11institutetext: Leiden Observatory, Leiden University, PO Box 9513, 2300 RA
Leiden, The Netherlands 22institutetext: Leiden Institute of Chemistry,
Gorlaeus Laboratories, Leiden University, Einsteinweg 55, 2333 CC Leiden, The
Netherlands 33institutetext: Max-Planck-Institut für Extraterrestrische
Physik, Giessenbachstrasse1, 85748 Garching, Germany 44institutetext:
Department of Space, Earth and Environment, Chalmers University of Technology,
Onsala Space Observatory, 43992, Onsala, Sweden

# OH mid-infrared emission as a diagnostic of H2O UV photodissociation

I. Model and application to the HH 211 shock

Benoît Tabone 11 Marc C. van Hemert 22 Ewine F. van Dishoeck 11 3 3 John H.
Black 44

###### Abstract

Context. Water is an important molecule in interstellar and circumstellar
environments. Previous observations of mid-infrared (IR) rotational lines of
OH toward star-forming regions suggest that OH emission may be used to probe
the photodissociation of water.

Aims. Our goal is to propose a method to quantify H2O photodissociation and
measure the local ultraviolet (UV) flux from observations of mid-infrared OH
lines.

Methods. Cross sections for the photodissociation of H2O resolving individual
electronic, vibrational, and rotational states of the OH fragment are
collected. The state distribution of nascent OH following H2O
photodissociation is computed for various astrophysically relevant UV
radiation fields (e.g., a single Ly$\alpha$ line or a broadband spectrum).
These distributions are incorporated in a new molecular excitation code called
GROSBETA, which includes radiative pumping, collisional (de)excitation, and
prompt emission (i.e., following the production of OH in excited states). The
influence of the photodissociation rate of H2O, the spectral shape of the UV
radiation field, the density, the temperature of the gas, and the strength of
the IR background radiation field on the integrated line intensities are
studied in detail. As a test case, our model is compared to Spitzer-IRS
observations at the tip of the HH 211 bow-shock.

Results. The OH rotational line intensities in the range
$9-16\leavevmode\nobreak\ \mu$m, covering rotational transitions with
$N_{\text{\text{up}}}=18$ to $45$, are proportional to the column density of
H2O photodissociated per second by photons in the range
$114-143\leavevmode\nobreak\ $nm (denoted as $\Phi^{\tilde{B}}$) and do not
depend on other local properties such as the IR radiation field, the density,
or the kinetic temperature. Provided an independent measurement of the column
density of water is available, the strength of the local UV radiation field
can be deduced with good accuracy, regardless of the exact shape of the UV
field. In contrast, OH lines at longer far-IR wavelengths are primarily
produced by IR radiative pumping and collisions, depending on the chemical
pumping rate defined as $\mathcal{D}^{\tilde{B}}=\Phi^{\tilde{B}}/N($OH$)$ and
on the local physical conditions ($n_{\rm{H}}$, $T_{\rm{K}}$, IR radiation
field). Our model successfully reproduces the OH mid-IR lines in the
$10-16\leavevmode\nobreak\ \mu$m range observed toward the tip of the HH 211
bow-shock and shows that the jet shock irradiates its surroundings, exposing
H2O to a UV photon flux that is about $5\times 10^{3}$ times larger than the
standard interstellar radiation field. We also find that chemical pumping by
the reaction H2 \+ O may supplement the excitation of lines in the range
$16-30\leavevmode\nobreak\ \mu$m, suggesting that these lines could also be
used to measure the two-body formation rates of OH.

Conclusions. The mid-IR lines of OH constitute a powerful diagnostic for
inferring the photodissociation rate of water and thus the UV field that water
is exposed to. Future JWST-MIRI observations will be able to map the
photodestruction rate of H2O in various dense ($n_{\rm{H}}$$\gtrsim 10^{6}$
cm-3) and irradiated environments and provide robust estimates of the local UV
radiation field.

###### Key Words.:

Stars: formation – molecular processes – radiative transfer – ISM:
astrochemistry – Individual: HH 211

##  1 Introduction

Oxygen is the third-most abundant element in the interstellar medium (ISM)
after hydrogen and helium. Water is an important oxygen-bearing molecule in
the formation of stars and planets.Water ice promotes the coagulation of dust
grains, from dense clouds to planet-forming disks (Chokshi et al., 1993; Testi
et al., 2014; Schoonenberg & Ormel, 2017), and water vapor is an important
coolant of the gas in warm molecular environments (Neufeld et al., 1995),
particularly in embedded protostellar systems (Karska et al., 2013).
Ultimately, the delivery of water to young planets and small bodies conditions
the emergence of life as we know it (Chyba & Hand, 2005). Following the trail
of water and its associated chemistry from clouds to young forming planets is
a major goal of astrochemistry (van Dishoeck et al., 2014).

Infrared (IR) observations, supplemented by chemical modeling, have
demonstrated that abundant ice is built at the surface of grains in cold dense
clouds, locking in $\sim 25\%$ of the oxygen (Cuppen & Herbst, 2007; Boogert
et al., 2015). En route to the protostellar disk, condensed H2O is released
into the gas phase by passive heating from the accreting protostar (Ceccarelli
et al., 1996) and jet-driven shocks (e.g., Flower & Pineau des Forêts, 2012).
Under these dense ($n_{\rm{H}}$$\gtrsim 10^{6}$ cm-3) and warm ($T\gtrsim 200$
K) conditions, oxygen chemistry is then driven by fast neutral-neutral
reactions and photodissociation. In this context, the hydroxyl radical (OH) is
a key reactive intermediate between atomic oxygen and water. The available
gas-phase atomic oxygen is converted into OH by the reaction O +
H${}_{2}\rightarrow$ OH + H, followed by the formation of water through OH +
H${}_{2}\rightarrow$ H2O + H. OH is destroyed by the latter reaction, by the
former reaction in the backward direction, and by UV photodissociation. Water
is also destroyed either by the backward reaction H2O + H $\rightarrow$ OH +
H2 or by photodissociation leading mostly to OH.

The observations of warm H2O and OH vapor toward embedded protostars
culminated with the Herschel Space Observatory (van Dishoeck et al., 2011),
which outperformed previous far-IR and submillimeter observatories (e.g.,
SWAS, Odin, ISO-LWS). The detection of far-IR OH lines in inner envelopes and
outflows (Wampfler et al., 2010, 2011, 2013; Goicoechea et al., 2015),
supplemented by similar detections in prototypical photodissociation regions
(Goicoechea et al., 2011; Parikka et al., 2017) and extragalactic sources
(Sturm et al., 2011; González-Alfonso et al., 2012) evidenced the presence of
an active warm oxygen chemistry in dense interstellar environments. One of the
most striking results is the relatively low abundance of gaseous water found
toward most of the protostellar sources, which contrasts with the abundant
water ice supplied by the infalling envelope. In particular, multiple outflow
components, which dominate H2O emission, exhibit abundances ranging from
$10^{-7}$ to $10^{-5}$ (Kristensen et al., 2012; Nisini et al., 2013;
Kristensen et al., 2017). Water vapor from envelopes and perhaps embedded
disks is typically traced by H${}_{2}^{17}$O and H${}_{2}^{18}$O lines and is
best observed by Herschel-HIFI and (sub)millimeter interferometers (NOEMA,
SMA, ALMA). While Visser et al. (2013) show that water abundance can be as
high as $10^{-4}$, most of the protostellar systems exhibit lower water
abundances, typically ranging between $10^{-7}$ and $10^{-5}$ (Jørgensen & van
Dishoeck, 2010; Persson et al., 2012, 2016; Harsono et al., 2020). Following
the dissipation of the envelope, near- and mid-IR observations from space
(e.g., Spitzer) and from the ground (e.g., VLT-CRIRES) have unveiled abundant
hot water vapor in the planet-forming regions of Class II disks ($\lesssim
10\leavevmode\nobreak\ $au, Pontoppidan et al., 2014) but surprisingly dry
Herbig disks (Pontoppidan et al., 2010; Fedele et al., 2011; Walsh et al.,
2015).

The solution to this conundrum of low water abundance may lie in the impact of
the ultraviolet (UV) radiation field produced by the accreting young star or
strong jet shocks ($V\gtrsim 40\leavevmode\nobreak\ $km s-1). Observations of
hydrides provide evidence of the driving role of UV irradiation in the
chemistry of protostars (Benz et al., 2016). It has been proposed that the low
abundance of H2O in protostellar outflows is the result of the UV
photodissociation of H2O vapor (Nisini et al., 2002, 2013; Karska et al.,
2014). The low water abundance in embedded protostellar disks is generally
attributed to the freeze-out of water (Visser et al., 2009). However, recent
constraints on the thermal structures of Class 0 disk-like structures may
contradict this interpretation (van ’t Hoff et al., 2020) and require the
efficient destruction of H2O or a large opacity of the dust in the far-IR.

The impact of UV photodissociation is generally explored by means of
astrochemical models (van Kempen et al., 2010; Visser et al., 2012; Panoglou
et al., 2012; Yvart et al., 2016). However, the strength of the local UV
radiation field incident on disks, envelopes, and outflows is poorly known,
due to the complex sources of UV emission and the unknown level of dust and
gas attenuation on the line of sight, severely limiting the diagnostic
capabilities. Assessing the exact role of UV photodissociation of water from
astrochemical models based only on molecular abundances remains highly
uncertain.

Alternatively, the excitation of molecular species provides in some cases a
direct access to their formation and destruction routes. In irradiated
regions, IR lines of H2 and CO from ro-vibrationally excited levels trace the
UV pumping followed by radiative decay (Krotkov et al., 1980; Black & van
Dishoeck, 1987) and have been used to probe the photodissociation of these
species (Röllig et al., 2007; Thi et al., 2013). The excitation of reactive
species for which the collisional time scale is comparable to the destruction
time scale, carry also key information on their formation and destruction
rates through a process called ”chemical pumping” (Black, 1998). So far, the
incorporation of a state-to-state chemistry in models (i.e., models
considering the state distribution of the reagents and the products) have been
carried out only for a handful of molecular ions relevant for diffuse ISM and
early universe (Godard & Cernicharo, 2013; Stäuber & Bruderer, 2009; Coppola
et al., 2013).

Mid-infrared observations with Spitzer-IRS toward protostellar outflows and
young disks evidenced the presence of a superthermal population of
rotationally excited OH with energies up to $E_{\text{up}}\simeq 28000$ K
(Tappe et al., 2008, 2012; Carr & Najita, 2014). Two decades of extensive
quantum calculations supplemented by laboratory experiments have permitted the
direct interpretation of these observations as the smoking gun of H2O
photodissociation. Specifically, seminal works have shown that H2O
photodissociation through the $\tilde{A}$ excited electronic state of H2O
($\lambda\gtrsim 143$ nm) produces OH in vibrationally hot but rotationally
cold states (e.g., Hwang et al., 1999a; Yang et al., 2000; van Harrevelt & van
Hemert, 2001), whereas photodissociation through the $\tilde{B}$ state
($114\lesssim\lambda\lesssim 143$ nm) produces OH in high rotational states
with levels up to $N\simeq 47$ ($E\simeq 45000$ K, Harich et al., 2000; van
Harrevelt & van Hemert, 2000). The rotationally excited lines of OH would thus
probe the radiative cascade of OH products following their formation via H2O
photodissociation, a process called ”prompt emission”. If consistently
modeled, rotational lines of OH can thus give a direct access to the
photodissociation of H2O. However, a detailed modeling that includes chemical
data accumulated over the past decades is lacking.

Probing H2O photodissocation in astrophysical environments may also lead to
unique constraints on the local UV irradiation field that H2O is exposed to.
The far-UV radiation field is a key parameter that controls the chemical,
physical, and dynamical evolution of circumstellar regions. It regulates the
overall chemistry of disk upper layers (e.g., van Dishoeck et al., 2006; Walsh
et al., 2012) and outflows (Panoglou et al., 2012; Tabone et al., 2020), the
thermal structure of disks (Gorti & Hollenbach, 2004; Bruderer et al., 2012;
Woitke et al., 2016), and the coupling between the gas and the magnetic field
(e.g., Gammie, 1996; Wang et al., 2019). The impact of the UV radiation field
depends not only on attenuation processes, but also on its spectral shape.
While the interstellar radiation field exhibits a broadband emission down to
91 nm (Habing, 1968; Draine, 1978; Mathis et al., 1983), the UV radiation
emitted by accreting nascent stars (Bergin et al., 2003; Herczeg et al., 2004;
Yang et al., 2012; Schindhelm et al., 2012) or by strong jet shocks ($\gtrsim
40$ km s,-1 Raymond, 1979; Dopita & Sutherland, 2017) is often dominated by
Lyman-$\alpha$ emission, leading to the selective photodissociation of species
that exhibit large photodissociation cross sections at
$121.6\leavevmode\nobreak\ $nm.

To date, few robust diagnostics have been proposed to directly access the
local UV radiation field. H2 and CO lines excited by UV pumping probe only UV
photons in narrow lines at $<114\leavevmode\nobreak\ $nm that are rapidly
attenuated and are not indicative of the broadband UV spectrum relevant for
the other UV photoprocesses. Chemical diagnostics have also been proposed but
they rely either on H2 UV pumping (e.g., CN formed from excited H2, Cazzoletti
et al., 2018) or depend on elemental abundance ratios (e.g., hydrocarbons,
Bergin et al., 2016).

In this work, we explore the potential of the OH mid-IR lines for probing H2O
photodissociation and the local UV radiation field under a broad range of
physical conditions representative of dense irradiated environments, such as
molecular shocks, circumstellar media, and prototypical photodissociation
regions (PDRs). To reach this goal, results of quantum mechanical calculations
resolving the electronic, vibrational, and rotational states of the OH product
following H2O dissociation at different UV wavelengths are collected. The
distribution of the OH fragments following H2O photodissociation by UV fields
of various spectral shapes are derived. The emerging line intensities are then
calculated using GROSBETA, a new molecular excitation code that includes
prompt emission, radiative decay, collisional (de)excitation, and IR radiative
pumping in a slab approach. For sake of conciseness, ro-vibrational lines of
OH are not studied in this work.

In Sect. 2, we present the OH model and the basics of GROSBETA. The
competition of the different excitation processes on the OH line intensities
are studied in detail for H2O photodissociation at Ly$\alpha$
($\lambda=121.6\leavevmode\nobreak\ $nm) and extended to other UV radiation
fields in Sect. 3. In Sec. 4, a method to observationally derive the amount of
H2O photodissociated per unit time, the photodissociation rate of H2O, and the
strength of the local UV radiation field is proposed and applied to Spitzer-
IRS observations of the tip of the HH 211 protostellar bow-shock. Our findings
are summarized in Sect. 5.

