syn-pdfQA/01.1_Input_Files_Non_PDF/research articles/2510.23000v1.tex

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\begin{document}

\title{Spatially Resolved, Multiphase Mass Outflows of the Seyfert 1 Galaxy NGC 3227}

\author[0000-0001-7238-7062]{Julia Falcone}
\affiliation{Department of Physics and Astronomy, Georgia State University, 25 Park Place, Atlanta, GA 30303, USA}
\email{jfalcone2@gsu.edu}

\author[0000-0002-6465-3639]{D. Michael Crenshaw}
\affiliation{Department of Physics and Astronomy, Georgia State University, 25 Park Place, Atlanta, GA 30303, USA}
\email{dcrenshaw@gsu.edu}

\author[0000-0002-4917-7873]{Mitchell Revalski}
\affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA}
\email{mrevalski@stsci.edu}

\author[0000-0002-3365-8875]{Travis C. Fischer}
\affiliation{AURA for ESA, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA}
\email{tfischer@stsci.edu}

\author[0000-0001-8658-2723]{Beena Meena}
\affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA}
\email{bmeena@stsci.edu}

\author[0009-0005-3001-9989]{Maura Kathleen Shea}
\affiliation{Department of Physics and Astronomy, Georgia State University, 25 Park Place, Atlanta, GA 30303, USA}
\email{mshea3@gsu.edu}

%\author[0000-0003-0483-3723]{Rogemar A. Riffel}
%\affiliation{Departamento de Física, CCNE, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS, Brazil}


\author[0000-0002-2713-8857]{Jacob Tutterow}
\affiliation{Department of Physics and Astronomy, Georgia State University, 25 Park Place, Atlanta, GA 30303, USA}
\email{jtutterow1@gsu.edu}

\author[0000-0003-3401-3590]{Zo Chapman}
\affiliation{College of Computer Science, Georgia Institute of Technology, 266 Ferst Drive, Atlanta, GA 30332, USA}
\email{zoechapman147@gmail.com}


% \author[0009-0005-2145-4647]{Madeline Davis}
% \affiliation{Department of Physics and Astronomy, College of Charleston, 66 George Street,
% Charleston, SC 29424, USA}
% \email{mdavis299@gsu.edu}

\author[0009-0005-2145-4647]{Kesha Patel}
\affiliation{Department of Physics and Astronomy, Emory University, 400 Dowman Drive,
Atlanta, GA 30322, USA}
\email{kesha.patel@emory.edu}


\correspondingauthor{Julia Falcone}
\email{jfalcone2@gsu.edu}

%\author[0000-0000-0000-0000]{...}
%\affiliation{Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30302-4106, USA}

%\collaboration{1}{(...)}
    
\begin{abstract}
We present spatially resolved mass outflow rates of the ionized and molecular gas in the narrow line region of the Seyfert 1 galaxy NGC 3227. Using long-slit spectroscopy and [O~III] imaging from from Hubble Space Telescope's Space Telescope Imaging Spectrograph and Apache Point Observatory’s Kitt Peak Ohio State Multi-Object Spectrograph, in conjunction with Cloudy photoionization models and emission line diagnostics, we find a peak ionized mass outflow rate of $\dot M_{\text{ion}} =$ $19.9\pm9.2$ M$_\odot$ yr$^{-1}$ at a distance of $47\pm6$ pc from the supermassive black hole (SMBH). Using archival data from the Gemini-North Near-infrared Field Spectrograph measuring H$_2$~$\lambda2.1218$ $\mu$m emission, we find a maximum peak warm molecular outflow rate of $\dot M_{\mathrm{H_2}} \le 9 \times 10^{-4}$ M$_\odot$ yr$^{-1}$ at a distance of $36\pm6$ pc from the SMBH. Using archival data from the Atacama Large Millimeter/submillimeter Array measuring CO(2-1) emission, we find a maximum peak cold molecular gas mass outflow rate of $\dot M_{\mathrm{CO}} \le$ $23.1$ M$_\odot$ year$^{-1}$ at a distance of $57\pm6$ pc from the SMBH. For the first time, we calculate spatially resolved gas evacuation timescales for the cold molecular gas reservoirs ostensibly sourcing the outflows, and find that evacuating gas to $\sim$400 pc from the SMBH occurs on timescales of $10^{6.0} - 10^{7.6}$ years. These results indicate that the multi-phase AGN outflows are effective in clearing the inner few hundred parsecs of NGC 3227's gas content on timescales that may set the AGN duty cycle of $10^5 - 10^8$ years.
\end{abstract}

\keywords{Active galactic nuclei (16) -- AGN host galaxies (2017) -- Seyfert galaxies (1447) -- Emission line galaxies (459) -- Galaxy winds (626) -- Galaxy kinematics (602) -- Supermassive black holes (1663)}


%\keywords{galaxies: active — galaxies: individual (NGC 3227, NGC 3226) ― galaxies: Seyfert ― ISM: jets and outflows}

\section{Introduction}

At the center of nearly every galaxy lies a supermassive black hole (SMBH). In the nearby Universe, a small fraction (5--10\%) of SMBHs are active galactic nuclei (AGN), which are actively accreting surrounding gas and facilitating a complicated feedback process between the SMBH and its host galaxy. This dynamic occurs as enormous amounts of energy are released from the accretion process, resulting in outflows of gas that can evacuate reservoirs of potential star-forming gas from the galactic bulges \citep{ciotti01, heckman14, piotrowska21, booth09, angles17} and potentially suppress star formation within the galaxy \citep{fischer17, fischer18, revalski21, venturi21}. 

\begin{figure}[t]
\centering  
\includegraphics[width=0.9\linewidth]{fig/paper_2_ngc_3227_w_judy_schmidt.pdf}
\caption{Optical image of NGC~3227 (left of center) and NGC~3226 (above center) taken by the Sloan Digital Sky Survey with \textit{ugriz} filters. The inset shows a color-composite image of the inner region surrounding NGC 3227's AGN, where red colors are from the HST WFC3/UVIS F658N filter, green colors are from the HST WFC3/UVIS F547M filter, and blue colors show F550M and F330W filters from Hubble's Advanced Camera for Surveys High Resolution Channel. Image credit: NASA / ESA / Judy Schmidt.} 
\label{fig: 3227}
\end{figure}


A category of outflow known as AGN winds operate across a wide range of spatial (sub-parsec to kiloparsec) scales and different gas phases. The propagation of these winds has been extensively studied \citep{antonucci85, pedlar93, nelson00, pogge88, travisthesis}. Geometric and kinematic models show that on large scales, the ionized gas travels along a biconical geometry with the central vertex of the bicone coinciding with the AGN \citep{crenshaw00a, crenshaw00b}. 
%The winds are likely instrumental towards understanding the symbiotic evolutionary relationship between SMBHs and their host galaxies \citep{ferrarese00, kormendy13}, and may ultimately play a key role in better understanding the various processes affecting galaxy evolution and galactic structure \citep{okamoto05, croton06, angles17}.   

To study the feedback mechanisms operating within galaxies, we perform studies on Seyfert galaxies \citep{seyfert43}, which are nearby ($z \leq 0.1$), and contain moderate-luminosity AGN ($L_{bol} \approx 10^{43} - 10^{45}$ erg s$^{-1}$).  Within Seyfert galaxies, we can study the impact of outflows and feedback by focusing on the kinematics of the narrow emission line region (NLR), which is composed of ionized gas at distances 1--1000 pc from the SMBH and has hydrogen densities ranging from $n_\mathrm{H} \approx 10^2 - 10^6$ cm$^{-3}$ \citep{revalski22}.  NLR outflows are a form of AGN winds that result from direct ionization and removal of reservoirs of cold molecular gas in the central regions of the galaxy, which would otherwise be available for star formation or fueling the SMBH \citep{storchi10, muller11, travisthesis, fischer17, king15, meena23}. 

Spatially resolved studies of NLR outflows are crucial because they connect the processes at the smallest scales (parsecs from the SMBH) to those on the scale of the galactic bulge (kiloparsecs) and beyond, and effectively lay the foundation for mechanisms that facilitate AGN feedback processes and affect galaxy evolution \citep{okamoto05, croton06, angles17}.
Specifically, the gas is likely pushed away from the nucleus via radiative driving \citep{proga00, das07, ramirez12, meena21}, in which the AGN-induced radiative acceleration and gravitational deceleration from the galaxy and SMBH control the velocity and physical extent of the NLR outflows.  

However, to more deeply understand the interplay between the outflows and the host galaxy, it is necessary to analyze not only the gas kinematics, but also the mass outflow rates that detail how the gas is evacuated by the outflows and subsequently distributed into the interstellar medium (ISM) \citep{dallagnol21, davies20, esposito24, revalski18a, revalski18b, revalski21, revalski22, trindade21}. The spatially resolved mass outflow rate directly addresses this question by quantifying the feedback as a function of distance from the SMBH, ultimately revealing crucial information such as the rates and timescales upon which this gas is evacuated from the NLR. With this information, we can characterize the effectiveness of the outflows in evacuating the cold gas reservoirs situated within the NLR, which informs us about the properties of the AGN phase lifetime (also known as the duty cycle).

Historically, studies of Seyfert galaxies have found a wide range of mass outflow rates and energetics over the spatial extent of the NLR \citep{barbosa09, storchi10, riffel09, muller11}. However, as a result of limited spatial resolution, these studies have predominantly produced global mass outflow rates represented by single (or occasionally a few) values. These studies are beneficial in that their results can be obtained relatively quickly; however, they rely on assumptions about the gas density and distribution that can significantly overestimate the outflow rates \citep{karouzos16, bischetti17}. To accurately account for the spatial distribution of the ionized gas, we utilize high-resolution ($<$0\farcs3) observations to calculate spatially resolved mass profiles and mass outflow rates. 

% \begin{figure*}[t]
% \label{fig: contour plots}
% \centering  
% \subfigure{\includegraphics[width=0.48\linewidth]{fig/OIII contours paper 2.pdf}\label{fig: OIII contours}}
% \hspace{5mm}
% \subfigure{\includegraphics[width=0.45\linewidth]{fig/ALMA plot paper 2.pdf}\label{fig: ALMA contours}}
%  \caption{(a) A contour plot of a continuum-subtracted 6\farcs5 $\times$ 6\farcs5 HST WFC3 F502N image of the nucleus, which shows the small-scale structure of the [O~III]~$\lambda5007$ emission. The solid cyan lines represent the outline of the KOSMOS slit oriented along the galactic minor axis, which we use in our analysis. The dashed solid lines represent the KOSMOS slits oriented along the galactic major (PA = 150\arcdeg) and outflow (PA = 190\arcdeg) axes. (b) A contour plot of the ALMA CO(2-1) flux density, originally shown in \cite{alonso19}. The physical orientation is the same as in (a). In both, the star in the center shows the location of the bright Seyfert 1 continuum source, which corresponds to the AGN.}

% \end{figure*}





Recent studies of spatially resolved mass outflow rates for ionized and molecular gas have uncovered radial trends in the distributions of the mass, kinetic energy, and their outflow rates \citep{garcia14, crenshaw15, morgianti15, alonso19, bischetti19, zanchettin21, ramos22, revalski18b,revalski21, revalski22, revalski25, marconcini25}. However, most work on spatially resolved mass outflow rates has focused on a single gas phase (typically ionized or molecular), which provides an incomplete picture of the multiphase AGN winds \citep{cicone18}. With a multiphase, multiscale study, we can better understand the effectiveness of the feedback mechanisms based on how the various phases compare with one another throughout the NLR \citep{fluetsch19, shimizu19, garcia21, ramos22, zanchettin23, travascio24, speranza24, esposito24}.

This paper continues the work of \cite{falcone24}, hereafter referred to as Paper I. In Paper I, we performed a kinematic analysis on the ionized, neutral, and warm molecular gas in the Seyfert 1 galaxy NGC~3227 ($z$ = 0.003859), shown in Figure \ref{fig: 3227}. This nearby ($D = 23.7 \pm 2.6$ Mpc; \citealp{tonry01, blakeslee01}), SAB(s)a-type galaxy \citep{devaucouleurs91} has a weakly-barred structure with an interior spiral and strong evidence of interaction with its dwarf elliptical companion NGC~3226 \citep{rubin68, mundell95}. We used the ionized kinematics to determine the orientation of the outflowing bicone and conclude that radiative driving is the dominant acceleration mechanism for the NLR outflows in NGC~3227. 
This paper utilizes those findings to develop spatially-resolved mass and mass outflow rate profiles for three gas phases in NGC 3227, from which we will estimate the rate at which gas evacuation from the nuclear region occurs.
%further develop our understanding of the gas motions within NGC~3227, which allows us to paint a more detailed picture of the AGN feedback paradigm. 

%In this study, we utilize the results of Paper I to model the gas masses and outflow rates in the cold molecular, warm molecular, and ionized gas phases for NGC 3227. This study joins a small but growing number of studies focused on spatially resolved mass outflow rates \citep{revalski18b, revalski21, revalski22}, even fewer of which are multiphase \citep{esposito24}. By implementing photoionization modeling to reduce assumptions and biases

%As these reservoirs are foundational to the fuelling of both feedback processes and star formation processes, understanding their relationship to 

%The timescales over which these winds operate, which depends on the rate of accretion into the SMBH, vary from days to hours (cite) to scales the order of $\sim 10^5$ years \citep{king15, schawinski15}.


%The propagation of the winds has been extensively studied \citep{antonucci85, pedlar93, nelson00, pogge88, travisthesis}, and geometric and kinematic models show that the ionized gas travels along a biconical geometry, with the central vertex of the bicone intersecting the AGN. 

\section{Observations}
\label{sec: obs}
We use a combination of photometric and spectroscopic data with high ($\le$ 0\farcs 3) spatial resolution to map the ionized, warm molecular, and cold molecular gas. To characterize the ionized gas, we utilize the Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) to analyze [O~III]~$\lambda5007$ images that map the entirety of the NLR \citep{amanda24, marinelli24}. We use a F547M image, which has a spectral range of 5060–-5885 \AA, as the continuum to subtract from the line emission of the F502N image, which has a spectral range of 4969–-5044 \AA. Both images were obtained from HST program ID 16246 (PI: M. Revalski). We complement these photometric data with spectroscopy we obtained from the Kitt Peak Ohio State Multi-Object Spectrograph (KOSMOS) on the ARC 3.5-meter telescope at Apache Point Observatory,which has an observed 3$\sigma$ sensitivity of $\sim 3.48 \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ \AA $^{-1}$. Further details on the WFC3 and KOSMOS data and reduction techniques, along with those from HST's Space Telescope Imaging Spectrograph (STIS) that we used for developing the kinematic models, are provided in Paper I. 

\begin{figure*}
\centering 
\subfigure{\includegraphics[width=0.42\linewidth]{fig/annuli/ALMA_annuli.pdf}\label{fig: ALMA annuli}}
\subfigure{\includegraphics[width=0.39\linewidth]{fig/annuli/NIFS_H2_annuli.pdf}\label{fig: NIFS annuli}} 
%\vspace{.2cm} 
\subfigure{\includegraphics[width=0.5\linewidth]{fig/annuli/WFC3_annuli.pdf}\label{fig: WFC3 annuli}}
 \caption{Multiphase flux maps of the central regions in NGC~3227. In each map, north is up and east is to the left, and the white star marks the position of the SMBH. (a) A map of the ALMA CO(2-1) flux density, originally shown in \cite{alonso19}. (b) A map of the NIFS H$_2$~$\lambda2.1218$ $\mu$m emission. (c) A map of a continuum-subtracted HST WFC3 F502N image of the nucleus, which shows the small-scale structure of the [O~III]~$\lambda5007$ emission. The heavy cyan lines represent the outline of the KOSMOS slit oriented along the galactic minor axis, which we use in our analysis. The light solid lines represent the KOSMOS slits oriented along the galactic major (PA = 150\arcdeg) and outflow (PA = 190\arcdeg) axes. In all three plots, the maps are overlaid with annuli that we describe in Section \ref{sec: annuli} and contours that better reveal the morphology of the flux distributions.}
 %\caption{Our annuli overlaid on top of nuclear images of (a) CO(2-1) (ALMA), (b) H$_2$~$\lambda2.1218$ $\mu$m (NIFS), and (c) [O~III] emission (HST WFC3  with the F502N filter). In each subfigure, the white star marks the position of the AGN. The bisecting line runs along the major axis of the galaxy. The annuli widths are 0\farcs102, which is the spatial sampling of the HST STIS spectra that we use in our determination of the velocity law for the ionized gas. The orientation and spatial scale are the same as outlined in Figure \ref{fig: 3227}.}
 \label{fig: annuli}

 \end{figure*}


Figure \ref{fig: WFC3 annuli} shows the flux distribution for the ionized gas using the [O~III] emission from the WFC3 image, which has a measured 3$\sigma$ sensitivity of $ \sim2.52 \times 10^{-18}$ erg s$^{-1}$ pix$^{-1}$. In this study, we use the KOSMOS slit that runs along the minor axis  of the host galaxy (PA = 240\textdegree). We choose this orientation because of the thick dust lane in the galactic disk (see Figure \ref{fig: 3227}), which intersects the NLR outflows roughly along the major axis of the galaxy, and provides a clear divide between the obscured emission to the SW of the AGN and the mostly unobscured emission to the NE of the AGN.  

The disk of the galaxy is inclined to us at an angle of $i=$ 48\arcdeg \citep{devaucouleurs91, ho97again, xilouris02}. The position angle (PA) of the galactic disk is 150\arcdeg\ (Paper I), which we choose to be the major axes of our annuli. We use Equations 1 and 2 from \cite{revalski18b} to deproject the distance and velocity of the gas along these axes. With our adopted orientation ($i=48$\arcdeg, $\phi = 90$\arcdeg), our intrinsic distances are $\approx$~1.5 times larger than the observed distances along the minor axis. Thus, although we observe emission lines that meet our signal-to-noise (S/N) threshold of S/N $\ge$ 3 out to 2\farcs57 along the minor axis, which equates to 296 pc using a scale factor of 115 pc arcsec$^{-1}$ on the plane of the sky, the deprojected distance is 442 pc.

