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\author{Donald G. Luttermoser \thanks{Visiting Astronomer, National Solar 
Observatory, operated by the Association of Universities for 
Research in Astronomy, Inc., under cooperative agreement with the National 
Science Foundation.}\\*[2mm]
East Tennessee State University,\\*[2mm]
Department of Physics and Astronomy,\\*[2mm]
Johnson City, TN 37614}
 
\date{\today}
 
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\begin{abstract}
  
Non-LTE calculations of semiempirical chromospheric models are presented 
for 30~g~Her (M6~III).  This star is one of the coolest ($\teff = 3250$~K)
SRb (semiregular) variable stars and has a mass perhaps as great as
4~M$_{\odot}$.  Chromospheric features 
we have observed in its spectrum include Mg~II h \& k; C~II] UV0.01, which is 
sensitive to electron density; Mg~I $\lambda$2852; Ca~II H, K \& IRT; Ca~I 
$\lambda$4227 \& $\lambda$6573; Al~II] UV1; and H$\alpha$.  We pay special 
attention to fitting the C~II intersystem lines and the Mg~II resonance lines 
but use all the other features as constraints to some extent.  The equations 
of radiative transfer and statistical equilibrium are solved 
self-consistently for H~I, H$^{-}$, H$_{2}$, He~I, C~I, C~II, Na~I, Mg~I, 
Mg~II, Al~I, Al~II, Ca~I, and Ca~II with the equivalent two-level technique.  
To simplify these calculations, a one-dimensional, hydrostatic, 
plane-parallel atmosphere is assumed.  
 
We investigate 10 separate ``classical'' chromospheric models, differing most 
importantly in total mass column density above the temperature minimum.
Synthetic spectra from these models fit some but not all of the 
observations.  These comparisons are discussed in detail.  However, we find 
that no single-component classical model in hydrostatic equilibrium is able 
to reproduce both the Mg~II line profiles and the relative strengths of the 
C~II] lines.  In all these models, chromospheric emission features are formed 
relatively close to the star ($\lappxe 0.05 R_{\star}$).  The circumstellar 
environment has a thick, cool 
component overlying the Mg~II emission region, which is relatively static and 
very turbulent.  Finally, we find that thermalization in the Mg~II h \& k 
lines in the coolest giant stars is controlled by continuum absorption from 
Ca~I $4p ^{3}$P$^{\circ}$ bound-free opacity and not collisional 
de-excitation as is the case for warmer K giants.

\vspace*{10mm}  % Insert a 10mm space (the "*" forces LaTeX to do this).

\noindent\underbar{Subject headings}: stars: chromospheres --- stars: 
late-type --- stars: individual (30 Herculis) 

\end{abstract}

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\section{INTRODUCTION}
  
In contrast to photospheres, stellar chromospheres are regions where the 
temperature increases with height, attaining values in the range $\teff 
\lappxe T \lappxe 10,000$~K as the result of mechanical heating.  Knowledge 
of the source of this mechanical heating in most stars is still preliminary
in nature, but it is generally agreed that the heating 
probably depends on magnetic fields in main-sequence stars and giants earlier 
than $\sim$ K2 (\eg Rutten 1987), while for cooler giants and supergiants, it 
may be accomplished by non-magnetic processes (Middelkoop 1982; Schrijver 
1987), for instance by long-period (perhaps in conjunction with short-period) 
acoustic waves (\eg Bowen 1988; Cuntz \& Stencel 1991; Ulmschneider 1991).
  
The chromospheres of cool stars reveal themselves through emission features 
of singly ionized and neutral metals in the violet and ultraviolet --- 
especially the 
resonance lines of Mg~II and Ca~II.  Mg~II h and k show chromospheric emission 
to a greater degree than Ca~II H and K because the higher abundance of 
ionized magnesium ensures a higher excitation rate, and because there is less 
photospheric continuum at the wavelength of Mg~II than at Ca~II in cool stars.
Many lines of other common chemical elements are detected, too, although 
these tend to be weaker because of lower abundances (\eg Al~II), 
unfavorable excitation potentials (\eg Si~II), or complicated spectra 
which spread the emission over many lines (\eg Fe~II).  Of particular 
importance is the C~II] UV0.01 intersystem multiplet, whose line ratios are 
sensitive to electron density (Stencel \ea 1981), and the optically thin 
Al~II] line at $\lambda$2669.  Representative ultraviolet spectra of cool 
giants are given by Ayres \ea (1986) for $\alpha$~Boo (K1~III), Eaton \& 
Johnson (1988) for several M giants, and Carpenter \ea (1988) for 
$\gamma$~Cru (M3~III).
  
