\section{Introduction}
\subsection{General Background}
\label{sec:introdution}
Active Galactic Nuclei (AGN) are powered by accretion of matter onto Supermassive Black Holes (SMBHs), with masses $\geq$ 10$^{6}$ ${\rm M\textsubscript{\(\odot\)}}$. This process of feeding the SMBH creates large amounts of electromagnetic radiation from the accretion disk of the black hole, which can accelerate winds, and may produce feedback \citep[e.g.,][]{begelman2004a}. These winds, if producing efficient feedback, evacuate the bulge of the host galaxy, quenching star formation, which produces the well-known relationship between the mass of the SMBH and the mass of the bulge \citep[e.g.,][]{gebhardt2000a}. AGN feedback is observed in two forms: highly-collimated jets and AGN-driven winds. Even though jets can be powerful enough to drive feedback, they only occur in only 5-10\% of the AGN population \citep{rafter2009a}.  Meanwhile, the fact that winds are prevalent in most AGN \citep[e.g.,][]{mullaney2013a, genzel2014a, woo2016a} suggests that they could be the principal feedback mode. \par 

While the details of AGN classification and structure have been reviewed numerous times \citep[e.g.,][]{peterson1997a, netzer2015a}, it is useful to briefly summarise them. Generally, AGN are classified via their optical emission. Type 1 AGNs have broad permitted lines, with full widths at half maximum (FWHM) $>$ 1000 ${\rm km~s^{-1}}$, and narrow forbidden lines, with FWHM $\leq$ 1000 ${\rm km~s^{-1}}$. Type 2s have narrow permitted and forbidden lines, and continua typically dominated by starlight from the host galaxy. The broad lines observed in Type 1s are formed in a region called the broad line region (BLR), which extends from 10s-100s light days from the AGN and possesses a typical electron density of $n_{H} > 10^{8}$ ${\rm cm^{-3}}$. Meanwhile, the narrow components are formed in the narrow-line region (NLR), which extends 10s - 100s pcs from the AGN, and its typical $n_{H}$ is $10^{2} - 10^{6}$ ${\rm cm^{-3}}$. The NLR kinematics is consistent with combination of radial outflows (winds) and rotation \cite[e.g.,][]{fischer2017a}. \par 


Ionised mass outflows are frequently observed as X-ray and UV absorption lines, originating in gas within hundreds of parsecs from the central source \citep[e.g.,][]{crenshaw2003a, crenshaw2012a, king2015a}, or optical emission-lines in the NLR of AGNs \citep[e.g.,][]{crenshaw2005a, muller2011a, fischer2013a, nevin2016a}. Spectroscopic observations have shown that the SMBH can drive the outflowing clouds to hundreds of parsecs with velocities up to $\sim$ 0.2c, in the case of broad absorption line (BAL) QSOs \citep[e.g.,][]{rodriguezhidalgo2011a, hamann2013a}, and up to $\sim$ 0.25c, in the case of UFOs \citep{tombesi2010a, tombesi2010b, tombesi2015a}. \par 

In order to investigate how effective AGN feedback is on galactic-bulge scales, as required in a star-formation quenching, negative feedback scenario, it is important to quantify the mass outflows properties and their impact on the host galaxy. The physical properties of the gas, such as mass, mass outflow rates and kinetic energy can be estimated by photoionisation models \citep[e.g.,][]{crenshaw2007a}, while emission-line studies, particularly those utilising high spatial-resolution \citep[e.g.,][]{fischer2017a, trindadefalcao2021a, revalski2021a}, provide the most accurate method to calculate critical quantities of these winds, such as the masses, velocities, outflow rates, and kinetic energies, as a function of distance from the SMBH.\par

The strength of the AGN-driven winds can be quantified in the form of kinetic luminosity, $\dot E(r) = \frac{1}{2} \dot M_{out} v^2$, where the mass outflow $\dot M_{out}(r) = 4\pi rN_{H}\mu m_{p}C_{g}v_{r}$, and $r$ is the radial distance from the SMBH, $N_{H}$ is the column density, $\mu$ is the mean mass per proton, in this case =1.4, $m_{p}$ is the proton mass, $C_{g}$ is the global covering factor of the gas, and $v_{r}$ is the radial velocity. Studies for efficient feedback require $\dot E(r) \sim 0.5\% - 5\%$ of $L_{bol}$, the bolometric luminosity of the AGN \citep{dimatteo2005a, hopkins2010a}. In addition, it is well-known that the amount of kinetic energy deposited into the host galaxy rises rapidly with the velocity of the gas, since $\dot E \propto v^{3}$.\par 

