating with half coming from
grains smaller than ${\sim}15$ Å. Note that the electrons contributed from
PAHs are generally an insignificant fraction of the free electron abundance
since the PAH fractional abundance $(n_{\rm PAH}/n\sim 10^{-7})$ is much less
than that of ${\rm C^{+}}$ $(n_{\rm C^{+}}/n\sim 10^{-4})$.

The heating rate per unit volume is given by $n\Gamma_{\rm
PE}=10^{-24}Gn\epsilon$ erg $\rm cm^{-3}$ ${\rm s^{-1}}$ with $\epsilon$ from
Bakes & Tielens (1994) (hereafter BT94), given by

$\epsilon=\frac{4.87\times 10^{-2}}{1.0+4\times
10^{-3}\gamma^{0.73}}+\frac{3.65\times 10^{-2}(T/10^{4})^{0.7}}{1+2\times
10^{-2}\gamma}\,.$ (8)

The second term is only important at high temperatures where a higher
recombination rate of electrons with ionized small grains and PAHs results in
lower positive charge and higher $\epsilon$. In addition, for gas temperatures
$T\gtrsim 3\times 10^{3}$ K and $\gamma\gtrsim 10^{4}$, recombination may
become a significant gas cooling process. The photoelectric heating rate is
often reported as the net heating minus the cooling. Wolfire et al. (2008)
introduced a parameter $\Phi_{\rm PAH}{\sim}0.5$, that modifies the charge
parameter as $\gamma^{\prime}=\gamma/\Phi_{\rm PAH}$ to account for a reduced
recombination rate on grains, needed to explain the observed column densities
of C in the diffuse ISM. The theoretical efficiency produces good agreement
with gas temperatures, thermal pressures, and [C II] cooling rates in diffuse
gas (Wolfire et al., 1995a, 2003, Jenkins & Tripp, 2011, Gerin et al., 2015)
where $T{\sim}100$ K, $n_{e}{\sim}0.01$, $G_{0}{\sim}1.7$, and
$\gamma{\sim}1.7\times 10^{3}$ ${\rm K^{1/2}}$ ${\rm cm^{3}}$. The agreement
with observed fine-structure line intensities and line ratios in dense PDRs is
usually quite good (e.g., Stacey et al., 1991, Hollenbach & Tielens, 1999).
The analytic fit for $\epsilon$ given by Weingartner & Draine (2001c)
(hereafter WD01) for a B0 star radiation field is a factor of ${\sim}1.7$
lower than BT94 at $\gamma{\sim}10^{5}$ ${\rm K^{1/2}}$ ${\rm cm^{3}}$ and
${\sim}4.1$ lower at extreme values of $\gamma{\sim}10^{6}$ ${\rm K^{1/2}}$
${\rm cm^{3}}$. At low values of $\gamma{\sim}10^{2}$ ${\rm K^{1/2}}$ ${\rm
cm^{3}}$, WD01 is a factor 2.6 higher than BT94. The $\epsilon$ given by the
WD01 fit is nearly the same as that derived from the Meudon PDR code (Section
5.1) at the cloud surface for an incident Draine (1978) field. Although the
shape of various heating efficiencies that are in use are quite similar, there
are differences by factors of a few especially for high $\gamma$. Observations
such as those discussed next might help to constrain the efficiency.

If [C II] is the dominant coolant and the integrated far-infrared (FIR) dust
continuum is a measure of the incident radiative energy, then the ratio of [C
II] intensity to FIR intensity ([C II]/FIR) provides an observational check of
the theoretical $\epsilon$. However, observations of several sources,
including dense PDRs and diffuse gas, of [C II]/FIR (Figure 3) present a
puzzling result (Salas et al., 2019, Pabst et al., 2022)111111Here FIR is
defined as the integrated 40 $\mu$m-500 $\mu$m continuum. Another measure in
use is the total infrared (TIR) 3 $\mu$m-1100 $\mu$m continuum (Dale & Helou,
2002). ${\rm TIR}{\approx}2\times{\rm FIR}$ depending on grain temperature..
When compared to the BT94 efficiency, observational points fall below the
theoretical curve. This may be the result of optical photons contributing to
grain heating that do not heat the gas, a substantial contribution from [O I]
cooling, or lower heating rates in dense PDRs where the PAH distribution is
uncertain.

Figure 3: Observations of the [C II] intensity to infrared dust continuum
intensity, [C II]/FIR, versus charge parameter $\gamma=G_{0}T^{1/2}/n_{e}$
$({\rm K^{1/2}\,cm^{3}})$. The curve overlaid on the observations is the
heating efficiency $\epsilon$ from Bakes & Tielens (1994). Points show a mix
of dense PDRs and diffuse gas lines-of-sight. OMC1 includes [O I]. See Salas
et al. (2019) and Pabst et al. (2021, 2022) for source descriptions. Figure
adapted with permission from Pabst et al. (2022) ©ESO.

The photoelectric heating diminishes with depth into the cloud mainly due to
dust opacity which reduces the integrated FUV radiation field. Reddening of
the spectral energy distribution also reduces $\epsilon$ by ${\sim}25$% at
$A_{\rm V}=1$ and $\gamma=10^{4}$. For a normally incident B0 star spectrum
and $R_{V}=3.1$ grains121212The total-to-selective extinction $R_{\rm
V}=A_{\rm V}/(E_{\rm B-V})$ has a typical value of 3.1 in the diffuse ISM
while molecular clouds can have larger $R_{\rm V}{\sim}5.5$., the FUV field
falls approximately as $\propto\exp(-1.8A_{V})$. A detailed treatment of the
FUV radiation transfer (Le Petit et al., 2006, Röllig et al., 2013) and of the
grain charge on a distribution of grains allows for different spectral energy
distributions, angle of incidence, and different grain size distributions,
than those used for the analytic expressions (van Hoof et al., 2004). A cooler
incident spectrum (Spaans et al., 1994), shifts the grain charge to more
neutral grains but ultimately produces less efficient heating since there are
fewer high energy photons. Spaans et al. (1994) provide a simple formula that
“corrects” equation 8 for the cooler radiation field and also quantitatively
assesses the effect on photodissociation and photoionization. Increasing
$R_{\rm V}$ shifts the grain size distribution to larger grains, which have
lower heating efficiency due to grain charging, and lower yield, and allows
for increased penetration of FUV radiation into the cloud (Weingartner &
Draine, 2001c, Abel et al., 2008).

Additional sources of heat come from H2 formation, photodissociation, and
collisional de-excitation of excited rovibrational levels (Sternberg &
Dalgarno, 1989). The formation of ${\rm H_{2}}$ on grains releases $4.48$ eV
of binding energy. The distribution of the energy is uncertain and depends on
the formation process. For the the Eley-Rideal (ER) formation mechanism (see
Section 3.2) ${\sim}0.6$ eV goes into kinetic energy of the molecule which can
heat the gas, 2.7 eV goes into internal rotational and vibrational excitation,
and most of the remainder goes into grain heating (Sizun et al., 2010). For
the Langmuir-Hinshelwood (LH) mechanism equipartition is generally assumed
with ${\sim}1.5$ eV going to the grain, kinetic energy, and excitation. These
“formation pumped” excited $(v,J)$ levels can be collisionally de-excited at
densities above the critical density $n\gtrsim n_{\rm cr}\approx 1.1\times
10^{5}/\sqrt{T_{3}}$ ${\rm cm^{-3}}$ where $T_{3}=T/(1000\,{\rm K})$ (Röllig
et al., 2006), and contribute to gas heating. The total formation heating is
the sum of the translational heating and de-excitation of excited levels. The
photodissociated H2 attains ${\sim}0.5$ eV in kinetic energy that also adds to
gas heating (Stephens & Dalgarno, 1973, Abgrall et al., 2000).

The ${\rm H_{2}}$ rovibrational levels can also be excited by FUV pumping, and
contribute to gas heating, called vibrational heating, through collisional de-
excitation of the excited levels (Sternberg & Dalgarno 1989, Burton et al.
1990 see also Röllig et al. 2006 for an analytic fit to the Sternberg &
Dalgarno 1995 ${\rm H_{2}}$ model). At the cloud edge, the formation and
vibration heating can dominate that of photoelectric heating (Burton et al.,
1990, Le Bourlot et al., 2012, Röllig et al., 2013). There, the gas
temperature and the FUV field are high and the destruction and formation cycle
is rapid thereby increasing the formation heating. In addition, rates of
formation go as $R_{\rm H_{2}}\propto nn_{\rm H}$ which is always higher at
the surface. When the density is greater than the critical density for de-
excitation, since there are 9 pumps for every formation, the vibrational
heating by FUV pumping tends to dominate. At lower densities, the formation
(translational) heating can be important.

Cosmic-ray ionization of H, ${\rm H_{2}}$, and He creates secondary electrons
that heat the gas by similar processes discussed for X-ray photoelectrons
(Section 4.2). Typical heating rates are $Q{\sim}10$ eV per ion pair in
diffuse molecular gas and $Q{\sim}13$ eV per ion pair in molecular clouds with
the heating rate per unit volume given by $n\Gamma=\zeta_{\rm p}Qn$ (Glassgold
et al., 2012). For cosmic-ray ionization rates typical of those found in the
Galactic disk, cosmic-ray heating is less important than photoelectric heating
at the cloud surface even for radiation fields as low as the interstellar
field, but can be important in cloud interiors that are shielded from FUV
radiation. For higher cosmic-ray ionization rates as found in the central
molecular zone near the Galactic center (${\sim}1-10\times 10^{-14}$ ${\rm
s^{-1}}$; Le Petit et al., 2016) or supernovae remnants (${\sim}10^{-10}$
${\rm s^{-1}}$; Priestley et al., 2017), the cosmic-ray heating can dominate
that of photoelectric heating even close to the cloud surface (Bayet et al.,
2011).