This paper represents the first attempt to form a detailed semiempirical 
model of the chromospheric structure of a cool ($\teff < 3500$~K), 
oxygen-rich giant star --- specifically, g~Her (M6~III).  Such stars 
represent fertile new ground for chromospheric models in that the electron 
density and heating mechanisms may well be different than in warmer stars.
In fact, line ratios within the C~II] intersystem multiplets in g~Her and 
2~Cen both indicate electron densities about an order of magnitude lower than 
in K giants (Eaton \& Johnson 1988).  
Such cool stars are usually variable, and pulsations are thought 
to contribute to heating the chromospheric gas (\eg Bowen 1988; Cuntz \&
Stencel 1991; Eaton, Johnson, \& Cadmus 1990).  Indeed, it has been suggested 
that the structure of cool M giants may be quite different than that of K and 
warm M giants, with weaker chromospheres, much lower ionization of all 
elements, and extensive circumstellar dust formation (Stencel, Carpenter, \&
Hagen 1986).  The mechanism of chromospheric heating, uncertain at present, 
may also be different in these cooler giants (Ulmschneider 1991; Dupree 1991; 
Cuntz \& Stencel 1991).
 
Section 2 of this paper discusses the observations used for comparison 
with our theoretical models.  Unlike the TX~Psc model, which was based on 
fitting the noisy profile of only the Mg~II h line and the integrated flux of 
C~II] UV0.01, our model of g~Her is derived from a multitude of 
high-resolution line profiles in the ultraviolet, optical, and infrared
regions.  Section 3 describes the procedure used in the NLTE calculations and 
applies them to g~Her, while \S 4 discusses the implications of the 
resulting model.  We conclude the paper with \S 5 by presenting the 
problems encountered with the modeling and summarizing possible solutions to 
the discrepancies between the synthetic and observed spectra.
      
\section{OBSERVATIONS}
  
The star 30 g~Her (M6 III) is a semi-regular (SRb) variable with V=5.04 
(B--V=1.52, U--B=1.17).  Ultraviolet fluxes are barely bright enough for 
high-resolution, long-wavelength spectra with IUE, since the star is very
inactive.  The period of the star is listed as 89 
days in the \textit{General Catalog of Variable Stars} (Kukarkin \ea 1976), and 
robotic photometry over the last 6 years shows changes in visual magnitude 
of $\sim$ 0.3 mag on a period of roughly this length superimposed on slower 
variations of close to 1.0 mag in V (Henry \& Baliunas 1992).  The visual 
spectrum is dominated by the typical TiO 
bands characterizing the M spectral type, but, unlike Mira variables, the star 
has never been reported to produce perceptible hydrogen-line emission at any 
phase.  Its metallicity seems to be roughly solar, although carbon is depleted 
by a factor of two with respect to iron (Smith \& Lambert 1985).  It has been 
suggested (Smith \& Lambert 1985; Judge \& Stencel 1991) that g~Her is an 
AGB (asymptotic giant branch) star from its position in the H-R diagram, a
possible slight enhancement of s-process elements, and its IRAS colors (\ie 
AGB stars show evidence for dust; indeed, the IRAS LRS Spectral Atlas (Olnon \&
Raimond 1986) shows g~Her to have weak dust features).
 
The semiempirical models described here are constrained by two 
high-resolution IUE spectra recording the line profiles of the C~II] UV0.01 
intersystem lines near 2325~\AA, the Mg~II h and k lines near 2800~\AA, and 
the Mg~I resonance line at 2852~\AA.  We have also used the overall appearance 
of an observed low-dispersion IUE spectrum to help constrain the angular size
of the star from comparisons to the synthetic spectra.  Table 1 lists the the 
properties of the spectra used.  The IUE data were reduced with the standard 
IUESIPS software at NASA's Goddard Space Flight Center; we have calibrated 
them with the calibrations of Cassatella \& Harris (1983) and Cassatella, 
Ponz, \& Selvelli (1983).  The integrated multiplet fluxes in these spectra 
were 6.6, 2.4, 71.8, and 2.5 $\times$ 10$^{-13}$ ergs~s$^{-1}$~cm$^{-2}$ for 
C~II] UV0.01, Al~II] UV1, Mg~II UV1, and Mg~I UV1, respectively.  The k/h 
ratio for Mg~II is 1.43, rather typical for K and M giants (Stencel \ea 1980).  
We have used LWP~13442 for the Mg~II lines, since 
they were saturated in LWP~13443, and LWP~13443 for the weaker features, 
C~II], Mg~I, and Al~II.  Table 2 gives some useful measurements of the 
spectra.  For the observations, we have listed integrated flux at the Earth 
in the Mg~II h and k lines, the flux at the Earth in the lines of the 
C~II intersystem multiplet, and the ratios of fluxes in other emission lines 
to Mg~II.  The emission line fluxes for the resonance transitions listed
represent the chromospheric portion of the line (\ie the ``emission cores'').
For the theoretical models, we give astrophysical flux at the star, along with 
the same ratios.
 