%We choose this orientation because the plane of the galaxy intersects the NLR outflows at an orthogonal angle along our line of sight (LOS), which  creates a clear demarcation between the emission to the NE of the AGN, which is visible, and that to the SW, which is obscured.
%We choose this slit orientation, which is nearly orthogonal to the direction of the NLR outflows, because in Paper I we describe our belief that a ring of circumnuclear molecular gas lies along the major axis, and we wish to study how gas in that ring is expelled. Although we use two separate instruments for photometric and spectroscopic data, integral field unit (IFU) observations that capture the entirety of the NLR at high spatial resolution would also be sufficient.

We consider the H$_2$ molecule to be representative of the molecular gas population due to its dominant abundance in the ISM. Although H$_2$ is not directly emissive at radio or sub-mm wavelengths, detectable transitions occur in near-IR wavelengths, including a strong emission-line at $\lambda2.1218$~$\mu$m. To characterize the warm molecular gas, we use archival observations of H$_2$~$\lambda2.1218$ $\mu$m emission from the \textit{K}-band of the Near-infrared Field Spectrograph (NIFS) at Gemini North (Program ID: GN-2016A-Q-6), which possesses a measured 3$\sigma$ sensitivity of $\sim3.78 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ \AA $^{-1}$. An in-depth description of these data is given in \cite{riffel17}, and the reduction processes of these data are described in Paper I. Each NIFS data cube covers $3\arcsec \times 3\arcsec$, which equates to an area of $\sim 345$~pc $\times $ 345 pc.
%at the distance to NGC~3227 of 23.7 Mpc \citep{tonry01, blakeslee01}.

Because cold H$_2$ is not directly detectable in emission, we employ CO measurements as a tracer (see the review by \citealp{bolatto13}). To characterize the cold molecular gas, we use the ALMA CO(2-1) observations and associated calibrations described in detail in \cite{alonso19}. These observations, part of project 2016.1.00254S (PI: A. Alonso-Herrero), were obtained with band 6 at frequencies of the CO(2-1) transition (229.8 GHz) and the submillimeter continuum ($\sim$231 GHz, or 1.3 mm) at a bandwidth of 1.875 GHz \citep{alonso19} and with an observed 3$\sigma$ sensitivity of $\sim3.7 \times 10^{-4}$ Jy km s$^{-1}$ beam$^{-1}$. The resulting reduction yields a data cube with a beam size of 0\farcs161 $\times$ 0\farcs214, corresponding to a physical resolution of 15.5 $\times$ 24.6 pc. Although the field of view for these measurements are $\sim26\arcsec$, the relevant data are largely contained within a 7$\arcsec$ radius.


\section{Spectroscopic Analysis}
In Paper I, we measured the kinematics of the ionized gas in the NLR of NGC~3227 using multi-component Gaussian fits of the H$\alpha$~$\lambda$6563 $+$ [N~II]~$\lambda\lambda$6548, 6583 and H$\beta$~$\lambda$4861 $+$ [O~III]~$\lambda\lambda$4959, 5007 emission lines in both KOSMOS and STIS long-slit spectra. We used these measurements to disentangle the components of rotation and outflows as a function of distance from the SMBH. In this paper, we expand our spectroscopic analysis by using the observed line ratios to determine sources of ionization along the KOSMOS slits and develop an extinction correction.


\begin{figure*}[t]
% \centering  
% \includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa150_kosmos_NII.pdf}
% \includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa150_kosmos_OI.pdf}
% \includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa150_kosmos_SII.pdf}
%  \vspace{.2cm}

% \includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa190_kosmos_NII.pdf}
% \includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa190_kosmos_OI.pdf}
% \includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa190_kosmos_SII.pdf}
%  \vspace{.2cm}

\includegraphics[width=0.35\linewidth]{fig/BPT_plots/ngc3227pa240_kosmos_NII.pdf}
%\hspace{.05mm}
\includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa240_kosmos_OI.pdf}
%\hspace{.05mm}
\includegraphics[width=0.32\linewidth]{fig/BPT_plots/ngc3227pa240_kosmos_SII.pdf}
 %\vspace{.2cm} 
 
\caption{BPT ionization diagrams for APO KOSMOS observations along the minor (PA = 240$^\circ$) axis. Positive distance refers to the north.}
\label{fig: BPT plots}
\end{figure*}

%utilizing our understanding of the outflowing kinematics to produce photoionization models and quantify the trends in ionized gas mass evacuation as a function of distance from the SMBH. 
%We can better comprehend the extent and magnitude of the ionized gas outflows through detailed analysis of the KOSMOS spectra, which we will utilize in the following section to produce photoionization models. 
%In this paper, we continue our analysis by determining the [O~III] luminosity as a function of distance using WFC3 images and by measuring the fluxes of other emission lines in the KOSMOS spectra using the [O~III] kinematic components as templates.  We use the KOSMOS long-slit spectra, rather than those from STIS, due to the former's higher signal-to-noise and coverage of both H$\beta$ and H$\alpha$ at the same position angle, despite the coarser resolution of KOSMOS compared to STIS. The ultimate goal of this analysis is to determine accurate spatially-resolved mass outflow rates in the manner of \cite{revalski22}.




% \begin{figure}[t]
% \centering  
% \includegraphics[width=\linewidth]{fig/velocity law.pdf}
% \caption{The velocity law that we use for the ionized gas in this work, which is explained in depth in Paper I. We utilize an empirical model of the velocity profile described in \cite{travisthesis} that follows a trend of linear acceleration up to a turnover radius ($26 \pm 6$ pc for NGC~3227), followed by linear deceleration out to the edge of the NLR bicone.} 
% \label{fig: velocity law}
% \end{figure}
% %




\subsection{Emission Line Fitting}
\label{sec: fitting}
%Our spectroscopic measurements allows us to better characterize various properties of the gas, including the kinematics, reddening, and source of its ionization, as functions of position. In order to do so, we use two different fitting routines on our spectra. This subsection describes our processes for fitting our spectra.

We use two tools to perform the spectral fitting on our data. First, we fit multi-component Gaussian profiles using the Bayesian Evidence Analysis Tool (BEAT; \citealp{fischer17}) routine. BEAT is a novel tool in its use of Bayesian statistics to algorithmically determine the optimal number of kinematic components for a given spectrum. Because each spectrum contains the composite kinematic elements from the rotation of the galaxy and the outflowing motions of the winds, using BEAT to disentangle these components is critical in quantifying the kinematic contribution from the outflows. We used BEAT extensively in Paper I, and we use it again in this work to determine a radial velocity trend for the NIFS H$_2$~$\lambda2.1218$ $\mu$m emission. We do not use BEAT on the ALMA data because we were given access to the velocity map for these data (see \S \ref{sec: ALMA outflow rate}).

%In this study we are primarily concerned with the line centroid, from which we calculate the redshift and thus velocity of the gas, and the emission line flux, which allows us to calculate line ratios and ascertain various qualities related to the gas density and temperature.




%\subsubsection{Spectral Fitting for Other Emission Lines}
%\label{sec: other spectral fitting}
%If we wish to fit several emission lines for a given spectrum, we do not employ BEAT, but instead implement a slightly different methodology. 

BEAT is optimized to fit only a few bright emission lines simultaneously, and those lines are typically close to each other in wavelength (such as H$\beta$ $\lambda$4861 and [O~III] $\lambda \lambda$4959, 5007, or H$\alpha$~$\lambda$6563 and [N~II]~$\lambda \lambda$6548, 6583). For fitting multiple emission lines simultaneously across a large wavelength range, we use an alternate procedure described in detail in \cite{revalski18a, revalski18b, revalski21, meena21} which aims to ensure that we fit the same kinematic components in each emission line for a given slit position.
This alternate routine uses BEAT fits of [O~III] $\lambda5007$ as a template to fit the narrow lines by adopting the velocities, positions, and widths of its components for the other lines while allowing the fluxes to vary to match the observed profiles. Nevertheless, they remain constrained by atomic line ratios such as those for the [N~II] $\lambda \lambda$6548, 6583 and [O~III] $\lambda \lambda$4959, 5007 doublets of 2.95 and 3.01, respectively \citep{osterbrock06}. We use this routine to fit the lines of H$\beta$~$\lambda$4861, H$\alpha$~$\lambda$6563, [N~II]~$\lambda \lambda$6548, 6583, [O~I]~$\lambda \lambda$6300, 6364, and [S~II]~$\lambda \lambda$6716, 6731. 




% \subsection{Deprojecting the Velocity Law}
% To obtain a more comprehensive picture of the kinematics, we must apply correction factors to deproject the observed velocities and positions. As noted in Paper I, most of the NLR emission arises in the intersection between the bicone of ionizing radiation and the galactic disk, where the ambient gas is located, and we therefore adopt the geometry of radial outflow of ionized gas along the disk in the manner of \cite{revalski25}.

% The disk of the galaxy is inclined to us at an angle of $i=$ 48\arcdeg \citep{xilouris02}. The position angle (PA) of the galactic disk is 150\arcdeg\ (Paper I), which we choose to be the major axes of our annuli. We use Equations 1 and 2 from \cite{revalski18b} to deproject the distance and velocity of the gas along these axes. With our adopted orientation ($i=48$\arcdeg, $\phi = 90$\arcdeg), our intrinsic distances are $\approx$~1.5 times larger than the observed distances along the minor axis. Thus, although we observe emission lines that meet our signal-to-noise (S/N) threshold of S/N $\ge$ 3 out to 2\farcs57 along the minor axis, which equates to 296 pc using a scale factor of 115 pc arcsec$^{-1}$ on the plane of the sky, the deprojected distance is 442 pc.


%We show our ionized gas velocity law in Figure \ref{fig: velocity law},  which linearly approximates the deprojected velocity of the gas as a function of deprojected distance from the nucleus.

\subsection{Ionization Source}
\label{sec: BPT}
One of the applications for myriad emission line measurements is to create Baldwin-Phillips-Terlevich (BPT) diagrams \citep{baldwin81, veilleux87} for NGC 3227. A BPT diagram is an important diagnostic tool that involves comparing line ratios of [O~III] $\lambda$5007/H$\beta$ to [N~II] $\lambda$6583/H$\alpha$, [O~I] $\lambda$6300/H$\alpha$, and [S~II] $\lambda$6730/H$\alpha$ to distinguish whether gas at a particular distance from the nucleus is ionized by the AGN, by star formation, or both.
We also use these emission-line fits to develop photoionization models for the AGN-ionized gas in the next section.

%This analysis allows us to trace the power and physical extent of the ionizing radiation as a function of distance from the nucleus, which allows us to better quantify the influence of AGN feedback processes within NGC 3227. 
BPT diagrams from the KOSMOS spectra for the slit along the minor axis are shown in Figure \ref{fig: BPT plots}. The demarcations that separate the various classifications are described in \cite{kewley01, kewley06} and \cite{kauffmann03}.
The BPT diagrams for the other two KOSMOS slits are discussed in the Appendix.
Our BPT diagrams reveal a gradient in ionization as we move from the NE side of the slit to the SW. To the NE of the SMBH, the gas is primarily ionized by the AGN (``Seyfert''), but at $\approx3''$ the ionization levels weaken and the emission shows traits representative of a low-ionization nuclear emission region (LINER). This turn into the LINER designation is likely attributable to the weakening influence of the AGN at large distances, possibly due to radiation absorption by gas at nearer distances to the SMBH. On the southern side of the SMBH, the ionization level drops with distance from the SMBH. At --4$\arcsec$ from the SMBH, star formation starts to dominate over the AGN ionization. The area of prominent star formation coincides with a H~II region visible in the inset of Figure \ref{fig: 3227}. 
%which exhibits color-composite data collected by HST's WFC3. 
The thick dust lanes, which are more pronounced in the SE region of the nucleus, harbor star forming clumps visible in this figure, and also act to obscure AGN-ionized emission from the other side of the disk.
%\todo{Say something about duty cycle, maybe? Reference Schawinski?}


% \begin{figure*}
% \centering 
% \subfigure{\includegraphics[width=0.33\linewidth]{fig/annuli/ALMA annuli.pdf}\label{fig: ALMA annuli}}
% \subfigure{\includegraphics[width=0.31\linewidth]{fig/annuli/NIFS H2 annuli.pdf}\label{fig: NIFS annuli}}
% \subfigure{\includegraphics[width=0.32\linewidth]{fig/annuli/WFC3 annuli.pdf}\label{fig: WFC3 annuli}}
%  \caption{Our annuli overlaid on top of nuclear images of (a) CO(2-1) (ALMA), (b) H$_2$~$\lambda2.1218$ $\mu$m (NIFS), and (c) [O~III] emission (HST WFC3  with the F502N filter). In each subfigure, the white star marks the position of the AGN. The bisecting line runs along the major axis of the galaxy. The annuli widths are 0\farcs102, which is the spatial sampling of the HST STIS spectra that we use in our determination of the velocity law for the ionized gas. The orientation and spatial scale are the same as outlined in Figure \ref{fig: OIII contours}.}
%  \label{fig: annuli}

%  \end{figure*}

\subsection{Accounting for Extinction}
\label{sec: extinction}

%  \begin{figure}[!htp]
% \centering  
% \includegraphics[width=0.8\linewidth]{fig/judy schmidt figure.png}
% \caption{Color-composite image of the inner region surrounding NGC 3227's AGN, where red colors are from the HST WFC F658N filter, green colors are from the HST WFC3/UVIS F547M filter, and blue colors show F550M and F330W filters from Hubble's Advanced Camera for Surveys High Resolution Channel. The black star shows the location of the nucleus. Image credit: Judy Schmidt.} 
% \label{fig: judy schmidt}
% \end{figure}

The dust lanes at NGC 3227's center, which result in extensive reddening, have been an area of study for over four decades \citep{cohen83, winge95, crenshaw01, gondoin03, mehdipour21}. It is therefore important to  carefully consider how reddening effects impact the observed optical emission lines and their luminosities, as the luminosity measurements are critical to our calculations of the mass outflows. 
%We describe two primary estimates of reddening that set upper and lower bounds.

%\subsubsection{E(B-V) = 0.18}
%\cite{crenshaw01} developed a reddening curve for NGC 3227 by using STIS spectra to compare the continuum flux of NGC 3227 to that of NGC 4151, which was assumed to be unreddened. They obtained a value of E(B-V) = 0.18 to be used across the area covered by the STIS slits, which extend to 52''. This value was a factor of 2 lower than that found by \cite{kraemer00}, who found E(B-V) = 0.4. However, \cite{kraemer00} utilized the Galactic extinction curve in their calculations, which is a very poor match to NGC 3227's extinction curve \citep{crenshaw01}. 

%\subsubsection{Variable E(B-V)}

We can determine the reddening $E(B-V)$ according to the H$\alpha$/H$\beta$ ratio for a given spectrum. Specifically, we utilize Equation 3 of \cite{revalski18a}:
\begin{equation}\label{eq: E(B-V)}
E(B-V) \equiv  -\frac{2.5\text{ log}\left( \frac{F_o}{F_i} \right)}{R_\lambda} = \frac{2.5 \text{ log} \left(\frac{(H\alpha/H\beta)_i}{(H\alpha/H\beta)_o} \right)}{R_{H\alpha} -R_{H\beta} }
\end{equation}
where $F_o$ and $F_i$ are the observed and intrinsic fluxes, respectively, and $R_\lambda$ is the reddening value at a particular wavelength. We assume an intrinsic H$\alpha$/H$\beta$ ratio of 2.90, in accordance with recombination properties \citep{osterbrock06}.  From the extinction curve for NGC~3227 shown in \cite{crenshaw01}, we determine $R_{H\beta} \approx  $  3.67 and $R_{H\alpha} \approx$ 2.50. Thus, we have a direct relationship between the observed H$\alpha$/H$\beta$ ratio and the color excess. By employing Gaussian line fitting techniques (see \S \ref{sec: fitting}), we measure the H$\alpha$ and H$\beta$ fluxes to derive values for E(B-V). 


Due to the strong presence of dust around the SMBH, the S/N of the H$\beta$ flux is drastically reduced to the extent that the reddening cannot be reliably determined at each location. \cite{schonell19} present a two-dimensional reddening map of NGC 3227 derived from Pa$\beta$/Br$\gamma$ line ratios obtained from NIFS data, but there are major spatial gaps in the data that cannot be accurately estimated through interpolation due to the clumpy nature of the data. Furthermore, low flux levels in the Pa $\beta$ and Br $\gamma$ emission to the NE and SW of the nucleus present uncertainty in the accuracy of their ratios.

We instead opt to estimate a value of $E(B-V)$ by summing all KOSMOS spectra along the minor axis within $\pm2''$ of the SMBH to create a composite spectrum, and perform a single fit of the H$\alpha$ and H$\beta$ fluxes to produce a global value for the reddening. This process yields a value of $E(B-V) = 1.07 \pm 0.12$. We apply this reddening value to all spectra, including those beyond $\pm2''$ because the intrinsic flux decreases significantly, resulting in regions of low S/N that are challenging to fit accurately and thus determine accurate reddening values in these regions. Our reddening value for the NLR of NGC~3227 is much larger than the value of $E(B-V) $= $ 0.18$ determined from the AGN continuum emission by \cite{crenshaw01}, which may indicate a smaller column of dust in the direct line of sight to the SMBH. Our reddening value of $E(B-V) = 1.07 \pm 0.12$  agrees well with the NLR values of \cite{schonell19} whose two-dimensional reddening map of NGC 3227 shows E(B-V) values near the central region in the range of 0.3 -- 2.2, and \cite{cohen83} who found E(B-V) = 0.94$~\pm~$0.23.
%, indicating a much smaller column of dust in the direct line of sight to the active SMBH. \todo{see schonell, cohen1983 for reddening value}


%consider a trend for reddening that varies as a function of distance. 

%The H$\alpha$/H$\beta$ ratio and the color excess as a function of distance along the minor axis is shown in Figure \ref{fig: HaHb and E(B-V)}. We see elevated levels of extinction within the inner 2'', with H$\alpha$/H$\beta$ ratios reaching as high as 28 \todo{check}. This corresponds to reddening values of $0.8-0.9$, which are higher than average values of \todo{figure this out}.  



