# Distinguishing Between Photoionized and Collisionally Ionized Gas in the
Circumgalactic Medium

Clayton Strawn1, Santi Roca Fàbrega2,3, Joel Primack1
1 Physics Department, University of California, Santa Cruz, CA 95064, USA
2 Universidad Complutense de Madrid, Departamento de Física de la Tierra y
Astrofísica, Madrid 28040, Spain
3 Universidad Nacional Autónoma de México, Instituto de Astronomía de
Ensenada, BC 22860, Mexico
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Most studies of highly ionized plasmas have historically assumed ions are
either in photoionization equilibrium, PIE, or collisional ionization
equilibrium, CIE, sometimes including multiple phases with different relevant
mechanisms. Simulation analysis packages, on the other hand, tend to use
precomputed ion fraction tables which include both mechanisms, among others.
Focusing on the low-density, high temperature phase space likely to be most
relevant in the circumgalactic medium, in this work we show that most ions can
be classified as ‘PI’ or ‘CI’ on an ion-by-ion basis. This means that for a
cloud at a particular point in phase space, some ions will be created
primarily by PI and others by CI, with other mechanisms playing only very
minor roles. Specifically, we show that ions are generally CI if the thermal
energy per particle is greater than $\sim 6$% of their ionization energy, and
PI otherwise. We analyse the accuracy of this ansatz compared to usual PIE/CIE
calculations, and show the surprisingly minor dependence of this conclusion on
redshift and ionizing background.

###### keywords:

Galaxies:Haloes – Quasars:Absorption Lines – Software:Simulations

††pubyear: 2022††pagerange: Distinguishing Between Photoionized and
Collisionally Ionized Gas in the Circumgalactic Medium–A

##  1 Introduction

The Circumgalactic Medium (CGM), the region of gas surrounding galaxies within
their virial radius $R_{\rm vir}$ remains an enigmatic puzzle for the study of
the evolution and development of galaxies and their environment. The existence
of significant amounts of gas in this region, and its significance for star
formation and structure formation has been well known for several decades. We
have seen that galaxies, on their own, do not contain most of the baryons in
the standard $\Lambda\rm{CDM}$ cosmology, and this problem is known as the
missing baryon problem (see Tumlinson et al., 2017, and references therein).
Observations suggest that a significant fraction of the cosmic baryon budget
is in the CGM of galactic haloes (Werk et al., 2013; Werk et al., 2014). The
CGM also contains a large fraction, or perhaps even a majority, of the metals
created in the Universe, as only about 20 – 25 percent of the created metals
remain in the galaxy in the form of stars, dust, and ISM gas. The mechanisms
by which gas flows into and out of galaxies, while understood through broad-
strokes ‘bathtub’ models, have many details that are difficult to fully work
out, such as the interaction between cool, inflowing streams,and hot, metal-
rich outflows driven by supernovas or active galactic nuclei, as well as the
effect of the interplay of this interaction with the contribution from virial
shock heating (Birnboim & Dekel, 2003; Dekel & Birnboim, 2008). Also uncertain
are the effects of magnetic fields (Nelson et al., 2021), cosmic rays (Hopkins
et al., 2020), and thermal instabilities (Mandelker et al., 2020a, b).

Due to its low density and high degree of ionization, it is very difficult to
observe the CGM through emission except in very nearby galaxies or the Milky
Way (Gupta et al., 2012; Fang et al., 2015; Lehner et al., 2020). Instead, the
primary way by which we study the CGM in the modern era is through absorption
line spectroscopy. Background objects, mainly quasars, are used as light
sources and their spectra are analysed to identify absorption lines and
therefore detect what kind of gas is in the intervening clouds. The increased
signal to noise of this kind of data, especially in recent years with the
deployment of the Cosmic Origins Spectrograph (COS) on Hubble Space Telescope
(Tumlinson et al., 2011; Werk et al., 2013; Werk et al., 2014), can give a
very sensitive picture of a multi-component cloud of different ions. However
the relatively small number of bright quasars means they rarely give multiple
glimpses into the same galaxy, though there have been a few examples of
multiple-detections either coincidentally (Lehner et al., 2015; Bowen et al.,
2016) or using strong gravitational lensing to probe the same quasar in
multiple places (Lopez et al., 2018; Okoshi et al., 2019). With the successful
launch of the James Webb Space Telescope in 2022, the new instruments should
be able to acquire much better studies of the CGM, both in absorption and
emission using the new MOS and IFU instruments (Kutyrev et al., 2008).

To extract maximally useful information from the observations we do have, we
often try to fit each velocity component of the gas as a separate phase.
However, line confusion, saturation, and noise limits sometimes mean that
there is some ambiguity about the possible mechanisms that can come into play.
At CGM densities (which we will take to mean
$10^{-8}$cm${}^{-3}<n<10^{2.5}$cm-3), the primary mechanisms for ionizing a
particle are photoionization and collisional ionization. In this low-density
regime, recombination is effectively dominated by radiative recombination, and
therefore only the photoionization and collisional ionization rates change
with position in phase space. However, at the low temperature, high-density
end, heavier elements can start to see the effects of other mechanisms, as
detailed in Appendix A.

Photoionization, or PI, is where atoms absorb incoming photons from a variety
of sources, including the metagalactic background radiation, emission lines
from gas clouds, and radiation from stellar, AGN, and supernova sources. This
state, assuming ions come to equilibrium, is called photoionization
equilibrium, or PIE. In the CGM, outside of the region very near the galaxy
(Sternberg et al., 2002; Sanderbeck et al., 2018), the most important source
is the metagalactic UV background. Generally, the most common background used
by the community is that of Haardt & Madau (2012), hereafter HM12, and we also
use that here unless otherwise specified. However, other UV background models
have also gained attention in recent years (e.g., Faucher-Giguère et al.,
2009; Kuhlen & Faucher-Giguère, 2012; Faucher-Giguère, 2020).

In the PI case, the ionization level is almost a pure function of density with
minimal temperature dependence. Effectively, denser clouds have fewer ionizing
photons per particle, thus stabilizing with a higher fraction of low
ionization states compared to high ionization states. So, using PIE, fitting
the detected ions in a given component gives a good estimate of the density of
the gas, which can be combined with the hydrogen column density to get an
estimate of the metallicity, while the absorption line widths can give an
estimate for the temperature.

Alternatively, atoms can be ionized through collisional ionization, or CI.
When they collide, some of their kinetic energy is transferred to their
internal electron structure, giving the electron(s) enough energy to escape.
If only this mechanism is relevant, it is called collisional ionization
equilibrium, or CIE. In this case, the ionization level is a pure function of
temperature, and at higher temperatures, a greater proportion of gas is in
high ionization states versus low ones. In CIE, fitting the detected ions
gives an estimate of the temperature of the cloud, which when combined with
the equivalent widths of the lines and relative amounts of different metal and
hydrogen species, can give a good sense of the overall phase of the gas.

