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

Dense cores are the places where stars are formed. Their physical
conditions determine when the gravitational collapse starts and how
it proceeds \citep{diFrancesco_2007-PPV}. 
Observations of molecular lines allow us to extract information not only 
about the physical structure of dense cores but also about their
chemical composition \citep{Bergin_Tafalla_2007-Review}.

When high densities and low temperatures are present, gas-phase
molecules can be depleted because of accretion onto the dust grains. 
The effect of depletion is commonly observed in \ce{CO} (and {isotopologues}),
and it is identified as a strong decrease in the \ce{CO} abundance towards 
the central part of a dense core 
\citep[e.g.,][]{Caselli_1999-L1544_Depletion,Tafalla2004-Cores}.
Surprisingly, some nitrogen-bearing species, such as \ce{NH3}\xspace, \ce{N2H+}\xspace and
\ce{CN}, do not present strong evidence for depletion 
\citep{Tafalla2004-Cores, Crapsi2007-L1544, 2008A&A...480L...5H,
2010A&A...513A..41H}. 
However, it is theoretically expected that all species, 
other than pure hydrogen species and helium, should be depleted by at 
least two orders of magnitude in the central 
region of a dense, starless core; this region can be called the zone of ``complete depletion'' \citep{Walmsley2004-L1544_Full_Depletion}.
In the past, modeling of single dish observations have provided indirect
evidence for depletion of \ce{N2H+}\xspace in the L183 and L1544 dense cores
\citep{2005A&A...429..181P, Pagani2007-L183_Depletion_N2Hp, Redaelli2019-N2Dp_freezeout}.
Only recent observations with ALMA, resolving the densest 
parts of the prestellar core L1544, show evidence for depletion of
deuterated ammonia, \ce{NH2D} \citep{Caselli2022-L1544_NH3_Depletion}.


To validate the concept of complete depletion, it is important to 
investigate if also the common isotopolog of ammonia, \ce{NH3}\xspace, is also depleted 
at high densities. The starless core H-MM1 \citep{2004ApJ...611L..45J,
2011A&A...526A..31P} is an excellent target for this purpose.  
H-MM1 lies in relative isolation on the outskirts of the L1688 region in the nearby Ophiuchus molecular cloud. 
{The distance to L1688 is $\approx$138.4$\pm$2.6\,pc \citep{2018ApJ...869L..33O}}.
Ammonia line emission in the L1688 
region has been previously studied with single-dish observations
\citep{Friesen2017,Harju2017-HMM1_NH2D,Chen2019-Droplets,Auddy2019-Bfield,
Choudhury2020-Letter,Choudhury_2021-L1688_Map}. 
These studies show a clear transition from the supersonic turbulence to a
subsonic regime at the boundary of the H-MM1 core, and that the interior
parts of the core are characterized by a {high degree} of deuteration in
\ce{NH3}\xspace \citep{Harju2017-HMM1_NH2D}. This suggests that the densities and
temperatures in the central region are appropriate for \ce{NH3}\xspace depletion. 

High-resolution \ce{NH3}\xspace observations are also useful for the
interpretation of recent ALMA observations that revealed the presence
of a methanol (\ensuremath{\rm CH_3OH}) sheath around the dense core \citep{Harju2020-HMMS1}. Of the methanol
desorption mechanisms discussed in \cite{Harju2020-HMMS1}, shocks
and grain-grain collisions induced by vigorous turbulence or shear
instability should be associated with increased kinetic temperatures or
velocity dispersion in the methanol emission region. Ammonia provides
excellent means to determine these quantities and thereby test
these desorption scenarios.

Here we present \ce{NH3}\xspace (1,1) and (2,2) observations of the starless core
H-MM1 with the Karl G. Jansky Very Large Array (VLA), which we combine
with previous Robert C. Byrd Green Bank Telescope (GBT) observations
\citep{Friesen2017}. 
The acquired data allow us to study the \ce{NH3}\xspace emission at high angular 
resolution to resolve the possible \ce{NH3}\xspace depletion zone and to improve our knowledge of the origin of methanol.

\section{Data}

\subsection{VLA}
We conducted VLA observations of H-MM1 on 2020 January 9, 12, 14, and 18
in the D-array configuration (project 19B-178, PI: Pineda). 
We used the high-frequency K-band receiver and configured the WIDAR
correlator to observe the \ce{NH3}\xspace (1,1) and (2,2) lines with 8 and 4\,MHz
bandwidth windows, respectively. 
Each window has a channel separation of 5.208\,kHz ($\approx$0.065\,\ensuremath{\rm km\,s^{-1}}\xspace).
The quasar 3C 286 is used as the flux calibrator, J1256-0547 as the
bandpass calibrator, and J1625-2527 as the phase calibrator. 

The calibration was performed using the VLA pipeline with the Common
Astronomy Software Applications package (CASA) version 5.6.2-2
\citep{McMullin2007-CASA}.  
This included Hanning-smoothing, and therefore the effective spectral resolution is 0.13\,\ensuremath{\rm km\,s^{-1}}\xspace. 
The imaging used the \verb+tclean+ command with natural weighting and \verb+uvtaper+ 
to increase the signal to noise ratio, a restoring beam of 6\arcsec, 
and with multi-scale clean (with scales 0, 6, 18, and 54\arcsec 
and a \verb+smallscalebias+ parameter of 0).
We also included the \ce{NH3}\xspace single dish data obtained with the Robert C.
Byrd Green Bank Telescope \citep{Friesen2017} as a model image to recover
the extended emission 
{using the \texttt{startmodel} option in \texttt{tclean}.
This method initializes the clean model image as the 
single dish data.}
The final noise level achieved in the data cubes is
3.5\,m\ensuremath{\rm Jy\,beam^{-1}}\xspace per channel, which corresponds to $\approx$0.21 K per channel.
{We estimate our absolute calibration uncertainty to be 10\%.}

\begin{figure*}
    \centering
    \includegraphics[height=6.71cm]{H-MM1_NH3_TdV}
    \includegraphics[height=6.71cm]{H-MM1_NH3_22_TdV}
    
    \caption{Integrated intensity maps of the \ce{NH3}\xspace (1,1) and (2,2) main
    line components are shown in {the} left and right panels, respectively.
    The contours are for \ce{NH3}\xspace (1,1) in both panels. The contour levels
    start from 1.5\,\ensuremath{\rm K\,km\,s^{-1}}\xspace and the spacing is 0.75\,\ensuremath{\rm K\,km\,s^{-1}}\xspace. The 
    beam size and the scale bar are shown at bottom left and right, respectively.}
    \label{fig:NH3_TdV}
\end{figure*}

The integrated intensity maps of the main hyperfine component 
of the \ce{NH3}\xspace (1,1) and (2,2) lines are shown in Fig.~\ref{fig:NH3_TdV}.