% Table 1   --  Here's an example on how you do a table.

\begin{table}
\caption{\textbf{Summary of g Her Observations}}
\begin{center}
\begin{tabular}{clcccc}
\hline  
Wavelengths& Date   &           & Camera   & Resolving & Exposure  \\
           & (U.T.) & Telescope & Sequence & Power $\lambda / \Delta\lambda$ 
 & Time (\textit{min}) \\
\hline
2500--3200 \AA & 16 June 1988   & IUE    & LWP 13441 &    500  &   5 \\
2500--2900 \AA & 16 June 1988   & IUE    & LWP 13442 & 10,000  &  40 \\
2300--3100 \AA & 17 June 1988   & IUE    & LWP 13443 & 10,000  & 880 \\
3922--3975 \AA & 13 August 1989 & McMath & SCCD 530  & 21,000  & 150 \\
8488--8590 \AA & 14 August 1989 & McMath & SCCD 547  & 23,000  &   2 \\
4181--4248 \AA & 27 August 1989 & McMath & SCCD 538  & 21,000  &   2 \\
6540--6600 \AA & 20 April 1992  & McMath & SCCD 517  & 45,000  &   5 \\
\hline
\end{tabular}
\end{center}
\end{table}
 
% Table 2  - Here's a table where we have shrunk down the font in order to fit

\begin{table}
\footnotesize\baselineskip=8pt
\caption{\textbf{Integrated Fluxes in Emission Lines$^{\mbox{a}}$}}
\begin{center}
\begin{tabular}{ccccccccccc}
\hline  
   Line                                  & Observed$^{\mbox{b}}$  & 
  T2   &   T3  &  T4  &  T5     &  T6    &  T7    &  T8    &  T9     &  T10 \\
\hline
Mg II h\&k                               & 7.2E-12 & 
  ---  &  896. &  713. & 6.96E3 & 1.3E5  & 5.89E6 &  7.2E4 &  8.05E3 & 1.99E4\\
C II $\lambda$2324                       & 1.1E-13 & 
  2.61 &  9.99 &  33.2 &  45.0  &   306. & 7.86E4 &   292. &  39.9   &  152. \\
C II $\lambda$2325                       & 3.1E-13 & 
  4.75 &  41.9 &  182. &  240.  & 1.58E3 & 1.60E5 & 1.60E3 &  215.   &  879. \\
C II $\lambda$2327                       & 1.6E-13 & 
  3.65 &  22.9 &  88.3 &  124.  &   637. & 6.37E4 &   664. &  112.   &  423. \\
C II $\lambda$2328                       & 8.6E-14 & 
  2.69 &  10.3 &  34.5 &  46.8  &   318. & 8.04E4 &   302. &  41.5   &  158. \\
\hline
  Line Ratios$^{\mbox{c}}$  &   &   &   &    &    &    &    &    &   &  \\
\hline
C II]/Mg II                              &  0.092 & 
  ---  &  1.02 & 0.498 &  0.068 &  0.069 &  0.067 &  0.039 & 0.053   &  0.084\\
Al II/Mg II                              &  0.033 &
   d   &    d  &    d  &   d    &  0.024 &  0.021 &  0.025 & 0.023   &  0.034\\
Mg II (k/h)                              &   1.43 & 
  ---  &  1.94 &  3.46 &  1.83  &  1.79   &  1.49 &  1.83  &  1.88   &  1.79\\
Mg I/Mg II                               &  0.035 &  
  ---  &  ---  &  ---  &  0.045 &  0.023 &  0.050 &  0.028 & 0.038   &  0.029\\
              &   &   &   &    &    &   &    &    &    &    \\
Ca I/H$\alpha^{\mbox{e}}$                &  1.12  & 
 0.017 & 0.017 & 0.017 &  0.041 &   d    &  12.   &  0.059 & 0.031   &  0.72\\
\hline
\multicolumn{11}{l}{Notes:}\\
%
\multicolumn{11}{l}{$^{\mbox{a}}$ Integrated fluxes in erg s$^{-1}$ cm$^{-2}$. 
An absorption line is indicated with a dash (note that all the lines for}\\
\multicolumn{11}{l}{\hspace*{5mm}model T1 were in absorption).  Also note that 
all fluxes for T1 through T4 are determined under}\\
\multicolumn{11}{l}{\hspace*{5mm}the assumption of CRD.  For T5--T10, 
resonance line fluxes are calculated with PRD.}\\
%
\multicolumn{11}{l}{$^{\mbox{b}}$ Divide this column by $(5.876 \times 
10^{-18}) \alpha_{\star}^{2}$, where $\alpha_{\star}$ is the angular diameter 
of the star in milli-arcsec, to}\\
\multicolumn{11}{l}{\hspace*{5mm}convert observed fluxes to model fluxes.}\\
%
\multicolumn{11}{l}{$^{\mbox{c}}$ C~II] $\equiv$ C~II] UV0.01; Mg~II $\equiv$ 
Mg~II h \& k; Ca~II $\equiv$ Ca~II H \& K; Ca~I $\equiv$ Ca~I $\lambda$6573.}\\
%
\multicolumn{11}{l}{$^{\mbox{d}}$ Line not calculated for this model.}\\
%
\multicolumn{11}{l}{$^{\mbox{e}}$ Ratio between the line center fluxes of
these absorption lines.  For model T10, this ratio gives 0.052 (no Kurucz}\\
\multicolumn{11}{l}{\hspace*{5mm}opacities), 0.48 (Kurucz opacities included), 
and 0.72 (Kurucz+McMath convolution).}
\end{tabular}
\end{center}
\end{table}

