\section{INTRODUCTION}\label{intro}

In large part, we are aware of the principal processes that create free
electrons in otherwise completely neutral parts of the interstellar medium
(ISM) in the disk of our Galaxy.  However, for some of the contributing
factors, ones that either enhance or diminish the relative ionization of the
gas, there is a need to validate our understanding of their strengths.  Most
of these processes are well understood qualitatively, but one major
quantitative uncertainty is the effectiveness of extreme
ultraviolet (EUV) and soft X-ray photons
in ionizing the gas.  Two factors contribute to this uncertainty: one is the
difficulty in measuring the fluxes of diffuse photons with energies above the
ionization potential of hydrogen (13.6~eV) but below about 100~eV, and the
other is an overall assessment of how well these photons can penetrate the
neutral regions, which depends on the porosity of the gas structures and the
distribution of radiation sources.

Our ultimate goal is not only to understand these processes better, but also
to obtain an estimate for the fractional amount of the gas that is in an
ionized state.  This information is relevant to gauging the strength of
heating of the gas due to the photoelectric effect from dust grains
irradiated by starlight, since the grain charge, which regulates its rate,
depends on the electron density (Weingartner \& Draine 2001b).
While generally considered to be less important than the photoelectric
effect, other means of creating thermal energy, such as the heating caused by
secondary electrons from cosmic ray and X-ray ionizations and the dissipation
of Alfv\'en waves and magnetosonic turbulence (Kulsrud \& Pearce 1969; McIvor
1977; Spangler 1991; Minter \& Spangler 1997; Lerche et al. 2007), depend on
the relative fractions of electrons.  Cooling of the gas through the
collisional excitation of the fine-structure levels of C$^+$, Si$^+$ and
Fe$^+$ or, at temperatures approaching $10^4\,$K various metastable levels
and the L$\alpha$ transition of hydrogen, are likewise governed by the degree
of partial ionization  (Dalgarno \& McCray 1972). In the following
paragraphs, we consider three different pathways for creating small amounts
of ionization in the mostly neutral ISM.

Except for the interiors of dense clouds where there is significant
extinction, all of the space in the disk of our Galaxy is exposed to
ultraviolet starlight photons that are capable of ionizing atoms that have
first ionization potentials less than that of hydrogen (13.6~eV).  Since the
recombination rates of the ions are slow relative to their ionization rates,
the concentrations of the ionized states of these atoms are strongly
dominant.  Thus, it is a simple matter to add together the contributions from
various elements that are able to supply free electrons.  The only
uncertainty here is an accounting of the strengths of depletions from the gas
phase caused by these elements condensing into solid form onto dust grains.
These strengths vary collectively for all of the elements from one region to
the next.  If we take such variations into account (Jenkins 2009), we
can state that most of the gas will have free electron contributions that
should be somewhere in the range $n({\rm M}^+)=0.8-1.7\times 10^{-4}n({\rm
H}_{\rm tot})$, where $n({\rm M}^+)$ is the number density of heavy elements
that are capable of being ionized\footnote{This range was computed from the
solar abundances and representative values of the gas fractions $[{\rm
X}_{\rm gas}/{\rm H}]_1$ and $[{\rm X}_{\rm gas}/{\rm H}]_0$ listed in
Jenkins (2009) for elements that can be ionized by
starlight, except that the gas-phase abundance of C was lowered by a factor
of 0.43, in accord with the recommendation by Sofia et al. (2011) that
earlier determinations of $N({\rm C~II})$ were systematically too high by a
factor of 1/0.43.} and $n({\rm H}_{\rm tot})$ is the total density of hydrogen
in both neutral and ionized forms.

As cosmic ray particles collide with gas atoms in the Galaxy, they heat and
ionize the ISM.  We are unable to observe the flux of the lowest energy
particles because we are shielded from them by the heliospheric magnetic
field, and extrapolations from the observed higher energy flux distributions
are uncertain (Spitzer \& Tomasko 1968).  Nevertheless, from
measurements of the relative abundances of trace molecular species, the
cosmic ray ionization rates $\zeta_{\rm CR}\approx 0.5-9\times 10^{-16}{\rm
s}^{-1}$ seems to be the most plausible range for the general ISM
 (Wagenblast \& Williams 1996; Liszt 2003; Indriolo et al. 2007; Neufeld et
al. 2010; Indriolo \& McCall 2012), although details in the chemical models
may introduce some uncertainty [cf. Le Petit et al. (2004) and Shaw et al.
(2008)].  The chemical models of Bayet et al. (2011) suggest that in some
particularly active regions the ionization rates may increase to $\zeta_{\rm
CR} > 1\times 10^{-14}{\rm s}^{-1}$.

We now consider a third mechanism for ionizing the gas, one that is harder to
quantify than the other two.  EUV and soft X-ray
radiation can ionize atoms, through both the action of primary
photoionizations and by creating a cascade of energetic, secondary
photoelectrons that can collisionally ionize other atoms.  Estimates of the
effectiveness of these agents are difficult to synthesize, since there are
many complicating factors.  At high energies, much of the radiation arises
from cooling, very hot ($T>10^6\,$K) gas coming from recently shocked regions
within the disk and halo of our Galaxy.  For energies slightly above about
100~eV, the photons can survive a journey through the neutral medium up to
about $N({\rm H}^0)=2\times 10^{19}\,{\rm cm}^{-2}$, and this penetration
depth progressively increases with energy.  Supplementing this ionizing
radiation is that coming from several kinds of sources that are embedded
within the neutral medium.  These include stars over a wide range of spectral
types on the main sequence, active X-ray binaries, and the plentiful, but
faint white dwarf (WD) stars.

\section{STRATEGY OF THE INVESTIGATION}\label{strategy}

The objective of the current study is to use specialized observations to help
resolve the uncertainties that were mentioned above and gain a quantitative
insight on how effectively the neutral regions are partially photoionized by
EUV and X-ray radiation.  We do this by repeating a method developed by Sofia
\& Jenkins (1998, hereafter SJ98), who examined
interstellar UV absorption features that could be seen in the spectra of
background stars so that they could compare the abundance of the neutral form
of argon, which is highly susceptible to being photoionized, to that of
hydrogen, which has an ionization potential very close to that of argon but
with a markedly lower ionization cross section.  We propose that it is safe
to assume that the abundance of argon (both neutral and ionized) relative to
that of hydrogen should be equal to the solar ratio. For instance, SJ98
presented arguments that support the principle that argon in the gas phase is
not likely to be depleted by being incorporated into an atomic matrix within
interstellar dust grains.  We will reinforce this idea with some indirect
observational evidence in Section~\ref{reference_abund}.  Thus, we operate on
the principle that any deficiency of neutral argon (Ar~I) below our
expectation for the amount of H~I that is present may be considered to arise
from the conversion of Ar to its ionized form, which is invisible.

The investigation by SJ98 covered an extremely limited number of target stars
observed with the {\it Interstellar Medium Absorption Profile Spectrograph\/}
({\it IMAPS\/}) (Jenkins et al. 1996).  Later, Jenkins et al.
(2000b) and Lehner et al. (2003)
reported on observations of absorption features of Ar~I observed by the {\it
Far Ultraviolet Spectroscopic Explorer\/} ({\it FUSE\/}) (Moos et al. 2000;
Sahnow et al. 2000) toward collections of WD stars inside and
slightly beyond the edge of the Local Bubble,\footnote{The Local Bubble is an
irregularly shaped region with an unusually low average density with a radius
of about 80~pc that is approximately centered on the Sun  (Vergely et al.
2010; Welsh et al. 2010; Reis et al. 2011).  It contains small, partly
ionized clouds immersed in a much lower density medium (Redfield 2006;
Redfield \& Linsky 2008).} in order to infer ionization conditions of clouds
subjected to the characteristic radiation field in our immediate
neighborhood.  Our current goal is to extend our reach well beyond the stars
surveyed in these two studies, again by using {\it FUSE\/} spectra, so that
we can sample regions of more typical densities well outside the Local
Bubble.  We do this by downloading from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science
Institute a large collection of {\it FUSE\/} spectra
of hot subdwarf stars that are situated several hundred pc away from us, well
beyond the boundary of the Local Bubble.

Our ultimate objective is to compare the neutral fractions of Ar and H, as
had been done in past investigations.  However, a conventional approach of
simply deriving the two column densities and dividing one by the other is not
easily achievable with the data in this survey for two reasons.  First, it is
difficult to measure $N$(Ar~I) because the lines are saturated, but only
moderately so, and recorded at low resolution.  Second, the amount of H~I on
a sight line can usually be determined from the damping wings of L$\alpha$,
but when the interstellar column density is low and there is a significant
stellar L$\alpha$ absorption, one must know the star's effective temperature
and surface gravity and then create a model for the stellar feature, against
which the interstellar feature is superimposed.  Also, as emphasized by Sofia
et al. (2011) measurements of $N$(H~I) using the
damping wings of L$\alpha$ can give a misleading outcome if one does not know
about and correct for the effects of the velocity structures of the gas.
This problem is probably most severe for low column density cases in the
present study.

We can overcome the difficulties mentioned above if we replace H with O as
the comparison element.  In the wavelength coverage of {\it FUSE\/} there are
a large number of O~I absorption features, and these lines cover a broad
range of transition probabilities.  Most important, the strengths of the O~I
features are comparable to the one available feature of Ar~I.\footnote{There
are two transitions of Ar~I in the {\it FUSE\/} wavelength coverage.  We can
use only the one at 1048.220$\,$\AA\ because the 1066.660$\,$\AA\ line has
interference from a pair of strong stellar Si~IV lines at nearly the same
wavelength.}  By a simple comparison of the strength of this Ar~I line to
those of O~I, we can determine $N({\rm Ar~I})/N({\rm O~I})$.  We describe
this process in more detail in Section~\ref{derivation}.

The ionization fraction of O is strongly locked to that of H through a strong
charge exchange reaction\footnote{We add a caution that deviations from a
nearly one-to-one relationship for the ionization fractions of O and H can
occur at low temperatures because the ionization potential of O is slightly
higher than that of H ($\Delta E/k = 229\,$K) for the lowest fine-structure
level in the O~I ground state.  However, since the ionization fractions are
small and most of the gas we are considering is at temperatures much higher
than $\Delta E/k$, this deviation is generally small enough to ignore.
Nevertheless, we performed explicit calculations of the O and H ionization
fractions, as described later in Section~\protect\ref{charge_exchange}.}
 (Field \& Steigman 1971; Chambaud et al. 1980; Stancil et al. 1999).  Thus
we can use O as a proxy for H.  The only shortcoming of this tactic is that O
can be depleted in the ISM, but the depletion factors are not very large in
the regimes of low densities considered here  (Cartledge et al. 2004, 2008;
Jenkins 2009).

Section~\ref{obs} of this paper describes the selection of archival {\it
FUSE\/} spectra and how they were processed to yield useful presentations of
the absorption features for measurements of equivalent widths.
Our method of interpreting the spectra to yield the ionization of Ar relative
to that of O is presented in Section~\ref{analysis}.  Section~\ref{beyond}
contains a short digression on how we verify that the target stars are beyond
the edge of the Local Bubble. In Section~\ref{fundamentals} we outline the
basic equations that take into account the processes that influence the
partial ionizations of Ar, relative to those of H and O.  The equations
presented in this section are virtually identical to those outlined by SJ98,
but with some new refinements (i.e., a few reactions that were not included
earlier).  We consider the creation of free electrons from the starlight
ionization of heavy elements and the effects from cosmic rays as processes
that are mostly understood and already accounted for, and view the actions
arising from EUV and X-ray ionizations as the principal unknowns whose
strengths are to be determined.  In Section~\ref{known_photoionization} we
make a prediction for the degree of ionization produced by known sources of
radiation, but find that in order to satisfy the general outcomes for our
measured ratios of Ar~I to O~I, an extraordinarily low volume density of
hydrogen $n({\rm H}_{\rm tot})$ is required.  In order to obtain the same
results for higher densities, we must propose a means of achieving higher
levels of ionization.  In Section~\ref{additional_photoionization} we propose
two possibilities: (1) there is a large residual ionization left over from
effects of radiation emitted by nearby, but now extinct supernova remnants (SNRs)
over the past several Myr or (2) the neutral medium is porous enough to allow
external, low-energy photons to penetrate the gas with less than the expected
amount of attenuation.  Section~\ref{discussion} presents an overview of the
implications of our results on an assortment of physical processes and
various other kinds of observations that depend on electron fractions in the
diffuse, neutral medium.  The paper ends with a summary of the main
conclusions (Section~\ref{summary}).

Appendices to this paper give descriptions of various atomic processes that
were incorporated into the calculations, but at a level of detail that most
readers may wish to ignore.  A general section on the ionizations arising
from secondary electrons (Appendix~\ref{gamma_s}) is broken into two
subsections: one treats the effects from electrons liberated by the
ionizations of H and He (Section~\ref{electrons_H_He}), while the second one
discusses primary and Auger electrons created by the inner shell X-ray
ionizations of heavy elements (Section~\ref{heavy_elem}).
Appendix~\ref{gamma_he} gives the equations for evaluating the effects from a
multitude of different kinds of ionizing photons that arise from the
recombination of He$^{++}$ and He$^+$ ions with free electrons.  Finally,
Appendix~\ref{cr_ioniz} describes how we can estimate the rates of cosmic ray
ionization of H$^0$, He$^0$ and Ar$^0$ from the observed rates that apply to
molecular hydrogen in dense clouds.

\section{OBSERVATIONS}\label{obs}

\subsection{Target Selection}\label{tgt selection}

Our objective was to make use of target stars that represented intermediate
cases between nearby WD stars, whose sight lines are entirely or
heavily influenced by conditions in the Local Bubble, and the much more
distant hot main-sequence, giant or supergiant stars that can create their
own enhanced ionizations in atypical concentrations of gas associated with
their formation.  Hot subdwarf stars represent a class of objects that fall
into this intermediate category.  They have distances that are of order a few
hundred parsecs from us, which reduces the contribution from material in the
Local Bubble to a very minor level.  Since they are old, they have had
adequate time to escape from their progenitorial gas clouds, and thus their
locations are essentially random and should show no preference for dense gas
complexes.  They have the additional advantage that they are bright enough to
yield good quality spectra, but they are not so bright that they exceed the
maximum allowed count-rate levels for {\it FUSE}.

In an initial screening of prospective targets, we examined the quick-look
plots of all stars classified as sdO and sdB spectral types in the MAST
archive of {\it FUSE\/} data.  In this step, we rejected all spectra that
either seemed to show very strong stellar spectral features (8 stars) or that
had an inadequate signal-to-noise ratio ($S/N$) at wavelengths in the
vicinity of 920$\,$\AA\ (70 stars), which is where most of the O~I lines are
situated.  A few further rejections were made after the spectra were
downloaded and found to have observing anomalies (an extraordinarily large
number of missing observations caused by channel misalignments: 3 stars),
strong stellar features that were not evident in the quick-look plots (3
stars), or molecular hydrogen lines that were strong enough to seriously
compromise the lines that we wanted to measure (2 stars).  The lack of stars
with exceptionally strong H$_2$ features helped to eliminate sight lines that
penetrate dense, cool gas clouds.

\subsection{Creation of the Spectra}\label{creation}

All of the downloads of the calibrated {\it FUSE\/} data from MAST occurred
well after the final pipeline reductions were performed for the archive with
CalFUSE version 3.2.3 (Dixon et al. 2007).  For every target, we
accepted data from all of the available observing sessions (identified by
unique archive root names) but rejected any subexposures that had an
extraordinarily low count level caused by a channel misalignment during the
observation.  We used exposures obtained during both orbital day and night.
Normally, one must be cautious about observations of features for either O~I
or N~I because they can be filled in by diffuse telluric emission lines
during daytime observing.  However the O~I transition strengths considered
here are so weak that the telluric contributions are insignificant.

All subexposures and spectral channels that passed our initial screening were
coadded with weight factors based on the inverse squares of their respective
values of $S/N$ for intensities smoothed over a wavelength interval of
0.12$\,$\AA\ (or 9 independent spectral elements -- this ensures that weights
are not strongly influenced by random noise excursions).  Before this
coaddition took place, we examined some strong interstellar features and
aligned the individual spectra in wavelength against a preliminary coaddition
with no wavelength shifts.  This process enabled us to virtually eliminate
any degradation in resolution caused by drifts of the spectra in the
wavelength direction from one subexposure to the next.  However, there can
still be overall small systematic errors in radial velocity of about
$10\,{\rm km~s}^{-1}$; see Appendix~A of Bowen et al. (2008).  In a few
cases, the spectral $S/N$ values were too low to allow such shifts to be made
with much confidence, even after the intensities were smoothed with a median
filter for viewing.  Such spectra were combined without any shifts.  For
every target, two combined spectra were created: one was made up with shifts
appropriate to the spectral region covering the Ar~I line at 1048$\,$\AA,
while the other had differentials that were optimized for the wavelengths
that covered the weakest O~I lines near 920$\,$\AA.

\section{ANALYSIS}\label{analysis}

\subsection{Equivalent Widths and Their Errors}\label{EW}

We measured equivalent widths of the Ar~I and O~I lines by integrating
intensity deficits below best-fitting Legendre polynomials for the continua
defined from intensities at locations somewhat removed from the features.
Special precautions were made to account for various sources of error, which
are important for later analysis stages that assign relative weights to
different measurements and also for the estimates of the ultimate errors in
the results.  First, we accounted for the direct effect that random
count-rate variations can have on the equivalent width outcome (Jenkins et
al. 1973).  Next, the weakest lines are subject to uncertainties arising from
improper definitions of the continua.  To construct the probable errors, we
evaluated the expected formal errors in the polynomial coefficients, as
described by Sembach \& Savage (1992), and then we multiplied them by 2 in
order to make an approximate allowance for additional uncertainties caused by
the arbitrariness in selecting the most appropriate polynomial order.  To
find the effects of these continuum errors on our measurements, the
equivalent widths were re-evaluated using the probable excursions of the
continua on either side of the preferred ones.  Errors in the background
subtraction in the {\it FUSE\/} data processing are small compared to the
other errors.