There is much debate over which mechanisms are relevant and for which clouds
of gas, and many different assumptions have been made to account for one, or
the other, or both. At a superficial level, this difficulty is exacerbated by
the fact that either mechanism taken by itself will clearly lead to pressure-
balanced states in the CGM, which range from low-density hot gas to high-
density cool gas, to contain higher and lower ions, respectively. However, to
analyse observations in a sophisticated way by including both mechanisms can
be very difficult, due to having more difficulty efficiently constraining
either density or temperature. If the mechanism cannot be assumed, then both
variables need to be decided by the noisy properties of the lines themselves,
and the appearance and relative strength of different ions cannot be used to
constrain either quantity.

In modern galaxy simulations, by contrast, determining the phase of gas in the
simulated CGM is not itself a challenge, though correctly evolving the phase
remains quite unsolved, with different codes leading to vastly different
results even with the same initial conditions (e.g. the agora project, Roca-
Fàbrega et al., 2021, Roca-Fàbrega et al. in prep., Strawn et al. in prep.).
Since in a simulation the full physical state of every parcel of gas is easily
available, there is no need to assume only one mechanism is relevant. Software
codes like trident (Hummels et al., 2016), pygad (Röttgers et al., 2020), and
others (e.g. Churchill et al., 2014; Churchill et al., 2015) simply
interpolate pre-made tables from Cloudy (Ferland et al., 1998; Ferland et al.,
2013, 2017) to determine ionization fractions as a function of both
temperature and density, without any need to explicitly reference the two
mechanisms. However, we believe there is still some value in defining certain
ions to be created predominantly through PI or CI, incorporating some of
breakthroughs in simulation studies. In particular in Roca-Fàbrega et al.
(2019) and Strawn et al. (2021), we used a definition of PI-dominated and CI-
dominated gas to distinguish O vi states in a simulation. This definition led
to the discovery within the cosmological simulation of a thin CI-O vi
interface layer on the edge of cool, inflowing PI-O vi clouds.

This definition of PI and CI leads to an ion-by-ion distinction where some
ions are predominantly determined by PI mechanisms, and others are
predominantly determined by CI mechanisms. By splitting up these two types of
ions, the weakness of using a full PI and CI model (requiring both temperature
and density of a cloud be determined by the absorption line shapes) can be
mostly alleviated, as one can use a rough temperature to determine which
mechanism is most relevant for a component, and then constrain the density
with PI ions, and/or the temperature with CI ions.

This paper is organized as follows. In Section 2 we discuss the definitions of
PI and CI gas for different ions, and show which ions are in which state as a
function of density and temperature throughout phase space. In Section 3, we
analyse the effect of changing the extragalactic background, either with HM12
but at different redshifts or by arbitrarily modifying the central powerlaw of
HM12 according to the procedure outlined in Haislmaier et al. (2021), to show
that none of these changes meaningfully affect the distinctions used here. In
Section 4, we discuss the physical principle at play here, and show that for
each ion, where the cutoff between CI and PI occurs depends on the ratio of
its ionization energy to the average energy per particle at that temperature.
In Section 5 we outline the effect of using PIE for some regions of phase
space, and CIE for others, depending on the detected ions. Finally, we
summarize our conclusions in Section 6.

##  2 Ion-by-ion Definition of CI and PI

We generate all the data necessary for this definition from the software
Cloudy111We use version Cloudy 17.03 for the data in this work, see Ferland et
al. (2017). Cloudy is an open-source spectral synthesis code which simulates
the state of gas under many different astronomically-relevant physical
conditions. It is at the basis of many, if not most, gas physics packages used
in modern simulation codes. Among its many other uses in the community, of
particular interest to us is that it tracks the distribution of each element
into different ions as a function of density, temperature, time, and location
relative to different ionizing sources, and intervening absorbers.

The way Cloudy is used in simulations of the circumgalactic medium, as in
trident, pygad and all their dependent papers, e.g. Peeples et al. (2019);
Strawn et al. (2021) and Röttgers et al. (2020), respectively, is to always
make a few simplifying assumptions that allow a small number of pregenerated
tables to define the ion fractions in each cell (or, in each smoothed gas
particle) of the simulation. There are two common assumptions that go into
this simulation.

First, it is assumed that the different ions are always in ionization
equilibrium, where the rate of particles entering a particular ionization
species through ionization of lower states and through recombination from
higher states is equal to the rate of particles leaving that state through
further ionization or recombination into lower states. In other words, it is
assumed that ionization equilibration, in all areas of phase space, takes
place much faster than changes in temperature or density. This approximation
is generally fairly good (see, for example, Ji et al., 2019) as long as there
are not too extreme of energy events, such as near AGN, that can leave some
‘fossilized’ ionization for thousands of years directly in the path of an AGN
jet, even long after the AGN has shut off (Oppenheimer & Schaye, 2013). In any
case, analysis of simulations through post-processing cannot be done without
this assumption, or an equivalent non-equilibrium ion fraction table, which is
a function of recent temperature and/or density. In principle ion fractions
could be tracked over time as separate fields within the simulation, and thus
evolve from the ionization and recombination rates directly, but this
calculation would greatly increase computational time and expense, and require
significantly finer timesteps than are possible today.

Second, it is assumed that ionizing radiation is uniform and isentropic.
Uniformity is enforced by inserting a constant ionizing background radiation
spectra, which does not depend on any local sources or effects. The most
common background used is HM12, though other possible backgrounds are also
relevant. When relatively close to a galaxy, especially if it is undergoing a
starburst or AGN activity, this approximation is not very good (Sternberg et
al., 2002; Sanderbeck et al., 2018). However, an improved schema which is not
yet in widespread use but may be soon would be to use some precomputed Cloudy
tables with different, realistic backgrounds according to the approximate
distance to the galaxy centre (e.g., Gnedin & Hollon, 2012; Kannan et al.,
2014, 2016).

Isentropy is essentially a claim that the CGM is optically thin, and therefore
the direction from which a photon comes has no impact on its penetration into
the material. Gas in the CGM is usually fairly low density (typical number
densities are between $10^{-7}$ and $10^{-1}$ cm-3), and so this assumption
should have little impact on our results. However, there are other regions of
the CGM which might be higher density and therefore self-shielded (O’Meara et
al., 2013; Altay et al., 2011). While this effect is generally accounted for
directly in subgrid models for heating and cooling (see, e.g. Kim et al.,
2016), it is not accounted for by our ion fraction grid.