Figure 1 has nothing to do with this paper, I just wanted to show you how to
include an encapsulated postscript file of a figure drawn in IDL into a 
\LaTeX\ file.  The synthetic flux uses a compilation of
bound-bound opacities described by Avrett, Machado, \& Loeser (1986) which 
are based upon the line list of Kurucz \& Peytremann (1975).  It has been 
noted by Malagnini \ea (1992) that synthetic spectra based upon these older 
Kurucz line data reproduce the ultraviolet flux of cool stars better than the
recent Kurucz line data.  As a further check of the angular size of g~Her, we 
altered the synthetic flux scale factor of our photospheric model 
until a best fit was obtained via a minimization of the residuals between the
observed and synthetic fluxes.  This fit gives an angular diameter of 28.0 mas 
and we have used this scale factor for our first attempts at fitting the 
profiles.  Finally, all synthetic spectra are shifted in wavelength by the 
photospheric radial velocity of 3 km~s$^{-1}$ (Hoffleit 1982) for comparison 
with observations.
 
% Figure 1 - how to include an encapsulated postscript file.

\begin{figure}
\vspace*{3.2in}         % Make a 3.5 inch space for the figure.
\caption{This figure has nothing to do with this paper.}
% This next command communicates information with the "dvips" software:
%  "offsets" are in units of points, "scales" are in percentages (80=80% size)
\special{psfile=temptex.eps hoffset=0 voffset=0 hscale=80 vscale=80}
\end{figure}

\section{RADIATIVE TRANSFER CALCULATIONS}
 
In constructing the chromospheric models, we followed the technique described 
by Luttermoser \ea (1989).  First we obtain the plane-parallel, 
radiative-equilibrium model photosphere from the ``Indiana-grid'' of 
atmospheres with the closest match to the characteristics of g~Her.  For this 
study, we selected a model with solar abundance, $\teff$ = 3200~K, and 
$g$ = 1 cm s$^{-2}$ from Brown \ea (1989).  Since this photospheric model 
extends up only to a column mass $m$ of 6.4 $\times$ 10$^{-2}$ g~cm$^{-2}$, 
where the resonance lines of interest would still be very optically thick, we 
first extend the model out to a column mass of approximately 10$^{-14}$ g 
cm$^{-2}$ (see Luttermoser \& Johnson 1992).  Note here that we did \underbar
{not} reconverge this extended photospheric model under the assumption of 
radiative equilibrium; we merely \underbar {extrapolated} the 
temperature-density gradient from the top of the 
published model.  A synthetic spectrum calculated for this model was then used 
to set the flux scale factor between the observed and synthetic spectra,
hence angular size of the star.  We then increased the 
temperature as a function of height at an arbitrary depth in this extended 
photosphere to simulate a chromosphere.  For all these calculations, we
initially assumed a low microturbulent velocity of 3 km~s$^{-1}$ throughout 
the atmosphere, roughly the photospheric value (Smith \& Lambert 1985; Tsuji 
1991).  
 