##  2 Model

###  2.1 OH model

####  2.1.1 Energy levels

Figure 1: OH model adopted in this work. a) $X^{2}\Pi$ and $A^{2}\Sigma^{+}$
electronic levels split into vibrational levels and further split into
rotational levels labeled by $N$ (left ladder corresponding the
OH(X${}^{2}\Pi$)($\varv=0$) state). The red line indicates the dissociation
energy of OH(X${}^{2}\Pi$). b) Structure of the rotational ladders of
OH(X${}^{2}\Pi$) within a vibrational state that gives rise to mid- and far-IR
lines. Each $N$ level is split by the spin-orbit coupling and the
$\Lambda$-doubling. The two spin-orbit states are labeled by the $\Omega$
quantum number and the $\Lambda$-doubling states are labeled by their
$\epsilon=e/f$ spectroscopic parity. Radiative transitions included in our
model and emerging from the four $N$-levels are also depicted by arrows. There
are three kinds of transitions: intra-ladder rotational transitions (blue and
red arrows), cross-ladder transitions (orange, with $\Delta N=+1$, and green,
with $\Delta N=0,2$), and $\Lambda$-doubling transitions (purple). c)
Optically thin LTE spectrum of OH at $T_{K}=750$ K. The color code is the same
as panel b) and is repeated in Fig. 4 and 9. The upper $N$ level is indicated
for some transitions. The $\Lambda$-doubling lines are too weak and at longer
radio wavelengths to appear in this spectrum. Ro-vibrational lines are at
shorter wavelengths than shown here ($\lambda\lesssim 2.7\leavevmode\nobreak\
\mu$m).

H2O photodissociation can produce OH with high rotational and vibrational
quantum numbers in the $X^{2}\Pi$ ground and the $A^{2}\Sigma^{+}$ first
excited electronic states. Our OH model includes the energy levels provided by
Brooke et al. (2016) and Yousefi et al. (2018) and made available on the
EXOMOL database111http://www.exomol.com/, Tennyson et al. (2016). Figure 1-a
shows the electronic, vibrational, and rotational levels included in the
model. Regarding the ground electronic state OH($X^{2}\Pi$), vibrational
levels up to $\varv=13$ were included. We retained rotational levels that are
stable against dissociation and in particular included energy levels above the
dissociation energy of the OH($X$) that are stabilized by the centrifugal
barrier (Chang et al., 2019). This corresponds to a quantum number of $N=54$
in the $\varv=0$ state, where $N$ denotes the rotational quantum number
associated with the motion of the nuclei. Each rotational level is further
split into two spin-orbit manifolds corresponding to projected quantum numbers
of the sum of the electronic orbital and spin angular momenta of $\Omega=3/2$
and $\Omega=1/2$ (see Fig. 1-b). This results in two rotational ladders with
fine-structure levels characterized by a total angular momentum quantum number
of $J=N+1/2$ and $J=N-1/2$, respectively. Lastly, each rotational level
($N,\Omega$) is further split by the $\Lambda$-doubling into two levels
labeled by their spectroscopic parity index222The spectroscopic parity index
is related to the parity $p=+/-$ by the relation $\epsilon=p(-1)^{J-1/2}$,
where $\epsilon=1$ for $e$ states and $\epsilon=-1$ for $f$ states. $e/f$. Our
model includes the fine structure and $\Lambda$-splitting of the
OH($X^{2}\Pi$)($\varv,N$) states. Hyperfine structure is not considered in
this work.

Regarding the OH($A^{2}\Sigma^{+}$) state, levels with $\varv\geq 2$ are
dissociative (van Dishoeck & Dalgarno, 1983) and the $\varv=0$ with $N\geq 26$
and the $\varv=1$ with $N\geq 17$ levels are predissociated (Yarkony, 1992).
Therefore, we limited the OH($A$) levels to lower $N$ of the $\varv=0,1$
states. The two rotational ladders emerging from the spin-orbit coupling of
the OH($A^{2}\Sigma^{+}$)($\varv$) states were also taken into account. In the
following, the electronic, vibrational, rotational, fine-structure, and parity
states of OH are labeled as $\Lambda$, $\varv$, $N$, $\Omega$, and $\epsilon$,
respectively.

####  2.1.2 Radiative transitions

Our OH model includes 54276 radiative transitions with their corresponding
Einstein-$A$ coefficients from Brooke et al. (2016) and Yousefi et al. (2018).
The mid- and far-IR lines of OH originate from radiative transitions occurring
within vibrational states ($\Delta\varv=0$) of the OH($X$) state (Fig. 1-c).
Figure 1-b shows the radiative transitions that connect levels within a
vibrational level of the $X^{2}\Pi$ ground state. The radiative transitions
with the highest Einstein-$A$ coefficients are the intra-ladder rotational
transitions $N\rightarrow N-1$ (or equivalently $J\rightarrow J-1$) that
preserve the e/f parity. These transitions give rise to lines from the mid- to
the far-IR (see Fig. 1-c). The cross-ladder transitions connecting the
$\Omega=1/2$ and $\Omega=3/2$ states are of two kinds: $J$-conserving ($\Delta
N=-1$) and e/f conserving transitions, and $J\rightarrow J\pm 1$ ($\Delta
N=0,-2$) and e/f parity changing transitions. Einstein-$A$ coefficients of the
latter two kinds are at least an order of magnitude lower than intra-ladder
transitions and can be used to measure the opacity of the intra-ladder lines.
Lastly, the $\Lambda$-doubling transitions, that lie from centimeter to sub-
millimeter wavelengths are also considered in this work. Rovibrational
transitions follow the same selection rules and any rovibrational transition
accompanied by any change of vibrational quantum number
$\varv^{\prime}\rightarrow\varv^{\prime\prime}$ is included in this work.
Rovibrational lines $\varv^{\prime}=1\rightarrow\varv^{\prime\prime}=0$
typically lie at near-IR wavelengths, around 2.7 $\mu$m. Transitions
connecting to the OH($A^{2}\Sigma^{+}$) state are electronic dipole allowed at
near UV wavelengths.

###  2.2 Excitation by H2O photodissociation

Figure 2: State distribution of the OH product following H2O photodissociation
at two photon wavelengths. a) At $\lambda=166$ nm, H2O dissociating via its
first excited $\tilde{A}$ electronic state and producing OH in rotationally
cold but vibrationally hot states. b) At $\lambda=121.6$ nm, H2O dissociating
via its second excited $\tilde{B}$ electronic state and producing OH in
rotationally hot but vibrationally cold states. Vibrational quantum numbers
are color coded as indicated in the top panel. The distributions within
OH(X${}^{2}\Pi,\varv$) and OH(A${}^{2}\Sigma^{+},\varv$) are plotted in solid
and dashed lines, respectively, as a function of the rotational quantum number
$N$. Only vibrational levels that contribute at least 0.1$\%$ to the
population of nascent OH are shown.

The OH state-specific formation rate associated with the photoprocess

$\text{H}_{2}\text{O}+h\nu\rightarrow\rm{OH}(\lambda,\Lambda,\varv,N,\Omega,\epsilon)+\rm{H}$
(1)

can be expressed as

$k(\Lambda,\varv,N,\Omega,\epsilon)=\int_{\lambda}\eta(\lambda,\Lambda,\varv,N,\Omega,\epsilon)\bar{\sigma}(\lambda)I(\lambda)d\lambda$
(2)

and measured in cgs units in s-1. In this equation,
$\eta(\lambda,\Lambda,\varv,N,\Omega,\epsilon)$ is the probability of forming
OH in a given state following H2O photodissociation by a photon of wavelength
$\lambda$, $\bar{\sigma}$ is the total photodissociation cross section ([cm2])
of H2O $\rightarrow$ OH + H at that wavelength, and $I(\lambda)$ is the photon
flux averaged over all incidence angles ([photon cm-2 s-1 cm-1]).

We assumed that the fine structure and $\Lambda$-doubling states of OH are
equally populated by the photoprocess (1) so that $\eta$ does not depend on
$\Omega$ and $\epsilon$. It is also convenient to define the nascent state
distribution of the OH fragments as

$f(\Lambda,\varv,N)=\frac{k(\Lambda,\varv,N)}{\bar{k}},$ (3)

where $\bar{k}$ is the total rate for the photdissociation process (1). For a
monochromatic UV radiation field emitting at a wavelength $\lambda$,
$f_{i}=\eta(\lambda,i)$.

The absorption of a UV photon in the range $143$ to $190$ nm excites H2O in
its first excited singlet state $\tilde{A}$ (first absorption band) and leads
to a direct dissociation to OH + H. In this work, we adopted the OH state-
specific cross sections computed by van Harrevelt & van Hemert (2001) using a
wave packet approach. These data successfully reproduce the total
photodissociation cross section of water in the considered energy range as
well as the OH distributions measured by time-of-flight spectroscopy
techniques (Yang et al., 2000). Figure 2-a shows the distribution of the OH
fragments following photodissociation by a photon of wavelength $\lambda=166$
nm, where the cross section of the first absorption band peaks. Throughout the
first absorption band, OH is produced in a rotationally cold ($N\lesssim 9$)
but vibrationally hot state (see also Appendix A). The proportion of
vibrationally excited states increases with photon energy.

The absorption of a UV photon in the second absorption band ($\tilde{B}$ state
of H2O) results in a nonadiabatic transition, leading to OH in its ground
electronic state and in a direct dissociation forming electronically excited
OH. The second absorption band produces a broad continuum bump in the
photodissociation cross section of H2O in the range $114$ to $143$ nm.
However, shortward of $124.6$ nm, this dissociation channel coexists with the
third and fourth absorption bands that result in sharper peaks in the
photodissociation cross section (see Fig. 15). As argued by van Harrevelt &
van Hemert (2008), the $\tilde{C}$ and $\tilde{D}$ states are bound states and
predissociated by the $\tilde{B}$ state. We consequently assumed that these
channels lead to the same state distribution of the OH fragments as that via
the $\tilde{B}$ state at the same photon energy. This is further supported by
the agreement between experiments at $\lambda\leq 124$ nm (e.g., Ly$\alpha$,
Harich et al., 2000) and theoretical work considering only the fragmentation
dynamics from the $\tilde{B}$ state (van Harrevelt & van Hemert, 2000, 2008).
In the following, we use the term ”photodissociation through the $\tilde{B}$
state” to refer to photodissociation in the range $114$ to
$143\leavevmode\nobreak\ $nm. In order to obtain the relevant OH
distributions, we repeated the quantum wave packet calculations as described
in van Harrevelt & van Hemert (2000), but using the Dobbyn and Knowles
potential (Dobbyn & Knowles, 1997) instead of the Leiden potential since it
was found that it gave better agreement with experiments (Fillion et al.,
2001). In this wavelength range, H2O photodissociation also leads to O with a
branching ratio computed by van Harrevelt & van Hemert (2008). We used the
total photodissociation cross section measured by Mota et al. (2005) that
includes features due to the $\tilde{C}$ and $\tilde{D}$ states of H2O (see
Fig. 15). The photodissociation cross section of H2O $\rightarrow$ OH + H was
then derived by taking into account the branching ratio of the OH forming
channel as described by Heays et al. (2017).

As an example, we show in Fig. 2-b the distibution of OH products at
Ly$\alpha$ wavelength ($\lambda=121.6$ nm). In contrast to photodissociation
by lower energy photons, photodissociation in the range $114$ to $143$ nm
produces OH($X^{2}\Pi$) with low vibrational excitation but high rotational
excitation. OH is produced preferentially in the ground vibrational and
electronic state, and the resulting rotational distribution within this state
peaks around $N=45$, corresponding to an energy level of $\simeq 45000$ K. As
shown in Appendix A (see Fig. 13), the peak in the rotational distribution
increases with the photon energy from $N\simeq 35$ up to $N\simeq 49$. The
other vibrational states follow a similar rotational distribution as that of
the $\varv=0$ state, but their relative contribution is much smaller, with
less than 5$\%$ of OH produced in each of the OH$(X)(\varv\geq 1)$ states at
Ly$\alpha$. A fraction of OH is also produced in the excited electronic state
OH($A$) for $\lambda<137\leavevmode\nobreak\ $nm. The rotational distributions
of the $\varv=0,1$ states exhibit the same pattern as that of OH($X$) but
shifted toward lower $N$ numbers.

Shortward of $\lambda=114$ nm, photodissociation proceeds via even more
excited electronic states of H2O and systematic quantum calculations are
lacking. Time-of-flight spectroscopy indicates that at $\lambda=115$ nm OH
distributions are vibrationally hotter than those computed through the
$\tilde{B}$ but still rotationally hot (Chang et al., 2019). Photodissociation
shortward of $\lambda=114$ nm may also produce preferentially atomic oxygen
instead of OH. In this work, we neglected the contribution of OH produced
through photodissociation at wavelengths shorter than 114 nm.

Our adopted chemical dataset pertains only to water in the rotational ground
state (”cold water”). Experimental data suggest that photodissociation of
”warm water” could result in a small change in the rotational distribution of
OH in the range $N\simeq 32-40$ (Hwang et al., 1999b). This change would
however affect only lines coming from $N\gtrsim 32$ in the
$9-10.5\leavevmode\nobreak\ \mu$m range (see Sec. 3). Additionally, the
adopted dataset does not include the fine-structure and the $\Lambda$-doubling
states of the OH products. Recent theoretical investigations suggest that
photodissociation through the $\tilde{B}$ state produces OH preferentially in
the $A^{\prime}$ symmetric states, that corresponds to the ($\Omega=1/2,e$)
and ($\Omega=3/2,f$) states (Zhou et al., 2015). This asymmetry is also
expected to depend on the rotational state of the parent H2O, which is not
considered here. New quantum calculations resolving the fine-structure and the
$\Lambda$-doubling state and including the rotational state of the parent H2O
are needed to study in detail the $A^{\prime}/A^{\prime\prime}$ asymmetry. Our
predictions presented in Sec. 3 remain valid by summing the line intensities
over the fine-structure and $\Lambda$ doublets of each $N_{\rm{up}}$ line.

###  2.3 OH excitation

When radiative processes dominate, the emitted spectrum of OH will be governed
by the state-specific dissociation rates of H2O (see Eq. (2)) and by the
spontaneous transition probabilities in the subsequent cascade of vibronic and
rotational transitions. This is called the prompt spectrum of OH. In
astrophysically relevant conditions, excited states of OH can be populated in
steady-state by various other collisional and radiative processes, and these
excited molecules also contribute to an observable spectrum. Thus the true
population of excited OH differs from the nascent population.