With these assumptions made, Cloudy can create accurate tables of ion fraction
as a function of temperature, density, and ionizing background. A contour plot
of these fractions with the background from HM12 at redshift $z=0$ is shown in
Fig. 1. We focus here on some of the species that are particularly well-
studied in observations and simulations, because they have very strong lines
due to their lithium-like nature (3 electrons remaining), although their
significant differences in charge means they have a broad range of ionizing
potentials. Besides these lithium-like ions, the most commonly studied other
ions in observation are low ions (neutral, singly or doubly ionized). These
ions are often not studied in cosmological simulations due to the expected
cloud sizes being too small to effectively resolve (Peeples et al., 2019,
Hummels et al., 2019; but see also Nelson et al., 2021).

Figure 1: Contour plots for C iv, O vi, Ne viii, and Mg x. Each has a CI peak
(upper right) and a PI peak (lower left). The definition in Section 2
distinguishes the two via the black line: CI above and PI below.

Each contour plot can easily be broken up into a PI region, on the lower left
side, and a CI region, on the upper right side. The PI gas is weakly dependent
on temperature and strongly dependent on density, while the CI region, is
totally independent of density (above some critical density), and depends
strongly on temperature.

While the two peak regions are easily identified by eye, researchers in the
CGM are still interested in giving a rigorous definition that correctly
categorizes the mechanisms in regions far from both peaks. Doing so has proven
somewhat difficult, despite many attempts over the last few years. Among
others, these include separating the regions by density (Faerman et al.,
2020), analyzing the gas PI and CI timescales (Churchill et al., 2015),
separating by temperature (Sanchez et al., 2019), and artificially restricting
to only two obvious options and then making a binary judgement (Stern et al.,
2018).

In Strawn et al. (2021), we showed, based on the work of Roca-Fàbrega et al.
(2019), that there is a straightforward way to define the contributions of the
two phases based on a physical argument, rather than purely on these contour
plots. This argument allows the definition to clearly extend to regions far
from both peaks, such as in the transition region between them or the high-
density, low-temperature corner. The process begins with fixed-temperature
fraction-density curves, as seen in Fig. 2. The density dependence can be
interpreted as follows: photon density per particle increases as density
decreases, here meaning when tracking these graphs from right to left. Thus an
increase of ion fraction when moving leftward indicates ions being created
through photoionization, and a decrease indicates ions being destroyed through
photoionization.

Figure 2: Ion fraction vs density for all species of four atoms: Carbon, Neon,
Oxygen, and Magnesium. All four images are at a fixed temperature of
$T=10^{5.5}$ K. The linestyle reflects what mechanism the algorithm in Section
2 identifies as dominant, with dashed indicating PI, dot-dashed indicating CI,
and solid indicating transitionary. For transitionary ions, the transition
density is also shown with a vertical dotted line and star of the same color.
Negligible species at this temperature (i.e., neutral) are not plotted.

There are three possible shapes of fraction-density graphs at fixed
temperature at CGM densities. They are characterized by the existence, or
nonexistence, of a maximum density, and by a flat shelf at high density, which
we will refer to as the ‘CI base’.

  * •

‘Collisionally Ionized’: First, the graph can stabilize to a CI base at high
density, and always decreases in fraction with decreasing density. This gas is
called ‘Collisionally ionized’ as PI processes only destroy, rather than
create, this ion at this temperature.

  * •

‘Photoionized’: Second, they can fail to stabilize to a CI base at high
density. Instead, with decreasing density, they first increase up to a
maximum, and then decrease back to zero. This gas is called ‘Photoionized’ as
PI processes create all of this ion at this temperature.

  * •

‘Transitionary’: Third, they can have both a maximum and a CI base, meaning
they stabilize at high density, but still increase from that value as density
decreases. We call this gas ‘transitionary’ and the ‘transition point’ is
defined as the density where the CI base is equal to 50 percent of the total
(the rest coming from PI). At densities on the right of the increase, the ion
is created mostly through CI, and on the left, the ion is created mostly
through PI.

The whole of phase space can thus be characterized for each ion (with some
subtlety needed for neutral, singly-ionized, and fully ionized states, see
Section 2.1): at some temperatures, an ion can be characterized as fully
created through PI, at other temperatures, it can be characterized as fully
created through CI, and at still other temperatures, it is primarily PI below
and CI above a particular density. In Fig. 3, we show the full z=0
distribution for all species of the first 12 elements. As expected, every ion
is PI at low temperatures, CI at high temperatures, and transitionary at
intermediate temperatures. However, the change does not happen at the same
temperature for all ions. Higher ionization states remain primarily PI at much
higher temperatures than lower ionization states, so in a single cloud of gas,
some low ions can be present that were created through CI while high ions are
also present that were created through PI.

Figure 3: PI vs CI cutoffs for all species of the first 12 metals. Above each
line, the indicated ion is CI, and below the line, it is PI. Some cutoffs are
overlapping, and are shown with slight offsets to see multiple colors at once.
Lower ions always have cutoffs at lower temperatures than high ions, but a
nonsequential colormap is used to facilitate identifying ions of interest.
Neutral and singly ionized states do not appear for the bottom row, see
Section 2.1 and Appendix A for details.

###  2.1 Neutral, Singly-ionized, and Fully ionized states

The above analysis can be expanded to capture the basic structure of atoms
even where the ionization state is harder to define. For neutral atoms, there
is first of all a semantic point. By definition, neutral atoms are not
ionized, so in principle they are not ‘created’ through PI or CI structures,
rather it would be the absence of either. We will still use the terms ‘PI’ and
‘CI’ for neutral atoms, but by this we simply mean ‘as if the fraction was
determined through PIE’ or ‘as if the fraction was determined through CIE’,
respectively.

The more serious problem is that, given a specific temperature, at no density
could there possibly be a maximum for the neutral fraction. This is because
the neutral fraction at all temperatures should be a monotonically increasing
function of density. A ‘PI’ state for a neutral atom requires the highest
density to be counted as a maximum, and a ‘Transition’ state is impossible.
So, we will define the difference between CI and PI to simply check for the
existence of a CI base on the high-density end which remains sufficiently
flat.

The fact that there is no possibility of a transition temperature for neutral
species also leads to complications for the first-ionized state (C II, O II,
etc.), because all transitions out of the neutral state will directly enter
the singly-ionized state, no matter which mechanism is used. In other words,
the neutral atom can only have a CI base if the singly-ionized state also has
a CI base at the same time. Thus, we add some additional considerations to the
algorithm that if either the singly-ionized or neutral state is PI, the other
will be also. For most atoms, this algorithm gives a transition for singly
ionized states around T$=10^{4.1}$ K.