Because the ionization and excitation in chromospheric gases may depart wildly
from the predictions of LTE (local thermodynamic equilibrium), a
self-consistent solution to the coupled equations of statistical equilibrium
and radiative transfer are required for a model atom of the species to be
considered.  These calculations were performed with the 
radiative transfer code PANDORA (\eg Vernazza, Avrett, \& Loeser 1973 (VAL I), 
1978 (VAL II), 1981 (VAL III); Avrett \& Loeser 1992) in a horizontally 
homogeneous, plane-parallel geometry.   Under these assumptions, the 
radiative transfer equation is written as,
\begin{equation}
\mu \frac{d I_{\nu}}{d \tau_{\nu}} = I_{\nu} - S_{\nu},
\end{equation}
where parameters above have their usual meaning, and the quantities shown 
include all processes (continuum and line) taking place at that frequency.  
Statistical equilibrium is formulated as 
\begin{equation}
n_{i} \sum_{j=1 (\neq i)}^{N}\, [P_{ij} + P_{i\kappa}] = \sum_{j=1 (\neq
i)}^{N}\, [n_{j}P_{ji} + n_{\kappa}P_{\kappa i}]
\end{equation}
and
\begin{equation}
n_{\kappa} \sum_{i=1}^{N} P_{\kappa i} = \sum_{i=1}^{N} n_{i}P_{i\kappa},
\end{equation}
where $n_{i}$ represents the number density in bound level $i$ (up to a maximum
of $N$ levels in the representative atomic model), $n_{\kappa}$ represents the 
continuum (\ie ion) number density, and $P$ represents the total transition 
rate (radiative $R$ plus collisional $C$) between the respective states.
Equations (1)-(3) are solved iteratively with the equivalent two-level
approach.  With this method, emission and absorption of photons in a line are 
expressed in terms of a net radiative bracket $\rho_{ji}$ defined by the net 
radiative rate equation 
\begin{equation}
n_{j} A_{ji} \rho_{ji} = n_{j} ( A_{ji} + B_{ji} \overline{J}_{ji} ) - 
n_{i} B_{ij}  \overline{J}_{ji},
\end{equation}
where $A_{ji}$, $B_{ji}$, and $B_{ij}$ are the Einstein coefficients for
spontaneous emission, stimulated emission, and absorption respectively, and
$\overline{J}$ is the total mean intensity integrated over the normalized 
absorption coefficient in the $ji$ transition.  It can be shown
that the line source function can be written as
\begin{equation}
S_{ji} = \frac{2 h \nu_{ji}^{3} / c^{2}}{(g_{j}/g_{i})(n_{i}/n_{j}) - 1},
\end{equation}
and the net radiative bracket as
\begin{equation}
\rho_{ji} = 1 - \frac{\overline{J}_{ji}}{S_{ji}},
\end{equation}
where $g$ represents the statistical weight of the level.
 
We calculated the line profiles for 10 separate chromospheric temperature 
distributions resembling ``classical'' chromospheric models -- a rapid rise 
in temperature at the top of the photosphere followed by a linear increase
in temperature as a function of column mass to some maximum value near 
10,000~K at the top of the chromosphere.  Properties of
all ten models are given in Table 3.  Our first chromosphere (T1) introduced a 
temperature rise at $T = 2119$~K and $\log m = -1.23$ (or $z = -1.52 \times 
10^{7}$ km, where $z$ increases into the star, and $z = 0$ corresponds to 
$\tau$[5000~\AA] = 1) with a linear temperature increase with respect to 
$\log m$ up to the top of the model, where 
$T = 12,000$~K at $\log m = -13.9$ ($z = -1.85 \times 10^{8}$~km).  This model 
was unsatisfactory in that it produced pure absorption features for Mg~I UV1, 
Mg~II h \& k, Ca~II H \& K, and C~II] UV0.01.  We next enhanced the 
temperature in the lower chromosphere ($-9.34 < \log m < -1.68$ [g~cm$^{-2}$] 
or $-1.02 \times 10^{8} < z < -1.70 \times 10^{7}$ km) by an average of 1000~K 
(chromosphere T2).  The temperature enhancement at the base of the 
chromosphere increased the pressure scale height, which in turn expanded the 
top of the chromospheric model out to $-2.21 \times 10^{8}$ km.  This model 
produced weak C~II] features with an integrated flux at Earth of $1.6 \times 
10^{-14}$ ergs~s$^{-1}$cm$^{-2}$ (for an angular diameter of 28 mas).  Our 
resonance lines of interest remained pure absorption features.
 