In this work, the OH level populations and the line intensities were computed
with a new code named GROSBETA (Black et al. in prep.). This code is based on
a single-zone model, following the formalism presented in van der Tak et al.
(2007) and implemented in the RADEX
333https://home.strw.leidenuniv.nl/~moldata/radex.html code. Under the
assumption of statistical equilibrium, the local population densities, denoted
$n_{i}$ $[\text{cm}^{-3}]$, are given by

$\frac{dn_{i}}{dt}=\sum_{j\neq i}P_{ji}n_{j}-n_{i}\sum_{j\neq
i}P_{ij}+\sum_{k}F_{i}^{k}-\sum_{k^{\prime}}n_{i}\mathcal{D}_{i}^{k^{\prime}}=0,$
(4)

where $F_{i}^{k}$ and $n_{i}\mathcal{D}_{i}^{k^{\prime}}$ are the formation
and destruction rates ($[\text{cm}^{-3}\text{s}^{-1}]$) for level
$i=(\Lambda,\varv,N,\Omega,\epsilon)$ and are associated with the chemical
reactions labeled $k$ and $k^{\prime}$, respectively. The $P_{ij}$ are the
radiative and collisional transition probabilities $i\rightarrow j$ given by

$P_{ij}=\cases{A}_{ij}+B_{ij}\bar{J}_{\nu_{ij}}+C_{ij}&(E_{i}>E_{j})\\\
B_{ij}\bar{J}_{\nu_{ij}}+C_{ij}(E_{i}<E_{j}).{}$ (5)

Here, $A_{ij}$ and $B_{ij}$ are the Einstein coefficients of spontaneous and
induced emission, $C_{ij}$ are the collisional rates [s-1], and
$\bar{J}_{\nu_{ij}}$ is the mean specific intensity at the frequency of the
radiative transition $i\rightarrow j$ averaged over the line profile. As in
van der Tak et al. (2007), the contribution of the lines to the local
radiation field was computed following the escape probability method for a
uniform sphere setting the line width to $\Delta V=2$ km s-1. When all other
physical parameters (e.g., density and temperature) are fixed, the solution
depends only on the value of the ratio $N($OH$)/\Delta V$.

The collisional (de)excitation rates in Eq. (5) were computed considering
collision with H2 and He for the OH levels within the $\varv=0$ state using
the rate coefficients from Offer et al. (1994) and Kłos et al. (2007),
respectively. The rate coefficients computed by Offer et al. (1994) that
include levels up to $N=5$ have been extrapolated to higher $N$ numbers (see
Appendix B).

The induced radiative rates in Eq. (5) depend upon mean intensities
$\bar{J}_{\nu}$ at the frequencies of all OH transitions. Thus the excitation
model requires a specified ambient radiation field. Pure rotational and
vibration-rotation transitions involve infrared radiation, which we
parametrized as a blackbody intensity (Planck function) at a temperature
$T_{\rm{IR}}$ times a geometrical dilution factor $W$. For illustration, we
took $W=1$. This gives a simple parametrization of the IR radiation field and
a reasonably good proxy for the local radiation field in the far-IR regime
relevant for the IR radiative pumping of pure rotational lines of the
OH($X^{2}\Pi$)($\varv=0$) state. The impact of a more complex and realistic IR
radiation field on the line intensities is briefly discussed in Sec. 3.1.2.
Electronic transitions $A-X$ respond to the near-ultraviolet. In this work, we
neglected the impact of the near-UV radiative pumping.

The impact of formation and destruction of OH on its level population is
described by the last two terms in Eq. (4). When activation barriers can be
overcome ($T_{\rm{K}}\gtrsim 200$ K), OH is primarily produced by the reaction
of O atoms with H2 and is rapidly converted into H2O by the reaction with H2.
In turn, OH can also be regenerated from H2O by UV photodissociation or by the
reaction with atomic hydrogen. Photodissociation of OH, which is generally
slower than H2O photodissociation (depending on the spectral shape of the UV
radiation field, see Heays et al., 2017), can also limit the abundance of OH.
The formation and destruction routes are described by the state specific rates
$F_{i}^{k}$ and $\mathcal{D}_{i}^{k^{\prime}}$, respectively. Their relative
importance depends on the exact physical conditions. A complete model coupling
chemistry and excitation of OH is beyond the scope of the present paper and
for the sake of generality and conciseness, we made a number of simplifying
assumptions to focus on the impact of H2O photodissociation on OH excitation
in various environments with a limited number of free parameters. The possible
impact of O+H2 on the excitation of OH is discussed in Sec. 4.2.1. In this
work, we assumed that chemical steady-state holds, so that

$F\equiv\sum_{k}\sum_{i}F_{i}^{k}=\sum_{i}n_{i}\mathcal{D}_{i},$ (6)

where $F$ is the total formation rate of OH ([cm-3 s-1]), and
$\mathcal{D}_{i}\equiv\sum_{k}\mathcal{D}_{i}^{k}$ the total destruction rate
of level $i$ ([s-1]). We also neglected any state selective destruction of OH
assuming that OH is destroyed at the same rate for each state denoted here as
$\mathcal{D}$ (i.e., $\mathcal{D}_{i}$ does not depend on
$i$)444$\mathcal{D}_{i}$ is the probability of OH in a given state to be
destroyed [in s-1]. Consequently, this assumption leads to
$\mathcal{D}=\mathcal{D}_{i}$ and not $\mathcal{D}_{i}=\mathcal{D}/n$, with
$n$ the number of OH levels included in the model, as stated in Stäuber &
Bruderer (2009).. Finally, we assumed that only H2O photodissociation leading
to OH modifies the level population of OH, with a state specific rate of
$F_{i}=n($H2O$)k(i)$, where $n$(H2O) is the number density of H2O and $k(i)$
the state specific formation rate defined in Eq. (2).

Equation (4) can then be rewritten as an equation on the column density of OH
in the levels $i$ denoted as $N_{i}$, upon which line intensities depend.
Assuming constant concentrations and excitation conditions along the line of
sight, this yields

$\sum_{j\neq i}P_{ji}N_{j}-N_{i}\sum_{j\neq
i}P_{ij}+\Phi\times\left(f_{i}-\frac{N_{i}}{N(\rm{OH})}\right)=0,$ (7)

with $f_{i}$ the state distribution of the OH fragments as defined in Eq. (3)
and

$\Phi\equiv N(\text{H}_{2}\text{O})\leavevmode\nobreak\ \bar{k}$ (8)

the column density of H2O photodissociated per unit time ([cm-2 s-1]). The
state distribution $f_{i}$ depends on the spectral shape of the UV radiation
field. In this work, we explore UV radiation fields of various shape that are
given in Fig. 15, including narrow-band and broadband UV spectra.

Equation (7) shows that $\Phi$ is the relevant parameter that ultimately
controls the impact of prompt emission on the OH line intensities. The
relation between the strength of the UV radiation field and $\Phi$ depends on
its spectral shape. For a Ly$\alpha$ UV radiation field

$\Phi=1.8\times
10^{4}G_{0}\frac{N(\rm{H}_{2}\rm{O})}{10^{13}\rm{cm}^{2}}\leavevmode\nobreak\
\rm{cm}^{-2}\leavevmode\nobreak\ \rm{s}^{-1},$ (9)

and for a Draine ISRF,

$\Phi=5.9\times
10^{3}G_{0}\frac{N(\rm{H}_{2}\rm{O})}{10^{13}\rm{cm}^{2}}\leavevmode\nobreak\
\rm{cm}^{-2}\leavevmode\nobreak\ \rm{s}^{-1},$ (10)

where $G_{0}$ is the intensity integrated between 91 and 200 nm, in units of
the Draine (1978) radiation field ($2.6\times 10^{-6}$ W m-2). In this work we
adopt $\Phi=10^{7}\leavevmode\nobreak\ \rm{cm}^{-2}\leavevmode\nobreak\
\rm{s}^{-1}$ as a fiducial value and explore a broad range of values (see
Table 1).

$\Phi$ can also be written as

$\Phi\equiv\mathcal{D}N(\text{OH}),$ (11)

where $\mathcal{D}$ ([s-1]) is generally referred to as a destruction rate
(see e.g., van der Tak et al., 2007; Stäuber & Bruderer, 2009). Since chemical
steady-state is assumed, $\mathcal{D}$ is also connected to the formation rate
of OH. If other chemical reactions participate to the formation and
destruction of OH (see Eq. (4)) and if these additional reactions form and
destroy OH in proportion to their local population densities, then Eq. (7)
still holds but $\mathcal{D}$ would not be associated with a destruction rate.
$\mathcal{D}$ should rather be associated with a chemical pumping rate equal
to the formation rate of the species by the considered process (in cm-3 s-1)
divided by its particle number density.

The line intensities were computed taking into account optical depth effects
under the large velocity approximation. When calculating the line intensities,
we assumed that the IR continuum background interacting with the gas is not
along the same line of sight as the observations are taken. That is, the IR
field contributes to the radiative pumping but not to the line formation.
Throughout this work, we present line intensities $I$ in erg s-1 cm-2, which
are sometimes called emergent fluxes. That is, our $I$ is related to the
specific intensity in a line $I_{\nu}$ in erg s-1 cm-2 Hz-1 sr-1 by
$I=4\pi\int I_{\nu}d\nu$. One can recover the observed line flux by the
relation:

$F=\frac{I}{4\pi}\Delta\Omega,$ (12)

where $\Delta\Omega$ is the solid angle subtended by the source.

In other words, the populations and the line intensities of OH were computed
from statistical equilibrium calculations involving prompt emission, IR
radiative pumping and collisional excitation. Our model is controlled by six
parameters summarized in Table 1: the spectral shape of the UV radiation field
that determines the distribution of nascent OH denoted as $f_{i}$, the column
density of H2O photodissociated per unit time denoted as $\Phi$, the column
density of OH $N($OH$)$, the proton density $n_{\text{H}}$, the kinetic
temperature $T_{\rm{K}}$, and the temperature of the IR radiation field
$T_{\rm{IR}}$.

Table 1: Parameters of the model and their fiducial values.

Parameter | Units | Range | Fiducial
---|---|---|---
Shape of the UV field | . | see text | Ly$\alpha$
$\Phi\equiv N(\text{H}_{2}\text{O})k$ | cm-2 s-1 |  $10^{4}$-$10^{10}$ | $10^{7}$
$N(\text{OH})$ | cm-2 | $10^{12}-10^{16}$ | $10^{14}$
$n_{\text{H}}$ | cm-3 |  $10^{5}$-$10^{13}$ | $10^{7}$
$T_{\text{K}}$ | K | $-$ | 500
$T_{\text{IR}}$ | K | 50, 120 | 120
$\mathcal{D}^{(1)}$ | s-1 |  $10^{-12}$-$10^{-2}$ | $10^{-7}$
555(1) the chemical pumping rate is not considered as a free parameter here
since $\mathcal{D}\equiv\Phi/N($OH$)$.

##  3 Results

Figure 3 illustrates the difference between OH spectra following H2O
photodissociation through the $\tilde{A}$ and $\tilde{B}$ state.
Photodissociation via the H2O $\tilde{A}$ state leads to an OH spectrum
dominated by far-IR lines coming from low rotational levels. H2O
photodissociation does not impact the mid- and far-IR spectrum since only
collisions and IR pumping contribute to the excitation of those lines. In
contrast, photodissociation through the H2O $\tilde{B}$ state produces
additional lines lying in the mid-IR coming from high-$N$ states ($15\lesssim
N\lesssim 45$). These lines are produced by H2O photodissociation. In this
section, we shall identify the physical quantities that can be retrieved from
the intensities of the mid- and far-IR lines. To do so, an in-depth study of
the excitation mechanisms is provided in the case of photodissociation by
Ly$\alpha$ photons (121.6 nm). The impact of a broadband UV radiation field,
for which photodissociation proceeds through both the H2O $\tilde{A}$ and
$\tilde{B}$ states is then studied.

Figure 3: Infrared spectra of OH following H2O photodissociation at two photon
wavelengths computed with GROSBETA. Top: At $\lambda=166$ nm, H2O dissociating
via its first excited $\tilde{A}$ electronic state and producing rotationally
cold OH. The OH infrared spectrum is dominated by far-IR lines excited by
collisions and IR radiative pumping. Bottom: At $\lambda=121.6$ nm, H2O
dissociating via its second excited $\tilde{B}$ electronic state and producing
rotationally hot OH. OH lines in the mid-IR with $N_{\text{up}}=20-45$ trace
H2O photodissociation through this electronic state. The density, temperature
of the background radiation and kinetic temperature are fixed to their
fiducial values (see Table 1) and $\Phi=10^{10}$
cm${}^{-2}\leavevmode\nobreak\ $s-1. In this figure, all lines are colored
red, even if coming from cross-ladder and intra-ladder transitions.

###  3.1 Lyman alpha

Figure 4: Infrared spectra of OH depending on the column density of H2O photodissociated per unit time $\Phi$ and on $N($OH$)$ computed with GROSBETA for a Ly$\alpha$ UV radiation field. The density, temperature of the background radiation and kinetic temperature are fixed to their fiducial values (see Table 1). Following the color code used in Fig. 1, intra-ladder rotational lines are in red and blue and cross-ladder lines are in orange and green. $\Lambda$-doubling lines are too weak to appear here. The weak mid-IR lines lying in the range $\sim$ 10-13 $\mu$m are intra-ladder rotational lines within the OH($X$)($\varv=1$) state. The $N_{up}$ rotational number of the upper energy level is indicated for selected lines. Table 2: Rotational lines used as a template for the different excitation regimes. Transition | $\lambda_{ij}$ | $E_{i}$ | $A_{ij}$
---|---|---|---
${}^{2}\Pi_{\Omega^{\prime}}(N^{\prime},\epsilon^{\prime})\rightarrow$ ${}^{2}\Pi_{\Omega^{\prime\prime}}(N^{\prime\prime},\epsilon^{\prime\prime})$ | ($\mu$m) | (K) | (s-1)
${}^{2}\Pi_{1/2}(30,f)\rightarrow$ ${}^{2}\Pi_{1/2}(29,f)$ | 10.8 | 22600 | 3.5(2)
${}^{2}\Pi_{1/2}(5,f)\rightarrow$ ${}^{2}\Pi_{3/2}(3,f)$ | 24.6 | 875 | 3.8(-2)
${}^{2}\Pi_{3/2}(10,f)\rightarrow$ ${}^{2}\Pi_{3/2}(9,f)$ | 27.4 | 2905 | 2.0(1)
Figure 5: OH line intensities and the associated excitation processes as a
function of $\Phi$ and $n_{\rm{H}}$ for various values of $N($OH$)$ and
$T_{\rm{IR}}$. The color indicates the value of $N($OH$)$ and the line style
the value of $T_{\rm{IR}}$ as defined in the right panels. The black circle
indicate the fiducial model. The processes that dominate the excitation of the
lines depending on the explored parameters are indicated along each curve.
Left: Intra-ladder rotational line at 10.8 $\mu$m coming from a high-$N$ level
($N=30$, $E_{\text{up}}=22600$ K). This line depends only on $\Phi$. Right:
Cross-ladder rotational line at 24.6 $\mu$m coming from a low-$N$ level
($N=6$, $E_{\text{up}}=875$ K). This line traces the bulk population of OH. As
such, it depends on $N($OH$)$, $T_{\rm{IR}}$ and $n_{\rm{H}}$ but does not
depend on $\Phi$.

