text:
\section{Introduction}
\label{sec: intro}

Emission and absorption line studies of the 21 cm hyperfine spin-flip transition of atomic hydrogen (H{\small I}) show that cold ($T<200$ K) dense clumps are commonly embedded within extended warm ($T\sim 8000$ K) diffuse atomic clouds.
The cold neutral medium (CNM) clumps are observed as narrow absorption line features embedded within broad emission line envelopes tracing the warm neutral medium (WNM) component
(e.g., in Galactic observations: \citealt{Dickey1978, Kulkarni1987, Colgan1988, Garwood1989, Kolpak2002, Dickey2003, Heiles2003, Roy2013, Stanimirovic2014a, Patra2018, Murray2018},
and in external galaxies:
\citealt{Braun1992, Dickey1993, Mebold1997, Dickey2000, Marx-Zimmer2000, Warren2012, Bolatto2013, Herrera-Camus2017}).
  Remarkably, for the characteristic thermal pressures prevailing in the interstellar gas, theory predicts that a multiphase medium is possible in which the CNM and WNM can coexist \citep[][]{Field1969, Wolfire1995, Liszt2002, Wolfire2003}.
The multiphase nature of the H {\small I} gas is also clearly observed in numerous hydrodynamical simulations of the interstellar medium (ISM)  \citep[e.g.,][]{Hu2016, Valdivia2015, Richings2016, Hill2018}.

In the standard theory for the thermal structures of interstellar H{\small I}, the WNM is cooled by electron impact Ly$\alpha$ emission \citep{Gould1970, Dalgarno1972}, and the cold medium by metal line fine structure cooling, mainly neutral impact \cii 158 $\mu$m emission \citep{Dalgarno1964}.  For a given heating rate and at sufficiently low densities metal line cooling becomes very inefficient and the gas heats up to temperatures ($\sim 8000$ K) at which the Ly$\alpha$ emission limits further heating.  At high densities the metal line cooling becomes effective and the temperature drops to values characteristic of the fine-structure energy level spacings ($\sim 100$ K). Because of the particular shape of the combined Ly$\alpha$ plus fine-structure cooling function (we discuss this in detail), an intermediate range of densities and associated pressures exists at which the gas can be either warm or cold.

Hydrodynamical simulations show that CNM forms at sites where flows of WNM collide and compress the gas \citep[e.g., ][]{Hennebelle1999, Kritsuk2002, Audit2004, Vazquez-Semadeni2006, Gazol2013, Saury2014, Gazol2016a}.
The formation of the dense CNM phase and the multiphase structure of the ISM may play a key role in controlling the star-formation properties of galaxies.
For example,
\citet{Elmegreen1994, Schaye2004} have argued that the
 low pressure prevailing at the outskirts of galaxy discs do not allow the formation of the cold phase resulting in a shut-off of star-formation.
The multiphase phenomena where CNM and WNM phases coexist at the same pressure may provide a feedback loop for star-formation in galaxies
\citep{Corbelli1988, Parravano1988, Parravano1989, Ostriker2010, Kim2011, Ostriker2011}.
If the star-formation rate drastically increases,  the high heating rate induced by the stellar UV prevents the formation of the CNM phase, which suppresses
star-formation.

Previous computations of the H{\small I} WNM/CNM thermal phase structures have generally assumed that metals dominate the CNM cooling. Then, as would be expected, as the metal abundance is reduced higher particle densities are required to compensate for the reduced cooling, and the multiphase behavior is predicted to occur at higher gas densities and thermal pressures \citep{Wolfire1995, Liszt2002, Walch2011a}.
The central question we address in this paper is how WNM/CNM multiphase behavior of atomic hydrogen (H{\small I}) gas is affected by H$_2$ cooling and heating processes as the metallicity becomes vanishingly small.

H$_2$ cooling in the context of primordial and/or low-metallicity gas and first star-formation has been considered by many authors over many years \citep[e.g.,][]{Saslaw1967,
Lepp1984,
Shapiro1987,
Haiman1996a,
Tegmark1997,
Omukai2000,
Abel2000,
Yoshida2006,
Glover2007b,
Schneider2012,
Wolcott-Green2017}.
It has long been known that in the absence of metals, just small amounts of H$_2$ can enable
cooling to a few 100 K (see for example the recent study by \citealt{Glover2014}) thereby controlling the fragmentation and dynamical properties of collapsing star-forming clouds in the low-metallicity regime. Less attention has been given to the specifics of WNM to CNM transitions and possible multiphase behavior at very low-metallicities. For example, in their study of protogalactic disks \citet{Norman1997} presented pressure versus density thermal phase diagrams for atomic gas over a range of redshifts and associated metallicities.
In their models
multiphase behavior disappears at high redshifts (see their Fig.~5). However it remains unclear what are the individual roles of the metallicity and radiation field when H$_2$ processes begin playing a role, and how these processes affect the phase structure.  Similarly, \citet{Inoue2015} carried out numerical hydrodynamical simulations of isobarically contracting gas clouds and studied the effects of H$_2$ cooling on thermal instabilities.  They found that at sufficiently low metallicities H$_2$ acts to smooth out thermal instabilities (see their Fig.~2).  But why?

The goal of our paper is to study the competing microphysical processes that control the phase structures of atomic gas, into the regime of vanishing metallicity. We address the following questions.
How are the WNM and CNM phases  affected by the onset of H$_2$ cooling (and heating)?
Is multiphase gas possible when H$_2$ replaces metal fine-structure line cooling?
If not, why not?
In this study we present numerical computations of the (equilibrium) thermal states, with UV radiation intensity, cosmic-ray (or X-ray) ionization rate, and metallicity, as distinct parameters, and with the addition of the H$_2$ processes in a step by step fashion.
This enables an analytic description of the behavior which we also present.

The structure of our paper is as follows. In \S \ref{sec: model} we present the governing equations, and discuss the chemical and thermal processes we include.
We begin with a discussion of the solar metallicity model in \S \ref{sec: Z=1}.
We discuss low metallicity behavior in \S \ref{sec: low Z} and \ref{sec: Parameter Study}, down to effectively pristine H$_2$ dominated gas.
We characterize the critical densities or pressures above which the gas may cool below $\sim 10^4$~K and condense into intermediate ($\sim 500$ K) or cold ($\sim 50$~K) phase (depending on metallicity), and whether two-phase media can exist, as function of metallicity, UV intensity and CR ionization rate.
We summarize in \S \ref{sec: conclusions}.

\section{Model Ingredients}
\label{sec: model}

We compute the temperatures and gas phase chemical compositions of the neutral atomic hydrogen (H{\small I}) components of the interstellar medium (ISM) of galaxies
assuming thermal equilibrium and chemical steady state. We use the results to construct pressure versus density phase diagrams for the H{\small I} gas. We consider standard heating and cooling mechanisms, with the addition of molecular hydrogen (H$_2$) processes that become important and eventually dominant in the limit of low metallicity. We investigate the role of the assumed far-UV radiation intensities and cosmic ray-ionization rates on the coupled thermal and chemical properties of the H{\small I} gas. We study the behavior of the thermal phase structures for standard ISM conditions into the limit of vanishing metallicity.

\subsection{Basic Equations}
\label{sub: basic equations}

In thermal equilibrium,
\begin{equation}
\label{eq: cooling-heating}
L(n,T; x_i) =  G(n,T; x_i)  \ ,
\end{equation}
where $G$ and $L$ are the total heating and cooling rates per unit volume (erg cm$^{-3}$ s$^{-1}$). The rates depend on the total hydrogen volume density $n$ (cm$^{-3}$), the gas temperature $T$ (K), the dust-to-gas abundance ratio, and the gas phase abundances of the chemical species, $x_i \equiv n_i/n$, especially for C$^+$, O, electrons, and H$_2$.
For heating we include far-UV photoelectric (PE) ejection of electrons from dust grains and polycyclic aromatic hydrocarbons  (PAHs),
and cosmic-ray (CR) and/or X-ray ionization of the hydrogen atoms. For cooling, we include H{\small I} Ly$\alpha$ line-emission excited by electron impacts, and fine-structure \cii (157.7$\mu$m), \oi (63.2$\mu$m, 145.5$\mu$m),  and [C{\small I}] (609.1$\mu$m, 370.4$\mu$m) line emissions excited primarily by collisions with the neutral hydrogen atoms. We also include energy losses due to electron recombination with dust grains and PAHs, and due to gas-dust collisions.
As discussed by \citet[][hereafter, \citetalias{Wolfire2003}]{Wolfire2003} these are the dominant heating and cooling mechanisms for thermally stable and unstable warm/cold ($\sim10^4$ to $\sim100$ K) interstellar H{\small I} for standard ISM conditions
\footnote{Metastable line emission (e.g.~[OI] $6302$ ${\rm \AA}$) also cools the WNM, but its contribution is subdominant (\citetalias{Wolfire2003}, see also Fig 30.1 in \citealt{Draine2011}).}.

Importantly, we also incorporate heating and cooling processes involving H$_2$.
This includes heating via the release of binding energy during H$_2$ formation, kinetic energy imparted to the H-atoms produced by H$_2$ photodissociation, and internal H$_2$ energy released to the gas via collisional deexcitation of UV excited states (we refer to this latter process as "UV pumping heating"). Cooling is mainly by H$_2$ ro-vibrational line-emissions excited by collisions with electrons and neutrals.
We also include cooling by
isotopic HD line emissions, and due to dissociation of the H$_2$ in collisions with thermal electrons and H-atoms,
although these are minor.
In the Appendix we present detailed discussions and expressions for the various heating and cooling processes we have included in our computations.
The roles and effects of H$_2$ heating and cooling processes have not been considered in previous analytic studies of the multiphase structure of the neutral ISM (\citealt{Field1969, Wolfire1995, Liszt2002}; \citetalias{Wolfire2003}).
We will show how the H$_2$ heating and cooling processes influence the thermal structures as the heavy element abundances become small.

Eq.~(\ref{eq: cooling-heating}) is coupled to the equilibrium rate equations for the chemical abundances,
\begin{equation}
\label{eq: chem rates}
\sum_{j>l} x_j x_l \ n k_{jl,i}  +  \sum_{j} x_j q_{j,i} \ = \ 0 \ .
\end{equation}
This is a set of $N$ equations, equal to the number of species in the chemical network.
The first sum is over the formation and destruction rates (s$^{-1}$) of species $i$ via two-body reactions involving species $j$ and $l$, where
$k_{jl,i}$ are the temperature-dependent
reaction rate coefficients (cm$^3$ s$^{-1}$).
The second sum is over the formation and destruction rates of species $i$ due to ionization or dissociation of species $j$ by either cosmic-rays or UV photons, and the $q_{j,i}$ are the rates (s$^{-1}$) for these processes.
For $j=i$, the coefficients $k_{jl,i}$ and $q_{j,i}$ are negative and represent the destruction of species $i$ in reactions with species $l$, and by UV and CR dissociations and ionizations. For $j\neq i$ the coefficients are positive and represent the formation channels for species $i$.
Eqs.~(\ref{eq: chem rates}) are augmented by the (dependent) conservation equations for the total elemental abundances and electric charge.
These are
\begin{equation}
\label{eq: conserv}
    X_{m} = \sum_i \alpha_{i, m} x_{i} \ ,
\end{equation}
where $X_{m}$ is the total elemental abundance of element $m$, and $\alpha_{i, m}$ is the number of atoms of element $m$ in species $i$.
Eq.~(\ref{eq: conserv}) also describes the electric charge conservation equation, with $\alpha_{i, m}$ representing the number of positive charges in species $i$, and with $X_{m}=0$.
In total, there are $M+1$ conservation equations, where $M$ is the number of elements in the chemical network.
We discuss our chemical network in \S \ref{sub: numerical method}.