\subsection{Total {\ce{H2}\xspace} Column Density}

The H-MM1 core appears as an absorption feature in the 8 $\rm \mu m$ map
measured by the InfraRed Array Camera (IRAC) of the {\it Spitzer Space
Telescope}. 
We use this to derive a high-resolution $\ce{H2}\xspace$ column
density map of the core. The method is described in Appendix \ref{sec:scatter} of
\cite{Harju2020-HMMS1}, and briefly summarized here. 
The method also uses
the 850 $\rm \mu m$ emission map from SCUBA-2 \citep{Pattle:2015bq}, and the
dust color temperature map, $T_{\rm C}$, derived from {\it Herschel}/SPIRE maps
\citep{Harju2017-HMM1_NH2D}. 
We assume the dust opacity model for unprocessed dust grains with thin ice mantles by
\cite{1994A&A...291..943O}. This model has a dust emissivity index of
$\beta=2.0$ over {\it Herschel}/SPIRE wavelengths (250--500 $\rm \mu m$).
The absorption cross-sections per unit mass of gas at 8 $\rm \mu m$ and
850\,$\rm \mu m$ are $\kappa_{8\,\rm \mu m}=8.86\, \ensuremath{\rm cm^2}\xspace\,\rm g^{-1}$ and
$\kappa_{850\,\rm \mu m}=1.13\times10^{-2}\, \ensuremath{\rm cm^2}\xspace\,\rm g^{-1}$,
respectively. The {\it Spitzer} map is smoothed to the resolution of the
VLA ammonia map, $6\,\arcsec$, before the derivation of the column
density map. The resulting $N(\ce{H2}\xspace)$ distribution is shown in
Fig.~\ref{fig:column_maps_H2}. 

\cite{Lefevre2016} pointed out that scattering by dust particles may bias 
column density derived from $8\,\rm \mu m$ absorption. 
This issue is discussed in Appendix \ref{sec:scatter}. 
Simulations using different dust models
show that the effect of scattering is likely to be small compared to the
uncertainties related to the intensities of the foreground and background
emission components, and thermal dust emission at 8 $\rm \mu m$. 
The 8 $\rm \mu m$ background intensity in the region of H-MM1 is
$\sim$10 $\ensuremath{\rm MJy\,sr^{-1}}\xspace$, whereas the scattered intensity according to our
simulations should stay below $0.1\,\ensuremath{\rm MJy\,sr^{-1}}\xspace$ (see Appendix \ref{sec:scatter}).

\begin{figure}[htb]
    \centering
    \includegraphics[width=\columnwidth]{H-MM1_N_H2}
    \caption{Dust derived total \ce{H2}\xspace column density map of the H-MM1 region. 
    The map shows a peak close to the center of the image. 
    The resolution and scale bar are shown at bottom left and right, respectively.
    The contours show the \ce{NH3}\xspace (1,1) integrated intensity, as in Fig~\ref{fig:NH3_TdV}.
    \label{fig:column_maps_H2}}
\end{figure}

\section{Observational Results}
\subsection{Line Fits} \label{sec:linefit}

We simultaneously fit the \ce{NH3}\xspace(1,1) and (2,2) line profiles with \verb+pyspeckit+ using the \verb+cold-ammonia+ model \citep{Friesen2017}, which gives a centroid velocity, \ensuremath{V_{\rm LSR}}\xspace, velocity dispersion, \ensuremath{\sigma_{\rm v}}\xspace, kinetic temperature, \ensuremath{T_{\rm k}}\xspace, excitation temperature, \ensuremath{T_{\rm ex}}\xspace, and total column density of \ce{NH3}\xspace, $N(\ce{NH3}\xspace)$, with an assumed ortho-to-para ratio of 1 \footnote{If the column density of para-\ce{NH3}\xspace is desired then a correction factor of 0.5 on the \ce{NH3}\xspace column densities and fractional abundances should be applied}. 
This model assumes {that
only} the rotational levels (1,1) and (2,2) are populated, 
which is appropriate for the low temperatures seen in this region.
We adopt a number of criteria to ensure that the different fit results are accurate; 
this approach is similar to what was used in \cite{Pineda2021-Ion_Neutrals}.
The mean value of the kinetic temperature (\ensuremath{T_{\rm k}}\xspace) in pixels with 
uncertainty less than 1\,K is 11\,K. 
For pixels with uncertainties in \ensuremath{T_{\rm k}}\xspace larger than 1\,K we re-run the fit, but with a fixed \ensuremath{T_{\rm k}}\xspace of 11\,K.
The kinematic parameters, \ensuremath{V_{\rm LSR}}\xspace and \ensuremath{\sigma_{\rm v}}\xspace, are usually well determined even when the 
kinetic and excitation temperatures are poorly constrained, thanks to the many hyperfine components. 
We discard all velocity determinations if the uncertainty in \ensuremath{V_{\rm LSR}}\xspace or \ensuremath{\sigma_{\rm v}}\xspace is larger 
than 0.02\,\ensuremath{\rm km\,s^{-1}}\xspace and 0.015\,\ensuremath{\rm km\,s^{-1}}\xspace, respectively.
We consider that the \ensuremath{T_{\rm ex}}\xspace and \ensuremath{T_{\rm k}}\xspace are well determined only when their derived uncertainties are smaller than 1\,K, 
and that the column density is well constrained where \ensuremath{T_{\rm ex}}\xspace and \ensuremath{T_{\rm k}}\xspace are well constrained.
The centroid velocity and the velocity dispersion maps are shown in Fig.~\ref{fig:NH3_velo}, 
the \ce{NH3}\xspace column density is shown in Fig.~\ref{fig:column_maps}, 
and the excitation and kinetic temperature maps are shown in Fig.~\ref{fig:TexTk}.
The results from the \ce{NH3}\xspace line fits are discussed below. 

\begin{figure*}[th]
    \centering
    \includegraphics[height=6.595cm]{H-MM1_NH3_Vlsr}
    \includegraphics[height=6.595cm]{H-MM1_NH3_dv}
    \caption{Centroid velocity and velocity dispersion maps derived from \ce{NH3}\xspace are shown in {the} 
    left and right panels, respectively. 
    The velocity fields show an elongated redshifted feature connecting the south-east edge and the
    central region of the core.
    The contours and the markers are as in Fig~\ref{fig:NH3_TdV}.
    \label{fig:NH3_velo}}
\end{figure*}


\begin{figure}[thb]
    \centering
    \includegraphics[width=\columnwidth]{H-MM1_NH3_Ncol}
    \caption{\ce{NH3}\xspace column density map.
    The central part of the map (where \ce{H2}\xspace peaks)
    shows a nearly constant \ce{NH3}\xspace column density,
    while the peak is located in the northern part of the core. 
    The contours and the markers as in Fig~\ref{fig:NH3_TdV}.
    \label{fig:column_maps}}
\end{figure}