As seen in Sect. 2.2, photodissociation of H2O by Ly$\alpha$ photons produces
OH in high rotational states (Fig. 2-b). Figure 4 illustrates the influence of
the column density of H2O photodissociated per unit time denoted as $\Phi$ and
of the column density of OH, $N($OH$)$. The results of a more systematic
exploration of the parameter space are presented in Fig. 5 by focusing on the
intensities of three representative rotational lines that will be observed by
JWST-MIRI (see Table 2).

####  3.1.1 High-$N$ lines: Prompt emission

Figure 6: Normalized photon intensity of the OH($X^{2}\Pi$)($\varv=0$) intra-
ladder mid-IR lines as a function of the quantum number of the upper energy
level $N_{\text{up}}$. The photon intensities are computed for the fiducial
values of the parameters and divided by $\Phi$ (see Eq. (13)). Circle and
triangle markers correspond to lines belonging to the $\Omega=1/2$ and
$\Omega=3/2$ ladders, respectively. $\Lambda$-doublets are indiscernible in
this plot. In this regime, rotational levels are only populated by the
radiative cascade of OH photofragments and $\mathcal{I}_{N,\Omega,\epsilon}$,
depends only on the spectral shape of the adopted radiation field. The solid
line is an analytical model that assumes that the intra-ladder lines within
the $\varv=0$ state are powered by the radiative cascade of the OH fragments
produced in the $\varv=0$ state. The dotted line is an analytical model that
assumes that any OH produced in a vibrational state instantaneously decays
toward the ground vibrational state with negligible change in its rotational
number (see Appendix C).

The mid-IR spectrum ($\lambda\lesssim 20\leavevmode\nobreak\ \mu$m) is
dominated by pure intra-ladder rotational lines emerging from levels with high
rotational quantum number ($N\geq 14$), corresponding to upper energies
$>5000$ K. Figure 4 shows that in this wavelength range, the line intensities
are higher for a larger value of $\Phi$ (top and middle panels) but do not
depend on $N($OH$)$ (top and bottom panels). The relative intensities of the
intra-ladder lines depend neither on $N($OH$)$ nor on $\Phi$ so we focus on
the intensity of the ${}^{2}\Pi_{1/2}(30,f)\rightarrow$
${}^{2}\Pi_{1/2}(29,f)$ line at 10.8 $\mu$m (see Table 2) as a proxy for the
intensities of the mid-IR lines. Figure 5 (left panels) shows that the
absolute line intensity is directly proportional to $\Phi$ and does not depend
on other parameters such as $N($OH$)$, $T_{\rm{IR}}$ or $n_{\rm{H}}$. This is
one of the most fundamental properties of the high-$N$ lines that makes them
an unambiguous diagnostic of H2O photodissociation.

This very simple result points toward a simple excitation process. Due to the
high energy of the upper levels, IR radiative pumping does not contribute to
the excitation of these lines. De-excitation by stimulated emission by the IR
background is also negligible as long as the photon occupation number is much
smaller than unity in the mid-IR domain, a condition that is fulfilled over
the full parameter space. Due to the very high critical densities of these
levels ($n_{\rm{crit}}\gtrsim 10^{13}$ cm-3), collisional (de)excitation is
also negligible. Instead, the level populations, and the intensity of the
lines coming from those levels, are set by the radiative cascade following the
formation of OH in high-$N$ states by H2O photodissociation. The line
intensities, or equivalently the number of radiative transitions per unit time
between two excited levels, are then directly proportional to the formation
rate of OH in higher excited states, which is proportional to the
photodissociation rate of H2O. The proportionality between the mid-IR line
intensity and $\Phi$ is thus a direct consequence of the radiative cascade.

In order to analyze the relative intensity of the mid-IR lines, it is then
convenient to define the normalized and dimensionless line intensity

$\mathcal{I}_{N,\Omega,\epsilon}=\frac{I^{\Omega,\epsilon}_{N\rightarrow
N-1}}{h\nu\Phi},$ (13)

where $I^{\Omega,\epsilon}_{N\rightarrow N-1}$ is the integrated intensity (in
erg s-1 cm-2) of the $N\rightarrow N-1$ line within the ladder $\Omega$ and
the parity $\epsilon$, and $\nu$ is the frequency of the line. As long as the
population of the high-$N$ levels is set by the radiative cascade,
$\mathcal{I}_{N,\Omega,\epsilon}$ depends only on the distribution of nascent
OH, or equivalently on the spectral shape of the UV field, and not on $\Phi$
or on any other physical parameter. Figure 6 shows that
$\mathcal{I}_{N,\Omega,\epsilon}$ depends mostly on $N$ and very little on
$\Omega$ and $\epsilon$. This is due to the fact that in our model, levels are
assumed to be populated by H2O photodissociation regardless of their $\Omega$
and $\epsilon$ states. In the following, we thus omit the reference to
$\Omega$ and $\epsilon$ and note the normalized line intensity as
$\mathcal{I}_{N}$. $\mathcal{I}_{N}$ increases with decreasing $N$, with a
stiff rise between $N=47$ and $N=35$. This feature is also visible in Fig. 4
where the line intensities increase with wavelength between $9$ and
$10\leavevmode\nobreak\ \mu$m. For $22\leq N\leq 32$, $\mathcal{I}_{N}$ is
rather constant, corresponding also to a rather flat mid-IR spectra between 10
and 20 $\mu$m (see Fig. 4). This results in suprathermal excitation
temperatures that vary between 2300 K for the lines at $\lambda\simeq
20\leavevmode\nobreak\ \mu$m up to 13000 K for the lines at $\lambda\simeq
10\leavevmode\nobreak\ \mu$m.

The normalized intensity $\mathcal{I}_{N}$ is directly related to the
distribution of nascent OH. Interestingly, $\mathcal{I}_{N}$ can be
interpreted as the probability that a photodissociation event H2O
$\rightarrow$ H + OH eventually leads to a radiative decay $N\rightarrow N-1$
via the radiative cascade. As such, $\mathcal{I}_{N}$ is necessarily smaller
than unity. Because we assume the spin-orbit and $\Lambda$-doubling states to
be equally populated by H2O photodissociation, $\mathcal{I}_{N}\lesssim 1/4$.
Owing to the selection rules, the rotational cascade within the $\varv=0$
state is dominated by $N\rightarrow N-1$ transitions. Moreover, OH produced in
an electronic level is shown to rapidly decay toward OH($X^{2}\Pi$) with
little change of rotational number. Consequently, the formation of an OH
fragment in an OH($\Lambda,\varv=0,N^{\prime}$) state eventually leads to an
intra-ladder transitions $N\rightarrow N-1$, with $N\leq N^{\prime}$. We plot
in Fig. 6 (solid line) an analytical prediction of $\mathcal{I}_{N}$ assuming
that transitions $N\rightarrow N-1$ are supplied by the radiative decay of the
OH fragments produced in the $\varv=0$ states (see model in Appendix C). This
analytical expression relies only on the distribution of nascent OH denoted as
$f_{i}$. The model reproduces the increase in $\mathcal{I}_{N}$ with deceasing
$N$ well, showing that its global variation with $N$ is mostly due the fact
that more and more OH fragments are added to the $N\rightarrow N-1$ rotational
cascade as $N$ decreases. The radiative cascade should then rather be seen as
a ”radiative river” that grows by it tributaries. In particular, the steep
increase in $\mathcal{I}_{N}$ from $N=47$ to $35$ is due to the fact that most
of the OH fragments are produced with these $N$-quantum numbers (see Fig.
2-b).

We also note that for $N<35$, our analytical model progressively
underestimates $\mathcal{I}_{N}$. This is due to the contribution of the OH
fragments produced in vibrationally excited states that are not included in
our first analytical model. In Fig. 6 (dashed line) we show the analytical
prediction of $\mathcal{I}_{N}$ assuming that any OH produced in an
OH($\Lambda$)($\varv,N$) state immediately decays toward the
OH($X^{2}\Pi$)($\varv=0,N-1$) state. The model overestimates $\mathcal{I}_{N}$
showing that vibrationally excited states tend to undergo rotational
transitions within $\varv\geq 1$ vibrational states before decaying to the
$\varv=0$ state.

In other words, the spectral shape of the normalized intensity
$\mathcal{I}_{N}$ is set by the population of nascent OH following
photodissociation of H2O. Our two simple analytical models, which rely only on
the knowledge of the distribution of the OH fragments $f_{\text{i}}$, allow
one to bracket $\mathcal{I}_{N}$.

####  3.1.2 Low-$N$ lines

Figure 4 shows that, in contrast to the mid-IR lines, the far-IR lines
($\lambda>40\leavevmode\nobreak\ \mu$m) do not depend on the column density of
H2O photodissociated per unit time $\Phi$ but on $N($OH$)$. The same
conclusions apply to the cross-ladder transitions apparent from
20$\leavevmode\nobreak\ \mu$m to 114$\leavevmode\nobreak\ \mu$m. All these
lines arise from $N\lesssim 6$ levels ($E_{\rm{up}}\lesssim 1200$ K) and trace
the bulk population of OH that is not excited by the radiative cascade from
high-$N$ levels. Because of the high optical depth of the intra-ladder lines,
we focus in the following on the optically thin cross-ladder transition at
$24.6\leavevmode\nobreak\ \mu$m that arises from a $N=5$ level (see Table 2).
Figure 5 (right) shows for a broader range of parameters that the intensity of
this line does not depend on $\Phi$ but on $N($OH$)$, $T_{\rm{IR}}$, and
$n_{\rm{H}}$. Prompt emission is always negligible and the line intensity is
the result of a competition between collisional (de-)excitation and IR
radiative pumping.

Figure 5 (lower right panel) highlights three distinct excitation regimes as a
function of density. At low density, the intensity does not depend on the
density but on $T_{\rm{IR}}$. In this regime, labeled by ②, levels are
exclusively populated by IR radiative pumping. Because of our specific choice
of the IR radiation field, all low-energy levels are thermalized to the same
excitation temperature equal to $T_{\text{IR}}$ and the intensities of the
optically thin lines are

$I_{i\rightarrow j}\simeq
A_{ij}g_{i}\frac{N(\text{OH})}{Q(T_{\text{IR}})}\frac{hc}{\lambda_{ij}}e^{-E_{i}/k_{B}T_{\text{IR}}},$
(14)

where $Q(T_{\text{IR}})$ is the partition function of OH. The intensity of the
line at $24.6\leavevmode\nobreak\ \mu$m is simply proportional to $N($OH$)$
and increases with $T_{\rm{IR}}$. For an IR radiation field that deviates from
a blackbody, the excitation of the OH($X^{2}\Pi$)($\varv=0,N$) levels is more
complex. In this general case, the radiation brightness temperature
$T_{\rm{rad}}$ varies with wavelength across the far-IR spectrum of OH
($\lambda\lesssim 120\leavevmode\nobreak\ \mu m$). As a rule of thumb, the
line intensity is then bracketed between our predictions with an undiluted
blackbody at a value of $T_{\rm{IR}}$ that lies between the minimum and
maximum values of $T_{\rm{rad}}$.

At intermediate density, the intensity increases with $n_{\rm{H}}$. For our
fiducial values of $T_{\rm{IR}}$ and $T_{\rm{K}}$, it corresponds to densities
between 108 and 1010cm-3. In this regime, labeled by ③, collisions contribute
to the excitation of the levels. However, line intensities also depend on
$T_{\rm{IR}}$, showing that IR radiative pumping is also relevant. Since the
density is smaller than the critical density of the upper energy level
($n_{\rm{H}}$$\sim 10^{11}$cm-3), the de-excitation of the level is via
radiative decay. Interestingly, the critical density above which collisions
contribute to the excitation of the levels depends on $T_{\rm{IR}}$ and on
$T_{\rm{K}}$. It is lower for a lower $T_{\rm{IR}}$(Fig. 5, bottom right) or
for a higher $T_{\rm{K}}$ (not shown here). By comparing the rates of
collisional excitation with the IR radiative pumping rate, collisions take
over from IR radiative pumping in the excitation of a given $N$-level for

$n_{\text{H}}\gtrsim n_{\gamma}(\nu_{\rm{N}})\leavevmode\nobreak\
n_{\text{crit}}\leavevmode\nobreak\ e^{h\nu_{\rm{N}}/k_{B}T_{\rm{K}}}\simeq
n_{\text{crit}}\leavevmode\nobreak\
e^{h\nu_{\rm{N}}/k_{B}(1/T_{\rm{K}}-1/T_{\rm{IR}})},$ (15)

where $\nu_{N}$ denotes the frequency of the $N\rightarrow N-1$ radiative
transition, $n_{\gamma}$ is the photon occupation number, and where we assume
$h\nu_{\rm{N}}\gg T_{\rm{IR}}$ for the second term. With $n_{\text{crit}}$
having a weak dependence on $T_{\rm{K}}$, the critical density above which
collisions take over from IR radiative pumping depends on the contrast between
the temperature of the gas and that of the radiation field.

Finally, above the critical density of the upper energy level
($n_{\text{crit}}\simeq 10^{11}$cm-3), the line intensity converges toward its
LTE value. In this regime, labeled by ④, collisions control the excitation and
the de-excitation of the level and Eq. (14) gives the intensity of the
optically thin line by substituting $T_{\rm{IR}}$ by $T_{\rm{K}}$.

One has to keep in mind that when IR radiative pumping is relevant for the
excitation of a level, the geometry of the source turns out to be of great
importance for the formation of the line coming from this level. We recall
that the line intensities presented in this work are computed assuming that
the IR background does not contribute to the line formation process. If the
background IR field does contribute to the observed emission, lines can be
weaker or seen in absorption (see OH lines observed by Wampfler et al., 2010,
2013). Collisions can also play a role in the formation of the line even when
negligible in the excitation of the levels. Models incorporating specific
source geometries are beyond the scope of this work and have already been
explored to analyze low-$N$ OH lines observed by Herschel toward protostars
and photodissociation regions (Goicoechea et al., 2011; Wampfler et al.,
2013).