\subsection{Parameters}
\label{sub: params}

The gas temperature $T$, and species abundances $x_i$, depend on the gas density, $n$,  the far-UV radiation intensity, $I_{\rm UV}$, the CR ionization rate, $\zeta_{\rm cr}$, and the metallicity $Z'$. We define these parameters below. Additional physical parameters affecting the thermal and chemical solutions are listed in Table \ref{table: params}.

\begin{table*}[]
\caption{Parameters}
\centering
\begin{tabular}{l l l l}
\hline\hline
notation  & fiducial value /  range & meaning & comments
  \\ [0.5ex]
   \hline
$Z'$ & from 10$^{-5}$ to 3 & metallicity relative to Galactic & see \S \ref{sub: params: Z and DGR}, \S \ref{sec: low Z} \\
$I_{\rm UV}$ &  0.1, 1, 10  & the 6-13.6 eV UV intensity, relative to Galactic & see \S \ref{sub: params: IUV}, \S \ref{sub: var UV zeta} \\
$\zeta_{-16}=\frac{\zeta_{cr}}{10^{-16} {\rm s^{-1}}}$ &  0.1, 1, 10 & the total (primary+secondary) ionization rate per H atom  &  see \S \ref{sub: params: zeta}, \S \ref{sub: var UV zeta} \\
     \\[0.2ex]
   \hline

   $A_{\rm C}$, $A_{\rm O}$ & 1.4 and 3.2 $\times 10^{-4}$   & the carbon and oxygen solar elemental abundances  & see \S \ref{sub: params: Z and DGR} and Eq.~(\ref{eq: C and O abundances}) \\
      $\delta_{\rm C}$, $\delta_{\rm O}$ & 0.53 and 0.41   & the carbon and oxygen dust depletion factors & see \S \ref{sub: params: Z and DGR} and Eq.~(\ref{eq: C and O abundances}) \\
$Z'_d$ &  & the DGR relative to Galactic & see \S \ref{sub: params: Z and DGR} and Eq.~(\ref{eq: Zd-Z}) \\
$\zeta_p$ &  $\zeta_{cr}/(1+\phi_s)$ & the {\it primary} ionization rate per H atom & see \S \ref{sub: params: zeta} and Eq.~(\ref{eq: phi_s})  \\
$\phi_s$ &  $\approx 0.6$ & the number of secondary electrons produced per ionization & see \S \ref{sub: params: zeta} and Eq.~(\ref{eq: phi_s})  \\

$D_0$ &  $5.8 \times 10^{-11}$~s$^{-1}$ & the H$_2$ photodissociation rate at $I_{\rm UV}=1$  & see \S \ref{sub: params: IUV}, \S \ref{sub: var UV zeta}  \\
$D_-$ &  $5.6 \times 10^{-9}$~s$^{-1}$ & the H$^-$ photodetachment rate at $I_{\rm UV}=1$  & see \S \ref{sub: params: IUV}, \S \ref{sub: var UV zeta}  \\
$R$ &  (cm$^{3}$ \ s$^{-1}$) & the H$_2$ formation rate coefficient  & see \S \ref{sub: params: R_form}\\
$x_i=n_i/n$ &  & the species abundance, relative to the hydrogen nucleon density  & see \ref{sub: basic equations} and \ref{sub: params: R_form} \\
$L$, $G$ &  (erg \ cm$^{-3}$ \ s$^{-1}$) & the cooling and heating rates per unit volume  & see \S  \ref{sub: basic equations} and \ref{sub: app_heat_cool}
\\
$\mathcal{L}_i(T)$ &  (erg \ cm$^{3}$ \ s$^{-1}$) & the cooling efficiency of species $i$. For $n/n_{\rm crit} \ll 1$, $L_i=x_i n^2 \mathcal{L}_i(T)$ & see \S \ref{sub: app_heat_cool} \\
$\Lambda_i(T)$ &  (erg \ s$^{-1}$) & the cooling rate per particle.
For $n/n_{\rm crit} \gg 1$, $L_i=x_i n \Lambda_i(T)$ &  see \S \ref{sub: app_heat_cool} \\
\hline
\end{tabular}
\label{table: params}
\end{table*}

\subsubsection{Metallicity and dust-to-gas ratio}
\label{sub: params: Z and DGR}

The metallicity, i.e.~the gas phase abundances of the heavy elements, and the dust abundance, are crucial parameters for the thermal balance.
The gas phase carbon and oxygen abundances determine the efficiency of metal-line cooling, and the dust-to-gas mass ratio determines the efficiency of PE heating and dust recombination cooling.
We assume that the total {\it gas-phase} abundances of the carbon and oxygen species are given by
\begin{equation}
\label{eq: C and O abundances}
    X_{\rm m} = A_{\rm m} Z' \left( 1- \delta_{\rm m} \times (Z'_d/Z') \right)  \ ,
\end{equation}
where $m=$C, or O.
In these expressions $Z'$ is the normalized metallicity such that $Z'=1$ corresponds to the Solar elemental abundances for which $A_{\rm O}=5.4 \times 10^{-4}$ and $A_{\rm C}=3.0 \times 10^{-4}$ \citep{Asplund2009}, $\delta_{\rm C}$ and $\delta_{\rm O}$ are dust depletion factors, and $Z'_d$ is the normalized dust-to-gas ratio.
We adopt
\begin{equation}
\label{eq: Zd-Z}
Z_d' =
\begin{cases}  Z' & \text{for } Z' \geq Z_0' \\
 Z_0' \ (Z'/Z_0')^{\alpha} & \text{for } Z' < Z_0' \ ,
\end{cases}
\end{equation}
with $Z_0'=0.2$, and $\alpha=3$.
This broken power-law relation is suggested by recent observations of low metallicity galaxies \citep{Remy-Ruyer2013}. Here, $Z'_d=1$ corresponds to a standard ISM dust-to-gas ratio of $6.2 \times 10^{-3}$ \citep[][Table 6, the BARE-GR-S model]{Zubko2003}.

For the dust depletion factors, we adopt
$\delta_{\rm C}= 0.53$ and $\delta_{\rm O}= 0.41$.
With these values,  we recover the characteristic interstellar Galactic carbon and oxygen gas-phase abundances $X_{\rm C} = 1.4 \times 10^{-4}$, $X_{\rm O}=3.2 \times 10^{-4}$ at $Z'=Z'_d=1$ \citep{Cardelli1996, Sofia1997, Meyer1997}.
On the other hand, at very low metallicities, when $Z'_d/Z' \ll 0.2$, the dust depletion terms vanish, and all the available carbon and oxygen remain in the gas-phase, with $X_{\rm C} = A_{\rm C}Z'$, $X_{\rm O}=A_{\rm O}Z'$.

\subsubsection{Cosmic-ray and X-ray ionization}
\label{sub: params: zeta}
Cosmic rays (or X-rays) ionize atomic and molecular hydrogen,
\begin{align}
\label{reac: H CR}
{\rm H \ + \ cr \ } & \rightarrow {\rm \ H^+ \ + \ e} \\
{\rm H_2 \ + \ cr \ } & \rightarrow {\rm \ H_2^+ \ + \ e} \\
{\rm H_2 \ + \ cr \ } & \rightarrow {\rm \ H^+ \ + \ H \ + \ e}
\end{align}
producing energetic non-thermal electrons.
The electrons generated by these primary ionizations lose energy through coulomb scatterings that heat the gas, and through further atomic and molecular excitations and secondary ionizations \citep{Dalgarno1999a}.
The total (primary+secondary) ionization rate per hydrogen atom is $\zeta_{\rm cr}=\zeta_p \times (1+\phi_s)$, where $\zeta_p$ is the primary ionization rate and $\phi_s \sim 0.7$ is the number of secondary electrons produced per primary ionization.
The cosmic-ray (CR) ionization rate determines the CR heating rate
and the electron abundance (see Eqs.~\ref{eq: xe} and \ref{eq: CR heating}, below).
For typical Galactic values, $\zeta_{\rm cr} \approx 10^{-16}$~s$^{-1}$ \citep{Indriolo2012, Tielens2013}, and we define
the normalized CR ionization rate, $\zeta_{-16} \equiv \zeta_{\rm cr}/(10^{-16} \ {\rm s^{-1}})$.
Our fiducial value is $\zeta_{-16} =1$. We study the effects of varying $\zeta_{-16}$ in \S \ref{sub: var UV zeta}.

X-rays can play an important role in providing an additional source of  ionization, affecting the chemistry and the heating rate.
We do not explicitly include X-ray ionization in this study,
however,
these may be simulated by choosing an appropriate values for $\zeta_{\rm cr}$ and $\phi_s$.

\subsubsection{Interstellar far-UV radiation field}
\label{sub: params: IUV}

The far-UV radiation intensity determines the efficiency of photoelectric heating, the H$_2$ photodissociation rate, and the production rate of the C$^+$ coolant.
\citet{Draine1978, Draine2011} estimated the local UV interstellar radiation field (ISRF), and obtained
$u_{\rm UV,0}=0.056$ eV cm$^{-3}$, for the energy density over the  6-13.6 eV band (=1.69 times the \citealt{Habing1968} estimate).
We adopt the Draine spectral shape and allow for variations in the UV intensity by multiplying the overall spectrum with a normalized intensity parameter, $I_{\rm UV}$, such that $u_{\rm UV} = u_{\rm UV,0} I_{\rm UV}$.
The UV intensity determines the PE heating rate through Eq.~(\ref{eq: p.e. heating}).
For a unit Draine ISRF, the free-space (unattenuated) H$_2$ photodissociation rate is
$D_0= 5.8 \times 10^{-11}$~s$^{-1}$ \citep{Sternberg2014}.
At low metallicity, a harder spectrum may be more typical, since the first generations of stars (pop-III stars) are theorized to be very massive, with stellar effective temperatures as high as 10$^5$ K \citep[e.g.,][]{Cojazzi2000, Barkana2001, Abel2002, Yoshida2012}.

Longer wavelength photons, down to 0.75 eV, play a role in removing the H$^-$ anion
via photodetachment \citep{Miyake2010}. H$^-$ is a critical intermediary for H$_2$ formation at low metallicity (see \S \ref{sub: params: R_form}, below).
\citet{Mathis1983} estimated the Galactic ISRF within this band as a combination of diluted blackbodies at 3000, 4000 and 7500 K.
For an ISRF combined of the Draine field over the 6-13.6 eV band, and with the Mathis field over the 0.75-6 eV band, the H$^-$ photodetachment rate is $D_- =2.7 \times 10^{-7}$ s$^{-1}$.
In contrast, for a 10$^5$ K blackbody (BB) spectrum (normalized such that the photon density in the 6-13.6 eV band equals that of the Draine field), $D_- = 2.8 \times 10^{-9}$ s$^{-1}$ (\citealt{Bialy2015a}; see their Table 2).
For our fiducial models we seek a compromise between these two extreme cases and extrapolate the \citet{Draine1978} spectrum (see their Eq.~11) down to 0.75 eV.
The resulting H$^-$ photodetachment rate is $D_- = 5.6 \times 10^{-9}$~s$^{-1}$.
This value is close to the 10$^5$ K BB value which represents the pristine metallicity gas, but is $\sim 2$ orders of magnitude smaller than the Draine+Mathis value, which represents the Galaxy and a high metallicity ISM.
For moderately low metallicities $D_-$ may range between these two extremes, and will depend on the form of the initial mass function (IMF).
We explore the effect of variations in $I_{\rm UV}$ and $D_-$ in \S \ref{sub: var UV zeta}.

The free-field photodissociation rate, $D_0$ may be strongly reduced in the interiors of a cloud due to
dust absorption and H$_2$ self-shielding, leading to partial or complete H {\small I}-to-H$_2$ conversion \citep[][]{VanDishoeck1986, Draine1996, McKee2010, Sternberg2014, Bialy2016a}.
In this paper we focus on the thermal properties of the {\it global atomic} ISM, rather than the structure of the H {\small I}-to-H$_2$ transition or the interior shielded molecular gas.
Thus, we always use the free-space UV field and do not include radiation attenuation in our models.
For the WNM shielding is in any case not significant (except for extremely large shielding columns).
For the CNM, our model applies to the cloud outskirts that are exposed to the free-field UV.

\subsubsection{H$_2$ Formation}
\label{sub: params: R_form}

At sufficiently high metallicity, the H$_2$ is formed predominantly on dust grains.
We adopt
\begin{equation}
\label{eq: RZ}
R_{d} =3 \times 10^{-17} \left( \frac{T}{ \rm 100 \ K }\right)^{1/2} Z_d' \ \ {\rm cm}^3 {\rm s}^{-1} \ \ \ .
\end{equation}
Sophisticated models of the H$_2$ formation process on  dust-grains \citep[e.g.,][]{Cazaux2002,LeBourlot2012, Bron2014} derived more complicated dependences on gas temperature, as well as on other physical parameters (e.g., through the dust temperature which in turn depends on the UV radiation intensity). However for our purposes such a simple description is sufficient, because as we show below, H$_2$ affects the thermal structure only at low metallicities where its formation is dominated through gas-phase processes.

 At low metallicities ($Z' \lesssim 0.1$), H$_2$ is formed mainly through gas-phase reactions, with the most important one being the H$^-$ formation route \citep{McDowell1961, Peebles1968, Hirasawa1969, Galli1998a}.
 Although, the H$_2$ formation rate is obtained through the numerical solution to the chemical network (as discussed in \S \ref{sub: basic equations}), it is useful to obtain approximate analytic equations for the effective gas-phase formation rate coefficient, and the resulting H$_2$ fractional abundance.