\subsection{Radial Velocity and Velocity Dispersion}
The velocity
centroids (Fig.~\ref{fig:NH3_velo}-left) range from 
$\approx$4.0\,\ensuremath{\rm km\,s^{-1}}\xspace to $\approx$4.4\,\ensuremath{\rm km\,s^{-1}}\xspace across the core. 
Prominent in this distribution is a narrow feature at velocities $>$4.3\,\ensuremath{\rm km\,s^{-1}}\xspace that seems 
to connect the South-East end and the center of the core. 
The velocity dispersion (Fig.~\ref{fig:NH3_velo}-right) is small across the map.
The narrowest line, $\approx$0.09\,\ensuremath{\rm km\,s^{-1}}\xspace, is found in the northern part of the core, whereas the largest
values, $>$0.15\,\ensuremath{\rm km\,s^{-1}}\xspace, lie in the southeast, in the region of the highest \ensuremath{V_{\rm LSR}}\xspace  values. 
The locations of large \ensuremath{\sigma_{\rm v}}\xspace
form a narrow strip, but the orientation of this 
strip deviates from that of the red-shifted $V_{\rm LSR}$ feature.

\begin{figure*}[hb]
    \centering
    \includegraphics[height=7.75cm]{H-MM1_NH3_Tkin}
    \includegraphics[height=7.75cm]{H-MM1_NH3_Tex}
    \caption{Kinetic and excitation temperature maps for H-MM1 are shown in {the} 
    left and right panels, respectively.
    The contours and the markers are as in Fig~\ref{fig:NH3_TdV}.
    These maps only include pixels where 
    the uncertainty is smaller than 1\,K. \label{fig:TexTk}}
\end{figure*}


\subsection{Kinetic Temperature and Excitation Temperature}

The \ensuremath{T_{\rm k}}\xspace distribution is almost flat in the central region, 
but it shows a tendency to increase toward the core boundaries (Fig.~\ref{fig:TexTk}).  
The average temperature in the 
core center is 11\,K, and it rises to $\approx$12\,K at lower $\ce{NH3}\xspace$ and $\ce{H2}\xspace$ column densities (Fig.~\ref{fig:Tk_vs_N_NH3}). 
The \ensuremath{T_{\rm ex}}\xspace of the $\ce{NH3}\xspace(1,1)$ line varies from $\approx$5\,K at the core boundaries to 
$\approx$11\,K in the central regions. 
This indicates that the line is thermalized in the densest parts of the core {(see also Appendix~\ref{sec:Tex_Tk_H2})}.

\begin{figure*}[htb]
    \centering
    \includegraphics[width=0.495\textwidth]{H-MM1_NH3_Tk_Ncol_KDE}
    \includegraphics[width=0.495\textwidth]{H-MM1_NH3_Tk_H2_KDE}
    \caption{The Kernel Density Estimates (KDE) of 
    the kinetic temperature as a function of \ce{NH3}\xspace and \ce{H2}\xspace 
    column density are shown in {the} left and right panels, respectively. 
    The black horizontal segments show the mean values within the column density bins. The gas temperature drops from $\approx$12 K down to
    $\approx11$ K towards the highest column densities in the core. 
    \label{fig:Tk_vs_N_NH3}
    }
\end{figure*}


\subsection{Ammonia Abundance\label{sec:X_calc}}
\begin{figure*}[htb]
    \centering
    \includegraphics[height=7.75cm]{H-MM1_NH3_v_H2_KDE}
    \includegraphics[height=7.75cm]{H-MM1_NH3_H2_ratio_H2col}
    \caption{Depletion of \ce{NH3}\xspace in the center of H-MM1.
    {\it Left:} KDE comparison between \ce{NH3}\xspace and \ce{H2}\xspace column densities. 
    This presents a positive correlation below $N(\ce{H2}\xspace)<2.6\times
    10^{22}$ cm$^{-2}$, whereas for higher $N(\ce{H2}\xspace)$ values the \ce{NH3}\xspace
    column density is nearly constant or decreasing to a small degree. 
    The black horizontal segments show the mean values within the 
    $N(\ce{H2}\xspace)$ bins. The red line shows a broken straight line fit to 
    the data (see text).
    {\it Right:} Map of the fractional \ce{NH3}\xspace abundance,
    $X(\ce{NH3}\xspace)=[\ce{NH3}\xspace]/[\ce{H2}\xspace]$, calculated as the ratio between the
    column densities. The \ce{NH3}\xspace abundance drops significantly in the
    central region, and the distribution is asymmetric so that 
    ammonia is more abundant on the Western side of the core as
    compared to the Eastern side. 
    The contours {show the \ce{H2}\xspace column density 
    starting at $1\times 10^{22}$ \ensuremath{\rm cm^{-2}}\xspace 
    and with a spacing of $1\times 10^{22}$ \ensuremath{\rm cm^{-2}}\xspace.
    The beam size and scale bar are shown}
    as in Fig~\ref{fig:NH3_TdV}.
    \label{fig:NH3_abundance}}
\end{figure*}

An important quantity is the molecular fractional abundance of \ce{NH3}\xspace with respect {to} \ce{H2}\xspace,
$X(\ce{NH3}\xspace) = [\ce{NH3}\xspace]/[\ce{H2}\xspace]$.
This quantity is key to accurately determine total {\ce{H2}\xspace} column densities from
\ce{NH3}\xspace observations {when no \ce{H2}\xspace column density map is
available}, 
while it is also important to constrain chemical models \citep[e.g.,][]{Sipila2019-NH3_no_freezeout}.  
The $N(\ce{NH3}\xspace)$ versus $N(\ce{H2}\xspace)$ relation is expected to be linear 
where $X(\ce{NH3}\xspace)$ is constant.  

The comparison of all pixels with an accurate \ce{NH3}\xspace and \ce{H2}\xspace column density determinations is presented in Fig.~\ref{fig:NH3_abundance}-left.
It shows a region with a linear rise followed by a sharp change to a nearly constant $N(\ce{NH3}\xspace)$ as the $\ce{H2}\xspace$ column density increases. 
We have fitted a broken straight line of the form 
\begin{equation}
    N(\ce{NH3}\xspace)[x] = 
    \begin{cases}
    a\cdot x + b & x<x_0 \\
    c\cdot x + (a-c)\cdot x_0 + b & x\ge x_0
    \end{cases}
\end{equation}
to the data points using \verb+emcee+ \citep{EMCEE}.
The results are listed in Table~\ref{tab:X_NH3_fit} and 
shown by the red curve in Fig.~\ref{fig:NH3_abundance}-left, 
while a more detailed 
description of the fit is given in Appendix~\ref{sec:X_fit}.


\begin{deluxetable}{ccc}
\tablecaption{Summary of Linear fit \label{tab:X_NH3_fit}}
\tablewidth{0pt}
\tablehead{
\colhead{Parameter} & \colhead{Value} & \colhead{Unit} 
}
\startdata
$a$ & $1.975_{-0.005}^{0.005}$ & $\times10^{-8}$\\
$b$ & $1.121_{-0.008}^{0.008}$ & $\times10^{14}$ cm$^{-2}$\\
$x_0$ & $2.575_{-0.004}^{0.003}$ & $\times10^{22}$ cm$^{-2}$\\
$c$ & $-0.110_{-0.005}^{0.005}$ & $\times10^{-8}$ \\
\hline
\enddata
{\tablecomments{Due to the systematic calibration uncertainties of the \ce{NH3}\xspace observations, 
we list a 10\% systematic uncertainty to $a$ and $c$ in addition to the statistical one in the text.}}
\end{deluxetable}


A different approach to estimate the molecular fractional abundance is 
to determine the fractional abundance at every pixel by 
taking the ratio of both column density estimates (see the map
in Fig.~\ref{fig:NH3_abundance}-right). 
Here we see a higher $X(\ce{NH3}\xspace)$ 
on the outskirts of the dense core, while in the central region there is
a sharp fall-off in the determined fractional abundance. 
Both the decreasing tendency in the $N(\ce{NH3}\xspace)$ versus $N(\ce{H2}\xspace)$
correlation at high column densities in Fig.~\ref{fig:NH3_abundance}-left
and the minimum in the fractional abundance seen in
Fig.~\ref{fig:NH3_abundance}-right indicate that ammonia is strongly
depleted in the center of the core. 