\begin{figure}
  \centering
  \includegraphics[width=8cm, height=5.3cm]{ngc1068_bright_profiles_rot-1.png}
\caption{Emission-line brightness profiles from STIS long-slit G430L and G750L spectra of
NGC 1068 \citep[][]{crenshaw2000b} ($\approx$ 50pc/$\arcsec$). The coronal lines [Fe~VII] 6087\AA, [Fe~X] 6375\AA, [Fe~XI] 7892\AA, and [Fe~XIV] 5303\AA~
reveal highly-ionised emission-line knots up to 2$\arcsec$ from the central AGN, often at different relative strengths compared to the [O~III] 5007\AA~knots.}
\label{fig:NGC1068_spectrum}
\end{figure}


\begin{figure*}
  \centering
 \begin{minipage}[b]{0.483\textwidth}
  \includegraphics[width=8cm, height=5.3cm]{Ion_abundances_2.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.483\textwidth}
  \includegraphics[width=8cm, height=5.3cm]{Ion_abundances_20,5_2.png}
 \end{minipage}\qquad 
\caption{\textbf{\textit{Left figure:}} Cloudy model predictions for the fractional abundances of relevant ionisation states of iron and silicon and H- and He-like oxygen and neon. Also, marked in black, we present the log$U$ and fractional abundances for which models predict the maximum fluxes for the X-ray emission-lines O~VII -f 22.1\AA, O~VIII Ly-$\alpha$, and Ne~IX -f 13.7\AA~ (as discussed in Section \ref{sec:study_footprints}). The model assumptions are: abundances (1.4x solar) and spectral energy distribution (SED) from \citealt{trindadefalcao2021a}, and constant column density N$_{H}$ = 10$^{21.5}$ cm$^{-2}$ \citep[e.g.,][]{kraemer2020a, trindadefalcao2021b}. As shown here, optical emission lines from ions such as Fe~X, Fe~XI, Fe~XIV, and IR emission lines from ions such as Si~X will be formed in the X-ray emitting gas and will act as X-ray ``footprints". \textbf{\textit{Right figure:}} Same as left figure, but with a model column density of N$_{H}$ = 10$^{20.5}$ cm$^{-2}$.}
\label{fig:cloudy}
\end{figure*}




The studies of mass outflow in nearby AGN show that, even though these outflows are very massive and inject considerable amounts of kinetic energy in the bulge of the host galaxy, they do not extend far enough to clear the bulge of gas \citep{fischer2018a} and also lack the power to do so \citep{trindadefalcao2021a}, since their $\dot E(r)/L_{bol}$ ratio does not reach the required 0.5\% for efficient feedback in AGNs radiating at near their Eddington limit \citep[e.g.,][]{hopkins2010a} . Therefore, these results suggest that winds of optical emission-line gas are not an efficient form of AGN feedback \citep[e.g.,][]{trindadefalcao2021a}). However, these results do not take into account the role of higher-ionization gas in this process of AGN feedback. For instance, X-ray winds in the form of UFOs \citep[e.g.,][]{tombesi2010a} appear to be able to drive large-scale outflows in Ultra-Luminous InfraRed Galaxies (ULIRGs) \citep{tombesi2015a}, which means that it is possible that X-ray winds are responsible for the dynamic effects we observe in nearby AGNs \citep{trindadefalcao2021b}.\par 

Extended soft X-ray emission, co-located with the [O~III] emission-line gas, was detected by \textit{Chandra}/Advanced CCD Imaging Spectrometer (ACIS) in nearby Seyfert galaxies \citep{ogle2000a, young2001a}. In addition, \textit{Chandra} imaging has been used to map the NLR X-ray emission in several Seyferts \citep[e.g.,][]{bianchi2010a, gonzales2010a, wang2011a, wang2011b, wang2011c, maksym2019a}. By isolating bands dominated by specific emission-lines, these authors were able to derive constrains on the structure of the X-ray emission-line regions. Notably, \citet{wang2011a} and \citet{maksym2019a} suggest that there is evidence for shocks, which indicates interaction of the X-ray with the ISM of the host galaxy.\par 

Even though it possible to obtain data with good spectral resolution with \textit{Chandra}/HETG \citep[e.g.,][]{kallman2014a, kraemer2020a}, as well as kinematics and detailed physical insights, it is challenging to obtain detailed spatially-resolved information from these data. For instance, in their study of NGC 4151, \citet{kraemer2020a} were not able to get any accurate kinematic profiles from the HETG data, even for such a nearby AGN. Therefore, we have no means to determine what, if any, role this high ionization gas plays in the process of AGN feedback.\par