Gas collisions with grains can be either a heating or cooling process
depending on if the gas temperature is lower or higher than the grain
temperature. Grain temperatures are determined from radiative equilibrium
using a full radiation transfer treatment with a distribution of grain sizes
(e.g., Röllig et al., 2013), or with a simple fitted formula (Hollenbach et
al., 1991, Hocuk et al., 2017). The gas is usually warmer than grains on PDR
surfaces to $A_{\rm V}{\sim}3$ but can be cooler than the IR heated grains at
greater depths. Collisional de-excitation of fine-structure levels that are
pumped by the infrared continuum radiation can also lead to gas heating. This
is most likely to occur for the [O I] 63 $\mu$m transition which can become
optically thick to the continuum radiation and is at a wavelength where the
dust continuum is strong.

###  3.4 Gas Cooling

In the atomic gas, cooling proceeds mainly through collisional excitation and
radiative de-excitation of atomic fine-structure levels. The dominant coolants
are [C II] 158 $\mu$m and [O I] 63 $\mu$m, and 145 $\mu$m line emission. As
${\rm C^{+}}$ recombines to C the fine-structure lines of [C I] 370, 609
$\mu$m become important. In molecular gas rotational transitions of CO
dominate the cooling (e.g., Neufeld & Kaufman, 1993). For high density and
high FUV field PDRs, [Si II] 35 $\mu$m and ro-vibrational transitions of ${\rm
H_{2}}$ can contribute to gas cooling (Abel et al., 2005, Kaufman et al.,
2006). The calculation for the level populations are carried out in
statistical equilibrium using non-LTE rate equations. Radiative pumping, self-
absorption, and line transfer is generally handled using an escape probability
formalism. The escape probability depends on $\tau_{\rm lu}$, the cloud
geometry, and whether a microturbulent or large velocity gradient (LVG)
approach is used. The line optical depth integrated from the cloud surface
$z=0$ to depth $z$ is given by:

$\tau_{\rm lu}=\int_{0}^{z}dz\frac{A_{\rm ul}c^{3}}{8\pi\nu_{\rm
ul}^{3}}n_{\rm u}\left[\frac{n_{\rm l}g_{\rm u}}{n_{\rm u}g_{\rm
l}}-1\right]\frac{1}{\delta v_{\rm D}}\,,$ (9)

where $n_{\rm u}$ and $n_{\rm l}$ are the densities of the species in the
upper and lower levels, and $\delta v_{\rm D}$ is the Doppler line width which
includes both thermal and turbulent broadening.

In a microturbulent model the velocity gradient is small compared to $\delta
v_{\rm D}$ and the full column of the species along a line of sight
contributes to the optical depth. In a large velocity gradient (LVG) model
(e.g., RADEX, van der Tak et al. 2007) the velocity gradient, $dv/dz$, is
sufficiently large so that the integration is limited to the velocity coherent
length $\delta v_{\rm D}/(dv/dz)$. Most PDR models adopt a microturbulent
approach because the physical size of the emitting region is narrow and the
velocity gradient is small compared to the line width. The $\beta_{\rm esc}$
for a microturbulent, semi-infinite, plane-parallel layer can be found in
TH85. See also Draine (2011) and Tielens (2021) for a derivation of escape
probability formalism used in line radiation transfer. At the cloud surface
$\beta_{\rm esc}=0.5$ and decreases as $1/\tau_{\rm lu}$ at large $\tau_{\rm
lu}$. In spherical geometry, the escape probability in microturbulent gas is
given in Stoerzer et al. (1996) and in the LVG limit in Goldsmith et al.
(2012).

It is important to use up-to-date collisional excitation rates in order to
find the gas cooling and emitted line intensities. The LAMBDA (van der Tak et
al., 2020) and BASECOL (Ba et al., 2020) databases are on-going efforts to
provide rates on-line in a standard format. The dominant collision partners
are usually atomic hydrogen near the PDR surface and molecular hydrogen at
larger depth. Electron impact excitation may also be important for molecular
ions (Hamilton et al., 2018). At intermediate depth both atomic and molecular
hydrogen can contribute to the collisional excitation rates.

Several excitation rates have been calculated separately for collisions with
Ortho and Para ${\rm H_{2}}$. Collision rates involving Ortho ${\rm H_{2}}$
are generally faster than those for Para ${\rm H_{2}}$ since the quadrupole
interaction averages to zero for Para, although the difference is much greater
for rotational transitions than for fine-structure transitions. Ortho to Para
conversion of H2 should be accounted for in the chemical network (e.g.,
Sternberg & Neufeld, 1999, Bron et al., 2016). Wiesenfeld & Goldsmith (2014)
note that the collisional rates for H2 with ${\rm C^{+}}$ are quite close to
those for H with ${\rm C^{+}}$, contrary to the factor of 2 difference
expected previously. Impacts of electrons with cations (e.g., ${\rm C^{+}}$)
may also be important for sufficiently high electron fraction. For
$T{\sim}300$ K, excitation of ${\rm C^{+}}$ by $e^{{\rm-}}$ and ${\rm H}$ are
comparable at $x_{e}=\gamma^{\rm H}/\gamma^{e}{\sim}5\times 10^{-3}$. See
Lique et al. (2018) for a recent derivation of the de-excitation coefficients
of O I by $\rm H_{2}$, H, and He, but contact the authors for a corrected data
set and extended to 8000 K.

##  4 The Physics of XDRs

###  4.1 1-D Structure

The 1-D structure of X-ray dominated regions (XDRs) differs from that of PDRs
in two important ways: XDRs are characterized by much larger column densities
($N\approx 10^{22}\rm cm^{-2}$) of warm gas ($T\approx 100-500$ K), and they
have a peculiar enhanced abundance of molecular ions. The physical mechanisms
causing the different 1-D structure of XDRs vs PDRs are linked to the deep
penetration of X-rays and to the production of photoelectrons following
primary ionizations. By analogy with PDRs, where the thermal and chemical
conditions are determined by $G_{0}/n$, the XDR structure can be parameterized
in terms of $H_{\rm X}/n$, namely the ratio between the energy deposition rate
per particle:

$H_{\rm X}=\int_{E_{\rm min}}^{E_{\rm max}}\sigma_{\rm pe}(E)F_{\rm X}(E)dE,$
(10)

and the gas density $n$. This follows from equating the heating and molecular
destruction rates induced by X-ray photons (which are proportional to $nH_{\rm
X}$), to the cooling and molecular formation rates (which are proportional to
$n^{2}$). At equilibrium, the thermal and chemical conditions of XDRs are thus
governed by $H_{\rm X}/n$. In equation 10, $F_{\rm X}(E)$ is the local photon
energy flux per unit energy interval, that is generally assumed to be a power-
law, $F_{\rm X}(E)=F_{0}(E/{\rm 1\,keV})^{-\alpha}$ (see Section 5.2 for a
discussion regarding this and other functional forms), and $\sigma_{\rm
pe}(E)$ is the photoelectric absorption cross section per hydrogen nucleus
(Morrison & McCammon, 1983, Wilms et al., 2000). Given that $\sigma_{\rm
pe}(E)$ goes roughly as $E^{-3}$ (Maloney et al., 1996), the lowest energy
photons are attenuated more than higher energy photons with increasing column
density. The gas is optically thick to X-rays with $E\leq E_{0}$ where the
energy $E_{0}$ for which $\tau(E)=1$ depends on the column density. For
molecular clouds with $N\approx 10^{22}\rm\,cm^{-2}$ nearly all photons with
$E\lesssim 1$ keV are absorbed before the cloud center is reached. Considering
a flux $F_{\rm X}$ from $E_{\rm min}=1\,\rm keV$ to $E_{\rm max}=100\,\rm keV$
the energy deposition rate can be written as $H_{\rm X}=4.8\times
10^{-24}F_{X}/(N/10^{22}{\rm cm^{-2}})^{0.9}$ (see Maloney et al., 1996, for
the complete derivation). Note its slow decline with column density as opposed
to the FUV induced photoelectric heating in PDRs (see Section 3.3) that is
instead confined in a much thinner layer by the exponential attenuation of the
FUV flux due to dust. In XDRs the X-ray energy that is converted to heat and
then re-radiated as line cooling – i.e., the heating efficiency – is larger
(up to $\epsilon\approx 1$, Maloney et al., 1996) than in PDRs
($\epsilon\approx 10^{-2}-10^{-3}$, Hollenbach & Tielens, 1999, and discussion
in Section 3.3). The chemistry is also affected up to high column densities
(see Section 4.4) by ion-molecule reactions initiated by X-rays and this
results in different emergent line emission and ratios that can be then used
to distinguish XDRs from PDRs (see Section 5.3.2).

Figure 4: Schematic 1-D structure of an X-ray dominated region (XDR) as a
function of the column density $N$. The ratio of the local X-ray energy
deposition rate per particle to the total density, $H_{\rm X}/n$, decreases
with increasing column density (see Section 4). We assume no incident FUV
photons, a 1-100 keV flux $F_{X}\approx 100\rm\,erg\,s^{-1}\,cm^{-2}$, and gas
density $n=10^{5}\rm\,cm^{-3}$. The schematic highlights the major
heating/cooling processes and the approximate temperature, electron fraction,
and chemical composition as a function of depth.