\section{Discussion}

\subsection{Gas Kinematics}


The velocity dispersion, \ensuremath{\sigma_{\rm v}}\xspace, derived from \ce{NH3}\xspace
observations changes only little within the core and indicates subsonic
non-thermal motions.  The H-MM1 core is thus a ``coherent core'' as
termed by \cite{1998ApJ...504..207B} and \cite{Goodman_1998-coherence} 
\citep[see also][]{Pineda:2010jq}. The kinematics of H-MM1 and other coherent 
cores in the Ophiuchus region have been previously studied using observations 
with the Green Bank Telescope (GBT) by \cite{Auddy2019-Bfield}, 
\cite{Chen2019-Droplets}, and \cite{Choudhury2020-Letter}. 
The thermal r.m.s. 
velocity of \ce{NH3}\xspace molecules at 11\,K is $73\,\rm m\,s^{-1}$ in one dimension. The 
broadening owing to the channel width of the Hanning smoothed spectra, 
$130\,\rm m\,s^{-1}$, corresponds to an instrumental $\sigma$ of $52\,\rm m\,s^{-1}$ 
(see Appendix A of \citealt{Choudhury2020-Letter}). So, the median 
velocity dispersion, 
$\ensuremath{\sigma_{\rm v}}\xspace \sim 110\,\rm m\,s^{-1}$, from the present VLA observations indicates 
a median non-thermal dispersion of $\sigma_{\rm NT} \sim 63\,\rm m\,s^{-1}$, 
or a median sonic Mach number of 
$\sim 0.3$. This value is approximately half of that derived by \cite{Choudhury2020-Letter} 
for the narrow $\ce{NH3}\xspace$ line component from GBT observations with an angular 
resolution of $31\arcsec$. 

\begin{figure}[ht]
    \centering
    \includegraphics[width=\columnwidth]{H-MM1_NH3_dv_CH3OH}
    \caption{Velocity dispersion of $\ce{NH3}\xspace$ lines (color map) and the distribution of bright 96.7\,GHz methanol emission (contours) in H-MM1. The contour levels go from 0.8 to 4.0 K\,$\ensuremath{\rm km\,s^{-1}}\xspace$ by steps of 0.8\,K\,$\ensuremath{\rm km\,s^{-1}}\xspace$ {and alternating between
    blue and purple}.     \label{fig:sigv+meth}}
\end{figure}

The velocity dispersion attains values clearly higher than the median
in a narrow region in the southeastern part of the core. This region
overlaps partially with the southern ``tail'' of the bright $\ensuremath{\rm CH_3OH}$
emission mapped by \cite{Harju2020-HMMS1}. The integrated intensity contours of four methanol lines at 96.7 GHz are overlaid onto the $\ce{NH3}\xspace$ \ensuremath{\sigma_{\rm v}}\xspace map in Fig.~\ref{fig:sigv+meth}. 
However, as can be seen in Fig.~\ref{fig:sigv+meth}, methanol emission 
arises mostly from regions where the dispersion is close to the average. 
Therefore, the present data do not support the idea that efficient $\ensuremath{\rm CH_3OH}$ desorption is only associated with enhanced turbulence or large velocity fluctuations, lending credence to the chemical desorption scenario of methanol formation  at the edge of the catastrophic CO {freeze-out zone} \citep{Vasyunin2017-COMs}. 
Perhaps it is worth to notice, though, that the most quiescent region in the northern part of the core with
$\ensuremath{\sigma_{\rm v}}\xspace \sim 90\,\rm m\,s^{-1}$ (implying $\sigma_{\rm NT} < 10\,\rm m\,s^{-1}$) is
clearly devoid of $\ensuremath{\rm CH_3OH}$.

The elongated redshifted feature seen in the velocity centroid map at  $\ensuremath{V_{\rm LSR}}\xspace > 4.3\,\ensuremath{\rm km\,s^{-1}}\xspace$ seems to connect the compact dense core with the outer regions.
This feature bears resemblance to the streamer detected using gas tracers around binary protostar Per-emb-2 \citep{Pineda2020-Streamer}. 
We will discuss the core kinematics and the nature of this possible streamer in detail in a separate paper. 
 
 
\subsection{Gas Temperature}

The gas temperature shows only a small drop toward the core
center. The temperature 11\,K in the inner parts is consistent with
previous gas and dust temperature determinations using GBT and
{\it Herschel} data 
\citep{Harju2017-HMM1_NH2D,Auddy2019-Bfield,Choudhury2020-Letter}. 
The temperature profile in
H-MM1 is very different from that in the well-studied prestellar core
L1544, where the central temperature reaches down to $\approx 6$\,K
\citep[][]{Crapsi2007-L1544}, 
or the starless core CB17, where the central temperature 
reaches down to $\approx 9.5$\,K \citep{Spear2021-CB17}.
The main difference is that H-MM1 is located in
an environment with strong radiation field  \citep[e.g.,][]{2013MNRAS.428.2617R}. 
Far-infrared radiation which penetrates the core is
absorbed by dust particles, and this heating is conveyed to gas by
gas-dust collisions. The density in the core, in excess of
$10^5\,\rm cm^{-3}$, is high enough to make gas-dust thermal coupling
efficient \citep{2001ApJ...557..736G,2002A&A...394..275G,2019ApJ...884..176I}.