####  3.1.3 Intermediate-$N$ lines

Figure 7: Transition from prompt emission to thermal or radiative excitation
as revealed by the intensity of the intermediate $N=10$ rotational line at
27.4 $\mu$m. The dominant excitation processes are indicated by ①,②,③,④ as
defined in Fig. 5. a) Line intensity normalized by $N($OH$)$ as a function of
$\mathcal{D}\equiv\Phi/N(\text{OH})$. The transition between radiative pumping
and prompt emission depends on $\mathcal{D}$ and on $n_{\rm{H}}$. In
particular, one can define a critical pumping rate $\mathcal{D}_{\rm{crit}}$
below which thermal or radiative excitation processes take over from prompt
emission. b) Line intensity normalized by $N($OH$)$ as a function of
$n_{\rm{H}}$ for $\mathcal{D}=10^{-9}\text{s}^{-1}<\mathcal{D}_{\rm{crit}}$.
c) Schematic parameter space indicating the dominant excitation process of the
$N=10$ line as a function of $n_{\rm{H}}$ and $\mathcal{D}$. We note that
whereas the specific values of $\mathcal{D}$ and $n_{\rm{H}}$ reported in the
x- and y-axis are for the $N=10$ line, this schematic view remains valid for
any rotational line. The prompt emission region is delimited by
$\mathcal{D}_{\rm{crit}}$ (green line). The regime ① corresponds to the
excitation regime of high-$N$ line whereas the regimes ②, ③ and ④ correspond
to the excitation regime of low-$N$ lines. Panels a) and b) are cuts in the
parameter space and are indicated by red, magenta and blue lines. The
temperature of the IR background, the kinetic temperature and the column
density of OH are fixed to their fiducial values. The other parameters are
indicated in each panel.

We have shown in Section 3.1.1 that the high-$N$ rotational lines excited by
prompt emission are proportional to $\Phi$ and can thus be used to probe the
photodissociation of H2O. In contrast, low-$N$ lines do not depend on $\Phi$
and trace the bulk population of OH. It is thus of a great importance to
determine if a line coming from an intermediate-$N$ level is indeed excited by
prompt emission or by other processes such as IR radiative pumping or
collisions.

Figures 7-a and b show that the intermediate$-N$ line at
27.4$\leavevmode\nobreak\ \mu$m, which originates from a $N=10$ level, shares
features of low-$N$ and high-$N$ lines. This complex excitation pattern is the
result of the competition between prompt emission, IR radiative pumping and
collisions. Figure 7-c summarizes the dominant excitation processes as a
function of the density $n_{\rm{H}}$ and the chemical pumping rate
$\mathcal{D}=\Phi/N(\rm{OH})$ for the fiducial values of $T_{\rm{IR}}$ and
$N($OH$)$.

For high values of the chemical pumping rate, the line intensity normalized by
$N($OH$)$ is proportional to $\mathcal{D}$ (Fig. 7-a). This corresponds to the
right part of the parameter space shown in Fig. 7-c. In this regime, the
excitation of the upper energy level is dominated by prompt emission (regime
①) and we recover the result obtained for the high-$N$ lines that the line
intensity is proportional to $\Phi=\mathcal{D}N($OH$)$ (Sec. 3.1.1). Figure
7-a shows that below a critical value of the chemical pumping rate, denoted
here as $\mathcal{D}_{crit}$, the intensity does not depend on $\mathcal{D}$
as IR radiative pumping or collisions take over from prompt emission. This
corresponds to the left region of the parameter space (Fig. 7-c). The
excitation of the line as a function of the density then follows a pattern
similar to that found for low$-N$ lines (Fig. 7-b). At low density, the
excitation is dominated by IR radiative pumping (regime ②) whereas at higher
density collisions progressively take over (regime ③). We find that for the
fiducial values of $T_{\rm{IR}}$ and $T_{\rm{K}}$, collisions contribute to
the excitation of the line above $n_{\rm{H}}$$\simeq 10^{8}$ cm-3. However, we
recall that the effect of collisions on levels with $N\gtrsim 5$ is highly
uncertain due to the lack of quantum calculations of collisional rate
coefficients for these levels. In other words, the parameter space is divided
into two regions: a region dominated by prompt emission for which results
found in Sec. 3.1.1 apply, and a region dominated by other excitation
processes for which results found in Sec. 3.1.2 apply.

The boundary between the two regions is defined by $\mathcal{D}_{crit}$ (green
line Fig. 7-c). Its value depends on the parameters that control the thermal
and radiative excitation of OH, namely $n_{\rm{H}}$, $T_{\rm{K}}$, and
$T_{\rm{IR}}$. For example, Fig. 7-a shows that $\mathcal{D}_{crit}$ increases
from $\sim 2\times 10^{-9}$ to $\sim 2\times 10^{-8}$ s-1 by increasing the
density from $n_{\rm{H}}$=$10^{7}$ to $10^{9}$ cm-3. Since
$\mathcal{D}_{crit}$ quantifies the competition between thermal or radiative
excitation and prompt emission, it also depends on the upper energy level of
the line. We derive in Appendix E simple estimates of
$\mathcal{D}_{\rm{crit}}$ as a function of the excitation conditions
($n_{\rm{H}}$, $T_{\rm{IR}}$, and $T_{\rm{K}}$) for any rotational level. In
particular, we show that the schematic view of the parameter space proposed in
Fig. 7-c remains valid for the low-$N$ lines for which the boundary is shifted
to the right, and for high-$N$ lines, for which the boundary is shifted to the
left by orders of magnitudes.

####  3.1.4 Summary

The integrated intensities of the lines coming from high-$N$ levels ($N\gtrsim
20$) are proportional to the column density of H2O photodissociated per unit
time and do not depend on other physical parameters. The shape of the mid-IR
spectrum is only set by the distribution of the nascent OH and we define a
normalized intensity of the intra-ladder lines, denoted as $\mathcal{I}_{N}$
(Fig. 6). The cross-ladder and intra-ladder rotational lines coming from
low-$N$ levels ($N\lesssim 6$) are populated by IR radiative pumping or
collisions and are thus tracing the bulk population of OH. Intermediate-$N$
lines can be either excited by prompt emission, IR radiative pumping and/or
collisions (see Fig. 7-c).

###  3.2 Other UV radiation fields

####  3.2.1 State distribution of the OH fragments

Figure 8: Rotational distribution of the OH fragments following H2O photodissociation by UV radiation fields of different shapes. The impact of three radiation fields is explored: a) a radiation field representative of an accreting T Tauri star dominated by a Ly$\alpha$ emission line, b) an ISRF radiation field and c) a blackbody radiation field at 10 000 K. Vibrational quantum number are color coded as indicated in the top panel. The distribution is summed over the two electronic states. Only vibrational levels that contribute to at least 1.5$\%$ to the population of nascent OH are shown. Table 3: Branching ratio between H2O photodissociation leading to OH through the $\tilde{A}$ state ($\lambda\gtrsim 143$ nm) and the $\tilde{B}$ state ($114\lesssim\lambda\lesssim 143$ nm) for UV radiation fields of various spectral shapes. Radiation field | $\tilde{A}$ | $\tilde{B}$
---|---|---
Ly $\alpha$ | 0$\%$ | 100$\%$
T Tauri | 13$\%$ | 87$\%$
ISRF | 46$\%$ | 54$\%$
Blackbody 1000K | 82$\%$ | 18$\%$

Figure 8 shows the state distribution of the OH fragments following H2O
photodissociation by UV radiation fields of various spectral shapes. As shown
in Section 3.1.1, the OH($A$)($\varv,N$) states decay toward the ground
electronic state with little change of $N$. We consequently plot only the sum
of the rotational distribution of OH($X$) and OH($A$) as defined by

$\tilde{f}_{i}=f(X,\varv,N)+\frac{1}{2}f(A,\varv,N),$ (16)

where the factor $\frac{1}{2}$ stands for the different degeneracies between
OH($A$)($\varv,N$) and OH($X$)($\varv,N$) states. The distribution of nascent
OH computed from Eqs. (2) and (3) results from photodissociation at various
wavelengths. As shown in Sec. 2.2, the distribution of the OH fragment
$\eta(\lambda,i)$ depends markedly on the UV wavelength (see Fig. 13). One of
the most prominent differences is between photodissociation longward of
$\lambda=143$ nm, that produces OH($X$) in low-$N$ states, and
photodissociation shortward of this value that produces OH($X$) in high-$N$
states and a small fraction of electronically excited OH($A$) with $N\leq 27$.
The distribution of OH following H2O photodissociation by a broad UV radiation
field reflects the relative contribution of the different photodissociation
channels. The fraction of photodissociation that proceeds through the two
channels is given in Table 3.

We first study the effect a radiation field representative of the UV spectrum
emitted by an accreting T Tauri star that includes a UV continuum plus
emission lines (see Fig. 15). As shown by Bergin et al. (2003) and Schindhelm
et al. (2012), Ly$\alpha$ emission dominates over the continuum emission with
$\sim 90\%$ of H2O photodissociation done by Ly$\alpha$ photons. It results in
a state distribution that is similar to that produced by a pure Ly$\alpha$
radiation field (Fig. 8-a). Contribution of H2O photodissociation through the
H2O $\tilde{A}$ state ($\lambda\geq 143$ nm), that represents $\sim 10\%$ of
the total photodissociation rate, is however seen at $N\leq 6$.

In contrast, a standard UV interstellar radiation field (ISRF) has a smooth
and rather flat spectral distribution (Fig. 15). H2O photodissociation then
proceeds through a broad wavelength range and results in a distribution of OH
fragments that exhibits features of both photodissociation through the H2O
$\tilde{A}$ and $\tilde{B}$ state (Fig. 8-b). Photodissociation through the
$\tilde{B}$ state creates a peak in the rotational distribution around $N=41$
similar to that produced by photodissociation by Ly$\alpha$, though somewhat
smoother. OH($A$) states are produced with a broad range of rotational quantum
numbers, which results in a small bump in the rotational distribution that is
perceptible in the range $N=10-27$. We note that a significant fraction of the
OH($A$) products are dissociative and therefore not visible in Fig. 8-b.
Photodissociation through the H2O $\tilde{A}$ state, that represents $46\%$ of
the total photodissociation rate, yields to a prominent peak at low-$N$
numbers.

A UV radiation field with a blackbody shape at $T=10000$ K has the steepest UV
slope. Most of H2O photodissociation proceeds through the $\tilde{A}$ state
($82\%$). The resulting distribution of nascent OH is then dominated by
low-$N$ numbers whereas the bump at $N\simeq 40$ is reduced accordingly (Fig.
8-c).

In other words, the state distribution of OH following H2O photodissociation
by a broad UV spectrum exhibits a bump at high-$N$ number and a peak at
low-$N$ number. The amount of OH produced with high-$N$ numbers is globally
proportional to the fraction of photodissociation occurring through the
$\tilde{B}$ state ($\lambda<143$ nm).

####  3.2.2 Mid-IR lines

Figure 9: OH mid-infrared spectrum for various UV spectra computed with
GROSBETA for $\Phi=10^{10}$ cm-2 s-1. Other parameters are constant and equal
to their fiducial values given in Table 1. In this regime, the intensity of
intra-ladder lines (red and blue) are proportional to $\Phi$ and do not depend
on other parameters such as $T_{\rm{IR}}$ or $n_{\rm{H}}$. Figure 10:
Normalized photon intensity of the OH($X^{2}\Pi$)($\varv=0$) intra-ladder mid-
IR lines as a function of the quantum number of the upper energy level
$N_{\text{up}}$. The intensities are computed for the fiducial values of the
parameters (see Table 1) and normalized by the column density of water
photodissociated via its $\tilde{B}$ state (see Eq. (18)). Circle and triangle
markers correspond to lines belonging to the $\Omega=1/2$ and $\Omega=3/2$
ladders, respectively. $\Lambda$-doublets are indiscernible in this plot. In
this regime, rotational levels are only populated by the radiative cascade of
OH photofragments and $\mathcal{I}^{\tilde{B}}_{N,\Omega,\epsilon}$, depends
only on the shape of the adopted radiation field.

As shown in the case of H2O photodissociation by a Ly$\alpha$ radiation field,
mid-IR lines are proportional to $\Phi$. The same conclusion applies for any
UV radiation field and we show in Fig. 9 the mid-IR spectrum emitted by OH for
the UV radiation fields explored in this work, for the same value of $\Phi$.

The mid-IR line intensities depend markedly on the shape of the radiation
field. The main difference resides in the absolute line intensity between $10$
and $\sim 20\leavevmode\nobreak\ \mu$m. For example, line intensities are
$\sim 5$ times weaker for a blackbody at 104 K than for a T Tauri radiation
field. As seen in the case of a Ly$\alpha$ radiation field, mid-IR lines are
fueled by the radiative decay of OH produced in high-$N$ states. The absolute
mid-IR line intensities are then proportional to the fraction of OH produced
in a rotationally excited state, which is also proportional to the column
density of H2O photodissociated through $\tilde{B}$ state ([cm-2 s-1]) as
defined by

$\Phi^{\tilde{B}}\equiv N(\text{H}_{2}\text{O})\int_{114\leavevmode\nobreak\
nm}^{143\leavevmode\nobreak\ nm}\sigma(\lambda)I(\lambda)d\lambda.$ (17)

This is further shown in Fig. 10, where the line intensities normalized by
$\Phi^{\tilde{B}}$ as defined by666In the case of the photodissociation by a
Ly$\alpha$ radiation field studied in Sec. 3.1, $\Phi=\Phi^{\tilde{B}}$ and
$\mathcal{I}_{N}=\mathcal{I}^{\tilde{B}}_{N}$

$\mathcal{I}^{\tilde{B}}_{N,\Omega,\epsilon}\equiv\frac{I^{\Omega,\epsilon}_{N\rightarrow
N-1}}{h\nu\Phi^{\tilde{B}}}$ (18)

are the about same for the different UV fields (within 20$\%$). In other
words, the absolute intensity of mid-IR lines traces the amount of H2O
photodissociated through the $\tilde{B}$ state ($\lambda<143$ nm). It follows
that the absolute intensity is, to a first approximation, only proportional to
$\Phi^{\tilde{B}}$, regardless of the exact shape of the UV radiation field.

The other difference resides in the relative intensity of the lines, that
reveals the precise shape of the rotational distribution of nascent OH
fragments. This is best seen in Fig. 10. The increase in the line intensities
from $N_{up}=46$ down to $N_{up}\sim 35$ is steeper for T Tauri and Ly$\alpha$
radiation fields than for the other two. These differences are due differences
in the exact shape of the rotational distributions of the OH($X$) fragments
shown in Fig. 8. In the case of T Tauri and Ly$\alpha$ radiation fields, the
peak at high-$N$ is more pronounced than for the broadband UV spectra. The
differences in the rotational distributions at lower $N$, that are mostly due
to differences in the OH($A$) distributions, result in minor differences in
the line intensities in the range $N_{up}=25-15$.