This formation channel is initiated by radiative attachment
\begin{equation}
\label{reac: H- formation}
{\rm H \ + \ e \ \rightarrow \ H^- + \nu} \ ,
\end{equation}
followed by associative detachment
\begin{equation}
\label{eq: H2 form gas reac}
{\rm H^{-} \ + \ H \ \rightarrow \ H_2 \ + \ e } \ \ \ .
\end{equation}
Assuming that the electron abundance is set by a balance between CR ionization of the atomic hydrogen and proton-electron recombination, we have
\begin{equation}
\label{eq: xe}
x_{\rm e} = \left( \frac{\zeta}{\alpha_B n} \right)^{1/2} = 2.7 \times 10^{-3} T_3^{3/8}  \zeta_{-16}^{1/2} \left( \frac{n}{10 \ {\rm cm^{-3}}} \right)^{-1/2}\ ,
\end{equation}
where $T_3 \equiv T/({\rm 10^3 \ K})$
and $\alpha_B$ is the case-B recombination rate coefficient.
The effective H$_2$
formation rate coefficient through this channel is
\begin{equation}
\label{eq: R gas phase}
R_{\rm -} \equiv x_{\rm e} k_{\ref{reac: H- formation}} \eta \approx 1.9 \times 10^{-18} \  T_3^{1.02} \eta \ \zeta_{-16}^{1/2} \left( \frac{n}{10 \ {\rm cm^{-3}}} \right)^{-1/2} \ {\rm cm^3 \ s^{-1}}
\end{equation}
 where
$k_{\ref{reac: H- formation}}$
is the radiative attachment rate coefficient\footnote{The rate coefficients are based on UMIST 2012 \citep{McElroy2013}.
See Table \ref{table: reduced netwrok} for a summary of the most important reactions and rates. See online material for the full network.} (Eq.~\ref{reac: H- formation}).
The H$_2$ formation sequence is moderated mainly by photodetachment,
\begin{equation}
\label{reac: H- photodetachment}
{\rm H^- \ + \nu \ \rightarrow \ H \ + \ e} \ \ \ .
\end{equation}
and also by mutual neutralization with protons,
\begin{equation}
\label{reac: mutual neutralization}
{\rm H^- \ + H^+ \ \rightarrow \ H \ + \ H} \ \ \ .
\end{equation}
In Eq.~(\ref{eq: R gas phase}) we define the branching ratio
\begin{equation}
\label{eq: eta}
\eta = \left( 1+ \frac{D_- I_{\rm UV}}{k_{\ref{eq: H2 form gas reac}} n} + \frac{k_{\ref{reac: mutual neutralization}} x_{\rm H^+}}{k_{\ref{eq: H2 form gas reac}}} \right)^{-1}
\end{equation}
for H$_2$ formation via associative detachment versus H$^-$  photodetachment and mutual-neutralizaion.
For $D_-=5.6\times 10^{-9}$ s$^{-1}$
photodetachmnet becomes important when
 $n/I_{\rm UV} \lesssim 1$~cm$^{-3}$, while mutual-neutralization
 is important when
$n/\zeta_{-16} \lesssim 0.01$~cm$^{-3}$.
As we show below, H$_2$ plays an important role in cooling and heating at densities $\gg 1$ cm$^{-3}$.
Thus, generally, $\eta \approx 1$ is a good approximation \footnote{For high $D_-$ values, $\eta$ may become smaller than unity at the cooling point, reducing the H$_2$ abundance. We discuss this in \S \ref{sub: var UV zeta}.}.

Assuming that H$_2$ destruction is dominated by photodissociation,
we obtain an analytic approximation for the H$_2$ fractional abundance at low metallicities,
\begin{equation}
\label{eq: xH2 gas}
x_{\rm H_2} =  \frac{x_{\rm e} k_{\ref{reac: H- formation}} n}{D_0 I_{\rm UV}}  \eta = 3.3 \times 10^{-7} T_3^{1.01} \eta \zeta_{-16}^{1/2}I_{\rm UV}^{-1} \left( \frac{n}{10 \ {\rm cm^{-3}}} \right)^{1/2}  \ .
\end{equation}
We find Eq.~(\ref{eq: xH2 gas}) to be in good agreement with the numerical solution for the chemical network, over the gas densities and temperatures in our parameter space.

\begin{figure*}[t]
	\centering
	\includegraphics[width=1\textwidth]{plots/fig1_Z1.png}
	\caption{
The temperature, pressure, and cooling and heating rates per hydrogen nucleus as functions of the volume density, assuming solar metallicity and $I_{\rm UV}=\zeta_{-16}=1$.
The thin diagonal (horizontal) lines in the left panels indicate the minimum and maximum pressures for a multiphase medium.
The dashed red and blue curves in the left panels show models with $I_{\rm UV} = \zeta_{-16} = 0.1$ and 10, respectively.
		}
		\label{fig: Tn_Z1}
\end{figure*}

\subsection{Numerical Method}
\label{sub: numerical method}

Given a set of parameters, $I_{\rm UV}$, $\zeta_{\rm cr}$, and $Z'$, we solve for the equilibrium temperature and species abundances as functions of the hydrogen density, $n$.
We form a logarithmic grid, spanning $\log T=1$ to $4.2$ and $\log n=-2$ to 6, with $100 \times  90$ sampling points.
For each volume density in the grid, we calculate the species abundances coupled with the net cooling, $L-G$, as functions of the temperature, and find the temperature(s) for which the thermal equilibrium condition (Eq.~\ref{eq: cooling-heating}) is satisfied.
This gives us the $T(n)$ curves shown in Figs.~\ref{fig: Tn_Z1} and \ref{fig: Tn_diff_Z} below. We compute the equilibrium thermal pressure (cm$^{-3}$~K) by simply multiplying by the density, i.e., $P(n)=nT(n)$. For any density, heating exceeds cooling for pressures less than $P(n)$, and cooling exceeds heating for pressures greater than $P(n)$.

For our chemical network (Eq.~\ref{eq: chem rates}),
we consider a set of 294 two-body gas-phase reactions among 34 species, composed of hydrogen, helium, carbon, and oxygen bearing atomic and molecular species.
We adopt the rate coefficients suggested by the UMIST 2012 database \citep{McElroy2013}, except for the H$_2$ collisional dissociation for which we adopt the \citet{Martin1996} rate (including both collisional dissociation and dissociative tunneling).
We also include the  formation of H$_2$ on dust grains, CR reactions and photo-reactions.
The full reaction list is provided in the supplement materials (online version).
The dominant photorates are also listed in \citealt[][see their table 2]{Bialy2015a}.

In total we obtain a set of 34 coupled non-linear algebraic equations for the 34 species. The equations depend on $T$ and $n$, as well as on the additional physical parameters, $I_{\rm UV}$, $\zeta_{\rm cr}$, and $Z'$.
We replace five of these equations with the element conservation (H, He, C, and O), and charge conservation equations. For any density $n$ there is always a single unique solution for the temperature $T$ and abundances $x_i$.
Rather than solving the equilibrium algebraic equations (e.g., via Newton-Raphson iteration), we instead integrate the time-dependent rate equations until a time, $t$, where $dx_i/dt=0$ and equilibrium is reached.
We find this approach to be more robust.

To improve computational speed, we also consider a reduced chemical network,
which allows us to calculate the steady-state temperature as a function of metallicity and volume densities (or pressures) over a very well sampled grid, as presented in Figs.~\ref{fig: T_2d} and  \ref{fig: T_2d_vars}.
The reduced network is discussed in the Appendix.

\section{Thermal Structure at Solar Metallicity}
\label{sec: Z=1}

In this section we first present results for our solar metallicity reference models ($Z'=Z'_d=1$).
The solid curves in Fig.~\ref{fig: Tn_Z1} show the temperature (upper left panel), the thermal pressure (lower left panel), and the cooling and heating rates per hydrogen nucleus (right panels), as functions of the hydrogen nucleus gas density $n$, assuming $I_{\rm UV}=\zeta_{-16}=1$.
The dashed curves in the left panels, located above and below the solid curve, are for $I_{\rm UV}=\zeta_{-16}=10$ and 0.1, respectively.

First we focus on the behavior at low to intermediate densities, $n \lesssim 10^3$ cm$^{-3}$, where H$_2$ cooling and heating processes are subdominant.
In this regime, our results
essentially reproduce those presented by \citet{Wolfire1995} and \citetalias{Wolfire2003}.
The behavior is controlled by the temperature dependence of the Ly$\alpha$ and fine-structure metal-line cooling functions. We display these cooling functions in the upper panel of Fig.~\ref{fig: cooling functions}. The black curve shows the exponentially steep electron impact excitation Ly$\alpha$ cooling rate (erg s$^{-1}$) per hydrogen nucleus. The blue curves show the much flatter metal cooling rates for $Z'=1$, 10$^{-2}$, and 10$^{-4}$, assuming all of the gas-phase carbon is C$^+$, and the oxygen is in neutral atomic form. The metal cooling is controlled by electron and neutral-hydrogen impact excitation of the fine-structure levels. The solid and dashed curves in Fig.~\ref{fig: cooling functions} are for $x_{\rm e}=10^{-4}$ and 10$^{-2}$ respectively. For the high $x_{\rm e}$, and when $T \lesssim 500$ K, metal line cooling is dominated by electron impact excitation.
The shapes of these curves are of the utmost importance.

Cooling becomes inefficient compared to heating as the density becomes small.
For $n \lesssim 0.1$~cm$^{-3}$, the gas becomes warm, and Ly$\alpha$ dominates the cooling.
In this regime the temperature depends weakly on density and remains close to 10$^4$~K.
This is because for $T \lesssim 10^4$~K Ly$\alpha$ cooling is exponentially sensitive to the temperature (see Fig.~\ref{fig: cooling functions}), and any increase in the cooling rate due to an increase in gas density requires just a small decrease in temperature to balance the heating rate. At $n \approx 1$ cm$^{-3}$,
the temperature decreases to $\sim 6 \times 10^3$~K, at which point energy losses by the \cii and \oi emissions become comparable to Ly$\alpha$, as seen in Fig.~\ref{fig: Tn_Z1} and Fig.~\ref{fig: cooling functions}.
Above this density the metals dominate the cooling and the temperature falls abruptly.
The sharp drop occurs because at these high temperatures (and as long as $T \gtrsim 300$~K, i.e.~above the transition energies of the fine-structure lines) metal cooling is a weak function of temperature (Fig.~\ref{fig: cooling functions}). Any increase in density then requires a large decrease in temperature to balance the heating rate.
We refer to the density at which metal cooling takes over as the ``cooling point density", and we provide analytic estimates in \S 4.
Below a few 100~K, the metal line cooling becomes exponentially temperature-sensitive (Fig.~\ref{fig: cooling functions}), and the equilibrium $T(n)$ curve flattens again.

In Fig.~\ref{fig: Tn_Z1}, the low-density flat part of the $T(n)$ curve followed by the sharp temperature drop gives rise to a local maximum, $P_{\rm max}$, in the $P(n)$ curve.
In general, a local maximum such as this occurs if and where the temperature starts decreasing more rapidly than $1/n$.
For our fiducial model with $I_{\rm UV}=\zeta_{-16}=1$, we find $P_{\rm max}=7.6 \times 10^3$ cm$^{-3}$ K occurring at a density $n_{\rm max}=1.1$ cm$^{-3}$.  For $n<n_{\rm max}$ the gas is warm with $T_{\rm WNM}=6000$ to $10^4$~K. This is the classical warm neutral medium (WNM), and $P_{\rm max}$ is the maximum pressure possible for the WNM.
Starting from cold gas at high densities, the steep rise of the $T(n)$ curve as the density is {\it reduced
} gives rise to a local minimum, $P_{\rm min}$, in the pressure versus density curve.
In general, a local minimum occurs if and where the temperature starts rising more rapidly than $1/n$ as the density is reduced.
For our fiducial model $P_{\rm min}=2.6 \times 10^3$ cm$^{-3}$ K at $n_{\rm min}=7.3$ cm$^{-3}$. For $n>n_{\rm min}$, the gas remains cold with $T_{\rm CNM} \lesssim 300$ K.  This is the cold neutral medium (CNM) and $P_{\rm min}$ is the minimum pressure possible for the CNM.
We show $P_{\rm min}$ and $P_{\rm max}$ in lower (upper) panels of Fig.~\ref{fig: Tn_Z1} as the pair of thin horizontal (diagonal) lines.

Our Fig.~\ref{fig: Tn_Z1} reproduces the well-known result \citep{Field1969, Wolfire1995} that in the transition from Ly$\alpha$ to metal-line cooling the temperature falls more rapidly than $1/n$ over a significant density interval from $n_{\rm max}$ to $n_{\rm min}$. This enables multiple solutions for the gas temperature between $P_{\rm min}$ and $P_{\rm max}$ for densities ranging from $n_{\rm min}T_{\rm CNM}/T_{\rm WNM}$ to $n_{\rm max}T_{\rm WNM}/T_{\rm CNM}$.  For densities between $n_{\rm max}$ and $n_{\rm min}$ the gas is thermally unstable and this is the UNM. For isobaric density perturbations $\delta n$ in the UNM, the gas either cools to the stable CNM branch (for positive perturbations), or heats to the stable WNM branch (for negative perturbations).  For thermal pressures between $P_{\rm max}$ and $P_{\rm min}$ the gas may become multiphased with CNM condensations coexisting within an enveloping WNM at the same thermal pressure.