\subsection{Ammonia Depletion}\label{ss:depletion}

Figure~\ref{fig:X_NH3_KDE} shows the Kernel Density Estimates (KDE) 
of the fractional \ce{NH3}\xspace abundance derived.
The mean  fractional \ce{NH3}\xspace abundance, $[\ce{NH3}\xspace]/[\ce{H2}\xspace]_{int}$, derived as the ratio between 
$N(\ce{NH3}\xspace)$ and $N(\ce{H2}\xspace)$ is $2.7\times10^{-8}$ (blue marker in Fig.~\ref{fig:X_NH3_KDE}), 
which is comparable to the values derived in other regions.  
However, this abundance value is 
higher by a factor of 3.5 compared to that derived from the single dish 
observations ($[\ce{NH3}\xspace]/[\ce{H2}\xspace]_{SD} = 7.6\times10^{-9}$, \citealt{Harju2017-HMM1_NH2D}), 
which is shown with the brown marker in Fig.~\ref{fig:X_NH3_KDE}.
The fractional \ce{NH3}\xspace abundance derived from the linear fit 
($[\ce{NH3}\xspace]/[\ce{H2}\xspace]_{fit}={(}1.975{\pm0.005)}\times 10^{-8}${, $\pm$10\% systematic})
is shown by the red marker in Fig.~\ref{fig:X_NH3_KDE}. 
The fractional \ce{NH3}\xspace abundance derived for H-MM1 
using the interferometric data is comparable 
to those derived from 
modelling single dish observations of L1498 and L1517 \citep[][]{Tafalla2004-Cores},
see Table~\ref{tab:X_NH3}.
This important difference between the interferometric and single dish results 
is due to the size of the depletion region, which is comparable 
to the single dish beam size. 
This highlights the crucial importance of 
deep spatially resolved maps of dense cores to study the depletion regions.


\begin{figure}[ht]
    \centering
    \includegraphics[width=\columnwidth]{H-MM1_X_NH3_H2_kde}
    \caption{{Probability density function estimated with} 
    KDE of the fractional \ce{NH3}\xspace abundance towards H-MM1, $[\ce{NH3}\xspace]/[\ce{H2}\xspace]$.
    The value estimated from single dish observations \citep{Harju2017-HMM1_NH2D} 
    is shown by the vertical brown line, $[\ce{NH3}\xspace]/[\ce{H2}\xspace]_{SD}=7.6\times10^{-9}$. The mean value of the column density ratio is marked with a 
    vertical blue line, $[\ce{NH3}\xspace]/[\ce{H2}\xspace]_{int}=2.7\times10^{-8}$,
    and the slope of the linear fit is marked with a red line,
    $[\ce{NH3}\xspace]/[\ce{H2}\xspace]_{fit}=
    {(}1.975{\pm0.005)}\times 10^{-8}$ 
    {($\pm$10\% systematic)}.
    \label{fig:X_NH3_KDE}}
\end{figure}


The fact that $\ce{NH3}\xspace$ is depleted in the central parts of the core is
evident from Fig.~\ref{fig:NH3_abundance}. The failure of
positive correlation between $N(\ce{NH3}\xspace)$ and $N(\ce{H2}\xspace)$ at high column
densities (Fig.~\ref{fig:NH3_abundance}-left), and the deficiency of
ammonia in the core center (Fig.~\ref{fig:NH3_abundance}-right), cannot
be explained by extremely high optical thicknesses. The optical thicknesses of
the four satellite groups of \ce{NH3}\xspace (1,1) are clearly below 1 even toward the
center of the core, implying that the drop in the derived ammonia abundance is 
not an artefact caused by radiative transfer effects.

According to the two-slope fit presented in Sect.~\ref{sec:X_calc}, the break-point
until which the fractional ammonia abundance is $1.975\times10^{-8}$,
occurs at the column density $N(\ce{H2}\xspace)=2.575\times10^{22}\,\ensuremath{\rm cm^{-2}}\xspace$.
We constructed a three-dimensional model of the core by fitting a
Plummer-type function to its horizontal column density profiles 
(cuts parallel {to} the right ascension axis). 
{The inversion method is adopted from
\cite{2011A&A...529L...6A}, and it is explained 
in more detail in Appendix~{\ref{sec:density_model}.}
Using column densities derived
from $8\,\rm \mu m$ absorption, the core model does not include the ambient
cloud visible in the {\it Herschel}/SPIRE maps of this region. We assume
that the offset, $N(\ce{NH3}\xspace)\sim10^{14}\,\ensuremath{\rm cm^{-2}}\xspace$, in the $N(\ce{NH3}\xspace)$ vs.
$N(\ce{H2}\xspace)$ correlation shown in Fig.~\ref{fig:NH3_abundance} is caused by
ammonia residing in this ambient cloud.  
The peak
density of the core model is $n(\ce{H2}\xspace)=1.3\times10^{6}\,\rm cm^{-3}$. 
The relationship between the densities and column densities in this
model {is} shown in Fig.~\ref{fig:col2dens}. In this correlation, we
only include cells that have densities higher than
$5\times10^4\,\rm cm^{-3}$. 
{The choice of the threshold density is based on the high
excitation temperatures of the {\ce{NH3}} lines ($>8$\,K, Fig.\,5-right),
which indicate that the emission is dominated by high-density gas.
According to the excitation calculations presented in 
\cite{Shirley2015-molecules} (their Fig.\,2) for
gas at $T_{\rm kin}=10$\,K, the $T_{\rm ex}$ of the $\ce{NH3}\xspace(1,1)$
transition exceeds 8\,K at the density $n(\ce{H2}\xspace)\sim
5\times10^4\,\rm cm^{-3}$. Using a lower density threshold would decrease
the mean densities but leave the power law, ${\bar n} \propto N^{0.8}$, unchanged.}


Line segments through the core having a column density around
$N(\ce{H2}\xspace)=2.6\times10^{22}\,\ensuremath{\rm cm^{-2}}\xspace$ where $X(\ce{NH3}\xspace)$ turns
down, have an average density of $(2.1\pm0.1)\times10^{5}\,\rm cm^{-3}$.
One can therefore assume that the depletion of $\ce{NH3}\xspace$ starts to take
effect at this density, whereas at lower densities the fractional
ammonia abundance is nearly constant, $2\times10^{-8}$. 


\begin{figure*}[htb]
    \centering
    \includegraphics[height=6cm]{hmm1_col2dens}
    \includegraphics[height=6cm]{hmm1_x_vs_dens}
    \caption{{\it Left:} Gas density, $n(\ce{H2}\xspace)$, as a function of the
    $\ce{H2}\xspace$ column density in a 3-D model of the H-MM1 core (see text).
    The shaded regions show the densities between the 5th and 95th percentile 
    (pale blue) and 16th and 84th percentile (turquoise) within a certain column
    density bin. The red line shows a linear fit to the weighted mean
    densities,  $\bar{n}(\ce{H2}\xspace)$, where each line of sight (LOS) is weighted 
    according to standard deviation of the density ($1/\sigma_n^2$) along that
    LOS. The error bars correspond to the weighted sigmas of the sample densities. 
    The median density in each $N(\ce{H2}\xspace)$ bin is {shown} with small 
    horizontal bars. 
    {\it Right:} Fractional $\ce{NH3}\xspace$ abundance as a function of gas
    density in H-MM1. The relationship is derived by fitting a broken
    line to mean $\ce{NH3}\xspace$ column densities below and above the turning
    point $N(\ce{H2}\xspace)=2.6\times10^{22}\,\ensuremath{\rm cm^{-2}}\xspace$ (see Sect.\,\ref{sec:X_calc}), and using a
    linear fit to $\bar{n}(\ce{H2}\xspace)$ vs. $N(\ce{H2}\xspace)$ on the left. Also shown
    are predictions from our chemistry model including chemical 
    desorption (CD, diamonds) and CD turned off 
    (plus signs; see text). 
    \label{fig:col2dens}}
\end{figure*}