####  3.2.3 Low and intermediate-$N$ lines

Regarding far-IR lines and cross-ladder mid-IR lines, that emerge from
$N\lesssim 6$ levels, we find that the integrated intensities depend neither
on the shape of the UV radiation field nor on $\Phi$. This is a surprising
result since UV radiation fields with a significant flux longward of $143$ nm,
produce OH with low rotational quantum numbers. Our finding indicates that the
contribution of this rotationally cold population of nascent OH to the
excitation of low-$N$ levels within $\varv=0$ is negligible compared to
infrared pumping or collisions. The only population of nascent OH that affects
the intensity of the rotational lines are the one produced with a high
rotational number, typically $N\gtrsim 15$. However, we note that for much
lower IR radiation fields and lower densities, these lines might be fueled by
prompt emission (see Fig. 7-c, regime ①). In that case, we expect to have a
competition pattern between excitation via the H2O $\tilde{A}$ state and the
$\tilde{B}$ state.

As studied in the case of photodissociation by a Ly$\alpha$ radiation field,
intermediate-$N$ lines can be excited either by prompt emission or IR
radiative pumping and/or collisions. For a given set of physical parameters
$\\{$$T_{\rm{IR}}$, $T_{\rm{K}}$, $n_{\rm{H}}$$\\}$, the transition between
prompt emission and the other excitation processes is controlled by the
chemical pumping rate $\mathcal{D}\equiv\Phi/N(\rm{OH})$ as summarized in Fig.
7-c. The analysis proposed in Sec. 3.1.3 can be generalized to any UV
radiation field by simply substituting $\mathcal{D}$ by
$\mathcal{D}^{\tilde{B}}$ as defined by

$\mathcal{D}^{\tilde{B}}=\Phi^{\tilde{B}}/N(\text{OH}).$ (19)

We note that for the explored UV radiation fields, this adaptation has little
impact on the transition between prompt emission and the other excitation
processes.

##  4 Discussion and application to HH 211

In this section, we present a method for observationally inferring the
photodissociation rate of H2O and deducing the local UV radiation field. As an
illustration, our model is applied to the Spitzer-IRS observations of the apex
of the HH 211 bow-shock published by Tappe et al. (2008).

###  4.1 Diagnostics

####  4.1.1 Column density of H2O photodissociated per second

Our results show that the absolute intensities of the intra-ladder lines in
the mid-IR are proportional to the column density of H2O photodissociated per
second via the $\tilde{B}$ state, denoted as $\Phi^{\tilde{B}}$ (see Fig. 10).
To our knowledge, the only alternative process that can also excite the
rotational levels of energy $E_{\text{up}}\geq 20000$ K is the electron impact
dissociation of H2O (Beenakker et al., 1974; Bodewits et al., 2019). In some
situations, such as disks around young stellar objects where there might be a
source of energetic electrons ($\gtrsim 10$ eV), the high-$N$ OH lines would
thus trace the destruction of H2O via both e-impact and UV photodissociation.
Complementary diagnostics, such as lines emerging from the triplet electronic
states of H2 or CO that can be excited efficiently by electron impact but not
by UV absorption, can help to determine if OH lines might also trace the
electron impact dissociation of H2O. In this context, the predictions
presented in this work remain valid as long as the destruction rate of H2O by
UV photodissociation dominates over destruction by e-impact, a condition that
is generally satisfied in irradiated regions.

Consequently, the mid-IR lines of OH($X$)($\varv=0$) provide a robust
measurement of the column density of H2O photodissociated in the range $114$
to $143\leavevmode\nobreak\ $nm. This key quantity can be observationally
derived using the relation

$\Phi^{\tilde{B}}=\frac{I_{N,\Omega,\epsilon}}{h\nu\mathcal{I}^{\tilde{B}}_{N,\Omega,\epsilon}},$
(20)

with $\mathcal{I}^{\tilde{B}}_{N,\Omega,\epsilon}$ provided in Fig. 10 and in
Appendix D. As shown above, $\mathcal{I}^{\tilde{B}}_{N,\Omega,\epsilon}$
depends little on the spectral shape of the UV radiation field, and, in the
absence of any other information, $\Phi^{\tilde{B}}$ can still be derived with
a high accuracy ($\sim 10\%$). Interestingly, the time scale associated with
the radiative cascade is about 10 ms, which is short compared with relevant
dynamical time scales. Therefore, the value of $\Phi^{\tilde{B}}$ should be
considered as an instantaneous measurement of the photodissociation activity.
If the emitting region is unresolved, the derived value of $\Phi^{\tilde{B}}$
depends on the assumed source size as for any determination of a column
density. Strictly speaking, the total flux integrated over a region on the sky
gives the total number of H2O molecules photodissociated per second in that
region.

####  4.1.2 Spectral shape of the UV radiation field

The small deviations in the relative line intensities could be used to
constrain the spectral shape of the radiation field (see Fig. 9 and 10). In
particular, our results show that a Ly$\alpha$ dominated radiation field
produces a steeper increase in the photon line intensities with decreasing
$N_{up}$. However, detecting these small differences in the intensities in the
range $9$ to $11\leavevmode\nobreak\ \mu$m requires an accurate relative flux
calibration. In fact, the $10\leavevmode\nobreak\ \mu$m silicate feature plays
a significant role in the dereddening of the fluxes at visual extinctions
$A_{\rm{V}}\gtrsim 10$. In that perspective, the relative intensity of the
lines between $\simeq 9$ and $10\leavevmode\nobreak\ \mu$m in regions with
high extinction should be interpreted with caution since the exact shape of
the extinction curve varies significantly with $A_{\rm{v}}$ (Chapman et al.,
2009; McClure, 2009; Hensley & Draine, 2020). We posit that a significant
extinction in the mid-IR hampers the diagnostic capabilities of the OH lines
to securely constrain the shape of the local UV radiation field. The impact of
the rotational excitation of the parent H2O may also complicate the
interpretation of the relative line intensities and detailed quantum
calculations are needed to propose robust diagnostics based on the exact shape
of the OH mid-IR spectrum.

####  4.1.3 Photodissociation rate of H2O

Our work shows that the detection of mid-IR lines of OH is prime evidence for
the presence of H2O, in particular in environments where the H2O column
density is limited by strong UV fields. If an independent measurement of the
local UV radiation field can be observationally obtained, the column density
of H2O that it exposed to the UV field can be deduced as

$N(\text{H}_{2}\text{O})=\frac{\Phi^{\tilde{B}}}{k_{\tilde{B}}},$ (21)

where $k_{\tilde{B}}$ is the photodissociation rate in s-1 associated with H2O
$\rightarrow$ OH + H through the H2O $\tilde{B}$ state.

Conversely, the OH emission can be used to measure the photodissociation rate
$k_{\tilde{B}}$ by measuring the column density of water
$N(\rm{H}_{2}\rm{O})$:

$k_{\tilde{B}}=\frac{\Phi^{\tilde{B}}}{N(\text{H}_{2}\text{O})},$ (22)

where $\Phi^{\tilde{B}}$ is measured from the mid-IR lines of OH (see Eq.
(20)). This quantity being a ratio between a column density and
$\Phi^{\tilde{B}}$, it does not depend on the assumed source size.

The measurement of $N(\rm{H}_{2}\rm{O})$ is challenging and may be the major
source of uncertainty in the determination of $k_{\tilde{B}}$. Besides the
uncertainties that plague the determination of the column densities, the
variation of the UV field and of the number density of H2O along the line of
sight may be an important bias. Indeed, irradiated environments exhibit a
layered structure with physical conditions that vary steeply with distance,
due to the attenuation of the radiation field or shocks. Because the OH mid-IR
lines are optically thin and depend little on the shape of the UV radiation
field, our predictions can be directly extended to any physical structure with

$I_{N}=\mathcal{I}^{\tilde{B}}_{N}\int_{z}n_{\text{H}_{2}\text{O}}(z)k_{\tilde{B}}(z)dz,$
(23)

with $n_{\text{H}_{2}\text{O}}(z)$ the local number density of H2O and
$k_{\tilde{B}}(z)$ the local photodissociation rate of H2O through the
$\tilde{B}$ state. $\Phi^{\tilde{B}}$ being a quantity integrated along the
line of sight, a spatial variation of the UV radiation field and of
$n_{\text{H}_{2}\text{O}}(z)$ does not affect its measurement. On the
contrary, $k_{\tilde{B}}$ is a local quantity and Eq. (22) should be applied
with caution. Considerations on the geometry and on the viewing angle of the
source may also help.

####  4.1.4 Local UV field

Our model shows that the OH mid-IR lines trace the photodissociation of H2O by
photons in the range $114-143\leavevmode\nobreak\ $nm. As such, mid-IR OH
lines carry information on the local UV radiation field in this range. In the
following, we note $F_{UV}^{\tilde{B}}$, the photon flux integrated between
$114$ and $143$ nm, which is $F_{UV}^{\tilde{B}}=6\times 10^{7}$ cm-2 s-1 for
a Draine radiation field. The photodissociation rate $k_{\tilde{B}}$ is the
integral over the wavelength of the local UV radiation field multiplied by the
photodissociation cross section:

$k_{\tilde{B}}=\int_{114\leavevmode\nobreak\ \rm{nm}}^{143\leavevmode\nobreak\
\rm{nm}}I(\lambda)\sigma(\lambda)d\lambda.$ (24)

Strictly speaking, the conversion between $k_{\tilde{B}}$ and the local UV
flux $F_{UV}^{\tilde{B}}$ should then rely on the knowledge of the shape of
the UV radiation field. However, for the considered radiation fields, the
intensity weighted cross section averaged between $114$ and $143$ nm varies
only by a factor of $\sim 3$ and is about $\sigma\simeq 6\times 10^{-18}$ cm2.
Thus, Eq. (24) can be approximated by

$F_{UV}^{\tilde{B}}\simeq\frac{k_{\tilde{B}}}{6\times
10^{-18}\leavevmode\nobreak\ \rm{cm}^{2}},$ (25)

with a typical uncertainly of about a factor of three.

###  4.2 Application to HH 211

Figure 11: Comparison of the Spitzer-IRS mid-IR spectrum of the tip of HH 211
outflow from Tappe et al. (2008) (black lines) and of a synthetic GROSBETA
spectrum for OH (in red) and H2O (in blue). The best fit parameters are
indicated in the top right parts of the panels. The GROSBETA spectra are
plotted assuming a source size equal to the extraction aperture
$\Delta\Omega=1.9\times 10^{-9}$ sr. The rotational quantum number
$N_{\text{up}}$ are indicated for some OH lines. The model reproduces well the
OH lines at 10-16 $\mu$m, but underproduces them at longer wavelengths.

The Spitzer-IRS observations of the apex of the young protostellar jet HH 211
provide one of the best examples of highly rotationally excited OH. Figure 11
shows the unique sequence of superthermal OH emission lines unveiled from
$10\leavevmode\nobreak\ \mu$m down to $30\leavevmode\nobreak\ \mu$m coming
from rotational levels between $N=34$ to $N=9$. The association between OH
mid-IR emission and ongoing H2O photodissociation is further supported by the
detection of a compact H$\alpha$ emission by Walawender et al. (2006), tracing
a strong UV emitting shock in the close vicinity of the OH emission.
Complementary observations of H2 rovibrational emission (2.12 $\mu$m, Hirano
et al., 2006) and in the sub-millimeter domain unveiled a complex structure of
multiple molecular bow-shocks encompassing the strong shock (Tappe et al.,
2012).

####  4.2.1 Diagnostics from $N>20$ lines

From the Spitzer-IRS spectrum published by Tappe et al. (2008), we measure a
flux of the OH line at 14.6 $\mu$m of $F=3.5\times 10^{-15}$ erg s-1 cm-2 in
an aperture of $\Omega=1.9\times 10^{-9}$ sr. This line emerges from the
$N=20$ level of the OH($X$)($\varv=0$) state and corresponds to an intensity
integrated over solid angle of $I=2.3\times 10^{-5}$ erg s-1 cm-2 (see Eq.
(12)). Figure 10 indicates that the conversion factor for each component of
the quadruplet is $\sim 0.2$, regardless the spectral shape of the radiation
field. At Spitzer-IRS spectral resolution ($R\simeq 600$ for the short–high
module), the $\Lambda$-doublet and the fine-structure is however not
spectrally resolved. Equation (20) then yields a column density of H2O
photodissociated per unit time of

$\Phi^{\tilde{B}}=\frac{I}{4\times
0.2h\nu}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>10^{8}}\leavevmode\nobreak\ \rm{cm}^{-2}\leavevmode\nobreak\ \rm{s}^{-1},$
(26)

where the factor 4 stands for the sum over the four components of quadruplet,
and $\nu$ is the frequency of the line. This corresponds to a total number of
H2O molecule photodissociated per second of $4\times 10^{41}$ molecule s-1.

Figure 11 (top panel) compares a synthetic spectrum of OH produced by a
Ly$\alpha$ radiation field with the Spitzer-IRS spectrum between 10 and 16
$\mu$m. We recall that in this wavelength range, the spectrum depends only on
$\Phi^{\tilde{B}}$ and not on $n_{\rm{H}}$, $T_{\rm{K}}$ or $T_{\rm{IR}}$. Our
model reproduces well the sequence of the lines coming from $N=31$ down to
$N=14$.

The higher three lines lying in the range 10-10.5$\leavevmode\nobreak\ \mu$m
are overestimated by the model. The extinction correction ($A_{\rm{v}}=10$
mag, Walawender et al., 2006) cannot explain this discrepancy since the
extinction curve used by Tappe et al. (2008) tends to overestimate the
extinction at 10 $\mu$m. An ISRF UV radiation field, which produces relatively
weaker line fluxes shortward of 11 $\mu$m does not significantly improve the
fit. Alternatively, we posit that the rotational state of the parent H2O could
modify the rotational distribution above $N\simeq 30$. JWST observations with
a higher signal-to-noise ratio, combined with modeling including the
rotational state of the parent H2O are required to clarify this.