In general, a 1-D gas slab (Figure 4) illuminated by a nearby X-ray source
with no attenuating foreground gas or dust has an external ionized layer.
Figure 4 spans the $10^{-29}\leq{\rm
H_{X}}/n<10^{-25}\rm{erg\,s^{-1}\,cm^{3}}$ range extending from substantially
ionized gas to cold molecular gas. The corresponding effective ionization
parameter ($\xi_{eff}$), which is related to the energy deposition rate as
${\rm H_{X}}/n=3.8\times 10^{-25}\xi_{eff}\rm\,erg\,s^{-1}\,cm^{3}$, ranges
from $10^{-4}\leq\xi_{eff}\leq 1$. If the slab is shielded from the direct
light of the X-ray source by e.g., a circumnuclear torus or a shadowing gas
cloud, then the external H II region will not be present. The external H II
region is followed by an FUV-produced PDR only if the ratio of the X-ray flux
to the gas density is low and the X-rays do not dominate over FUV heating at
low column densities. Note that FUV photons can be produced internally
following the various degradation paths of the initial X-ray photon and this
is discussed in detail in Section 4.2. At high column densities, and for high
X ray incident flux (so as to dominate cosmic rays) regardless of the presence
or not of an outer H II/PDR layer, X-rays completely dominate the heating and
the chemical composition, and this is the actual XDR. A peculiar feature of
XDRs is the much less abrupt $\rm C^{+}/C/CO$ transition as compared to PDRs.
This is produced by the slow decrease of $H_{\rm X}/n$, and by the internally
generated FUV photons resulting from collisions with secondary electrons which
maintain fairly constant $\rm C^{+}$ and C abundance until the CO dominates at
large column densities $N\approx 10^{22}-10^{23}\rm\,cm^{-2}$ (Meijerink &
Spaans, 2005).

An important consideration regarding the X-ray photoelectric cross-section,
$\sigma_{\rm pe}$, is that despite being much less abundant than H and He,
heavy elements (C, O, Mg, etc) dominate $\sigma_{\rm pe}$ above $0.5$ keV
(Wilms et al., 2000), and hence they are the major sources of photoelectrons.
Note that these elements are readily incorporated into dust grains, but there
remains a significant fraction of C and O (mostly in the form of CO), and
noble elements, in the gas phase. As such, the deposition of X-ray energy
occurs through absorption by both gas and dust. Bethell & Bergin (2011)
provide a simple polynomial fit to the X-ray photoelectric cross-section for a
mixture of gas and dust, with specific focus on protoplanetary disks. For
energies below 1 keV the gas is the main opacity source while at energies
$E>1$ keV the metals in dust grains dominate the total opacity. Rab et al.
(2018) also include the contribution of PAHs, concluding that they play a
negligible role in the X-ray radiative transfer.

Figure 5: Flow chart of the energy deposition and loss routes in XDRs for the
ionized, atomic, and molecular gas. The different mechanisms driving the
energy deposition depend on the ionized fraction ($x_{e}$) and H2 fraction
($x_{\rm H_{2}}$) of the gas. Figure adapted with permission from Maloney et
al. (1996), ©AAS.

###  4.2 Gas Heating

The XDR heating is produced by the degradation of the incident X-ray photon
energy through several channels. The first step is the production of primary
photoelectrons and secondary electrons as follows: The X-rays ionize heavy
elements preferentially by removing a K-shell electron. The vacancy is then
filled by a cascade of radiative (fluorescent) and non-radiative Auger
transitions. These primary photoelectrons are typically energetic enough
($E\sim$ keV) to induce secondary ionizations resulting in the ejection of
(non-thermal, secondary) electrons ($E\sim$ 8 eV) which, in turn, play a key
role in all the subsequent heating processes. Given that each primary
photoelectron loses about 20-30 eV per H ionization it produces $\approx 35$
secondary ionizations in addition to the initial ionization by the X-ray
photon. The next steps involve the interactions of the primary and secondary
electrons with the gas (see flowchart in Figure 5). Glassgold & Langer (1973),
Cravens & Dalgarno (1978), Dalgarno et al. (1999), and Glassgold et al. (2012)
carried out an extensive analysis of the energy loss of primary and secondary
electrons in a H, He, $\rm H_{2}$ mixture. They derived the heating produced
by elastic collisions with ambient thermal electrons (Coulomb heating), along
with a detailed treatment of the dissociation, vibrational and rotational
heating, and an in-depth analysis of the _chemical heating_ i.e., that
resulting from exothermic dissociative recombination reactions between
electrons and molecular ions. Each heating mechanism dominates in different
conditions depending on the electron and H2 fractions ($x_{e}$, and $x_{\rm
H_{2}}$). The first branching is set by $x_{e}$ (see Figure 5): if the
electron fraction is relatively high ($x_{e}>0.1$) nearly all the primary
photoelectron energy goes into heating through Coulomb interactions between
the secondary electrons and the ambient thermal electrons (Swartz et al.,
1971, Dalgarno et al., 1999). By contrast, in the low-ionization limit
($x_{e}\ll 1$) only $\approx 10\%$ of the primary photoelectron energy goes
into Coulomb heating while $\approx 50\%$ is expended in excitation processes,
and the remainder $40\%$ in ionization processes (Glassgold et al., 2012). To
summarize, the XDR heating due to Coloumb interactions can be parameterized in
terms of the energy deposition rate $H_{\rm X}$, as $n\Gamma_{\rm C}=\eta_{\rm
c}\,n\,H_{\rm X}$, where $\eta_{\rm c}=\eta_{\rm c}(x_{e},x_{\rm H_{2}})$ is
the Coloumb heating efficiency (Shull & van Steenberg, 1985, Dalgarno et al.,
1999, Meijerink & Spaans, 2005, Glassgold et al., 2012). The excitation and/or
ionization heating are instead mainly influenced by the H2/H I ratio because
$\rm H_{2}$, having a wide variety of energy levels, offers more channels for
energy loss than those of pure atomic hydrogen. In atomic gas ($x_{\rm H2}\ll
1$), the secondary electrons collisionally excite Lyman-$\alpha$ (Shull & van
Steenberg, 1985) and once their energy drops below $E_{Ly\alpha}=10.2\,\rm
eV$, all the remaining electron energy goes into heating through elastic
scattering with H and thermal electrons (Dalgarno et al., 1999). In this case
the X-ray heating efficiency is only 12%, whereas in molecular gas it can be
up to 50% at very high densities (Glassgold et al., 2012). If the H2 fraction
is high, $x_{\rm H2}\approx 0.5$, the initial energy is deposited as heat into
the gas through several processes. Among the possible mechanisms there is the
excitation of the rovibrational levels of the $\rm H_{2}$ molecules and the
electronic excitation of H2 followed by fluorescence to $v,J$ states. If the
gas density exceeds the critical densities of such transitions ($n\gtrsim
10^{4}\rm\,cm^{-3}$), then the collisional de-excitation (Tiné et al., 1997)
results in net heating (Meijerink & Spaans, 2005, Glassgold et al., 2012).
Additionally, heating is produced by the $\rm H_{2}$ excitation to
dissociative states (Dalgarno et al., 1999, Glassgold et al., 2012) which
injects energetic H atoms that then thermalize. A peculiarity of XDRs is the
abundant presence of $\rm H_{2}^{+}$ produced in $\rm H_{2}$ secondary
ionizations. These molecular ions react with $\rm H_{2}$ to produce $\rm
H_{3}^{+}$. The $\rm H_{3}^{+}$ may further undergo an exothermic dissociative
recombination with an electron or a proton-transfer reaction as e.g., in $\rm
H_{3}^{+}+O\rightarrow OH^{+}+H_{2}$. The molecular ions can undergo an
exothermic dissociative recombination thus adding to gas heating. If the
density is high enough for collisional de-excitation of vibrationally excited
H2, then in the limit of $x_{e}\approx 0$ and $x_{\rm H_{2}}\approx 1$, the
total maximum heating is Q=18.7 eV per ion pair.

###  4.3 Gas Cooling

The deep penetration of X-ray photons heats the gas to high column densities.
In thermal equilibrium, the heating is balanced by cooling, and the neutral
and molecular gas stays warm to high column densities with the heating,
cooling and temperature a function of $H_{\rm X}/n$ (see Figure 6a. Also shown
for comparison is a typical PDR in Figure 6b). One peculiarity of XDR cooling
compared to PDRs is the overall higher ratio of the energy radiated as line
emission over that emitted as infrared continuum (Maloney et al., 1996). That
is, gas cooling/grain cooling is higher than in PDRs. While in PDRs almost all
the FUV photon energy absorbed by dust is re-irradiated in the infrared, in
XDRs about half of the energy is deposited into the gas and thus the cooling
lines can carry a large fraction of the total deposited energy (Maloney et
al., 1996, Meijerink et al., 2007). For high $H_{\rm X}/n$, the predominantly
ionized gas ($x_{e}\gtrsim 0.1$) is characterized by $T\approx 10^{4}\,\rm K$,
as sketched in Figure 4 and shown by the temperature profile in Figure 6a.
Thermal collisions can thus excite Ly$\alpha$ (e.g., Sternberg & Dalgarno,
1989), and forbidden lines such as [C I] $\lambda\lambda$9823,9850 (e.g.,
Escalante et al., 1991), and [O I] $\lambda\lambda$6300,6363 (e.g., Störzer &
Hollenbach, 2000) that dominate the cooling. The [Fe II] 1.26 and 1.64 $\mu$m
lines are also efficient coolants for high X-ray illumination, temperature,
and density as the upper state of the 1.64 $\mu$m line lies about
$10^{4}\rm\,K$ above the ground state. As $H_{\rm X}/n$ decreases, the $\rm
H_{2}$ abundance increases while the gas temperature remains warm ($T\approx
1000\,\rm K$). These conditions favor the excitation of $\rm H_{2}$
rovibrational transitions that significantly contribute to the cooling
(Sternberg & Dalgarno, 1989, Neufeld & Kaufman, 1993, Le Bourlot et al., 1999,
Spaans & Meijerink, 2008, Glover & Abel, 2008, Lique, 2015). The range of
$H_{\rm X}/n$ values for which vibrational lines dominate the cooling are
rather narrow because the first two vibrational levels ($v=1$ and $v=2$) lie
at $\approx 6000\rm\,K$ and $\approx 12000\rm\,K$ above the ground state,
respectively (see Fig. 3 in Shaw et al., 2005). In particular, for
$T>2000\,\rm K$ the $\rm H_{2}$ 1-0 S(1) 2.122 $\mu$m transition significantly
contributes to the total cooling, while once the temperature falls below
$T\approx 1000\rm\,K$ the $\rm H_{2}$ cooling is dominated by rotational lines
in $v=0$: 0-0 S(0) 28.22 $\mu$m, 0-0 S(1) 17.03 $\mu$m, 0-0 S(2) 12.28 $\mu$m,
and 0-0 S(3) 9.66 $\mu$m.131313 The H2 spectroscopic notation reads as
follows: the first two numbers refer to the vibrational level transition, the
letter indicates the branch (S corresponding to transitions between rotational
states with $\Delta J=+2$), and the number in parenthesis is the rotational
quantum number of the final state. For instance 1-0 S(1) stands for:
($v=1,J=3$) $\rightarrow$ ($v=0,J=1$). A recent analytic approximation of the
$\rm H_{2}$ cooling function can be found in Moseley et al. (2021).