\begin{deluxetable}
{
l
c
c
c
c
c
c
c
c
c
c
c
c
c
c
}
\tabletypesize{\scriptsize}
\rotate
\tablecolumns{14}
\tablewidth{0pt}
\tablecaption{Equivalent Widths\tablenotemark{a}\label{EW_table}}
\tablehead{
\colhead{} & \colhead{Ar I\tablenotemark{b}} &
\multicolumn{12}{c}{O~I\tablenotemark{b,c}}\\
\cline{3-14}
\colhead{} & \colhead{1048.220} & \colhead{919.917} & \colhead{922.200} &
\colhead{925.446} &
\colhead{916.815} & \colhead{930.257} & \colhead{919.658} & \colhead{921.857}
&
\colhead{924.950} & \colhead{950.885} & \colhead{976.448} & \colhead{948.686}
&
\colhead{971.738}\\
\colhead{Star Name} & \colhead{2.440} & \colhead{$-0.788$} &
\colhead{$-0.645$} &
\colhead{$-0.484$} & \colhead{$-0.362$} & \colhead{$-0.301$} &
\colhead{$-0.137$} &
\colhead{$-0.001$} & \colhead{0.155} & \colhead{0.176} & \colhead{0.509} &
\colhead{0.778} & \colhead{1.052}\\
\colhead{} & \colhead{\nodata} & \colhead{11.6} & \colhead{15.3} &
\colhead{20.6} &
\colhead{25.0} & \colhead{27.8} & \colhead{34.5} & \colhead{40.4} &
\colhead{46.9} &
\colhead{49.0} & \colhead{\nodata} & \colhead{\nodata} & \colhead{\nodata}\\
\colhead{(1)}& \colhead{(2)}& \colhead{(3)}& \colhead{(4)}&
 \colhead{(5)}& \colhead{(6)}& \colhead{(7)}& \colhead{(8)}& \colhead{(9)}&
\colhead{(10)}& \colhead{(11)}& \colhead{(12)}& \colhead{(13)}&
\colhead{(14)}
}
\startdata
2MASS15265306\\
~~~~~+7941307\dotfill&$  77\pm 18$
&$(  14\pm 16)$&$(  -4\pm 16)$&$  29\pm 12$&       \nodata&$  24\pm 22$&$
41\pm 22$
&$  39\pm 19$&$  84\pm 11$&$  59\pm 15$&$  97\pm 12$&$ 111\pm 11$&$( 120\pm
12)$\\
AA Dor\dotfill&$  60\pm 10$
&$  30\pm 18$&       \nodata&$  56\pm 13$&       \nodata&$  75\pm 31$&$
51\pm 37$
&       \nodata&$  93\pm 13$&$  73\pm 21$&$( 121\pm 19)$&       \nodata&$(
225\pm  9)$\\
AGK+81 266\dotfill&$  76\pm 19$
&$  22\pm 15$&$  44\pm 15$&$  33\pm 15$&       \nodata&$  60\pm 19$&$  76\pm
15$
&$  87\pm 18$&$  99\pm 14$&$ 122\pm 14$&$( 116\pm 18)$&$( 102\pm 15)$&$(
125\pm 14)$\\
BD+18 2647\dotfill&$  43\pm 18$
&$  22\pm 13$&       \nodata&$  37\pm 11$&       \nodata&$  26\pm 16$&$
39\pm 18$
&       \nodata&$  78\pm 11$&$  68\pm 11$&$(  63\pm 13)$&$( 131\pm 12)$&$(
159\pm 11)$\\
BD+25 4655\dotfill&$  33\pm 18$
&$   3\pm 12$&       \nodata&$  18\pm 12$&       \nodata&$  22\pm 11$&$
21\pm 12$
&       \nodata&$  53\pm 11$&$  41\pm 12$&$  64\pm 13$&$  77\pm 12$&$(  75\pm
11)$\\
BD+28 4211\dotfill&$  31\pm  8$
&$  14\pm  3$&$  27\pm  5$&$  24\pm  3$&$  25\pm  4$&$  37\pm  4$&$  34\pm
4$
&$  35\pm  4$&$  42\pm  3$&$  51\pm  3$&$(  42\pm  4)$&$(  83\pm  4)$&$(
62\pm  3)$\\
BD+37 442\dotfill&$ 166\pm  7$
&       \nodata&$  61\pm 28$&$ 107\pm 16$&       \nodata&$  95\pm 24$&
\nodata
&$ 166\pm 17$&$ 128\pm 16$&       \nodata&       \nodata&       \nodata&
\nodata\\
BD+39 3226\dotfill&$  54\pm  3$
&$(  21\pm  8)$&$(  81\pm  8)$&$  34\pm  7$&$  40\pm  8$&$  48\pm 10$&$
47\pm  8$
&$  38\pm  8$&$  52\pm  7$&$  59\pm  7$&$  54\pm  7$&$  69\pm  8$&$(  73\pm
7)$\\
CPD$-$31 1701\dotfill&$  16\pm  8$
&$  11\pm  5$&       \nodata&$  19\pm  4$&       \nodata&$  20\pm 11$&$
39\pm  6$
&       \nodata&$  45\pm  4$&$(  28\pm  5)$&$(  45\pm  7)$&$(  68\pm  5)$&$(
91\pm  4)$\\
CPD$-$71D172\dotfill&$  34\pm 10$
&$  31\pm 13$&$  34\pm 13$&$  52\pm 12$&       \nodata&$  57\pm 13$&$  47\pm
15$
&$  36\pm 18$&$  67\pm 12$&$  67\pm 13$&$(  61\pm 13)$&$(  93\pm 13)$&$(
83\pm 14)$\\
EC11481$-$2303\dotfill&$ 132\pm 16$
&$(  39\pm 17)$&       \nodata&$  75\pm 16$&       \nodata&$  92\pm 16$&$
99\pm 17$
&$  80\pm 34$&$ 145\pm 15$&$ 121\pm 17$&$ 153\pm 17$&$ 146\pm 16$&$ 177\pm
16$\\
Feige 34\dotfill&$  36\pm 17$
&$(   9\pm 13)$&       \nodata&$  13\pm 13$&       \nodata&$  28\pm 14$&$
38\pm 13$
&       \nodata&$  42\pm 13$&$  46\pm 14$&$  41\pm 21$&$  86\pm 19$&
\nodata\\
HD$\,$113001\dotfill&$ 115\pm 20$
&$(  28\pm 10)$&       \nodata&$(  50\pm 10)$&       \nodata&$(  53\pm 15)$&$
63\pm 10$
&       \nodata&$  96\pm 10$&$  93\pm 12$&$ 100\pm 13$&$ 125\pm 11$&$ 128\pm
10$\\
JL 119\dotfill&$ 112\pm  5$
&$  52\pm 14$&$  39\pm 27$&$  89\pm  8$&       \nodata&$ 103\pm 15$&$  83\pm
25$
&$  80\pm 23$&$ 138\pm  7$&$ 111\pm 19$&$( 169\pm  7)$&$( 180\pm  6)$&$(
174\pm  7)$\\
JL 25\dotfill&$  77\pm 19$
&$  29\pm 17$&       \nodata&$  43\pm 17$&       \nodata&$  72\pm 22$&$
29\pm 17$
&       \nodata&$  69\pm 17$&$  94\pm 17$&       \nodata&$ 111\pm 18$&$
106\pm 17$\\
JL 9\dotfill&$ 100\pm 10$
&$  46\pm 12$&       \nodata&$  67\pm  9$&       \nodata&$  95\pm 16$&$
87\pm 14$
&       \nodata&$ 106\pm  9$&$ 116\pm  9$&       \nodata&$ 129\pm 11$&$(
142\pm 12)$\\
LB 1566\dotfill&$  57\pm  6$
&$  32\pm 13$&       \nodata&$  51\pm 11$&       \nodata&$  56\pm 14$&$
48\pm 15$
&       \nodata&$  81\pm 11$&$  73\pm 19$&$(  98\pm 11)$&$( 108\pm 14)$&$(
111\pm 11)$\\
LB 1766\dotfill&$  67\pm 16$
&$  -3\pm 13$&       \nodata&$  61\pm 12$&       \nodata&$  58\pm 14$&$
38\pm 14$
&       \nodata&$  98\pm 11$&       \nodata&$ 120\pm 11$&$( 148\pm 10)$&$(
168\pm 10)$\\
LB 3241\dotfill&$  56\pm  9$
&$  29\pm  8$&$  25\pm  9$&$  54\pm  8$&       \nodata&$  53\pm  8$&$  52\pm
8$
&$  97\pm  8$&$  82\pm  7$&$ 106\pm  7$&$( 112\pm  7)$&$( 123\pm  7)$&$(
128\pm  7)$\\
LS 1275\dotfill&$  62\pm 15$
&$  29\pm 10$&$  42\pm 12$&$  45\pm  9$&$  52\pm 14$&$  65\pm 13$&$  46\pm
10$
&$(  99\pm 13)$&$  58\pm  9$&$  74\pm 10$&$  80\pm 10$&$  89\pm 11$&$(  89\pm
9)$\\
LSE 234\dotfill&$ 115\pm 12$
&$(  47\pm 16)$&$  72\pm 17$&$  76\pm 15$&       \nodata&$  94\pm 16$&$
85\pm 18$
&$ 123\pm 17$&$ 107\pm 15$&$ 122\pm 15$&       \nodata&$ 154\pm 18$&$ 130\pm
15$\\
LSE 259\dotfill&$ 149\pm  6$
&       \nodata&       \nodata&$  88\pm 14$&       \nodata&$ 134\pm 20$&
\nodata
&       \nodata&$ 123\pm 13$&$ 156\pm 13$&       \nodata&       \nodata&
\nodata\\
LSE 263\dotfill&$ 119\pm 17$
&$(  49\pm 14)$&       \nodata&$  73\pm 12$&$ 101\pm 19$&$  80\pm 15$&$
76\pm 14$
&       \nodata&$  91\pm 13$&$  76\pm 13$&$  95\pm 17$&$ 139\pm 14$&$ 159\pm
12$\\
LSE 44\dotfill&$  93\pm 12$
&$  56\pm 11$&$  68\pm 12$&$  68\pm 10$&       \nodata&$  78\pm 15$&$  61\pm
15$
&$  94\pm 12$&$  92\pm  9$&$ 132\pm 10$&       \nodata&$( 140\pm 10)$&$(
107\pm 11)$\\
LSII+18 9\dotfill&$  91\pm 15$
&       \nodata&       \nodata&$  54\pm 21$&$  66\pm 38$&$  73\pm 23$&$
72\pm 21$
&       \nodata&$  78\pm 21$&$ 125\pm 21$&       \nodata&       \nodata&$(
119\pm 21)$\\
LSII+22 21\dotfill&$  45\pm 12$
&$  27\pm 10$&       \nodata&$  35\pm  9$&$  40\pm 15$&$  52\pm 13$&$  44\pm
11$
&$( 116\pm  8)$&$  63\pm  8$&$  74\pm  9$&$(  91\pm  8)$&$( 100\pm  9)$&$(
119\pm  8)$\\
LSIV+10 9\dotfill&$ 179\pm 12$
&$(  58\pm 10)$&       \nodata&$  88\pm  9$&       \nodata&$ 106\pm 18$&$
88\pm 12$
&       \nodata&$ 152\pm  9$&$ 170\pm 10$&$ 231\pm  9$&       \nodata&$(
192\pm 10)$\\
LSS 1362\dotfill&$  88\pm  3$
&$(  39\pm 11)$&$  52\pm 14$&$  59\pm  8$&       \nodata&$  78\pm 18$&$
64\pm 20$
&$(  65\pm  8)$&$(  68\pm 10)$&$ 134\pm  7$&       \nodata&       \nodata&
\nodata\\
MCT 2005\\
~~~$-$5112\dotfill&$ 103\pm  8$
&       \nodata&       \nodata&$  74\pm 11$&       \nodata&$  75\pm 18$&
\nodata
&       \nodata&$ 113\pm  9$&$ 135\pm 12$&$( 168\pm 15)$&$( 150\pm 13)$&$(
124\pm 18)$\\
MCT 2048\\
~~~$-$4504\tablenotemark{d}\dotfill&$ 122\pm 12$
&       \nodata&$  86\pm 21$&$  83\pm 15$&       \nodata&$  61\pm 59$&
\nodata
&$  90\pm 28$&$ 118\pm 14$&$ 117\pm 17$&$ 149\pm 14$&$ 174\pm 16$&$( 172\pm
15)$\\
NGC6905\\
~~~star\tablenotemark{e}\dotfill&$ 147\pm  8$
&$(  45\pm 39)$&       \nodata&$ 101\pm 39$&       \nodata&$  83\pm 43$&$
62\pm 46$
&       \nodata&$ 139\pm 38$&$ 167\pm 38$&$ 205\pm 39$&       \nodata&$(
154\pm 39)$\\
PG0919+272\dotfill&$  55\pm 27$
&$  34\pm 17$&$  31\pm 16$&$  43\pm 16$&$  45\pm 27$&$  83\pm 17$&$  59\pm
17$
&$  64\pm 16$&$  81\pm 15$&$  65\pm 17$&$  87\pm 15$&$ 108\pm 16$&$( 106\pm
15)$\\
PG0952+519\dotfill&$  52\pm  3$
&$  19\pm  8$&       \nodata&$  36\pm  7$&       \nodata&$  43\pm  8$&$
53\pm  9$
&       \nodata&$  74\pm  7$&       \nodata&$( 125\pm  6)$&$( 154\pm  8)$&$(
148\pm  6)$\\
PG1032+406\dotfill&$  -21\pm 27$
&$   3\pm 23$&$  11\pm 19$&$  15\pm 18$&$(  95\pm 24)$&$  20\pm 19$&$  -2\pm
36$
&$  36\pm 18$&$  15\pm 17$&$  72\pm 17$&$  46\pm 20$&$  70\pm 18$&$(  94\pm
17)$\\
PG1051+501\dotfill&$ 165\pm 15$
&       \nodata&       \nodata&$  81\pm 30$&       \nodata&$  64\pm 33$&
\nodata
&       \nodata&$ 128\pm 27$&$ 142\pm 28$&       \nodata&       \nodata&$
185\pm 28$\\
PG1230+068\dotfill&$ 102\pm 15$
&$  40\pm  8$&$  45\pm 11$&$  72\pm  7$&       \nodata&$  82\pm 16$&$  86\pm
9$
&$  79\pm 13$&$ 110\pm  7$&$  88\pm  8$&$ 105\pm 18$&$ 126\pm  7$&$( 139\pm
7)$\\
PG1544+488\dotfill&$  84\pm  3$
&$  26\pm 11$&$  25\pm 13$&$  56\pm  8$&$  76\pm 34$&$  69\pm 10$&$  74\pm
13$
&$  89\pm 11$&$  91\pm  7$&$  97\pm  9$&$( 124\pm 11)$&$( 132\pm  7)$&$(
151\pm  8)$\\
PG1605+072\dotfill&$ 131\pm 15$
&       \nodata&       \nodata&$  72\pm 21$&       \nodata&$  99\pm 25$&
\nodata
&       \nodata&$ 136\pm 20$&$ 163\pm 20$&$ 229\pm 20$&$( 166\pm 22)$&$(
186\pm 20)$\\
PG1610+519\dotfill&$ 122\pm  9$
&       \nodata&       \nodata&$  81\pm 12$&       \nodata&       \nodata&
\nodata
&       \nodata&$ 150\pm  9$&       \nodata&$ 256\pm  5$&       \nodata&$(
228\pm  7)$\\
PG2158+082\dotfill&$ 117\pm  7$
&       \nodata&       \nodata&$ 100\pm 19$&       \nodata&$  97\pm 26$&
\nodata
&       \nodata&$ 121\pm 19$&$ 175\pm 17$&       \nodata&$( 154\pm 22)$&
\nodata\\
PG2317+046\dotfill&$  68\pm 20$
&$  49\pm 18$&$  29\pm 31$&$  61\pm 19$&       \nodata&$  50\pm 24$&$  28\pm
21$
&$  48\pm 33$&$  71\pm 18$&$ 135\pm 18$&       \nodata&$ 132\pm 19$&$(  81\pm
19)$\\
Ton 102\dotfill&$  51\pm 13$
&$  21\pm 15$&$  22\pm 16$&$  26\pm 16$&       \nodata&$  39\pm 16$&$  34\pm
16$
&$  79\pm 16$&$  42\pm 17$&$  63\pm 16$&$  72\pm 15$&       \nodata&
\nodata\\
Ton S227\dotfill&$  80\pm 25$
&$  27\pm 10$&$  26\pm 11$&$  43\pm 10$&       \nodata&$  21\pm 20$&$  27\pm
11$
&$  30\pm 12$&$  58\pm  9$&       \nodata&$  85\pm 11$&$ 111\pm 12$&$ 121\pm
9$\\
UV0904$-$02\tablenotemark{f}\dotfill&$  69\pm  4$
&$(  30\pm  9)$&       \nodata&$  43\pm  8$&       \nodata&$  59\pm 12$&$
52\pm 10$
&       \nodata&$  67\pm  7$&$  65\pm 11$&$  75\pm  7$&$  84\pm  8$&$(  89\pm
7)$\\
\enddata
\tablenotetext{a}{Equivalent widths are given in m\AA.  Values given in
parentheses indicate
lines that were not used in the linear fits.}
\tablenotetext{b}{Numbers given below indicate (1) wavelengths in \AA, (2)
line strengths in
terms of $\log (f\lambda)$ taken from Morton (2003), and (3) the
equivalent width in m\AA\
for $N({\rm O~I})=10^{16}\,{\rm cm}^{-2}$ and $b=6\,{\rm km~s}^{-1}$.}
\tablenotetext{c}{{}Lines are arranged in order of increasing strength.}
\tablenotetext{d}{Name recognized by SIMBAD: 2MASS J20515997-4042465.}
\tablenotetext{e}{Central star of the planetary nebula NGC 6905.}
\tablenotetext{f}{Name recognized by SIMBAD: 2MASS J09070812-0306139.}
\end{deluxetable}

\clearpage
The sources of error mentioned above are straightforward to evaluate and
would apply to just about any measure of an equivalent width.  However, with
the subdwarf stars, we must also contend with the confusion produced by
stellar lines.  We made no attempt to model such features, because in order
to do so we would need to know the details of the stellar parameters for each
star.  Instead, we regarded the influence of stellar features as random
sources of error in our line measurements.  In order to estimate the
amplitude of such errors, which can vary markedly from one star to the next
and can change with wavelength, we measured for each target the variance of a
large number of equivalent width measurements of imaginary, fake lines at
wavelengths similar to the ones under study, but that were displaced away
from known real interstellar lines, both atomic and molecular.  This variance
was then used as a guide for estimating the errors that should arise from
stellar features.

Figure~\ref{sample_spectra} shows samples of spectra covering the relevant
wavelength regions for two stars.  These two cases illustrate strong
differences in the degree of interference from stellar lines.  The first
example, AGK+81~266, has many stellar lines that can either add an apparent
absorption to an interstellar line or distort the continuum level that is
measured on either side of the line.  For this target, these effects dominate
over other sources of error and create $1\sigma$ uncertainties in $W_\lambda$
equal to 18.5 and 14.2$\,$m\AA\ for the Ar~I and O~I lines, respectively.
This star also exhibits molecular hydrogen features of moderate strength, but
here the lines are not strong enough to compromise the measurements of the
atomic lines.  A far more favorable case for measuring interstellar features
is presented by the star UV0904$-$02.  Here, uncertainties produced by random
stellar features should create errors of only 3.6$\,$m\AA\ for the Ar~I line
and 6.6$\,$m\AA\ for the O~I lines.

All of the errors discussed in this section were combined in quadrature to
synthesize the overall errors in the equivalent widths.  Values for the
equivalent widths of all lines and their associated uncertainties are listed
for each of our targets in Table~\ref{EW_table}.  The columns in this table
are arranged in a sequence from the weakest to the strongest lines.

\subsection{Derivations of [Ar~I/O~I] Values and Their
Uncertainties}\label{derivation}

\subsubsection{Reference Abundances}\label{reference_abund}

Detailed discussions on various methods of measuring the protosolar and
B-star abundances of Ar have been presented by Lodders (2008) and Lanz et
al. (2008).  We adopt a mean value for the recommended outcomes of the two,
$\log ({\rm Ar/H})+12=6.60\pm 0.10$.  (Since both determinations might be
subject to common errors, the error of the mean is not reduced below the
0.10$\,$dex errors specified by each of them.) This value is higher than the
solar photospheric value proposed by Asplund et al. (2009), and it remains so
even after one applies a correction for gravitational settling of $+0.07\,$dex
 (Lodders 2003) to obtain a protosolar value of $\log ({\rm
Ar/H})+12=6.47\pm 0.13$.  For O, we take the solar photospheric value given
by Asplund et al. (2009) and again apply a
$+0.07\,$dex settling correction to get $\log ({\rm O/H})+12=8.76\pm 0.05$.
This value agrees remarkably well with the measurement $\log ({\rm
O/H})+12=8.76\pm 0.03$ obtained for B-stars by Przybilla et al.
 (2008).

We must now consider the prospect that some of the Ar and O atoms are
incorporated into solid form within or on the surfaces of dust grains, and
this effect might be large enough to distort our findings on the differences
in ionization.  Unfortunately, we have no direct information about the
depletion of gas-phase Ar, since ionization corrections (the object of the
present study) can influence the outcome.  While in principle it would be
beneficial if we could study $N({\rm Ar~I})$ along sight lines that penetrate
dense media, where depletions are likely to dominate over ionization effects,
this is not possible because the absorption lines are far too saturated (much
more so than in the current study).

\onecolumn
\begin{figure}
\vspace*{-2cm}
\epsscale{.95}
\plotone{sample_spectra.eps}
\caption{Examples of {\it FUSE\/} spectra, where signals from all detector
channels have been combined, for the two stars AGK+81~266 (top group of
panels) and UV0904$-$02 (bottom group).  Various interstellar features are
indicated.  The two panels showing the Ar~I feature best illustrate the large
contrasts in the strengths of unidentified stellar features that can
interfere with the interstellar ones.\label{sample_spectra}}
\end{figure}
\twocolumn

SJ98 presented a number of
theoretical arguments that suggested Ar is not appreciably depleted in
the low density ISM that we can observe.  However, it would be good to
confirm this outlook by some independent, experimental means.  Fortunately,
krypton is an element that can be observed in the ISM, and, like argon, is
chemically inert.  It would be reasonable to expect that the capture of Kr
onto interstellar dust grains, if it happens, would be similar to that of Ar.
An advantage of studying Kr is that its interstellar features are weak
 (Cartledge et al. 2008), which means that they can be used to measure
column densities over sight lines that have high values of $n({\rm H}_{\rm
tot})$ where element depletions should be generally very strong.  As with Ar,
Kr has a photoionization cross section that is substantially larger than that
of H (Sterling 2011).  Thus, any simple measure of the deficiencies
of this element in the low density ISM could simply be a product of it being
more easily photoionized than H.

A way to overcome the confusion from the offset produced by ionization is to
compare differential capture rates of elements onto grains as the conditions
that favor grain formation change.  For instance, Jenkins (2009) has
determined that for sight lines with $N({\rm H}^0)>10^{19.5}$, where
ionization corrections should be small, when $5\times 10^5$ O atoms are
removed from the gas phase, $1(\pm 1)$ atom of Kr vanishes.  Since O is more
abundant than Kr by a factor of $2.5\times 10^5$, any relative decrease in
the abundance of Kr in the gas phase would be about half that of O (but the
errors allow for this factor to range from zero to being equal to that of
oxygen).  If we accept the idea that Ar depletes in the same manner as Kr,
probably to within a factor of $\sqrt{m_{\rm Kr}/m_{\rm Ar}}$, and that in low
density media O shows very low depletions ($<0.1\,$dex) (Jenkins 2009), it is
reasonable to adopt a gas-phase abundance ratio that is virtually the same as
the protosolar ones, $\log ({\rm Ar/O})_\odot=-2.16\pm 0.11$.  If indeed
there is some mild depletion of O due to the formation of silicate dust
grains, we may understate the strength of the ionization of Ar.

\subsubsection{Interpretation of Line Strengths}\label{line_strengths}

We adopt the premise that the distribution of radial velocities of the
neutral argon atoms is identical to that of neutral oxygen (but this is not
exactly correct; we will revisit this issue later). Figure~\ref{oi_fits}
shows examples of some standard curve-of-growth plots for the O~I lines
appearing in the spectra of the same two targets that were featured in
Fig.~\ref{sample_spectra}, AGK+81~266 and UV0904$-$02.  The former of the two
illustrates an average amount of line saturation for relevant features, while
the latter represents an extreme case of saturation caused by a low overall
dispersion of radial velocities.  In principle, we could have derived values
for $N$(O~I) from the best-fit curves of growth shown by the dashed curves in
the figure panels and then assume that $N$(Ar~I) follows from the equivalent
width of the one available line at 1048.220$\,$\AA\ assuming the same
velocity dispersion parameter $b$ as that found for O~I. Instead, we used a
much simpler approach that sidesteps the goal of deriving explicit values of
$N$ and $b$ (whose errors are strongly correlated) and proceeds directly to
an answer for just the ratio of the two column densities.  An advantage here
is that we can make a straightforward analytical determination of the
uncertainty of the outcome based on the errors of some linear fitting
coefficients.

The comparison of Ar~I to O~I is based on the following principle.  If one
could imagine the existence of a hypothetical O~I line with a transition
strength $\log (f\lambda)_{\rm O~I}$ that is just right to produce a value of
$W_\lambda/\lambda$ that exactly matches that of the Ar~I line, one could
then derive the deficiency of Ar~I with respect to its expectation based on
O~I, which we denote in logarithmic form as [Ar~I/O~I].  This quantity yields
the logarithm of the ratio of the two neutral fractions relative to the solar
abundance ratio and is given by the relation
\begin{eqnarray}\label{ar_o_eqn}
[{\rm Ar~I/O~I}]&=&\log (f\lambda)_{\rm O~I}-\log (f\lambda)_{\rm Ar~I}-\log
({\rm Ar/O})_\odot\nonumber\\
&=&\log (f\lambda)_{\rm O~I}-0.28\pm 0.11,
\end{eqnarray}
where $\log (f\lambda)_{\rm Ar~I}=2.440\pm 0.004$ (Morton 2003).