Fig.~\ref{fig: Tn_Z1} shows that for $n \gtrsim 5 \times 10^3$ cm$^{-3}$, the temperature
{\it increases} with density, as H$_2$ heating processes start contributing.
In this regime the H$_2$ heating rate increases more steeply than $n^2$ (see Eq.~\ref{eq: H2 heating} in the limit $n/n_{\rm crit}<1$) leading to the temperature increase with $n$.
We stress that the H$_2$ heating processes play this role so long as the gas is primarily atomic. If any self-shielding induces an HI-to-H$_2$ transition, the H$_2$ heating will be strongly suppressed. For solar metallicity a total (atomic plus molecular) column as small as $N\sim 10^{17}$ cm$^{-2}$ is sufficient to induce conversion to molecular form for $n \gtrsim 10^4$ cm$^{-3}$ (and $I_{\rm UV}=1$). However, at low metallicities, much large columns, $N\sim 10^{21}$ cm$^{-2}$, are required \citep{Bialy2016a}

At still higher densities (beyond the considered parameter space) the hydrogen undergoes a conversion into H$_2$ and the atomic carbon and oxygen convert into CO and H$_2$O. These molecular formation processes become efficient as the temperature exceeds few hundred Kelvins, which triggers the neutral-neutral chemical network of H$_2$O \citep{Sternberg1995, VanDishoeck2013a, Bialy2015b}.
As our focus is on the (predominantly) atomic ISM, we restrict ourselves to densities $n \lesssim 10^5$ cm$^{-3}$.

The dashed curves in the left panel of Fig.~\ref{fig: Tn_Z1} are the equilibria for $I_{\rm UV}=0.1$ and 10, assuming a constant ratio $\zeta_{-16}/I_{\rm UV}=1$.
For all values of $I_{\rm UV}$ in Fig.~\ref{fig: Tn_Z1} photoelectric heating dominates and is proportional to $I_{\rm UV}$.
Thus, when $I_{\rm UV}$ is increased (decreased) by a factor of 10, the $T(n)$ curves shift to the right (left), by a factor of 10.
As a result, the pressure range within which the gas may be multiphased shifts to higher or lower values in proportion to $I_{\rm UV}$.
For example, for $I_{\rm UV}=10$, $(P_{\rm max}, P_{\rm min})= (7.7, 2.9) \times 10^4$ cm$^{-3}$ K.
For $I_{\rm UV}=0.1$, $(P_{\rm max}, P_{\rm min})= (7.5, 2.7) \times 10^2$ cm$^{-3}$ K.

At solar metallicity, H$_2$ cooling and heating become important only in the very dense CNM,
but as we discuss below, at lower metallicities, H$_2$ cooling starts to play a role also at low-to-intermediate densities, in the WNM, affecting the WNM-to-CNM transition.

\section{Thermal Structure at Low Metallicity}
\label{sec: low Z}

In this section we investigate the thermal structures as the metallicity is lowered. We first consider the transition from photoelectric to cosmic-ray dominated heating as $Z'$ is reduced, for a range of ratios $\zeta_{-16}/I_{\rm UV}$. We then solve the equation of thermal equilibrium, in two steps. First, we lower the metallicity and solve for $T$ with the exclusion of the H$_2$ heating and cooling processes. Then we solve again with the H$_2$ included. This procedure enables us to distinguish the effects of diminished metal line cooling from the appearance of the H$_2$ heating and cooling processes that eventually dominate the behavior at very low metallicity.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.5\textwidth]{plots/fig2_CR_vs_photoheating.pdf}
	\caption{
Contours at which CR heating equals PE heating in the $I_{\rm UV}/\zeta_{-16}-Z'$ parameter space. The solid, dashed and dotted curves are for $T=10^4, 10^3$ and 10$^2$ K respectively, and the different colors correspond to different values of $n/\zeta_{-16}$, from  $n/\zeta_{-16}=10$ (left most curves) to $n/\zeta_{-16}=0.1$ (rightmost curves; see legend).
		}
		\label{fig: CR_vs_PE}
\end{figure}

\subsection{PE versus CR}
\label{sub: PE versus CR}

Near solar metallicity ($Z'\sim 1$) photoelectric (PE) heating usually dominates, and the PE heating rate decreases in proportion to the dust-to-gas ratio $Z_d'$.  At sufficiently low $Z'$ heating by cosmic-ray ionization becomes competitive and dominates.
In Fig.~\ref{fig: CR_vs_PE} we show the boundary lines at which the PE and CR heating rates are just equal, in the $I_{\rm UV}/\zeta_{-16}$ versus $Z'$ plane, for $n/\zeta_{-16}$ ratios equal to 0.1, 1, and 10 cm$^{-3}$.
For  $\zeta_{-16}/I_{\rm UV} =1$ the transition from PE to CR heating occurs between $Z'=0.3$ and 0.1, depending on density and temperature.
With decreasing $\zeta_{-16}/I_{\rm UV}$, the transition metallicity is lowered, but not by much.
This is because below $Z'=0.2$ the dust-to-gas ratio falls super-linearly with metallicity (see Eq.~\ref{eq: Zd-Z}).
Conversely, for sufficiently large  $\zeta_{-16}/I_{\rm UV}$ ($\gtrsim 100$), CR heating may dominate even at solar metallicity.

\subsection{Step-1: Reducing the Metallicity}
\label{sub: H2 off}

We now consider the thermal structures at reduced metallicity, with the exclusion of the H$_2$ heating and cooling processes in {\it Step-1}. Results for $T(n)$ and $P(n)$ are shown as the dashed curves in Fig.~\ref{fig: Tn_diff_Z}
(top and middle panels)
for $Z'$ from 3 to $10^{-5}$.
In these computations we have assumed
$I_{\rm UV}=\zeta_{-16}=1$.

Fig.~\ref{fig: Tn_diff_Z} shows that as $Z'$ is lowered to $0.1$, ($P_{\rm max}$,$P_{\rm min}$) drop to $(4.6,0.43) \times 10^{3}$ cm$^{-3}$ K. These drops are due to the more rapid reduction in $Z'_d$ and associated PE heating rate, compared to the diminished metal line cooling rate that decreases with $Z'$.
The reduced heating relative to cooling  at $Z'=0.1$,
reduces the minimum CNM density by a factor of 2.6 (compared to $Z'=1$), to $n_{\rm min}=2.8$ cm$^{-3}$.
The maximal density of the WNM is also lower, with $n_{\rm max}=0.71$ cm$^{-3}$.
As the metallicty is reduced further CR heating takes over, and the gas then remains warm to ever higher densities. This is expected given the reduction in the cooling efficiencies for a fixed CR heating rate that is independent of metallicity. Importantly, multiphase behavior is maintained, with $P_{\rm min}$ and $P_{\rm max}$ shifted to higher pressures.

The behavior in the CR heating regime may be analyzed straightforwardly as follows.
As we discussed in \S \ref{sec: Z=1}, as the density is increased in the warm phase, metal line-emission energy losses start contributing at a temperature of $\sim 6\times 10^3$~K. The cooling point temperature is nearly independent of $Z'$, and we denote the corresponding density as $n_{{\rm cool},Z}$
\footnote{Here $n_{{\rm cool},Z}$ is the transition point for Ly$\alpha$-to-metal cooling. Later we also introduce $n_{\rm cool, H_2}$ for the transition point from Ly$\alpha$-to-H$_2$ cooling.}.
For $n>n_{{\rm cool},Z}$, the temperature drops sharply as Ly$\alpha$ becomes inefficient and metal cooling dominates.
To obtain an analytic formula for $n_{{\rm cool},Z}$ we equate the heating and metal line cooling rates at $T_0\equiv 6\times 10^3$~K. We assume heating by CR ionization.
This gives
\begin{equation}
\label{eq: point of LyA=OI low Z}
n_{{\rm cool},Z} = \frac{\zeta_{\rm p} E_{\rm cr}}{A_{\rm O} \mathcal{L}_{Z}(T_0) Z'} \
 \approx 4.0 \left( \frac{10^{-2}}{Z'} \right) \zeta_{-16} \ {\rm cm^{-3}} \ ,
\end{equation}
where we have defined the Solar metallicity fine-structure line cooling efficiency (erg s$^{-1}$ cm$^3$) per oxygen atom $\mathcal{L}_Z \equiv \mathcal{L}_{\rm [OI]} +  (x_{\rm C^+}/x_{\rm O}) \mathcal{L}_{\rm [CII]}$ \footnote{As shown in Figs.~\ref{fig: Tn_Z1} and \ref{fig: Cool_heat_diff_Z}, [CII] always dominates over [CI] cooling.
}.
In Eq.~(\ref{eq: point of LyA=OI low Z}) $\zeta_p$ is the primary ionization rate, and $E_{\rm cr}$ is the mean deposition energy per ionization (see Eq.~\ref{eq: CR heating}).
In the numerical evaluation we used $\mathcal{L}_Z(T_0)= 7.2 \times 10^{-23}$
~erg~s$^{-1}$~cm$^3$,
$x_{{\rm C^+}}/x_{\rm O}=A_{\rm C}/A_{\rm O}=0.56$, $A_{\rm O}=5.4 \times 10^{-4}$ (assuming no depletion), and $E_{\rm cr}=15.2$~eV, $\phi_s=0.57$.
These values are appropriate for $x_{\rm e} = 8.2 \times 10^{-3}$, the equilibrium electron fraction at the transition point at $Z'=0.01$.
For $Z'=10^{-2}$
Eq.~(\ref{eq: point of LyA=OI low Z}) gives, $n_{{\rm cool},Z}=4.0$~cm$^{-3}$, decreasing linearly with $Z'$, in good agreement with the temperature drop points shown for $T(n)$ in Fig.~\ref{fig: Tn_diff_Z} (dashed curves). For $Z'\gtrsim 0.1$ PE heating contributes in addition to CR, and the cooling densities are larger than given by Eq.~(\ref{eq: point of LyA=OI low Z}).

When the H$_2$ processes are neglected (or negligible) the sharp drop in temperature induced by the transition from Ly$\alpha$ to metal line cooling gives rise to the multiphased $P(n)$ curves, and then $n_{{\rm cool},Z}$ is approximately equal to $n_{\rm max}$ the maximum density for which a WNM is possible. Thus,
\begin{equation}
\label{eq: point of LyA=OI low Z P}
P_{\rm max} \simeq 1.1 n_{{\rm cool},Z} T_0 \
 = 2.6 \times 10^4 \left( \frac{10^{-2}}{Z'} \right) \zeta_{-16} \ {\rm cm^{-3} \ K} \ .
\end{equation}
{We assume a cosmological helium abundance, $A_{\rm He}=0.1$.
Eq.~(\ref{eq: point of LyA=OI low Z P}) is consistent with our numerical results in Fig.~\ref{fig: Tn_diff_Z}.

The scalings given by Eqs.~(\ref{eq: point of LyA=OI low Z}) and (\ref{eq: point of LyA=OI low Z P}) are in agreement with previous models \citep[][\citetalias{Wolfire2003}]{Wolfire1995,Liszt2002}.
As we now discuss, when H$_2$ is included this simple picture is greatly altered.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.5\textwidth]{plots/fig3_cooling_functions.png}
	\caption{
Top: The various components of the cooling function per hydrogen nucleus, $L/n$ (erg s$^{-3}$), Ly$\alpha$, metal ([CII], [OI]), and H$_2$ cooling, as functions of temperature.
We assume $n=1$ cm$^{-3}$, $x_{\rm H^+}=x_{\rm e}=10^{-4}$ (solid) or $10^{-2}$ (dashed), and $x_{\rm H_2}=10^{-6}$.
Typically, for $T \ll 10^4$ K the density is high and $x_{\rm e}$ is small.
For Ly$\alpha$ cooling we always assume $x_{\rm e} =10^{-2}$.
The three sets of metal cooling curves are for $Z'=1, 10^{-2}$, and $10^{-4}$.
Bottom: The H$_2$ cooling  efficiency $\mathcal{L}_{\rm H_2}$ (erg cm$^3$ s$^{-3}$) by atomic hydrogen excitation (based on data from \citealt{Lique2015}).
Recall, for low densities, $L_{\rm H_2}=x_{\rm H_2} n^2 \mathcal{L}_{\rm H_2}$.
The contributions from decays of individual ro-vibrationnal H$_2$ levels, $(v,J)$, and their energies above ground (in Kelvin) are indicated.
		}
		\label{fig: cooling functions}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[width=0.5\textwidth]{plots/fig4_T_P_x_vs_n_different_Z_N0.png}
	\caption{
Temperature, thermal pressure, H$_2$ abundance,
and the C, C$^+$ and CO abundances relative to carbon nuclei abundance,
as functions of  the gas hydrogen density, for various metallicity values, from $Z'=3$ down to 10$^{-5}$.
The dashed curves in the top panels are for models that exclude H$_2$ cooling and heating processes.}
		\label{fig: Tn_diff_Z}
\end{figure}

\subsection{Step-2: Inclusion of H$_2$ Processes}
\label{sub: H2 ON}

The red curves in the upper panel of Fig.~\ref{fig: cooling functions} show the H$_2$ cooling function.
In the lower panel we show the contribution of individual
H$_2$ rotational and vibrational levels to H$_2$ cooling, for excitation by atomic hydrogen.
As we discuss below, the fact that (unlike metal cooling) the H$_2$ cooling function never saturates with temperature all the way until Ly$\alpha$ becomes operational, smooths out of the multiphase phenomena and eventually eliminates it entirely at low metallicity.