% Table 3 - another table, note the use of "multicolumn"

\begin{table}
\caption{\textbf{Chromospheric Models Considered}}
\begin{center}
\begin{tabular}{ccccccc}
\hline  
Model & $T_{min}^{a}$ & $\log m_{min}^{a}$ & $T_{knee}^{b}$ & $\log 
m_{knee}^{b}$ & $n_{e}$(C~II])$^{c}$ & $H$ (km)$^{d}$ \\
\hline
T1 & 2119 & --1.23 &  --- &   ---  &          ---       & $1.70\times 10^{8}$\\
T2 & 2119 & --1.23 & 3500 & --1.68 & $2.9\times 10^{6}$ & $2.06\times 10^{8}$\\
T3 & 2140 & --0.92 & 5500 & --2.28 & $9.3\times 10^{6}$ & $2.21\times 10^{8}$\\
T4 & 2140 & --0.92 & 6200 & --2.28 & $8.9\times 10^{6}$ & $2.31\times 10^{8}$\\
T5 & 2156 & --0.70 & 6800 & --2.28 & $1.6\times 10^{7}$ & $2.35\times 10^{8}$\\
T6 & 2264 & --0.03 & 5800 & --1.23 & $9.1\times 10^{6}$ & $1.67\times 10^{8}$\\
T7 & 2426 &   0.68 & 6920 & --1.23 & $1.2\times 10^{9}$ & $2.16\times 10^{8}$\\
T8 & 2223 & --0.25 & 6200 & --1.68 & $9.1\times 10^{7}$ & $2.02\times 10^{8}$\\
T9 & 2156 & --0.70 & 6800 & --2.28 & $1.4\times 10^{7}$ & $2.45\times 10^{8}$\\
T10 & 2156 & --0.70 & 7000 & --2.28 & $3.4\times 10^{7}$ & $2.23\times 10^{8}$\\
\hline
\multicolumn{7}{l}{Notes:}\\
%
\multicolumn{7}{l}{$^{\mbox{a}}$ Temperature (in K) and the logarithm of the
column mass (in g cm$^{-2}$)}\\
\multicolumn{7}{l}{\hspace*{5mm}of the atmospheric position that separates the 
photosphere from the}\\
\multicolumn{7}{l}{\hspace*{5mm}chromosphere.}\\
%
\multicolumn{7}{l}{$^{\mbox{b}}$ Temperature (in K) and the logarithm of the
column mass (in g cm$^{-2}$)}\\
\multicolumn{7}{l}{\hspace*{5mm}of the atmospheric position where the 
chromospheric temperature}\\
\multicolumn{7}{l}{\hspace*{5mm}profile flattens.}\\
%
\multicolumn{7}{l}{$^{\mbox{c}}$ Electron density (in cm$^{-3}$) at the depth 
of formation of the C~II] UV0.01}\\
\multicolumn{7}{l}{\hspace*{5mm}features.}\\
%
\multicolumn{7}{l}{$^{\mbox{d}}$ Thickness of the chromosphere (in km) from the
temperature minimum}\\
\multicolumn{7}{l}{\hspace*{5mm}to the outermost region.}
\end{tabular}
\end{center}
\end{table}
  
\section{DISCUSSION}
 
Because of discrepancies among the results for C~II], Mg~II, and H$\alpha$, 
we have been unable to fit even the best available chromospheric diagnostics 
with a single plane-parallel, hydrostatic, homogeneous \textit{classical} 
chromospheric model.
These discrepancies had the following form:  In order to get the electron 
density high enough with a traditional model, the lines must form at such 
high mass column density that H$\alpha$ becomes much too strong, some 
photospheric lines develop unobserved emission cores, and the wings of Mg~II h 
and k become much too broad.  Other discrepancies include the ratios of 
emission lines.
While C~II], Al~II], Mg~II, and Mg~I are formed in roughly the same part of the 
chromosphere, the emission portion of the Mg~I and Mg~II features form over a
range from $4.7 \times 10^{-4} \le m \le 5.9 \times 10^{-2}$ g cm$^{-2}$ and 
$3.5 \times 10^{-6} \le m \le 27$ g cm$^{-2}$ respectively.  Indeed, the peak
flux of the Mg~I and Mg~II line originate from the same depth ($m = 2.1 \times
10^{-2}$ g cm$^{-2}$) as the formation depths of the C~II] and Al~II] 
emission lines and the core of H$\alpha$.  This coincidence arises quite 
naturally in this one-dimensional, hydrostatic modeling, for we need to find 
a temperature rise to produce both Mg~II and Mg~I emission.  If we raise the 
temperature too far out, the density is too low for Mg~I to have sufficient 
opacity for emission features to form at the typical chromospheric 
temperatures (\ie 5000-7000 K), whereas Mg~II still possesses sufficient 
opacity, due to its slightly higher number density at these
temperatures as compared to Mg~I, to produce emission.  We cannot start the
temperature rise too deep since observed ``photospheric'' absorption lines 
(\ie Mg~I $\lambda$4571) go into emission.  Due to these constraints, the
temperature rise must begin around $m = 0.1-0.3$ g cm$^{-2}$ and increase
dramatically to temperatures between 5000-7000 K at around 0.02 g cm$^{-2}$.  
Such a sharp temperature rise suggests that a shock structure may exist in 
this region of the atmosphere.
 