\onecolumn
\begin{figure}
\epsscale{1.25}
\plottwo{oi_fit_AGK+81_266.eps}{oi_fit_UV0904-02.eps}
\caption{Examples of how the values of [Ar~I/O~I] are derived for the two
stars chosen for Fig.~\protect\ref{sample_spectra}.  The equality shown in
each $x$-axis label applies to the horizontal projection of the Ar~I line
strength onto the O~I curve of growth, as expressed in
Eq.~\protect\ref{ar_o_eqn}.  We performed weighted least-squares linear fits
for the logarithms of $W_\lambda/\lambda$ for lines of O~I that are most
influential in establishing the trends ({\it solid lines\/}) for a comparison
with the measurements of $\log W_\lambda/\lambda$ of the Ar~I line.  Values
for $c_0$ and $c_1$ given in Eq.~\ref{linear_trend} that define these trends
are shown in the boxes.  The best-fit curves of growth together with their
$b$ values are also shown, but they are not used in the derivations.  O~I
lines that had strengths that were well above or below those in the important
portion of curve of growth were not included in the derivation of the best
fits, and they are indicated here as ``(not used).''  These unused lines are
identified with parentheses around the equivalent width values listed in
Table~\protect\ref{EW_table}.\label{oi_fits}}
\end{figure}
\twocolumn

To determine the strength of the hypothetical O~I line, we perform a weighted
least-squares linear fit for $\log (W_\lambda/\lambda)$ vs. the quantity on
the right-hand side of Eq.~\ref{ar_o_eqn} (the abscissa for each plot in
Fig.~\ref{oi_fits}) for an appropriate selection of O~I lines.  [As with the
Ar~I line the $f$-values of the O~I transitions are from Morton
 (2003).] The lines that are chosen for this fit are ones
that are situated not too far from the horizontal projection (shown by dotted
lines in the figure) of $\log (W_\lambda/\lambda)$ for the single line of
Ar~I onto the trend for the O~I lines.  With this restricted fit, we define a
simple relationship in $\log (W_\lambda/\lambda)$ that is a good
approximation to a relevant portion of the curve of growth.

In the two panels of Fig.~\ref{oi_fits}, these best-fit linear trends are
shown by the straight solid lines.  They depict the relation between $y=\log
W_\lambda/\lambda$ and $x=\log (f\lambda)-0.28$ according to the equation
\begin{equation}\label{linear_trend}
y=c_0+c_1(x-x_0),
\end{equation}
where
\begin{equation}\label{x_0}
x_0=\sum_i x_i\sigma (y)_i^{-2}\Big/ \sum_i \sigma (y)_i^{-2}
\end{equation}
represents a zero reference point that produces a vanishing covariance for
the errors in the fitting coefficients $c_0$ and $c_1$.  A nominal value on
the $x$ axis for the projection of $y_{\rm Ar~I}$ onto the linear relation is
given by
\begin{equation}\label{aro_solution}
x_{\rm Ar~I}=x_0+{y_{\rm Ar~I}-c_0\over c_1}
\end{equation}
In the fraction part of this equation, the numerator and denominator have
errors $(\sigma(y_{\rm Ar~I})^2+\sigma(c_0)^2)^{0.5}$ and $\sigma(c_1)$,
respectively.  A conventional approach for deriving the error of the quotient
is to add in quadrature the relative errors of the two terms, yielding the
relative error of the quotient.  However, this scheme breaks down when the
error in the denominator is not very much less than the denominator itself.
A more robust way to derive the error of a quotient has been developed by
Geary (1930); for a concise description of this method see
Appendix~A of Jenkins (2009).  We use this method here; it
is effective as long as there is little chance that the denominator minus its
error could become very close to zero or be negative, i.e., ${\rm
denom.}/\sigma({\rm denom.}) \gtrsim 3$.

The dotted lines in Figure~\ref{oi_fits} show schematically how the best fit
values and the error ranges for [Ar~I/O~I] are derived.  Note that the
horizontal and vertical segments for the error limits do not exactly
intersect the best linear trend for the O~I lines because the error analysis
allows for the uncertainty for the location of this line.  (But we point out
that the intersections for the worst possible error in one direction for
$y_{\rm Ar~I}$ do not occur at the locations for the worst possible
deviations in the opposite direction for the trend line.)  Also, the final
errors for $[{\rm Ar~I}/{\rm O~I}]=x_{\rm Ar~I}$ are not symmetrical about
the best values.  On average, the upward error bounds are about 75\% as large
as the negative ones.

The difference in atomic weights of Ar and O will cause the thermal
contributions to the Doppler broadenings of these two elements to differ from
each other.  The impact of this effect on our results for the column density
ratios is small however.  For instance, if we consider that we are viewing
absorption lines arising from the warm, neutral medium (WNM) and there were
no bulk motions of the gas, the line broadening parameters $b_{\rm therm.}$
caused by thermal Doppler broadening for $T=7000\,$K would be 2.7 and
$1.7\,{\rm km~s}^{-1}$ for O~I and Ar~I, respectively.  For a typical
observation, such as the one for AGK+81~266 illustrated in the left-hand
panel of Fig.~\ref{oi_fits}, we find that the observed curve of growth for
the O~I lines, yielding an apparent $b_{\rm obs.}=9.9\,{\rm km~s}^{-1}$,
indicates that kinematic effects arising from turbulent motions (or multiple
velocity components) should be the most important contribution to the
broadening since $b_{\rm turb.}=\sqrt{b_{\rm obs.}^2-b({\rm O~I})_{\rm
therm.}^2}=9.53\,{\rm km~s}^{-1}$ is only slightly smaller than $b_{\rm
obs.}$.  The curve of growth that characterizes the saturation of the Ar~I
line would conform to a slightly lower velocity dispersion parameter compared
to that observed for O~I, $b_{\rm obs.}=\sqrt{b_{\rm turb.}^2+b({\rm
Ar~I})_{\rm therm.}^2}=9.68\,{\rm km~s}^{-1}$,  because the higher atomic
weight of Ar causes the thermal contribution to be smaller.  For the observed
equivalent width of the Ar~I line, the error in the ratio of column densities
caused by our assumption that the values of $b_{\rm obs.}$ of the two
elements are identical will create an underestimate of $[{\rm
Ar~I/O~I}]=-0.012\,$dex.  A similar calculation for the more extreme line
saturation exhibited by UV0904$-$02 shown in the right-hand panel of
Fig.~\ref{oi_fits} indicates that the perceived outcome for [Ar~I/O~I] could
be too low by $-0.12\,$dex.  Cases showing this much saturation are rare for
our collection of sight lines.

If the kinematic line broadening is not an approximately Gaussian form that
one would expect from pure turbulent broadening, but instead results from
distinct and well-separated narrow components, the errors in [Ar~I/O~I] could
be larger than those evaluated above.  However, the recordings of Ar~I lines
for 9 different stars made by {\it IMAPS\/} at a resolution of $4\,{\rm
km~s}^{-1}$ that were shown by SJ98 reveal profiles that, while not exactly
Gaussian, are nevertheless generally smooth and devoid of any narrow spikes
that are isolated from each other.  Thus, the numerical estimates presented
in the above paragraph should be reasonably accurate.

One might question whether or not errors in the adopted $f$-values could
cause global systematic errors in the evaluations of [Ar~I/O~I].  While
Morton (2003) listed a very small uncertainty for the
$f$-value of the Ar~I line at 1048.220$\,$\AA, he did not specify errors for
the O~I lines.  Nevertheless, empirical evidence from high quality {\it
FUSE\/} observations of WD stars by H\'ebrard et al. (2002),
Sonneborn et al. (2002) and Oliveira et al.
(2003) all showed curves of growth that are
remarkably well behaved for the same lines that are used in the present
study.  While one could still pose the objection that all of the O~I lines
could collectively have a systematic error of a certain magnitude, and yet
could still yield acceptable curves of growth, this seems unlikely: the
values of [O~I/H~I] derived by Sonneborn et al. (2002) and Oliveira et al.
(2003) are generally consistent with those found elsewhere in the
ISM based on measurements of the
intersystem O~I line at 1355.6$\,$\AA\ (Jenkins 2009).
[H\'ebrard et al. (2002) did not attempt to measure
$N$(H~I).]

\subsubsection{Outcomes}\label{outcomes}

Table~\ref{aro_results} shows the outcomes of our analysis for all of the
targets in the survey, along with the applicable Galactic coordinates and
apparent magnitudes of the stars.  The last two columns present some cautions
in numerical form.  First, Column (9) lists for the fraction part of
Eq.~\ref{aro_solution} the denominator divided by its error.  If this number
is less than about 3, the upper limit for [Ar~I/O~I] should not be trusted.
Second, Column (10) lists the probability of obtaining a worse fit of the O~I
lines to the linear trend (i.e., a higher value for $\chi^2$), given the
errors that we derived. A plot of the frequency of all of these numbers shows
a distribution that is consistent with a uniform distribution between 0 and
1, which indicates that our error estimates for $W_\lambda$ of the O~I lines
are neither too conservative nor too generous.  Thus, any individual case
where this probability is low should not be considered as a real anomaly.

\begin{figure}[h!]
\epsscale{1.0}
\plotone{sorted_results.eps}
\caption{Outcomes for [Ar~/O~I] sorted according to their best
values.\label{sorted_results}}
\end{figure}
\begin{figure}[h!]
\vspace{-.3cm}
\plotone{sky_display.eps}
\caption{Locations of the stars and their values of [Ar~I/O~I] shown on an
Aitoff projection of Galactic coordinates.  The sizes of the black circles
indicate the values of [Ar~I/O~I] according to the key shown at the
bottom.\label{sky_display}}
\end{figure}

Figure~\ref{sorted_results} shows a graphic representation of all of the
results and their uncertainties.  They were ranked and then arranged in order
of the best values of [Ar~I/O~I] to make it easier to see the dispersion of
results and also to show that the more extreme deviations often represent
cases where the errors are somewhat larger than normal.
\onecolumn
{\setlength{\textheight}{9in}
\begin{deluxetable}{
l
c
c
c
c
c
c
c
c
c
}
\tabletypesize{\footnotesize}
\tablecolumns{10}
\tablewidth{0pt}
\tablecaption{Results for [Ar I/O I]\tablenotemark{a}\label{aro_results}}
\vspace{-.5cm}
\tablehead{
\colhead{} & \multicolumn{2}{c}{Galactic Coord.}\tablenotemark{b} &
\colhead{} & \colhead{} & \multicolumn{3}{c}{[Ar I/O I]}\\
\cline{2-3} \cline{6-8}
\colhead{Star} & \colhead{$\ell$} & \colhead{$b$} &
\colhead{} & \colhead{} & \colhead{Lower} & \colhead{Best} &  \colhead{Upper}
&
\colhead{Denom. Val.}&\colhead{Prob. of}\\
\colhead{Name} & \colhead{(deg.)} & \colhead{(deg.)} &
\colhead{$m_B$\tablenotemark{b}} &
\colhead{$m_V$\tablenotemark{b}} & \colhead{Limit} & \colhead{Value} &
\colhead{Limit} & \colhead{/Error\tablenotemark{c}} &
\colhead{Worse Fit}\\
\colhead{(1)}& \colhead{(2)}& \colhead{(3)}& \colhead{(4)}&
 \colhead{(5)}& \colhead{(6)}& \colhead{(7)}& \colhead{(8)}& \colhead{(9)}&
\colhead{(10)}
}
\startdata
2MASSJ15265306+7941307&$115.06$&$34.93$&$11.30$&$11.60$&$-0.45$&$-0.11$&$0.14
$&$4.74$&$0.26$\\
AA
Dor\dotfill&$280.48$&$-32.18$&$10.84$&$11.14$&$-1.18$&$-0.76$&$-0.51$&$2.60$&
$0.70$\\
AGK+81
266\dotfill&$130.67$&$31.95$&$11.60$&$11.94$&$-0.69$&$-0.47$&$-0.30$&$5.25$&$
0.89$\\
BD+18
2647\dotfill&$289.48$&$80.14$&$11.48$&$11.82$&$-1.14$&$-0.63$&$-0.34$&$3.76$&
$0.66$\\
BD+25
4655\dotfill&$81.67$&$-22.36$&$9.39$&$9.68$&$-1.25$&$-0.44$&$-0.02$&$4.68$&$0
.80$\\
BD+28
4211\dotfill&$81.87$&$-19.29$&$10.17$&$10.51$&$-1.03$&$-0.69$&$-0.44$&$7.42$&
$0.07$\\
BD+37
442\dotfill&$137.07$&$-22.45$&$9.69$&$9.92$&$-0.32$&$-0.20$&$-0.02$&$2.54$&$0
.04$\\
BD+39
3226\dotfill&$65.00$&$28.77$&$9.89$&$10.18$&$-0.49$&$-0.28$&$-0.10$&$3.42$&$0
.69$\\
CPD$-$31
1701\dotfill&$246.46$&$-5.51$&$10.22$&$10.56$&$-1.69$&$-1.04$&$-0.70$&$4.68$&
$0.42$\\
CPD$-$71
172\dotfill&$290.20$&$-42.61$&$10.90$&$10.68$&$-2.60$&$-1.46$&$-0.94$&$2.28$&
$0.77$\\
EC11481-2303\dotfill&$285.29$&$37.44$&$11.49$&$11.76$&$-0.53$&$-0.18$&$0.11$&
$4.81$&$0.43$\\
Feige
34\dotfill&$173.32$&$58.96$&$10.84$&$11.18$&$-1.16$&$-0.42$&$0.02$&$3.43$&$0.
82$\\
HD113001\dotfill&$110.97$&$81.16$&$9.65$&$9.65$&$-0.26$&$0.23$&$0.63$&$4.12$&
$0.42$\\
JL
119\dotfill&$314.61$&$-43.36$&$13.22$&$13.49$&$-0.63$&$-0.53$&$-0.45$&$5.68$&
$0.16$\\
JL
25\dotfill&$318.63$&$-29.17$&$13.09$&$13.28$&$-0.94$&$-0.27$&$0.20$&$3.54$&$0
.29$\\
JL
9\dotfill&$322.60$&$-27.04$&$12.96$&$13.24$&$-0.68$&$-0.43$&$-0.22$&$5.12$&$0
.24$\\
LB
1566\dotfill&$306.36$&$-62.02$&$12.81$&$13.13$&$-0.89$&$-0.65$&$-0.48$&$3.06$
&$0.82$\\
LB
1766\dotfill&$261.65$&$-37.93$&\nodata&$12.34$&$-1.03$&$-0.62$&$-0.34$&$4.71$
&$0.07$\\
LB
3241\dotfill&$273.70$&$-62.48$&$12.45$&$12.73$&$-0.86$&$-0.69$&$-0.55$&$7.98$
&$0.01$\\
LS
1275\dotfill&$268.96$&$2.95$&$10.94$&$11.40$&$-1.01$&$-0.44$&$0.00$&$4.69$&$0
.77$\\
LSE
234\dotfill&$329.44$&$-20.52$&\nodata&$12.63$&$-0.77$&$-0.34$&$0.00$&$3.70$&$
0.38$\\
LSE
259\dotfill&$332.36$&$-7.71$&\nodata&$12.54$&$-0.40$&$-0.26$&$-0.12$&$2.52$&$
0.08$\\
LSE
263\dotfill&$345.24$&$-22.51$&$11.40$&$11.30$&$-0.27$&$0.07$&$0.37$&$6.32$&$0
.24$\\
LSE
44\dotfill&$313.37$&$13.49$&$12.21$&$12.45$&$-0.73$&$-0.52$&$-0.34$&$5.53$&$0
.11$\\
LSII +18
9\dotfill&$55.20$&$-2.65$&$11.81$&$12.13$&$-0.65$&$-0.40$&$-0.21$&$2.43$&$0.6
6$\\
LSII +22
21\dotfill&$61.17$&$-4.95$&$12.23$&$12.58$&$-1.06$&$-0.67$&$-0.41$&$4.04$&$0.
91$\\
LSIV +10
9\dotfill&$56.17$&$-19.01$&$11.71$&$11.98$&$-0.18$&$-0.10$&$-0.03$&$11.01$&$0
.21$\\
LSS
1362\dotfill&$273.67$&$6.19$&$12.27$&$12.50$&$-0.60$&$-0.53$&$-0.48$&$7.05$&$
0.65$\\
MCT
2005$-$5112\dotfill&$347.71$&$-32.68$&$15.20$&\nodata&$-0.63$&$-0.47$&$-0.36$
&$3.82$&$0.46$\\
MCT
2048$-$4504\dotfill&$1.10$&$-39.51$&$14.85$&$15.20$&$-0.52$&$-0.28$&$-0.09$&$
4.97$&$0.97$\\
NGC6905
star\dotfill&$61.49$&$-9.57$&$16.30$&$14.50$&$-0.46$&$-0.26$&$-0.12$&$2.56$&$
0.75$\\
PG0919+272\dotfill&$200.46$&$43.89$&$12.27$&$12.69$&$-2.24$&$-0.89$&$-0.16$&$
3.83$&$0.64$\\
PG0952+519\dotfill&$164.07$&$49.00$&$12.47$&$12.80$&$-0.60$&$-0.51$&$-0.44$&$
5.11$&$0.90$\\
PG1032+406\dotfill&$178.88$&$59.01$&$10.80$&\nodata&$-3.79$&$-1.84$&$-0.41$&$
2.71$&$0.30$\\
PG1051+501\dotfill&$159.61$&$58.12$&$14.59$&\nodata&$-0.01$&$0.28$&$0.59$&$2.
77$&$0.61$\\
PG1230+068\dotfill&$290.42$&$68.85$&$12.25$&\nodata&$-0.57$&$-0.23$&$0.06$&$7
.58$&$0.03$\\
PG1544+488\dotfill&$77.54$&$50.13$&$12.80$&\nodata&$-0.48$&$-0.40$&$-0.34$&$5
.41$&$0.56$\\
PG1605+072\dotfill&$18.99$&$39.33$&$13.01$&$12.84$&$-0.52$&$-0.36$&$-0.24$&$5
.31$&$0.88$\\
PG1610+519\dotfill&$80.50$&$45.31$&$13.73$&\nodata&$-0.53$&$-0.44$&$-0.36$&$1
0.93$&$0.22$\\
PG2158+082\dotfill&$67.58$&$-35.48$&$12.66$&\nodata&$-0.91$&$-0.64$&$-0.49$&$
2.78$&$0.16$\\
PG2317+046\dotfill&$84.84$&$-51.10$&\nodata&$12.87$&$-1.42$&$-0.79$&$-0.37$&$
4.20$&$0.04$\\
Ton
102\dotfill&$127.05$&$65.78$&$13.29$&$13.54$&$-0.89$&$-0.45$&$-0.15$&$2.96$&$
0.31$\\
Ton
S227\dotfill&$201.39$&$-77.81$&$11.60$&$11.90$&$-0.34$&$0.12$&$0.44$&$9.06$&$
0.36$\\
UV0904$-$02\dotfill&$232.98$&$28.12$&$11.64$&$11.96$&$-0.39$&$-0.19$&$-0.03$&
$3.78$&$0.93$\\
\enddata
\tablenotetext{a}{$[{\rm Ar~I/O~I}]\equiv {N({\rm Ar~I})/N({\rm O~I})\over
({\rm Ar/O})_\odot}$.  The range of possible values shown here do not include
an overall systematic error that arises from the uncertainties in the
constants that are included in Eq.~\protect\ref{ar_o_eqn}, which amounts to a
combined error of 0.11~dex.}
\tablenotetext{b}{Coordinates and apparent magnitudes were supplied by the
SIMBAD database.}
\tablenotetext{c}{The relevance of this quantity is discussed in the text
that follows Eq.~\protect\ref{aro_solution}.  If the listed value is less
than about 3, the positive value of the error quotient may be misleading.}
\end{deluxetable}
}
\twocolumn

Figure~\ref{sky_display} shows the locations of the targets in the sky and
their respective values of [Ar~I/O~I].  While no obvious regional trends seem
to be evident, we can perform a test to determine whether or not the
variability of the outcomes exceeds what we would have expected from our
errors (assuming that they are correct).  We do this by computing a value for
$\chi^2$, where the choice for the error of each case depends on whether a
test value is above or below the measurement outcome.  This procedure
properly takes into account the asymmetries of the errors.  Adopting this
method, we find that a minimum $\chi^2$ of 66.9 for the 44 measurements
occurs at a value $[{\rm Ar~I/O~I}]=-0.438$.  This minimum for the $\chi^2$
with 43 degrees of freedom is greater than what we would have expected from
our errors alone (the probability of obtaining by chance a higher value for
$\chi^2$ is only 1\%).  If we were to propose that the real variability
across the sky is $\sigma=0.11~{\rm dex}$, and we add this value in
quadrature to all of the experimental errors, the minimum $\chi^2$ drops to
42.4, which makes the probability of a worse fit equal to 50\%.  The location
of this new minimum is at $[{\rm Ar~I/O~I}]=-0.427$.

The discussion above has considered only the random fluctuations arising from
either the measurements or the true variability in [Ar~I/O~I] in different
sight lines.  We must not lose sight of the fact that there can be an overall
systematic error of 0.11~dex for the entire collection.  This global error
arises from the uncertainties of the value for $\log ({\rm Ar/O})_\odot$ that
went into Eq.~\ref{ar_o_eqn}.  It is much larger than the error in the
weighted mean value for all of the measurements.