The solid curves in Fig.~\ref{fig: Tn_diff_Z} show our results for the equilibrium $T(n)$ and $P(n)$ with the H$_2$ processes included in our {\it Step-2}. Again, these are heating via H$_2$ formation, photodissociation, and UV-pumping, and cooling via ro-vibrational line emissions, and collisional dissociation.
The various heating and cooling rates for $Z'=10^{-1,-2...,-5}$, are shown in Fig.~\ref{fig: Cool_heat_diff_Z}.

The efficiencies of the molecular processes depend on the H$_2$ abundance, $x_{\rm H_2}$, that we compute self-consistently with the temperature, for a given gas density and metallicity. In Fig.~\ref{fig: Tn_diff_Z} we show $x_{\rm H_2}$ versus $n$, for $Z'$ from 3 to 10$^{-5}$.
At all metallicities, H$_2$ destruction by photodissociation is very efficient and the gas remains predominantly atomic.
The H$_2$ abundance is higher at high metallicities as the H$_2$ is formed efficiently on dust-grains.
For $Z'\lesssim 0.1$, the H$_2$ is formed much less efficiently in the gas phase.
The resulting H$_2$ abundances then range from $10^{-7}$ to $10^{-5}$, consistent with Eq.~(\ref{eq: xH2 gas}).
Interestingly, it is in the low metallicity limit where $x_{\rm H_2}$ is very low, that H$_2$ plays the most significant role in heating and cooling the gas.
At high $Z'$ molecular hydrogen is more abundant but metal line cooling is very efficient and induces a WNM-to-CNM transition at densities well below the densities at which H$_2$ cooling starts playing a role. However, at sufficiently low $Z'$, H$_2$ cooling must eventually dominate over metals.

\subsubsection{The onset of H$_2$ cooling at low metallicity}
When the H$_2$ processes are included the $T(n)$ and $P(n)$ curves in Fig.~\ref{fig: Tn_diff_Z} are altered in several ways. First, at sufficiently low metallicity it is H$_2$ rather than the metals that compete with Ly$\alpha$ at the ``cooling point" temperature of $\sim6\times 10^3$~K below which Ly$\alpha$ becomes inefficient.
This is also illustrated in the left panels of Fig.~\ref{fig: Cool_heat_diff_Z}.
We define $Z'_{\rm cool,H_2}$ as the critical metallicity at which this switch from metals to H$_2$ occurs at the cooling point. Numerically we find $Z'_{\rm cool,H_2}\approx 10^{-2}$.
The cooling point density is then shifted to a low value of $\approx 5$~cm$^{-3}$, independent of metallicity (see Figs.~\ref{fig: Tn_diff_Z} and \ref{fig: Cool_heat_diff_Z}).
The H$_2$ abundance at this point is $x_{\rm H_2} \approx 10^{-6}$.

For an analytic estimate we define the gas density for the H$_2$ cooling point where
\begin{align}
\label{eq: point of LyA=H2, 0}
n \mathcal{L}_{\rm H_2} x_{\rm H_2} = \zeta_p E_{\rm cr}\ \ \ .
\end{align}
We again assume heating by cosmic-ray ionization and $\mathcal{L}_{\rm H_2}$ is the H$_2$ cooling efficiency (erg s$^{-1}$ cm$^3$).
With Eq.~(\ref{eq: xH2 gas}) for $x_{\rm H_2}$ with $\eta=1$, we obtain
\begin{equation}
\label{eq: point of LyA=H2}
n_{\rm cool,H_2} = \left( \sqrt{\frac{\alpha_B}{\zeta}} \frac{D_0 I_{\rm UV}}{k_{\ref{reac: H- formation}}} \ \frac{E_{\rm cr} \zeta_p}{\mathcal{L}_{\rm H_2}} \right)^{2/3}
\approx 5.0 \ \zeta_{-16}^{1/3} I_{\rm UV}^{2/3} \ \ \ {\rm cm^{-3}} \ \ .
\end{equation}
Plugging back into Eq.~(\ref{eq: xH2 gas}) we get the H$_2$ abundance at the cooling point,
\begin{equation}
\label{eq: xH2_cool}
x_{\rm H_2, cool} = 1.4 \times 10^{-6} \left( \frac{\zeta_{-16}}{I_{\rm UV}} \right)^{2/3} \ .
\end{equation}
In the numerical evaluations in Eqs.~(\ref{eq: point of LyA=H2} and \ref{eq: xH2_cool}) we use
$\mathcal{L}_{\rm H_2} = 2.1 \times 10^{-22}$~erg cm$^3$ s$^{-1}$ at $T_0$, and $E_{\rm cr}$ and $\phi$, as in Eq.~(\ref{eq: point of LyA=OI low Z}).
The corresponding pressure is
\begin{equation}
\label{eq: point of LyA=H2, P}
P_{\rm cool,{\rm H_2}} \approx 1.1 n_{\rm cool,H_2} T_0 \approx 3.3 \times 10^4 \ \zeta_{-16}^{1/3} I_{\rm UV}^{2/3} \ {\rm cm^{-3} \ K} \ \ \ .
\end{equation}
Unlike the Ly$\alpha$-to-metal transition which is accompanied by a maximum in the $P-n$ curve (Eq.~\ref{eq: point of LyA=OI low Z P}), the Ly$\alpha$-to-H$_2$ transition does not induce such a maximum but only a kink in the $P-n$ curve (see Fig.~\ref{fig: Tn_diff_Z}).

The critical metallicity below which H$_2$ cooling becomes important
is
\begin{equation}
\label{eq: Z_c_H2}
Z'_{\rm cool,H_2} \equiv \frac{\mathcal{L}_{\rm H_2} x_{\rm H_2}}{\mathcal{L}_{z} A_{\rm O}} \approx 8.0 \times 10^{-3} \left( \frac{\zeta_{-16}}{I_{\rm UV}} \right)^{2/3} \ .
\end{equation}
As we discuss in \S \ref{sec: Parameter Study}, this
formula is in good agreement with our numerical results.
Importantly, the temperature decreases more gradually for transitions from Ly$\alpha$ to H$_2$ cooling. Because of the contributions from many lines in the ro-vibrational ladder (as illustrated in the bottom panel of Fig.~\ref{fig: cooling functions}), the H$_2$ cooling function remains sensitive to temperature in the entire range $10^2< T < 10^4$~K.
This is illustrated in the upper panel of Fig.~\ref{fig: cooling functions} where the red curves show the much steeper H$_2$ cooling function compared to the flat metals function (blue curves). For this reason, the temperature drop at $n_{\rm cool,H_2}$ is moderated and does not produce a local maximum in the $P(n)$ curve. Most generally, {\it the transition from Ly$\alpha$ to H$_2$ cooling does not result in a multiphased behavior.}

\begin{figure*}[t]
	\centering
	\includegraphics[width=1\textwidth]{plots/fig5_Heat_cool_diff_Z.png}
	\caption{
Cooling (left) and heating (right) rates per hydrogen nucleus (erg s$^{-1}$), as functions of  the gas hydrogen density, at various metallicities.
}
		\label{fig: Cool_heat_diff_Z}
\end{figure*}

While H$_2$ cooling does not induce a multiphase, Fig.~\ref{fig: Tn_diff_Z} shows that multiphased behavior continues to be produced by metal-line cooling, although over a much narrower pressure range, down to a metallicity as low as $10^{-4}$.
This multiphase is induced by the transition from H$_2$ to metal cooling (rather than Ly$\alpha$ to metals) and occurs for a lower cooling point temperature of $T_0\approx 600$ K. So long as the heating is by CR ionization the cooling point density at which the sharp temperature drop occurs is still given by Eq.~(\ref{eq: point of LyA=OI low Z}) but evaluated at the lowered $T_0$. Thus, in this regime $n_{{\rm cool},Z}$ continues to vary as $1/Z'$, but the density prefactor in Eq.~(\ref{eq: point of LyA=OI low Z}) is reduced.
For example, for $Z'=10^{-3}$, $n_{{\rm cool},Z}=40$ cm$^{-3}$, leading to a pressure maximum of $P_{\rm max}\approx 1.1 n_{{\rm cool},Z}T_0=7.6 \times 10^{4}$ cm$^{-3}$~K for the WNM (see Fig.~\ref{fig: Tn_diff_Z}).

Because H$_2$ cooling reduces the temperature of the WNM, the multiphase strip is now narrower than at solar metallicity, and continues to shrink with decreasing metallicity.
For example, while at solar metallicity the multiphase range is wide, with $\delta_n \equiv (n_{\rm max}-n_{\rm min})/n_{\rm min}=5.4$ and $\delta_P \equiv (P_{\rm max}-P_{\rm min})/P_{\rm min}=1.9$,
for $Z'=10^{-3}$ we have $\delta_n=3.8$ and $\delta_P=0.88$.
For $Z'=10^{-4}$, the multiphase phenomena is already extremely suppressed, with $\delta_n=1.6$ and $\delta_P=0.05$.

\subsubsection{The onset of H$_2$ heating at extremely low metallicity}
For still lower metallicities ($Z' \leq 10^{-5}$) sharp gas cooling and associated multiphased structures disappear entirely.
The density at which metal cooling can balance CR heating increases as $1/Z'$,
and at sufficiently low metallicities this density becomes so high that H$_2$ heating becomes operational and dominant. In this limit metal cooling becomes ineffective. This is  because H$_2$ heating includes a dependence on the molecular formation rate (see Eq.~\ref{eq: H2 heating}) so that the heating rate increases more rapidly than $n^2$. However, metal cooling varies no faster than $n^2$, and so cannot compete. In this limit H$_2$ is the dominant coolant, multiphase behavior does not occur, and the gas is driven to
intermediate, thermally stable, temperatures of $\sim 600$~K.
This behavior is illustrated in Figs.~\ref{fig: Tn_diff_Z} and \ref{fig: Cool_heat_diff_Z}.

We obtain an analytic estimate for the critical metallicity, $Z'_{\rm heat,H_2}$, below which metal line cooling is no longer effective as follows. We first estimate the density $n_{\rm heat,H_2}$ at which H$_2$ and CR heating are equal. For $n>n_{\rm heat,H_2}$ we are in the H$_2$ heating regime. The critical $Z'_{\rm heat,H_2}$ is then the metallicity at which the metal cooling density, $n_{{\rm cool},Z}$, as given by Eq.~(\ref{eq: point of LyA=OI low Z}), is at least as large as $n_{\rm heat,H_2}$.  For $n_{{\rm cool},Z} \gtrsim n_{\rm heat,H_2}$ metal line cooling is ineffective because it cannot compete with H$_2$ heating.

Equating the H$_2$ and CR heating rates gives,
\begin{align}
\label{eq: n_crit H2 heating}
n_{\rm heat,H_2} &= \left( \frac{ \zeta_p E_{\rm cr} n_{\rm crit}}{E_{\rm H_2,2} k_{\ref{reac: H- formation}} \sqrt{\zeta/\alpha_B}} \right)^{2/3} \\
&= 7.2 \times 10^3 \ \zeta_{-16}^{1/3} \ T_3^{-1.08} \ {\rm cm^{-3}} \ ,
\end{align}
where we used our H$_2$ heating expression, Eq.~(\ref{eq: H2 heating}), assuming UV pumping heating dominates (but still in the $n<n_{\rm crit}$ limit), and used Eq.~(\ref{eq: R gas phase}) for the effective gas-phase formation rate coefficient $R_-$.
    Next,
equating the metal cooling density $n_{{\rm cool},Z}$ (Eq.~\ref{eq: point of LyA=OI low Z}) with $n_{\rm heat,H_2}$ above gives,
\begin{equation}
\label{eq: Z_h_H2}
Z'_{\rm heat,H_2} = 1.2 \times 10^{-5} \  \zeta_{-16}^{2/3} \ .
\end{equation}
In the numerical evaluation we adopted a metal cooling efficiency $\mathcal{L}_z=1.5 \times 10^{-23}$ erg cm$^{3}$ s$^{-1}$ at $n= 10^4$ cm$^{-3}$, $T=10^3$~K.
For $Z' \lesssim Z_{\rm heat,H_2}$
 the gas never undergoes a second metal-line cooling phase, and there is no multiphase structure.
 This formula is in agreement with the numerical results shown in Fig.~\ref{fig: Tn_diff_Z}.

Eq.~(\ref{eq: Z_h_H2}) is a strict requirement on the metallicity, expressing the metallicity below which a multiphase {\it cannot} possibly occur.
In practice, as discussed above, much before the metallicity becomes that low, the multiphase region may already become very narrow, so that any thermal instability effects may become negligible.