\subsection{Electron Densities}
 
The observed C~II] line ratios indicate the electron density should be
increased in our model by an order of magnitude.  The hydrogen density is
quite low at the formation depth of these lines, and one would have to increase 
the temperature at this depth to well over 10,000 K to produce the observed 
ratios.  This, of course, would produce enormous Balmer emission lines (as seen
in Mira variable stars) which have not been seen in this star.  This 
discrepancy can 
also be remedied by moving the chromosphere in to lower depths, but then one
runs into the problems mentioned above.  There is the possibility that the 
C~II] ratios are unrepresentative of this star because of noise in the IUE 
spectra near 2325~\AA\ (GHRS observations of these very cool giants would be
invaluable), but the appearance of this multiplet here is very similar to that 
in the M5~III star 2~Cen (Eaton \& Johnson 1988), the only other M giant this
cool observed in the C~II] lines.  At the formation depth of the C~II]
lines and the peak flux of the Mg~II lines, H~I contributes 64.0\% of the
electrons followed by O~I (20.6\%), C~I (8.5\%), Ne~I (1.1\%), and N~I (1.0\%). 
Both hydrogen and carbon are handled in NLTE, while the rest are in LTE.  
Carbon is 96.8\% ionized at this depth and $n_{\kappa}/n_{\kappa}^{*} \approx 
1$, indicating that we cannot increase the electron density at these depths 
with carbon.  Hydrogen is 64\% ionized at this region and is severely 
underionized with respect to LTE ($\sim3$ orders of magnitude).  Luttermoser 
\& Johnson (1992) have shown that the Lyman lines act as a \textit{drain} in 
the chromosphere of late-type giants and that PRD gives the same level/ion 
densities as CRD at these depths.  Hence, we cannot enhance the electron 
density at this region of the atmosphere from a better treatment of hydrogen
(\ie including more levels will slightly enhance the Lyman-line drain).  One 
might expect oxygen to have similar ionization properties as hydrogen, since 
it has nearly the same ionization potential and has strong UV resonance lines. 
Radiative transfer in the oxygen atom is very complicated however.  Field \& 
Steigman (1971) have shown that the charge-exchange reaction rate O +
H$^{+}$ $\Leftrightarrow$ O$^{+}$ + H can dominate the ionization equilibrium
of oxygen in low density environments. This charge-exchange reaction for our 
model in the Mg~II emission region would give $n(\mbox{O}^{+})/n(\mbox{O}) 
\approx \frac{9}{8} n(\mbox{H}^{+})/n(\mbox{H})$ --- oxygen underionized with 
respect to LTE which would substantially reduce the electron density.  Haisch 
\ea (1977), however, have shown the importance of the Ly$\beta$-pumped 
fluorescence mechanism to the excitation of the 1302--1306~\AA\ resonance lines 
of the O~I atom.  The $^{3}P_{2}$ -- $^{3}D$ excitation at 1025.77~\AA\ via 
Ly$\beta$ photons, followed by an immediate ionization out of the $^{3}D$ 
state from the intense far-IR photospheric radiation field of this star, 
might give rise to an overionization of O~I with respect to LTE.  Such a 
mechanism could enhance the electron density in the lower chromosphere and 
rectify the C~II] line ratio problem; however, Carlsson \& Judge (1993) have
determined that this scenario will not work.  Unfortunately, we cannot further
test this mechanism since such a calculation involving fluorescence is 
currently not possible with PANDORA.  The likely explanation to the electron
density discrepancy probably involves our omission of hydrodynamic processes,
which will enhance the gas density at these depths, and possible
inhomogeneities in the atmospheric structure (see Ayres 1990; J\o rgensen \& 
Johnson 1991; and Wiedemann \& Ayres 1991).  Finally, an increase in the
surface gravity by an order of magnitude would help to rectify this problem.
 
\subsection{Ca II H \& K}
 
The chromospheric emission features in the Ca~II H \& K line cores of g~Her 
(and most cool giant stars) are observed to be too weak relative to Mg~II h \& 
k.  From this study, we can identify several reasons for this inconsistency.  
(1) The thermalization of photons scattering in the line wings is dominated by 
collisional de-excitation (\ie depth-dependent PCS) instead of continuum 
processes (\ie depth-independent PCS) as was the case for the Mg~II lines.  
This results in the Ca~II chromospheric emission flux being weaker and broader 
than would be the case for depth-independent PCS.  (2) The enhanced 
microturbulent velocity in the lower chromosphere further broadens the Ca~II 
emission, depressing the peak flux further.  
(3) Boesgaard \& Hagen (1979) have also suggested that the
weakness could further be the result of increased attenuation in a wind that 
absorbs over the entire range of velocity of the Ca~II emission features.  We 
do not place too much weight on this suggestion, since it is difficult to 
understand why a large velocity gradient presents itself in the formation 
region of the Ca~II cores (\ie the H$_{3}$ and K$_{3}$ features) and is not 
evident in the Mg~II line cores which are formed slightly higher in the 
atmosphere.  Instead, we simply claim that the chromospheric velocity field 
shifts the K$_{3}$ and H$_{3}$ features slightly blueward to help hide some of 
the chromospheric emission.  (4) We also need to rely on overlying absorption 
from neutral metals to explain the weakness of the features.
 