\section{ARE THE STARS BEYOND THE BOUNDARY OF THE LOCAL
BUBBLE?}\label{beyond}

As discussed in Section~\ref{strategy}, our objective is to sample
interstellar material that is beyond the edge of the Local Bubble, enough so
that our measurements are not heavily influenced by the very low density gas
within this cavity.  In principle, we could compare the three-dimensional locations of the
target stars with maps that outline the boundary of the Local Bubble
 (Vergely et al. 2010; Welsh et al. 2010; Reis et al. 2011) to indicate
whether or not we are primarily sampling gas in the surrounding denser
medium.  However, the distances to our targets are uncertain, which makes
this approach unworkable.  Instead, we adopt a definition of the boundary
proposed by Sfeir et al. (1999) (and one that was also used by Lehner et al.
(2003)), who linked its location to the sudden onset of Na~I D-line
absorption that crossed a threshold $W_\lambda({\rm D2})=20\,$m\AA.  This
threshold is equivalent to $N({\rm H~I})\approx 2\times 10^{19}{\rm cm}^{-2}$
 (Ferlet et al. 1985), which in turn corresponds to $N({\rm O~I})$ that is
slightly greater than $10^{16}{\rm cm}^{-2}$.  Lehner et al.  (2003) found
that a typical velocity dispersion parameter $b({\rm O~I})=6\,{\rm km~s}^{-1}$
occurred inside the Local Bubble.  Using these two parameters for O~I, $(N,
~b)=(10^{16}{\rm cm}^{-2},~6\,{\rm km~s}^{-1})$, we can compute the equivalent
widths for all of the O~I lines when the boundary is crossed.  The third row
of numbers in the column headings in Table~\ref{EW_table} shows the values of
$W_\lambda$ (in m\AA) for all but the strongest three lines.  By comparing
these values with the entries that show our measurements, particularly the
weaker but securely measured ones, we can ascertain that our targets are
beyond the edge of the Local Bubble.

\section{INTERPRETATION: FUNDAMENTAL PROCESSES AND
EQUATIONS}\label{fundamentals}

In this section, we address the basic physical processes that relate our
findings on [Ar~I/O~I] to the ionization balance of the gas and the resulting
degree of partial ionization.  Our discussion about the means of ionizing the
atoms will focus mainly on the primary ionization by photons, along with the
effects of collisional ionizations caused by secondary electrons that
originate from these primary ionizations.  These two processes are the most
important sources of ionization, and they represent one side of the balance
between recombinations with free electrons and charge exchange reactions
between various constituents of the medium.  For completeness, we will also
cover other means of ionizing atoms and creating free electrons, such as
cosmic ray ionizations, the ionizations of inner shell electrons of heavy
elements by X-rays, the creation of ionizing photons when helium ions
recombine, and the nearly complete ionization of many elements that have
first ionization potentials below that of hydrogen.

\subsection{Direct and Secondary Ionizations of H and Ar}\label{ionization}

\subsubsection{Photoionization}\label{photoionization}

The primary photoionization cross sections for neutral Ar are larger than
those for H by about one order of magnitude at low energies, and the ratio
increases substantially at higher energies.  Various secondary ionizing
processes initiated by the primary photoionizations of H and He likewise have
a stronger effect on Ar than on H.  This contrast in ionization rates is the
fundamental tool that we use in the interpretation of the Ar data to quantify
the photoionization of H and the subsequent creation of free electrons.  For
the collective effect of all of these ionization channels, we can construct a
simple formalism based on arguments created by SJ98.  They defined a quantity
based on ionization rates $\Gamma$ and recombination coefficients $\alpha$
for the two elements,
\begin{equation}\label{p_ar}
P_{\rm Ar}= {\Gamma({\rm Ar}^0)\alpha({\rm H}^0,T)\over \Gamma({\rm
H}^0)\alpha({\rm Ar}^0,T)}~.
\end{equation}
SJ98 considered only primary photoionizations of these two elements.  Going
beyond their development, we construct a more comprehensive picture by
considering some refinements in the calculations of $\Gamma$ for both ${\rm
H}^0$ and ${\rm Ar}^0$.

First, we start with the primary photoionization rates $\Gamma_p=\int
F(E)\sigma(E)dE$, where $F(E)$ is the ambient photon flux as a function of
energy $E$ and $\sigma(E)$ is the photoionization cross section for either
H$^0$
 (Spitzer 1978, pp. 105-106) or Ar$^0$ (Marr \& West 1976).
Next, we include secondary ionizations with rates $\Gamma_s$ that are created
by the collisions from energetic electrons that are liberated by the primary
photoionizations of H and He.  Added to this are the effects from photons
with energies above about 300$\,$eV, which can interact with the abundant
heavy elements in the ISM to produce additional energetic electrons that will
ionize H and Ar with a rate that we identify as $\Gamma_{s^\prime}$.  These
electrons arise from the primary ionizations of the inner electronic shells,
and they are supplemented by one or more additional electrons from the Auger
process.  Finally, we must acknowledge that recombinations of singly- and
doubly-ionized He ions with electrons create additional photons when
de-excitation occurs in the lower stage of ionization, either He$^0$ or
He$^+$.  The importance of not overlooking secondary electrons and
recombination photons from the He ionizations is underscored by the fact that
while the abundance of He is only 1/10 that of hydrogen, its primary
ionization cross section is 6 to 100 times that of H over the energy range 25
to 4000$\,$eV.  Most of the recombination photons are capable of ionizing
both H and Ar, and they supplement the other sources of ionization with rates
$\Gamma_{He^0}$ and $\Gamma_{He^+}$.  In short, we consider that the total
ionization rates for the two elements in Eq.~\ref{p_ar} each consist of five
contributions,
\begin{equation}\label{Gammas}
\Gamma=\Gamma_p+\Gamma_s+\Gamma_{s^\prime}+\Gamma_{He^+}+\Gamma_{He^0}
\end{equation}
The details of how we compute $\Gamma_s$, $\Gamma_{s^\prime}$,
$\Gamma_{He^+}$ and $\Gamma_{He^0}$ are discussed in Appendices~\ref{gamma_s}
and \ref{gamma_he}.

The quantity $P_{\rm Ar}$ defined in Eq.~\ref{p_ar} provides a means for
evaluating how large the neutral fraction of Ar should be relative to that of
H according to the formula
\begin{equation}\label{ar_h}
[{\rm Ar~I/H~I}]=\log\left[ {1+n({\rm H}^+)/n({\rm H}^0)\over 1+P_{\rm Ar}
n({\rm H}^+)/n({\rm H}^0)}\right] .
\end{equation}
For a more accurate formulation of [Ar~I/H~I] that will be developed later in
Section~\ref{equilibrium}, we will introduce a more refined parameter
$P^\prime_{\rm Ar}$ that will be based on a calculation that is more
elaborate than the one shown in Eq.~\ref{p_ar}.  This new parameter will be
substituted for $P_{\rm Ar}$ in Eq.~\ref{ar_h}.  Under most conditions, the
differences between $P^\prime_{\rm Ar}$ and $P_{\rm Ar}$ are small.

Figure~\ref{p_ar_plot} shows for both H and Ar the monoenergetic cross
sections for primary ionization (dashed lines), the effective enhancements
arising from the secondary ionizations $\Gamma_s$ (dash-dot lines) for
$x_e=0.05$, and the additional effects from $\Gamma_{s^\prime}$,
$\Gamma_{He^+}$ and $\Gamma_{He^0}$, all of which give total ionization rates
shown by the solid lines. For H, secondary ionizations outweigh the primary
ones for photon energies above about 100~eV.  By contrast, we find that for
Ar the enhancement at these energies is small compared to its much higher
primary cross section.  A more narrowly defined version of $P_{\rm Ar}$,
which we denote as $P_{\rm Ar}(E)$, applies to an irradiation of the gas by
photons of a given energy $E$, rather than from a combination of fluxes over
a broad energy range.  The definition of $P_{\rm Ar}(E)$ is illustrated in
the upper panel of the figure, and its dependence on $E$ is shown in the
bottom panel.

\onecolumn
\begin{figure}
\epsscale{.8}
\plotone{p_ar_plot.eps}
{\renewcommand{\baselinestretch}{.95}
\caption{{\it Upper panel:\/} Various combinations of effective
photoionization cross sections for neutral hydrogen and argon as a function
of energy $E$.  The dashed lines depict the absorption cross sections for the
primary ionization rates $\Gamma_p$ that would apply to monoenergetic
radiation at the energy $E$ shown on the $x$-axis.  The effective cross
sections that arise by including the additional effects from ionizations by
secondary electrons produced by H and He (related to $\Gamma_s$) are shown by
the dot-dash lines.  These secondary ionization efficiencies were calculated
according to the principles outlined in Section~\protect\ref{electrons_H_He}
of the Appendix for an electron fraction $x_e=0.05$.; see
Eqs.~\protect\ref{Hphi} and \protect\ref{Hephi}.  These two ionization
processes can be further enhanced by photons that come from recombinations of
He ions with free electrons (Appendix~\protect\ref{gamma_he}) and electrons
from inner shell ionizations of heavy elements
(Appendix~\protect\ref{heavy_elem}), leading to total effective cross
sections depicted by the solid lines.  Across the top of the plot are markers
showing the energies at which half-intensity penetration depths for the
photons occur for various hydrogen column densities, assuming the gases have
a solar abundance ratio for various elements.  The quantity $P_{\rm Ar}(E)$
defined in Eq.~\ref{p_ar} for a specific energy $E$ is approximately equal to
the ratio of the total effective photoionization cross sections that give
rise to $\Gamma({\rm H}^0,~{\rm Ar}^0)$ defined in Eq.~\protect\ref{Gammas},
since the recombination rates for the two elements are about the same for all
temperatures.  {\it Bottom panel:\/} A plot of $P_{\rm Ar}(E)$ as a function
of energy.\label{p_ar_plot}}}
\end{figure}
\twocolumn

\subsubsection{Other Ionization Mechanisms}\label{other_ionization}

As discussed earlier in Section~\ref{intro}, neutral atoms in the ISM are
subjected to ionization by cosmic ray particles.  These ionizations, with a
rate $\zeta_{\rm CR}$, add to the effects of primary and secondary
ionizations from photons discussed above.  To obtain proper values of
$\zeta_{\rm CR}$ that apply to the diffuse medium, we must apply corrections
to the values of $\zeta_{\rm CR}$ that were measured for the more dense media
that have appreciable concentrations of molecules.  The details of this
computation are discussed in Appendix~\ref{cr_ioniz}.

In the WNM, the main source of free electrons is from the
ionization of H and He.  However, a small number of additional electrons
arise from other atoms that can be ionized by starlight photons less
energetic than the ionization potential of H.  For the combined effect from
these elements, we adopt an estimate $n({\rm M}^+)$ equal to $1.5\times
10^{-4}$ times the density of hydrogen based on the assumption that the gas
we are viewing is in a regime where the element depletions are relatively
modest.

\subsection{Recombination}\label{recombination}

The radiative recombination coefficients for free electrons and ions to
create neutral hydrogen and singly-ionized helium, $\alpha({\rm H}^0,T)$ and
$\alpha({\rm He}^+,T)$, are taken from
Spitzer (1978, pp. 105-107).  For $\alpha({\rm H}^0)$, we
excluded recombinations to the lowest electronic level, since they generate
Lyman limit photons that can re-ionize hydrogen atoms over a short distance
scale.  Recombination coefficients for He$^0$ were taken from Aldrovandi \&
P\'equignot (1973) and those for Ar$^0$ from Shull \& Van Steenberg
 (1982).  The results for Ar$^0$ given by
Shull \& Van Steenberg (1982) agree well
with the radiative recombination coefficients listed by Aldrovandi \&
P\'equignot (1974).  The minimum
temperature where dielectronic recombination for Ar$^0$ becomes important is
$2.5\times 10^4\,$K
 (Aldrovandi \& P\'equignot 1974), which is above temperatures that we
will consider, hence we can ignore this process.

In addition to recombining with a free electron, an ion can also be
neutralized by colliding with a dust grain and removing an electron from it
 (Snow 1975; Draine \& Sutin 1987; Lepp et al. 1988; Weingartner \& Draine
2001a).  The operation of this effect on protons is important for regulating
the fraction of free electrons in the cold neutral medium (CNM), but it is of
lesser significance for the WNM [cf. Figs. 16.1 and 16.2 of
Draine (2011)], which probably dominates the sight lines in our study.  The
rate constant $\alpha_g$ for this process is normalized to the local
hydrogen density $n({\rm H_{tot}})\equiv n({\rm H}^0)+n({\rm H}^+)$, and it
depends on several physical parameters that influence the charge on the
grains, such as the electron density $n(e)$, the rate of photoelectric
emission that is driven by the intensity $G$ of the local radiation field
between 6 and 13.6~eV, and the temperature $T$.  For our equilibrium
equations in Section~\ref{equilibrium}, we have adopted parametric fits for
$\alpha_g({\rm H}^0,n(e),G,T)$ and $\alpha_g({\rm He}^0,n(e),G,T)$ from
Weingartner \& Draine  (2001a).  They do not supply fit coefficients
for $\alpha_g({\rm Ar}^0,n(e),G,T)$, but since the ionization potential of
Ar$^0$ is close to that of H$^0$, it is reasonable to adopt the hydrogen rate
coefficient and divide it by the square-root of the atomic weight (40) of Ar.
Throughout our analysis, we set $G=1.13$, which is the value recommended for
the general ISM by Weingartner \& Draine (2001a).

\subsection{Charge Exchange}\label{charge_exchange}

In addition to the recombination processes mentioned in the previous section,
charge exchange reactions with neutral hydrogen can also lower the ionization
state of an atom.  With reference to such reactions for element X, ${\rm
X}^++{\rm H}^0\rightarrow {\rm X}^0+{\rm H}^+$ and ${\rm X}^{++}+{\rm
H}^0\rightarrow {\rm X}^++{\rm H}^+$, we adopt the notation $C^\prime({\rm
X}^+,T)$ and $C^\prime({\rm X}^{++},T)$ for the respective rate constants.
For our calculations of $C^\prime({\rm He}^+,T)$, $C^\prime({\rm
He}^{++},T)$, $C^\prime({\rm Ar}^+,T)$, and $C^\prime({\rm Ar}^{++},T)$, we
adopted Kingdon \& Ferland's (1996) fits to the
calculations from various sources (see their Table~1 for the coefficients and
references).

Since the ionization potential of neutral oxygen is close to that of
hydrogen, the rate constant for charge exchange of these two species is large
and reverse endothermic reaction is not negligible, except at very low
temperatures.  The rate constant $C({\rm O}^+,T)$ for the reaction ${\rm
O}^0+{\rm H}^+\rightarrow {\rm O}^++{\rm H}^0$ can be obtained from $C^\prime
({\rm O}^+,T)$ by the principle of detailed balancing.  However, in doing so,
one must treat the three fine-structure levels of the ground state of O$^0$
separately, since their energy separations are comparable to the differences
in the ionization potentials of H and O.  The large rate constants in both
directions (Stancil et al. 1999) assures that the ionization fraction
of O is locked very close to a value 8/9 times that of H for $T\gtrsim
10^3\,$K.  This is a key principle that allows us to use O as a substitute
for H in the present study or Ar vs. H fractional ionizations.

For completeness, we should also consider charge exchange reactions between
Ar and He, even though the abundance of He is much lower than that of H.  The
charge transfer recombination reaction ${\rm Ar}^{++}+{\rm He}^0\rightarrow
{\rm Ar}^++{\rm He}^+$ has a rate constant $D^\prime({\rm
Ar}^{++},T)=1.3\times 10^{-10}{\rm cm}^3{\rm s}^{-1}$ according to Butler \&
Dalgarno (1980), which they claim to be
constant for $T\geq 10^3\,$K.  The charge transfer ionization reaction with
He$^+$, i.e., ${\rm Ar}^0+{\rm He}^+ \rightarrow {\rm Ar}^++{\rm He}^0$, has
a rate constant $D({\rm Ar}^+,T)<10^{-13}{\rm cm}^3{\rm s}^{-1}$ according to
Albritton (1978), so we will ignore this process in the
equation for the ionization balance for Ar (Eq.~\ref{equi1} below).

\subsection{Equilibrium Equations}\label{equilibrium}

In a medium where both hydrogen and helium are partly ionized, the densities
of an element X in its 3 lowest levels of ionization X$^0$, X$^+$ and
X$^{++}$ are governed by the equilibrium equations\footnote{The
development here follows that given by Eqs. 12--19 of SJ98, except that we
have added the He charge exchange recombination reaction as an additional
channel for reducing the ionization of element X in its doubly charged form.
We have also added cosmic ray ionizations and have implicitly included the
various kinds of secondary photoionizations in the definition of $\Gamma$, as
indicated in Eq.~\protect\ref{Gammas}.  We have corrected
Eqs.~\protect\ref{equi1} and \protect\ref{n0} here to include a missing
$n({\rm H^+})$ term, which was a typographical oversight in the equations of
SJ98.}
\begin{eqnarray}\label{equi1}
&&\Big[ \Gamma({\rm X}^0)+\zeta_{\rm CR}({\rm X}^0)+C({\rm X}^+,T)n({\rm
H^+})\Big] n({\rm X}^0)\nonumber\\
&=&\Big [\alpha({\rm X}^0,T)n(e)+C^\prime({\rm X}^+,T)
n({\rm H}^0)\nonumber\\
&+&\alpha_g({\rm X}^0,n(e),G,T)n({\rm H_{tot}})\Big] n({\rm X}^+)
\end{eqnarray}
and
\begin{eqnarray}\label{equi2}
&&\Gamma({\rm X}^+)n({\rm X}^+)=\Big[ \alpha({\rm X}^+,T)n(e)+C^\prime({\rm
X}^{++},T)n({\rm H}^0)\nonumber\\
&+&D^\prime({\rm X}^{++},T)n({\rm He}^0)\Big] n({\rm
X}^{++})
\end{eqnarray}
where $\Gamma({\rm X}^y)$ is the photoionization rate of element X in
its ionization state $y$ (neutral, +, or ++) and $\alpha({\rm X}^y,T)$ is the
recombination rate of the $y+1$ state with free electrons as a function of
temperature $T$.  The simultaneous solution to these two equations yields the
fractional abundances in the three ionization levels
\begin{eqnarray*}\label{n0}
f_0({\rm X},T) \equiv {n({\rm X}^0)\over n({\rm X}^0)+n({\rm
X}^+)+n({\rm X}^{++})}=\Bigg( 1\, +
\end{eqnarray*}
\begin{mathletters}
\begin{equation}
{\Big[\Gamma({\rm X}^0)+\zeta_{\rm CR}({\rm X}^0)+C({\rm
X}^+,T)n({\rm H^+})\Big]
\Big[\Gamma({\rm X}^+)+Y\Big]\over \Big[\alpha({\rm X}^0,T)n(e) +
C^\prime({\rm X}^+,T)n({\rm H}^0)+\alpha_g({\rm X}^0,n(e),G,T)n({\rm
H_{tot}})\Big]Y}\Bigg)^{-1}
\end{equation}
with
\begin{equation}
Y=\alpha({\rm X}^+,T)n(e)+C^\prime({\rm X}^{++},T)n({\rm
H}^0)+D^\prime({\rm X}^{++},T)n({\rm He}^0)~,
\end{equation}
\end{mathletters}
\begin{equation}\label{n++}
f_{++}({\rm X},T) \equiv {n({\rm X}^{++})\over n({\rm X}^0)+n({\rm
X}^+)+n({\rm X}^{++})}={1-f_0({\rm X},T)\over 1+Y/\Gamma({\rm X}^+)}~,
\end{equation}
and
\begin{equation}\label{n+}
f_+({\rm X},T)\equiv {n({\rm X}^+)\over n({\rm X}^0)+n({\rm X}^+)+n({\rm
X}^{++})}=1-f_0({\rm X},T)-f_{++}({\rm X},T)~.
\end{equation}

\subsection{Electron Density and the Ionization Fractions of H and
He}\label{electron_density}

Before the ionization fractions of Ar can be derived, we must determine not
only the ionization balance of hydrogen, but also that for helium.  We do
this by solving
Eqs.~\ref{n0}--\ref{n+} (substituting He for X and eliminating the $D^\prime$
term) along with the equation for the hydrogen ionization balance,
\begin{mathletters}
\begin{equation}\label{H_ioniz_balance}
{n({\rm H}^+)\over n({\rm H}^0)}={\Gamma({\rm H}^0)+\zeta_{\rm CR}({\rm
H}^0)+Z\over \alpha({\rm
H}^0,T)n(e)+\alpha_g({\rm X}^0,n(e),G,T)n({\rm H_{tot}})-Z}~,
\end{equation}
with
\begin{equation}\label{Z}
Z=0.1n({\rm H}^0)\big[C^\prime({\rm He}^+,T)f_+({\rm He},T)+C^\prime({\rm
He}^{++},T)f_{++}({\rm He},T)\big]~,
\end{equation}
\end{mathletters}
and the constraints
\begin{equation}\label{He_total}
n({\rm He}^0)+n({\rm He}^+)+n({\rm He}^{++})=0.1n({\rm H}^0)\left[1+{n({\rm
H}^+)\over n({\rm H}^0)}\right]
\end{equation}
and
\begin{equation}\label{n(e)}
n(e)=n({\rm H}^+)+n({\rm He}^+)+2n({\rm He}^{++})+n({\rm M}^+)~.
\end{equation}

Since the hydrogen ionization balance depends on $n(e)$ which in turn is
influenced by the ionization fractions of He (which are also influenced by
$n(e)$), we must solve
Eqs.~\ref{H_ioniz_balance}-\ref{n(e)} iteratively to obtain a solution.  We
found that these equations converged very well if we kept $n({\rm H_{tot}})$
and the ionization rate of H pegged to a certain value.\footnote{Fixing
$n(e)$ instead of $n({\rm H_{tot}})$ produces unstable, oscillating solutions
under certain circumstances.}  Starting with a zero helium ionization rate,
the iterations progressed slowly to successively higher rates until the
final, correct value was reached and the ionization fractions had stabilized.