\section{Parameter Study}
\label{sec: Parameter Study}

\begin{figure*}[t]
	\centering
	\includegraphics[width=1\textwidth]{plots/fig6_2D_T_n_vs_nZ_and_cooling.png}
		\includegraphics[width=1\textwidth]{plots/fig6_2D_Tn_vs_ZP.png}
	\caption{
Top: The temperature as a functions of metallicity and gas density.
In the left panel, the solid contours delineate the regions where Ly$\alpha$-, metal-, and H$_2$-cooling become dominant. The dashed contours delineate PE-, CR-, and H$_2$-heating dominated regions.
The contours in the right panel are the cooling times in Myrs.
Bottom: The temperature (left) and gas density (right) as a functions of metallicity and pressure.
The WNM, CNM and multiphase regions are indicated.
A 3D interactive surface plot is presented in the online version (see also
\href{https://plot.ly/~sb2580/47}{https://plot.ly/~sb2580/47}).
		}
		\label{fig: T_2d}
\end{figure*}

In this section, in Figs.~\ref{fig: T_2d} and \ref{fig: T_2d_vars}, we present results of our thermal phase computations in the form of equilibrium temperature contour plots in the metallicity versus density ($Z'-n$) plane, as well as temperature and density contour plots in the metallicity versus pressure ($Z'-P$) plane.
We investigate the effects of variations in $\zeta$ and $I_{\rm UV}$.
We also discuss the time scales relevant for our assumptions of thermal and chemical equilibrium.

\subsection{Contour plots}
\label{sub: contourplots}

The top panels of Fig.~\ref{fig: T_2d} show the equilibrium gas temperature in the metallicity versus density ($Z'-n$) plane, for $\zeta_{-16}=I_{\rm UV}=1$. The colors indicate the temperature, ranging from cold $\sim 10^2$~K (blue), to intermediate $\sim 10^3$~K (yellow), to warm $\sim 10^4$~K (red).
In the upper-right panel we
overplot contours of the cooling times.
 The cooling times range from $\sim 1-10$ Myrs in the WNM, depending on the metallicity,
down to 0.1-0.4 Myrs in the CNM at low $Z'$, and down to less then 0.01  Myrs at high $Z'$.
At higher metallicities the increase in the heating rate due to the onset of PE heating, is accompanied by appropriately higher cooling rates.
For the CNM, the cooling rate naturally increases with increasing metallicity, as \cii and \oi dominates the cooling.
In the WNM, where Ly$\alpha$ dominates, the cooling rate increases via a slight increase in the equilibrium temperature.
Since at low $Z'$ a large portion of the parameter space is WNM, or the intermediate $\approx 10^3$ K medium (at very low $Z'$), with cooling times 1-10 Myrs, the gas may potentially be out of thermal equilibrium, depending on the magnitude of the local dynamical time (e.g., the sound crossing time, turbulent time, free-fall time, etc.).
We further discuss the cooling and chemical time in the Appendix.

In the upper-left panel the various curves delineate the dominating heating and cooling processes that operate in different regions of parameter space. For heating, these are (a) photoelectric (PE) heating, above the dashed curve from the WNM into the CNM, (b) H$_2$ heating, to the right of the dashed curve at high density in the CNM at moderately reduced metallicity, and again at high densities and very low metallicities in the intermediate temperature zone, and (c) cosmic-ray (CR) heating everywhere else. For cooling these are (a) Ly$\alpha$ to the left of the solid curve in the low density WNM, (b) metals to the right of the solid curve into the CNM, and (c) H$_2$ cooling below the solid curve in the lower right portion of parameter space.

For any $Z'$ the gas temperature decreases monotonically with increasing $n$, with sharp drops in $T$, from the WNM to CNM (red to blue), occurring near the Ly$\alpha$/metal or H$_2$/metal cooling point boundaries. The WNM-to-CNM transition points depend on metallicity. The intermediate temperature zone appears at very low $Z'$ and high density.

For $Z'\gtrsim 0.2$ PE heating dominates over CR ionization in the WNM. Because PE heating and metal cooling both vary linearly with $Z'$ (see Eq.~\ref{eq: Zd-Z}), in this regime the Ly$\alpha$ to metal cooling point runs vertically (independent of $Z'$) at an approximately constant density of $\approx 0.5$~cm$^{-3}$. Below $Z'\approx 0.2$ CR heating takes over, and the cooling point then moves diagonally, with $n_{{\rm cool},Z}\propto 1/Z'$, consistent with Eq.~(\ref{eq: point of LyA=OI low Z P}) for a cooling point temperature $T_0=6\times 10^3$~K appropriate for the transition from Ly$\alpha$ to metal cooling. Below a metallicity of $8\times 10^{-3}$, H$_2$ cooling (rather than metals) takes over from Ly$\alpha$ in the WNM. This is  consistent with our analytic estimate (Eq.~\ref{eq: Z_c_H2}) for $Z'_{\rm cool, H_2}$. The Ly$\alpha$ to H$_2$ boundary line is vertical (independent of $Z'$) and occurs well inside the WNM at a density $\approx 3$~cm$^{-3}$ consistent with our Eq.~\ref{eq: point of LyA=H2} for $n_{\rm cool,H_2}$.
Below $Z'_{\rm cool, H_2}\approx 8\times 10^{-3}$ the WNM-to-CNM boundary remains diagonal with $n_{{\rm cool},Z}\propto 1/Z'$ as given by Eq.~(\ref{eq: point of LyA=OI low Z P}), but with a lower $T_0$ (i.e., $T_0=5000$ to $600$ K, for $\log Z'=-2.3$ to $-4.7$), as appropriate for the transition from H$_2$ to metal cooling.

For $Z'< Z'_{\rm heat,H_2}\approx 10^{-5}$ and $n> n_{\rm heat,H_2} \approx 10^4$~cm$^{-3}$, H$_2$ heating dominates over CR ionization, and therefore metal cooling becomes ineffective.  The critical values $Z'_{\rm heat,H_2}$ and $n_{\rm heat,H_2}$ occur at the intersection of the CR-to-H$_2$ heating and H$_2$-to-metal cooling boundary curves in Fig.~\ref{fig: T_2d}, and are consistent with our analytic estimates given by Eqs.~(\ref{eq: n_crit H2 heating}) and (\ref{eq: Z_h_H2}).
In this portion of parameter space an intermediate equilibrium temperature of $\sim 10^3$~K is set by the balance of H$_2$ heating and cooling.

The lower panels of Fig.~\ref{fig: T_2d} show the temperature (left) and density (right) in the pressure versus metallicity ($P-Z'$) plane (again for $\zeta_{-16}=I_{\rm UV}=1$). The regions of pure CNM (cold/dense) and WNM (warm/diffuse) are indicated, as is the intermediate temperature H$_2$ zone at very low metallicity and high density. The multiphased region where the CNM and WNM overlap occurs between the solid black curves, from $P_{\rm min}$ (to the left) to $P_{\rm max}$ (to the right).

The position of the multi-phased zone in Fig.~\ref{fig: T_2d} is determined primarily by (a) the switch from PE to CR heating, (b) the dependence of the metal cooling point on $Z'$, and (c) the onset of H$_2$ cooling and heating at low metallicity.
For $Z'$ down to $\sim 0.1$ PE heating dominates, and the multiphased zone widens and occurs at reduced values of $P_{\rm min}$ and $P_{\rm max}$. The width is maximal at $Z' \approx 0.1$ where PE heating is least effective compared to metal cooling. For $Z'\lesssim 0.1$ CR heating takes over (independent of metallicty) and $P_{\rm min}$ and $P_{\rm max}$ increase as the transition to metal cooling requires larger densities.

As the metallicity is lowered below $Z'_{\rm cool,H_2}\approx 8 \times 10^{-3}$ the multiphased zone become narrower due to H$_2$ cooling in the WNM.
At $Z' \approx 10^{-4}$ there still exists a (small) temperature jump  at the WNM to CNM boundary, but the pressure range of the multiphase zone is already negligible.
At $Z' < Z'_{\rm heat,H_2}\approx 10^{-5}$ metal cooling becomes ineffective, the temperature then decreases smoothly with increasing pressure, from the $\sim10^4$~K Ly$\alpha$ cooled WNM, to the $\sim 10^3$~K H$_2$ cooled intermediate temperature zone.

\subsection{Varying $\zeta$, $I_{\rm UV}$ and $D_-$}
\label{sub: var UV zeta}

\begin{figure*}[t]
	\centering
	\includegraphics[width=1\textwidth]{plots/fig7_2D_T_vs_Zn_vars.png}
		\includegraphics[width=1\textwidth]{plots/fig7_2D_T_vs_ZP_vars.png}
	\caption{
As Fig.~\ref{fig: T_2d}, but assuming variations in $\zeta$ and $I_{\rm UV}$.
The contours, and shaded regions have the same meaning as in Fig.~\ref{fig: T_2d}.
The dotted curves in the lower panels are the \citetalias{Wolfire2003} formula for $P_{\rm min}$ (Eq.~\ref{eq: Pmin Wolfire}).
		}
				\label{fig: T_2d_vars}
\end{figure*}

In Fig.~\ref{fig: T_2d_vars} we explore the thermal structures for varying UV intensities and cosmic-ray ionization rates.
The left column shows the cases $\zeta_{-16}=0.1$ and 10, assuming $I_{\rm UV}=1$.
The middle column is for $I_{\rm UV}=0.1$ and 10, with $\zeta_{-16}=1$.
The right column shows models in which $I_{\rm UV}$ and $\zeta_{-16}$ scale together by $\pm 1$dex relative to our fiducial case $\zeta_{-16}=I_{\rm UV}=1$.
This last set reflects a physical scenario in which the massive stars that produce the energetic UV photons, are also the sources of the cosmic-rays originating in the remnants of core-collapse supernovae.
For the $Z'-n$ plots, the PE/CR/H$_2$ and Ly$\alpha$/metals/H$_2$ heating and cooling boundaries are shown as the solid and dashed curves, as in the upper left panel of Fig.~\ref{fig: T_2d}. For the $Z'-P$ plots, the various phases including the multiphased zones are indicated as in the lower panels of Fig.~\ref{fig: T_2d}.

In the lower panels we also show, as dotted white curves, a comparison to the \citetalias{Wolfire2003} formula for $P_{\rm min}$ as a function of $I_{\rm UV}$, $Z'$, $Z'_d$, and $\zeta_{-16}$,
\begin{equation}
\label{eq: Pmin Wolfire}
    P_{\rm min} = 8500 I_{\rm UV} \frac{Z_d'/(Z' f)}{1+3.1(I_{\rm UV}Z_d'/\zeta_{-16})^{0.365}} \ {\rm cm^{-3} \ K} \ ,
\end{equation}
(their Eq.~33) over its validity range (their Eqs.~35-38).
We include the factor $f \equiv (1-\delta_{\rm C} Z_d'/Z')/(1-\delta_{\rm C})$ to correct\footnote{While \citetalias{Wolfire2003} use elemental abundances similar to ours for $Z'=1$, they do not
have a metallicity dependent depletion in their model, and $f$ corrects their formula for this effect.} for the metallicity dependent depletion onto dust (i.e., our Eq.~\ref{eq: C and O abundances}).
For $Z_d'$ we use our Eq.~(\ref{eq: Zd-Z}).
The agreement is not perfect due to the different treatment of cosmic-ray and X-ray heatings, and PAH chemistry.
However, overall, the general trends with $Z'$, $\zeta$ and $I_{\rm UV}$ are recovered.

The left-panels of Fig.~\ref{fig: T_2d_vars} show that as $\zeta$ increases, the WNM, CNM and the multiphase regions, are all shifted to higher densities and pressures. Within the CR-heating region (below the almost horizontal dotted line)
higher densities are required for cooling to offset an increasing CR heating rate.
Within the PE-heating region,
as $\zeta$ increases, the electron abundance increases, the dust-grains are less positively charged, and the release of photoelectrons from the dust-grains into the gas is more efficient (i.e., see Fig.~1 in \citealt{Wolfire1995}).
In the middle panels, where $I_{\rm UV}$ varies (with $\zeta$ kept constant), the shift is observed only at high metallicities ($Z' \gtrsim 0.1$) where PE heating dominates, as expected.

The critical metallicity at which H$_2$ cooling becomes important, $Z'_{\rm cool, H_2}$, increases
with $\zeta$ and decreases with $I_{\rm UV}$.
When $\zeta$ and $I_{\rm UV}$ scale together, $Z'_{\rm cool, H_2}$ remains unchanged.
This is because the H$_2$ abundance (a) increases with $\zeta$, as the gas phase formation is more efficient, and (b) decreases with $I_{\rm UV}$, as photodissociation is more efficient.
The observed dependence of $Z'_{\rm cool, H_2}$ on $\zeta$ and $I_{\rm UV}$ is in good agreement with Eq.~(\ref{eq: Z_c_H2}).
The location of the Ly$\alpha$-to-H$_2$ transition (the vertical white contour) is in good agreement with Eq.~(\ref{eq: point of LyA=H2}).
For example, looking at the left panels of Fig.~\ref{fig: T_2d_vars} we see that as $\zeta_{-16}$ increases from 0.1 to 10 (with $I_{\rm UV}=1$ kept constant), the Ly$\alpha$-to-H$_2$ transition shifts from $n \approx 2$ to 8 cm$^{-3}$.
On the other hand, when $I_{\rm UV}$ varies from 0.1 to 10 (and $\zeta_{-16}=1$ is kept constant, middle panels), the transition point shifts from $\approx 0.7$ to $\approx 20$ cm$^{-3}$.
Finally, when both $\zeta$ and $I_{\rm UV}$ vary together from 0.1 to 10, the transition point shifts by two orders of magnitude, from $\approx 0.4$ to 40  cm$^{-3}$.