The broken-line fit to the $N(\ce{NH3}\xspace)$ vs. $N(\ce{H2}\xspace)$ 
{distribution} (Sect.~\ref{sec:X_calc})
and the inversion to mean densities described above, give the
relationship $X(\ce{NH3}\xspace) \propto n^{-1.1}$ at high densities. The
dependence of the fractional $\ce{NH3}\xspace$ abundance on the density is
shown in Fig.~\ref{fig:col2dens}-right. 
The density threshold adopted in the conversion from column densities to average volume densities affects the break-point density in this diagram.
{Applying no threshold density would shift the break point to $1.0\times10^5\,\rm cm^{-3}$. 
However, the slope of the decreasing abundance
would remain the same. We note that the pivotal density as
well as the fractional $\ce{NH3}\xspace$ abundance at low densities are likely
to depend on the temperature, the external radiation field, and the
chemical age of the core, and they probably change from core to
core.} 

Also shown in {Fig.~\ref{fig:col2dens}-right} are data points
from a series of one-point chemistry models at different densities with the gas and dust temperatures fixed to 11\,K. The chemistry code is described in Sect.~\ref{ss:modeling}, and the simulations used to investigate the density dependence are described in Sect.~\ref{ss:density_dependence}. 

\begin{deluxetable}{cccc}
\tablecaption{Summary of Reported \ce{NH3}\xspace Abundances\label{tab:X_NH3}}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{Method} & \colhead{[\ce{NH3}\xspace/\ce{H2}\xspace]} 
& \colhead{Ref.} 
}
\startdata
H-MM1 & Int. Col.Rat. & $2.7\times10^{-8}$ & This work\\
H-MM1 & Int. Linear Fit & $(1.975\pm0.005)\times10^{-8}$ & This work\\
\hline
H-MM1 & SD Col.Rat. & $(0.76\pm0.02)\times10^{-8}$ & 1\\
L1498 & SD Mod. & $3.4 \times10^{-8}$ & 2\\
L1517B& SD Mod. & $2.8 \times10^{-8}$ & 2\\
L1688 & SD Col.Rat. & (0.3--3)$\times10^{-8}$ & 3\\
L1544\tablenotemark{a} & Int. Mod. & (0.3--0.8)$ \times10^{-8}$ & 4\\
\enddata
\tablenotetext{a}{An abundance profile is reported.}
\tablerefs{
(1) \cite{Harju2017-HMM1_NH2D}, 
(2) \cite{Tafalla2004-Cores},
(3) \cite{Friesen2017}, 
(4) \cite{Crapsi2007-L1544}.
}
\tablecomments{
SD = Single Dish data used; 
Int. = Interferometric data used.;
Col.Rat. = Column Density Ratio used to determine fractional abundance;
Mod. = fractional abundance constrained via radiative transfer modelling.
}
\end{deluxetable}


\subsection{Chemistry of ammonia}
\label{ss:ammonia_chemistry}

The formation of ammonia in the interstellar gas is 
recapitulated in, for example, \cite{2015A&A...576A..99R} and
\cite{Sipila15b}. Once the \ce{NH+} ion is available, a sequence of
reactions with \ce{H2}\xspace lead quickly to the ammonium ion, \ce{NH4+},
which is the principal precursor of \ce{NH3}\xspace. Electron
recombination of \ce{NH4+} and smaller cations give the
nitrogen hydrides \ce{NH}, \ce{NH2}  and \ce{NH3}\xspace.  Ammonia is
expected to be the most abundant of these in dense cores.
The pivotal ion, \ce{NH+}, is formed by the charge transfer reaction
\ce{NH + H+ -> NH+ + H}
or by the reaction 
\ce{N+ + ortho-H2 -> NH+ + H}.
In dark cloud 
conditions, the \ce{N+} ion forms by the dissociative charge 
transfer reaction 
\ce{N2 + He+ -> N+ + N + He}
\citep{2010A&A...513A..41H, 2020A&A...643A..76H}.

In the gas, nitrogen hydrides are constantly converted to
cations through charge or proton exchange reactions, and returned to
neutral species in recombination reactions with electrons.
The dominant ions reacting with ammonia are \ce{H+} and \ce{H3+}
with approximately equal destruction rates, and the ionic species
formed in these reactions, \ce{NH3+} and \ce{NH4+}, are quickly
returned to ammonia; \ce{NH3+} reacts with \ce{H2}\xspace to form
\ce{NH4+}, which gives mainly \ce{NH3}\xspace in recombination with  electrons
(see Fig.~2 in \citealt{Sipila15b}).

Neutral nitrogen hydrides, as well as nitrogen atoms and 
molecular nitrogen, are also likely to be adsorbed onto dust
grains, and this should eventually diminish the abundances of
ammonia and related species, unless they are returned into the
gas to the same degree. The accreted \ce{N} atoms, and the \ce{NH} and 
\ce{NH2} radicals, are efficiently hydrogenated to \ce{NH3}\xspace,
and most nitrogen in interstellar ices is probably bound to
ammonia. As a consequence, ammonia is much more abundant in the
icy mantles of dust grains than in the gas phase, and the ice
mantles constitute the biggest ammonia reservoir in dense
molecular clouds. In quiescent, shielded regions, the most
important mechanisms releasing part of this ammonia probably 
are the cosmic-ray-induced desorption 
\citep[e.g.,][]{1985A&A...144..147L} and reactive desorption 
\citep[e.g.,][]{2006FaDi..133...51G}. At very high densities, where the accretion timescale is short, the gas-phase abundance of ammonia probably depends on the balance between accretion and desorption.  


\subsection{Chemical Modeling}
\label{ss:modeling}

To aid the interpretation of the present observational results which
indicate strong depletion of ammonia, we simulated the chemistry of 
\ce{NH3}\xspace in H-MM1 using our rate-equation gas-grain astrochemical model
\citep[e.g.,][]{Sipila15b,Sipila19b}. In brief, the model solves a set of
rate equations connecting chemical reactions in the gas phase and on
grain surfaces. Several desorption mechanisms are considered: thermal
desorption, cosmic-ray induced desorption, photodesorption, and chemical
desorption. We employ a one-dimensional physical model that was
constructed by extracting density and dust temperature cuts through the
density peak of the three-dimensional model described in
Sect.\,\ref{ss:depletion} and making these cuts symmetric with respect to
the density peak by averaging. The dust temperature distribution of the
three-dimensional model was calculated by exposing the density model to
an external radiation field composed of a) diffuse isotropic
interstellar radiation, and b) radiation from B-type stars located 
$\sim$1\,pc west of the core. The intensities of these components were
adjusted until the 850\,$\rm \mu m$ surface brightness map from the model
agreed with the SCUBA-2 map and the average dust temperature map agreed
with the color temperature $T_{\rm C}$ map derived from {\it Herschel}. We 
discretized the physical model into a series of points, and ran the
chemical simulations separately in each point to obtain time- and 
radially-dependent \ce{NH3}\xspace abundance profiles. We employ here the same
initial chemical abundances and other chemical model parameters as in
\citet{Harju2017-HMM1_NH2D}. The physical model is shown in
Fig.\,\ref{fig:HMM1_physicalModel}, and the initial abundances 
are listed in Table~\ref{tab:initialabundances}.