%An additional consideration for the reddening is found in the extensive work of \cite{crenshaw01}, who developed a reddening curve for NGC 3227 by using STIS spectra to compare the continuum flux of NGC 3227 to that of NGC 4151, which was assumed to be unreddened. They obtained a constant value of E(B-V) = 0.18, but because their work represents an area only $15 \times 20$ pc in size, it is insufficient to use in this study.
%This value was a factor of 2 lower than that found by \cite{kraemer00}, who found E(B-V) = 0.4. However, \cite{kraemer00} utilized the Galactic extinction curve in their calculations, which is a very poor match to NGC 3227's extinction curve \citep{crenshaw01}. Nevertheless, because the value from \cite{crenshaw01}  only represents an area $15 \times 20$ pc in size, it is insufficient to use in this study.

%Figure \ref{fig: E(B-V) ratio} shows that the E(B-V) values reach as high as 1.2, and do not overlap with the value of E(B-V) = 0.18 reported by \cite{crenshaw01} at any point. 

%If this was simply due to the fading power of the NLR outflows with distance, we would expect to see symmetry with the data on the northern end, however the ionization power increases sharply from $\sim 2'' - 4''$. Instead, the disparity is likely due to the presence of thick dust lanes in the galactic disk that significantly obscure emission to the south of the SMBH. In Paper I, we described the challenges of characterizing the gas kinematics in this region as a result of the heavy dust presence
%They reveal that the gas is primarily ionized by the AGN along the major (PA = 150$^\circ$) and outflowing (PA = 190$^\circ$) axes out to about 6$"$ (or about 700 pc). However, along the minor axis (PA = 240$^\circ$), gas is ionized instead by star formation at about 6$''$ SW, which coincides with the location of a prominent HII region visible in the SE region. Along the outflowing axis, there are a few points close to or within the LINER region, but these may be due to filtering of ionizing photons at closer distances. Future work will concentrate on these effects and use these lines to create photoionization models and mass profiles for determining mass outflow rates.


\section{Photoionization Models}

%As mentioned in Section \ref{sec: obs}, the spatially resolved measurements are in the direction of the galactic minor axis, along the KOSMOS slit at PA = 240\textdegree\ shown in Figure \ref{fig: OIII contours}. %We choose this orientation because of the thick dust lane of in the galactic disk that provides a clear divide between the obscured emission to the SW of the AGN and the unobscured emission to the NE of the AGN.  
To calculate spatially resolved mass outflow rates using the techniques of \cite{revalski22}, we use the Cloudy code \citep{ferland13, chatzikos23}, which produces photoionization models for a given range of input parameters. This section describes our methodology to identify trends in these parameters as a function of distance, constrained by the emission-line fluxes, which allows us to generate models along the KOSMOS slit.


\subsection{Defining Extraction Annuli}
\label{sec: annuli}

Because we are calculating spatially resolved measurements of gas mass and mass outflow rate, we must extract fluxes from the images at equal distances along the inclined disks in Figure \ref{fig: annuli}. To do so, we utilize the Elliptical Panda routine within the SAOImage DS9 image software \citep{joye03}. We construct a series of concentric semi-ellipses centered on the nucleus \citep{revalski21}, with spacings equal to the spatial sampling of the HST spectra (2 pixels, or 0\farcs10156). The ellipticity of each annulus is calculated from our adopted inclination of 48\arcdeg\ \citep{xilouris02}. The ellipses are bisected along the major axis so that we collect data in bins across the minor axis (which is to say, from the SW to the NE).  We create these annuli for the flux maps for all three of our data sets, as shown in Figure \ref{fig: annuli}. Figures \ref{fig: ALMA annuli} and \ref{fig: WFC3 annuli} show that the annuli do not cover the full extent of either the CO(2-1) or [O~III] emission, and this is because the physical extent of our analysis is limited by our adopted gas velocity laws which do not extend past 400 pc (as described in Paper I). 

%We use WFC3 photometry to convert the [O~III] fluxes to mass along the NLR. As in \cite{revalski21} and described in Section \ref{sec: obs}, we bisect annuli along the kinematic major axis so that we collect data in bins along the minor axis. 
%As previously mentioned, we choose this orientation because of the thick gas lanes that are most visible along the major axis, and which clearly divide the disk into a visible region to the NW and an obscured region to the SE. The inset of Figure \ref{fig: 3227} shows an image of the nucleus region, which clearly exhibits the impact of these dust lanes. 






\subsection{Generating Cloudy Models}
\label{sec: cloudy parameters}

The ionization parameter $U$ is the dimensionless ratio of the number of ionizing photons to hydrogen atoms at the face of a gas cloud \citep{osterbrock06}.
$U$ and the hydrogen number density, $n_{\text{H}}$, are crucial parameters for the Cloudy models needed to determine the ionized gas mass of the cloud for a given AGN continuum spectral energy distribution (SED) and column density ($N_{\text{H}}$).

In our models, we choose an upper boundary of log($N_{\text{H}}$) = 24 cm$^{-2}$. The actual column of each model will be less than this value, and is established when the temperature drops to a value of 4000 K and the cloud becomes optically thick.

\cite{revalski22} show how the [O~III]/H$\beta$ emission line ratio varies primarily as a function of $U$ and secondarily with $n_{\text{H}}$. $U$ and $n_{\text{H}}$ are also related via the definition of the ionization parameter \citep{osterbrock06}:
\begin{equation}\label{eq: U_nH}
n_{\text{H}}=\left( \frac{Q(H)_{\text{ion}}}{4\pi r^2c \  U} \right)
\end{equation}
where $r$ is the radial distance of the gas from the SMBH, $c$ is the speed of light, and $Q(H)_{\text{ion}}$ is the number of ionizing photons s$^{-1}$ emitted by the AGN, given by $\int_{\nu_0}^{\infty}(L_\nu / h\nu)d\nu$, where $L_\nu$ is the luminosity of the AGN as a function of frequency (as denoted by the SED), $h$ is Planck's constant, and $\nu_0$ = 13.6 eV/$h$ is the ionization potential of hydrogen (\citealp{osterbrock06}, Section 14.3). We adopt a typical power-law SED that has been successfully utilized in previous studies \citep{kraemer00a, kraemer00b, kraemer09, revalski18a}. For $L_\nu \propto \nu^{\alpha}$, we choose slopes of $\alpha$ = --0.5 from 1 to 13.6 eV, $\alpha$ = --1.4 from 13.6 eV to 0.5 keV, $\alpha$ = ---1 from 0.5 to 10 keV, and $\alpha$ = --0.5 from 10 to 100 keV, with low- and high-energy cutoffs below 1 eV and above 100 keV, respectively.

We can therefore determine realistic values of $U$ and $n_{\text{H}}$ at each distance $r$ using the combined constraints of matching the observed [O~III]/H$\beta$ with photoionization models and satisfying Equation \ref{eq: U_nH}. 
%Various $U-n_H$ combinations will result in different [O~III]/H$\beta$ ratios because \todo{why? ask Mitch}. 
%Using the above two constraints, we find analytical relationships between $U$, $n_H$, distance, and the [O~III]/H$\beta$ ratio by computing Cloudy models for a wide range of log($U$), log(n$_H$), and distance values, and calculating [O~III]/H$\beta$ ratios that result from the simulations. 
%If we generate a wide spread of models that vary widely in their $U$ and $n_H$ values, we can compare the models' [O~III]/H$\beta$ ratios to those that we receive with our data. 
%The model with the closest fit will therefore be the most representative of the physical conditions for the spectrum at a given location.
In this study, we vary log($U$) from 0 to $-$4.2 in increments of 0.2, and for each value of log($U$) we also vary log($n_{\text{H}}$) from 1.5 to 4.7 cm$^{-3}$ in increments of 0.1. 


\begin{figure*}[btp]
\centering  
\subfigure{\includegraphics[width=0.3\linewidth]{fig/OIII_Hb_flux_vs_radius_along_minor_axis.pdf}\label{fig: OIII-Hb kosmos}}
\hspace{2mm}
\subfigure{\includegraphics[width=0.305\linewidth]{fig/logU-OIII_matching.pdf}\label{fig: matching OIII-Hb}}
\hspace{2mm}
\subfigure{\includegraphics[width=0.325\linewidth]{fig/density_and_U_with_distance_minor_axis_new}\label{fig: U and nH vs r}}
 \caption{(a) The observed [O~III]~$\lambda$5007/H$\beta$~$\lambda$4861 ratio as a function of distance along the kinematic minor axis, which has been corrected for projection effects. The shaded region represents the uncertainty in the observed line ratio.
 (b) The predicted [O~III]~$\lambda$5007/H$\beta$ ratio (curves) as a function of $U$ and $n_\mathrm{H}$ from a grid of Cloudy models. The point on each curve represents the $U$ and $n_\mathrm{H}$ that also satisfy Equation \ref{eq: U_nH} at a given distance (in this example, we chose 1\farcs34\ or 154 pc). The point that best matches the observed [O~III]/H$\beta$ ratio at this distance (dotted line) is selected as the correct $U$ and $n_{\text{H}}$ for this location. In practice, we use a finer grid of $U$ and $n_{\text{H}}$, only showing three curves here for illustrative purposes.
% For each value of n$_\mathrm{H}$ with a corresponding $r$ and $U$, we run a Cloudy model to generate [O~III]/H$\beta$ ratios. The curves show the relationship between $U$ and the modeled [O~III]/H$\beta$ ratios over a wide range of radii for a given n$_\mathrm{H}$. For a given distance, the points mark the [O~III]/H$\beta$ for their corresponding densities, using Equation \ref{eq: U_nH} to solve for $U$ under these conditions. The dashed line represents the [O~III]/H$\beta$ ratio from the KOSMOS data plotted in (a).
 (c) log($U$) (red, solid line) and log($n_\mathrm{H}$) (blue, dashed line) as functions of distance along the slit, where the shaded regions represent their respective uncertainties.}
 %The density as a function of distance along the slit, color coded by the corresponding ionization parameter log($U$).}

\end{figure*}


%\todo{why didn't I vary but maintain U-nH relationship?} 
%To understand what these relationships look like from a visual perspective, Figure \ref{fig: matching OIII-Hb} shows how [O~III]/H$\beta$ ratios vary with $U$ when n$_H$ is constant.

We determine the [O~III]~$\lambda$5007 and H$\beta$~$\lambda$4861 emission line fluxes for each spectrum along our KOSMOS slit, which has a spatial scale of 0.257 arcsec pixel$^{-1}$, or 29.5 parsecs pixel$^{-1}$. To obtain these fluxes, we fit the spectra using the Gaussian line fitting techniques described in \S \ref{sec: fitting}. We show the [O~III]/H$\beta$ emission line ratio as a function of distance in our KOSMOS slit in Figure \ref{fig: OIII-Hb kosmos}. This ratio decreases with increasing distance from the SMBH, consistent with the trend of decreasing AGN ionization seen in the BPT diagrams (Figure \ref{fig: BPT plots}).


By utilizing the information from Figure \ref{fig: OIII-Hb kosmos}, we can determine the $U$-$n_{\text{H}}$ pair whose model's corresponding [O~III]/H$\beta$ ratio most closely matches that given by KOSMOS for a particular distance, as shown in Figure \ref{fig: matching OIII-Hb}. If we choose to look at a distance of +1\farcs34 as an example, we first use Figure  \ref{fig: OIII-Hb kosmos} to find the observed [O~III]/H$\beta$ ratio at that distance. That value is represented in Figure \ref{fig: matching OIII-Hb}  as the dashed line. 

To use Equation \ref{eq: U_nH} as a constraint we estimate $Q(H)_{\text{ion}}$ by taking the AGN continuum luminosity of Mrk~78, which is comparable to that of NGC~3227, and scaling it according to the relative ratio of bolometric luminosities between NGC~3227 and Mrk~78.  For Mrk~78 we use $L_{bol} = 7.9\times 10^{45}$  erg s$^{-1}$ and $Q(H)_{ion} = 3.8 \times 10^{54}$ photons s$^{-1}$ \citep{revalski21}, and for NGC 3227 we use $L_{bol} = 2.25\times 10^{44}$ erg s$^{-1}$ (Paper I). This results in $Q(H)_{ion} = 1.1 \times 10^{53}$ photons s$^{-1}$ for NGC~3227, which is applied to all distances from the nucleus.

In Figure \ref{fig: matching OIII-Hb}, we plot curves showing the Cloudy-predicted [O~III]/H$\beta$ ratios as a function of $U$ for three different densities. The points on these curves represent the $U$-$n_{\text{H}}$ pairs that satisfy Equation~\ref{eq: U_nH} at the chosen distance.
The $U$-$n_{\text{H}}$ pair that gives the lowest residual between the modeled and observed [O~III]/H$\beta$ ratio is chosen. 
We also note how Figure \ref{fig: matching OIII-Hb} shows that, regardless of the density, there is a degeneracy in the [O~III]/H$\beta$ ratios for log($U$) $\ge -1$. We therefore exclude any points at those $U$ values, consistent with our detection of strong low-ionization lines that would not be present at high values of $U$. In the example of Figure \ref{fig: matching OIII-Hb} where we are only choosing between three values of $n_{\text{H}}$, the orange curve which shows $n_{\text{H}}$ = 10$^3$ cm$^{-3}$ and log(U) = $-$2.88 would be considered the optimal model for this location. In practice, we use a finer grid as described above.


Figure \ref{fig: U and nH vs r} shows the results after completing this analysis for each spectrum along the KOSMOS slit. The plot shows how $n_H$ and $U$ vary with each other as a function of distance, with errors calculated according to uncertainties in the observed [O~III]/H$\beta$ ratios. The residuals  between the observed [O~III]/H$\beta$ ratios and the best models at each distance are low, usually only 0.1--0.4. The density is highest at the nucleus, reaching log($n_{\text{H}}$) $\approx 4.5$ cm$^{-3}$, and then drops off steeply on either side in a nearly symmetrical manner. A similar pattern is observed in the $U$ distribution as well, although its peak is at +0.5\arcsec NE, possibly due to the [O~III]/H$\beta$ ratio in Figure \ref{fig: OIII-Hb kosmos} peaking around that distance. We can compare these trends to those in \cite{revalski22}, whose study of six Seyfert galaxies shows trends in $U$ and $n_{\text{H}}$ that are also generally symmetric with densities decreasing with distance from the SMBH.
%On the other hand, \textit{U} is not symmetric about the nucleus, and instead peaks at a distance $\sim$0\farcs5 NE from the AGN. One possible explanation is that the strong presence of dust near the center is blocking some of the ionizing photons that would otherwise be prominent in the nucleus.




% \begin{figure}[t]
% \centering  
% \includegraphics[width=\columnwidth]{fig/OIII Hb flux vs radius along minor axis.pdf}\label{fig: OIII-Hb kosmos}
% %\subfigure[]{\includegraphics[width=0.24\linewidth]{fig/U-nH relation.png}\label{fig: U-nH relation}}
%  \caption{(The [O~III] $\lambda$5007 / H$\beta$ $\lambda$ 4861 ratio as a function of distance along the kinematic minor axis, which has been corrected for projection effects.}

% \end{figure}

% \begin{figure}[t]
% \centering  
% \includegraphics[width=0.8\columnwidth]{fig/logU-OIII matching.pdf}\label{fig: matching OIII-Hb}
%  \caption{For each value of n$_\mathrm{H}$ with a corresponding $r$ and $U$, we run a Cloudy model to generate [O~III]/ H$\beta$ ratios. The curves show the relationship between $U$ and the modeled [O~III]/ H$\beta$ ratios over a wide range of radii for a given n$_\mathrm{H}$. The points mark the [O~III]/ H$\beta$ ratio at a specific distance, using the $U$ value solved in Figure X. The dashed line represents the [O~III]/ H$\beta$ ratio from the KOSMOS data.}

% \end{figure}



%\subsection{Ionized Gas Velocity Law}
%\label{sec: velocity law}
%To calculate the mass outflow rates of the ionized gas, we must approximate the velocity of the gas as a function of distance from the nucleus. Kinematic and geometric models of observations show that the AGN-driven winds flow outward along a biconical geometry, with the central vertex of the bicone intersecting the AGN \citep{antonucci85, pedlar93, nelson00, pogge88, travisthesis, revalski18b}. In Paper I, we updated the orientation for the biconical structure for NGC 3227 which had previously been studied by \cite{travisthesis}, and found that it has an inclination of 40$^{+5}_{-4}$\arcdeg pointing SW along our line of sight and a maximum height of 150 pc. The maximum height effectively marks the extent to which the gas can be driven outwards unless there are other sources of acceleration that are present. 

%\cite{travisthesis} show that Seyfert NLR regions follow an empirical velocity trend, wherein the velocity profile starts at 0 km s$^{-1}$ in the nucleus and increases linearly until a specified turnover radius. In Paper I we used extensive modeling to determine that the turnover radius for NGC 3227 is 26$^{+6}_{-6}$ pc, at which location the velocity reaches a maximum of 600 km s$^{-1}$.  After reaching this peak, the velocity linearly decelerates until it reaches the edge of the bicone at $\sim$400 pc, which can be calculated from the bicone height and outer opening angle, which we established to be 68$^{+1}_{-1}$\arcdeg in Paper I. 
%If we assume that the NLR outflows traveling along the edge at this angle, then we can use the bicone height in conjunction with an opening angle of $68$\arcdeg to determine an edge of 400 pc. 
%The bicone orientation is such that the projection of its edge onto the plane of the disk varies widely with position angle. Along the minor axis, which is the focus of this study, the edge of the bicone reaches 268 pc. Combining these elements, we formulate a model for the velocity law for NGC 3227, which is shown in Figure \ref{fig: velocity law}. 




%\subsection{Generating Cloudy Models}
%\label{sec: cloudy models}
We choose to create Cloudy models, which assume a slab geometry, at evenly spaced distances separated by 0\farcs10156, as described in \S \ref{sec: annuli}. We create models in both the NE and SW directions, moving along the KOSMOS slit oriented on the minor axis as shown in Figure \ref{fig: WFC3 annuli}. We interpolate along the trends in [O III]/H$\beta$, \textit{U}, and $n_{\text{H}}$ as a function of distance as shown in Figures \ref{fig: OIII-Hb kosmos} and \ref{fig: U and nH vs r}, which allow us to generate unique parameters for each model. We create models extending to the edge of the bicone at $\sim$400 pc, where the velocity law goes to zero. We use the solar abundances as given in \cite{grevesse10}, which is a stored abundance set in Cloudy. 



%As we discuss in Section \ref{sec: ionized mass}, the ionized gas mass at a given location depends on the ratio of observed to model brightnesses. 