Alternatively, a model for X-ray gas, characterised by a log$U$\footnote{$U= \frac{Q}{4\pi n_{H}r^{2}c}$, where, $n_{H}$ is the hydrogen number density, $r$ is the
distance to the ionising source, $c$ is the speed of light, and the ionising luminosity, $Q = \int_{\nu_{0}}^{\infty} L_{\nu} \,d\nu \ $, where $h\nu_{0}$ = 13.6eV.} = 0.0 predicts a strong emission from optical and IR lines (see Section \ref{sec:study_footprints}), which can be considered as "footprints" of the X-ray wind \citep[e.g.,][]{porquet1999a}. Since these footprint lines could be detected using ground-based
near-infrared telescopes \citep[e.g.,][]{rodriguez-ardila2011a, lamperti2017a} or the \textit{James Webb Space Telescope} and \textit{HST}/STIS, it provides an
opportunity to accurately constrain the kinematics of the X-ray wind.




\subsection{Mrk 34: a case for X-ray feedback}
In our most recent paper \citep[][]{trindadefalcao2021b}, we analysed the dynamics of mass outflows in the NLR of Mrk 34, a Type 2 quasar (QSO2) ($z$ = 0.051), which possesses a L$_{bol} = 2.6\times 10^{45}$~erg~$s^{-1}$ \citep[][]{trindadefalcao2021b}, and a black hole mass range of log~$M\textsubscript{\(\odot\)}$~=~$6.8-7.5$ \citep[e.g.,][]{revalski2018a}.\par 

We presented evidence for the presence of X-ray winds in the NLR of this QSO2 by performing an analysis based on the kinetic energy density of the [O~III]-emitting gas and the X-ray winds, and also by suggesting that high velocity gas is being accelerated in-situ via entrainment by X-ray winds \citep[for details see][]{trindadefalcao2021b}. Even though we were able to get some constraints on the properties of the X-ray wind for this target, our analysis used the \textit{Chandra} imaging technique described above, which means that the spectral resolution was too coarse to measure kinematics. \par 

In addition, we used Cloudy photoionisation models \citep{ferland2017a} to calculate the integrated fluxes for some of the X-rays footprints, i.e., [Si~X] 1.43 $\mu$m and [Fe~X] 6375\AA. After comparing our results to the measured values, we found that our integrated and radial flux distributions for the [Fe~X] emission-line agree very well with the Apache Point Observatory (APO) data for Mrk 34 \citep[][]{revalski2018a}, which not only supports our parameterisation of the X-ray winds, but also confirms the ability of the X-ray footprint lines to be used to constrain the X-ray wind dynamics.\par 

Even though our X-ray models provided predictions for the [Fe~X] that were consistent with the optical spectra, as mentioned above, our parameterisation of the physical properties of the X-ray winds were determined indirectly, by its inferred dynamical effects and the distributions of the [O~III] gas in the NLR of Mrk 34. Therefore, although we were able to obtain self-consistent results, this does not present itself as a well-constrained model of the X-ray wind.\par 

In the present paper, we model the X-ray footprint lines using Cloudy photoionisation models, which allows us to construct radial flux profiles for these lines. We then compare our predictions to the results of \citet{kraemer2020a} for NGC 4151. In section \ref{sec:kraemer_results} we review the methodology and results of \citet{kraemer2020a} for the study of the mass outflows of the X-ray emission-line gas in NGC 4151. In section \ref{sec:models} we present our photoionisation modeling analysis.  In section \ref{sec:footprints} we present a comparison between our modeling results and the results of \citet{kraemer2020a}. Finally, in sections \ref{sec:discussion_1} and \ref{sec:discussion} we discuss our results and present our conclusions, respectively.