\begin{table}[t]
	\centering
	\caption{Initial chemical abundances (with respect to $n_{\rm H}$).}
	\begin{tabular}{cc}
		\hline
		\hline
		Species & Abundance\\
		\hline
		$\rm H_2$ & $5.00\times10^{-1} \,^{(a)}$\\
		$\rm He$ & $9.00\times10^{-2}$\\
		$\rm C^+$ & $7.30\times10^{-5}$\\
		$\rm N$ & $5.30\times10^{-5}$\\
		$\rm O$ & $1.76\times10^{-4}$\\
		$\rm S^+$ & $8.00\times10^{-8}$\\
		$\rm Si^+$ & $8.00\times10^{-9}$\\
		$\rm Na^+$ & $2.00\times10^{-9}$\\
		$\rm Mg^+$ & $7.00\times10^{-9}$\\
		$\rm Fe^+$ & $3.00\times10^{-9}$\\
		$\rm P^+$ & $2.00\times10^{-10}$\\
		$\rm Cl^+$ & $1.00\times10^{-9}$\\
		\hline
	\end{tabular}
	\tablenotetext{a}{The initial $\rm H_2$ ortho/para ratio is $1 \times 10^{-3}$.}
	\label{tab:initialabundances}
\end{table}

\begin{figure}
\centering
        \includegraphics[width=1.0\columnwidth]{pNH3vsH2}
    \caption{Physical model of H-MM1 adopted in chemical simulation: 
    \ce{H2}\xspace number density (green) as well as the gas (blue) and dust (red) 
    temperatures as functions of the radial distance from the center.}
    \label{fig:HMM1_physicalModel}
\end{figure}

Based on the derived abundance profiles, we then simulated the \ce{NH3}\xspace 
(1,1) and (2,2) lines using the radiative transfer code LOC \citep{Juvela20}. 
The line profiles and the column density were convolved to
the angular resolution of the observations (6$\arcsec$).
In \citet{Harju2017-HMM1_NH2D}, we derived a best-fit simulation time of
$3 \times 10^5 \, \rm yr$ based on fitting multiple deuterated ammonia
lines. For consistency we adopt the same time here. 


\begin{figure*}
\centering
        \includegraphics[width=2.0\columnwidth]{pNH3_MAP_PAPER}
    \caption{Simulated p\ce{NH3}\xspace column density (left), 
    and integrated intensity ((1,1) and (2,2); middle and right panels, respectively) maps. 
    The white circle in each panel indicates a region of 2000\,au diameter.}
    \label{fig:HMM1_model}
\end{figure*}


Figure~\ref{fig:HMM1_model} shows the total \ce{NH3}\xspace column density
(summed over ortho and para \ce{NH3}\xspace) and integrated intensity maps
obtained at this time step. The plots are displayed using the same
scaling for the color bar as in
Figs.\,\ref{fig:NH3_TdV} and \ref{fig:column_maps}, and we have added
$10^{14} \,\ensuremath{\rm cm^{-2}}\xspace$ to the simulated column density to account for
the ammonia in the outer cloud that is not included in the core model
(see Sect.\,\ref{ss:depletion}). The match between the model
and the observations is quite good in the innermost region of the maps,
which represents the volume immediately surrounding the \ce{H2}\xspace column
density peak in H-MM1, even if the model somewhat under-produces the
emission in both the (1,1) and the (2,2). Although not very clearly
visible in the plot, both the column density and the integrated intensity
maps present a dip toward the center of the core. Naturally, the model
does not recover the morphology of the observed column density or
integrated intensity maps outside the central region as we are using a
spherically symmetric 1D model. The strength of the line emission is
sensitive to uncertainties in the gas temperature in the outer core where
the lines are produced.

The fractional \ce{NH3}\xspace abundance is higher near the western edge of
the core than at the eastern edge (Fig.~\ref{fig:NH3_abundance}-right). 
This asymmetry can
probably be attributed to the fact that the temperature gradient is
steeper on the western side owing to intense radiation from that
direction. We examined this hypothesis by using two one-dimensional
models, which we call ``east'' and ``west'', with the density and
temperature profiles corresponding to the eastern and western sides of
the core (instead of a symmetrized profile used above). The steeper
rise of the temperature as a function of the distance from the core
center in the ``west'' model was also accompanied with a steeper
increase of the \ce{NH3}\xspace abundance as compared to the ``east'' model. 




\subsection{Density dependence of the ammonia abundance}
\label{ss:density_dependence}

As discussed in Section~\ref{ss:ammonia_chemistry}, the dominant
gas-phase reactions involving ammonia produce closely related molecular
ions that are quickly returned back to \ce{NH3}\xspace. Therefore, the diminishing
ammonia abundance at high densities must ultimately be caused by
adsorption of \ce{NH3}\xspace and related species, mainly \ce{NH}, \ce{N}, and \ce{N2},
onto grains. The rate at which an atom {or} molecule accretes onto grains
is proportional to its average thermal speed and the total surface 
area of dust grains.  The latter can be expressed as the
product of the grain surface area per H atom, $\sigma_{\rm H}$ , and
the total hydrogen density, $n_{\rm H}$ ($\sim 2\times n(\ce{H2}\xspace)$). 
{In other words, the accretion rate is proportional to the density.}

We examined the dependence of $X(\ce{NH3}\xspace)$ and related species on
$n(\ce{H2}\xspace)$ by employing our full chemical network. This was done by
running a series of one-point models at different densities with the
gas and dust temperatures fixed to 11\,K. Ammonia abundances were
extracted at the simulation time $3\times10^5$\,yr. Two different
set-ups were used: one with chemical desorption (CD) and another with
CD turned off. The results are shown in Fig.~\ref{fig:model_x_vs_n}. 
One can see that at densities above $\sim 10^5\,\rm cm^{-3}$, $X(\ce{NH3}\xspace)$
follows the decreasing abundances of \ce{N} and \ce{N2}. The slope of the
$\log X(\ce{NH3}\xspace)$ vs. $\log n(\ce{H2}\xspace)$ curve is $-1.04\pm0.01$ in the model
with CD, and a little steeper, $\sim -1.15\pm0.04$, when CD is turned off. 
The dominance of accretion leads to a density
dependence close to $n^{-1}$ for neutral species, whereas 
electrons and light ions (\ce{H+} and \ce{H3+}) show a shallower
power law, $\sim n^{-0.44}$ in our models. The latter is determined by
the competition between the cosmic ray ionization and neutralization in 
recombination reactions.  

\begin{figure*}[htb]
    \centering
    \includegraphics[height=6cm]{x_vs_dens_nocd}
    \includegraphics[height=6cm]{x_vs_dens_cd}
    \caption{Modeled fractional abundances of selected species as functions of the volume density at the time $3\times10^5$\,yr after the beginning of the simulation. 
    {\it Left:} A model without chemical desorption (CD). 
    {\it Right:}  A model with CD turned on, assuming that 1\% of exothermic surface reactions will lead to the evaporation of the product. 
    \label{fig:model_x_vs_n}}
\end{figure*}

According to our model, most of ammonia is to be found on dust grains, where it reaches a
fractional abundance of $\sim10^{-4}$ relative to $\ce{H2}\xspace$, and stays
almost constant to the end of the simulation.  
Replenishment of the
gas with ammonia from grains is significant when CD is
effective. Consequently, $X(\ce{NH3}\xspace)$ decreases more slowly and at higher
densities in the model with CD than in that without CD. 