After obtaining the final results for the coupled hydrogen and helium
ionization balances, we can solve for $f_0({\rm Ar})$ using
Eq.~\ref{n0}.  This result can then be used to derive a more accurate value
for $P_{\rm Ar}$,
\begin{equation}\label{Pprime_ar}
P^\prime_{\rm Ar}={n({\rm H}^0)\over n({\rm H}^+)}\left( {1\over f_0({\rm
Ar})}-1\right)~,
\end{equation}
which can be substituted for $P_{\rm Ar}$ in Eq.~\ref{ar_h} to obtain a
solution for [Ar~I/H~I] that makes use of all of the physical processes that
were discussed in Sections~\ref{ionization}--\ref{charge_exchange}.

\section{OUTCOME FROM KNOWN SOURCES OF
PHOTOIONIZATION}\label{known_photoionization}

\subsection{External Radiation}\label{external_radiation}

Over many decades, the diffuse, soft X-ray background has been measured by a
large number of different experiments [for a review, see McCammon \& Sanders
 (1990)].  Most of the emission below 1$\,$keV
arises from hot ($T>10^6\,$K) gas in the Galactic disk and halo, with
radiation from extragalactic sources dominating at higher energies
 (Chen et al. 1997; Miyaji et al. 1998; Moretti et al. 2009).  Much of the
literature on the diffuse radiation shows a distinction between contributions
from a local component with little foreground absorption and more distant
emissions with varying levels of absorption.  The local background was once
identified as having originated from hot gas in the Local Bubble (Sanders et
al. 1977; Hayakawa et al. 1978; Fried et al. 1980), but in recent years it
has been recognized to be strongly contaminated, or completely dominated, by
X-rays arising from charge exchange produced by the interaction of the solar
wind with incoming interstellar atoms (Cravens 2000; Lallement 2004; Pepino
et al. 2004; Koutroumpa et al. 2006, 2007, 2009; Peek et al. 2011; Crowder et
al. 2012).  For this reason, we ignore the weakly absorbed, nearly isotropic
portion of the X-ray background and focus our attention to the component that
exhibits a pattern in the sky that clearly shows absorption by gas in the
Galaxy.

Kuntz \& Snowden (2000) have performed a detailed
investigation of the nonlocal component, which they call the transabsorption
emission (TAE).  They describe the strength and spectral character of the TAE
in terms of emissions from optically thin plasmas at two different
temperatures.  They define a soft component that has a mean intensity over
the sky $I=2.6\times 10^{-8}{\rm erg~cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}$ over
the interval $0.1 < E < 2\,$keV and a spectrum consistent with the emission
from a plasma at a temperature $T=10^{6.06}\,$K, and this flux is accompanied
by a hard component with $I=8.5\times 10^{-9}{\rm erg~cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}$ over the same energy interval with $T=10^{6.46}\,$K.
To translate the sum of these two components into a distribution of the
photon flux as a function of energy, $F(E)$, we calculate synthetic flux
representations using the CHIANTI database and software (Version~6.0)
 (Dere et al. 1997, 2009), after normalizing the
emission measures  to give the intensities stated above (we find that ${\rm
EM}=10^{16.37}$ and $10^{15.81}{\rm cm}^{-5}$ for the soft and hard
components, respectively).  We supplement the TAE result with an underlying
power-law extragalactic emission of the form $10.5\,{\rm phot~cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}{\rm keV}^{-1}E({\rm keV})^{-1.46}$ (Chen et al. 1997).

The ISM is opaque to X-rays at the lowest energies.  The energies at which
half of the X-rays are absorbed for various column densities are shown in the
top portion of Figure~\ref{p_ar_plot}, which were derived from the
calculations of Wilms et al. (2000).  At energies of
around 100$\,$eV where the ISM is neither completely opaque or transparent
for $N({\rm H}^0)\approx {\rm few}\,\times 10^{19}{\rm cm}^{-2}$,
uncertainties in the layout of emitting and absorbing regions make it
difficult to calculate with much precision how far the X-rays can penetrate
the typical gas volumes that were sampled in our survey of Ar~I and O~I.
Thus, rather than implement an elaborate attenuation function that would be
difficult to explain (and perhaps not especially correct at our level of
understanding), we apply a simplification that all of the X-rays are
transmitted above some threshold energy and none below it.  The threshold
that we adopted was 90$\,$eV, on the assumption that in some directions the
gas can view the unattenuated X-ray sky through $N({\rm H}^0)$ slightly less
than $10^{19}{\rm cm}^{-2}$.

The upper panel of Figure~\ref{flux_plot} shows our synthesis of the sum of
the extragalactic power-law emission and TAE synthesis described above.  For
the purpose of calculating $\Gamma$ for various elements, we make use of only
the flux depicted by the dark trace in the figure, i.e., that which starts at
the cutoff energy (90$\,$eV) and ends at an energy beyond which no
appreciable additional ionization occurs.

\subsection{Internal Radiation Sources}\label{internal_radiation}
Embedded within the ISM are sources of EUV and X-ray radiation that can make
additional contributions to $\Gamma$.  We can make estimates for their
average space densities and the character of their emissions, but one
uncertainty that remains is how well the ensuing photoionizations are
dispersed throughout the ambient gas.  At one extreme representing minimum
dispersal, we envision the classical Str\"omgren Spheres that surround
sources that are not moving rapidly and that emit most of their photons with
barely enough energy to ionize hydrogen.  These photons have a short mean
free path in a neutral medium.  Under these circumstances, the zone of
influence of the source is sharply bounded, and the resulting ionization is
nearly total inside the region and zero outside it.  At the other extreme,
one can imagine that the photons, ones that have relatively high energy, can
travel over a significant fraction of the inter-source distances before they
are absorbed.  In addition, the sources themselves could move rapidly enough
that they never have a chance to establish a stable condition of ionization
equilibrium.  (This issue will be investigated quantitatively in
Section~\ref{recomb_time}.)  These conditions could lead to the ionization
being more evenly distributed and not necessarily complete.  If the sources
have large enough velocities, we can even imagine a picture where there is a
random network of ``fossil Str\"omgren trails'' (Dupree \& Raymond 1983) that
ultimately might blend together.  More extreme manifestations of such trails
in denser media may have already been discovered by McCullough \& Benjamin
(2001) and Yagi et al.  (2012), who observed faint, but straight and narrow
lines of H$\alpha$ emission in the sky (but were unable to identify the
sources that created them).

It is difficult to establish where in the continuum between the two extremes
for the dispersal of ionization the true effects of embedded EUV and X-ray
sources are to be found.  For our treatment of the influence of these
production sites for ionizing radiation, we will adopt the simplified premise
that all of their photons are available to create a uniform but weak level of
ionization everywhere.  This picture is not entirely correct, since one
should expect that very near the sources some fraction of the ionizing
photons are ``wasted'' by creating localized regions with much higher than
usual levels of ionization.  Such regions dissipate most of their ionization
rapidly because their recombination times are short.  For this reason, our
making use of a calculated average production rate of ionizing photons per
unit volume will lead to an equilibrium equation that overestimates the
space- and time-averaged level of ionization.  (Later, it will be shown that
this overestimate is of no real consequence because we obtain an answer that
is still below that needed to explain the overall average ionization level.)

In the sections that follow, we consider three classes of sources that are
randomly distributed throughout the neutral ISM and can in principle help to
ionize it: main-sequence stars, active X-ray binaries, and WD stars.
Luminous, early-type stars contribute large amounts of ionizing radiation,
but most of their radiation completely ionizes the surrounding media and
makes the gas virtually invisible in the Ar~I and O~I lines.  These stars
also tend to be clustered inside the dense clouds of gas that led to their
formation.

\onecolumn
\begin{figure}
\epsscale{0.7}
\plotone{flux_plot.eps}
\caption{{\it Upper panel:\/} External fluxes arising from the known X-ray
background radiation (solid line) and a hypothetical, time-averaged flux
(over a recombination time of $\approx 1\,$Myr) from 3 supernova remnants at
a distance of 100$\,$pc (dashed line).  The gray portion of the solid curve
is in an energy range where the opacity to X-rays is high, and thus this
radiation is not likely to penetrate into much of the gas that we observe.
{\it Lower panel:\/} Internal average rates per unit volume for the injection
of ionizing photons by the coronae of main-sequence stars (MS), active
binaries (AB) and the photospheres of WD stars
(WD).  (A machine readable table of all 5 flux distributions vs. $E$
shown in this figure is available in the online version of the
{\it Astrophysical Journal.})\label{flux_plot}}
\end{figure}
\twocolumn

The question may arise as to whether or not, by considering the embedded
sources as a separate contribution that adds to the external radiation
background discussed earlier, we are possibly ``double counting'' some of the
photons that could ionize the ISM.  Kuntz \& Snowden (2001) have computed the
probable relative contribution of Galactic point sources that were not
explicitly taken out of their measurements of the diffuse X-ray background,
and they concluded that this contamination was, at most, only about $2-10\%$
of the radiation that was thought not to arise from the Local Bubble.  Within
their highest energy bands ($0.73-2.01\,$keV), they state that the
contamination could be as high as 51\%.

\subsubsection{Main-sequence Stars}\label{ms}

For various kinds of point sources, the ratio of the emission of X-rays to
photons in the $V$ band is usually defined by the relation
\begin{equation}\label{f_X/f_V}
\log(f_X/f_V)=\log f_X+0.4m_V+5.37~,
\end{equation}
where $f_X$ is the apparent X-ray flux of the source over a specified energy
interval, expressed in ${\rm erg~cm}^{-2}{\rm s}^{-1}$, and $m_V$ is its
apparent visual magnitude.\footnote{Numerical
values for this formula are not universal, since the adopted X-ray energy
interval depends on which instrument was used.  For instance, surveys that
used the {\it Einstein Observatory\/}, e.g. those reported by Maccacaro et
al. (1988) or Stocke et al. (1991), used the passband $0.3<E<3.5\,$keV for
$f_X$, whereas those based on the {\it ROSAT\/} All-Sky Survey (RASS)
 (Krautter et al. 1999; Ag\"ueros et al. 2009) measured $f_X$ over
the interval $0.1<E<2.4\,$keV.}  The total output of X-rays from a source with an
absolute magnitude $M_V=0$ should be $5.11\times 10^{34}(f_X/f_V)\,{\rm
erg~s}^{-1}$.  Within each spectral class, there is a large dispersion in the
measured values of $\log(f_X/f_V)$, typically of order 1$\,$dex, which is
probably attributable to differences in stellar rotation velocities
 (Audard et al. 2000; Feigelson et al. 2004), variations in
foreground absorption by the ISM, and time variability of the X-ray emission.
While the most significant X-ray flares from stars can create spectacular
increases in flux, their time-averaged effect has been estimated by Audard et
al. (2000) to amount to only about 10\% of the steady
emission.  On the basis of white-light monitoring of dwarf stars, Walkowicz
et al. (2011) found that the duty cycle of
flaring events is only of order a few percent.

For any given stellar spectral class with a characteristic absolute magnitude
$M_V$ that spans a range $\Delta M_V$ along the main sequence and has
luminosity function $\phi(M_V)\,{\rm stars~mag}^{-1}{\rm pc}^{-3}$, the
energy density of X-rays per unit volume is given by
\begin{eqnarray}\label{F(X)}
&&F(E)_{\rm MS}=1.74\times 10^{-21}\phi(M_V)\Delta M_V\nonumber\\
&\times& 10^{\log (f_X/f_V)-0.4M_V}{\rm erg~cm}^{-3}{\rm s}^{-1}~.
\end{eqnarray}
If we use the mean values of $\log (f_X/f_V)$ for different spectral classes
listed by Ag\"ueros et al. (2009) for the X-ray band $0.1<E<2.4\,$keV,
obtain values of $M_V$ for these classes from Schmidt-Kaler
(1982), define a luminosity function for stars in the
disk of our Galaxy from the formula given by Bahcall \& Soneira
(1980), and then sum the results over all
spectral types A-M, we obtain an average energy density equal to $1.90\times
10^{-28}{\rm erg~cm}^{-3}{\rm s}^{-1}$.  While we acknowledge that the
spectral character of coronal emissions can vary for different stars along
the main sequence, in the interest of simplicity we adopted for all cases a
spectrum based on the differential emission measure (DEM) for the coronal
emission from the quiet Sun, as defined in the CHIANTI database.  To
obtain a final photon emission rate per unit volume and energy in the ISM, we
made use of this spectrum and normalized its energy output over the
$0.1-2.4\,$keV band to the energy density factor given above.  The result is
shown by the spectrum labeled ``MS'' in the lower panel of
Figure~\ref{flux_plot}.

\subsubsection{Active Binaries and Cataclysmic Variables that Emit
X-rays}\label{ab}

At high Galactic latitudes, most of the X-ray radiation above several keV
originates from extragalactic sources.  Near the direction toward the
Galactic center, however, Revnivtsev et al. (2009)
found that at 4$\,$keV about half of the X-ray emission arises from point
sources in the Galaxy that can be resolved by the {\it Chandra X-ray
Observatory\/}, with the remainder coming from either a diffuse Galactic (hot
gas) emission or an extragalactic contribution.  To estimate the average
emission per unit volume from these sources, we integrate the $2-10\,$keV
X-ray emission over the entire luminosity function $\phi(\log L_{2-10\,{\rm
keV}})_{\rm AB}$ specified by Sazonov et al. (2006)
to obtain the total energy output
\begin{eqnarray}\label{L_AB}
&&L(2-10\,{\rm keV})_{\rm AB}\nonumber\\
&=&\int_{22.5}^{34.0} L_{2-10\,{\rm keV}}\phi(\log
L_{2-10\,{\rm keV}})_{\rm AB}\nonumber\\
&\times& d\,(\log L_{2-10\,{\rm keV}})= 2.13\times
10^{38}{\rm erg~s}^{-1}
\end{eqnarray}
for all of the sources in our Galaxy.\footnote{Note that the upper bound for
the integration is at $10^{34}{\rm erg~s}^{-1}$.  While the energy output
over the whole Galaxy for neutron star and black hole X-ray binaries with
individual outputs $L_{2-10\,{\rm keV}}>10^{34}{\rm erg~s}^{-1}$ is
substantial, these objects are so few in number that they can no longer be
considered as embedded sources.  The brightest such object in the sky,
Sco~X$-$1, is at a distance of 2.8$\,$kpc and creates a flux at our location of
only $2.8\times 10^{-7}{\rm erg~cm}^{-2}{\rm s}^{-1}$ over the $2-10\,$keV
band (Grimm et al. 2002), which is substantially lower than the
extragalactic background flux integrated over the whole sky.} We may convert
this value to an average volume emissivity by multiplying it by the stellar
mass density at our location ($0.04\,M_\odot~{\rm pc}^{-3}$) divided by
the total stellar mass of the Galaxy ($7\times 10^{10}M_\odot$), where
both of these numbers were those adopted by Sazonov et al. (2006), and
ultimately obtain a value $4.15\times 10^{-30}{\rm erg~cm}^{-3}{\rm s}^{-1}$.
 To define the spectral shape for the radiation emitted by these sources, we
used the CHIANTI software to compute the emission from a plasma with
an average of the DEM functions expressed by Sanz-Forcada et al. (2002,
2003)\footnote{These authors describe their DEM functions in terms of
$N(e)N({\rm H}^+)dV/d(\log T$), whereas the convention in CHIANTI
assumes that the DEM is defined in terms of $N(e)N({\rm H}^+)dV/dT$.}
that were constructed from their {\it Extreme-Ultraviolet Explorer\/}
observations of various active
binary sources.  This spectrum was then normalized such that the flux in the
$2-10\,$keV band matched the volume emissivity described above.  The emission
from active binaries that we derived $F(E)_{\rm AB}$ is shown by the curve
labeled ``AB'' in the lower panel of Figure~\ref{flux_plot}.

\subsubsection{White Dwarf Stars}\label{wd}

WD stars that are hot enough to emit significant fluxes in the EUV
spectral range are much less numerous than the main-sequence stars considered
in Section~\ref{ms}.  However their photospheres generate outputs in the EUV
region that exceed by far the coronal emissions from individual main-sequence
stars.  This fact is demonstrated by the actual observations of the local EUV
sources compiled by Vallerga (1998), where he found that a
vast majority of the detected objects were nearby hot WD
stars.\footnote{The spectrum shown by Vallerga (1998)
indicates that the $m_V=1.5$ B2~II star Adhara ($\epsilon$~CMa) dominates the
local flux at energies below the He~I ionization edge at 24.6$\,$eV.  This is
an atypical situation, since $N({\rm H~I})\lesssim 10^{18}{\rm cm}^{-2}$
toward this star (Gry \& Jenkins 2001).}  Krzesinski et al.
 (2009) have measured the luminosity function for
DA WDs $\phi(M_{\rm bol})_{\rm WD}$, expressed in terms of (${\rm
stars}~M_{\rm bol}^{-1}{\rm pc}^{-3}$), in our part of the Galaxy using
results from the {\it Sloan Digital
Sky Survey Data Release 4\/} (SDSS DR4) database.  If we combine this information with the
theoretical computations of stellar atmosphere fluxes $F_\lambda$ (equal to 4
times the Eddington flux $H_\lambda$), expressed in the units ${\rm
erg~cm}^{-2}{\rm s}^{-1}{\rm cm}^{-1}$ (Rauch 2003; Rauch et al. 2010),
convert it to a physical flux at the stellar surface ${\mathcal F}(E)=
7.74\times 10^7\pi F_\lambda E({\rm eV})^{-3}{\rm phot~cm}^{-2}{\rm s}^{-
1}{\rm eV}^{-1}$, and assume each star has a radius $r_*=0.014r_\odot$
(Liebert et al. 1988), we can obtain a total emissivity per unit volume,
\begin{eqnarray}
&&F(E)_{\rm WD}=4\pi r_*^2 (1.65\times 10^{-34})\nonumber\\
&\times& \int{\mathcal F}(E)\phi(M_{\rm
bol})_{\rm WD}d\, M_{\rm bol}~{\rm phot~cm}^{-3}{\rm s}^{-1}{\rm eV}^{-1}~.
\end{eqnarray}
A conversion from $M_{\rm bol}$ to $T_{\rm eff}$ is shown in the plot of the
WD luminosity function presented by Krzesinski et al. (2009).  The model
fluxes for stars with $T_{\rm eff}<40,000\,$K were assumed to arise from
stars with pure hydrogen atmospheres, but stars with temperatures above this
limit are known to have significant abundances of metals in their atmospheres
because radiative levitation can overcome diffusive settling (Dupuis et al.
1995; Marsh et al. 1997; Schuh et al. 2002).  Thus, for $T_{\rm eff}=50,000\,
$K and above, we used the fluxes for model atmospheres with [$X$]~=~[$Y$]~=~0 and
$[Z]=-1$, which have significant sources of opacity that reduce the
radiation at energies $E>54\,$eV.

As a check on the calculations described above, we can compute the X-ray
energy outputs as a function of $T_{\rm eff}$ over the $0.1-0.28\,$keV range,
synthesize a luminosity function as a function of $L_X$ in this band, and
then compare the results to X-ray luminosity function derived by Fleming et
al. (1996) from a RASS survey of WDs.  Our
source densities at the high end of the luminosity distribution compare
favorably with the distribution shown by Fleming et al., but we predict a
somewhat greater number of sources with $L_X\lesssim 10^{31}{\rm erg~s}^{-1}$
due to a large space density of stars with $30,000\leq T_{\rm eff}\leq
40,000\,$K.

\subsection{Interpretation of Internal
Ionizations}\label{interpretation_internal}

The treatments of the ionizations that arise from external and internal
sources differ in a fundamental way.  External radiation above some energy
threshold is regarded as not being consumed by ionizing the gas, i.e., it is
assumed to be unattenuated and thus each gas constituent is ionized at an
appropriate rate $\Gamma$, as defined in Eq.~\ref{Gammas}, that is simply
proportional to $\int \sigma(E)F(E)dE$.  Here, $\sigma(E)$ is the effective
cross section for the combination of the different ionization channels, as
depicted for Ar and H by the solid lines in Fig.~\ref{p_ar_plot}.

As indicated in the beginning of Section~\ref{internal_radiation}, for the
internally generated radiation we switch to a very different concept and
adopt the simplified premise that all of the photons emitted by embedded
sources are used up by ionizing the various gas constituents that surround
them.  Thus the primary ionization rate for each kind of atom (or ion) $X$ is
given by
\begin{equation}\label{Gamma_pX}
\Gamma_p(X)=\int F(E)y(X,E)n(X)^{-1}dE
\end{equation}
where $F(E)$ in this equation is the sum of all of the photon generation
rates per unit volume specified in Sections~\ref{ms} through \ref{wd}.  Each
of the different species represented by $X$ must compete with others for the
photons that are consumed.  Thus the equation includes a term $y(X,E)$, which
is a sharing function for the ionization rate that is represented by the
relative probability that any photon with an energy $E$ will interact with a
given species $X$,
\begin{equation}\label{y(X,E)}
y(X,E)={n(X)\sigma(X,E)\over\sum\limits_{X^\prime}n(X^\prime)\sigma(X^\prime,
E)}~,
\end{equation}
where $n(X)$ is the number density of $X$, $\sigma(X,E)$ is the
photoionization cross section of $X$ at an energy $E$ (likewise for
$X^\prime$), and the sum in the denominator covers all of the major species
competing for photons, i.e., $X^\prime={\rm H}^0$, ${\rm He}^0$, ${\rm
He}^+$, and heavy elements whose inner shells respond to the more energetic
X-rays.  For either the external or internal radiations, the secondary
ionization rates $\Gamma_s$ and $\Gamma_{s^\prime}$ follow in proportion to
the primary ones according to the descriptions given in Appendix~\ref{gamma_s}.
The ionizations from helium recombinations $\Gamma_{\rm He}^+$ and
$\Gamma_{\rm He}^0$ are treated as internal sources of ionization, and their
rates are driven by the local densities of He$^{++}$, He$^+$, and electrons,
as described in Appendix~\ref{gamma_he}.