Figure 6: ($a$) Gas temperature profile (dashed line) of a typical XDR
characterized by a gas density $\log(n/{\rm cm^{-3}})=5$ as a function of
$H_{\rm X}/n$. The major XDR gas coolants as a function of $H_{\rm X}/n$ are
plotted with colored lines. Figure adapted with permission from Maloney et al.
(1996), ©AAS. ($b$) Gas and grain temperatures as a function of $A_{\rm V}$
are shown as dashed lines for a PDR with $\log(n/{\rm cm^{-3}})=4$ and
$G_{0}=10^{3}$. Major PDR gas coolants are indicated with colored lines. The
gas temperatures are generally higher than the grain temperatures and grains
act to cool the gas. At large depth warm grains can heat the gas. PDR models
from https://dustem.astro.umd.edu with freeze out turned off.

Deeper into the XDR the temperature decreases ($T\lesssim 500\rm\,K$) and the
[O I] 63 $\mu$m, [Si II] 35 $\mu$m, and [C II] 158 $\mu$m lines become the
major coolants (Maloney et al., 1996, Meijerink et al., 2007). The [O I] 63
$\mu$m cooling ($n_{\rm cr}\approx 5\times 10^{5}$ cm-3, $\Delta
E/k=228\rm\,K$) is particularly efficient because warm temperatures ($T\gtrsim
100$) remain at high column densities where the oxygen is neutral. XDRs are
thus characterized by high [O I]/[C II] ratios ($\gtrsim 10$) as compared to
values $\lesssim 10$ in PDRs (Maloney et al., 1996, Hollenbach & Gorti, 2009).
XDRs also feature high [Si II]/[C II] ratios (Meijerink et al., 2007)
reflecting the high $\rm Si^{+}$ abundance that results both from X-ray
induced secondary ionizations and from those produced by Lyman and Werner
photons following H2 excitation. As $H_{\rm X}/n$ declines further, the
neutral carbon [C I] 370 $\mu$m and 609 $\mu$m fine-structure transitions
become important coolants (Bisbas et al., 2021). Enhanced [C I] emission as
compared to PDRs is another peculiarity of XDRs as abundant C coexists with C+
through a thick layer in XDRs (see Section 4). In the same range of $H_{\rm
X}/n$ significant cooling is produced by high-$J$ CO rotational transitions
which are exceptionally bright in XDRs, boosted by the warm ($T\approx
200\rm\,K$) temperature at high column densities maintained by X-ray
penetration (Meijerink et al., 2007, Spaans & Meijerink, 2008). By contrast,
in typical PDRs most of the CO is present beyond the H/H2 transition at much
lower temperatures ($T\approx 20$ K), which do not allow for bright high-$J$
emission. Other coolants of the warm molecular gas in XDRs are $\rm H_{2}O$,
$\rm HCN$, $\rm OH$ rotational lines. Goldsmith & Kauffmann (2017) found that
electron excitation of HCN, HCO+, CN, and CS might be important. This might be
especially relevant when using HCN/HCO+ ratios as a diagnostic for inferring
the presence of XDRs in external galaxies (see Section 5.3.2). In the lowest
$H_{\rm X}/n$ regime the gas-grain collisions become an important cooling
mechanism as the gas is warmer than the dust due to the effect of X-ray
penetration and can therefore be cooled by collision with lower temperature
dust grains (Maloney et al., 1996).

###  4.4 Chemistry

X-ray dominated regions are characterized by a peculiar chemistry initiated by
the secondary ionizations resulting from the ejection of primary
photoelectrons, and by the effect of internally generated Lyman and Werner
photons arising from the decay of H2 excited states. These processes are
generally more important than the effect of direct X-ray ionizations and
dissociations. Numerous studies have explored the effects of X-ray irradiation
on the chemistry of the ISM (Maloney et al., 1996, Meijerink & Spaans, 2005,
Stäuber et al., 2005) assuming steady-state for the temperature and the
abundances of the different species. Maloney et al. (1996) considered a
chemical network focused primarily on carbon and oxygen, whereas the Meijerink
& Spaans (2005) network included all species with sizes up to 3 atoms and some
of 4 atoms and adopted the UMIST database for the chemical reaction rates. The
chemical signature of X-ray induced processes can be retrieved from the
abundances of key species. We highlight below and in Table 1 the major
formation/destruction pathways that differ from PDRs.

The Hydrogen and Helium chemistry can be summarized as follows: $\rm
H_{2}^{+}$ (He+), resulting from H2 (He) collisional ionization produced by
primary electrons trigger an X-ray characteristic chemistry which is similar
to that induced by cosmic rays. These ions, whose abundance is particularly
enhanced in XDRs, together with $\rm H_{3}^{+}$ resulting from reactions of H2
with $\rm H_{2}^{+}$, efficiently exchange charge with neutral constituents
(e.g., O, CO, OH, $\rm H_{2}O$) producing (molecular) ions.

The O-bearing molecular ions (OH+, $\rm H_{2}O^{+}$ and $\rm H_{3}O^{+}$) are
overall enhanced in gas irradiated by X-rays (Maloney et al., 1996, Meijerink
& Spaans, 2005). Among them, $\rm OH^{+}$ is produced either through the
reaction of atomic oxygen with $\rm H_{3}^{+}$, or between ionized oxygen and
H2. Note that the presence of ionized oxygen where the H2 is abundant is due
to charge transfer reactions between O and H+ following X-ray induced
ionizations of H. OH+ can also be formed in reactions between He+ and $\rm
H_{2}O$. The OH+ forms $\rm H_{2}O^{+}$ and, subsequently, $\rm H_{3}O^{+}$ by
reactions with H2. For $\rm H_{3}O^{+}$, models predict an order of magnitude
greater abundance in XDRs than in PDRs. The $\rm H_{3}O^{+}$/$\rm H_{2}O$
ratio is as large as $10^{-2}$ in XDR models, while the ratios in PDRs are
generally $10^{-3}$ or less (van der Tak et al., 2008). This enhancement has
been observed in relation to a strong X-ray illumination in the center of AGN
host galaxy NGC 1068 (e.g., Aalto et al., 2011).

A feature of carbon chemistry in XDRs, which is also shared with environments
characterized by high cosmic-ray fluxes (Bisbas et al., 2021), is the lack of
the well defined C+/C/CO stratification that characterizes PDRs (Meijerink &
Spaans, 2005, Meijerink et al., 2007). The C+ and C coexist through the XDR
layers because the internally produced FUV photons cause CO dissociation and C
ionization much deeper than in PDRs. In XDRs there is another viable path for
the production of C+ through charge exchange reactions involving He+ which,
for high X-ray fluxes, dominates over the carbon photoionization induced by
FUV photons. The abundance of CO+ is enhanced in regions affected by strong UV
and X-ray radiation fields (Wolfire et al., 1995b). At low $T$, C is destroyed
by reactions with abundant molecular ions such as $\rm HCO^{+}$ and $\rm
HOC^{+}$ which are enhanced by the reactions of $\rm H_{2}^{+}$ and $\rm
H_{3}^{+}$ with CO, and of C+ with $\rm H_{2}O$ (Lepp & Dalgarno, 1996).