%For each half-annulus, we create a photoionization model using Cloudy \citep{ferland13} that matches the physical conditions of the emitting clouds in our spectra. For our input parameters, we use values for U and $n_{\text{H}}$ that we interpolate as described in Section \ref{sec: cloudy parameters}. 



% \begin{figure}
% \label{fig: HaHb and E(B-V)}
% \centering  
% \subfigure{\includegraphics[width=\linewidth]{fig/E(B-V) ratio with HaHb color.pdf}\label{fig: HaHb ratio}}
%  \caption{The scatter plot shows the H$\alpha$/H$\beta$ ratio as a function of distance along the minor axis, where negative distance is in the SE direction. The colors represent corresponding E(B-V) values from Equation \ref{eq: E(B-V)}.}

% \end{figure}



\section{Mass Outflow Rates and Evacuation Timescales: Techniques}



From the Cloudy models, we can determine mass outflow properties by extracting the model H$\beta$ model fluxes to use alongside our observed [O~III] fluxes. Although we divide our bins into hemispheric annuli as shown in Figure \ref{fig: annuli}, we are primarily concerned with radial trends under a general assumption of axial symmetry. Thus, in our subsequent calculations of mass and mass outflow trends, we perform the calculations for each semi-annulus and sum azimuthally. 



The mass outflow rate ($\dot{M}_{\mathrm{out}}$) for each of our three phases can be calculated via
\begin{equation}\label{eq: Mdot}
\dot{M}_{\mathrm{out}} = \frac{Mv}{\delta r}
\end{equation}
where $M$ is the mass in each annulus, $v$ is the deprojected velocity which has been corrected for the effects of inclination and position angle, and $\delta r$ is the deprojected width of each annulus. The deprojection factor is the same for the warm and cold molecular gas as for the ionized gas because the available evidence indicates that the molecular disk lies in the plane of the galaxy \citep{alonso19}. Although the mass and velocity for each gas phase are dependent on several different factors, the deprojected width of each annulus is a constant value of $\delta r$ for all gas phases of 0\farcs102. The following subsections describe the methodology utilized to obtain the mass and velocity measurements for each of the three phases.

%
% \begin{figure*}[t]

% \centering  
% \subfigure{\includegraphics[width=0.34\linewidth]{fig/ALMA plots/ALMA velocity field original.pdf}}
% \subfigure{\includegraphics[width=0.32\linewidth]{fig/ALMA plots/ALMA barolo field recreation.pdf}}
% %\vspace{-.2cm}
% \subfigure{\includegraphics[width=0.32\linewidth]{fig/ALMA plots/velocity subtracted recreation.pdf}}
% \caption{Our recreation of the CO(2-1) residual velocity field, as shown in Figure 7 of \cite{alonso19}. These maps show (a) the observed CO(2-1) mean velocity field, (b) our recreation of the $^\mathrm{3D}$BAROLO model of the rotating disk, and (c) the resulting residual velocity field after subtracting an additional 30 km~s$^{-1}$ offset as detailed in \S \ref{sec: ALMA outflow rate}.}
% \label{fig: barolo}
% \end{figure*}
%
\subsection{Cold Molecular Gas }
\subsubsection{Mass Calculation}
%We consider the H$_2$ molecule to be representative of the molecular gas population due to its dominant abundance in the ISM. However, because cold H$_2$ is not directly detectable in emission, we employ CO measurements as a tracer (see the review by \citealp{bolatto13}). In this work, we use the ALMA CO(2-1) data presented extensively in \cite{alonso19}, who mapped the emission for the central region of NGC 3227. 
To convert the CO(2-1) flux to H$_2$ mass, we follow \cite{alonso19}, who referred to Equation 2 of \cite{sakamoto99}:
\begin{equation} \label{eq: sakamoto}
\begin{split}
\left( \frac{M_{\mathrm{H}_2}}{M_\odot} \right) = 1.18 \times 10^4 \times \left( \frac{D}{\mathrm{Mpc}} \right)^2 \left( \frac{S_{\mathrm{CO(1-0)}}}{\text{Jy km s}^{-1}} \right)\\
 \times \left[ \frac{X}{3.0 \times 10^{20} \text{ cm}^2 \text{(K km s}^{-1})^{-1}} \right]
\end{split}
\end{equation}
where $D$ is the distance, $S_{\mathrm{CO}(1-0)}$ is the total CO(1-0) line flux, and $X$ is the CO-to-H$_2$ conversion factor. \cite{alonso19} estimate a brightness temperature ratio, $R_{21}$ = CO(1–0)/CO(2–1) = 1, based on an average value for spiral galaxies. For the CO-to-H$_2$ conversion factor, they use a value of $X = 2 \times 10^{20}$ cm$^{-2}$ (K km s$^{-1})^{-1}$ \citep{bolatto13}.  

We repeat this process, but we choose a conversion factor described in \cite{sandstrom13}, who perform a spatially-resolved study on the CO-to-H$_2$ conversion factor with local spiral galaxies and assume $R_{21}$ = 0.7. We choose this value over the one given in \cite{bolatto13} because the latter is ideal for galaxies like the Milky Way, whereas the former is a comprehensive study of $X_{\mathrm{CO}}$ in extragalactic sources. \cite{sandstrom13} solve for the conversion factor using the formula $X_{\mathrm{CO}}  = \alpha_{\mathrm{CO}} \times (4.6 \times 10^{19})$, where $ \alpha_{\mathrm{CO}}$ relates the CO flux to the H$_2$ surface brightness.
%and is also known in the literature as the CO-to-H$_2$ conversion factor. For consistency in this work, we choose the CO-to-H$_2$ conversion factor to refer to $\alpha_{\mathrm{CO}}$. 
\cite{sandstrom13} find that the $\alpha_{\mathrm{CO}}$ profile is generally flat past $0.2r_{25}$ at a value of 3.1, where $r_{25}$ is the B-band isophotal radius at 25 mag arcsec$^{-2}$. At distances closer than $0.2r_{25}$, values for $ \alpha_{\mathrm{CO}}$ decrease significantly.

In NGC 3227, the value of r$_{25}$ is $91\farcs57 \pm 18\farcs72$ \citep{robinson21}. Our maximum distance of 2\farcs8 is only 2.7\% of $r_{25}$. \cite{sandstrom13} direct that for distances closer than 0.1$\times r_{25}$, the average $\alpha_{\mathrm{CO}}$ is a factor of two lower than the value of 3.1 associated with the remainder of the radius. Thus, when we factor that into our calculations, we find $X_{\mathrm{CO}} = 7.19 \times 10^{19}$ cm$^{-2}$ (K km s$^{-1}$)~$^{-1}$. 

%When we factor this value for $X_{\mathrm{CO}}$ into Equation 2 from \cite{sakamoto99},  we receive an H$_2$ mass distribution shown in Figure X. Our total H$_2$ mass is $3.16 \times 10^8$ M$_\odot$. \cite{davies06} estimate the molecular gas mass from the 1–0 $S$(1) line in the innermost 0\farcs80 to be between $4 \times 10^7$ M$_\odot$ and $(2-8) \times 10^8$ M$_\odot$, which is within agreement of our results. Hicks et al 2009 find the molecular gas mass to be between $2.2 - 24.2 \times 10^8$ M$_\odot$.  

% \begin{figure}[!htp]
% \centering  
% \includegraphics[width=0.8\linewidth]{fig/NIFS plots/NIFS H2 annuli.pdf}
% \caption{The annuli that we use for the spatially-resolved mass profiles, shown as white rings, are overlaid on the H$_2$ $\lambda2.1218$ $\mu$m emission distribution. The X in the center marks the location of the SMBH. The orientation of these annuli are the same as those used for the [O III] and CO(2-1) flux distributions, shown in Figures \ref{fig: WFC3 annuli} and \ref{fig: ALMA annuli}.} 
% \label{fig: NIFS annuli}
% \end{figure}

\subsubsection{Mass Outflow Rate Calculation}
\label{sec: ALMA outflow rate}
To estimate the velocities associated with the CO(2-1) emission, we replicate the process of creating a rotation-subtracted CO(2-1) velocity field as described in detail in \cite{alonso19}. To isolate the outflows and noncircular kinematics in NGC 3227, \cite{alonso19} modeled the CO(2-1) data with $^{\mathrm{3D}}$BAROLO \citep{diteodoro}, which approximates the rotating galactic disk, and created a CO(2-1) residual mean velocity map by subtracting the modeled rotational kinematics from the CO(2-1) mean velocity field. For our study, we manually recreated the $^{\mathrm{3D}}$BAROLO model and subtracted the kinematics from the CO(2-1) mean velocity field to produce our own velocity residual plot, which is a nearly identical copy to the version shown in Figure 7 of \cite{alonso19}. 

Subtracting the $^{\mathrm{3D}}$BAROLO model yields a map of residual kinematics. These kinematics may represent outflows, or they may be associated with other local processes such as streaming motions or bar funneling. \cite{alonso19} has shown outflows to be dominant in the innermost 0\farcs2, and to be present alongside rotation out to 0\farcs5. Based on the observed kinematics and the assumption of a decelerating velocity law (as we have employed in our ionized data), we expect the outflows to extend in some form out to distances of $1''$.  Because of uncertainties in the origins of these motions, our analysis of the cold molecular gas represents \textit{maximum} possible values in the outflow properties. We hope to incorporate a spectral decomposition of these data, which would help disentangle the kinematic sources, into future analyses of NGC~3227. 

In our calculation of the maximum mass outflow rate, we interpret the residual kinematics as exclusively outflowing material  although later in our analysis, we will also assume the other extreme that none of the cold molecular gas near the nucleus is outflowing. In their analysis, \cite{alonso19} find contributions from structures such as a large-scale stellar bar to be insufficient to drive gas to the observed locations in position-velocity (p-v) diagrams. Rather, they find the kinematics can be most accurately replicated when with a model uses the $^{\mathrm{3D}}$BAROLO model as a base and adds a noncircular radial velocity component (i.e., outflows) and a nuclear warp in the galactic disk. However, their model without the warp produces similar results in its ability to reach specific locations in the p-v diagrams. Therefore, although \cite{alonso19} do not find outflowing motions to exclusively define the kinematics, our assumption is reasonable for the determining the maximum outflow rate.


However, there is a systemic velocity of $\sim$ 30 km s$^{-1}$ that remains in both our CO(2-1) velocity residual and the version in \cite{alonso19}. This systemic velocity causes an offset in the velocity field which creates a substantial asymmetry in the residual velocities. The cause of this systemic velocity may be attributable to inflowing streaming motions resulting from a large-scale stellar bar \citep{davies14}, which correlate with expected kinematics for H$_2$ and CO(2-1) emission \citep{alonso19}. Thus, in order to further isolate the residual kinematics from other sources of motion near the nucleus, we subtract an additional 30 km s$^{-1}$ from the residual velocity map to produce the version we use in our analysis. The resulting velocity law for the CO(2-1) data is shown in Figure~\ref{fig: ALMA and NIFS velocity laws}.

\begin{figure}[t]
\centering  
\subfigure{\includegraphics[width=\linewidth]{fig/ALMA_NIFS_OIII_velocity_law.pdf}}

\caption{The velocity laws for the ALMA CO(2-1) (blue solid line), NIFS H$_2$ $\lambda2.1218$ $\mu$m (red dot-dashed line), and HST [O~III] $\lambda 5007$ (gold dashed line) data that are used to calculate the maximum mass outflow rates for the cold molecular and warm molecular gas, and the approximate mass outflow rate for the ionized gas.}
\label{fig: ALMA and NIFS velocity laws}
\end{figure}


\subsection{Warm Molecular Gas}
\subsubsection{Mass Calculation}
%Although H$_2$ is not directly emissive at radio or sub-mm wavelengths, detectable transitions occur in near-IR wavelengths, including a strong emission-line at $\lambda2.1218$~$\mu$m. As in the previous section, we use H$_2$ as a proxy for the gas population, although in this case we use the transition at $\lambda2.1218$ $\mu$m as a proxy for the warm ($\sim$2000 K; \citealt{bianchin22}) molecular gas.

%Our most direct measurement of the warm H$_2$ gas kinematics in NGC 3227 comes from archival observations in the \textit{K} band from the Near-infrared Field Spectrograph (NIFS) at Gemini North (Program ID: GN-2016A-Q-6; \citealp{riffel17}). These integral field unit (IFU) observations obtained with adaptive optics have an angular resolution of $\sim $0\farcs1 over a 3\farcs0 $\times$ 3\farcs0 field of view, which allows the \textit{K} band to measure H$_2$ $\lambda2.1218$ $\mu$m emission at superb resolution. In Paper~I, we describe the \textit{K} band data reduction and analyze the H$_2$ kinematics. Here, we advance that analysis further by calculating the warm molecular gas mass as a function of distance. 

To calculate the gas mass from the H$_2$ $\lambda2.1218$ $\mu$m emission in the NIFS data cube, we employ Equation~6 from \cite{storchi09}:
\begin{equation} \label{eq: storchi H2 mass}
\begin{split}
M_{\mathrm{H_{2}}}  &= \frac{2m_\mathrm{p} \ F_{\mathrm{H}{_\lambda}2.1218} \ 4 \pi D^2}{f_{\nu=1, J=3}A_{S(1)} h\nu}\\
 & = 5.0776 \times 10^{13} \left( \frac{F_{\mathrm{H}{_\lambda}2.1218}}{\mathrm{erg \ s}^{-1} \mathrm{ \ cm}^{-2}} \right) \left( \frac{D}{\mathrm{Mpc}} \right)^2 
\end{split}
\end{equation}
where $m_\mathrm{p}$ is the proton mass, F$_{\mathrm{H}{_\lambda}2.1218}$ is the line flux, $D$ is the distance to the galaxy, $f_{\nu=1, J=3}$ is the population fraction, $A_{S(1)}$ is the transition probability, and the resulting $M_{\mathrm{H}_2}$ is in units of solar masses. \cite{storchi09} assume a vibrational temperature of $T_{\mathrm{vib}}$ = 2000 K, yielding $f_{\nu=1, J=3} = 1.022 \times 10^{-2} $ and $A_{S(1)} = 3.47 \times 10^{-7}$ s$^{-1}$. The orientation of our semi-annuli relative to the flux density of the H$_2$ $\lambda2.1218$ $\mu$m emission is shown in Figure \ref{fig: NIFS annuli}.

%Mention failed Muller-Sanchez equation? 
\subsubsection{Mass Outflow Rate Calculation}
We operate under the same assumption described in \S \ref{sec: ALMA outflow rate} that we are computing the \textit{maximum} warm molecular mass outflow rate. 

The rotation-subtracted velocity map of these H$_2$ data are shown in Figure 10 of Paper I. We recreate the process of overlaying the semi-annuli on this map in the same manner shown in Figure \ref{fig: NIFS annuli}, and taking the average velocity in each of those wedges. The resulting velocity law is shown in Figure \ref{fig: ALMA and NIFS velocity laws}.
Interestingly, both cold and warm molecular gas phases show velocity turnovers at similar radii (40 -- 60 pc) as the ionized gas phase, although with much smaller velocity amplitudes and much quicker returns to zero velocity at $\sim$100 pc.
We employ Equation~\ref{eq: Mdot} to solve for the outflow rate as a function of distance.


\subsection{Ionized Gas Mass}
\label{sec: ionized mass}
\subsubsection{Mass Calculation}
We calculate the luminosities of our [O~III] $\lambda 5007$ semi-annuli through the formula
\begin{equation}\label{eq: luminosity}
L(\lambda 5007) = 4 \pi D^2 F_{\lambda 5007} \times 10^{0.4 \times E(B-V) \times R_{5007}}
\end{equation}
where \textit{D} is the distance to the galaxy, $ F_{\lambda 5007}$ is the intrinsic [O~III] $\lambda5007$ flux given by the HST WFC3 data, and $R_{5007}$ is the reddening value for $\lambda$5007. \S \ref{sec: extinction} describes our methodology to determine values for $E(B-V)$ and $R_{5007}$. We use constant values of $R_{5007} = 3.65$ and $E(B-V) =1.07 \pm 0.12$. 

To convert $L_{5007}$ to a mass, we utilize the equation \citep{peterson97, crenshaw15}:

\begin{equation}\label{eq: Mslit}
M = N_{\mathrm{H}} \mu m_p \left( \frac{L(H\beta)}{F(H\beta)_m}\right)
\end{equation}
where $N_\mathrm{H}$ is the model hydrogen column density (which we obtain from the Cloudy results), $\mu$ is the mean mass per particle (which is 1.40 for solar abundances), $m_p$ is the mass of a proton, $L(H\beta)$ is the luminosity of H$\beta$ that we obtain from our conversion of the [O~III] luminosity, and $F(H\beta)_m$ is the H$\beta$ model flux that we obtain from the Cloudy models. We convert $L(\lambda 5007)$ to $L(H\beta)$ at each annulus by utilizing our interpolation of the [O~III]/H$\beta$ flux ratios shown in Figure \ref{fig: OIII-Hb kosmos}.  

\subsubsection{Mass Outflow Rate Calculation}
As shown in Paper I (see Figure 13 in that paper), the ionized gas in NGC~3227 is completely dominated by outflow to a distance of at least 400 pc from the SMBH.
To calculate the ionized mass outflow rate, we employ Equation \ref{eq: Mdot} where the mass at each distance is calculated in the previous subsection. \cite{travisthesis} show that Seyfert NLRs often follow an empirical velocity trend, wherein the velocity profile starts at $\sim$0 km s$^{-1}$ in the nucleus and increases roughly linearly until it reaches a turnover radius, at which point the velocity declines until it approaches systemic at the full extent of the NLR. In Paper I, we found that the NLR of NGC~3227 follows this trend and we used extensive modeling to determine that the turnover radius for NGC 3227 is 26$^{+6}_{-6}$ pc, at which location the velocity reaches a maximum of 600 km s$^{-1}$.  After reaching this peak, the velocity linearly declines until it reaches the full extent of the bicone at $\sim$400 pc, as shown in Figure \ref{fig: ALMA and NIFS velocity laws}. To calculate the ionized mass outflow rate, we interpolate at each distance from our velocity law.

%Figure X shows the resulting mass profile, as well as the mass profile that results from summing together the two halves of each annulus.