\subsection{Predicted Level of Ionization}\label{predicted_level}

Given the computed rates of ionization in the previous sections, an
evaluation of the electron fraction $x_e=n(e)/n({\rm H}_{\rm tot})$ will
depend on both the temperature $T$ and density $n({\rm H}_{\rm tot})$ for the
gas.  This fraction is higher than $n({\rm H}^+)/n({\rm H}^0)$ because some
electrons arise from the ionization of He.  The coupling of the H and He
ionization fractions is governed by Eqs.~\ref{H_ioniz_balance} through
\ref{n(e)}.  Our observational constraint, which must ultimately agree with
the ionization calculations, arises from the values for [Ar~I/O~I], which
respond to $n({\rm H}^+)/n({\rm H}^0)$  in accord with Eq.~\ref{ar_h}.  While
the use of $P_{\rm Ar}$ in this equation will give an approximate value for
[Ar~I/O~I], a more accurate result emerges by replacing $P_{\rm Ar}$ by
$P^\prime_{\rm Ar}$, the derivation of which was described in
Sections~\ref{equilibrium} and \ref{electron_density}.  Our goal will be to
explore parameters that will match a computed value for [Ar~I/O~I] to the
representative observed value $[{\rm Ar~I/O~I}]=-0.427\pm0.11$.

We have no direct knowledge about the local values of $n({\rm H}_{\rm tot})$
that apply to the gas in front of the stars in this survey.  We must
therefore rely on general estimates that have appeared in the literature.  We
can draw upon two resources.  First, surveys of 21-cm emission indicate the
amounts of H~I in the disk at a Galactocentric distance of the Sun, but
difficulties in interpreting the outcomes arise from self-absorption effects
and ambiguities in distinguishing between cold, dense clouds
(CNM) and their surrounding WNM.  Ferri\`ere (2001) lists values
for $n({\rm H}^0)_{\rm WNM}$ that are in the range $0.2-0.5\,{\rm cm}^{-3}$.
Dickey \& Lockman (1990) state a value of
$0.57\,{\rm cm}^{-3}$, but it is not clear whether this excludes
contributions from the CNM.  Kalberla \& Kerp (2009) estimate that the
midplane density of the WNM is $0.1\,{\rm cm}^{-3}$, but this low value is
difficult to reconcile with measurements of the mean thermal pressures $nT=
3800\,{\rm cm}^{-3}$K by Jenkins \& Tripp  (2011) and a general recognition
that $T_{\rm WNM}<10^4\,$K.

\begin{deluxetable}{
l
c
c
c
c
c
c
}
\tablecolumns{7}
\tablewidth{0pt}
\tablecaption{Ionization Rates from Known
Sources\tablenotemark{a}\label{known_source_rates}}
\tablehead{
\colhead{Ionization} & \colhead{Rate} & \multicolumn{5}{c}{Relative
Contribution\tablenotemark{b}}\\
\cline{3-7}
\colhead{Source\tablenotemark{c}} & \colhead{($10^{-17}\,{\rm s}^{-1}$)} &
\colhead{$\Gamma_p$} & \colhead{$\Gamma_{s,{\rm H}^0}$} &
\colhead{$\Gamma_{s,{\rm He}^0}$} & \colhead{$\Gamma_{s,{\rm He}^+}$} &
\colhead{$\Gamma_{s^\prime}$}\\
\colhead{(1)}& \colhead{(2)}& \colhead{(3)}& \colhead{(4)}&
\colhead{(5)}& \colhead{(6)}& \colhead{(7)}
}
\startdata
$\Gamma({\rm H}^0)_{\rm ext.}$& 5.26& 0.20& 0.22& 0.52& 0.013& 0.048\\
$\Gamma({\rm H}^0)_{\rm MS}$& 6.72& 0.26& 0.044& 0.57& 0.11& 0.012\\
$\Gamma({\rm H}^0)_{\rm AB}$& 0.649& 0.035& 0.015& 0.52& 0.22& 0.21\\
$\Gamma({\rm H}^0)_{\rm WD}$& 2.19& 0.92& 0.030& 0.044& 6.3e-3& 9.4e-6\\
$\Gamma_{He^+}({\rm H}^0)$& 0.166&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^0}({\rm H}^0)$& 4.94&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\zeta_{\rm CR}({\rm H}^0)$& 12.3&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm H}^0)$& {\bf
32.3}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm He}^0)_{\rm ext.}$& 31.4& 0.87& 0.035& 0.084& 2.2e-3& 7.5e-3\\
$\Gamma({\rm He}^0)_{\rm MS}$& 25.7& 0.80& 0.012& 0.15& 0.028& 3.0e-3\\
$\Gamma({\rm He}^0)_{\rm AB}$& 0.861& 0.33& 0.011& 0.36& 0.16& 0.14\\
$\Gamma({\rm He}^0)_{\rm WD}$& 17.0& 0.99& 4.5e-3& 5.9e-3& 8.6e-4& 1.2e-6\\
$\Gamma_{He^+}({\rm He}^0)$& 0.916&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^0}({\rm He}^0)$& 7.10&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\zeta_{\rm CR}({\rm He}^0)$& 14.4&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm He}^0)$& {\bf
97.3}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm He}^+)_{\rm ext.}$&
16.9&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm He}^+)_{\rm MS}$& 8.32&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm He}^+)_{\rm AB}$&
0.0978&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm He}^+)_{\rm WD}$&
0.224&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^+}({\rm He}^+)$& 0.317&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm He}^+)$\tablenotemark{d}& {\bf
25.9}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm Ar}^0)_{\rm ext.}$& 251.& 0.94& 0.018& 0.042& 1.1e-3& 3.9e-3\\
$\Gamma({\rm Ar}^0)_{\rm MS}$& 82.2& 0.77& 0.014& 0.18& 0.035& 3.9e-3\\
$\Gamma({\rm Ar}^0)_{\rm AB}$& 10.5& 0.77& 3.6e-3& 0.12& 0.053& 0.049\\
$\Gamma({\rm Ar}^0)_{\rm WD}$& 45.3& 0.99& 5.6e-3& 8.2e-3& 1.2e-3& 1.8e-6\\
$\Gamma_{He^+}({\rm Ar}^0)$& 1.35&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^0}({\rm Ar}^0)$& 93.6&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\zeta_{\rm CR}({\rm Ar}^0)$& 113.&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm Ar}^0)$& {\bf
597.}&\nodata&\nodata&\nodata&\nodata&\nodata\\[5pt]
$\Gamma({\rm Ar}^+)_{\rm ext.}$&
123.&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm Ar}^+)_{\rm MS}$& 47.2&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm Ar}^+)_{\rm AB}$& 1.57&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma({\rm Ar}^+)_{\rm WD}$& 49.0&\nodata&\nodata&\nodata&\nodata&\nodata\\
$\Gamma_{He^+}({\rm Ar}^+)$& 0.969&\nodata&\nodata&\nodata&\nodata&\nodata\\
Total $\Gamma({\rm Ar}^+)$\tablenotemark{d}& {\bf
222.}&\nodata&\nodata&\nodata&\nodata&\nodata\\
\enddata
\tablenotetext{a}{The internal rates (with subscripts MS, AB and WD) are
based on a volume density $n({\rm H}_{\rm tot})=0.5\,{\rm cm}^{-3}$.  Such
rates scale inversely with density (although not exactly so, because the
secondary ionization efficiencies change when $x_e$ changes).}
\tablenotetext{b}{Fractions of the values shown in column (2).  See
Eqs.~\protect\ref{Gammas}, \protect\ref{Hphi}, \protect\ref{Hephi},
\protect\ref{auger_H}, and \protect\ref{auger_He}.}
\tablenotetext{c}{Meaning of the subscripts that follow the different forms
of $\Gamma$: ext.~=~external X-ray background radiation
(Section~\protect\ref{external_radiation}), MS~=~embedded main-sequence stars
(Section~\protect\ref{ms}), AB~=~embedded active binary stars
(Section~\protect\ref{ab}), and WD~=~embedded WD stars
(Section~\protect\ref{wd}).}
\tablenotetext{d}{The total ionization rates for the ions He$^+$ and Ar$^+$
do not include the ionizations from secondary electrons or cosmic rays.
Hence these totals underestimate the true rates.}
\end{deluxetable}
Heiles \& Troland (2003) estimate
that an overall average $\langle n({\rm H}^0)_{\rm WNM})\rangle=0.28\,{\rm
cm}^{-3}$ translates into local densities of $0.56\,{\rm cm}^{-3}$ if the WNM
has a volume filling factor of 0.5.  A second method for estimating $n({\rm
H}^0)_{\rm WNM}$ is to note where the WNM branch of the theoretical thermal
equilibrium curve of Wolfire et al. (2003) intersects
the average thermal pressure $p/k=3800\,{\rm cm}^{-3}$K of Jenkins \& Tripp
 (2011).  This exercise yields a value of
$0.4\,{\rm cm}^{-3}$.

Table~\ref{known_source_rates} shows a detailed accounting of the ionization
rates from the different radiation sources, under the condition that the
value of $x_e$ corresponds to what the equilibrium equations yield for a WNM
at $T=7000\,$K.  The entries in this table reveal that for neutral hydrogen
the ionization caused by all of the sources of EUV and X-ray photons result
in a combined total rate $\Gamma({\rm H}^0)$ that is about 1.6 times the rate of
ionization by cosmic rays $\zeta_{\rm CR}$.  For neutral He and Ar, the
photon ionization rates are several times higher than those from cosmic rays.
The changing patterns in the distribution of different ionization mechanisms
revealed in Columns (3) to (7) in the table reflect differences in the
distribution of fluxes with energy shown in Fig.~\ref{flux_plot}.  For
sources with hard spectra, about half of the hydrogen ionization arises from
the ionization of He$^0$, which produces energetic electrons that can cause
secondary collisional ionizations, i.e., the process associated with
$\Gamma_{s,{\rm He}^0}({\rm H}^0)$.  It is only for the very soft spectrum of
the collective radiation from WD stars that we find that
$\Gamma_p({\rm H}^0)$ strongly dominates over the other ionization routes.

The ionization rates and concentrations of various primary constituents are
coupled to each other by the network of reactions described in
Section~\ref{fundamentals} and Appendices \ref{gamma_s} through
\ref{cr_ioniz}.  The first column of Table~\ref{properties_table} lists
various quantities of interest, and their derived values under the assumption
that $n({\rm H_{tot}})=0.50\,{\rm cm}^{-3}$ and $T=7000\,$K are given in
Column (2). (The remaining two columns of this table will be discussed later
in Section~\ref{additional_photoionization}.)

Figure~\ref{sdosdb_plot} shows the predicted outcomes for [Ar~I/O~I] after we
perform the calculations, again using the equations and reaction rates given
in Section~\ref{fundamentals} and the Ar ionization rates derived from the
information in Sections~\ref{external_radiation} and
\ref{internal_radiation}.  The upper curve in this figure that represents
$n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$ is clearly inconsistent with our
findings for [Ar~I/O~I].  At $T=7000\,$K, a lower value $n({\rm H}_{\rm
tot})=0.14\,{\rm cm}^{-3}$ is consistent with the upper error bound for
[Ar~I/O~I].  At lower temperatures, however, the predictions for [Ar~I/O~I]
once again are found to be considerably above the observed values. It is not
until a density of $0.09\,{\rm cm}^{-3}$ is reached, a value that is
unacceptably low, that the predicted ionization conditions for $T=7000\,$K
fit comfortably with the nominal value from the observations.  Even here,
however, this result is not fully satisfactory because a good fraction of the
WNM is known to be at temperatures well below 7000$\,$K, where the gas is
thermally unstable (Heiles \& Troland 2003).

Our overall conclusion is that the ionization rates arising from what we
consider to be known sources of ionizing radiation are not able to maintain a
level of ionization in the WNM that is consistent with our low observed
values for [Ar~I/O~I].  Our quest to resolve this problem by exploring some
possible supplemental means for ionizing the medium will be addressed later
in Section~\ref{additional_photoionization}.

\begin{figure}[h!]
\epsscale{1.0}
\plotone{sdosdb_plot.eps}
\caption{Calculated values for $10^{\rm [Ar I/O I]}$ for three different
densities $n({\rm H}_{\rm tot})$ and a range of different gas temperatures
$T$ shown on the $x$ axis, computed using estimates for the ionization rates
from known sources of radiation, both external and internal.  A general value
for $10^{\rm [Ar I/O I]}$ that arose from the survey is indicated by the
horizontal line, and the shaded band shows the range of possible
systematic errors.\label{sdosdb_plot}}
\end{figure}
\begin{deluxetable}{
l
c
c
c
}
\tablewidth{0pt}
\tablecaption{Gas Properties\tablenotemark{a}\label{properties_table}}
\tablehead{
\colhead{} & \colhead{Without SNR} & \colhead{With SNR} & \colhead{Greater
Low}\\
\colhead{Quantity} & \colhead{Contrib.\tablenotemark{b}} &
\colhead{Contrib.\tablenotemark{c}} & \colhead{Energy
Penetration\tablenotemark{d}}\\
\colhead{(1)} &\colhead{(2)} &\colhead{(3)} & \colhead{(4)}
}
\startdata
$P_{\rm Ar}$\tablenotemark{e}\dotfill&13.1&19.3&12.6\\
$P^\prime_{\rm Ar}$\tablenotemark{f}\dotfill&13.4&22.8&15.9\\
$n(e)\,({\rm cm}^{-3})$\dotfill&0.018&0.033&0.052\\
$n({\rm H}^0)\,({\rm cm}^{-3})$\dotfill&0.48&0.48&0.46\\
$n({\rm H}^+)\,({\rm cm}^{-3})$\dotfill&0.015&0.025&0.036\\
$n({\rm He}^0)\,({\rm cm}^{-3})$\dotfill&0.047&0.042&0.035\\
$n({\rm He}^+)\,({\rm cm}^{-3})$\dotfill&3.0e-03&8.1e-03&0.014\\
$n({\rm He}^{++})\,({\rm cm}^{-3})$\dotfill&2.1e-05&2.1e-04&1.0e-03\\
Total $\Gamma({\rm H}^0)\,({\rm s}^{-1})$\dotfill&3.2e-16&9.2e-16&2.1e-15\\
$\ell_c$ ($10^{-27}{\rm erg~s}^{-1}{\rm
H~atom}^{-1})$\tablenotemark{g}\dotfill&4.3&6.2&8.5\\
\enddata
\tablenotetext{a}{For $n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$ and
$T=7000\,$K.}
\tablenotetext{b}{Applies to the photoionization rates expressed in
Table~\protect\ref{known_source_rates} from the known sources only.}
\tablenotetext{c}{The radiation from known sources is supplemented by the
average flux over 1$\,$Myr from 3 supernova remnants created by explosions
with energies $E_{\rm SN}=3\times 10^{50}\,{\rm erg}$ in a medium with
$n({\rm H})=1.0\,{\rm cm}^{-3}$ located at a distance of 100$\,$pc.  This
supplemental radiation is shown by the dashed line in the upper panel of
Fig.~\protect\ref{flux_plot}.}
\tablenotetext{d}{Allows for a low energy cutoff of 60$\,$eV instead of
90$\,$eV for the external radiation background; see
Section~\protect\ref{penetration}.}
\tablenotetext{e}{As defined in Eq.~\protect\ref{p_ar}.}
\tablenotetext{f}{As defined in Eq.~\protect\ref{Pprime_ar}.}
\tablenotetext{g}{Cooling rate from radiation by C$^+$ in its excited
fine-structure state.}
\end{deluxetable}

\subsection{Dependence of Recombination with Time}\label{recomb_time}

An argument that helps to support the concept that the internal sources
spread their ionization rather evenly throughout the medium is that they move
at a rate that makes their local residence time short compared to a
characteristic $e$-folding time $t_{\rm recomb.}$ for the decay of the proton
density from an initial high value to some end equilibrium state $n({\rm
H}^+)_{\rm eq.}$ as $t\rightarrow\infty$,
\begin{equation}\label{t_recomb}
t_{\rm recomb.}=-{n({\rm H}^+)-n({\rm H}^+)_{\rm eq.}\over dn({\rm H}^+)/dt}~,
\end{equation}
where
\begin{eqnarray}\label{dnpdt}
&&dn({\rm H}^+)/dt=-\Big[\alpha({\rm H}^0,T)n(e)\nonumber\\
&+&\alpha_g({\rm
H}^0,n(e),G,T)n({\rm H_{tot}})\nonumber\\
&+&\Gamma({\rm H^0})\Big] n({\rm
H}^+)+\Gamma({\rm H}^0)n({\rm H_{tot}})~.
\end{eqnarray}
This expression overlooks the complications arising from charge exchange
reactions, and when we evaluate the trend of $n({\rm H}^+)$ with time we
assume that $n(e)$ is always equal to $1.2n({\rm H}^+)$, as is the case when
the gas is in a steady-state ionization condition.  Another simplification is
that $T$, and hence $\alpha({\rm H}^0,T)$, remains constant.\footnote{A lack
of variation in $T$ is probably a safe assumption for isobaric recombination,
but in the isochoric case $T$ may deviate to lower values at intermediate
times (Dong \& Draine 2011), and this would increase the value of
$\alpha({\rm H}^0,T)$ and make the recombination more rapid.  This could be
of importance for very large regions of space ionized by the radiation from
old SNRs, which will be considered later in
Section~\protect\ref{previous_SNR}.}  When the gas is highly ionized,
$\Gamma({\rm H}^0)$ is not equal to the value that we computed for $n({\rm
H}^+)_{\rm eq.}=0.015\,{\rm cm}^{-3}$ because the secondary ionization
processes change with $n(e)$ and depend on the strength of the helium
ionization.  However, we can ignore this complication that occurs at early
times because the recombination terms in Eq.~\ref{dnpdt} are considerably
larger than the terms involving $\Gamma({\rm H}^0)$.  The ionization rate
influences the character of the decay only when the gas is weakly ionized.
Hence there is no harm in declaring that at all times $\Gamma({\rm
H}^0)=3.23\times 10^{-16}{\rm s}^{-1}$, as stated in
Table~\ref{known_source_rates}. We can adopt for the final state
(equilibrium) densities the values $n({\rm H}^+)_{\rm eq.}=0.015\,{\rm
cm}^{-3}$ and $n(e)_{\rm eq.}=0.018\,{\rm cm}^{-3}$ listed in Column (2) of
Table~\ref{properties_table}.

The change of $t_{\rm recomb.}$ with time and the relaxation of $n({\rm
H}^+)$ from a fully ionized condition to $n({\rm H}^+)_{\rm eq.}$ for $n({\rm
H_{tot}})=0.50\,{\rm cm}^{-3}$ and $T=7000\,$K is illustrated in
Fig.~\ref{ioniz_decay}.  Values for $t_{\rm recomb.}$ start at 0.13$\,$Myr
for an initial $n({\rm H}^+)=0.50\,{\rm cm}^{-3}$ and increase to 1.0$\,$Myr
when $n({\rm H}^+)$ reaches $0.036\,{\rm cm}^{-3}$ at $t=1.77\,$Myr, beyond
which $t_{\rm recomb.}$ very gradually climbs toward a steady value of
1.68$\,$Myr and the subsequent decay in $n({\rm H}^+)$ toward $n({\rm
H}^+)_{\rm eq.}=0.015\,{\rm cm}^{-3}$ is almost purely exponential.

\begin{figure}[h!]
\epsscale{1.0}
\plotone{ioniz_decay.eps}
\caption{The behavior of $t_{\rm recomb.}$ (upper solid curve) and $n({\rm
H}+)$ (lower solid curve) with time $t$, starting from a fully ionized
condition to an equilibrium state $n({\rm H}+)_{\rm eq.}$ for $n({\rm
H_{tot}})=0.50\,{\rm cm}^{-3}$ and $T=7000\,$K. Values of $t_{\rm recomb.}$
are defined by Eq.~\protect\ref{t_recomb}, and $n({\rm H}+)$ as a function of
$t$ can be found by solving the differential equation for $dn({\rm H}+)/dt$
given in Eq.~\protect\ref{dnpdt}.  By solving a similar equation for Ar$^+$
and comparing it to that for H$^+$, we obtain the trend for [Ar~I/H~I] shown
by the dashed line.\label{ioniz_decay}}
\end{figure}

Of the three different classes of internal sources, WD stars emit
the softest radiation, which means the influences of their ionizations can be
more sharply bounded than those of the others.  While this consideration may
present a challenge to our assumption about the uniformity of ionization in
space, we note that these stars have observed radial velocity dispersions of
about $25\,{\rm km~s}^{-1}$ (Falcon et al. 2010), or rms speeds of
$43\,{\rm km~s}^{-1}$, which are consistent with the transverse velocity
dispersion measurements reported by Wegg \& Phinney (2012).  This indicates
that WDs typically move about 76$\,$pc during the time interval that
$n({\rm H}^+)$ decays from 0.50 to $0.036\,{\rm cm}^{-3}$.  This amount of
travel is considerably larger than the radius of a Str\"omgren Sphere
\begin{equation}\label{r_S}
r_{\rm S}=\left( {3r^2_*\int_{13.6\, {\rm eV}}^\infty F(E)_{\rm WD}dE\over
n({\rm H})^2\alpha({\rm H}^0,T)}\right)^{1/3}
\end{equation}
that would be established if a WD star were stationary.  (Recall
that $F(E)_{\rm WD}$ is the flux emitted at the surface of the star with a
radius $r_*$.)  A typical value for $r_{\rm S}$ in the WNM would be about
1.8$\,$pc; this value applies to a WD star with $r_*=0.014r_\odot$
that has $T_{\rm eff}=50,000\,$K and is situated within a gas with a density
$n({\rm H_{tot}})=0.50\,{\rm cm}^{-3}$.

One can view the dispersal of ionization caused by the star's motion in the
context of the simplified description of ``cometary H~II regions'' described
by Raga (1986); the dimensionless elongation parameter in this
development $\beta=3v/[n({\rm H_{tot}})\alpha({\rm H}^0,T)r_{\rm S}]=14$ if
$r_{\rm S}=1.8\,$pc and $v=43\,{\rm km~s}^{-1}$, which would result in a tube
of ionization with a radius $\sim 0.5r_{\rm S}$ and a very long tail.  The
fact that the ionizations produced by WD stars could be highly
diluted because they are distributed over large volumes could explain why a
sensitive survey of H$\alpha$ emission from regions around such stars carried
out by Reynolds (1987) yielded only one detection out of
nine targeted regions surrounding these stars.