However, neutral and singly ionized states for ‘heavy elements’, here meaning
elements in the third row of the periodic table (here Na, Mg, Al, and Si) are
exceptions to this rule. In these cases new shapes not described in Section 2
can appear, and therefore cannot be categorized as PI or CI. This effect
arises because due to both their low ionization energy and their quantum
structure, these ions have a much more tenuous hold on their outer electrons
than the rest of the species often seen in the CGM. At low temperatures and
high densities, their ionization mechanism does not conform to the
classification algorithm, because a third mechanism, charge-exchange (Dopita &
Sutherland, 2003; Kallman et al., 2021), becomes comparable to
photoionization. This difficulty reiterates the fact that this classification
is only a simplification that applies at the low densities characteristic of
the CGM and with ions that have relatively tightly bound electrons. The lowest
ions for heavy elements therefore have no meaningful ‘PI dominated region’,
and cannot ‘transition’ from PI to CI. As such, they will be excluded from
analysis for the rest of the paper. See Appendix A for further details.
Unfortunately, this means one of the most commonly studied ions in absorption,
Mg ii (e.g. Nelson et al., 2021; Darekar et al., 2022, in simulations and
observations, respectively), cannot be simply classified as CI or PI.

Fully ionized states with no electrons remaining in principle also could
require a more subtle definition. Symmetrically to the neutral state, there
can be no maximum at any temperature, only a monotonic (now decreasing)
function with increasing density. However by taking the lowest-density
fraction as the maximum, we find this situation remains remarkably similar to
the standard case, and no special treatment is needed. The algorithm detects a
transition if a CI base exists, and, when moving from high density to low
density, the ion fraction increases by a factor of two or more from the CI
base to the maximum. The ion is CI at all densities if this change is never
that large, and PI at all densities if there is no base to speak of at that
temperature. We see in Fig. 3 that fully ionized states become created
entirely through CI at between $T=10^{6}$ K, for C vii, and $T=10^{7}$ K, for
Mg xiii. We will show in Section 4 that this outcome aligns perfectly with the
theoretical prediction of all other states, and thus we will not modify the
algorithm to account for these ions as we did for neutral and singly ionized
states.

##  3 Redshift and Background Dependence

One might anticipate that the above results are strongly dependent on ionizing
background radiation. After all, HM12 already shows an extremely wide range of
ionizing intensities with redshift, with basically no background radiation at
$z=10$, to a peak at $z=2$, and then winding back down to the value observed
today in nearby galaxies. One might expect that a stronger ionizing background
leads to a higher proportion of ions created through PI. But it turns out that
the ionizing background has a surprisingly small effect on the conclusion
about where in phase space an ion is created through primarily PI or CI
mechanisms, at least over the redshift range $z=0-4$. The reason is primarily
the fact that increasing the ionizing radiation, as long as it is at least
somewhat uniform and doesn’t have any outrageous spikes at particular
frequencies, always increases the number of ions photoionized into a state at
the same time as it increases the number further photoionized out of that
state. At a higher overall level of background radiation, the overall
ionization of the whole population of ionic species shifts to higher
ionization, but each individual species is only slightly moved, and only in a
very small region does it actually change the dominant mechanism.

We test this dependence in two ways, first by checking the results of this
procedure with tables generated by Cloudy with HM12 at $z=0$, $1$, $2$, $3$,
and $4$, and then by checking the results by arbitrarily varying the slope of
the ionizing background around the $z=0$ fiducial shape. The formula for this
modification is taken from Haislmaier et al. (2021), and was first used in
Crighton et al. (2015), (see also Fumagalli et al., 2016). At energies greater
than 1 Rydberg, the power-law slope of HM12 is approximately
$\alpha_{UV}=-1.41$. We replace the HM12 with a new background with
$\log_{10}F_{\nu}=f(E)$, where $f(E)$ is defined as

$f(E)=\begin{cases}H(E),&E\leq E_{0}\\\
H(E)+(\alpha_{UV}+1.41)\cdot\log_{10}(E/E_{0}),&E>E_{0}\\\ \end{cases}$ (1)

Here $H(E)$ is the base 10 logarithm of HM12 and $E_{0}$ is 1 Rydberg, or
$13.6$ eV. In Fig. 4 we show the difference in ionizing background caused by
varying these two quantities.

Figure 4: Top, evolution of the HM12 UVB with changes in redshift from $z=4$
to $z=0$. Bottom, effect of changes in UVB by artificially varying the
powerlaw slope $\alpha_{UV}$. In both panels, the black line is the fiducial
HM12 $z=0$ spectrum used throughout this paper except when otherwise
specified. Figure 5: Ionization mechanism vs temperature for all species of
Carbon, Oxygen, Neon, and Magnesium, computed on a grid of 0.1 dex in
temperature space. The colors besides red and orange show the mechanism at
redshift $z=0$ with the fiducial background. Red bars indicate grid points
which had any change from the $z=0$ mechanism at $z=1,2,3,$ or $4$., and
orange bars indicate the same, but for any change from the HM12
($\alpha_{UV}=-1.41$) mechanism at $\alpha_{UV}=-0.28,-0.835,-1.945,$ and
$-2.5$. The grayed-out region is not classified properly with this scheme, see
Appendix A. Even with significant changes in background, changes in mechanism
are shown to be minimal.

In Fig. 5 we show the overall distribution of ionization mechanism with
temperature at redshift $z=0$. All ions follow the same trend of being PI
below some temperature, transitionary for a small number of temperature steps
(sometimes zero), and then CI above. Of course, the CI fraction becomes
negligible at high enough temperature, and those regions are indicated in
gold. The red bars here indicate regions where the dominant mechanism is
changed with changes in redshift from $z=4$ to $z=0$. Effectively, the change
in the ionization mechanism’s temperature dependence is negligible, with the
differences confined mostly to a single data point per ion (spacing being 0.1
dex in temperature). Even this change generally only happens with low ions.
Magnesium is a heavy element in our schema, so Mg i and Mg ii at low
temperatures cannot be classified with either mechanism in this scheme (see
Appendix A), which is why that region is greyed out. The orange bars indicate
ions which have different mechanisms at the same temperatures with changes in
$\alpha_{UV}$. Again, differences are fairly rare, however in this case they
are more likely on high ions, presumably because the ionizing photons for
highly ionized species are more affected by the changes to $\alpha_{UV}$.
Affected species include, interestingly, the highest ions at the lowest
temperatures. This has the physically intuitive meaning that with very soft
backgrounds ($\alpha_{UV}=-1.945,-2.5$), there are not enough high-energy
photons to photoionize all the way to Mg xiii or Ne xi, and the PI ‘maximum’
the algorithm searches for is never detectable. It therefore classifies those
states as CI at low temperatures, because they are not PI-accessible with this
extremely soft background.