\section{CONCLUSIONS}
  
Luttermoser \& Johnson (1992) have already described the difficulties in
carrying out NLTE calculations in cool, low density, stellar atmospheres and
the techniques one must use to converge a solution using the equivalent
two-level approach.  Since one must attempt to
fit as many \textit{chromospheric indicators} as one can access, many runs must
be made for each atom and ion handled in NLTE.  We have obtained a large 
number of observational constraints for g~Her to carry out this analysis and
have found that semiempirical techniques involving ``classical'' models 
under the assumptions of one-dimensional, homogeneous geometries in 
hydrostatic equilibrium cannot reproduce all of the spectral features of this 
star.  We note here that we have begun to carry out preliminary calculations 
of a g~Her model with similar characteristics to hydrodynamic models of Mira 
variables (\ie a model with a permanent chromosphere, or ``calorisphere,'' and
outward propagating shocks, see Bowen 1988).  We will show in a subsequent 
paper that many of the ``chromospheric'' spectral features of this semiregular 
variable can be reproduced with such a model.  This 
suggests that future modeling of these late-type variable stars, both the 
Miras and semiregulars, require atmospheric modeling with dynamic processes 
(see Bowen 1988) and possibly inhomogeneous geometries (see J\o rgensen \&
Johnson 1991) included in the calculations.  Previous semiempirical 
chromospheric models of late-type giants, based on a limited set of spectral 
features (\eg the model for TX~Psc by Luttermoser \ea 1989), are not true 
representations of the thermal structure of these stellar atmospheres.
 
This investigation leads to the following conclusions.  (1) The chromospheric
structure of semiregular, late-type giant stars (g~Her as the prototype)
cannot be described by the classical, one-component model in hydrostatic 
equilibrium.  (2) Inconsistencies between ultraviolet emission features and 
infrared CO absorption suggest the outer atmosphere of these stars are 
inhomogeneous.  (3) Such inhomogeneities, however, cannot account for the
difficulty in fitting both the C~II] intersystem line ratios and the Mg~II
profiles --- we suggest that hydrodynamic processes (\ie shocks) may help
resolve this problem.  (4) Thermalization in the Mg~II h \& k lines in cool 
giant stars is controlled by continuum absorption and \underbar{not} 
collisional de-excitation as is the case for warmer K giants.  (5) The 
chromospheric emission features (\ie UV emission features) seen in these stars 
are formed relatively close to the star ($r \lappxe 0.05 R_{\star}$).  (6) The 
circumstellar environment includes a cool component that resides above the 
chromosphere, which is relatively static and very turbulent with a 
hydrogen column mass of $\sim3 \times 10^{20}$ cm$^{-2}$.
 
We thank Mr.\ Dave Jaksha of the National Solar Observatory for showing us how 
to use the McMath solar telescope and stellar spectrograph and for help with 
the preliminary data reductions of our ground-based observations.  We thank 
Dr.\ Frederick Walter for help in reading FITS-formatted tapes, and Drs.\ 
Eugene Avrett and Rudolf Loeser for encouraging us to use their PANDORA 
stellar atmospheres code.  Thanks are also extended to Dr.\ Philip Judge 
for helpful discussions and useful comments on the manuscript, and to 
an anonymous referee for additional comments.  Finally, we
wish to thank Drs.\ Sallie Baliunas, Louis Boyd, Russell Genet, and Greg Henry
for the photometric results of g~Her, which were obtained with the Smithsonian
0.25m APT, operated jointly by the Smithsonian Astrophysical Observatory,
Fairborn Observatory, and Tennessee State University and generously supported
by SAO Director Dr.\ Irwin Shapiro.  Use of the
Colorado IUE-RDAF, supported by NASA contract NAG 5-28731, is acknowledged.
This research was partially supported by NASA grants NAG 5-1181 and NAG
5-1777 to Iowa State University; NAG 5-1305 to the University of Colorado; 
NAG 5-182 to Indiana University; and NAG 5-1327 to East Tennessee State 
University.
 
% ******************* Insert your bibliograph here.

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