% \subsection{Mass Outflow Rate}
% We calculate the mass outflow rate ($\dot{M}_{\mathrm{out}}$) for each of the three phases using
% \begin{equation}\label{eq: Mdot}
% \dot{M}_{\mathrm{out}} = \frac{Mv}{\delta r}
% \end{equation}
% where $M$ is the mass in each annulus, $v$ is the deprojected velocity which has been corrected for the effects of inclination and position angle (see Section \ref{sec: cloudy models}), and $\delta r$ is the deprojected width of each annulus. Although the mass and velocity for each phase are dependent on several different factors, the deprojected width of each annulus is a constant value of $\delta r$ for all gas phases. For the CO(2-1) and [O III] emission, because there flux distribution is asymmetric about the galactic major axis, we  splice the annuli along this axis to create half-annuli in the northeast and southwest directions. Thus, we calculate the mass and mass outflow rate in each of these half-annuli.
%which we obtain from the velocity law shown in Figure \ref{fig: velocity law} and has been corrected for the effects of inclination and position angle (see Section \ref{sec: cloudy models})
%Figure X shows the resulting plot for the ionized mass outflow rate after, both as half-annuli and with the two halves of each annuli summed.




%\subsubsection{Mass Outflow Rate Estimate}
%To calculate the outflow rate, we use our ionized gas velocity law, shown in Figure \ref{fig: velocity law},  which linearly approximates the velocity of the gas as a function of distance from the nucleus. This velocity law, described in detail in Paper I, was developed by modeling the kinematics of the [O III] gas with data from the HST STIS.


\subsection{Timescale Calculation}
\subsubsection{Depletion Timescale}
\label{sec: depletion}
%Although the multiple gas mass outflow rates are crucial towards understanding the AGN feedback processes in this galaxy, the addition of CO(2-1) data adds another layer to analyze these outflows on a deeper level. 
Assuming that the cold molecular gas reservoir revealed in the CO(2-1) data shown in Figure \ref{fig: ALMA annuli} is the source of this galaxy's AGN outflows, we can use the mass outflow rates to calculate the time it will take to evacuate the reservoir. 
%heat the cold molecular gas to a given state and be accelerated over a fixed distance, such as the width of an annulus ($\Delta r$). 
In determining an evacuation timescale ($t_e$) for moving the gas to a fixed overall distance, we must consider both the depletion timescale ($t_d$) for the cold molecular gas to be removed and accelerated within its original annulus, and the crossing timescale ($t_c$) for gas in each phase to move across the remaining annuli, such that $t_e = t_d +t_c$.

%As NGC 3227 contains a large-scale stellar bar \citep{mulchaey97, schinnerer00, davies14, alonso19}, we expect that molecular gas will be inflowing concurrently as gas is outflowing. 
%In the inner kpc of the galactic nucleus, we expect to see bar-driven inflow \todo{look at Mike's paper}. 
For the purposes of this work, we are assuming a static environment in which gas is not driven inwards. This inflow rate has not been quantified for NGC 3227, but simulations have shown nuclear inflow rates to be highly variable and difficult to accurately quantify on our scales \citep{bournaud11, gabor13}. Thus, we opt to exclude the impact of inflows from our study, but future analyses of mass outflows may wish to consider this aspect.
%bar fueling timescales are generally on the order of $\sim10^8$ years \citep{combes00, silva22}, which is 1--2 orders of magnitude larger than the depletion timescales we will observe in Section \ref{sec: results}. We can therefore exclude a consideration of inflows via the stellar bar with minimal impact to the study, but future studies of feedback processes in this galaxy may wish to consider this aspect.

When considering how gas is propagated outwards, we have shown that most of the outflowing gas is created in situ, likely from radiation pressure on the local dusty molecular gas \citep[Paper I]{das07, fischer17, meena21, meena23}. Thus, we consider two extremes when making our calculations: 1) all of the outflowing gas continues to be pushed outwards from one annulus to the next, or 2) none of the outflowing gas in an annulus is found in subsequent outer annuli. Physically, the latter situation could occur if the gas is either ionized to a higher phase in the annulus, and therefore undetectable in the visible, if it is driven to lower outflow velocities at larger radii and is not decelerating, or if it decelerates and does not reach the next annulus. The second and third cases are likely happening to some extent because the mass outflow rate declines significantly at distances past the peak in this and other AGN \citep{revalski21}. Some portion of the first case is likely occurring as well for the ionized gas, because our radiative driving analysis in Paper I shows that high-velocity ($>$ 200 km~s$^{-1}$) ionized clouds at locations $>$ 100 pc in NGC~3227 were launched from distances $<$ 10 pc from the SMBH, indicating movement across many annuli. This scenario is likely true for the cold and warm molecular gas outflows as well. Thus, the real situation is likely between these two extremes, which we can use to calculate limits on the timescales.


\begin{deluxetable*}{lccccc}
\vspace{-0.5em}
\setlength{\tabcolsep}{0.18in}
\def\arraystretch{0.95}
\tablecaption{Properties of the Mass Outflow Rates}
\tablehead{
\colhead{Gas Phase} & \colhead{$\dot{M}_{\mathrm{peak}}$} & \colhead{$\dot{M}_{\mathrm{peak}}$ Distance} & \colhead{Extent of Outflows} &\colhead{Integrated Gas Mass \vspace{-.6em}} \\
\colhead{} & \colhead{(M$_\odot$ yr$^{-1}$)} & \colhead{(pc)} & \colhead{(pc)} &\colhead{(M$_\odot$) \vspace{-1.5em}} \\
%\colhead{(1)} & \colhead{(2)} & \colhead{(3)} &\colhead{(4)} & \colhead{(5)}
}
\startdata
Cold Molecular & $18.9 \pm 4.2$&  57 $\pm$ 6&  $92 \pm 6$& $(2.213 \pm 0.001) \times 10^8$\\ 
Warm Molecular& $(6 \pm 3)\times10^{-4}$ &  36 $\pm$ 6&  $164 \pm 6$& $(1.63 \pm 0.22) \times 10^3$\\ 
Ionized & $19.9 \pm 9.2$  &  47 $\pm$ 6&  $423 \pm 6$& $(7.7 \pm 0.6) \times 10^6$\\
\enddata
\tablecomments{A summary of the findings for each of the three phases studied in this work. The columns list (1) gas phase, (2) peak mass outflow rate for the azimuthally summed profiles (3) deprojected distance from the SMBH at which the peak mass outflow rate occurs, (4) total distance over which outflows are observed, and (5) the integrated gas mass up to the extents in column 4. \vspace{-1.7em}} 
\label{table: results}
\end{deluxetable*}

The depletion timescale $t_d$ for an environment where none of the outflowing gas moves to subsequent annuli, and therefore all created locally, can be simply calculated from $t_d = M_{\mathrm{H_2}} / \dot{M}_{\mathrm{out, \ total}}$, where $M_{\mathrm{H_2}}$ is the molecular mass for a given bin assuming that the ionization time scale in the inner regions of the galaxy is comparatively small and can be neglected \citep[see, e.g.][]{peterson13}, and $\dot{M}_{\mathrm{out, \ total}}$ is the sum of the individual mass outflow rates for the cold molecular, warm molecular, and ionized phases within a given bin. In this calculation, we further divide $\dot{M}_{\mathrm{out, \ total}}$ into two options: a maximum value, which incorporates the maximum mass outflow rates for the cold and warm molecular gas, and a minimum value, wherein the cold and warm molecular gas outflow rates are set to zero and the only contribution comes from the ionized gas outflows.

In situations where all of the gas is pushed outward to the next annulus, it is necessary to define the net ionized mass outflow rate for each phase, $\dot{M}_{\mathrm{net}}$,  as $\Delta \dot{M}_{\mathrm{out}}$ from one annulus to the next moving outward. This is because the outflowing gas mass being pushed forward from a previous annulus must still be carried forward in subsequent annuli, and so it must be considered when discussing the capacity to evacuate the gas.
The depletion timescale in this case is given by $t_d = M_{\mathrm{H_2}} / \dot{M}_{\mathrm{net, \ total}}$, where 
\begin{align*}
\dot{M}_{\mathrm{net, \ total}} = \dot{M}_{\mathrm{net,\ ion.}} + \dot{M}_{\mathrm{net, \ warm \ mol.}} + \dot{M}_{\mathrm{net, \ cold \ mol.}}. 
\end{align*}
We also calculate $\dot{M}_{\mathrm{net, \ total}}$ as a maximum and minimum value in the same way as described above for $\dot{M}_{\mathrm{out, \ total}}$. $\dot{M}_{\mathrm{net, \ total}}$ is not relevant and therefore not calculated when $\dot{M}_{\mathrm{out}}$ declines from one annulus to the next, because no molecular gas is being removed from that annulus.
%As the value of $\dot{M}_{\mathrm{out}}$ does not discriminate between gas that newly ionized and that which is carried over from previous annuli, the difference in $\dot{M}_{\mathrm{out}}$ from one annulus to the next, i.e. $\dot{M}_{\mathrm{net}}$, represents the amount of radiation that is left to evacuate the newly ionized H$_2$ gas in the current annulus, as the remainder in $\dot{M}_{\mathrm{out}}$ is used to continue the acceleration of gas from previous annuli. 

% \begin{figure*}
% \label{fig: mass and outflow profiles}
% \centering 
% \subfigure[]{\includegraphics[width=0.45\linewidth]{fig/mass profile.pdf}\label{fig: mass profile}}
% \subfigure[]{\includegraphics[width=0.45\linewidth]{fig/mass outflow rate.pdf}\label{fig: mass outflow rate}}
% %\subfigure[]{\includegraphics[width=0.45\linewidth]{fig/mass profile combined annulus.pdf}\label{fig: mass profile annulus}}
% %\subfigure[]{\includegraphics[width=0.45\linewidth]{fig/mass outflow profile annulus.pdf}\label{fig: mass outflow annulus}}

%  \caption{The ionized gas mass profile (left) and ionized gas mass outflow rate (right) profiles for NGC 3227, calculated from Cloudy models. }
%  %The top row shows how the values change on either side of the SMBH, while the bottom row present radial profiles.


% \end{figure*}

\subsubsection{Crossing \& Evacuation Timescales}
\label{sec: evacuation}
The time that is required to remove the gas to some outward boundary is known as the crossing timescale. Specifically, we define the crossing timescale $t_c$ as the time for the gas of a particular phase at a given distance to be pushed from that distance to the expected extent of the outflows, which we limit according to the extent of our velocity laws as shown in Figure \ref{fig: ALMA and NIFS velocity laws}. The crossing timescale for every individual annulus of each phase is calculated from $\Delta t_c = M / \dot{M}_{\mathrm{out}}$ ($ = \Delta r / v$), where $M$ is the gas mass of the given phase within that annulus and $ \dot{M}_{\mathrm{out}}$ is the outflow rate of the given phase within that annulus. The values are then cumulatively added starting from the edge of the outflows and moving inwards to that annulus. Each gas phase has its own crossing timescale according to its specific velocity law.

Finally, the crossing timescale is added to the depletion timescale to calculate the evacuation timescale $t_e$ for each phase at every annulus. The evacuation timescale describes the total amount of time for gas to be removed from the reservoir and expelled from within the inner regions surrounding the SMBH, extending $100 - 400$ pc depending on the phase.

It should be noted that although the topic of gas evacuation timescales has been the subject of much study \citep{cicone14, fiore17}, our work takes a slightly different approach because of our novel spatially resolved methodology. As a result, other works such as \cite{cicone14} use the term ``depletion timescale'' to refer to what we call the evacuation timescale because they rely on a single global value to describe the overall time required to evacuate the entire nuclear region.

%The evacuation timescale provides an extremely important estimate of the duration of the AGN's duty cycle, and can \todo{end better}

%\subsection{Kinetic Energy and Momentum?}
%\todo{include??}



\subsection{Sources of Uncertainty}

\subsubsection{Cold Molecular Gas}
\label{sec: molecular errors}
The primary source of random error that we consider in the cold molecular gas mass estimate is the uncertainty in the CO(2-1) flux measurements. We estimate the uncertainty by first designating a continuum region where there is minimal emission on the flux map. The uncertainty is calculated with the equation

\begin{equation}
\sigma_{\text{line}} = \sqrt{\frac{f_{\text{line}}}{f_{\text{cont.}}}} \times \sigma_{\text{cont.}}
\label{eq: unc}
\end{equation}
where $f_{\text{line}}$ is the average line flux per pixel, $f_{\text{cont.}}$ is the average continuum flux per pixel, and $\sigma_{\text{cont.}}$ is the standard deviation in the continuum region region.
%This uncertainty was measured by taking the standard deviation of a continuum region in the CO(2-1) flux field where there is minimal emission, and scaling it . 
We find that $\sigma_{\text{line}} \approx 0.00035$ Jy km s$^{-1}$ beam$^{-1}$, which yields a fractional uncertainty of $< 1$\% after propagation. We repeat this process on the velocity field to obtain an uncertainty in the mass outflow rates, and obtain a fractional uncertainty of $\sim1.5$\%. 

Additionally, there are potential systematic errors that depend on choices of conversion factors and can be used to scale our results accordingly.
The primary source of systematic uncertainty is the CO-to-H$_2$ conversion factor, $\alpha_{\mathrm{CO}}$, which can vary by as widely as 0.3~dex in the centers of galaxies \citep{sandstrom13}.  A few of the main factors contributing to this uncertainty pertains to weaknesses in the correlation between $\alpha_{\mathrm{CO}}$ and metallicity; uncertainty in $R_{21}$, the CO(2-1)/CO(1-0) brightness temperature ratio; and biases related to quantifying the dust-to-gas ratio. 
%Additionally, there are sizable uncertainties in the $R_{21}$ value, for which \cite{sandstrom13} believe that uncertainties are typically less than 0.2 dex. 
%We adopt 0.2 dex for the error of $R_{21}$ as an upper limit  \citep{sandstrom13}. 
\cite{sandstrom13} estimate 0.2~dex for the error of $R_{21}$. This estimation is in line with that presented by \cite{braine92}, who find $R_{21}$ = 0.78 for NGC 3227 as part of a survey carried out using the IRAM 30m telescope. 
Another potential systematic error is that of the distance to NGC 3227, which we take to be $D = 23.7 \pm 2.6$ Mpc \citep{tonry01, blakeslee01}.
%, which are so substantial that we opt to exclude them from our plots in Figures \ref{fig: ALMA mass} and \ref{fig: ALMA outflow rate}. Nevertheless, our trends are consistent with molecular outflows in comparable AGN (see Section \ref{sec: molecular results}).
%Furthermore, these errors represent extreme upper limits of uncertainty, and are far lower if less conservative estimations are applied.
We exclude systematic uncertainties from our plots with the understanding that future improvements could be used to scale our results accordingly.


\subsubsection{Warm Molecular Gas}
The errors in the warm H$_2$ mass result primarily from uncertainties in the fluxes measured in the NIFS data. As in the previous subsection, we use Equation~\ref{eq: unc} to estimate the uncertainty using the line fluxes and continuum region emission of the H$_2$ data, resulting in errors that range from 50--70\%.
%this uncertainty is measured by taking the standard deviation of a continuum region in the NIFS field where there is minimal emission. These errors range from 50--70\%, but the warm molecular gas.

In calculating the uncertainty in the mass outflow rate, we divide our H$_2$ velocity map into the same semi-annuli shown in Figure \ref{fig: NIFS annuli}, and find the velocity error for each annulus by taking the standard deviation of the velocity measurements within it.




\subsubsection{Ionized Gas}
The dominant source of error in the ionized gas mass and outflow rate calculations is the uncertainty in our estimate of the reddening, $E(B-V)$, which is due in most part to uncertainties associated with fitting the observed emission line fluxes for H$\alpha$ and H$\beta$ to determine a value for $E(B-V)$ as described in Section \ref{sec: extinction}. The errors in the fits are quite small, especially because the fitted spectrum covering $\pm 2''$ represents the sum of multiple spectra, which decreases the noise significantly. However, these values are compounded into large uncertainties for $L(\lambda5007)$ (see Equation \ref{eq: luminosity}).

The uncertainty in the extinction also contributes to the uncertainty in the conversion from [O III] $\lambda$5007 to H$\beta$ $\lambda$4861, which is needed to determine Cloudy input parameters (see Section \ref{sec: cloudy parameters}). In order to perform this conversion, we utilize the empirical $\lambda \lambda$5007/4861 relationship as a function of distance from the nucleus, which we obtain by comparing the emission line ratios of [O III] $\lambda$5007 and H$\beta$ $\lambda$4861 in the KOSMOS data (see Figure \ref{fig: OIII-Hb kosmos}). 
%The H$\beta$ emission is much more sensitive to dust effects than [O III], leading to variations in the [O III]/H$\beta$ ratio across the NLR. We factor this consideration into our calculations by propagating our uncertainties in the $E(B-V)$ value among our other uncertainties related to distance and flux measurement. 