We remind the reader however that this redshift independence does not mean
that there will be no evolution in the relative significance of the two
mechanisms with redshift or background when considering an individual ion. In
Roca-Fàbrega et al. (2019) and Strawn et al. (2021) we showed that changes in
redshift accompanied changes and even reversals in PI or CI dominance for O
vi. The difference here is that we are not analyzing a cosmological
simulation, but a simple grid in density-temperature space. Gas processes like
heating, cooling, and chemical evolution cause metals in the CGM to occupy
different regions of this graph with time, and thus affect the ‘dominance’ of
one mechanism in the region as a whole. In fact, the conclusions here suggest
that the change in PI-CI ratio over time is almost entirely due to evolution
in these processes, and not the increasing or decreasing strength of the
background with redshift.

Figure 6: The change with redshift in the transition densities for selected
ions. The vertical offset is just for visibility, but the temperature of the
transitionary points stays fixed with redshift. Different symbols show
different redshifts.

The density dependence on the ionizing background is somewhat more noticeable,
but only under the constraint that the cutoffs were already only weakly
dependent on density (i.e., they were density dependent only within a small
range of transitionary temperatures). In Fig. 6 we show the effect of redshift
on the density thresholds for the transitionary points. Tracing from $z=0$ to
$z=4$ we see that the density threshold for each ion drifts to the right from
$z=0$ to $z=2$, reflecting the increasing strength of the ionizing background
at that redshift, and then drifts back to the left from $z=2$ to $z=4$, as the
ionizing background decreases. A few species have their transition
temperatures changed, and so do not appear at all 5 redshifts. Vertical
offsets are added for increased visibility.

Note that the shapes are slightly different between different ions, so
wavelength-specific effects of the changes from $z=4$ to $z=0$ can cause minor
changes to their relative abundances, but the main effect of increased
background activity at $z\sim 2$ is just decreasing the effective density of
the gas for the purposes of ion fraction calculations. This change has
noticeable effects on ion fraction when an ion is PI, and basically has no
effect if the ion is CI.

##  4 Ionization Mechanism cutoffs as a function of ionization Energy

The very limited effects of redshift and ionizing background on key ionization
mechanism suggests that there is a physical reason for the cutoffs being where
they are which does not depend strongly on the ionizing background. In this
section we show that the PI-CI cutoff for an ion appears to be determined
primarily by the ion’s ionization energy.

Figure 7: Top: The ionization energies for all ions of the first 12 metals.
Bottom: The PI-CI cutoff temperatures for the same ions. Different ionization
levels are shown with different colors, different elements with different
symbols. Note that the PI-CI cutoff is defined for neutral atoms, but
ionization energy is not. Atoms which have no transition (see Appendix A) are
not shown.

In the top panel of Fig. 7 we show the ionization energies of all ionization
species for the first 12 metals, including those with the highest abundances
in the Universe generally and in the CGM. The quantum mechanical ionization
structure gives rise to multiple shells of increasing ionization energy from
the outside in, and clear gaps form between the energy required to leave an
outer shell and the energy required to leave the next innermost shell. In the
bottom panel of Fig. 7, we show the CI-PI cutoff for each ion. Pictured is the
minimum temperature level for a transition point, that is the first
temperature at which CI gas is a majority at any density. Changing the
threshold to represent the CIE peak or the first temperature which is CI at
all densities has only minor effects on this conclusion. The left axis is in
temperature units (K) and the right axis in units of energy per particle (eV),
following $\left<E\right>=\frac{3}{2}k_{b}T$. We see here that the same shell
structure is replicated in both.

Figure 8: Top: The lowest PI-CI cutoff temperature versus ionization energy
for all ions of the first 12 metals, except for those without transitions
(Appendix A). The boxed equation is the best-fitting power law (black line).
Bottom: The PI-CI cutoff temperatures divided by ionization energy for the
same ions. Shaded blue region shows prediction from equation (9). It is clear
that ionization mechanism linearly depends on ionization energy.

In Fig. 8 we compare the two numbers directly, and see that they have an
almost perfectly linear relationship. In the top panel, we see that the best-
fitting line in log-log space has a slope of 0.950. We fit the line to all
states except the singly ionized states, which appear to have a slightly
different relationship, which might be due to the fact that the singly-ionized
state has a threshold which is somewhat more challenging to define than the
further states, as described in Section 2.1. The linear relationship suggests
that the meaningful quantity of interest to determine whether an ion is PI or
CI is the ratio between the average energy per particle and the ionization
energy. We see in the bottom panel of Fig. 8 that over almost three orders of
magnitude in ionization energy, the threshold energy for being CI-dominated is
consistently around 4 – 7 percent of the average energy per particle,
increasing to up to $\sim 30$ percent for the singly ionized states. For
higher ions, this result aligns quite well with the predicted value (blue),
derived in Section 4.1.

###  4.1 Derivation of relationship between CI threshold and ionization energy

The law shown above in Fig. 8 can be approximately derived as a consequence
of, above all, the steeply exponential dependence of ion fraction on
temperature in CIE. In ionization equilibrium, the ionization rate of each
species is set equal to the recombination rate for the species ionized one
additional time. Generally, there are many mechanisms governing both of these
rates, including photoionization, collisional ionization, radiative
recombination, collisional recombination, and what are called ‘charge-exchange
reactions’ which change the ionization of two species simultaneously. We are
interested in the PI-CI cutoff, which takes place at the lowest temperature in
which CI fractions are relevant. In this regime, the density is taken to be
high enough that the metagalactic background is negligible, and there are no
significant local sources of ionization, so the photoionization term is
neglected. However since even the highest densities in the CGM are much less
than those studied in the ISM or solar system environment, we can also neglect
the contributions of collisional recombination and charge-exchange
interactions (House, 1964; Dopita & Sutherland, 2003). This density regime is
known as the ‘coronal approximation’, where ion ratios are determined by the
following equation:

$\displaystyle\frac{n_{j+1}}{n_{j}}$
$\displaystyle=\frac{C_{j,j+1}}{\alpha_{j+1,j}}\ ,$ (2)

where $n_{j}$ is the concentration of ions with $j$ electrons removed from the
neutral state, $C_{j,j+1}$ is the collisional ionization rate from state $j$
to state $j+1$, and $\alpha_{j+1,j}$ is the radiative recombination rate from
the higher state to the lower one.

Following House (1964), we will use the rates given by Allen (1961) and Elwert
(1952).