Table 1: Major formation (left column) and destruction (right column) pathways
for major chemical species in PDRs and XDRs.

| MAJOR FORMATION PATHWAYS | MAJOR DESTRUCTION PATHWAYS
---|---|---
| Hydrogen and Helium chemistry
H2 |  | \- Formation on dust grains
---
\- Gas phase reactions (e.g. H- \+ H $\rightarrow$ H2\+ $e^{-}$)
|  \- Secondary ionizations: H2\+ $e^{-}\rightarrow$ H${}_{2}^{+}$ \+ 2$e^{-}$
---
\- Reaction with H${}_{2}^{+}$ or O+
\- Reactions with O when T$>3000$ K
H${}_{2}^{+}$ |  \- Secondary ionizations: H2\+ $e^{-}$ $\rightarrow$ H${}_{2}^{+}$ \+ 2$e^{-}$ |  \- Reaction with H2 to form H${}_{3}^{+}$
H${}_{3}^{+}$ |  \- Reaction of H${}_{2}^{+}$ with H2 |  \- H${}_{3}^{+}$ \+ X $\rightarrow$ HX+ +H2 where X is O, CO, OH, H2O
He+ |  \- Secondary ionizations: He+ $e^{-}$ $\rightarrow$ He+ \+ 2$e^{-}$ |  \- Charge exchange reactions: He+ \+ XY $\rightarrow$ He + X + Y+
| Oxygen chemistry
O+ |  \- Charge exchange of O with H+ |  \- Reaction with H2: O+ \+ H2$\rightarrow$ OH+ \+ H
OH+ |  |  \- Ionized oxygen reaction with H2: O+ \+ H2$\rightarrow$ OH+ \+ H
---
\- Neutral oxygen reaction with H${}_{3}^{+}$: O + H${}_{3}^{+}$ $\rightarrow$
OH+ \+ H2
\- H2O reaction with He+: He+ \+ H2O $\rightarrow$ He + OH+ \+ H
|  \- Hydrogen abstraction: OH+ \+ H2$\rightarrow$ H2O+ \+ H
---
\- Dissociative recombination: OH$+$ \+ $e^{-}$ $\rightarrow$ O + H
OH |  |  \- H2O + $e^{-}$ $\rightarrow$ OH + H
---
\- Neutral-neutral reaction (high temperature): O + H2$\rightarrow$ OH
\- Photodissociation of H2O by internally generated FUV photons
\- At high temperature OH reacts with H2 to form H2O
H2O+, H3O+ |  | \- Hydrogen abstraction
---
\- Oxygen reaction with H${}_{3}^{+}$
\- H2O ionization by internally generated FUV photons
\- HCO+ \+ H2O $\rightarrow$ H3O+
\- H2O+ \+ H2$\rightarrow$ H3O+
\- Dissociative recombination of H3O+ with $e^{-}$
H2O |  |  \- Recombination of H3O+ with $e^{-}$
---
\- At high temperature OH reactions with H2
|  \- Reaction with HCO+ and H${}_{3}^{+}$ for T$>$100 K
---
\- At high temperature, reaction with H atoms
\- Dissociation by FUV photons
| Carbon chemistry
C |  \- Photodissociation of CO by internally generated FUV photons |  |  \- Photoionization by internally generated FUV photons
---
\- Reaction with O2 and HCO+, at T$<100$ K
C+ |  | \- C ionization by internally generated FUV photons
---
\- CO reactions with He+: He+ \+ CO $\rightarrow$ He + C+ \+ O
|  \- C+ \+ $e^{-}$ $\rightarrow$ C
---
\- C+ \+ H2O
CO+ |  | \- Electron impact ionization of CO
---
\- CO2 reaction with He+
\- Reactions of C+ with OH and O2, respectively
\- Reactions of CO+ with H2
HCO+ |  |  \- Reaction of CO with H${}_{3}^{+}$
---
\- Reaction of C+ with H2O
\- Reaction of CO+ with H2
|  \- Reaction with H2O
---
\- Electron recombination to form CO + H and OH + C
| Sulfur chemistry
S+ |  \- Photoionization of S due to the internally generated FUV photons |  \- Reaction with H2, OH, O2
S2+ | \- X-ray photoionization |  \- Reaction with H2
SO+ SH+ |  |  \- Reaction of S+ with H2
---
\- Reaction of S with H${}_{3}^{+}$ and HCO+
\- Reaction of S2+ with H2
\- Reaction of S+ with OH and O2
\- Electron recombination reactions
| Nitrogen chemistry
N+ |  |  \- X-ray ionization and secondary ionizations of atomic N
---
\- N2 reaction with He+: He+ \+ N2 $\rightarrow$ He + N+ \+ H
|  \- N+ reactions with H2 to produce NH+
---
| HCN
---
HNC
|  \- Dissociative recombination of HCNH+
---
(with approximately equal branching between HCN and HNC)
\- Reaction of CN with H2
|  \- HNC + H $\rightarrow$ HCN + H (when T$>200$ K)
---
\- Reactions with H${}_{3}^{+}$, H3O+, HCO+

Processes which are peculiar to XDRs are highlighted in boldface, species that
are enhanced in XDRs are highlighted in yellow. References: Maloney et al.
(1996); Yan (1997); Stäuber et al. (2005); Meijerink & Spaans (2005) Abel et
al. (2008) Notsu et al. (2021).

The sulfur chemistry is also affected by X-rays because the internally
produced FUV field dissociates SH, followed by direct X-ray photoionization of
sulfur, which enhance S2+ column densities with respect to PDRs. Reactions of
S+ with OH, and $\rm H_{3}^{+}$ or HCO+ with OH produce SO+ and SH+
respectively, which are also particularly abundant in XDRs. As outlined by
Abel et al. (2008) and Godard et al. (2012), SH+ can be formed also via
another channel involving S2+ and H2. If the branching ratio of S2+ \+
H2$\rightarrow$ SH+ exceeds 1%, compared to other reaction products, then the
double ionized chemistry will be the dominant pathway to SH+.

X-ray induced FUV photon production and the presence of He+ in XDRs influence
the nitrogen chemistry by $\rm N_{2}$ dissociation. The atomic nitrogen
initiates a series of reactions which ultimately enhance HCN abundance. In
particular N reacts with OH to form NO, which subsequently reacts with C to
form CN. HCN and HNC are formed in almost equal amounts through the
dissociative recombination of HCNH+, and through reactions of CN with H2. The
abundance difference between HCN and HNC are thus largely determined by the
selective destruction pathways of HNC which have relatively high activation
barrier. For this reason low HNC-to-HCN line ratios generally indicate warm
temperatures (e.g., Hacar et al., 2020). In XDRs, for high $H_{\rm X}/n$,
HNC/HCN is lower than in PDRs. At high column densities, where $H_{\rm X}/n$
is low (and so is the temperature), HNC/HCN ratios are equal or somewhat
higher than those of PDRs (Meijerink & Spaans, 2005).

The most recent and comprehensive discussion concerning the physics and
chemistry of water in star forming regions (van Dishoeck et al., 2021) is also
relevant for those interested in XDR chemistry because the impact of X-ray
irradiation on the H2O abundance is also addressed. In astrophysical
environments three pathways lead to $\rm H_{2}O$ (see Fig. 4 in van Dishoeck
et al., 2021, and references therein). The first route is through neutral-
neutral reactions $\rm O+H_{2}\rightarrow OH+H$, $\rm OH+H_{2}\rightarrow
H_{2}O+H$ that contribute to the water formation only at high temperatures
($T\gtrsim 250$ K; van Dishoeck et al., 2013), the second route involves ion-
molecule reactions (e.g., Hollenbach et al., 2009, Stäuber et al., 2005) that
are effective in diffuse and moderately ionized gas, and the third route is by
grain surface reactions (Notsu et al., 2021) because at low $T\approx 10-30$ K
water is efficiently formed by hydrogenation of oxygen atoms sticking onto
dust grains (Cuppen et al., 2010). Thus, accounting for thermal and non-
thermal gas-grain interactions and for grain-surface reactions is fundamental
(Meijerink et al., 2012, Notsu et al., 2021).

Gas-phase destruction of molecules by X-ray chemistry and X-ray-induced
photodesorption are important processes as well (e.g., Dupuy et al., 2018,
Notsu et al., 2021). Gas-phase water is mainly destroyed by ion-molecule
reactions (with X-ray boosted HCO+, H+, $\rm H_{3}^{+}$, He+) and by X-ray
induced photodissociation (to H+OH; Meijerink et al., 2012). This might
explain the low gas-phase abundance of warm water in the inner regions of
protostellar envelopes (Notsu et al., 2021). Meijerink et al. (2012) instead,
studied the possibility of enhanced gas phase $\rm H_{2}O$ abundance in X-ray
exposed environments using (bare) carbonaceous dust grains as a catalyst.
Outside the snowline, the gas-phase abundance of water is likely increased by
X-ray photodesorption from icy grains, although results are very sensitive to
the photodesorption rates assumed (see Notsu et al., 2021, which acknowledge
that they might be overestimated).

Thermal equilibrium and chemical balance are often assumed when comparing PDR
vs XDR chemical composition but time-dependence and non-equilibrium conditions
are often relevant in many astrophysical environments. For instance, short-
term X-ray flaring is common in young solar mass stars, and AGN activity
varies due to strong fluctuations in the super massive black hole accretion
rate. Meijerink et al. (2013) included time dependence in their XDR code and
found that tracers such as the HCO+/HCN ratio (see Section 5.3.2) are strongly
time-dependent (see also Harada et al., 2013). Strong evolutionary trends,
occurring over time scales $0.01-100$ Myr are also found for $\rm H_{3}O^{+}$,
CO, and $\rm H_{2}O$. These species reflect time dependent effects in the
ionization balance, the transient nature of the production of molecular gas,
and the freeze-out/sublimation of water (key to much of the grain surface
chemistry), respectively. Viti (2017) also addressed time-dependence in XDR-
like environments with UCLCHEM, focusing on the four most observed species
(CO, HCN, HCO+, and CS). Among them, HCN is the most affected by time-
dependence followed by CS. Recently, Waggoner & Cleeves (2019) addressed the
effect of time-dependence on water chemistry reporting a significant but
short-lived (days) boost in gas-phase $\rm H_{2}O$ abundance. Mackey et al.
(2019) presented a non-equilibrium XDR chemistry calculation using a
simplified chemical network of 17 species to study the time-dependent response
of a molecular cloud to X-ray flares, emphasizing the faster destruction of CO
by an internally generated FUV field as compared to H2. Using KROME (Grassi et
al., 2014), Liu et al. (2020) presented a time-dependent study of the
molecular chemistry of the Galaxy as resulting from the putative past AGN
activity of Sgr∗. In particular, H2O, CH3OH, and H2CO abundances are enhanced
with respect to the baseline model without X-ray irradiation both in the gas
phase and on the dust grain surface up to 10 Myrs after the turn off of the
X-ray source.