\section{PROPOSALS FOR ADDITIONAL
PHOTOIONIZATION}\label{additional_photoionization}

Given that the outcome for a reasonable value of $n({\rm H}_{\rm tot})$ and
the ionization rate from known sources discussed in
Section~\ref{known_photoionization} do not agree with our observations, we
must explore some alternatives for boosting the average rate of
photoionization in the WNM.  Two possibilities are discussed in the
following sections.

\subsection{Ionization from Previous Supernova Remnants}\label{previous_SNR}

A popular theme in the ISM literature of early 1970's was the consideration
that the low density portions of the ISM were ionized impulsively by UV and
X-ray radiation from supernovae and their remnants (SNRs), which represented
sources of ionization that had limited durations and that materialized at
random locations and times (Bottcher et al. 1970; Werner et al. 1970; Jura \&
Dalgarno 1972; Gerola et al. 1973; Schwarz 1973).  A particularly instructive
example that shows the wide variations in temperature and fractional
ionization can be seen in the results from a numerical simulation by Gerola
et al. (1974).  This topic has also been approached by Slavin et al. (2000),
who estimated the effects of irradiation of the neutral medium by SNRs, with
a special emphasis on its influence on gas in the Galactic halo. The spectral
character and strength of the emitted radiation from a remnant depends on a
combination of supernova energy and the density of the ambient medium
(Mansfield \& Salpeter 1974).  The emission of soft X-rays as a remnant
evolves can last up to a late phase when radiative cooling causes a dense
shell to form, which could be opaque to the lowest energy X-rays.  However
some of the radiation may continue to escape afterward if instabilities
cause the shell to break up and create gaps (Vishniac 1994; Blondin et al.
1998).  The magnitudes of such instabilities are uncertain, since they might
be diminished if either the ambient medium has a low density and high
temperature (Mac Low \& Norman 1993) or if the shock is partly stabilized by
an embedded transverse magnetic field
(T\'oth \& Draine 1993).

Chevalier (1974) has computed the strength and
distribution over energy for the radiation emitted by various models for
SNRs.  As a representative example, we can use the emission
over a time interval of $2.5\times 10^5\,{\rm yr}$ from his model that had an
energy $E_{\rm SN}=3\times 10^{50}\,{\rm erg}$ and was expanding into a
medium with an average density of $1\,{\rm cm}^{-3}$.  If we assume that
there was a succession of about 3 such remnants that occurred once every Myr
(i.e., of order $t_{\rm recomb.}$ defined in Eq.~\ref{t_recomb} for $n({\rm
H}^+)=0.036\,{\rm cm}^{-3}$; see Fig.~\ref{ioniz_decay}) and they were at a
distance of 100$\,$pc from some location in the ISM, the resulting
time-averaged flux shown by the dashed line in the upper panel of
Fig.~\ref{flux_plot} would increase the total ionization of H from
$\Gamma({\rm H}^0)=3.2\times 10^{-16}{\rm s}^{-1}$ for the steady ionization
sources to the much higher value of $9.2\times 10^{-16}{\rm s}^{-1}$ and
create conditions that would yield values for [Ar~I/O~I] that are virtually
the same as those depicted by the curve labeled $n({\rm H}_{\rm
tot})=0.14\,{\rm cm}^{-3}$ in Fig.~\ref{sdosdb_plot}, but in this case for a
density $n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$.  At $T=7000\,$K, the
densities of various constituents are represented by the numbers that are
shown in Column (3) of Table~\ref{properties_table}.  For this temperature,
the calculation of [Ar~I/O~I] is just consistent with the upper error bound
from the observations.

A stronger flux from SNRs would be needed to make the
calculated value for [Ar~I/O~I] match nominal value for the observed ones.
To obtain a curve that matches the lowest curve (labeled $n({\rm H}_{\rm
tot})=0.09\,{\rm cm}^{-3}$) in Fig.~\ref{sdosdb_plot} and still maintain an
assumed local density $n({\rm H}_{\rm tot})=0.50\,{\rm cm}^{-3}$, we would
require either an increase in the average energy of the supernovae (SNe), or the
rate would need to be increased from 3 to 8~SN~${\rm Myr}^{-1}$ at a distance
of 100$\,$pc.  In this circumstance, $n(e)=0.047\,{\rm cm}^{-3}$.

While one might be tempted to ask whether some average SN rate in our
location within the Galaxy (van den Bergh \& McClure 1994; McKee \& Williams
1997; Ferri\`ere 2001) is consistent with at least 3 SNe Myr$^{-1}$ at
separations of about 100$\,$pc from some representative location, it is
probably more realistic to draw upon actual estimates of the recent history
of explosions in the general neighborhood of the Sun, which have been
inspired by the evidence of how the ISM has been disrupted by the expansions
of the explosion remnants.  The Local Bubble (see footnote 2 in
Section~\ref{strategy}) and the neighboring radio emitting Loop~I
superbubble\footnote{Loop~I is coincident with the North Polar Spur of X-ray
radiation. Its center is estimated to be only 180 pc away from us  (Bingham
1967).}, which intersect one another (Egger \& Aschenbach 1995), are both
thought to have been created by a series of SN explosions that
occurred over the past 10-15$\,$Myr
(Ma\'iz-Apell\'aniz 2001; Bergh\"ofer \& Breitschwerdt 2002).  Fuchs
et al. (2006) analyzed the numbers and mass functions of stars within nearby
O-B associations and traced their motions back in time.  From this
information, they concluded that the Local Bubble was created by 14-20
SNe.  This estimate is consistent with the 19 SNe that
Breitschwerdt \& de~Avillez (2006) used in their hydrodynamic simulation of
the creation of the Local Bubble.

Within a shorter time frame, there is some evidence that atoms from a nearby
SN were deposited on the Earth some 2.8$\,$Myr ago, as revealed by an
enhancement of $^{56}$Fe in a thin layer within deep-sea ferromanganese crust
 (Knie et al. 2004).  Fitoussi et al. (2008)
attempted to replicate this result in a marine sediment, but the outcome did
not agree with the earlier result at the same level of significance.
However, there are alternative means for investigating past depositions of
SN elements in terrestrial samples (Bishop \& Egli 2011; Feige et al.
2012), and new results may emerge soon.

If we add to the Local Bubble contribution the past SNe in the Sco-Cen
association that created Loop~I (Iwan 1980; Egger 1998; Diehl et al. 2010),
one can imagine that, to within the uncertainties in the X-ray production
rates, the time-averaged level of ionizing radiation could conceivably be of
the same order of magnitude as that discussed in the preceding paragraphs.

A legitimate question to pose is whether or not, at some intermediate stage
of recombination, the value of [Ar~I/H~I] makes a brief excursion to a level
below the final equilibrium state.  If this were the case, we might be able
to relax the requirement for a high frequency of SN explosions.  The
recombination rates $\alpha({\rm H}^0,T)$ and $\alpha({\rm Ar}^0)$ are nearly
equal to each other, but the dust grain recombination rate $\alpha_g$ for Ar
should be much less than that for H because the thermal velocities of Ar are
lower.  Thus, at early times where recombinations still dominate over the
steady-state ionization rate but the electron density has decreased to a
point that neutralization of Ar and H by collisions with dust grains become
important, one could imagine that the difference in grain recombination rates
might be influential in allowing the H$^+$ to recombine more quickly than
Ar$^+$.  In order to investigate this possibility, we can evaluate the time
history of the Ar recombination using Eqs.~\ref{t_recomb} and \ref{dnpdt} but
with a substitution of ``Ar'' for each appearance of ``H.''  A resulting
comparison of the Ar recombinations to those of H shows that [Ar~I/H~I] shows
a steady decrease from an initial value slightly greater than 1.0 (when the
gas is just starting to recombine) to the final equilibrium state for this
quantity without any excursion to lower values.  This behavior is illustrated
by the dashed line in Fig.~\ref{ioniz_decay}.

\subsection{Penetration of the WNM by Low-energy Photons}\label{penetration}

Until now, we have regarded the emission of X-rays from the hot ($T\sim
10^6\,$K) gas in our Galaxy as an external source of ionizing radiation that
must penetrate the bulk of the WNM regions under study.  With this condition
in mind, we established a 90$\,$eV cutoff for the external radiation, below
which the X-rays were assumed to be absorbed in the outer layers of the
neutral gas.  However, some of the soft X-ray emitting gas may be threaded
through the WNM, a picture that is reminiscent of the network of hot
interstellar tunnels proposed by Cox \& Smith (1974).
Hot, interspersed gas could allow some or all of the neutral medium to have
access to lower energy radiation.  Observations indicate that O~VI, an
indicator of gas that is collisionally ionized at $T\sim 3\times 10^5\,$K,
can be seen in emission (Dixon et al. 2006; Otte \& Dixon 2006) and absorption
(Jenkins 1978a, b; Bowen et al. 2008) almost everywhere in the disk of the
Galaxy.  The simulations of SNe exploding at random in the Galactic
disk by Ferri\`ere  (1995) and de~Avillez \& Breitschwerdt (2005a, b) indicate
that this hot gas should be sufficiently pervasive and frothy that, even
though our measurements penetrate through column densities well in excess of
$10^{19}\,{\rm cm}^{-3}$, individual parcels of gas may be exposed to
adjacent sources of radiation through significantly lower column densities.
Indeed, there could be unusually strong X-ray emission at the boundaries
where the neutral regions come into contact with the hot gas, where charge
exchange reactions between neutrals and highly ionized species could produce
an enhancement of soft X-ray emission over that which is emitted by just the
hot gas (Wang \& Liu 2012).

It is clear from the gray parts of the external radiation spectrum depicted
in Fig.~\ref{flux_plot} that even a modest lowering of the cutoff below our
adopted 90$\,$eV threshold will create a significant increase in the
ionization of the WNM.  For instance, if the threshold were dropped to
60$\,$eV, an energy just below the strong M-line complex of emission features
from highly ionized Fe at 70$\,$eV, the level of ionization would increase to
an extraordinarily large total hydrogen ionization rate $\Gamma({\rm
H}^0)=2.1\times 10^{-15}{\rm s}^{-1}$.  For $n({\rm H}_{\rm tot})=0.50\,{\rm
cm}^{-3}$ and $T=7000\,$K, this increase would create a value for [Ar~I/O~I]
that is consistent with the upper error bound for the measured outcomes.  In
this particular case, most of the ionization is caused by photons with
energies near 70$\,$eV.  For this energy, $P_{\rm Ar}(E)$ is near its minimum
value (see the lower panel of Fig.~\ref{p_ar_plot} and the entry in Column
(4) of Table~\ref{properties_table}).  As a consequence, in order to match
our observations, Eq.~\ref{ar_h} shows us that the required $n({\rm H}^+)$
(and thus $\Gamma({\rm H}^0)$) must be increased beyond the value needed from
the time-averaged SNR radiation with its larger value of
$P_{\rm Ar}$.

A consideration that disfavors the presence of a strong ionizing flux at
energies in the vicinity of 70$\,$eV, at least at our location, is that it
does not seem to be present at anywhere near the intensity level shown in
Fig.~\ref{flux_plot}.  For instance, from observations by the {\it Cosmic Hot
Interstellar Plasma Spectrometer\/} ({\it CHIPS\/}) Hurwitz et al.
 (2005) stated an upper limit for the flux emitted by
the Fe lines near 70$\,$eV.  This limit was consistent with an emission
measure for a plasma with a solar abundance pattern at $T=10^6\,$K that is
less than 5\% of the value ${\rm EM}=10^{16.37}{\rm cm}^{-5}$ that we used
for reconstructing the emission spectrum from the soft component of the X-ray
background described in Section~\ref{external_radiation}.  Jelinsky et al.
 (1995) and Bloch et al. (2002)
similarly found flux upper limits at low energies (but not as stringent as
those from {\it CHIPS\/}) that were much lower than model predictions at this
energy that we obtained from fits to the observed radiation at higher
energies.  A marginal detection of the Fe emission complex toward high
Galactic latitudes by McCammon et al. (2002),
$F(E)=100\pm 50\,{\rm phot~cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}$ is also well
below the peak at 70$\,$eV that appears in our reconstructed flux.  The
weakness of these fluxes could be caused either by a deficiency of Fe below
the solar abundance ratio in the hot emitting gases or by our inability to
see hot gas regions whose low energy radiation is not absorbed by intervening
neutral material.

\section{DISCUSSION}\label{discussion}

We have now reached a point where it is appropriate to investigate the
consistency of the newly derived WNM ionization levels with other
observational and theoretical findings, along with some consequences of our
results on various relevant physical processes.

\subsection{Other UV Observations}\label{other_obs}

Along a number of sight lines, there have been detailed investigations of UV
absorption lines observed at high enough velocity resolutions to identify
which components came from either mostly neutral or mostly ionized clouds
 (Spitzer \& Fitzpatrick 1993, 1995; Fitzpatrick \& Spitzer 1994, 1997; Wood
\& Linsky 1997; Holberg et al. 1999; Welty et al. 1999; Jenkins et al. 2000a;
Gry \& Jenkins 2001; Sonnentrucker et al. 2002).  These investigations relied
either on the relative populations of the excited fine-structure level of
${\rm C}^+$ or the ratios
of ions to neutrals for various elements.
The ion-to-neutral ratios of different elements gave very different outcomes
for $n(e)$ (but systematically went up and down together from one velocity
component to the next); these disparities probably arose as a result of a
lack of a good understanding of the physical processes involved.  Welty et
al. (2003) have presented a good summary of the results
that exhibited this problem.

Values of $n(e)$ identified with the WNM by the investigators cited above
generally ranged between 0.04 and $0.12\,{\rm cm}^{-3}$, which seem to be
higher than our values given in Table~\ref{properties_table}.  A common theme
in the discussions of these results was that such high partial ionizations
could not be attributed to the known EUV, X-ray and cosmic ray ionization
rates -- a conclusion that is stated once again from our [Ar~I/O~I] results.

Electron densities for clouds embedded in the Local Bubble have an average
value of $0.11\,{\rm cm}^{-3}$ (Redfield \& Falcon 2008), a result
that once again relied on comparisons of excited C$^+$ to other species
(either unexcited C$^+$ or S$^+$).  The fractional ionization of these clouds
appears to be much higher than what we found for the general WNM outside the
Local Bubble, if one assumes that the characteristic total density $n({\rm
H}_{\rm tot})\approx 0.2\,{\rm cm}^{-3}$ within the clouds (Redfield \&
Linsky 2008). By solving for the time-dependent ionization in Eq.~\ref{dnpdt}
for $n({\rm H_{tot}})=0.2\,{\rm cm}^{-3}$, we find that the decay from a
fully ionized condition to an approximately half ionized state takes only
0.40$\,$Myr.  It is likely that this higher level of ionization might be
explained by supplemental radiation from evaporative boundaries that surround
the clouds (Slavin \& Frisch 2002), if indeed they are embedded in a very
hot, low density gas, or by the infiltration of some ionizing radiation from
$\epsilon$~CMa, which strongly dominates over other radiation sources within
the Local Bubble at low energies (Vallerga \& Welsh 1995).

\subsection{Pulsar Dispersion Measures}\label{pulsar_DM}

Cordes \& Lazio (2002) have derived characteristic
electron densities for three volumes within about 1$\,$kpc of the Sun.
Inside the Local Bubble, their model seems to best fit an average density of
only $\langle n(e)\rangle=0.005\,{\rm cm}^{-3}$, which is not surprising
because much of the volume has probably been cleared of material by SN
explosions, except for some isolated, warm clouds with an average filling
factor in the range $5.5-19$\% (Redfield \& Linsky 2008).  Regions
outside the Local Bubble have higher densities: two volumes that they studied
yield $\langle n(e)\rangle=0.012$ and $0.016\,{\rm cm}^{-3}$.  However, these
regions are identified with ellipsoidal volumes designated as either a
``local superbubble'' (LSB) or a ``low density region'' (LDR), so they too
could have significant voids that would dilute the apparent electron
densities.  The values of $\langle n(e)\rangle$ given for these regions could
be consistent with our measures of $n(e)$ given in the last two columns of
Table~\ref{properties_table} for the WNM if this medium had a filling factor
of about 1/3 in these regions and there were no significant contributions
from the warm ionized medium (WIM).

An independent analysis of dispersion measures was carried out by de Avillez
et al. (2012) for 24 pulsars with known distances
between 0.2 and 8$\,$kpc from the Sun and with $\vert z\vert<0.2\,$kpc. They
found a distribution of $n(e)$ outcomes that was consistent with a log-normal
distribution centered on $\log n(e)=-1.47$ and a dispersion $\sigma[\log
n(e)]=0.17$.  The fact that their representative values of $n(e)$ are higher
than the determination of Cordes \& Lazio (2002)
and close to our findings based on [Ar~I/O~I] may be accidental, since their
results may be strongly influenced by sight-line interceptions of fully
ionized regions.

\subsection{Emission Lines}\label{emission_lines}

Observations of diffuse line emissions in the sky most generally apply to
probing the physical conditions in fully ionized gases, either the bright
H~II regions around hot stars or the WIM
(Reynolds et al. 1977, 2002; Haffner et al. 1999; Madsen et al. 2006).  Most
of the emissions are dominated by contributions from species that are
expected to be abundant in such highly ionized media, and they should
overwhelm any contributions from the very dilute ionization in the WNM.
Nevertheless, there are a few cases where emission line fluxes from some
neutral species in the WIM are expected to be very weak, and thus a
contribution from the WNM might in principle be identified.  We need to
explore whether or not the predictions for line strengths from a medium with
our enhanced electron densities are not in serious violation of detections or
the upper limits for the fluxes.  We explore three such cases in the
subsections that follow.  The first test involving line emission from He
recombinations is especially important, because we predict that the fraction
of singly-ionized He could be as high as 16$-$28\% of the total amount of
helium.

\subsubsection{He $\lambda$5876 Recombination Radiation}\label{He_recomb}

Reynolds \& Tufte (1995) attempted to compare
the strength of the recombination line of He$^0$ at 5876$\,$\AA\ to that of
H$\alpha$ in parts of the sky away from well defined H~II regions. Their
motive was to determine the hardness of the radiation that maintains the
ionization in the WIM. While an explicit upper limit for the He recombination
line flux was not stated by Reynolds \& Tufte (1995), we estimate that the
flux in the two directions that they sampled was found to be less than 0.1$\,
$R (R~=~Rayleigh~=~$10^6/(4\pi) {\rm phot~cm}^{-2}{\rm sr}^{-1}{\rm s}^{-
1}$).

Using the line emission rates given by Benjamin et al. (1999) we find that
for $n(e)=0.04\,{\rm cm}^{-3}$, $n({\rm He}^+)=0.01\,{\rm cm}^{-3}$, and $T=
7000\,$K (representative values for the enhanced ionization cases presented
in the last two columns of Table~\ref{properties_table}) the emission should
be $3.0\times 10^{-17}{\rm phot~cm}^{-3}{\rm s}^{-1}$.  If we were to
propose that the emission is seen over a path of 200$\,$pc with no
extinction (i.e., imagine a length of 400$\,$pc with a filling factor for the
WNM of 50\%), we could expect to find an emission of $1500\,{\rm phot~cm}^{-
2}{\rm sr}^{-1}{\rm s}^{-1} = 0.02\,$R.  This expectation is well below the
sensitivity of the observations by Reynolds \& Tufte (1995), so our predicted
intensity from the enhanced electron density and ionization of He does not
violate their upper limit.

\subsubsection{Emission of [O~I] $\lambda$6300 from Electron
Collisions}\label{OI_excitation}

In order to determine the neutral fraction of H in the WIM, Reynolds et al.
 (1998) measured the strength of the [O~I]
$\lambda$6300 line emitted in parts of the sky that had a uniform, moderately
strong H$\alpha$ emission (but away from obvious H~II regions excited by
stars), much as they had done for the He recombination line discussed above.
In three different directions, they detected intensities of 0.2, 0.09, and
0.11$\,$R.

According to Federman \& Shipsey (1983),
electron collisions dominate over those by hydrogen for the excitation of the
$^1D_2$ state of neutral oxygen when the electron fraction exceeds
$1.5\times 10^{-4}$.  Hence, we can ignore hydrogen impacts.  From the
fitting formula of P\'equignot (1990) to the collision strengths computed
by Berrington \& Burke (1981), we derive a
collisional rate constant $C_e$ for electron excitations to be $8.0\times
10^{-11}{\rm cm}^3{\rm s}^{-1}$ at $T=7000\,$K.  If we again make the
conservative assumption expressed in Section~\ref{reference_abund} that at
the low densities of the WNM the depletion of O is negligible, we expect that
$n({\rm O}^0)=2.7\times 10^{-4}{\rm cm}^{-3}$ if $n({\rm H}^0)=0.47\,{\rm
cm}^{-3}$.  For $n(e)=0.04\,{\rm cm}^{-3}$, we expect the emissivity to be
equal to $n({\rm O}^0)n(e)C_e$ multiplied by a branching fraction 0.76
 (Froese Fischer \& Tachiev 2010) for the proportion of decays from
the $^1D_2$ level to the lower $^3P_2$ level.  This product
equals $6.6\times 10^{-16}{\rm phot~cm}^{-3}{\rm s}^{-1}$, which should
produce 0.41$\,$R over a path of 200$\,$pc.  This value is greater than the
three measurements by Reynolds et al. (1998).  The
magnitude of this violation is not large, considering the uncertainties of
our assumptions about the lack of depletion of O and the adopted path length
estimate.  Also, for temperatures less than 7000$\,$K, the expected strength
of the emission will be considerably less: for instance, at $T=5000\,$K, the
emission should be 4.5 times weaker than at 7000$\,$K.