$\displaystyle C_{j,j+1}$ $\displaystyle=2.47\cdot
10^{-8}A\zeta_{j}n_{e}\left(\frac{kT}{\varepsilon_{H}}\right)^{\frac{1}{2}}\left(\frac{\varepsilon_{H}}{\chi_{j}}\right)^{2}e^{\frac{-\chi_{j}}{kT}}$
(3)

where

$\displaystyle A$ $\displaystyle=3.1-\frac{1.2}{Z_{j}}-\frac{0.9}{Z_{j}^{2}}$
$\displaystyle\zeta_{j}$ $\displaystyle=\textrm{number of electrons in outer
shell,}$ $\displaystyle n_{e}$ $\displaystyle=\textrm{unbound electron
density,}$ $\displaystyle\chi_{j}$ $\displaystyle=\textrm{difference in
ionization energies, }\varepsilon_{j+1}\textrm{ and }\varepsilon_{j}$
$\displaystyle Z_{j}$ $\displaystyle=\textrm{ionic charge after ionization,}$
$\displaystyle\varepsilon_{H}$ $\displaystyle=\textrm{ionization energy of
hydrogen,}$

and

$\displaystyle\alpha_{j+1,j}$ $\displaystyle=5.16\cdot
10^{-14}f_{1}ngn_{e}\left(\frac{\varepsilon_{H}}{kT}\right)^{\frac{1}{2}}\left(\frac{\chi_{j}^{2}}{\varepsilon_{H}kT}\right)e^{\frac{\chi_{j}}{kT}}E_{1}\left(\frac{\chi_{j}}{kT}\right),$
(4)

where

$\displaystyle n$ $\displaystyle=\textrm{quantum number of ground state,}$
$\displaystyle E_{1}(x)$
$\displaystyle=\int_{x}^{\infty}\frac{e^{-t}dt}{t}\textrm{ is the first
exponential integral.}$

$f_{1}$ and $g$ are empirical factors of $O(1)$. The ion ratio is thus

$\displaystyle\frac{n_{j+1}}{n_{j}}=B\frac{\left(\varepsilon_{H}kT\right)^{2}}{\chi_{j}^{4}}\frac{e^{\frac{-2\chi_{j}}{kT}}}{E_{1}\left(\frac{\chi_{j}}{kT}\right)}\
,$ (5)

where the unitless value $B=4.79\cdot
10^{5}\left(\frac{A\zeta_{j}}{f_{1}ng}\right)$ is constant with respect to $T$
and $n_{e}$, and only mildly varies with species. We will treat it as a
constant of $O(10^{6})$ in this rough analytic calculation.

Successively applying equation (5) can give $n_{j}$ for each of an ion’s $k$
possible states in terms of $n_{0}$. Ion fractions can be found with

$\displaystyle f_{j}$
$\displaystyle=\frac{n_{j}}{n_{0}+n_{1}+\dots+n_{j}+\dots+n_{k}}$
$\displaystyle=\frac{1}{\frac{n_{0}}{n_{j}}+\frac{n_{1}}{n_{j}}+\dots+1+\dots+\frac{n_{k}}{n_{j}}}$
$\displaystyle\approx\frac{1}{\frac{n_{j-1}}{n_{j}}+1+\frac{n_{j+1}}{n_{j}}}\
,$ (6)

where the last line approximates the denominator in the neighborhood of the
CIE peak to be dominated by the $j-1$, $j$, and $j+1$ terms, which can be
considered as governing the behavior where the fraction is rising, flat, and
falling with increasing temperature (see the right-hand side of Fig. 1). In
Fig. 9, we show each of the ratios used in this approximation separately,
compared to the full ion fraction including all terms. This shows the ratio
$\frac{n_{j}}{n_{j-1}}$ term accurately tracks the rising ion fraction at
temperatures below the peak, while the ratio $\frac{n_{j}}{n_{j+1}}$ term
tracks, somewhat less effectively, the falling ion fraction at temperatures
above the peak.

Combining equations (5) and (6), and taking the low-T approximation so that
$E_{1}\left(\frac{\chi_{j}}{kT}\right)\approx e^{\frac{-\chi_{j}}{kT}}$ and
the left (rising) term of the denominator dominates, this equation simplifies
to

$\displaystyle f_{j}$ $\displaystyle\approx\frac{n_{j}}{n_{j-1}}$ (7)
$\displaystyle\approx C\left(j,T\right)e^{\frac{-\varepsilon_{j}}{k_{B}T}},$
(8)

where
$C\left(j,T\right)=\frac{B\left(\varepsilon_{H}kT\right)^{2}}{\chi_{j-1}^{4}}e^{\frac{\varepsilon_{j-1}}{kT}}$
encapsulates all dependence besides the (larger) exponential. We are
interested in the first temperature for which this fraction is non-negligible,
which takes place in our algorithm at around $f_{j}=10^{-8}$. Even if $B$ is
taken as constant, this equation is clearly transcendental and depends on both
$T$ and $\varepsilon_{j-1}$, and therefore cannot be solved analytically.
However, the dependence of the fraction on $C\left(j,T\right)$ is weak enough
that it is sufficient to note its order of magnitude in the relevant
temperature region, which is $C\left(j,T\right)=O(10^{3}-10^{8})$.

Figure 9: Demonstration of the three-term approximation used in equation (6)
for all species of oxygen. We see here $n_{j}$ vs $n_{j-1}$ and $n_{j+1}$,
written out with parentheses, e.g. $n(j)$, for legibility. These ratios are
compared to the actual ion fraction when using all nine terms (dotted line).
Each term is calculated using equation 5 with a constant $B=10^{6}$.

Then, we see that $f_{j}$ first passes $10^{-8}$ at roughly

$\displaystyle\frac{3}{2}kT$
$\displaystyle=\frac{0.65}{8+\log\left[C\left(j,T\right)\right]}\varepsilon_{j}$
$\displaystyle=\left(0.041-0.059\right)\varepsilon_{j},$ (9)

where the low end and the high end of this range involve taking
$C\left(j,T\right)=10^{8}$ and $C\left(j,T\right)=10^{3}$, respectively. This
prediction aligns remarkably well with the detected trend shown in Fig. 8,
which had cutoffs that varied between 4 percent and 7 percent of
$\varepsilon_{j}$.

##  5 Consequences for CGM Modeling and Interpretation

We will now analyse whether our physically motivated distinction between PI
and CI ions in a single gas parcel can give a meaningfully good approximation
for studies of the CGM. Essentially, older studies generally assume gas is in
either a PIE state (e.g. Stern et al., 2016), or a CIE state (e.g. Faerman et
al., 2017, 2020). The power of making this approximation is obvious: if a
component with multiple ions is a single phase in PIE, then the density can be
constrained not only by the equivalent widths, which depend on sensitive
geometric constraints and noisy spectral resolution, but also by the ratios
between different ions, which can be more robust. On the other hand, if this
component is in CIE, then instead the temperature can be constrained in an
analogous way, increasing the power of the line shape and structure to
determine the density independently. In recent years, more researchers are
becoming aware that both PI and CI mechanisms can make meaningful
contributions for almost every ion, depending on the gas phase it is found in.
While it is possible to attempt to constrain both the temperature and the
density via Voight profile $b$ parameters and equivalent widths, it is very
noisy and hard to sort accurately into phases. A more modern approach, which
might be used more commonly in the future, is that of Haislmaier et al.
(2021). They did not assume either CIE or PIE, but instead allowed multiple
phases to exist in the same components, using one or both mechanisms, and used
Monte Carlo simulation techniques to identify the best-fitting density and
temperature for each phase. However, this addition of multiple phases leads to
possibly unnecessarily increased complexity of the final state.