##  5 Using Models to Analyze Observations

###  5.1 Overview

Since the mid 70’s (e.g., Glassgold & Langer, 1975, Black & Dalgarno, 1977)
several groups have developed PDR codes that are either directly available for
download, or make their output available on-line. These include the model
derived from the original TH85 code (Kaufman et al., 1999, 2006, Neufeld &
Wolfire, 2016), the UCL-PDR code (Papadopoulos et al., 2002, Bell et al.,
2006, Priestley et al., 2017), CLOUDY (Abel et al., 2005, Ferland et al.,
2013, 2017), the Meudon PDR code (Le Petit et al., 2006, Le Bourlot et al.,
2012, Bron et al., 2014) and KOSMA-$\tau$ (Röllig et al., 2006, Cubick et al.,
2008, Röllig et al., 2013). Meijerink & Spaans (2005) emphasized modelling
XDRs, but we note that now also CLOUDY (Ferland et al., 2017) explicitly
handles XDR calculations. The majority of these models are 1-dimensional and
steady-state. We discuss multi-dimensional and time-dependent models in
Section 6. Specific PDR and XDR codes have also been developed for distinct
applications, especially for the study of protoplanetary disks. We will not
review the disk models here (see Bergin et al., 2007, Öberg & Bergin, 2021).

The considerable heterogeneity among models (including their geometry,
physical and chemical structure, choice of parameters) makes the comparison
between them challenging. Nevertheless, a number of benchmark models have been
created to understand where different results originate and, as much as
possible, converge on a common solution for a common input. Röllig et al.
(2007) give a detailed report of the 2004 Lorentz Center workshop comparing 10
different PDR codes. An additional Lorentz Center workshop on the CO ladder
from both PDR and XDR models was held in 2012 with results that can be found
on-line at https://markusroellig.github.io/research/CO-workshop/. Even using
consistent inputs and similar microphysics there is considerable range in the
outputs in the gas temperatures, abundances, and line intensities. From the
first workshop, for typical PDR conditions and neglecting the obvious
outliers, the [C II] line is most consistent between models and varies within
a factor of ${\sim}2-3$ depending on $G_{0}/n$. The next most consistent are
the [O I] lines (varying by a factor ${\sim}2-5)$ and the least consistent are
the [C I] lines. From the second workshop, the CO ladder line intensities can
vary by 10-100 between models for similar inputs. The largest differences are
likely due to differences in gas temperature where the lines are produced. The
gas temperatures typically vary between models by factors of 2-3 at $A_{\rm
V}\approx 1$, but are seen to vary by as much as a factor 4-10 at $A_{\rm
V}<1$, but also at $A_{\rm V}\approx 1$ between models with unconstrained
microphysics. The most extreme variations occur for the test run
$\chi=10^{5}$, $n=10^{5.5}$ ${\rm cm^{-3}}$ (see also Bruderer et al., 2012).
The temperature differences could be the result of a thermal instability that
rapidly drives gas temperatures from $T{\sim}2000$ K to ${\sim}8000$ K with
only minor differences in the heating/cooling rates. In addition, the details
of the ${\rm H_{2}}$ formation rates, vibrational heating rates, and
photoelectric heating rates can cause large differences in gas temperature and
are especially important for the prediction of ${\rm H_{2}}$ and high-J CO
line intensities. The conclusion from Röllig et al. (2007) is that the model
outputs for specific densities and radiation fields should not be considered
exact results but should instead be used as guides to the physical conditions.
The choice of a specific code to compare to observations should be motivated
by the physics and chemistry included in the code but also the characteristics
(in terms of geometry, density profile, etc.) most adapted to the emission
source. See Röllig et al. (2007) for a guide to PDR model characteristics
while noting that many codes have been in continuous update since the
workshops and another workshop would be well worth revisiting.

###  5.2 Main input parameters

An important parameter of the models is the incident radiation field. The
field can be adjusted in several ways including setting the shape, strength,
and geometry (isotropic, uni-directional, external vs internal source for
spherical clouds, or one-sided vs two-sided for slabs). Several scalings of
the FUV field are in use based on estimates of the local Galactic interstellar
radiation field (see Table 2). For comparison, it is convenient to refer to
the integrated energy density $u(\lambda)=4\pi J(\lambda)/c$, where
$J(\lambda)=1/(4\pi)\int I(\lambda)d\Omega$ is the mean intensity and
$I(\lambda)$ the specific intensity. The value of $J$ equals $I$ for an
isotropic field only. When integrating over the surface of a semi-infinite
cloud illuminated over $2\pi$ and taking into account backscattering from
grains, $J\sim 0.54I$ (Le Petit et al., 2006, Röllig et al., 2013).

The Habing field (Habing, 1968), noted as $G_{0}$, when integrated over a
range of energies 6 eV $<h\nu<13.6$ eV, has an energy density $u=5.33\times
10^{-14}$ erg ${\rm cm^{-3}}$ corresponding to an isotropic intensity of
$I=J=1.3\times 10^{-4}$ erg ${\rm cm^{-2}}$ ${\rm s^{-1}}$ ${\rm sr^{-1}}$.
The 1-D flux is the unidirectional flux equivalent to the total flux incident
on a sphere in the isotropic radiation field $=4\pi J=cu=1.6\times 10^{-3}$
erg ${\rm cm^{-2}}$ ${\rm s^{-1}}$. Other fields that are in use are the
Mathis field ($U$; Mathis et al. 1983, Weingartner & Draine 2001c) and the
Draine field ($\chi$; Draine 1978). In terms of the Habing field, the median
local Galactic interstellar radiation field is estimated to be
$G_{0}{\sim}1.6$ (Parravano et al., 2003), comparable to the Draine field,
while $G_{0}{\sim}10^{5}$ for the PDR behind the Trapezium cluster.

For XDR codes the incident radiation flux between 1-100 keV, $F_{X}$ (in $\rm
erg\,s^{-1}\,cm^{-2}$) is instead the required input parameter. For historical
reasons (Maloney et al., 1996), the impinging radiation is generally assumed
to follow a powerlaw distribution, $F_{\rm
X}(E)=F(0)(E/100{\rm\,keV})^{-\alpha}$, with a low energy cut-off at 1 keV. In
Meijerink et al. (2007), $\alpha=0.9$ is chosen, while CLOUDY has a default
$\alpha=0.7$. This is a good approximation of the X-ray regime of the spectral
energy distribution (SED) of AGN. Note that the 1-100 keV range is only a
fraction of the total AGN SED, which is also bright in the mid-infrared,
optical and ultraviolet. XDR models including the total AGN SED instead of the
standard power-law for the incident radiation, produce different kinetic
temperatures, mid-infrared line intensities, and low level populations of $\rm
H_{2}$ despite being normalized to have the same $F_{\rm X}$ between 1-100 keV
(Ferland et al., 2013). XDR models are also used to derive the gas conditions
in the vicinity of young stellar objects. The incident photon flux is then
expressed in terms of the spectrum of a thermal plasma $F_{\rm
X}(E,r)=F_{0}(r){\rm exp}(-E/kT_{X})$ in units of photons s-1 cm-2 eV-1, where
$r$ is the radius in the envelope from the central protostar, and $T_{X}$ is
the temperature of the X-ray emitting plasma. In this case, the spectrum goes
as an exponential and $H_{\rm X}/n$ falls off more steeply at high column than
a power-law.

Table 2: Conversion between FUV fieldsa

| Energy Density | Isotropic Intensity | 1-D Flux |  | Examples
---|---|---|---|---|---
Notation | (erg ${\rm cm^{-3}}$) | (erg ${\rm cm^{-2}}$ ${\rm s^{-1}}$ ${\rm sr^{-1}}$) | (erg ${\rm cm^{-2}}$ ${\rm s^{-1}}$) | Ref. | of codes
$G_{0}$ | $5.3\times 10^{-14}$ | $1.3\times 10^{-4}$ | $1.6\times 10^{-3}$ | [1] | UCL-PDR
$\chi$ | $8.9\times 10^{-14}$ | $2.1\times 10^{-4}$ | $2.7\times 10^{-3}$ | [2] | KOSMA-$\tau$
$U$ | $6.1\times 10^{-14}$ | $1.4\times 10^{-4}$ | $1.8\times 10^{-3}$ | [3] | Meudonb

aEnergy density, Isotropic Intensity, and 1-D flux for unit values of
radiation fields $G_{0}$, $\chi$, and $U$. Quantities are integrated over 6 eV
to 13.6 eV unless otherwise noted. The Meudon code assumes limits of 912 Å to
2400 Å or 13.6 eV to 5.166 eV, resulting in energy densities of $5.6\times
10^{-14}$ (erg ${\rm cm^{-3}}$), $1.05\times 10^{-13}$ (erg ${\rm cm^{-3}}$),
and $6.8\times 10^{-14}$ (erg ${\rm cm^{-3}}$) for $G_{0}$, $\chi$, and $U$
respectively.; bCan also use the Draine field. References: [1] Habing (1968);
[2] Draine (1978); [3] Mathis et al. (1983).

In addition to the radiation field, the cosmic-ray ionisation rate is also a
necessary input. This is usually given as the total (including secondaries)
ionization rate per H2, $\zeta_{\rm H_{2}}$, but may also be given as the
primary ionization rate per H, $\zeta_{\rm p}$, where the rate per H2 is
higher by a factor ${\sim}2.3$.

The abundances of gas phase metals, PAHs, and large grains affect the cooling
rates heating rates, and conversion between $A_{\rm V}$ and column density,
$N$. These initially vary linearly with metallicity, Z⊙, where Z⊙ is the
metallicity with respect to solar, but are observed to have different
dependencies for Z⊙$\lesssim 0.2$ (Gordon et al., 2003, Sandstrom et al.,
2010, Rémy-Ruyer et al., 2014) and the model results will depend on the
relative variations (see e.g., Röllig et al. 2006, and Jameson et al. 2018).
In light of other uncertainties a linear dependence with Z⊙ is often assumed.