\section{Mass Outflow Rates and Evacuation Timescales: Results}
\label{sec: results}


\subsection{Mass Profiles and Outflow Rates}
Figure \ref{fig: mass and outflow profiles} presents the spatially resolved mass and mass outflow rate profiles in the cold molecular, warm molecular, and ionized gas phases as a function of distance from the SMBH in NGC~3227 in bins of 11.5 pc, continuing under the assumption that the cold and warm molecular gas outflow rates represent upper limits. We include hatch markings for the outflow rates of the cold and warm molecular gas to indicate our uncertainty in the amount of gas present in the outflows.
The extents of these profiles are limited to those of the velocity profiles in Figure \ref{fig: ALMA and NIFS velocity laws}, which approach zero velocity in the rest frame of the galaxy at distances 100 -- 400 pc from the SMBH. The primary findings for the three phases are listed in Table \ref{table: results}.


\begin{figure*}[t]
\centering  

\subfigure{\includegraphics[width=0.47\linewidth]{fig/ALMA_plots/ALMA_H2_profile_with_errors.pdf}\label{fig: ALMA mass}}
\subfigure{\includegraphics[width=0.48\linewidth]{fig/ALMA_plots/ALMA_mass_outflow_rate_with_errors.pdf}\label{fig: ALMA outflow rate}}
\vspace{-.2cm}

\subfigure{\includegraphics[width=0.45\linewidth]{fig/NIFS_plots/NIFS_H2_mass.pdf}\label{fig: NIFS H2 mass}}
\subfigure{\includegraphics[width=0.51\linewidth]{fig/NIFS_plots/NIFS_H2_outflow_rate.pdf}\label{fig: NIFS H2 outflow rate}}
\vspace{-.2cm}

\subfigure{\includegraphics[width=0.47\linewidth]{fig/OIII_plots/mass_profile.pdf}\label{fig: OIII mass}}
\subfigure{\includegraphics[width=0.48\linewidth]{fig/OIII_plots/mass_outflow_rate.pdf}\label{fig: OIII outflow rate}}
\vspace{-.2cm}

\caption{The gas mass (left) and mass outflow rate profiles (right) as a function of distance for each of the three gas phases: cold molecular (top), warm molecular (middle), and ionized (bottom).  The warm and cold molecular gas outflow rates represent upper limits and the hatch markings represent the scope of our uncertainty, whereas the ionized gas rate represents actual values. Trends in the mass and outflow rate profiles are generally symmetric around the nucleus for the warm and cold molecular gas, but show strong asymmetries for the ionized gas due to significant extinction. 
%For the cold molecular gas, we opt not to plot errors for the mass and outflow profiles. This is due to the substantial uncertainties in the CO-to-H$_2$ conversion factor, $\alpha_{\mathrm{CO}}$, which result in uncertainties for the mass and outflow profiles of $\sim$~230\%. 
}
\label{fig: mass and outflow profiles}
\end{figure*}



\subsubsection{Cold and Warm Molecular Gas}
\label{sec: molecular results}

Figure \ref{fig: mass and outflow profiles} shows that the cold and warm molecular gas phases share similarities in the shapes and extents of their mass and maximum outflow rate profiles. The mass profiles rise sharply from the center before leveling out around $\sim$50 pc, reflecting the deficit of molecular emission seen near the nucleus in Figure \ref{fig: annuli}.
%The mass outflow profiles for both cold and warm molecular gas rise sharply from the center to peak at $\sim$50 pc on either side of the nucleus before declining rapidly, reflecting both the mass profiles and the peaks in the velocity laws at this distance.
%seen in Figure \ref{fig: velocity law}.
The dynamics between cold and warm molecular gas populations in the central regions of galaxies are still being understood, with some studies finding the warm H$_2$ acts as a thin shell or skin that surrounds the cold H$_2$ gas reservoir, heated by the AGN radiation \citep{storchi10, riffel21, bianchin22}, and other studies finding that the warm H$_2$ occupies cold molecular gas cavities \citep{rosario19, feruglio20}.  Our work under the assumption of observed outflows aligns with the former interpretation, and as such, we expect the mass of the warm H$_2$ emission to be magnitudes lower than that observed for the cold H$_2$ emission, as has been seen in other AGN \citep{dale05, muller06, bianchin22}. 
%and for the warm H$_2$ emission to extend slightly beyond the bounds of the cold H$_2$. 
We find this to be true: Figure \ref{fig: mass and outflow profiles} and Table \ref{table: results} show that the warm molecular gas mass is $\sim$5 orders of magnitude lower than the cold gas mass. 
%, and comparisons between \ref{fig: ALMA mass} and \ref{fig: NIFS H2 mass} reveal that the peak of the mass profile is 10-20 pc closer to the nucleus for the cold gas than for the warm gas.

\begin{figure*}[t]
\centering  

\subfigure{\includegraphics[width=0.47\linewidth]{fig/mass_outflow_rates_plotted_together.pdf}\label{fig: mass outflow rates plotted together}}
\hspace{3mm}
\subfigure{\includegraphics[width=0.47\linewidth]{fig/mass_outflow_rates_phases_combined.pdf}\label{fig: mass outflow rates phases combined}}

\caption{(a) The mass outflow rates from the three phases, as shown in Figure \ref{fig: mass and outflow profiles}, are plotted. We have plotted them as a function of radial distance from the center by summing the mass outflow rates at equal distances in either direction from the SMBH. The hatch markings represent the systemic uncertainty in estimating the contributions of the cold and warm molecular gas towards their outflows. (b) The composite mass outflow rates, comprising the sum of the ionized and molecular outflow rates from the left panel (black, solid curve) and only the ionized mass outflow rate (orange, dashed curve).}
\label{fig: composite mass outflow rate}
\end{figure*}



%For the cold molecular gas, we opt not to plot errors for the mass and outflow profiles. This is due to the substantial uncertainties in the CO-to-H$_2$ conversion factor, $\alpha_{\mathrm{CO}}$, which result in uncertainties for the mass and outflow profiles of $\sim$~230\%.
%Further discussion of our error propagation for the cold molecular gas is found in Section \ref{sec: molecular errors}.

Using observations of CO and HCN with the IRAM Plateau de Bure Interferometer at 0\farcs6 resolution, \cite{schinnerer00} found that NGC~3227 hosts a molecular ring of gas centered around the SMBH with a diameter of 3$''$ (345 pc) and with stronger emission on its eastern side.
%However, they also find counterrotation of the gas in the inner 1$''$, which is best explained by the presence of an inner molecular disk that starts $\sim$75 pc from the nucleus and warps to become nearly perpendicular at a distance $\sim$30 pc from the nucleus.  \cite{schinnerer00} believed that the warping is most likely due to gas pressure from the NLR as traced by the outflowing bicones, which exerts a torque upon the gas ring.
%Although \cite{schinnerer00} did not specify whether they thought the molecular ring was the site of outflowing gas,
Subsequent studies \citep{hicks08, davies14, schonell19} have sought to elucidate the role of the ring on the nuclear kinematics of NGC~3227, generally by characterizing the noncircular motions. \cite{alonso19} are the first to claim that these noncircular motions imply gas outflows, based on their high-resolution ALMA observations of CO, and this idea is supported by our spatial analysis of the cold and warm molecular kinematics in Paper I. 

Based on this finding of outflows in the molecular ring, if we assume that the entirety of the ring may be outflowing, Figure \ref{fig: ALMA outflow rate} shows that we obtain maximum cold molecular gas outflows that peak at $\le 18.1$ M$_\odot$ yr$^{-1}$ at 57 $\pm$ 6~pc NE and $\le 6.9$ M$_\odot$ yr$^{-1}$ at 34 $\pm$ 6 pc SW of the SMBH. A discussion of uncertainties for these values is found in \S \ref{sec: molecular errors}. These values are substantially larger than the values of 5 M$_\odot$ yr$^{-1}$ and 0.6 M$_\odot$ yr$^{-1}$ for the NE and SW ends, respectively, calculated by \cite{alonso19}.  However, it should be noted that in calculating the outflow rate using Equation \ref{eq: Mdot},  \cite{alonso19} confined the outflows to a 0\farcs2 square aperture whereas  Figure \ref{fig: ALMA outflow rate}  shows outflows that peak outside of this aperture and extend to almost 1\arcsec. 
%The substantial disparities between these values further underscores the need for spatially-resolved measurements of mass outflow rates. 




Figure \ref{fig: NIFS H2 outflow rate} shows that for the warm molecular gas, we see maximum outflow rates with one peak of $\dot{M}\mathrm{_{out}}\le 3.0 \times10^{-4}$ M$_\odot$ yr$^{-1}$ at a distance of 47 $\pm$ 6 pc SW of the nucleus, and another peak at $\dot{M}\mathrm{_{out}}\le 4.4 \times 10^{-4}$ M$_\odot$ yr$^{-1}$ at a distance of 47 $\pm$ 6 pc NE of the nucleus, which creates a combined maximum annular mass outflow rate of $\dot{M}\mathrm{_{out}}\le 7.4\times10^{-4}$ M$_\odot$ yr$^{-1}$ for this gas. The warm molecular gas mass and mass outflow rate are five orders of magnitude smaller than those of the cold molecular gas, which agrees with observations that the cold gas mass constitutes the majority of the gas mass in galaxies \citep{dale05, muller06}.  Warm molecular outflows are also detected in NGC~3227 by \cite{bianchin22}, whose study of the same NIFS H$_2$ $\lambda2.1218$ $\mu$m data reveals outflow rates of $\dot{M}\mathrm{_{out}}= 6.4 \times 10^{-4}$ M$_\odot$ yr$^{-1}$, which closely agrees with our values.




\subsubsection{Ionized Gas}
\label{sec: ionized results}
Figures \ref{fig: OIII mass} and \ref{fig: OIII outflow rate} show very significant ionized gas masses and mass outflow rates, but stark differences when comparing the NE region to the SW. This is due to the strong obscuration effects of dust, which are especially prominent in the SW (see the inset in Figure \ref{fig: 3227}). As such, the observed gas mass is likely undercounted, leading to an underestimate of the outflow rate in that region. Based on our model of the biconical outflows described in Paper I, we expect the outflow trends in the NE and SW to be fairly symmetric with one another. 

When the semi-ellipse annuli are azimuthally summed, our radial ionized mass outflow rate shown in Figure \ref{fig: mass outflow rates plotted together} reveals a trend that increases sharply to reach a peak of 19.9 $\pm$ 9.2 M$_\odot$ yr$^{-1}$ at 47 $\pm$ 6 pc, maintains an average value of 6  M$_\odot$ yr$^{-1}$ for the distance spanning $\sim$100--200 pc, and then steadily declines until it drops to 0 at $\sim 400$ pc. 
%The mass outflow rate extends to 280 pc due to the assumed linear velocity law, shown in Figure X. 
%In this model, we assume the gas is linearly accelerated by the radiation pressure and subsequently decelerated by gravity. In Paper I, we determined a model for the bicone with a height of 300 pc. When that height becomes projected along our line of sight in accordance with the galactic inclination, we arrive at a maximum projected height of 280 pc. As the bicone height controls the extent of the NLR outflows, it sets a maximum distance for the velocity law of 280 pc. The velocity also sets an upper limit on the distance for the mass outflow rate, on which it depends directly (see Equation \ref{eq: Mdot}). 

%When the annuli are summed and we compare the trend to those in Figure 11 in \cite{revalski21}, which graphs the spatially resolved gas-mass profiles for six galaxies, we find that it generally agrees with the other trends. 




In Figure \ref{fig: Mitch comparisons}, we compare our integrated ionized mass and mass outflow peak for NGC~3227 to those in Figure 5 of \cite{revalski25}, which also analyzes spatially-resolved emission to graph mass outflow rates for six galaxies alongside six galaxies previously studied in \cite{revalski21}. Our values for the peak ionized mass outflow rate and total integrated ionized mass are $\dot{M}$ = 19.9 $\pm$ 9.2 M$_\odot$ yr$^{-1}$ and log($M$) = $6.9 \pm 0.03$ M$_\odot$. We see that NGC~3227 generally follows the established trend in mass with bolometric luminosity, but it stands out as having the highest outflow rate of the sample despite a relatively modest bolometric luminosity (log$(L_{bol}) \sim  44.35$ erg s$^{-1}$; see Paper I). Nevertheless, its peak rate is close to that of NGC 788 at a similar luminosity. Similar positive trends connecting mass outflow rates to the bolometric luminosity have been previously studied \citep{cicone14, fluetsch19}, but the addition of spatially-resolved elements allows us to study these trends at a new level of detail.

% There may be a few factors that account for NGC~3227's high outflow rates. For instance, all AGN studied in \cite{revalski21, revalski25} are Type 2 Seyferts, while NGC~3227 is a Type 1 Seyfert. Whereas the Seyfert 2 bicones are generally pointed along the plane of the disk, resulting in instances of extending NLR structure \citep{fischer17, gnilka20}, the NLR bicone for NGC~3227 is pointed out of the plane of the disk as evidenced by the conical [O III] emission in Figure \ref{fig: OIII contours}. Additionally, although we did not find direct evidence of fueling flows in Paper I, we know that tidal interactions between NGC~3227 and its dwarf elliptical companion NGC 3226 are driving gas from NGC 3226 into NGC~3227. This tidal driving alongside a preponderance of gas and dust in the nuclear environment (Figure \ref{fig: 3227}) builds the image of a dense galactic center. Thus, when one envisions the ionizing radiation emanating into this environment, it is reasonable to expect that high levels of mass outflows may occur that are up to an order of magnitude larger than the peak outflow rates for other AGN of comparable bolometric luminosity.

% This line of reasoning explains why we are not concerned to see our cold molecular mass outflow rate agree with the sample presented by \cite{esposito24}, whereas our ionized outflow rates differ. Following our discussion in the previous subsection, \cite{esposito24} record an integrated ionized mass outflow rate of $0.076 \pm 0.017$ M$_\odot$ yr$^{-1}$ for NGC 5506, which is two orders of magnitude lower than our spatially-resolved rates for NGC~3227. The lower outflow rate likely occurs because NGC 5506 possesses an ionized mass of ${M} = 9.8 \times 10^{4}$ M$_\odot$, which is far lower than the ionized gas mass we find for NGC~3227 of ${M} = 7.9 \times 10^{6}$ M$_\odot$. This discrepancy can be reconciled if we assume that the ionized mass within the central few arcseconds is enlarged by tidal fueling flows.
% %Figure \ref{fig: mass outflow rates plotted together} shows the radial mass outflow rates for all our observed phases, which was calculated from summing the values at each equivalent distance away from the nucleus. This allows us to make a direct comparison with the results of \todo{Revalski+25}. 
% %However, whereas the mass profile for NGC~3227 agreed with the other galaxies, the mass outflow rate differs notably. 
% %\cite{revalski21} concerns the study of six nearby AGN: NGC 4151, NGC 1068, Mrk 3, Mrk 573, Mrk 78, and Mrk 34. The maximum outflow rate ($\dot{M}_{\mathrm{max}}$) among these galaxies is $12.45 \pm 2.72$ M$_\odot$ yr$^{-1}$, and corresponds to the brightest AGN (Mrk 34, $log(L_{bol}) \sim  46.2$). $\dot{M}_{\mathrm{max}}$ for NGC~3227 is 18 \todo{errors}, and it is noteworthy that NGC~3227, with $log(L_{bol}) \sim  44.35$ (see Paper I), is on the dimmer end of the galaxies in \cite{revalski21}, whose luminosities range $log(L_{bol}) \sim  43.9 - 46.2$. Indeed, based on the galaxies presented in this sample, we would expect the gas in NGC~3227 to possess a max outflow rate of 3 - 9 M$_\odot$ yr$^{-1}$. 

% We can also compare our results to those in \cite{bianchin22}, who find ionized outflows for the bicone model on the scale of 0.008 - 0.18 M$_\odot$ yr$^{-1}$. This is likely attributable to the fact that they use H II as the proxy for ionized mass rather than [O III], and their estimate of the outflowing ionized mass is derived from observations of the Pa$\beta$ using adjustments to Equation 5 in \cite{storchi09}. \todo{Discuss more-- is there a way to compare to what we have?}

% In Figure \ref{fig: Mitch comparisons}, we add NGC~3227 to the data points presented in \cite{revalski21, revalski25}, which graph the integrated gas mass and peak ionized mass outflow rate as a function of bolometric luminosity. \cite{revalski21} notes the positive correlation in both these trends, which is corroborated by the addition of NGC~3227. Similar positive trends connecting mass outflow rates to the bolometric luminosity have been previously studied \citep{cicone14, fluetsch19}, but the addition of spatially-resolved elements allows us to expand our knowledge of these trends to a new level of detail.


%Is there cause for alarm in seeing such an outlier as NGC~3227? \cite{ramos22} proposes that ionized outflows may, in conjunction with jets and winds, drag the molecular gas outwards assuming the NLR bicone bisects the plane of the cold molecular disk, which is indeed the case for NGC~3227. The idea that the cold molecular reservoirs may be propelled outwards by the ionized gas outflows is further supported by Figure \ref{fig: mass outflow rates plotted together}, which shows extremely similar trends in their mass outflow rates. If we believe that the cold molecular outflows may be impacted by the ionized outflows, perhaps the ionized gas is additionally accelerated by the cold molecular gas, leading to elevated ionized mass outflow rates.

%To answer this question, we investigate the potential influence of cold molecular outflows on the other galaxies presented in the student by \cite{revalski21}. We choose to exclude the brightest two AGN, Mrk 78 and Mrk 34, because their outflows extended far further than those of NGC~3227. In NGC 1068 the orientation of the bicone is tilted substantially away from the plane of the disk \citep{revalski21, cecil02}, so we can assume there is minimal propagation of the ionized outflows by the molecular outflows and vice versa. In Mrk 3, although the bicone squarely intersects the plane of the galactic disk along which the cold molecular outflows are likely oriented \citep{revalski21}, \cite{gnilka20} are unable to make assertions about molecular outflows because it is unable to be observed by ALMA. Additionally, Mrk 3 is the site of substantial tidal inflow from a companion, so even if there were observed molecular outflows, it might be hard to disentangle these kinematics from those due to streaming motions. Lastly, Mrk 573 is another AGN whose NLR bicone intersects the galactic disk \citep{fischer17}. Additionally, unlike the previous two galaxies, Mrk 573 shows evidence of molecular outflows moving at speeds of $\sim$100 km s$^{-1}$ at distances of up to 1$''$ from the nucleus. The fact that we observe coplanar ionized and molecular outflows in Mrk 573, and that it shows much lower levels of ionized outflows than NGC~3227, complicates this idea. Although the findings are inconclusive, this may be worth further study in the future. 

%However, NGC~3227 departs from from the trend of the three dimmer galaxies (NGC 4151, NGC 1068, and Mrk 3) in that while they show a mass increase for the first 100-200 pc before sharply decreasing (except for Mrk 3), the gas mass profile of NGC~3227 is fairly constant across the duration of the profile. There is a downturn in the mass profile that occurs at 250 pc, which is also reflected in the mass outflow profile. This is due to the [O III] emission on the SW end getting blocked out, 




%\subsubsection{Warm Molecular Gas}



%The molecular gas outflow rate differs from the outflow rate associated with the depletion timescale because the latter assumes ionization of the molecular gas occurs before driving, whereas the former does not.

%To convert the H$_2$ flux into mass, we utilize the relationship developed by \cite{muller06}, who found  $M_\odot \sim 2.5 \times 10^{-4} L_\odot$. 
%We recreate the radial annuli seen in Figures \ref{fig: WFC3 annuli} and \ref{fig: ALMA annuli} for the H$_2$ data, and sum the fluxes in each ring. We convert the fluxes to H$_2$ luminosity by utilizing the distance to the galaxy, $D = 23.7 \pm 2.6$ Mpc \citep{blakeslee01, tonry01}. 
%To calculate the mass outflow rate using Equation \ref{eq: Mdot}, we must also determine our velocity and $\Delta r$ values. The $\Delta r$ values are simply the widths of the annuli, which are each 11.5 pc as explained in Section \ref{sec: cloudy models}. The rotation-subtracted (i.e. outflow) velocities for the H$_2$ gas are shown in Figure 10 of Paper I and reproduced below \todo{?}. It is evident that the outflows are extraordinarily asymmetrical, with the location of the highest velocity oriented along the NLR outflow axis (PA = 240 deg from pointing south). 