Figure 10: Left: Comparison of a pure PIE, pure CIE, and combined (‘P+C’)
output for O vi. The ‘split’ approximation is defined to be PIE below the
cutoff, CIE above. Middle: comparison of ‘split’ (black lines) approximation
to Cloudy ion fraction table. This panel shows the logarithm of $f_{\rm
split}/f_{\rm P+C}$, so red indicates an overestimation, blue an
underestimation, and white approximately correct. Dark blue, however,
represents regions which are negligible in both approximations. Right: same as
middle, but showing the absolute difference $f_{\rm split}-f_{\rm P+C}$. This
figure shows our approximation is significantly better than naive assumption
of one or the other mechanism, although errors remain at the low-density CIE
end and high-density PIE end.

Using a strict definition of PI vs CI gas can allow the power of the old
approach, which used ion ratios to get relatively clean estimates of density
and temperature from PIE and CIE, respectively, to be incorporated into a
system where clearly both ionization mechanisms matter. A suggested workflow
for analysis of complex spectra with multiple ions is as follows. By getting a
rough estimate of the temperature, or even guessing a temperature and
iterating over multiple guesses, all ions can be assigned PI or CI prior to
fitting. Then, if the detected ions are PI, they can be fit to a particular
density, and if they are CI, they can be fit to a particular temperature. Ions
which are transitionary at this rough temperature can be ignored on this first
pass, to be fit later for additional precision.

In Fig. 10 we show the errors involved in this approach (and by extension,
involved in prior studies using pure PIE or pure CIE approaches). Here we
compare the actual distribution, from iterating over a grid of simulations run
through Cloudy, to the approximation we will call the ‘split’ distribution.
This distribution is created by assuming density-independent CIE above the
defining line, and (nearly) temperature-independent PIE below the line. PIE is
never fully temperature independent, rather it is assumed that the contours
follow straight powerlaws in log-log space with a slope of
$\gamma=\frac{5}{3}$. This dependence springs from the fact that the effective
absorption resonance gets larger with higher temperature, as the increased
Doppler broadening not only increases a species’ receptiveness to absorption
lines, but also to ionizing radiation. This dependence is quite consistent and
is generally included in PIE modeling by using the ionization parameter $U$.

On the left panel, we simply show these two distributions, the simulated and
the approximate one, and in the middle and right panels we show the relative
and the absolute difference, respectively. Red pixels on each graph indicate
where the ‘split’ distribution overestimates the ion fraction compared to the
simulation, and blue pixels show where it instead underestimates the real
distribution. Dark blue pixels represent where both approximations give
negligible values ($f<10^{-6}$), so the difference is not meaningful.

There are two major error-prone regions to be careful of with this approach.
The first is somewhat obvious. CIE is density independent, but as we saw in
Fig. 2, every CI ion except for fully ionized states collapses in fraction at
low enough density, at all temperatures. Every ion drops off as
photoionization destroys their numbers at low density, and our definition did
not give a role to PI there because it only destroys, and does not create,
that ion. So, the CI region on the left end of the graph vastly overestimates
each ion. The second is the PI region on the right hand side of the PIE peak.
Errors here are a result of the fact that we have defined the density cutoff
for transitionary gas to specifically refer to the point where, at a given
density, 50 percent of ions are created through each mechanism. Clearly this
approximation will lead to notable errors near this point, as it effectively
asserts that on the left of the 50 percent mark, 100 percent of the ions are
created through PI, and on the right it is 0 percent. Similarly, right below
the first transitionary temperature are usually several temperatures where
some ions are created through PI, but do not quite reach 50 percent, which is
also approximated as 0 percent PI.

This approximation of a CIE and PIE ‘split’ thus functions most effectively in
the low-density PIE limit, and the high-density CIE limit. But even in regions
with substantial errors, it remains a better approach than assuming a
mechanism which might be totally wrong, as is traditionally done by both
observers and modellers. It is also important to note that there are only
small regions in which high and low ions are likely to both coexist and be
created through different mechanisms, and that is precisely where the low-
density CIE regime, for the low ion, overlaps the high-density PIE regime, for
the high ion. Thus, unfortunately, this definition is most relevant exactly
where the errors are highest.

##  6 Summary and Conclusions

In this work we refined a novel definition of PI and CI gas, which has
previously been shown in early forms in Roca-Fàbrega et al. (2019) and Strawn
et al. (2021). This definition allows ions to be identified individually,
without assuming a universal ionizing mechanism – i.e., PIE or CIE – but also
without foregoing the lessons learned from those two regimes and demanding use
of an arbitrary 2D ion fraction table. Analyzing each ion at fixed
temperature, we define an ion at that temperature to be PI if its fraction-
density curve has a maximum and does not stabilize at high density, CI if it
stabilizes at high density and does not have a maximum, and transitionary if
it has both a maximum and a high-density shelf. In the transitionary case, the
majority of the ion is created through CI at high density, and PI at low
density. Further examination of the consequences of this definition using
Cloudy showed several key insights, which are often ignored in existing CGM
literature.

The main results are as follows:

  * •

Temperature Threshold: Most ions have only a few or no transitional
temperatures, covering less than 1 order of magnitude in temperature. A good
first approximation, then, is that the difference between PI-dominated and CI-
dominated gas is a temperature threshold. Higher ions have a transition at a
higher temperature, and thus in a single cloud at moderate temperature, high
ions can be created through PI and low ions through CI, which is the reverse
of what is often assumed.

  * •

Redshift Independence: Where each mechanism dominates in phase space has only
a slight dependence on redshift from $z=0$ to $z=4$, and on ionizing
background radiation more generally, including little change with variation of
the slope $\alpha_{UV}$ to include both a much harder and softer potential
background. Even though the fractions induced by PI change with redshift,
which mechanism dominates in a certain region barely changes in temperature at
all, and only changes moderately in density at the few transitionary
temperatures.

  * •

Ionization Energy Relation: The temperature cutoff for CI dominance is
strongly determined by ionization energy. Nearly all ions become CI-dominant
when the average kinetic energy per particle exceeds roughly six percent of
the ionization energy. This result is a straightforward consequence of the
steep temperature dependence of the coronal approximation used in CIE. The
exception to this trend is singly-ionized states, which require a larger
fraction.