\subsubsection{Emission of [N~I] $\lambda$5201 from Electron
Collisions}\label{NI_excitation}

An upper limit of 0.13$\,$R for the [N~I] $\lambda$5201 line was determined
for a single direction in the sky by Reynolds et al. (1977).  If we adopt
the same calculations as in the previous section for O~I, but make the
substitution that $n({\rm N}^0)=8.0\times 10^{-5}n({\rm H}^0)=3.8\times
10^{-5}{\rm cm}^{-3}$ (again, assuming no depletion), and $C_e=8.6\times
10^{-11}{\rm cm}^3{\rm s}^{-1}$ at $T=7000\,$K (Tayal 2006), we obtain an
emissivity equal to $1.3\times 10^{-16}{\rm phot~cm}^{-3}{\rm s}^{-1}$,
which yields an intensity of 0.08$\,$R over a 200$\,$pc path.  This value is
below the upper limit determined by Reynolds et al. (1977).

\subsection{Cooling Rates from Carbon Ions and Oxygen
Atoms}\label{cooling_rate}

A major coolant for the neutral ISM is the singly charged carbon ion
 (Dalgarno \& McCray 1972; Wolfire et al. 1995, 2003), whose $^2P_{3/2}$
excited fine-structure level in the ground electronic state can be excited
by collisions with electrons and hydrogen atoms.  The cross section for
excitation by electrons is substantially greater than that for neutral
hydrogen.  For this reason, the level of partial ionization is an important
factor in the excitation rate.  After excitation, an energy loss can occur
because the excited level can undergo a spontaneous radiative decay with a
transition probability $A_{21}=2.29\times 10^{-6}{\rm s}^{-1}$ to the lower
$^2P_{1/2}$ state (Nussbaumer \& Storey 1981), a process that liberates a
photon with a wavelength of $158\,\mu$m.  If we assume that the abundance
ratio $({\rm C}^+/{\rm H}^0)=9.5\times 10^{-5}$ (see footnote~1 in
Section~\ref{intro}), we can use the densities that we derived for $n({\rm
H}^0)$ and $n(e)$ along with the excitation cross section by H$^0$ impacts
computed by Barinovs et al.  (2005) and electron collision strengths of
Wilson \& Bell (2002) to compute the ${\rm C}^{+*}$ energy
loss rates $\ell_c$ per H atom.  These rates are given in the last row of
Table~\ref{properties_table}.  They do not change in direct proportion to
$x_e$ because about half of the excitations come from collisions with H
atoms, which are more numerous than the electrons.  In addition, the overall
cooling rate for the WNM does not increase in direct proportion to $\ell_c$
because it accounts for only about one-third of the total cooling.

Most of the remaining cooling comes from the fine-structure excitation of O~I
and the subsequent emissions at 44 and $63\,\mu$m. The O~I cooling is
insensitive to changes in $x_e$ because the collision rate constants for
electrons (Bell et al. 1998) are much less than those for neutral
hydrogen.  Using an extrapolation of the atomic hydrogen collisional rate
constants given by Abrahamsson et al. (2007)
above a temperature of $10^3\,$K, the spontaneous decay rates of the two
excited levels given by Galav\'is et al. (1997), and ${\rm O/H}=5.75\times
10^{-4}$, we find that for $n({\rm H}^0)=0.47\,{\rm cm}^{-3}$ and $T=7000\,$K
that the energy loss rate per H atom $\ell_o=1.13\times 10^{-26}{\rm
erg~s}^{-1}$.

The results for $\ell_c$ shown in the table are significantly lower than the
average value of $2\times 10^{-26}{\rm erg~s}^{-1}{\rm H~atom}^{-1}$ found
for low-velocity clouds by Lehner et al. (2004), who
measured the column densities of C~II$^*$ from spectra recorded by {\it
FUSE}.  [Lehner et al. (2003) obtained similar results
for sight lines toward WD stars within or just outside the Local
Bubble.]  Lehner et al. (2004) compared their results
with measurements of H$\alpha$ emission in the same directions, and they
concluded that about half of the C~II$^*$ that they detected came from fully
ionized gas (but with large variations from one sight line to the next).
This inference was supported by the fact that their values of $\ell_c$ were
slightly lower for sight lines with large values of $N$(H~I).  In principle,
one can gain an insight on the relative importance of H~II regions by
comparing the observed abundances of C$^{+*}$ to those of N$^{+*}$, since the
former can come from both neutral and ionized regions, while the latter
arises only from ionized regions.  Gry et al. (1992)
compared these two species in a comprehensive study of both the absorption
lines observed by the {\it Copernicus\/} satellite (Rogerson et al. 1973) and
the 158 and 205$\mu$m emission lines observed by the {\it COBE\/}
satellite (Wright et al. 1991).  They concluded that H~II regions were
responsible for a major portion of the C$^{+*}$ that was observed, but this
result is clearly dependent on the assumed ratio of atomic C to N in the H~II
regions.

An important advantage of the $\ell_c$ determinations synthesized from our
results for [Ar~I/O~I] is that they apply {\it only\/} to the WNM; hence they
give more accurate indications of the carbon cooling rates within this medium
without any contamination from H~II regions.  They do, of course, rely on the
value of assumed relative abundances of C and H in the gas phase.

\subsection{Heating Rates}\label{heating_rates}

\subsubsection{Thermal Time Constants}\label{thermal_time_constants}

The cooling time for a medium with $T=7000\,$K and thermal pressure
$nT=3800\,{\rm cm}^{-3}$K (Jenkins \& Tripp 2011) is about
4.1$\,$Myr [ (Wolfire et al. 2003); see their Eq.~(4)].  However, when
the medium is impulsively heated and ionized by the EUV and X-ray
illuminations from a SNR, the temperature can approach or exceed $10^4\,$K,
and the onset of L$\alpha$ cooling creates a dramatic increase in the overall
cooling rate (Dalgarno \& McCray 1972).  When this happens, the
thermal relaxation timescale becomes much shorter than the mean interval
between the bursts of radiation, each of which last only about one to a few
times $10^5\,$yr.  Thus, while we can compute a time-averaged heating rate
for the SNR illuminations that might explain our high levels of partial
ionization, we have no reason to expect that this average should be balanced
by the cooling rates that would apply for the medium at $T=7000\,$K.

\subsubsection{Secondary Electrons}\label{secondary_heating}

The same energetic electrons that are responsible for the secondary
ionizations $\Gamma_s$ and $\Gamma_{s^\prime}$ can also heat the gas through
collisions with other electrons.  As with the calculations of the efficiency
of secondary ionizations described in Appendix~\ref{gamma_s}, we use the
analytic approximations of Ricotti et al. (2002) for
the heating efficiencies, based on the numerical results for various
conditions obtained by Shull \& Van Steenberg (1985).  Aside from replacing
ionization efficiencies with heating efficiencies and multiplying by the
energies of the respective secondary electrons, the calculations here are
virtually the same as for the ionization rates.

Table~\ref{heating_rates_table} shows the outcomes for the evaluations of
secondary electron heating rates.  It is no surprise that there is a
substantial increase in the heating rates when we advance from the ionization
created by known sources to either of the two hypothetical enhanced
ionization examples that could explain the electron fractions indicated by
the observations of [Ar~I/O~I].

Our high result ($1.2\times 10^{-25}{\rm erg~s}^{-1}{\rm H~atom}^{-1}$) for
the heating in the regime where there is increased penetration of X-rays down
to 60$\,$eV creates a serious problem for this model, since this steady-state
rate is considerably larger than the corresponding cooling rate
$\ell_c+\ell_o=1.98\times 10^{-26}\,{\rm erg~s}^{-1}{\rm H~atom}^{-1}$ given
in Section~\ref{cooling_rate}.

\subsubsection{Cosmic Ray Heating of Electrons}\label{CRheating_elec}

In addition to ionizing the gas and creating secondary electrons that can
heat the gas, cosmic rays can also interact with free electrons in the medium
and heat them.  The heating rate per unit volume is approximately equal to
$A\zeta_{\rm CR}({\rm H}^0)n(e)$, where $A\approx 4.6\times 10^{-10}{\rm
erg}$
(Draine 2011, p. 338). With the electron densities listed in
Table~\ref{properties_table} and $\zeta_{\rm CR}=1.25\times 10^{-16}{\rm
s}^{-1}{\rm H~atom}^{-1}$, we find that for the three different electron
densities in this table the heating rates are $1.0\times 10^{-27}$ (for no
SNR contribution), $1.9\times 10^{-27}$ (with SNR contributions), and
$3.0\times 10^{-27}{\rm erg~cm}^{-3}{\rm s}^{-1}$ (for the lower energy
penetration example).  After dividing these numbers by values of $n({\rm
H}^0)$ given in Table~\ref{properties_table}, we find that these rates are
about 0.4 to 1.2 times the rate from secondary electrons liberated by the
cosmic rays, and 0.05 to 0.5 times the respective heating rates from the
secondary electrons generated by ionizations from the X-ray backgrounds.

\begin{deluxetable}{
l
c
}
\tablewidth{0pt}
\tablecaption{Secondary Electron Heating
Rates\tablenotemark{a}\label{heating_rates_table}}
\tablehead{
\colhead{} & \colhead{Rate}\\
\colhead{Source\tablenotemark{b}} & \colhead{($10^{-27}{\rm erg~s}^{-1}\,{\rm
H~atom}^{-1}$)}
}
\startdata
X-ray background (no SNR)\dotfill&4.1\\
X-ray background (plus SNR)\dotfill&50\\
X-ray background (down to 60$\,$eV)\dotfill&110\\
Main-sequence stars\dotfill&1.5\tablenotemark{c}\\
Active binaries\dotfill&0.063\tablenotemark{c}\\
WDs\dotfill&0.84\tablenotemark{c}\\
Cosmic rays\dotfill&4.9\tablenotemark{c}\\
He$^+$ recomb.\tablenotemark{d} (background with no SNR)\dotfill&0.059\\
He$^+$ recomb.\tablenotemark{d} (background plus SNR)\dotfill&1.2\\
He$^+$ recomb.\tablenotemark{d} (background down to 60$\,$eV)\dotfill&8.8\\
\enddata
\tablenotetext{a}{Heating by secondary electrons that are liberated by the
ionizations of H$^0$, He$^0$ and He$^+$, expressed in terms of an energy
dissipation rate per neutral H atom.  Direct interactions of cosmic rays with
free electrons produce an additional heating which is discussed in
Section~\protect\ref{CRheating_elec}.}
\tablenotetext{b}{The conditions for the top and bottom three rows correspond to those
for the last three columns in Table~\protect\ref{properties_table}.}
\tablenotetext{c}{To first order, the heating rates from internal sources and
cosmic rays should not change as the strength of the background radiation
increases above the basic rate from known sources.  In
reality, they increase by modest amounts ($\sim$30\%) when the values of
$x_e$ (which drive the heating efficiency) increase.}
\tablenotetext{d}{From secondary electrons that are produced by
$\Gamma_{He^+}({\rm He}^0)$ and $\Gamma_{He^+}({\rm H}^0)$.  Other
ionizations arising from helium recombinations do not produce electrons with
sufficient energy to cause any appreciable heating.}
\end{deluxetable}
\begin{deluxetable}{
l
c
c
c
c
c
c
c
c
c
c
c
}
\tablecolumns{12}
\tablewidth{0pt}
\tablecaption{Dust Grain Heating and Cooling
Rates\tablenotemark{a}\label{grain_rates_table}}
\tablehead{
\colhead{} & \multicolumn{3}{c}{Without SNR} & \colhead{} &
\multicolumn{3}{c}{With SNR} & \colhead{} & \multicolumn{3}{c}{Greater Low}\\
\colhead{$b_c$\tablenotemark{b}} &
\multicolumn{3}{c}{Contrib.\tablenotemark{c}} & \colhead{} &
\multicolumn{3}{c}{Contrib.\tablenotemark{c}} & \colhead{} &
\multicolumn{3}{c}{Energy Penetration\tablenotemark{c}}\\
\cline{2-4} \cline{6-8} \cline{10-12}
\colhead{$\times 10^5$} & \colhead{H} & \colhead{C} & \colhead{$\Delta$} &
\colhead{} & \colhead{H} & \colhead{C} & \colhead{$\Delta$} & \colhead{} &
\colhead{H} & \colhead{C} & \colhead{$\Delta$}\\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{} &
\colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{} & \colhead{(8)} &
\colhead{(9)} & \colhead{(10)}
}
\startdata
0.0&9.9&5.1&4.8&~&16.1&7.5&8.5&~&22.0&10.0&12.0\\
2.0&15.1&7.3&7.8&~&23.2&11.0&12.1&~&30.7&14.9&15.8\\
4.0&24.4&10.6&13.9&~&36.4&16.0&20.4&~&46.9&21.8&25.1\\
6.0&31.6&13.8&17.8&~&47.4&20.9&26.5&~&61.3&28.5&32.9\\
\enddata
\tablenotetext{a}{Expressed in units of $10^{-27}{\rm erg~s}^{-1}\,{\rm
H~atom}^{-1}$.  H~=~heating, C~=~cooling, and $\Delta={\rm H-C}$ (i.e., net
heating rate).  The values were computed for the average interstellar
radiation field (ISRF), as defined by Eq.~31 and Table~1 of Weingartner \&
Draine (2001b) (with a radiation intensity
$G=1.13$).  The adopted value of $R_V\equiv A_V/E(B-V)=3.1$.  The values for
the heating and cooling rates per H atom tabulated here were computed from
Eq.~44 (for grain heating) and Eq.~45 (for grain cooling) of Weingartner \&
Draine (2001b) (but without the $n({\rm H})$
factor) and the coefficients listed in their Tables~2 and 3.}
\tablenotetext{b}{The abundance of carbon, relative to hydrogen, in the
grains.  A high value of $b_c$ implies a population of dust grains that is
rich in polycyclic aromatic hydrocarbons (PAHs).}
\tablenotetext{c}{The same conditions as for Columns (2)-(4) in
Table~\protect\ref{properties_table}.}
\end{deluxetable}

\subsection{Dust Grains: Photoelectric Heating and Recombination
Cooling}\label{photoelectric_grain_heating}

When dust grains are illuminated by starlight, they emit photoelectrons,
which can heat the medium (Watson 1972; Draine 1978; Pottasch et al. 1979;
Bakes \& Tielens 1994; Weingartner \& Draine 2001b).  This heating is
partly offset by collisional cooling via grain-ion recombination (Draine \&
Sutin 1987).  The efficiencies of these mechanisms are regulated by the
charge of the grains, which in turn depends on $n(e)$, $T$ and $G$ (the
density of starlight).  Table~\ref{grain_rates_table} shows our
calculations of grain heating and cooling for the three cases (no SNR, with
SNR, and lower energy X-ray penetration) using the fitting formulae and
coefficients given by Weingartner \& Draine (2001b); see note~$a$ of the
table for details.

\placetable{grain_rates_table}

It is generally regarded that the photoelectric heating from grains is
greater than that from cosmic rays by about one order of magnitude
 (Draine 2011, p. 339) [or considerably more than this if the assumed value
for $\zeta_{\rm CR}({\rm H}^0)$ is lower than that adopted here, e.g.,
Wolfire et al. (1995, 2003)].
Any enhancement in $x_e$ tends to increase the heating rate from grains,
since the grains will be more negatively charged.  However, this increase is
not as strong as the effect of a greater electron fraction on the direct
cosmic ray heating, so the disparity between the two rates is decreased.

\section{SUMMARY AND CONCLUSIONS}\label{summary}

We have developed a means for deriving the representative rates of
ionization, and thus the resulting electron densities, along sight lines that
penetrate the WNM and that extend out to several
hundred pc from us, well beyond the edge of the Local Bubble. Our method
makes use of the fact that when a mostly neutral medium is exposed to the
ambient EUV and soft X-ray ionizing radiation, the argon atoms are far more
susceptible to being ionized than hydrogen atoms.  Thus, by comparing the
abundances of Ar~I to those of H~I, we gain an understanding of the strength
of the photoionization and secondary processes related to it.  For a number
of practical reasons, we find that it is desirable to use O~I as a proxy for
H~I.  The partial ionization of oxygen is strongly coupled to that of
hydrogen through a rapid charge exchange process.  Using this strategy, we
compare the column densities of Ar~I and O~I derived from absorption lines
seen in the {\it FUSE\/} spectra of 44 hot subdwarf stars.

Since the neutral forms of argon and oxygen are virtually absent in regions
that are fully ionized (either the prominent H~II regions around hot stars or
the much lower density but more pervasive WIM), our
probes sample only regions that have appreciable concentrations of H~I.  This
sets our measurements apart from conventional determinations of average
electron densities (e.g., pulsar dispersion measures, H$\alpha$ intensities,
C~II fine-structure excitation, etc.), which are strongly influenced by
contributions from the fully ionized regions.

We find that, on average, the abundance of neutral argon, relative to that of
neutral oxygen, is $[{\rm Ar~I/O~I}]=-0.427\pm 0.11\,$dex below what we would
expect if both species had no partial ionization and the solar abundance
ratio is a proper standard of comparison.  We interpret this deficiency in
terms of the greater susceptibility of argon to photoionization.  After
accounting for a broad range of processes that can modify the fractional
ionizations, we conclude that with known sources of ionizing radiation, both
external and internal, the only straightforward way to reconcile the large
deficiency of Ar~I is to propose that the neutral medium has a characteristic
density of only $0.09\,{\rm cm}^{-3}$, which is far below a generally
accepted value of about $0.5\,{\rm cm}^{-3}$.  At this latter, higher volume
density of hydrogen, known sources of ionization should produce an electron
density $n(e)=0.018\,{\rm cm}^{-3}$ and create a result $[{\rm
Ar~I/O~I}]=-0.14$, which is greater than all but a small fraction of the
measurements and well above the overall average value.

In order to achieve a result for [Ar~I/O~I] that is consistent with both our
observations and a density $n({\rm H~I})=0.5\,{\rm cm}^{-3}$, we must propose
nonconventional sources of additional ionization.  We discuss two such
possibilities; in reality we might be witnessing the outcome of some
combination of both of them working together.

The first explanation is that the shielding of the external, low energy X-ray
flux is less effective than expected: if we lower the cutoff energy from a
nominally expected 90$\,$eV to only 60$\,$eV, we can gain sufficient
additional ionizing photons from the external background to create a level of
ionization that is consistent with an upper bound for the value implied by
our observed [Ar~I/O~I]. To explain this lower shielding, one might envision
a nearly sponge-like topology for the WNM, where the internal holes and
channels are filled with hot, X-ray emitting gas.  Indeed, de Avillez et al.
 (2012) have performed hydrodynamical simulations
of the effects from random SN explosions and stellar winds in the
plane of the Galaxy, and the outcome of their model exhibits a complex,
turbulent entanglement of hot, warm and cold gas complexes.  Their
simulation, which includes non-equilibrium ionization calculations, yields an
average electron density $n(e)=0.04\,{\rm cm}^{-3}$, a value that is not far
removed from our determination for the WNM.  This near match may be
coincidental, however.  In the simulation, voids containing hot,
collisionally ionized gas with very low $n(e)$, in combination with dense
regions that are photoionized, may produce an outcome that is close to the
value for the WNM.

A serious shortcoming of the interpretation that we are viewing a
steady-state maintenance of a high level of ionization in the WNM is that the
calculated heating rate of $1.2\times 10^{-25}\,{\rm erg~s}^{-1}\,{\rm
H~atom}^{-1}$ from secondary electrons is unacceptably high -- much higher
than the cooling rate from atomic fine-structure excitations or
recombinations of ions onto dust grains.

Our second explanation is a time-dependent solution that can sidestep the
problem of overheating in the equilibrium case discussed above.  Over a
period of one to a few times $10^5\,$yr, an SNR can create a
burst of ionizing radiation in the X-ray region that can briefly elevate the
ionization of the ISM to levels well above normal out to distances of order a
few hundred pc away.  Initially, the recombination timescale is quite short,
but then it advances to about 1$\,$Myr when the level of ionization
approaches our observed average state.  For heating by secondary
electrons, the temperature should exhibit a similar rapid adjustment that is
followed by a much slower trend with time.  The radiation burst will indeed
create a very high heating rate, but as the gas temperature approaches or
exceeds $10^4\,$K, the abrupt onset of the very strong L$\alpha$ cooling will
dump most of the heat over a time scale that is considerably shorter than the
cooling time ($\sim 4\,$Myr) for the gas in its usual state.  A good
exposition of this quick thermal recovery to $T<10^4\,$K and a comparatively
more gentle relaxation in the heat loss and recombinations can be seen in
plots of $T$ and $x_e$ vs. time in Fig.~2 of Gerola et al. (1974).

We propose that the ionization imprint of an SN explosion on the WNM
can last well beyond the time that the remnant is recognizable in X-rays.  We
have good evidence that a number of SNe have exploded in our general
vicinity over the past $1-10\,$Myr.  As a consequence, if one considers that
the activity in our neighborhood has recently been higher than normal for our
Galaxy, the WNM out to several hundred pc from the Sun may be ionized to a
level that is somewhat higher than the low-density neutral gas at similar
galactocentric distances elsewhere in the Galactic plane.

\acknowledgments

This research was supported by a NASA Astrophysics Data Processing grant No.
NNX10AD44G to Princeton University.  All of the data presented in this paper
were obtained from the Mikulski Archive for Space Telescopes (MAST) at the
Space Telescope Science Institute (STScI). STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support for MAST for non-{\it HST\/} data is provided by the
NASA Office of Space Science via grant NNX09AF08G and by other grants and
contracts. The author acknowledges many useful discussions with B.~T.~Draine.
Drs. Draine and C.~Gry supplied suggestions for improvement after reading a
draft of this paper.  Some of the conclusions presented here relied on the
use of the CHIANTI database and software, which is a collaborative project
involving the NRL (USA), the Universities of Florence (Italy) and Cambridge
(UK), and George Mason University (USA); other conclusions made use of the
T\"ubingen Model Atmosphere Fluxes within the framework of the German
Astrophysical Virtual Observatory (GAVO).  The coordinates and apparent
magnitudes displayed in Table~\ref{aro_results} were provided by the
SIMBAD database, operated at CDS, Strasbourg, France.

{\it Facility:\/} \facility{FUSE}