$\Phi^{\tilde{B}}$ can then be used to derive $k_{\tilde{B}}$, provided the
column density of H2O is known. Rotational lines of H2O are detected from the
far-IR with Herschel-PACS to the mid-IR with Spitzer-IRS, spanning energy
levels from 110 up to 1800 K. The mid-IR lines have been analyzed using
GROSBETA and we derive a column density of $2.2^{+2.0}_{-1.2}\times 10^{13}$
cm2, a density of
$n_{\rm{H}}$$\simeq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}5\times
10^{7}}$ cm-3, and a temperature of $T_{\rm{K}}$$\simeq
1300\leavevmode\nobreak\ $K (see synthetic spectrum in Fig. 11, bottom panel).
Interestingly, the same model reproduces the excitation temperature derived
from the observations of far-IR lines ($T_{ex}\simeq 90$ K, Dionatos et al.,
2018). This suggests that both mid-IR and far-IR lines trace the same warm H2O
reservoir that is photodissociated. Assuming a common origin for the OH and
H2O lines, Eq. (22) yields

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{\tilde{B}}\simeq
9.5\pm 4\times 10^{-6}\leavevmode\nobreak\ \rm{s}^{-1}.}$ (27)

The strong shock revealed by compact H$\alpha$ emission is expected to produce
a UV field dominated by Ly$\alpha$ photons. Equation (24) then gives
$F_{UV}^{\tilde{B}}\simeq 8.5\times 10^{11}$ photon cm-2 s-1, which
corresponds to a UV photon flux $5\times 10^{3}$ times larger than that of the
Draine ISRF in the range $91.2-200$ nm. This value is well in line with models
of fast dissociative shocks with $V_{S}\simeq 50$ km s-1 and
$n_{\rm{H}}$$\simeq 3\times 10^{5}$cm-3 (Raymond, 1979; Lehmann et al., 2020),
assuming that H2O is produced in the warm postshock without further dilution
of the UV radiation field. An origin of OH and H2O emission in slower shocks
passively illuminated by the stronger dissociative shock is also possible (see
models of Melnick & Kaufman, 2015; Godard et al., 2019). Spatially resolved
observations with JWST-MIRI of both OH mid-IR lines and atomic lines that
trace fast dissociative shocks such as [NeII] at $12.8\mu m$ or [Ni II] at
$6.6\mu m$ (Hollenbach & McKee, 1989) are required to distinguish the two
possible scenarios.

####  4.2.2 Evidence of additional chemical pumping routes

Figure 11-b shows that the OH lines coming from lower $N$ levels are detected
with the Spitzer-IRS long-high module. In particular, the cross-ladder
transition at 28.9 $\mu m$ is well reproduced by our GROSBETA model with a
column density of $N($OH$)=2\times 10^{13}$cm-2, assuming a density of
$n_{\rm{H}}$$=5\times 10^{7}$cm-3 and a temperature of $T_{\rm{K}}$$=1300$ K
derived from the H2O lines. This model is illustrative since the emission of
the OH cross-ladder line might originates from a different layer of gas but
its suggests a $N($OH$)/N($H2O$)$ ratio of about $\simeq 0.9$. This value lies
in the upper part of the range $10^{-3}$ to $0.8$ typically derived toward
protostellar outflows, and is higher than those derived toward warm inner
envelopes (Wampfler et al., 2011; Goicoechea et al., 2012, 2015).
Interestingly, our derived ratio of $\simeq 0.9$ is closer to that estimated
in low $A_{\rm{V}}$ regions of the Orion Bar ($\gtrsim 1$, Goicoechea et al.,
2011). This further supports the driving role of UV photodissociation in
preventing the full conversion of OH into H2O and in maintaining a relatively
high $N($OH$)/N($H2O$)$ ratio under warm conditions. Interestingly, detailed
models of molecular shocks do predict that both, fast dissociative shocks and
passively irradiated slow shocks exhibit high $N($OH$)/N($H2O$)$ ratios
(Neufeld & Dalgarno, 1989; Godard et al., 2019; Lehmann et al., 2020).

For this set of GROSBETA parameters, the intra-ladder rotational lines
longward of $\lambda=18\leavevmode\nobreak\ \mu$m are underestimated by our
GROSBETA model by up to a factor of eight. As shown in Sec. 3.1.3, these
intermediate-$N$ lines ($9<N<15$) are excited either by prompt emission, or by
IR radiative pumping or collisions. In HH 211, the IR background detected by
Spitzer-IRS is too weak to have a significant impact on the excitation of
these levels. In fact, for the selected set of parameters, the excitation is
dominated by prompt emission (right part of the schematic diagram, Fig. 7-c).
One could invoke collisional excitation to take over from prompt emission and
increase the intensities of the intra-ladder transitions. This can be achieved
by either decreasing the chemical pumping rate $\mathcal{D}$ or by increasing
$n_{\rm{H}}$ and/or $T_{\rm{K}}$ (i.e., increasing $\mathcal{D}_{crit}$, see
Fig. 7-c). Because $\Phi=\mathcal{D}N($OH$)$ is constrained by the lines in
the range 10 to 16$\leavevmode\nobreak\ \mu$m, the intensity of the chemical
pumping rate $\mathcal{D}$ can only be decreased by increasing $N($OH$)$.
However, increasing $N($OH$)$, $n_{\rm{H}}$ or $T_{\rm{K}}$ would increase the
intensities of the cross-ladder transitions, which are already well reproduced
by our model. In other words, collisions or IR radiative pumping cannot
reconcile our models with observations.

Interestingly, in this wavelength range, $\Lambda$ doublets are spectrally
resolved and show an asymmetry between $A^{\prime}$ and $A^{\prime\prime}$
$\Lambda$-doublet states of about a factor of 1.5-2. As discussed by Carr &
Najita (2014), this further supports the minimal impact of collisional
(de)excitation since collisions might eliminate asymmetry in the population.
This suggests that the intermediate-$N$ transitions reveal another excitation
process that is not included in our model.

The reaction H2+O can produce OH in rotationally and vibrationally excited
states and supplement the excitation of intermediate-$N$ levels. In
particular, experimental data show that the reaction

$\rm{H}_{2}+O(^{1}D)\rightarrow OH+H$ (28)

produces OH with rotational number between $N=10$ and $25$ with a large
fraction produced in $\varv\geq 1$ and a propensity to the $A^{\prime}$ states
(Liu et al., 2000). Interestingly, the discrepancy between our predictions and
the Spitzer-IRS spectra increases from $N\sim 16$ down to $N\sim 9$. The case
of chemical excitation by reaction (28) can be treated in a similar way as we
described the excitation of OH following H2O photodissociation (see Sec.
3.1.1). In the case of H2O photodissociation, the line intensities are given
by the column density of OH formed by H2O photodissociation. Similarly,
reaction (28) will increase the line intensities in proportion to the number
of OH produced per second via this reaction. The discrepancy between the model
and the observations suggests that the formation rate of OH via Eq. (28) is
about 6 times larger than that via H2O photodissociation. Interestingly, at
the density derived by our non-LTE analysis of H2O lines, O(1D) would be
converted into OH instead of decaying via the radiative transition at
$\lambda=$6300$\leavevmode\nobreak\ \AA$. However, this scenario requires an
efficient production of O(1D), either by OH or H2O photodissociation, or by
collisional excitation. Still, the estimated column density ratio OH/H2O
appears to be too low for OH photodissociation to take over from H2O
photodissociation in the excitation of OH mid-IR lines. Moreover, at
Ly$\alpha$, only $\sim 10\%$ of the water photodissociation ends in O(1D)
(Slanger & Black, 1982; van Harrevelt & van Hemert, 2008), a fraction that is
too small for the required efficiency of O(1D) production. Alternatively, the
reaction between O(3P) and vibrationally excited H2, known to be present at
the tip of the bow-shock, could lead to OH in rotationally (and vibrationally)
excited states (Gorti et al., 2011; Carr & Najita, 2014). These additional
chemical pumping routes could also impact the excitation of even lower-$N$
lines. Further modeling combining OH excitation and chemistry is required to
clarify the excitation of the intermediate-$N$ lines.

###  4.3 Perspectives with JWST-MIRI

Figure 12: Expected OH mid-IR spectrum at JWST-MIRI resolving power ($R\simeq
2000-3000$). The parameters of the model correspond to those derived from the
Spitzer-IRS observations of HH 211. The weak lines around 9.8 $\mu$m are pure
rotational lines within the OH($X$)($\varv=1$) state.

JWST-MIRI will provide a unique view in the mid-IR regime by combining
spectroscopic and imaging capabilities (Rieke et al., 2015; Wright et al.,
2015). At the observed wavelength range ($5-28\leavevmode\nobreak\ \mu$m),
JWST will be able to detect pure rotational lines coming from $N\geq 10$ and
cross-ladder lines coming from $N\geq 5$, allowing for detailed studies of the
excitation of OH across the full rotational ladder. In this section, we focus
on the high-$N$ rotational lines.

Up to now, OH mid-IR lines tracing H2O photodissociation have been detected
toward three protostellar outflows (Tappe et al., 2012) and a large number of
protoplanetary disks (Pontoppidan et al., 2010; Salyk et al., 2011; Carr &
Najita, 2011, 2014). We predict that the jump in sensitivity will extend the
detectability of high-$N$ lines to other irradiated environments. Since the
line intensities depend on the local abundance of H2O and on the local
radiation field, only astrochemical models designed for specific sources can
robustly predict mid-IR line intensities. However, the detectability of the
mid-IR lines as a function of the physical conditions can be explored with
simple chemical arguments. The line emissivity per hydrogen atom depends on
$x($H2O$)G_{0}$, where $x($H2O$)$ is the H2O abundance with respect to the
total density of H atoms and $G_{0}$ the local UV flux (see Eq. (23)). In warm
and irradiated environments (fast shocks, disk surface layers, low
$A_{\rm{V}}$ depths of prototypical PDRs), one can assume that H2O is
destroyed by photodissociation and formed in the gas phase by two-body
reactions so that $x($H2O$)\propto$ $n_{\rm{H}}$$/G_{0}$. For cooler regions,
for which H2O vapor is produced by photodesorption or by gas-phase ion-neutral
routes, the scaling is expected to be different. Therefore, at least for warm
environments, the emissivity of the OH lines per total hydrogen is expected to
scale with the density. This may explain why superthermal OH emission has been
detected by Spitzer in dense environments and remains undetected is classical
photodissociation regions (Goicoechea et al., 2011). We thus posit that
high-$N$ lines could be detected by JWST-MIRI in more diffuse regions.
Observation of dense classical PDRs such as the Orion bar or NGC 2023 should
be considered as primary targets. In even more diffuse environments, such as
translucent clouds, we expect OH mid-IR lines to remain undetected where we
estimate $\Phi\simeq 10^{3}-10^{4}$ cm-2 s-1 (Flagey et al., 2013),
corresponding to line intensities as low as $\sim 10^{-12}-10^{-11}$ erg s-1
cm-2 sr-1.

The sensitivity of JWST will also allow us to probe the mid-IR emission in
deeply embedded protostars. At high extinction ($A_{\rm{V}}\gtrsim 20$), the
OH lines in the range 13-15 $\mu$m will be particularly suited to minimize the
IR extinction. The OH mid-IR emission can then constitute a unique tool to
unravel the poorly known physical and chemical structure of young protostellar
disks. Regarding protostellar jets and outflows, it will allow us to test if
the low water abundance derived from Herschel observations is due to UV
photodissociation or other physical processes (Karska et al., 2014).

The spatial resolution provided by mapping capabilities will also be crucial
to constrain the emitting region of OH and map the photodissociation of H2O.
For dense environments, maps of $N($H2O$)$ can be built from pure rotational
lines of H2O lying typically longward of $20\leavevmode\nobreak\ \mu$m
($E_{\rm{up}}\lesssim 1200\leavevmode\nobreak\ $K) or ro-vibrational lines and
used to construct maps of the local UV radiation field (see Eq. (25)).

The Medium Resolution Spectrometer (MIRI-MRS), with its relatively high
spectral resolution ($R\sim 2000-3000$) will provide a valuable astrophysical
view of H2O photodissociation that will challenge current quantum
calculations. Figure 12 shows a synthetic spectrum at MIRI-MRS spectral
resolution. First, MIRI-MRS will give access to the OH lines in the range
9-10$\leavevmode\nobreak\ \mu$m that are directly populated by H2O
photodissociation ($35<N_{\rm{up}}<46$, see also Fig. 8). The
$\Lambda$-doublets up to $N_{\rm{up}}=40$ and the fine-structure up to
$N_{\rm{up}}=29$ will be spectrally resolved. This information will help to
further constrain the shape of the UV radiation field and understand the
impact of the rotational state of the parent H2O. New quantum calculations,
including the rotational state of H2O, and the fine-structure and the
$\Lambda$-doubling of the OH product, are warranted.

##  5 Conclusion

In this work, we explore the potential of the OH($X$)($\varv=0,N$) mid-IR
emission to probe H2O photodissociation. To reach this goal, results from
quantum mechanical calculations resolving the electronic, vibrational and
rotational state of the OH product following H2O photodissociation at
different UV wavelengths are collected. The distribution of the OH
photofragments is then calculated for UV fields of various spectral shapes and
included in a new radiative transfer code called GROSBETA. The impact of
prompt emission (i.e., formation of OH in excited states following H2O
photodissociation), collisional excitation and radiative pumping is
extensively studied.

The main conclusions of this study are:

  1. 1.

The mid-IR emission of OH in the range 9-16 $\mu$m is an unambiguous tracer of
H2O photodissociation. The mid-IR emission is the result of the rotational
radiative cascade of OH photofragments within the $\varv=0$ state. As such,
the detection of these lines constitutes a prime evidence for the presence of
active H2O chemistry. In particular, the line intensities are directly
proportional to the column density of H2O photodissociated per second by
photon in the range $114-143$ nm, denoted as $\Phi^{\tilde{B}}$. The
conversion factors between the line intensities and $\Phi^{\tilde{B}}$ is
provided for each rotationally excited line and for UV radiation fields of
various shapes in Fig. 10 and in Appendix D. These conversion factors depend
little on the exact spectral shape of the UV radiation field.

  2. 2.

Provided a measurement of the column density of the irradiated water is known,
the photodissociation rate by photons in the range $114-143$ nm can be
inferred (see Eq. (22)). The UV flux can then be deduced with good accuracy,
regardless of the shape of the UV radiation field. Alternatively, provided an
estimate of the strength of the local UV radiation field is available, the
column density of H2O can be derived.

  3. 3.

The precise state distribution of the OH fragments depends on the spectral
shape of the UV radiation field and results in small differences in the shape
of mid-IR spectrum of OH. This suggests that the relative line intensities, if
accurately measured, can be used to derive constraints on the spectral energy
distribution of the UV radiation field.

  4. 4.

The lower rotational levels, probed either by cross-ladder rotational lines
shortward of $30\leavevmode\nobreak\ \mu$m, or by far-IR rotational lines, are
excited by IR radiative pumping or collisions. We provide criteria to
determine if a rotational level of an intermediate energy (typically
$E_{\text{up}}=3000$ K$,N\simeq 10$) is excited by prompt emission or thermal
processes.