  * •

‘Split’ distribution: An approximation which assumes ions are distributed into
roughly their CIE fractions above the cutoff temperature, and roughly their
PIE fractions below it, is an adequate approximation of the full, complex
distribution, especially at the high-density CIE end and the low-density PIE
end.

Future applications of this framework can be used for help interpreting
complex spectra such as the CASBaH survey (Prochaska et al., 2019). This
depiction could, for instance, radically improve the priors used for phase
assignment as in Haislmaier et al. (2021), which found both PIE and CIE
necessary to replicate CASBaH absorption components.

## Acknowledgements

Partial support for CS was provided by grant HST-AR-14578 to JP from the STScI
under NASA contract NAS5-26555 and from JP’s Google Faculty Research Grant. CS
also received support from the UCSC Science Internship Program (SIP) as well
as the ARCS Foundation. SRF acknowledges support from a Spanish postdoctoral
fellowship, under grant number 2017- T2/TIC-5592. SRF also acknowledges
financial support from the Spanish Ministry of Economy and Competitiveness
(MINECO) under grant number AYA2016-75808-R, AYA2017-90589-REDT and
S2018/NMT-429, and from the CAM-UCM under grant number PR65/19-22462. We would
like to thank Christophe Morriset for many helpful discussions of Cloudy and
the various mechanisms which it uses, and would also like to acknowledge the
many other developers of that software. We also benefited from helpful
discussions with J. Xavier Prochaska, Joe Burchett, Sandra Faber, David Koo,
Anatoly Klypin, Joanna Woo, and the SIP interns working with CS, Paul
Mayerhofer, Soumily Maji, and Antonio Man. Finally, we thank an anonymous
referee who helped us clarify several key points of the paper.

## Data Availability

Data underlying this article is available at
https://github.com/claytonstrawn/pi_or_ci. This site includes a notebook for
directly generating all images used in this paper, as well as scripts to apply
this definition to other studies.

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##  Appendix A Low Ions of Heavy Elements

In Sections 2, 3, and 4, the neutral and singly ionized species of Mg, Na, Al,
and Si were generally not shown. This omission is because at the low
temperatures which would presumably house the transition points for these
ions, they are not adequately characterized by the shapes described in Section
2. While some of them appear to follow similar patterns, and indeed the
algorithm as initially written does classify them as one or the other, they do
not have several of the features which should be present in either PI or CI,
and the temperatures of their ‘cutoffs’ were far out of line with all the
predictions in Section 4.1.

Figure 11: Like Fig. 2, but showing neutral (blue), singly-ionized (orange),
and doubly-ionized (green) species of the six largest atoms studied here.
Linestyle represents the naive sorting of each ion into PI (dashed), CI (dot-
dashed), or transitionary (solid). In each cell are 5 lines of each color,
representing increments of 0.2 dex in temperature.

To explore the new processes that appear, in Fig. 11 we analyse a large
portion of the ion fraction grid, showing fraction with density at a wide
range of temperatures. The leftmost two columns show F and Ne, which we will
consider ‘light elements’ because they are in the second row of the periodic
table. The rightmost four columns show Na, Mg, Al, and Si, which are ‘heavy
elements’, in the third row. In each panel, five lines are shown for each at
increments of 0.2 dex in temperature. While they are not labeled individually,
they do follow some expected trends, (i.e. neutral fractions always decrease
with increasing temperature, though see point (iii) below).

Essentially, the usual case (light elements) is naive PIE at $T<10^{4}$ K. At
high enough densities, ion fraction for the neutral state approaches 1, all
other states approach 0. Each state has a peak at some characteristic density,
and while the peaks are not always exactly the same height or width, each ion
is dominant around its own peak, with at least 50 percent of the total. At
higher temperatures, ions transition to CIE on the high-density side, with
characteristic flat shelves even for low ions.

In contrast, there are several strange behaviors for the heavy elements which
do not appear to follow the ‘universal’ patterns.

  1. 1.

Na i and Al i are never dominant over Na ii and Al ii, even at the highest
densities and lowest temperatures. This outcome is of course possible with a
CI classification, however Na can be seen to actually curve upwards at high
densities at around $10^{4}$ K, showing this ‘CI’ region is not at all
density-independent. Al i does the same when extended to higher densities,
though not shown here.

  2. 2.

Mg ii and especially Si ii have extremely wide peaks, even reaching a long
plateau before declining slightly at very high densities. This plateau is
effectively density independent, and therefore the high-density decline would
is not simply according to the decreasing strength of the ionizing background
as a ‘PI’ classification would assume, but due to the trade-off between the
photoionization mechanism and another mechanism.

  3. 3.

Mg i and Al i have large regions in temperature space where, while the shape
appears ‘CI’, changes in temperature have no effect on the fraction. The 7
lines for Al i between $10^{3}$ and $10^{4.5}$ K are all overlapping, and Mg i
fractions from $10^{3.5}$ and $10^{4.4}$ K have very little movement.

All these effects take place because the assumption that ions can be
categorized as a binary of ‘primarily PI’ and ‘primarily CI’ relies on the
fact that the no other mechanism is relevant, even though there are a variety
of both ionization and recombination processes studied in the literature and
implemented in Cloudy. In densities relevant to the CGM, usually the only
relevant ionization mechanisms are photo and collisional ionization, and the
only relevant recombination process is radiative recombination. Radiative
recombination cancels out the density dependence of collisional ionization
rates, leaving ‘CI’ ions completely density independent at fixed temperature,
while it does not cancel for ‘PI’ ions, giving rise to simplified peaks at
fixed temperature.

The other ionization processes relevant in astronomy include the Auger process
and charge transfer, while other recombination processes include dielectronic
processes, three-body recombination, and charge transfer (Ferland et al.,
1998; Dopita & Sutherland, 2003; Kallman et al., 2021). For the lowest
ionization states of heavy elements, specifically those with valence electrons
in the 3n shell, electrons are not tightly enough bound to the nucleus to
effectively resist these other processes. This susceptibility is not only
because they have low ionization energies, but also because their electrons
have larger average distance to the nucleus, and lower average speed, and so
are easier to interact with. A detailed study of this regime, including
analysis of whether this regime is relevant in the CGM at all, will almost
certainly be much more complicated than the PI and CI binary explored here,
and is left for future work.

The main use case of the approximation presented in Section 5 remains for ions
which are ionized more than once, including fully-ionized states. To some
extent (Fig. 8) even the lighter elements have difficulty following the trends
for neutral and singly ionized ions, however the definition at least is
coherent and consistent. For these heavy elements, it is neither, and they
effectively show the limits of where this approximation is appropriate.