The choice of geometry is also important to consider. Many models adopt a
plane-parallel slab of fixed width in terms of $A_{\rm V}$ (or with a stopping
condition provided as input), while others have spherical clouds with central
cavities, spherical clumps, or distributions of clumps. Several density
structures can be adopted, such as constant density, constant pressure, or a
user specified density law. The adopted model for the photoelectric heating is
also important and can result in significant temperature variations. Finally,
different models include different chemical networks and elemental abundances.
The species, reactions, and rates that are used have an impact on the
resulting predictions.

###  5.3 Diagnostic plots

Combinations of line intensities and dust continuum can be used to constrain
the physical conditions of the gas in PDRs and XDRs, as well as to distinguish
between excitation sources (PDR, XDR, shocks). We describe here how some of
the commonly observed line intensities and line ratios lead to predictions of
the incident radiation field strength and gas density in the PDRs and XDRs,
and help to distinguish between different heating processes.

####  5.3.1 Diagnostic to determine the gas density $n$, and radiation field
strength, $G_{0}$

Figure 7: Diagnostic contour plots of model integrated line intensities and
continuum ratios. The intensity units are erg ${\rm cm^{-2}}$ ${\rm s^{-1}}$
${\rm sr^{-1}}$ (rather than K km ${\rm s^{-1}}$). ($a$) [C II]/[O I] ratio
versus the ([O I]+[C II])/TIR ratio where [O I] refers to the 63 $\mu$m
transition. The [C II]/[O I] ratio is mainly sensitive to density while the
([O I]+[C II])/TIR ratio is sensitive to $G_{0}/n$. ($b$) [C II]/CO(6-5)
versus [C II]/TIR. The [C II]/CO(6-5) ratio is a strong density indicator
while [C II]/TIR is mainly sensitive to $G_{0}$. PDR models from
https://dustem.astro.umd.edu.

The PDR model used here is based on that described in Wolfire et al. (2010),
Hollenbach et al. (2012), and Neufeld & Wolfire (2016). We use a maximum
$A_{\rm V}=7$, freeze out turned off, and a power-law dependence of the
cosmic-ray ionization rate $\zeta_{\rm p}=2\times 10^{-16}(1+N/2\times
10^{21}\,{\rm cm}^{-2})^{-1}$. The [C II]/[O I] 63 $\mu$m ratio is a good
diagnostic for density since for $n\gtrsim 3000$ ${\rm cm^{-3}}$, the critical
density for [C II], this ratio will be a strong function of density, and for
$G_{0}\gtrsim 100$ [O I] cooling is important. The ratio ([C II] + [O I])/TIR
is a measure of the heating efficiency (Section 3.3) where TIR is the total (3
$\mu$m-$1.1\,{\rm mm}$) infrared integrated dust continuum. The combination of
these two ratios (Figure 7a), are useful to obtain physical conditions (gas
density and radiation field strength) directly from plotted observations.
Figure 7b shows [C II]/CO ($J=6-5$) versus [C II]/TIR and is an example that
is especially useful for high density regions where CO ($J=6-5$) emission is
produced. The CO ($J=6-5$) line is often near the peak of the CO ladder
(Figure 8) in extragalactic observations and is often the brightest (or even
the only) line observed. For the model plots, we assume the TIR intensity is
$2\times G_{0}$ where the factor of 2 accounts for dust heating by stellar
optical radiation and by EUV radiation emitted by the star that is converted
by line emission to optical radiation. The observed continuum intensity to
plot on the model grids should ideally be the TIR. However, in practice, the
integrated intensity depends on the wavelength range of the available
observations. For $\lambda\gtrsim 40$ $\mu$m, generally referred to as the FIR
continuum, ${\rm TIR/FIR}{\sim}2$ depending on the grain temperature. See
e.g., Dale & Helou (2002) to convert between TIR and FIR. Additional on-line
plots and model results can be found at https://dustem.astro.umd.edu or
https://ism.obspm.fr for the models included here and for the Meudon code
respectively. Both also include the KOSMA-$\tau$ models. The on-line tools can
analyze both pointed observations and maps.

Diagnostic plots typically use ratios of intensities rather than absolute
values. This is justified because the model outputs are for unit beam filling
factor, $f_{b}=1$. For unresolved sources, $f_{b}<1$ and using line ratios has
the advantage of cancelling the beam filling factor. However, if the emitting
regions for different lines have substantially different filling factors, then
the intensities used in the ratio should account for these differences as best
as possible (see e.g., Wolfire et al. 1990, Kaufman et al. 1999, for a typical
procedure). Once the physical conditions are estimated from diagnostic plots,
the beam filling factor can be derived by comparing the model with the
observed intensity, $f_{b}=I^{\rm obs}/I^{\rm model}$. Note that the filling
factor is less than 1 for unresolved sources but can be greater than 1 for
several PDRs along the line of sight. The covering factor, discussed by
Cormier et al. (2019), is the fraction of the H II region surrounded by
neutral PDR gas. This can be calculated using models that include both the
ionized and neutral gas (e.g., CLOUDY). In a sample of low-metallicity
galaxies, Cormier et al. (2019) find PDR covering factors $f_{b}=0.2-1$, with
a median value of 0.4, and show that the PDR covering factor decreases with
metallicity (see Section 8.4).

Simple, model independent estimates can be made for $G_{0}$, $n$, and $T$
(e.g., Pabst et al., 2017). From the distribution of OB stars on the sky, the
maximum $G_{0}$ is ${\sim}0.5L/(4\pi d^{2}1.6\times 10^{-3})$ with $d$ the
projected distance in cm between the source and PDR, and $L$ the stellar
luminosity in erg ${\rm s^{-1}}$. This maximum $G_{0}$ and the $G_{0}$
estimated from a PDR model are compared to determine the true (rather than
projected) distance between an FUV source and the illuminated PDR and thus
reveal the 3-dimensional geometry of a region. This method has been used for
example to establish that the PDRs in 30 Doradus that seem close to the
central cluster in projection are actually located at more than 40 pc away
from it (Pellegrini et al., 2011, Chevance et al., 2016), and to measure the
deprojected distance of the Trapezium stars to the Orion bar (0.33-0.45 pc,
Salgado et al., 2016). From the physical distance, $d$, between [C II] and CO
emission peaks observed in a resolved edge-on PDR, and assuming an $A_{\rm
V}{\sim}2$ between the ${\rm C^{+}/C}$ and CO transitions, the density can be
estimated from $n=1.9\times 10^{21}A_{\rm V}/d=3.8\times 10^{21}/d$ ${\rm
cm^{-3}}$ where $d$ is measured in cm. Pabst et al. (2017) use this method to
find $n{\sim}3\times 10^{3}-4\times 10^{4}$ ${\rm cm^{-3}}$ in the Horsehead
PDR and surrounding region. Finally, the gas temperature can be inferred from
the peak [C II] line brightness and estimate of gas density, and from H2 pure
rotational lines. Typical gas temperatures found are between 100 and 500 K
(e.g. Young Owl et al., 2002, Pabst et al., 2021).

####  5.3.2 Diagnostics to distinguish X-rays from other heating mechanisms

Differences in the temperature and chemical abundance structure between PDRs
and XDRs (Sections 3 and 4) can be leveraged to infer the presence of an XDR,
despite the challenge – as we will discuss later in this Section – of
distinguishing X-ray from cosmic-ray or shock heating.

Figure 8: Normalized CO(1-0) ladder models from CLOUDY in the PDR (left
panels) and XDR regime (right panels) for $n=\rm 10^{2.9}$ and $10^{5.1}\,\rm
cm^{-3}$. The variation in $G_{0}$ and $F_{\rm X}$, respectively, are
indicated in the colorbar. The dashed line indicates a thermalized CO ladder.
Figure adapted with permission from Pensabene et al. (2021), ©ESO.

The [Si II] 35 $\mu$m/[C II] 158 $\mu$m, [O I] 63 $\mu$m/[C II] 158 $\mu$m,
and [Fe II] 26 $\mu$m/[C II] 158 $\mu$m, ratios are all larger in XDRs than in
PDRs (Meijerink et al., 2007). The same holds true – albeit with caveats on
similar results from an enhanced cosmic-ray rate (Bisbas et al., 2017, 2021) –
for [C I]/CO and [C I]/[C II]. These ratios are often used as diagnostics in
the circumnuclear disk of AGN (e.g Izumi et al., 2020) and in high-$z$ quasar
host galaxies (e.g. Venemans et al., 2017, see Section 9).

The whole CO ladder of AGN host sources can be another valuable tool to infer
the presence of an XDR component contributing to the gas heating (Bradford et
al., 2009, van der Werf et al., 2010, Hailey-Dunsheath et al., 2012, Pozzi et
al., 2017, Mingozzi et al., 2018, Valentino et al., 2021) as XDRs are
characterized by large column densities of warm molecular gas, where high-$J$
CO lines ($J\gtrsim 8$) can be efficiently excited. For this reason the CO
Spectral Line Energy Distribution (SLED) resulting from XDR models peaks at
increasingly higher $J$ for increasing X-ray flux (Vallini et al., 2019, and
see Figure 8), even though CO emission can be suppressed by X-ray induced CO
dissociation (Section 4.4) at intermediate column densities (Kawamuro et al.,
2020). As a caveat, note that mechanical heating from shocks can boost
high-$J$ CO lines either on galactic scales (e.g. Meijerink et al., 2013,
Bellocchi et al., 2020, as resulting from outflows and/or merger activity) or
in resolved molecular complexes (Lee et al., 2016). Kazandjian et al. (2015)
show that the temperature of the molecular gas can be significantly increased
by a small amount of mechanical heating from stellar feedback in the form of
stellar winds and supernovae, although we note that the mechanical energy
input may be fairly localized in the star-forming regions and may be
dissipated in neutral atomic or ionized gas (Lancaster et al., 2021a, b), thus
having much less an impact on the bulk of the molecular gas. While the
emission from low-$J$ CO lines only moderately increases with mechanical
heating, high-$J$ CO line emission can increase by several orders of magnitude
in clouds with $n{\sim}10^{5}$ ${\rm cm^{-3}}$ and a galactic star formation
rate of 1 M⊙ yr-1. For spatially resolved observations towards dense
($n\gtrsim 10^{5}$ ${\rm cm^{-3}}$) clumps in molecular clouds, the
differentiation between PDRs and XDRs based on the peak of the CO ladder
breaks down as high-density PDRs produce bright high-$J$ CO lines (Burton et
al., 1990, Joblin et al., 2018, Wu et al., 2018). Note however that on
galactic scales, high density PDR contributions to high-$J$ CO emission are
generally diluted by low filling factors (Indriolo et al., 2017).