Inspection of reaction rates during our chemical simulation shows that
the density $5\times10^{4}\,\rm cm^{-3}$, where the \ce{NH3}\xspace abundance turns
down, is where the adsorption rate of ammonia exceeds the production
of \ce{NH+} ions in the gas. In the model with CD, desorption from
grains always gives more ammonia than ion-molecule reactions, but the
adsorption rate surpasses the desorption rate at a density of
$\sim 1\times10^{5}\,\rm cm^{-3}$. At still higher densities, above
$n(\ce{H2}\xspace)=2\times10^{5}\,\rm cm^{-3}$, a balance between the primary
production of gaseous ammonia and its adsorption is established: 
the adsorption rate is equal to the combined rates of desorption and
\ce{NH+} production up to the highest density modeled, $10^6\,\rm cm^{-3}$.
In our CD model, desorption accounts for $\sim 80\%$ of
ammonia production in this density range. 

The dominant surface reaction releasing ammonia into the gas in this
model is 
\ce{NH2^* + H^* -> NH3}.
Here we have indicated
species residing on the surface with asterisks. The mentioned reaction is the
end point of hydrogenation of N atoms trapped on grains. On the other
hand, a majority of \ce{NH2^*} molecules on the surface is formed by
dissociation of $\ce{NH3^*}\xspace$, either  by cosmic rays or by secondary photons
induced by these.  We can therefore present a simplified picture, where 
desorption is proportional to the density of ammonia residing on grains
and a desorption coefficient. This picture is also valid for direct
ammonia desorption caused by external agents, such as grain heating
through cosmic-ray hits. 


{The equilibrium condition between adsorption and desorption
can be written as $k_{\rm ads}\, n(\ce{NH3}\xspace) = k_{\rm des}\, n(\ce{NH3^*}\xspace)$,
where $k_{\rm ads}= n_{\rm H} \sigma_{\rm H} {\bar v}_{\ce{NH3}\xspace}$ is the
adsorption rate per molecule ($\rm s^{-1}$), ${\bar v}_{\ce{NH3}\xspace}$ is the
mean thermal speed of ammonia molecules, and $k_{\rm des}$ is the
desorption rate per molecule ($\rm s^{-1}$). The ammonia densities in the
gas and on grains (per cubic centimeter) have been denoted by $n(\ce{NH3}\xspace)$ and
$n(\ce{NH3^*}\xspace)$. Dividing the equilibrium equation by $n(\ce{H2}\xspace)$ and
re-arranging the terms one gets for the fractional ammonia abundance
$X(\ce{NH3}\xspace) = k_{\textrm{des}} \, X(\ce{NH3^*}\xspace)/(\sigma_{\rm H}\, {\bar v}_{\ce{NH3}\xspace}
\,n_{\rm H})$. According to our chemistry model, the ammonia abundance in ice 
exceeds the gas-phase abundance by several orders of magnitude in starless dense cores, 
and accretion of ammonia from the gas does not change $X(\ce{NH3^*}\xspace)$ significantly 
at this stage. Assuming that also the desorption rate is approximately constant, 
balance between adsorption and desorption leads to an $n^{-1}$ dependence of
$X(\ce{NH3}\xspace)$ in the gas. }

{As can be seen from Fig.~\ref{fig:col2dens}-right, both} 
model set-ups, with or without CD, fail to reproduce
the observed abundances at low densities. This probably depends on
the fact that the simulation starts from atomic gas composition in
physical conditions appropriate for a dense core. In reality, the
dense core material has already been processed in ambient molecular
cloud conditions with $n(\ce{H2}\xspace)\sim 10^3-10^4\,\rm cm^{-3}$, where ammonia
production on grains and in the gas has already started, and a 
substantial fraction of nitrogen is already incorporated in ammonia ice. 

\section{Conclusions}

We present high spatial resolution ammonia observations with the Very Large Array and 
Green Bank Telescope 
toward the starless core H-MM1 in Ophiuchus.
They indicate that ammonia has
a constant fractional abundance, $X(\ce{NH3}\xspace)={(}1.975{\pm0.005)}\times 10^{-8}$ 
{($\pm$10\% systematic)}, 
up to a column density of $N(\ce{H2}\xspace) = 2.575\times 10^{22}\,\ensuremath{\rm cm^{-2}}\xspace$,
beyond which the abundance starts to decrease.
This is direct evidence of the \ce{NH3}\xspace depletion from the gas phase, 
in a region of size comparable to the single dish beam size, which 
explains why the depletion region is only detected 
in deep high-angular resolution observations.
By
deriving a volume density model of the core we established that the
critical density where \ce{NH3}\xspace starts to disappear from the gas is
approximately $2\times10^5\,\rm cm^{-3}$ in this core, and at higher
densities $X(\ce{NH3}\xspace)$ follows the power-law $X(\ce{NH3}\xspace)\propto n^{-1.1}$. This
tendency at high densities is reproduced by chemical simulations, and
is consistent with the idea that adsorption onto grains is the main
mechanism {reducing the gas-phase abundances} of neutral molecules 
in the core center.  Our
chemistry simulations suggest that the break-point in $X(\ce{NH3}\xspace)$
versus $n(\ce{H2}\xspace)$ relationship marks the density where the adsorption
rate surpasses the production rate of gaseous ammonia through
ion-molecule reactions and desorption.  The density threshold for
depletion is likely to vary from core to core because it depends on
the temperature and the total grain surface area per hydrogen atom.

\acknowledgments
We thank the referee for their thoughtful comments which 
improved the paper.
JEP, JH, PC, OS, MJM, SP, and DMS acknowledge the support by the 
Max Planck Society. 
DMS is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2102405.

\facilities{VLA, GBT} 

\software{Aplpy \citep{aplpy,aplpy2019}, 
Astropy \citep{Robitaille_2013,Astropy2018}, 
Matplotlib \citep{Hunter_2007},
SciPy \citep{2020SciPy-NMeth},
pyspeckit \citep{pyspeckit},
spectral-cube \citep{spectral-cube},
EMCEE \citep{EMCEE},
corner  \citep{corner}}


\bibliographystyle{aasjournal}