The critical metallicity, $Z_{\rm heat, H_2}$, at which H$_2$ heating takes over, increases with $\zeta$, and is independent of $I_{\rm UV}$, as predicted by Eq.~(\ref{eq: Z_h_H2}).
As discussed in \S \ref{sub: H2 ON}, this metallicity is set by
the requirements that H$_2$ heating equal CR heating, which in turn must equal metal cooling, all of which are independent of $I_{\rm UV}$.
An increase in the CR intensity increases both the H$_2$ formation rate and the CR heating rate, but the latter is affected more strongly, and thus $Z_{\rm heat, H_2}$ increases with $\zeta$.

As discussed in \S \ref{sub: params: IUV}, the H$^-$ photodetachment rate at low metallicity is uncertain. We consider variations between $D_-=5.6 \times 10^{-9}$ s$^{-1}$ for the extrapolated Draine field (our fiducial models), up to $D_-=2.7 \times 10^{-7}$ s$^{-1}$ for a Draine+Mathis field.
To investigate the effect of a varying $D_-$, we have also computed a set of models with the higher Draine+Mathis $D_-$ rate.
The resulting temperature map is identical to that in Fig.~\ref{fig: T_2d}, with the only difference that the H$_2$ cooling line is shifted to a higher density, $n_{\rm cool, H_2}=15$ cm$^{-3}$, i.e., an increase by a factor of 3.5.
Since the $n_{\rm cool,Z}$ curve is unaffected by variations in $D_-$ (as it is set by metal cooling), the critical metallicity for H$_2$ cooling is reduced, and $Z_{\rm cool, H_2}$ is reduced to $10^{-3}$.
The shift to a higher cooling density is expected as the higher detachment rate reduces the H$_2$ formation rate, requiring a higher gas density for the onset of H$_2$ cooling.
Unlike $n_{\rm cool,H_2}$ and $Z_{\rm cool, H_2}$, the heating points $n_{\rm heat,H_2}$ and $Z_{\rm heat,H_2}$ are unaffected by the value of $D_-$, as they occur at very high densities where H$_-$ photodetachment is subdominant.

Quantitatively, the factor 3.5 shift in $n_{\rm cool, H_2}$ may be understood as follows.
For the higher $D_-$ rate, photodetachment dominates H$^-$ removal for densities $n < 10^2 I_{\rm UV} T_3^{0.39}$ cm$^{-3}$.
For such densities, $\eta \approx 0.1 T_3^{-0.39} I_{\rm UV}^{-1} (n/10{\rm cm^{-3}})$ (see Eq.~\ref{eq: eta}), and the H$_2$ abundance is
\begin{equation}
\label{eq: xH2 gas high D-}
x_{\rm H_2} =  \frac{x_{\rm e} k_{\ref{reac: H- formation}} k_{\ref{eq: H2 form gas reac}} n^2}{D_0 D_- I_{\rm UV}^2} = 3.3 \times 10^{-8} T_3^{0.62} \zeta_{-16}^{1/2} I_{\rm UV}^{-2} \left( \frac{n}{10\ {\rm cm^{-3}}}\right)^{3/2}  \ .
\end{equation}
At $T_3=6$ and $n=10$ cm$^{-3}$ (i.e., near the Ly$\alpha$-to-H$_2$ transition) this $x_{\rm H_2}$ is a factor of $\approx 20$ smaller than the corresponding value in Eq.~(\ref{eq: xH2 gas}), and has a steeper dependence on density.
Plugging this back into Eq.~(\ref{eq: xH2 gas high D-}), we see that $n$ will now enter with a power 5/2, thus the factor 20 lower $x_{\rm H_2}$, needs to be compensated by an increase in density by a factor of $\approx 20^{2/5} \approx 3.3$, in agreement with our numerical results.

Importantly, while $\zeta$, $I_{\rm UV}$ and $D_-$ affect the values of the critical densities and metallicities, the qualitative behaviour, the shift of the WNM-to-CNM transition to higher pressures  with decreasing metallicity, the quenching of the multiphase zone due to H$_2$ cooling at low $Z'$, and the final disappearance of the multiphase phenomenon at very low $Z'$, is robust.

\section{Summary and Discussion}
\label{sec: conclusions}

In this paper we have studied the thermal phase properties of atomic gas from high to vanishing metallicity, with special emphasis on understanding how multiphase behavior is altered when H$_2$ heating and cooling processes become dominant. As discussed in \S \ref{sec: intro}, the critical role of H$_2$ cooling at low metallicity has long been recognized in many studies.  However, multiphased CNM/WNM behavior into the low metallicity and primordial regimes has received less attention.

For solar metallicity, we reproduce the known result that a CNM phase cooled by \cii and \oi line emission, ($T_{\rm CNM} \sim 100$ K) and a WNM phase cooled by Ly$\alpha$ emission ($T \sim 6000$ K) phases can coexist at thermal equilibrium at the same thermal pressure (\citealt{Field1969, Wolfire1995, Liszt2002}, \citetalias{Wolfire2003}).
The WNM-to-CNM transition and the multiphase region shifts to higher densities and pressures with increasing
UV intensity, $I_{\rm UV}$, and/or increasing ionization rate, $\zeta$, as photoelectric (PE) heating becomes more efficient.

As the metallicity decreases, the metal cooling rate decreases.
However, as long as  $Z' \gtrsim 0.1$, PE heating continues to dominate so that the heating rate also decreases.
For sufficiently low metallicities, $Z_d'$ is expected to scale
superlinearly
with $Z'$ (in our model this occurs for $Z'<0.2$), PE heating then falls faster than metal cooling, as $Z'$ decreases, and the CNM is then colder and is less dense compared to  solar metallicity models.
The pressure range that allows a multiphase is then larger  compared to solar metallicity models.

For metallicities $Z' \lesssim 0.1$, the ionization of neutral gas by cosmic-rays (CR) becomes the dominant heating mechanism of the gas (the metallicity value depends on the CR ionization rate).
In this limit as the metallicity decreases, the metal cooling rate decreases, while heating remains constant, shifting the WNM-to-CNM transition to higher densities and pressures.
In this limit, and when H$_2$ heating and cooling processes are ignored, we find that the Ly$\alpha$-to-metal transition density (and associated WNM-to-CNM transition) scales as $\zeta/Z'$.

However, while previous analytic models of the H{\small I} phase structure often ignored H$_2$ cooling, we find that  H$_2$ cooling dominates and modifies the phase structure at
low metallicity (as we show H$_2$ cooling and heating also play a role at solar metallicity gas but only at very high densities, $\sim 10^6$ cm$^{-3}$, deep in the CNM).
The critical metallicity below which H$_2$ cooling becomes important is $Z'_{\rm cool,H_2} = 8.0 \times 10^{-3} (\zeta_{-16}/I_{\rm UV})^{2/3}$.
Below this metallicity, H$_2$ cooling reduces the temperature of the WNM, to temperatures as low as 600 K (the exact value depending on metallicity).
This lowers $P_{\rm max}$ and results in a narrower multiphase zone, both in pressure and density.

Finally, at extremely low metallicities $Z'<Z'_{\rm heat, H_2} = 1.2 \times 10^{-5} \zeta_{-16}^{2/3}$, the density at the WNM-to-CNM transition is already so high that H$_2$ heating starts contributing.
In this limit the thermal structure is determined only by H{\small I} (Ly$\alpha$) and H$_2$ cooling.
There is a single-phase solution at any pressure, as the temperature decreases smoothly with gas pressure,  from the $\approx 10^4- 6000$ K  WNM, down to $T \approx 600$ K, as determined by the balance of H$_2$ cooling and heating.

Gravitational collapse is favored in the cold-dense CNM, where the free-fall time is relatively short.
Our result that the transition from warm to cold gas occurs at ever increasing pressures as the metallicity is reduced (over the range $10^{-5} \lesssim Z' \lesssim 0.1$, for a constant $I_{\rm UV}$ and $\zeta$), suggests that galaxies  with metal-poor ISM (i.e., galaxies in early evolutionary stages or low mass dwarf galaxies), should have maintained very high interstellar pressures in order to allow the formation of a CNM phase.
For example, for our fiducial model of $I_{\rm UV}=\zeta_{-16}=1$, and for a metallicity $Z'=10^{-3}$ a minimum pressure, $P/k \approx 10^5$ cm$^{-3}$ K is required.
This requires abnormally large ISM column densities (for hydrostatic pressure equilibrium), deep gravitational potential wells of the dark-matter halos, or a highly pressurized hot ambient gas
that may provide the needed pressure support.

Alternatively, a WNM-to-CNM transition and a multiphase ISM may still occur at normal ISM pressures at low metallicity  if the heating rate is reduced.
This requires a reduced $\zeta$ and $I_{\rm UV}$ values, suggesting lower star-formation (supernovae) rates in low metallicity galaxies.
For example, for standard Galactic conditions ($Z'=I_{\rm UV}=\zeta_{-16}=1$), the multiphase occurs at $P\approx 3000$ cm$^{-3}$ K (Fig.~\ref{fig: T_2d}).
For  $Z'=10^{-3}$ the multiphase may still occur at a similar pressure if $I_{\rm UV}$ and $\zeta$ are reduced by a factor of 10 (Fig.~\ref{fig: T_2d_vars}, rightmost column, 3$^{\rm rd}$ row).
When connected to a theory of a self-regulated star forming ISM \citep{Ostriker2010}, this would imply that  the star-formation rate per unit gas column (i.e., the normalization of the Kennicutt Schmidt relation) should be lower at very low metallicities compared to solar (e.g., a factor of 10 lower for $Z'=10^{-3}$ compared to $Z'=1$).
If neither of the above mechanisms are sufficient to maintain the high pressure needed, the ISM would remain warm, at $\approx 10^4$ K.
In such a case, a different mode of star-formation would be expected, with only very massive clouds susceptible to gravitational collapse.

Previous studies have suggested that the atomic ISM of star-forming galaxies is naturally driven towards the multiphase pressure range, which provides a regulation mechanism for star-formation  \citep{Parravano1988, Parravano1989, Ostriker2010}.
The idea is as follows.
If the star-formation rate of a galaxy increases this leads to an increase in the UV intensity which in turn increase the PE heating rate.
The $P-n$ curve is then pushed up and to the right  (see Fig.~\ref{fig: Tn_Z1}, red-dashed curve), the gas is driven to the warm (WNM) phase, resulting in a decrease in the efficiency of star-formation, and vice versa (see Fig.~1 in \citealt{Ostriker2010}).
In the limit where CR dominates over PE heating, the regulation is also expected to hold,
since CRs
are accelerated in supernovae remnants, and thus the CR ionization rate is also likely proportional
 to the star-formation rate.

 If multiphase structure is indeed fundamental for star-formation regulation in galaxies, then our model provides a prediction for the thermal
 pressure of the atomic ISM in galaxies, as a function of metallicity, CR ionization rate and UV intensity (i.e., see Figs.~\ref{fig: T_2d}, \ref{fig: T_2d_vars}).
 This pressure may be then related to the star-formation rate and galaxy gas and dark matter halo, through the requirement of hydrostatic equilibrium.

At extremely low metallicities, the multiphase structure disappears, and the multiphase-regulation mechanism cannot hold.
This may imply a different (potentially a bursty) mode of star-formation for galaxies in this early evolutionary stage.
In this limit the dense gas remains warm, at $\approx 600$ K, and is cooled by H$_2$ cooling. This temperature is a factor of $\sim 10$ higher than the classic CNM at solar metallicity.
The corresponding \citet{Jeans1902} mass is a factor of $\approx 10^{3/2} \approx 30$ higher.
Stellar structures forming in this low metallicity regime are expected to be more massive compared to their high metallicity counterparts.

Our finding that the cooling of the atomic ISM at low metallicity (below $\approx 0.01$) is often dominated by H$_2$ line emission from the moderately warm phase, may open up a new observational window for the study of the low metallicity ISM, in local dwarfs and in galaxies of early evolutionary stages at high redshift.
This is particularly timely with the upcoming launch of the James Webb Space Telescope (JWST) which will be able to observe IR H$_2$ lines at extreme sensitivity.