\subsubsection{Composite Mass Outflows}
\label{sec: composite outflows}
Figure \ref{fig: mass outflow rates plotted together} plots the maximum mass outflow rates for the cold molecular, warm molecular, and ionized phases for complete annuli, where we use hatch markings to signify the systemic uncertainty related to the contributions of the cold and warm molecular gas towards the outflows. Our results show how the outflow rates of the cold molecular and ionized gas in this case are similar to each other, while the warm molecular gas outflow rate is more than four magnitudes lower.  
%The cold molecular gas has a peak outflow rate of $18.9 \substack{+42.8 \\ -14.1}$ M$_\odot$ yr$^{-1}$ at a distance of 57 $\pm$ 6 pc from the SMBH, and the warm molecular gas has a peak outflow rate of $(6 \pm 1)\times10^{-4}$ M$_\odot$ yr$^{-1}$ at a distance of 36$\pm$ 6 pc from the SMBH.

Studies have found that the cold molecular outflows tend to be larger than the ionized outflows by a factor of $10-10^3$ for $L_{AGN} \lesssim 10^{46}$ erg s$^{-1}$\citep{carniari15, fiore17, fluetsch21}, although others have found ionized outflow rates in galaxies that are 1--8 times larger than the molecular outflows \citep{venturi21}.  \cite{fluetsch19} find that for AGN with luminosities $L_{bol} \approx 10^{44}$ erg s$^{-1}$ like NGC~3227, molecular outflows are about an order of magnitude more powerful than the ionized outflows. Thus, NGC~3227 can be considered an outlier in this regard.
%Nevertheless, NGC~3227 is still unusual in its nearly equal ionized and cold molecular outflow rates.

%The mass outflow rates for a given phase depend largely on the gas mass of that phase. 
%There have been many attempts to derive a relationship between the cold molecular and ionized gas masses, with studies typically pointing to a cold gas mass that is 10$^5$ - 10$^7$ times larger than the ionized gas mass \citep{dale05,muller06, mazzalay13, schonell19}. We find that for NGC~3227, the cold molecular gas mass is $\approx 30$  times larger than the ionized gas mass. 
%However, as we have discussed in Section \ref{sec: ionized results}, NGC~3227 possesses an unusual amount of ionized gas near its nucleus, likely as a result of tidal fueling from NGC~3226, which may be increasing the outflow rates as a result. 
%Nevertheless, there is precedent for comparable outflows between cold molecular and ionized gas: in a study of the Teacup galaxy \cite{venturi21} find ionized outflows that are larger than the molecular outflows by a factor of 1--8.

Figure \ref{fig: mass outflow rates phases combined} shows the sum of the mass outflow phases, added for each azimuthally-summed radial bin, plotted alongside the ionized mass outflow rate. In doing so, we can estimate the impact of the outflows at their highest (with the presence of molecular outflows) and their lowest (without molecular outflows) values. The combination of the ionized and molecular outflows leads to a peak outflow rate of $\dot{M}\mathrm{_{out}}= 37.9 \pm 8.4$ M$_\odot$ yr$^{-1}$ at $54 \pm 6$ pc, which afterwards falls off steeply past 100 pc, where outflows of the cold molecular gas are no longer detected. Due to the lack of published material regarding multiphase mass outflow rates, it is challenging to contextualize our composite outflow values. Nevertheless, \cite{fluetsch19} record several Seyfert galaxies for which global ionized and molecular mass outflow rates combine to reach 10$^2$ - 10$^3$ M$_\odot$ yr$^{-1}$, although the contributions from the molecular outflows dominate over those from the ionized outflows. 

\begin{figure*}[t]
\centering  
\subfigure{\includegraphics[width=0.48\linewidth]{fig/comparisons_to_Mitch/comparison_to_Mitch_mass.pdf}\label{fig: Mitch mass comparison}}
\hspace{3mm}
\subfigure{\includegraphics[width=0.49\linewidth]{fig/comparisons_to_Mitch/comparison_to_Mitch_outflow.pdf}\label{fig: Mitch outflow comparison}}

\caption{The integrated ionized gas masses (left) and peak ionized outflow rates (right) of the AGN studied in \cite{revalski21} and \cite{revalski25}, alongside those for NGC~3227 presented in this work.}
\label{fig: Mitch comparisons}
\end{figure*}

%\todo{I'm trying to find other examples of composite mass outflow rates that I can compare this to but I'm not having much luck.}


% \subsection{Influence of Stellar Feedback?}
% It is well documented that feedback processes resulting in ionization and gas driving can result from AGN feedback, which is the focus of this work, and star formation feedback \citep{silk98, melioli15, drummond17}. Ideally, if we can ascertain an accurate measure of the star formation rate (SFR) in NGC~3227 or in the vicinity of its nucleus, we can quantity the extent to which gas driven outwards by the AGN outflows are being replenished. This would provide important information about the AGN's duty cycle, because an AGN can only stay active if there is sufficient material in its immediate surroundings on which it can feed. 



%\cite{schonell19} has calculated the SFR for the inner $3'' \times 3''$ of NGC~3227 to be $(0.9 \pm 0.2) \times 10^{-3}$  M$_\odot$ yr$^{-1}$, arriving at this value through calculating the mass accretion rate and the SFR surface density using NIFS data of NGC~3227 \citep{riffel17}. However, as we have shown with our BPT diagrams in Figure \ref{fig: BPT plots}, the AGN is the predominant source of ionization in this area. This finding is further confirmed by the fact that analysis of spectra outside the NLR show minimal signs of stellar absorption, implying that even beyond the immediate vicinity of the nucleus, the AGN is the preeminent source of ionizing radiation. Thus the SFR surface density, which is calculated from NIFS measurements of the Pa$\beta$ emission line, is contaminated by the AGN source and is not a reliable proxy for the SFR. A more accurate galactic SFR could be estimated if measurements were taken far beyond the bounds of NGC~3227's NLR bicone \todo{put bicone figure?}, where AGN contamination would be minimized. Alternatively, other studies have used measurements of warm dust with Herschel \citep{sturm11} to determine an estimate of SFR, but no such observations for NGC~3227 exist. Additionally, spectral energy distribution (SED) modeling in the IR would also reveal the heating sources of the dust \citep{garcia22}. Future studies on the star formation properties in NGC~3227's nucleus might wish to pursue one of these methodologies to achieve an estimation that effectively filters out the interference from the AGN.



% In the absence of available concrete measurements of the SFR on either galactic or nuclear scales, we can instead use other observations to obtain general estimates. Namely, \cite{riffel17} uses the same NIFS data as \cite{schonell19} to connect velocity dispersion maps to the presence of young stars. The fact that NGC~3227 displays a centrally-peaked velocity dispersion rather than patches of low velocity dispersion reveals a lack of young stars in NGC~3227's inner $3'' \times 3''$. 

% All these factors considered, we can confidently say that the SFR of NGC~3227 is very small compared to its mass outflow rates. Indeed, for a spiral galaxy with a stellar mass of $1.1 \times 10^9$ M$_\odot$ \citep{mundell95}, the SFR we expect to observe is approximately two magnitudes lower than the peak outflow rate of 20 - 40 M$_\odot$ yr$^{-1}$ \citep{renzini15}. Thus, there is no realistic expectation in which the gas being evacuated by the outflows is replenished by a new supply of stars. One potential reason the supply of stars is not being replenished is that the bicone opening intersects the plane of the galactic disk, so the gas that would form stars is being directly pushed away by the outflows. \cite{fiore17} showed that outflows can be powerful enough to deplete molecular gas reservoir, and this is corroborated by the ring of cold molecular gas that we see around the nucleus of NGC~3227 \citep{alonso19}. This is in contrast to other bicone orientations which are out of the plane of the disk, and so are not necessarily driving away the cold molecular gas reservoirs. 

\begin{figure*}[t]
\centering  
\subfigure{\includegraphics[width=0.465\linewidth]{fig/timescales/depletion_time_combined_phases.pdf}\label{fig: depletion timescale}}
\subfigure{\includegraphics[width=0.44\linewidth]{fig/timescales/crossing_time_for_all_3_phases.pdf}\label{fig: crossing time}}
\vspace{-.2cm}


\caption{(a) The depletion timescale for the molecular gas reservoir to be emptied by outflows in the three phases studied and (b) the crossing timescale for each of the three phases. The labels ``total transfer" and ``no transfer" refer to the extreme situations in which all or none of the gas mass is transferred from one bin to the next, respectively. The labels ``max'' and ``min'' refer to the presence and absence of warm and cold molecular outflows, respectively. The ``total transfer'' case has fewer points and a shorter maximum distance than the ``no transfer" because it can only be quantified for positive $\dot{M}_{\mathrm{net, \ total}}$ values at each distance.}
\label{fig: dep and cross timescales}
\end{figure*}

\begin{figure*}[t]
\centering  

\subfigure{\includegraphics[width=0.335\linewidth]{fig/timescales/evacuation_timescale_cold_molecular.pdf}\label{fig: evacuation cold molecular}}
\subfigure{\includegraphics[width=0.32\linewidth]{fig/timescales/evacuation_timescale_warm_molecular.pdf}\label{fig: evacuation warm molecular}}
\subfigure{\includegraphics[width=0.31\linewidth]{fig/timescales/evacuation_timescale_ionized.pdf}\label{fig: evacuation ionized}}


\caption{The evacuation timescales for (a) maximum cold molecular, (b) maximum warm molecular, and (c) ionized gas. The meanings of the ``total transfer," ``no transfer," ``min,'', and ``max'' labels are the same as in Figure \ref{fig: dep and cross timescales}.}
\label{fig: evac timescales}
\end{figure*}



\subsection{Resulting Timescales}

The depletion timescale as a function of distance from the SMBH can be calculated for all three phases combined from the cold molecular gas profile in Figure \ref{fig: ALMA mass} and the mass outflow rates in Figure \ref{fig: mass outflow rates phases combined}. This calculation is made for the two extremes described in Section \ref{sec: depletion} in which either all gas or no gas is propagated radially outwards from one annulus into the next. In Figure \ref{fig: depletion timescale}, the ``total transfer'' scenario yields higher timescales than the ``no transfer'' one because the net mass outflow rates $\dot{M}_{\mathrm{net}}$ are lower in the former. Within these two scenarios, we also note how the ``minimum'' cases, which exclude the molecular mass outflow rates from the calculations, result in higher timescales than the ``maximum'' cases, which include the molecular and ionized outflows. This is simply because without the molecular outflows contributing to the excavation of the reservoirs, it will take longer to deplete them. The depletion of the cold gas reservoir in each annulus occurs on timescales from $10^{4.5} - 10^{7.5}$ years, generally increasing with distance, with variations up to $\sim$1 dex depending on how much mass is transferred from one location to the next. It is also significant to note from Figure \ref{fig: depletion timescale} that the ionized outflows are the dominant drivers of outflowing gas in this AGN system, which sets it apart from the systems that are dominated by molecular outflows as discussed in the previous subsection.




Figure \ref{fig: crossing time} shows the crossing timescales, which depend on gas phase and span from $10^{5.3}$  -- $10^{7.0}$ years over the inner few hundred parsecs. As mentioned in \S \ref{sec: evacuation}, because the crossing timescale spans the distance from the current location to the edge of the outflows, $t_c$ will always increase as distance to the SMBH decreases. This figure shows that for the cold and warm molecular gas, $t_c$ increases significantly with decreasing distance from the SMBH. The crossing timescale of the ionized gas is fairly constant over most of the distance, ranging from $(2-5) \times 10^4$ years in each radial annulus until 350 pc, at which point the mass outflow rate declines significantly. 
%The difference between the crossing timescale for the ionized gas and those for the cold and warm molecular gas is likely attributable to substantial changes in the cold molecular mass profile \todo{???}
%The consistency in the timescale is evident when comparing \ref{fig: OIII mass} and \ref{fig: OIII outflow rate}, and observing similarities in shape of their trends.
%In Figure \ref{fig: crossing time}, we see the crossing time steadily increasing towards the center until a distance of 23 pc, at which point the crossing time jumps. Because the crossing time is simply a ratio of ionized slit mass to the mass outflow rate, we can attribute the high crossing time to the large central mass shown in Figure \ref{fig: OIII mass}. It should be noted according to Figure \ref{fig: OIII outflow rate} that the ionized gas outflow rate is also high in the inner 23 pc, 

Figure \ref{fig: evac timescales} shows the separate evacuation timescales for the three phases of gas. The cold and warm molecular evacuation timescales plotted represent the ``maximum'' cases, whereas the ``minimum'' cases of $\dot{M}_\text{mol}=0$ are not plotted. For the cold and warm molecular gas, In general, we observe timescales on the order of $10^{6.0} - 10^{7.6}$ years for the gas evacuation in the inner 400 pc to occur. The evacuation timescales for the warm and cold molecular gas are heavily influenced by longer crossing timescales at distances $<$ 50 pc, and by increasing depletion time scales at distances $>$ 50 pc. Because the crossing time for the ionized gas decreases slowly with distance, its evacuation timescales closely reflect the trends of its corresponding depletion timescales. The values at each annulus vary as widely as $\sim$1 dex between the ``total transfer'' and ``no transfer''  trends. Interestingly, the evacuation time scales span similar ranges (10$^6$ -- 10$^7$ years) in all three phases in the inner $\sim$100 pc, with the upper limit increasing to 10$^{7.6}$ years at larger distances for the ionized gas.

To contextualize these results, we can compare these timescales to the length of the AGN duty cycle. It should be noted that the literature is inconsistent when referring to the ``duty cycle'': in some contexts it refers to the total amount of time that an SMBH is active over its lifetime, while in other contexts it refers to the length of time in which an SMBH is continuously active, otherwise known as the ``AGN lifetime.'' In this work we adhere to the latter definition. 

Comparing our results to predictions of duty cycles is complicated because of large uncertainties. There are findings that the phase can last $10^7 - 10^9$ years \citep{martini01, marconi04}, but there are other suggestions that the longer phase is actually split into many shorter phases lasting $\sim 10^5$ years \citep{schawinski15, clavijo24}.  Furthermore, in certain situations some AGN-driven molecular outflows may continue to propagate $\sim10^8$ years after the AGN has turned off \citep{king11}. 

Nevertheless, our results show that removal of cold gas reservoirs due to mass outflows in the inner $\sim$400 pc of a moderate-luminosity Seyfert galaxy can be accomplished in  $10^{6.0} - 10^{7.6}$ years. This process can therefore be a mechanism for establishing the AGN duty cycle over these time scales, assuming no replenishment of cold gas during this time.
Other studies have identified AGN that may evacuate the molecular gas content within this time span \citep{sturm11, cicone14, fiore17, fluetsch19}. 

% We have shown in Paper I that the nuclear regions are dominated by outflowing rather than inflowing mechanisms \todo{what about Schinnerer warped disk?}, and the clear structure in the [O III] emission (see Figure \ref{fig: OIII contours}) exhibits a process of ongoing ionization.  



% Our spatially-resolved timescales allow us unprecedented insight into answering these crucial questions about the duty cycle. The changing evacuation timescale implies that the AGN's output is also changing as a function of time, represented by the distance. Because the crossing timescale is relatively small ($\sim$ 0.1 dex) and the radial cold molecular mass profile varies by 0.4 dex from 100-400 pc, variations in the depletion (and therefore evacuation) timescale depend largely on the extent to which material is being pushed by the radiation, and quantifying changes in these timescales may allow us to infer about periods of greater and lesser activity within the AGN's duty cycle.

% To investigate this idea, we can look to the evacuation timescales in Figure \ref{fig: evacuation timescale}. Let us look specifically at the case for "total transfer," which according to Section \ref{sec: depletion} describes the case in which gas pushed outwards in one bin is subsequently carried through the next. Thus, we are only concerned with situations where the radial ionized mass outflow rate (as shown in Figure \ref{fig: mass outflow rates plotted together}) is increasing with distance. This trend shows, upon first glance, a widely variable timescale that shoots up from 0-100 pc before decreasing from 120 - 180 pc, and then rising again to plateau from 180-200 pc. In theory, such an observation could be indicative of AGN 'flickering' \citep{schawinski15}, and that dips in the timescale correspond to periods of high AGN activity and peaks in the timescale correspond to periods of low AGN activity. In this context, phases of activity span $\sim$70 pc as judged by the spans of the plateaus, or $\sim$230 years when we consider light travel time over that distance. 

% However, we are hesitant to adopt this interpretation for a couple reasons, namely 1) we expect phases to last $\sim10^5$ years \citep{schawinski15}, and 2) our evidence for this 'flickering' is dependent upon a single data point at 100 pc, which interrupts an otherwise generally monotonic trend. Indeed, this monotonic trend is certainly observed in the "no transfer" scenario, wherein none of the gas from the bins are transferred into subsequent outer bins. In this trend which shows the timescale increasing as a function of distance from the nucleus, we can infer the presence of a continual "ramping up" phase, where the AGN is growing more powerful over time and accelerating material faster, which lowers the evacuation timescales as a result.