  5. 5.

As a test case, our model is applied to an irradiated region located at the
tip of the HH 211 protostellar jet. Our simple single-zone model reproduces
the Spitzer-IRS spectrum in the range 11-16$\leavevmode\nobreak\ \mu$m and
shows that water is exposed to a UV photon flux that is about $\sim 5\times
10^{3}$ times larger than the standard UV interstellar radiation field.
Discrepancies between our predictions and the observations at longer
wavelength suggest that additional chemical pumping processes such as the
reaction H2+O might contribute to the excitation of the rotational levels
between $N\sim 10-20$.

  6. 6.

JWST-MIRI will probe the rotational ladder of OH from $N=5$ up to $N\simeq 45$
corresponding to energy levels from $800\leavevmode\nobreak\ $K up to $45000$
K. We posit that the jump in sensitivity will unveil OH mid-IR emission in new
environments such as classical photodissociation regions (e.g., Orion bar) and
embedded protostars. Moreover, armed with good spectral resolution ($R\sim
2000-3000$), the MIRI-MRS observing mode will give access to even more excited
lines in the range $9-10\leavevmode\nobreak\ \mu$m and to the fine-structure
and $\Lambda$-doubling state distribution of the OH fragments.

Our work demonstrates the potential of OH mid-IR lines to probe H2O
photodissocation and the strength of the UV field. New quantum calculations
and astrochemical models taking into account the quantum state of the species
are warranted to leverage future JWST observations and unveil oxygen chemistry
through the excitation of OH.

###### Acknowledgements.

The authors thank the anonymous referee for their constructive comments. B.T.
would also like to thank B. Godard, A. Bosman and A. Faure for fruitful
discussions. This work is part of the research programme Dutch Astrochemistry
Network II with project number 614.001.751, which is (partly) financed by the
Dutch Research Council (NWO).

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##  Appendix A Distribution of OH fragments

Figure 13: Rotational distributions of nascent OH for various vibrational and
electronic states as a function of the photodissociation wavelength. The OH
distributions are summed over fine-structure and $\Lambda$-doubling states.
The difference in the rotational distribution between photodissociation
through the $\tilde{A}$ ($\lambda>143\leavevmode\nobreak\ $nm) and $\tilde{B}$
($\lambda<143\leavevmode\nobreak\ $nm) states of H2O is clearly visible.

In this appendix, we present the state distribution of OH following H2O
photodissociation as a function of the UV wavelength implemented in our model.
The data stem from the quantum calculations published in van Harrevelt & van
Hemert (2000) and van Harrevelt & van Hemert (2001). For the latter, the
calculations have been repeated using an another potential energy surface that
gives better agreement with experiments (see Sec. 2.2). As discussed in Sec.
2.2, we denote as $\eta(\lambda,\Lambda,\varv,N)$ the probability to form OH
in the state OH($\Lambda$)($\varv,N$) following H2O photodissociation by a
photon of wavelength $\lambda$. Figure 13 shows the rotational distributions
of OH as function of the photon wavelength for different electronic and
vibrational states. Figure 14 gives the vibrational distributions for each
electronic state. The distributions shown in Fig. 13 and 14 do not account for
the subsequent dissociation of the OH products. For example, the OH($A$)
rotational levels with $\varv\geq 2$ are dissociative and are thus discarded
in our excitation calculations.

In this work, the state distribution of the OH fragments following H2O
photodissociation by UV radiation fields of different shapes are computed
using Eq. (2) and (3). Figure 15 presents the spectral shape of the radiation
fields adopted in this work. The radiation fields representative of an
accreting T Tauri star and of the interstellar radiation field corresponds to
those used by Heays et al. (2017) and are available at
https://home.strw.leidenuniv.nl/~ewine/photo/. The UV radiation field named
”Ly$\alpha$” is a single Ly$\alpha$ line ($\lambda=121.567$ nm) with a Doppler
broadening parameter of $b=200$ km s-1. The total photodissociation cross
section of H2O leading to OH (Fig. 15) is collected by Heays et al. (2017)
from Fillion et al. (2003, 2004), Mota et al. (2005), and van Harrevelt & van
Hemert (2008).

Figure 14: Vibrational distributions of nascent OH for the two electronic
states of OH as a function of the UV wavelength. The distributions $\eta$ are
summed over rotational, fine-structure and $\Lambda$-doubling states. The
difference in the vibrational distribution between photodissociation through
the $\tilde{A}$ ($\lambda>143\leavevmode\nobreak\ $nm) and $\tilde{B}$
($\lambda<143\leavevmode\nobreak\ $nm) states of H2O is clearly visible.
Figure 15: Radiation fields and H2O photodissociation cross section for
photodissociation adopted in this work. Top: Spectral shape of the radiation
fields as a function of wavelength. The radiation fields have been scaled to
agree with the integrated energy-intensity of the Draine 1978 radiation field
between 91.2 and 200 nm, that is $2.6\times 10^{-6}$ W m-2. Bottom:
Photodissociation cross section of H2O producing OH adopted in this work
(black) and the extension to 911 Å (gray). The features due to
photodissociation through the $\tilde{A}$, $\tilde{B}$, $\tilde{C}$, and
$\tilde{D}$ states of H2O are also indicated.

##  Appendix B Collisional rate coefficients

The collisional rate coefficients between OH and H2 computed by Offer et al.
(1994) that include levels up to $N=5$ have been extrapolated to higher $N$
numbers assuming that the downward rates follow

$k_{N\rightarrow N^{\prime}}=g_{N^{\prime}}ae^{-b\Delta E/T_{K}},$ (29)

where $N$ and $N^{\prime}$ designate the rotational quantum number of the
upper and lower levels, $\Delta E$ the energy difference between these levels
and $g_{N^{\prime}}$ the degeneracy of the lower energy level. The parameters
$a$ and $b$ are the best fit coefficients of available data at temperature
$T_{K}$. Collisions with the ortho and para states of H2 have been considered
separately. In this work, $T_{K}\geq 500\leavevmode\nobreak\ $K and we adopted
the rate coefficient computed at a maximum kinetic temperature of
$T_{K}=300\leavevmode\nobreak\ $K by Offer et al. (1994) without any further
extrapolation. The relative proportion of ortho and para populations of H2 is
set to 3. We estimate that the extrapolated rate coefficients are accurate
within a factor of ten. Collisional (de)excitation involving vibrational and
electronic states is not considered here. Collisional excitation by electrons
and H can also be relevant in certain irradiated environments. However, to our
knowledge, the corresponding rate coefficients have not yet been computed:
Thus, we neglected the contribution of electrons and H to the excitation of
OH.

##  Appendix C Analytical model for mid-IR lines

Figure 16: Normalized photon intensity of the OH($X^{2}\Pi$)($\varv=0$) intra-
ladder mid-IR lines as a function of the rotational number of the upper energy
level. The intensities are computed for the fiducial parameters given in Table
1 and for three spectral shapes of the UV field (color coded), and divided by
$\Phi$ according to Eq. 13. Circle and triangle markers correspond to lines
belonging to the $\Omega=1/2$ and $\Omega=3/2$ ladders, respectively.
$\Lambda$-doublets are indiscernible in this plot. In this regime, rotational
levels are only populated by the radiative cascade of OH photofragments. The
solid line is the analytical model assuming that the intra-ladder lines within
the $\varv=0$ state are produced by the radiative cascade of OH fragments
produced in the $\varv=0$ state (Eq. (35)). The dotted line is the analytical
model that assumes that any OH produced in a vibrational state instantaneously
decays toward the ground vibrational state with a negligible change of its
rotational number (see Eq. (36)).

The link between the state distribution of OH fragments $f_{i}$ (Fig. 2 and 8)
and the variation of the line intensities with the upper $N$ number (see Fig.
6 and 9), quantified here by $\mathcal{I}_{N}$, can be clarified by a simple
analytical model of the radiative cascade. Neglecting collisional
(de)exciatation and radiative pumping, and assuming that the fraction of OH in
rotationally excited states constitutes a negligible fraction of the total
population of OH, the statistical equilibrium equation (see Eq. (7)) for
high-$N$ states yields

$\sum_{i>j}M_{ij}N_{j}+\Phi\times f_{i}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\ \leavevmode\nobreak\
M_{ij}=\cases{A}_{ij}&(E_{i}>E_{j})\\\ -A_{ij}(E_{i}<E_{j}),{}$ (30)

where $N_{i}$ is the column density of OH in the state $i$, and $A_{ij}$ the
Einstein-A coefficient of the spontaneous emission $i\rightarrow j$. This
equation being linear in $N_{i}$ and the right hand side term depending
linearly on $\Phi$, shows that $N_{i}$ is proportional to $\Phi$ and that the
relative population $N_{i}/N(\rm{OH})$ depends only on the Einstein-$A$
coefficients and on $f_{i}$ but not on $\Phi$. Consequently, the linear
relation between high-$N$ line integrated intensities and $\Phi$ shown in Fig.
5 (top right panel), as well as the fact that the global shape of the mid-
infrared spectrum is independent of $\Phi$, is a characteristic feature of the
radiative cascade.

The radiative matrix $M_{ij}$ in Eq. (30) can be greatly reduced by noting
that the radiative decays of the $N$ levels within the $X^{2}\Pi(\varv=0)$
state are dominated by pure rotational $N\rightarrow N-1$ transitions.
Moreover, excited electronic states are found to rapidly decay to the ground
electronic state with little change of their vibrational and rotational
quantum number. By neglecting the contribution of vibrationally excited levels
to the populations of OH($\varv=0,N$) and assuming that all OH photofragments
produced in the OH(A${}^{2}\Sigma^{+}$) ($\varv=0,N$) state decay immediately
to the OH($X^{2}\Pi$) ($\varv=0,N$) state, Eq. (30) applied to the
OH($X$)($\varv=0,N$) levels yields

$A_{N+1\rightarrow N}N_{N+1}-A_{N\rightarrow
N-1}N_{N}+\Phi\times\tilde{f}(N,\varv=0)\simeq 0,$ (31)

with

$\tilde{f}(N,\varv=0)=f(X,N,\varv=0)+\frac{1}{2}f(A,N,\varv=0),$ (32)

where the factor $\frac{1}{2}$ stands for the different degeneracies between
the OH($X$)($\varv,N$) and OH($A$)($\varv,N$) states. The solution of Eq. (31)
is then

$A_{N\rightarrow N-1}N_{N}\simeq\tilde{I}_{N}\Phi,$ (33)

where

$\tilde{I}_{N}\simeq\sum_{N^{\prime}\geq N}\tilde{I}(N^{\prime},\varv=0)$ (34)

is the probability to form OH in the ground vibrational state with a
rotational number larger than $N$. The high-$N$ lines are found to be
optically thin. Their integrated intensities are then given by

$I_{1}(N\rightarrow N-1)=\tilde{I}_{N}\frac{hc}{\lambda}\Phi.$ (35)

Thus, according to our analytical model, the normalized integrated photon
intensity $\mathcal{I}_{N}$ corresponds to $\tilde{I}_{N}$, the probability to
form an OH photofragment in a $\varv=0$ state with a rotational number
$N^{\prime}$ larger than $N$. In particular, the line intensities
$N\rightarrow N-1$ do not depend on the Einstein-$A$ coefficients. This
equation reflects the fact that any OH produced in an excited rotational level
$N^{\prime}$ will eventually decay through the $N\rightarrow N-1$ transition
with $N\leq N^{\prime}$. Because each rotational state of OH is split into two
spin-orbit substates that are further subdivided into two $\Lambda$-doubling
states, $\tilde{I}_{N}\leq 1/4$, as seen in Figs. 6 and 16.

As discussed in Sec. 3.1.1 and further shown in Fig. 16, our model
underestimates the line intensities computed with GROSBETA. This indicates
that the radiative decay from vibrationaly excited levels contributes to the
population of $X^{2}\Pi(N,\varv=0)$ levels. In order to take into account the
contribution of vibrationaly excited levels, we assume that any OH produced in
a vibrationally excited state immediately decays toward the
$X^{2}\Pi(\varv=0)$ state with no change of the rotational number. This leads
to a normalized integrated photon intensity of

$\tilde{I}_{N}\simeq\sum_{\varv\geq 0}\sum_{N^{\prime}\geq
N}\tilde{I}(N^{\prime},\varv).$ (36)

This model always overestimates the line intensities (dashed lines, Figs. 6
and 16). This is due to the fact that for $N\gtrsim 15$, Einstein-$A$
coefficients of rovibrational transitions are smaller than those of pure
rotational transitions. Thus, an OH fragment produced in a $\varv>0$ state
tends to decay through the intra-ladder transition within its nascent
vibrational state before decaying to the $\varv=0$ state with a lower $N$
number than its nascent one. Below $N\simeq 25$, the analytical model
reproduces well the computed intensities as all levels produced in higher $N$
numbers in $\varv>0$ have decayed toward the $\varv=0$ states. The lines
emerging from $N<25$ are thus tracing the production of OH in high rotational
states, even though the decay of electronically excited OH in the ground
vibrational state also contributes to the variation of $\mathcal{I}_{N}$ below
$N\simeq 25$.

##  Appendix D Conversion factors

Table 4 gives the line intensities of the OH mid-IR lines in the pure
radiative cascade regime normalized by $\Phi^{\tilde{B}}$ denoted as
$\mathcal{I}^{\tilde{B}}$ (see Eq. (18)). The intensity of the
$N_{up}\rightarrow N_{up}-1$ lines of the OH($X$)($\varv=0$) state are summed
over the four components of the quadruplet. $\mathcal{I}^{\tilde{B}}$ can be
used to convert any line intensity into a column density of H2O
photodissociated per second through the H2O $\tilde{B}$ state denoted as
$\Phi^{\tilde{B}}$ using Eq. (20).

Table 4: Conversion factors between the photon intensity integrated over solid angle (in photon cm-2 s-1) of the rotational lines $N_{up}\rightarrow N_{up}-1$ summed over the components of each quadruplet and $\Phi^{\tilde{B}}$ (molecule cm-2 s-1). $N_{up}$ | Ly$\alpha$ | T Tauri | ISRF | BB $10^{4}$ K
---|---|---|---|---
10 | 0.96 | 0.94 | 0.88 | 0.88
11 | 0.95 | 0.92 | 0.86 | 0.85
12 | 0.93 | 0.91 | 0.84 | 0.83
13 | 0.92 | 0.89 | 0.82 | 0.81
14 | 0.90 | 0.88 | 0.80 | 0.79
15 | 0.88 | 0.86 | 0.78 | 0.77
16 | 0.87 | 0.84 | 0.76 | 0.75