The use of line ratios involving high density tracers (such as HCN and HCO+
lines) as XDR diagnostics can be valuable but it is still debated given the
non-trivial effects of gas density, temperature, opacity, and time-dependence
(see Fig.1 in Viti, 2017). XDR models (Meijerink et al., 2007) indicate that
HCN/HCO+ abundance ratios exceed that of PDRs only when $F_{X}\gtrsim
100\rm\,erg\,s^{-1}\,cm^{-2}$, and $n\approx 10^{4}\,$${\rm cm^{-3}}$. Below
these limits, the enhanced abundance of HCO+ in XDRs (see Section 4.4) drives
the ratio below that of dense PDRs. Note that HCN emission can be boosted by
IR pumping (Costagliola et al., 2011, Martín et al., 2015, Vollmer et al.,
2017, Harada et al., 2018) and electron impact excitation in XDRs (Goldsmith &
Kauffmann, 2017). A combination of HCN/HCO+ ratios with HNC/HCN represents
another set of valuable diagnostics, as outlined in Baan et al. (2008), Loenen
et al. (2008) and recently in Krieger et al. (2020) where such diagrams have
been exploited to study the heating mechanisms in star forming regions within
NGC 253 (Figure 9). In particular the higher HNC/HCN abundance ratios in XDRs
as compared to PDRs at large column densities (Baan et al., 2008, Loenen et
al., 2008, and discussion in Section 4.4) make HCN/HNC intensity ratios in
XDRs somewhat higher than those in PDRs. A caveat is that, as for HCN/HCO+,
IR-pumping (Aalto et al., 2007) and shock heating could produce similar
effects (Cañameras et al., 2021). Meijerink et al. (2007) and Harada et al.
(2013) also suggest CN/HCN as a potential XDR-vs-PDR diagnostic because the
column density ratios for PDRs and XDRs are very different, ranging from
40-1000 in the XDRs, to 0.5-2 in PDRs. Nevertheless, only relatively modest CN
enhancements have been reported in circumnuclear disks of three AGN (Ledger et
al., 2021).

Figure 9: HCN, HNC, HCO+ intensity ratios used to infer excitation mechanisms.
Points with error bars are super star clusters in NGC 253 and indicate that
PDR chemistry is favored over XDR chemistry. Figure adapted with permission
from Krieger et al. (2020), ©AAS.

Observations of CH/CO ratios in four nearby AGN host galaxies (Rangwala et
al., 2014) reveal values $\approx 10$ times higher than expected in normal
galaxies, thus suggesting that CH/CO could have some potential as an XDR
diagnostic. Rangwala et al. (2014) supported this conclusion by noting that
Meijerink et al. (2007) models return CH abundances that are significantly
enhanced relative to other molecules in XDRs. A discussion concerning the PDR
vs XDR impact on the abundances of hydrides (CH+, OH+, H2O+, CH) in the
context of young stellar objects is presented in Benz et al. (2016). By
studying CH+/OH+, OH+/H2O+, and C+/CH+ Benz et al. (2016) find no chemical
evidence for X-ray induced chemistry and put tight constrains on the
(negligible) contribution of an XDR component to the observed emission.

Distinguishing an XDR from an environment characterized by high cosmic-ray
rates is instead rather difficult, both in external galaxies where
observations encompass very extended gas patches (Meijerink et al., 2006), and
in the envelopes of young stellar objects (Bruderer et al., 2009b). The non-
thermal electrons resulting from cosmic-ray and X-ray ionizations interact in
a similar way with the atomic or molecular gas (Glassgold et al., 2012), thus
producing comparable heating and chemistry (e.g., high [C I]/CO and [C I]/[C
II] ratios; Bisbas et al. 2017, 2021). For this reason, enhanced cosmic-ray
flux is sometimes used to “mimic” XDR-like conditions (Bayet et al., 2011,
Viti, 2017). The X-ray chemistry can be reproduced by means of an enhanced
cosmic-ray ionization rate with deviations by less than 25% (Bruderer et al.,
2009b). Meijerink et al. (2011) suggested that OH+/OH and $\rm H_{2}O^{+}$/H2O
ratios might help in distinguishing very high cosmic-ray rates from the
typical XDR conditions at $\approx 150$ pc from an AGN of $L=10^{44}\,\rm
erg\,s^{-1}$. A cloud of density $n=10^{5.5}\rm\,cm^{-3}$ at such distance
from the galaxy center experiences an impinging $F_{X}\approx
29\rm\,erg\,s^{-1}\,cm^{-2}$, and is expected to have OH+/OH and $\rm
H_{2}O^{+}$/H2O column density ratios as high as $(250-7.5)\times 10^{-4}$ and
$(64-3)\times 10^{-3}$, compared to $(3-5)\times 10^{-4}$ and $(2-14)\times
10^{-4}$ for high CR rates $\zeta_{\rm H_{2}}\approx(10^{-14}-10^{-13})\,\rm
s^{-1}$ (for $N=3\times 10^{22}-10^{24}\,\rm cm^{-2}$). However no attempts
were made to model the resulting line emission. Vallini et al. (2019) found
that cosmic rays can boost the mid/high-$J$ end of the CO ladder but not at
the level observed in XDRs (see also Rangwala et al., 2011, for Arp 220).

The effect of mechanical heating needs to be considered before determining
physical parameters of the gas even though estimating the contribution of
stellar winds/supernovae to the mechanical heating in external galaxies is
challenging. Meijerink et al. (2013) suggest that shock dominated ISM yields a
much larger (up to $\approx 10$ times) CO-to-continuum ratio than XDRs because
shocks heat the gas directly, with little heating of dust, and hence do not
contribute appreciably to the IR luminosity (Pellegrini et al., 2013). Note
that shock velocities $v_{\rm sh}<30$ km/s are required to preserve the CO
from dissociation (Hollenbach & McKee, 1989). Fast ($v_{\rm sh}>50$ km/s)
shocks destroy the CO (although it can reform in post shock gas to some extent
before getting too cold) and also produce optical and FUV photons which heat
the grains, providing IR continuum.

###  5.4 Origin of [C II]

The ionisation potential of neutral carbon C (11.3 eV) is lower than that of
neutral hydrogen (13.6 eV). As a result, the [C II] line can in principle
originate both from the ionized and the neutral gas. In the Milky Way and in
nearby galaxies, most studies have concluded that the vast majority of the [C
II] originates from the neutral regions (e.g., Oberst et al., 2011, Pineda et
al., 2014, Goicoechea et al., 2015, Croxall et al., 2017, Pabst et al., 2017,
2021). A low contribution from the ionized gas to the [C II] emission is also
found at high-redshift, both observationally (e.g., Stacey et al., 2010,
Gullberg et al., 2015) and in simulations (e.g., Olsen et al., 2015, Katz et
al., 2017, Pallottini et al., 2017, Lagache et al., 2018). Typically,
observations find the ionized gas contributes ${\sim}5-30$% to the [C II]
emission.

The component of the neutral phase of the ISM that dominates the [C II]
emission is still a matter of debate. Early suggestions were that [C II] is
associated with molecular clouds (Stacey et al., 1985, Shibai et al., 1991) or
the diffuse ISM (Bennett et al., 1994). The dominant component must depend on
the observing scale and emission source (Mookerjea et al., 2016). Observations
concentrating on Galactic star-forming regions generally find the dense,
intensely illuminated gas, dominating the emission. Analysis of the line
emission from unresolved normal galaxies finds that [C II] arises from PDRs of
moderately high FUV fields $G_{0}=10^{2}-10^{4.5}$ and densities
$n=10^{2}-10^{4.5}$ ${\rm cm^{-3}}$ (Malhotra et al., 2001) with similar range
found in resolved ($0.2-2.1$ kpc) star forming regions in local galaxies
(Sutter et al., 2021) and in low-metallicity dwarf galaxies (Cormier et al.,
2019). These are usually interpreted as the bright classic PDRs. Extended
mapping of [C II] in Orion (Goicoechea et al., 2015, Pabst et al., 2017,
2021), and a detailed PDR analysis of extragalactic H II regions (Abdullah et
al., 2017) finds that the extended, moderately illuminated, molecular cloud
surfaces can contribute significantly or dominate the emission. An analysis of
the COBE [C II] emission in ${\sim}400$ pc beams from the Orion-Eridanus
superbubble (Abdullah & Tielens, 2020) finds that molecular cloud surfaces up
to ${\sim}10$ pc from the stellar clusters dominate the emission with a
moderate field strength of $G_{0}{\sim}100$ and density $n{\sim}10^{3}$ ${\rm
cm^{-3}}$. Because these projected beam sizes are comparable to those of the
Herschel Space Observatory observations in nearby galaxies, Abdullah & Tielens
(2020) suggest that the [C II] emission in these observations may be similarly
dominated by moderately illuminated extended cloud surfaces. Such moderate
field strengths are also found for the Galactic [C II] emission using the all-