\acknowledgements
We thank John Black, Emeric Bron, Franck Le Pettit, Evelyne Roueff, Chris McKee, and Eve Ostriker,
for fruitful discussion and helpful suggestions.
 We thank the referee for constructive comments that improved our manuscript significantly.
SB thanks the Center for Computational Astrophysics at the Flatiron Institute for hospitality and funding where some of this research was carried out with AS.
This work was also supported by the German Science Foundation via DFG/DIP grant STE 1869/2-1 GE 625/17-1 at Tel Aviv University.

redpajama_set_name: RedPajamaArXiv


How do I delete a pod?
Deleting a pod from your account will completely disassociate this pod from your account. You will be able to add the pod back to your account or add it to another Plume account after deleting the pod.
Open the main menu by tapping the Ξ button on the bottom-left of the app home screen.
Tap on the Adaptive WiFi™ tab.
Select the pod you would like to delete by clicking on the ⋯ beside the name of the selected pod.
Select Delete this Pod... and then Delete Pod to confirm.
You can also go directly to the individual pod's menu by clicking it from the topology view.
Open the main menu by tapping the Ξ button on the top-left of the app home screen.
Select the pod you would like to delete by clicking on the ︙ beside the name of the selected pod.
Select Delete Pod and Delete to confirm.
You can also go directly to the pod's menu by clicking it from the topology view.
On Android, you can Delete all Pods from the Pods menu by tapping the ︙ on the top-right of the screen.


text:
Home Destinations Asia Pacific Soundtrack to Southeast Asia
Soundtrack to Southeast Asia
Explore the wealth of musical customs that still prevail in modern Asia
For travellers with an ear for music, an astounding variety of traditional genres are waiting to be discovered and enjoyed in the Southeast Asia region.
Every country boasts a rich tapestry of musical history that has given rise to a wide assortment of styles. There is also an incredible array of traditional instruments still being played, some of which date back over a thousand years.
The unique range of musical influences expressed across Southeast Asia is impressive indeed. Thanks to the close proximity of many countries in the region to one another and the vast ethnic and linguistic diversity this brings, an array of styles can be found and heard in much of the Asian music that survives today.
With so much variety, it's impossible to make a definitive list, but a few examples from specific countries in the region provide a suitable sampler.
Thailand Tunes
Indonesian Gongs
Vietnamese Percussion
A Few Hand-picked Venues
Thai classical music emerged in its present form from the court ensembles and repertoires that evolved within the royal courts of Thailand some 800 years ago.
Traditional Thai classical style is based on the oral tradition of performance, passed down from generation to generation, although modern Thai classical composers now write down their compositions and many have found fame through their work.
Classical Thai music is polyphonic, born from the layering of multiple individual sounds, instruments and voices, and each part improvises within the accepted parameters laid out along basic lines of harmony, or melody, referred to as paths. The tempo is generally steady using simple patterns and regular punctuation. There are three primary types of classical ensembles, the Piphat, Khrueang sai and Mahori; although they differ in significant ways, all three follow the same basic instrumentation.
There are many traditional Thai instruments used in this type of music, far too many to list here, but some of the main ones you still find being played include 'Ching' hand cymbals (small, thick cymbals, joined by a cord) used to set the background tempo, while the 'khrap' (a bundle of bamboo or hardwood sticks) create the primary beat references. Several kinds of small drums, or 'klong', are also used in these ensembles, outlining the basic rhythmic structure, which is punctuated by the striking of a 'mong'; an embossed hanging gong.
With performances common in hotels, restaurants and even night markets, travellers may get a chance to hear Thai classical music performance during a trip to the Kingdom. Those that want to learn about the tradition will find that most Thai musicians are more than happy to talk about their craft.
Gamelan, is traditional ensemble music from the Java and Bali regions of Indonesia, made up mostly of percussive instruments. It forms an important part of Southeast Asia's gong-chime musical culture and can be heard as background in most major tourist areas.
The word Gamelan, derives from the low Javanese word 'gamel'; a type of mallet used to strike percussive instruments. There are many varieties of Gamelan. In fact, no two ensembles are the same, and different regions, cultural contexts and collections of instruments, all form their own styles.
The Gamelan is made up of a range of traditional Indonesian instruments, primarily metallophones; a group of instruments formed by a series of tuned metal bars that are struck to make sounds. Indonesian traditional metallophones are numerous, yet most Gamelans will include a 'gendér' (made up of ten to fourteen metal bars of different notes) and a 'saron' (which has seven bronze bars placed over a resonating frame). The 'Kendhang' (a double-headed hand-played drum), which has played a role in Indonesian music for centuries, is also essential for the tempo and rhythm of Gamelan compositions.
The Gamelan is still regularly performed today around the Indonesian archipelago, often accompanying rituals and ceremonies. It is still considered culturally important, so much so that there is a Javanese saying 'It is not official until the gong is hung', whilst in Bali, almost all religious rituals include Gamelan performance.
Traditional Vietnamese instruments are also varied and numerous but can be grouped into three basic categories, stringed, wind and percussion instruments. The latter group includes drums, of various sizes, struck with a stick or beater, an assortment of xylophone style instruments, gongs, clappers and 'Song Loan'; a wooden block with a hollow centre containing a ball, played by the foot forcing the ball to beat the sides of the block.
Vietnamese folk music also comes in many different styles and variations, depending on the occasion or instruments used. Much of this folk music still plays an important part in Vietnamese life today, so it's easy to seek out and enjoy distinctive compositions, many of which have evolved with the changing times.
Chèo is a popular form of satirical musical theatre in Vietnam, often including dance, and was traditionally performed in outdoor public spaces by peasants. Its origins date back as far as the 12th century when the original ensembles consisted of a fiddle, flute and drum, and actors performed with sparse costumes and no props or scenery. Back then it was about creating a story through music, singing and dancing alone, but more modern interpretations are more elaborate; often performed by professionals in dedicated venues.
There are endless types of musical entertainment to be enjoyed in the Southeast Asia region. In fact, almost every province has its own distinctive sound and it is therefore worth doing some research to find out where you can listen to local music in a particular area. Below are a few hand-picked venues and events to include in a musical journey.
If you want to hear a Gamelan, then Radio Republik Indonesia, which is the state radio network, has regular performances and recordings played on their channels, for example the Pura Pakualaman Gamelan performs live every Minggu Pon (a day in the 35-day cycle of the Javanese calendar). In major Indonesian towns and cities, the RRI employ professional musicians and actors to broadcast a wide selection of Gamelan music and dramas. For more information check out their website: http://www.rri.co.id/home.html
If you find yourself in Vietnam, you do not want to miss a Water Puppet performance. The genre takes its musical influence from the Chèo style of traditional folk music, and combines it with puppet theatre, where the stage is the surface of a body of water. This enchanting form of entertainment brings traditional folk tales and traditions to life. Regular performances take place in major cities, but in Hanoi, the Thang Long Water Puppet Theatre offers particularly impressive shows.
It is becoming increasingly difficult to find full performances of Thai Classical Music other than those presented for tourists, especially with the growing popularity of Thai pop music and western music. If you want to see a live performance, then one of the best remaining venues is the Sala Chalermkrung Royal Theatre in Bangkok, which offers regular shows that combine traditional music and dance. For more information, and to view a full schedule, check out their website: http://www.salachalermkrung.com
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redpajama_set_name: RedPajamaCommonCrawl


Tether may currently be stable and may have seen its value return to be even with the US dollar, but this may not last long. The stablecoin held 94% of the stablecoin market at the beginning of 2018 – with only two other stablecoin competitors – and continues to lose ground. Now with eight serious stablecoins in the market, Tether's market dominance lies at only 74% and could fall even further.
It seems like everyone wants to launch a stablecoin these days. There is the USDCoin from Circle, Paxos, TrueUSD and even Gemini Dollars. Even the Huobi exchange has gotten in on the fun, launching its HUSD stablecoin. On the surface, they appear to be solid as they're backed by physical assets such as dollars or gold, but we have already witnessed how easily it is to see a stablecoin come unglued. Not too long ago, Tether saw its price break free from the US dollar, falling at one point to $94.
Tether has had to deal with a series of issues that may have contributed to its decline among stablecoins. It has repeatedly refused to release audits of its holdings, stating that it would be too difficult (even though other stablecoins readily acknowledge that they can provide the data). Its new banking partner, Deltec out of Brazil, is facing an investigation for its possible involvement in a money-laundering scheme. There have also been concerns raised that the stablecoin was used to manipulate Bitcoin Core (BTC) prices last year.
The case for stablecoins is tenuous, at best. While all cryptocurrency options are still young and the industry needs to be developed, there isn't much call for an option that can both show volatility on its own, as well as on the asset which backs it.
The post Tether continues to lose market dominance appeared first on CoinGeek.


Center First UMC Missions/Outreach Committee extends their gratitude and appreciation to everyone who entered the cook-off and to everyone who attended and voted for their favorite Soup and Chili! Thank you for making our first cook-off fundraiser a success!
Congratulations to our Soup & Chili Cook-Off 2019 Winners!
We began this year with prayer that God would open the doors and lead us to what mission He has planned for us to do this year. For this we say thank you Lord.
I would like to extend a big blessing from 2017 that truly stands out in my mind… Members were asked to donate to Faith United Methodist Church in Orange. $3,000.00 was donated!!! Thanks to everyone for your help with the Missions Committee. We have had a wonderful year, and God has blessed our program with so much support. It has been such a blessing to be your chairperson for the last several years. You have all made this program such a success.


Our vision is that all children and youth value and enjoy the lifelong benefits of healthy, active living. Since 1921, we have worked in partnership with school boards, public health, government, non-government organizations and companies to develop groundbreaking programs and services.
""Ophea is many things to us: a valuable partner to work with in supporting school health, and a trusted provider of quality and evidence-informed school health resources that we can use and confidently recommend to our local school partners.""
""I use the Ophea website as the primary resource for my Faculty of Education students. Being familiar with Ophea will make teaching easier and more effective for these teacher candidates in the future.""
""I think that [Ophea's capacity building project] really helped give me confidence… You know, they emphasized that just because you didn't go and do a Physical Education degree doesn't mean that you're not able to provide an effective [Health & Physical Education] program." "


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Adobe Premiere Pro: How To Make A Glitch Effect
outubro 31, 2022 abril 3, 2022 Por Writer
A glitch effect is a popular video editing effect to emulate. Beginners can use this easy-to-use effect in many different ways in a project like to help tell a story, make art, or make a transition. Even more fun is playing around with how the glitch looks when you make it.
Below, we show you how to make a glitch effect in Adobe Premiere Pro simple, step-by-step. You can use this method for almost any project. You can use this tutorial as a base and try different things to get your desired look. In this article, you'll also find some ideas for making your glitch effect even better.
How to Make a Glitch
Many people use After Effects or add-ons to make these effects. However, you can make a beautiful glitch effect with Premiere. Follow these steps to make a simple glitch effect. Make sure that the clip you want to use for your effect is on the timeline.
It will look like this: First, split the clip where you want the glitch to start. Then move the clips a little farther apart, as shown in the screenshot below. Right-click to choose and copy all the split-up clips to make a new layer. Then paste them on the V2 timeline above your original clips to make a copy of them.
Select one of the clips that have been copied and go to the Effect Controls panel. You can change how the clip looks. When you go to the Opacity value, could you set it to 50%, then click on it? As the last step, change the Position values in the Motion drop-down menu so that the overlay duplicate clip is slightly off the original.
Do this with the rest of the copies, too. You'll need to change the glitch's position and, if possible, its scale values to make it more interesting. To get different looks, you can also play around with Blend Mode. This is enough for a simple glitch effect. There are even more ways to make it better.
Changing the Glitch Effect, You Make
With a few effects in Premiere, you can make your glitch look even more real. Glitch effects can look a lot different with these effects. You can play around with them to get the look you want.
Effects in RGB
Play around with the RGB values of the video clips you copy and paste to make good glitch art. Doing this can make your VHS or camera look more like it's been messed up. I show you how to do this with the Arithmetic effect in this video.
Go to the Effects panel and look for the Arithmetic effect. You can also go to Video Effects > Obsolete > Arithmetic to look for the effect there.
When you make copies of parts of a video, apply this to those parts.
Under Arithmetic, change a color channel's value to make it look like it has glitched together. Also, change the drop-down menu for the Operator to Difference. A cleaner look can be achieved with this.
It will make the color more or less saturated if you change the Red value. This is a typical glitch effect. You can even play around with the other colors to see how they look different. You could play around with keyframe animations to make the RGB effect cut in and out. This would add to the overall glitch effect.
Effects of Wave Warp
Another way to make your footage look shaky is to use the Wave Warp distortion tool. To make the video look like a VHS scan line, you can use Effect Controls to make it look more like a glitch.
To start, you'll need to add an adjustment layer. This layer will apply the wave effect to both the original and the copies of the clips. If you want to make changes to your project, go to the bottom-right corner of the Project panel and choose New Item > Adjustment Layer. Move the layer you made from the Project panel to the V3 timeline above the cut-up clips.
Wave Warp is in the Effects panel. You can go to Video Effects > Distortion > Wave Warp to do this. Add it to your adjustment layer by moving it.
To do this, go to the Effect Controls panel and click on the Wave Warp drop-down menu. This is where you can change the effect until it looks the way you want it to. Try the Square option in the Wave Type drop-down to get a glitchy effect.
Wave Width and Wave Speed can also change how far apart each wave is. Keyframes could also be used to make the wave warp look more glitchy. To get a glitch that looks good in your video, you can play around with these options.
Make a Glitch Effect Video to Make an Exciting Video
People use a glitch effect on YouTube and TikTok videos to make them more visually interesting and interesting. It can also be utilized as a story-telling tool to get the effect of a broken camera or another device. Because there isn't a simple effect you can use to make a glitch effect in Premiere, it might not be obvious how to make one at first. But if you play around with some of the existing effects, it's not too hard.
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