% This trend shows that the predicted timescale is low (log($\tau$) $\sim$ 6.4) close to the nucleus, and then shoots up to log($\tau$) $\sim$ 7.25 at 100 pc, dips down to log($\tau$) $\sim$ 7 - 7.1 from 120 - 180 pc, and finally plateaus at log($\tau$) $\sim$ 7 .5 from 180-200 pc. The magnitude of these changes corresponds to the magnitude of the positive $\dot{M}_{\mathrm{net}}$ values. We propose that these timescale variations might be indicative of AGN 'flickering' \citep{schawinski15}, and that dips in the timescale correspond to periods of high AGN activity and peaks in the timescale correspond to periods of low AGN activity. In this context, phases of activity span $\sim$70 pc as judged by the plateaus from 0-50 pc, 100 - 170 pc, and 180 - 260 pc, or $\sim$230 years when we consider light travel time over that distance. This is substantially shorter than the 10$^5$ years predicted by \cite{schawinski15}



%\subsection{Implications for Feedback Processes}

%Ultimately, the goal of this study is to address the question of whether outflows are the source of effective feedback mechanisms in NGC~3227, which are responsible for The mass outflow rate is also helpful because it allows us to better quantify the potential of the outflows to impact the host galaxy. calculate the kinetic luminosity, $\dot{E} = \frac{1}{2}\dot{M}v^2$


\section{Discussion}

\subsection{NGC~3227's Cold Molecular Outflow Rates}

We can compare our current study to that done by \cite{esposito24}, who calculate spatially-resolved mass outflow rates for cold molecular and ionized gas in NGC 5506. This galaxy is a comparable analog to NGC~3227 because both galaxies are classified as Seyfert~1, are at relatively similar distance (21 Mpc and 26 Mpc for NGC~3227 and NGC 5506, respectively), and have similar bolometric luminosities ($\sim$2.25 $\times$ 10$^{44}$ erg~s$^{-1}$ and $\sim$1.3 $\times$ 10$^{44}$ erg s$^{-1}$ for NGC~3227 and NGC~5506, respectively). Using CO(3-2) measurements, they record a trend in the molecular gas mass outflow rate that peaks at $\dot{M}{_{\mathrm{out, max}} ^{\mathrm{mol}}}$ = 28 M$_\odot$ yr$^{-1}$ at a distance of 85 pc (the site of the galaxy's molecular ring) and decreases to single-digit values past 2$''$. Although our own CO(2-1) data does not extend that far, the similarities in scale and magnitude of our observed maximum molecular outflow trends are notable.

Our results also concur with those of \cite{ramos22}, who compile molecular mass outflow rates across a variety of AGN encompassing QSO2s, Seyferts, and ultra-luminous infrared galaxies (ULIRGs). Their Figure 18 shows a trend relating the outflow mass rate to the bolometric luminosity. Their outflow mass value for NGC~3227, which comes from \cite{alonso19}, falls far below the trend. However, if we compare the trend with the results presented in this work, where the molecular mass outflow rate is predominantly above 2 M$_\odot$ yr$^{-1}$ and peaks at 15.6 M$_\odot$ yr$^{-1}$, we see that NGC~3227 is much more agreeable with the other galaxies. This comparison further underscores the benefit of spatially resolved mass outflow analyses.

\subsection{NGC~3227's Elevated Ionized Outflow Rates}


%There may be a few factors that account for NGC~3227's high outflow rates. For instance, all AGN studied in \cite{revalski21, revalski25} are Type 2 Seyferts, while NGC~3227 is a Type 1 Seyfert. Whereas the Seyfert 2 bicones are generally pointed along the plane of the disk, resulting in instances of extending NLR structure \citep{fischer17, gnilka20}, the NLR bicone for NGC~3227 is pointed out of the plane of the disk as evidenced by the conical [O III] emission in Figure \ref{fig: OIII contours}. 
NGC~3227 is unusual in its elevated ionized outflow rates, and in \S \ref{sec: composite outflows} we describe how the ionized outflow rates are similar to the maximum cold molecular outflow rates between 40 -- 80 pc, despite evidence that molecular outflow rates are typically magnitudes higher than ionized outflow rates. 
%Although we did not find direct evidence of fueling flows in Paper I, it seems likely that tidal interactions between NGC~3227 and its dwarf elliptical companion NGC 3226 are driving gas from NGC 3226 into NGC~3227. This tidal driving may increase the gas density of the galactic center, but from summing the cold molecular gas masses in Figure \ref{fig: ALMA mass}, we find a total cold molecular gas mass for NGC~3227 of $2.2 \times 10^8$ M$_\odot$. Typical cold molecular gas mass estimates in nearby galaxies range from $10^8 - 10^{10}$ M$_\odot$ \citep{busch17}, implying that we do not observe unusually high levels of cold molecular gas in NGC~3227's nucleus. 
%Thus, when one envisions the ionizing radiation emanating into this environment, it is reasonable to expect that high levels of mass outflows may occur that are up to an order of magnitude larger than the peak outflow rates for other AGN of comparable bolometric luminosity.
We wish to better understand why our cold molecular mass outflow rates agree with the sample presented by \cite{esposito24}, whereas our ionized outflow rates differ. Following our discussion in the previous subsection, \cite{esposito24} record an integrated ionized mass outflow rate of $0.076 \pm 0.017$ M$_\odot$ yr$^{-1}$ for NGC 5506, which is two orders of magnitude lower than our spatially-resolved rates for NGC~3227. The lower outflow rate likely occurs because NGC 5506 possesses an ionized mass of ${M} = 9.8 \times 10^{4}$ M$_\odot$, which is far lower than the ionized gas mass we find for NGC~3227 of ${M} = (7.7 \pm 0.6) \times 10^6$ M$_\odot$. However, the integrated cold molecular mass estimates are very similar: in this work we record a molecular mass of $2.2 \times 10^8$ M$_\odot$ for NGC~3227, and \cite{esposito24} record a molecular mass of $1.75 \times 10^8$ M$_\odot$ for NGC 5506. Thus, the ionized and molecular mass estimates may explain the corresponding mass outflow rates.
%This discrepancy can be reconciled if we assume that the ionized mass within the central few arcseconds is enlarged by tidal fueling flows.
%Figure \ref{fig: mass outflow rates plotted together} shows the radial mass outflow rates for all our observed phases, which was calculated from summing the values at each equivalent distance away from the nucleus. This allows us to make a direct comparison with the results of \todo{Revalski+25}. 
%However, whereas the mass profile for NGC~3227 agreed with the other galaxies, the mass outflow rate differs notably. 
%\cite{revalski21} concerns the study of six nearby AGN: NGC 4151, NGC 1068, Mrk 3, Mrk 573, Mrk 78, and Mrk 34. The maximum outflow rate ($\dot{M}_{\mathrm{max}}$) among these galaxies is $12.45 \pm 2.72$ M$_\odot$ yr$^{-1}$, and corresponds to the brightest AGN (Mrk 34, $log(L_{bol}) \sim  46.2$). $\dot{M}_{\mathrm{max}}$ for NGC~3227 is 18 \todo{errors}, and it is noteworthy that NGC~3227, with $log(L_{bol}) \sim  44.35$ (see Paper I), is on the dimmer end of the galaxies in \cite{revalski21}, whose luminosities range $log(L_{bol}) \sim  43.9 - 46.2$. Indeed, based on the galaxies presented in this sample, we would expect the gas in NGC~3227 to possess a max outflow rate of 3 - 9 M$_\odot$ yr$^{-1}$. 

We also compare our results to those in \cite{bianchin22}, who find ionized outflows for the bicone model on the scale of 0.008 - 0.18 M$_\odot$ yr$^{-1}$. This is attributable to their calculation of the integrated ionized gas mass for NGC~3227, which they find to be $(0.19 - 3.72) \times 10^4$ M$_\odot$. This is over two orders of magnitude lower than our ionized gas mass of $(7.7 \pm 0.6) \times 10^6 $ M$_\odot$. Our mass estimate is likely higher because we factor the effects of reddening into the calculation, which increases the mass estimate tenfold. According to \cite{schonell19} and as shown in Figure \ref{fig: Mitch mass comparison}, ionized gas masses for Seyferts tend to be higher than $10^4$ M$_\odot$, and it is likely that NGC~3227 would follow this trend as well. 



\subsection{Influence of Stellar Feedback?}
It is well documented that feedback processes resulting in ionization and gas driving can result from AGN feedback, which is the focus of this work, and star formation feedback \citep{silk98, melioli15, drummond17}. If we utilize a measure of the star formation rate (SFR) within NGC~3227 to better understand the extent to which gas may be expelled due to star formation-driven outflows, we can further constrain the impact of the observed AGN-driven outflows.
%Ideally, if we can ascertain an accurate measure of the star formation rate (SFR) in NGC~3227 or in the vicinity of its nucleus, we can quantity the extent to which gas driven outwards by the AGN outflows are being replenished. This would provide important information about the AGN's duty cycle, because an AGN can only stay active if there is sufficient material in its immediate surroundings on which it can feed. 

\cite{schonell19} has calculated the SFR for the inner $3'' \times 3''$ of NGC~3227 to be $(0.9 \pm 0.2) \times 10^{-3}$  M$_\odot$ yr$^{-1}$, arriving at this value through calculating the mass accretion rate and the SFR surface density using NIFS data of NGC~3227 \citep{riffel17}. However, as we have shown with our BPT diagrams in Figure \ref{fig: BPT plots}, the AGN is the predominant source of ionization in this area. This finding is further confirmed by the fact that analysis of spectra outside the NLR show minimal signs of stellar absorption, implying that even beyond the immediate vicinity of the nucleus, the AGN is the preeminent source of ionizing radiation. Thus the SFR surface density, which is calculated from NIFS measurements of the Pa$\beta$ emission line, is contaminated by the AGN source and is not a reliable proxy for the SFR. A more accurate galactic SFR could be estimated if measurements were taken far beyond the bounds of NGC~3227's NLR bicone, where AGN contamination would be minimized. Alternatively, other studies have used measurements of warm dust with Herschel \citep{sturm11} to determine an estimate of SFR, but no such observations for NGC~3227 exist. Additionally, spectral energy distribution (SED) modeling in the IR would also reveal the heating sources of the dust \citep{garcia22}. Future studies on the star formation properties in NGC~3227's nucleus might wish to pursue one of these methodologies to achieve an estimation that effectively filters out the interference from the AGN.



In the absence of available concrete measurements of the SFR on either galactic or nuclear scales, we can instead use other observations to obtain general estimates. Namely, \cite{riffel17} uses the same NIFS data as \cite{schonell19} to connect velocity dispersion maps to the presence of young stars. The fact that NGC~3227 displays a centrally-peaked velocity dispersion rather than patches of low velocity dispersion reveals a lack of young stars in NGC~3227's inner $3'' \times 3''$. 

All these factors considered, we can confidently say that the SFR of NGC~3227 is very small compared to its mass outflow rates. Indeed, for a spiral galaxy with a stellar mass of $1.1 \times 10^9$ M$_\odot$ \citep{mundell95}, the SFR we expect to observe is approximately two magnitudes lower than the peak outflow rate of 20 -- 40 M$_\odot$ yr$^{-1}$ \citep{renzini15}. 
%Thus, there is no realistic expectation in which the gas being evacuated by the outflows is replenished by a new supply of stars. One potential reason the supply of stars is not being replenished is that the bicone opening intersects the plane of the galactic disk, so the gas that would form stars is being directly pushed away by the outflows. \cite{fiore17} showed that outflows can be powerful enough to deplete molecular gas reservoir, and this is corroborated by the ring of cold molecular gas that we see around the nucleus of NGC~3227 \citep{alonso19}. This is in contrast to other bicone orientations which are out of the plane of the disk, and so are not necessarily driving away the cold molecular gas reservoirs. 

\section{Conclusions}

We present spatially resolved gas mass and mass outflow rate profiles for NGC~3227 in the cold molecular, warm molecular, and ionized gas phases, with the warm and cold molecular rates representing upper limits in this system. 
This study joins a small but growing number of studies focusing on spatially resolved mass outflow rates \citep{revalski18b, revalski21, revalski22}, even fewer of which are multiphase \citep{esposito24}. 
Our conclusions are as follows:
\begin{enumerate}
    \item The maximum cold molecular, maximum warm molecular, and ionized gas show peak mass outflow rates of $23.1$ M$_\odot$ yr$^{-1}$, $9\times10^{-4}$ M$_\odot$ yr$^{-1}$, and $19.9 \pm 9.2$ M$_\odot$ yr$^{-1}$, respectively. In all three phases, the peaks occur 35 -- 60 pc from the nucleus. When summed, the maximum peak outflow rate is $37.9 \pm 8.4$ M$_\odot$ yr$^{-1}$ and occurs around $54 \pm 6$ pc, where bright knots of CO emission are located at the ends of a $\sim$1\arcsec\ molecular bridge that crosses the location of the SMBH.
    \item Enormous variations in the outflow rates as a function of distance (both radially and azimuthally) underscore the necessity of spatially-resolved trends rather than relying on single (i.e. global) values, particularly for understanding the locations and origins of the outflows and their role in depleting the gas reservoirs.
    \item The peak ionized outflow rate agrees well with the trends observed by \cite{revalski21, revalski25}, which establish positive correlations between bolometric luminosity and both ionized gas mass and peak ionized mass outflow rate.
    \item We find the ionized mass outflow rates and maximum cold molecular in NGC~3227 to be nearly equal with one another, which is in contrast with studies of most other AGN where molecular outflows are typically larger than ionized outflows by a factor of $10 - 10^3$. 
%We find that the discrepancy is likely due to the abundance of molecular gas in the nuclear region, which is likely caused by tidal streaming motions that funnel gas into the center.
    \item The molecular gas reservoirs are depleted and evacuated from the nuclear region on timescales of $10^{6.0} - 10^{7.6}$ years, which agrees with other studies of gas evacuation times. We show that the ionized outflows, rather than the molecular outflows, are the primary drivers of the gas excavation. This level of dominance by the ionized outflows is uncommon in AGN feedback systems.
    \item The gas evacuation occurs on similar timescales as predicted for the AGN duty cycle, which describes the length of time over which the AGN is continuously active. By comparing these timescales, we conclude that AGN outflows are an effective means of clearing the inner few hundred parsecs of cold gas reservoirs in this moderate luminosity Seyfert galaxy.
\end{enumerate}

\begin{acknowledgements}

The authors would like to thank the anonymous referee for the constructive and detailed feedback. J.F. would like to thank Dr. Almudena Alonso-Herrero for graciously sharing her ALMA data on NGC 3227. 

Much of the data presented in this work are based on observations with the NASA/ESA Hubble Space Telescope and were obtained from the Mikulski Archive for Space Telescopes (MAST), which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. These observations are associated with program No. 16246 (\href{https://archive.stsci.edu/proposal\_ search.php?mission=hst\&id=16246}{https://archive.stsci.edu/proposal\_ search.php?mission=hst\&id=16246}).
Support for program No. 16246 was provided through a grant from the STScI under NASA contract NAS5-26555. The specific observations used in this analysis can be accessed via DOI:  \dataset[10.17909/cm86-me24]{\doi{10.17909/cm86-me24}}.


This research has made use of NASA’s Astrophysics Data System. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
Some of the observations used in this paper were obtained with the Apache Point Observatory 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium.
%\clearpage
\end{acknowledgements}


\appendix
\label{sec: appendix}
\restartappendixnumbering
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\begin{figure*}[t]
\centering  
\includegraphics[width=0.34\linewidth]{fig/BPT_plots/ngc3227pa150_kosmos_SII.pdf}
\hspace{.05mm}
\includegraphics[width=0.31\linewidth]{fig/BPT_plots/ngc3227pa150_kosmos_NII.pdf}
\hspace{.05mm}
\includegraphics[width=0.31\linewidth]{fig/BPT_plots/ngc3227pa150_kosmos_OI.pdf}
%\vspace{4mm}
\includegraphics[width=0.34\linewidth]{fig/BPT_plots/ngc3227pa190_kosmos_SII.pdf}
\hspace{.05mm}
\includegraphics[width=0.31\linewidth]{fig/BPT_plots/ngc3227pa190_kosmos_NII.pdf}
\hspace{.05mm}
\includegraphics[width=0.31\linewidth]{fig/BPT_plots/ngc3227pa190_kosmos_OI.pdf}

\caption{BPT ionization diagrams for APO KOSMOS observations along the major (PA = 150\arcdeg, top) and outflowing (PA = 190\arcdeg, bottom) axes. Positive distance refers to the north. The presence of triangle and plus shapes alongside the circles in plots of the major axis indicate second and third components, respectively, in the Gaussian fits.}
\label{fig: other BPTs}

\end{figure*}

In \S \ref{sec: BPT}, we use BPT diagrams to analyze the ionization source along the galactic minor axis of NGC~3227. We use the KOSMOS spectrograph to take data along two other axes of importance: the galactic major axis, and the outflow axis along which the NLR outflows propagate most strongly. The orientation of these other two slits are shown in dotted lines in Figure {fig: OIII contours}. Although the analysis in this paper does not involve these two slits, this section of the Appendix discusses our interpretation of their associated BPT diagrams as shown in Figure \ref{fig: other BPTs}.




\section{Major Axis (PA = 150\texorpdfstring{\arcdeg}{})}
The major axis contains more kinematically complex regions than those observed along the minor axis, and consequently there are a number of three-component fits that are plotted in its BPT diagrams, which is seen in the top row of Figure \ref{fig: other BPTs}. The additional number of components can likely be attributed, at least in part, to the rotational motion which is much more visible along the major axis where it is at its highest projection along our line of sight, compared to the minor axis which suppresses our visibility of rotational motion.

Nevertheless, along this axis, we see clear indication that the gas is ionized by the AGN at virtually all recorded distances, which extend out to $6''$ NW. This may be surprising when comparing to the major axis slit placement in Figure {fig: OIII contours}, which appears to show minimal NLR emission along the major axis beyond $\sim2''$ NW. KOSMOS may have been able to pick up the emission at greater distances than WFC3 because of its larger spatial scale that surveys a greater area to create the spectrum at each slit position. %\todo{maybe? does this sound legit?}

\section{Outflows Axis (PA = 190\texorpdfstring{\arcdeg}{})}
The outflows axis is the orientation which most comprehensively envelopes the NLR outflows, as seen in Figure {fig: OIII contours}. Unlike the major and minor axes, the emission from the outflows axis is associated with LINER signatures at distances both NE and SW of the SMBH, according to the leftmost plot in the bottom row of Figure \ref{fig: other BPTs}.  However, the other two plots in the bottom row of Figure \ref{fig: other BPTs} show significantly less overlap with the LINER regime. 

As shown in Figure \ref{fig: 3227}, there is a strong presence of star-forming H~II regions in the immediate vicinity of the SMBH, represented in the figure's inset by red coloring. It is likely that contamination from this emission is creating this effect in the BPT diagram.




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\facilities{HST(STIS, WFC3), APO (KOSMOS), Gemini North (NIFS), ALMA}

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\software{Cloudy \citep{ferland13}, MultiNest \citep{feroz19}, Astropy \citep{astropy22}}



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%\appendix
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\end{document}