\section{The Study of Optical/IR X-ray Footprint Lines}
\label{sec:study_footprints}

As shown in Figure \ref{fig:NGC1068_spectrum} the STIS long-slit G430L and G750L spectra for NGC 1068 show spatially resolved emission from iron lines, such [Fe~VII] 6087\AA, [Fe~X] 6375\AA, [Fe~XI] 7892\AA, and
[Fe~XIV] 5303\AA (see Table \ref{tab:fluxes} for ionisation potentials). The fact that these lines show different spatial distributions than [O~III]
5007\AA~ means that they likely track different emission components \citep[e.g.,][]{kraemer2009a}. Figure \ref{fig:cloudy} shows the Cloudy model predictions for the relative abundance of the Fe, Si, and Ca ions, compared to H and He-like O and Ne, as a function of the ionisation parameter.
The fact that these lines are formed in gas over the same range in ionization state indicates
that they will be formed in gas that will also be the source of X-ray emission-lines, detected
in the HETG or \textit{XMM-Newton}/Reflection Grating Spectrometer spectra.
As noted above, these lines provide a footprint of the X-ray emitting gas in UV/optical/IR spectra that can be spatially-resolved, e.g., via \textit{Hubble Space Telescope} (HST) observations with STIS, \textit{James Webb Space Telescope} (JWST)/NIRSpec or even ground based
spectra \citep[e.g.,][]{may2017a} and, therefore, can potentially be used to measure the kinematics and, ultimately, the dynamics of the extended X-ray gas. \par 

It is important to distinguish between coronal lines and footprint lines. Coronal lines are emission-lines formed from very highly ionized atoms, and they possess this name because they were first observed in the solar corona \citep[e.g.,][]{mazzalay2010a}. These lines are collisionally excited forbidden transitions within low-lying levels of highly ionized species with ionization potentials (IP) $\geq$ 100 eV. They have been detected in the optical and IR spectra of all types of Seyfert galaxies, including type 1s, type 2s, and narrow-line Seyfert 1 galaxies \citep[e.g.,][]{seyfert1943a, nagao2000a, rodriguez-ardila2002a, landt2015a}, and represent one of the key gaseous components of the active nucleus. They appear to be approximately equally abundant in type 1 and type 2 AGNs \citep[e.g.,][]{rodriguez-ardila2011a}.\par 

 On the other hand, we define footprint lines to be a subset of "coronal" lines from ions with IP $>$ 138 eV, i.e., that of O~VII, which peak over the same range in ionisation parameter as the H and He-like ions. For instance, as shown in Figure \ref{fig:cloudy}, the fractional abundance for Fe~VII, a ionisation state which produces a coronal line, peaks at log$U\approx$ -1.0, i.e., far from where the X-ray lines peak (log$U\approx$ 0.0). However, the fractional abundance for Fe~X, a ionisation state which produces a footprint line, peaks at log$U\approx$ 0.0, i.e., at the same ionisation parameter as the X-ray lines. It is also important to note here that the peak ionic abundance corresponds to the peak flux for collisionally excited lines, like the footprint lines, but not the X-ray lines. These are primarily due to recombination in photo-ionised plasma, as shown in Figure \ref{fig:cloudy}, hence peak at somewhat higher
$U$ than where the ionic abundances is greatest.\par 

In addition, due to its high ionization the gas is optically thin to the ionizing radiation, which means that our results are not a strong function of the column density of the gas. For instance, in Figure \ref{fig:cloudy} we present the results for the Cloudy model predictions for the fractional abundances of relevant ionisation states of iron and silicon and H- and He-like oxygen and neon considering that the gas possesses a N$_{H}$ = 10$^{21.5}$ cm$^{-2}$ (left panel) and N$_{H}$ = 10$^{20.5}$ cm$^{-2}$ (right panel). As we can see, the ions from which the footprint lines originate and H- and He-like ions still coexist over the same range in ionisation parameter in a gas with a lower column density, which means that our results do not change over a factor of $\gtrsim$ 10 in N$_{H}$.







\section{NGC 4151}

\subsection{Previous Analysis of NGC 4151}
\label{previous_analysis}
NGC 4151 ($z$ = 0.0033 \citep{vaucouleurs1991a}) is a Seyfert 1.5 galaxy which possesses a black hole mass of log $M\textsubscript{\(\odot\)}$ = $7.55^{+0.051}_{-0.047}$ \citep{bentz2015a}. The most recent distance estimate is $\approx$ 15.8 $\pm$ 0.4 Mpc \citep[][]{yuan2020a}, which gives a scale of 70 pcs per arcsecond. \textit{Chandra}/HETG observations of NGC 4151 revealed soft X-ray emission extending $\sim$ 1.6 kpc from the nucleus \citep{ogle2000a}. Based on their analysis, \citet{wang2011a, wang2011b, wang2011c} conducted a detailed study of the target's X-ray emission and, by isolating regions of the spectrum, they were able to map the O~VII, O~VIII, and Ne~IX emission, which appeared to be spatially correlated with the optical [O~III] 5007\AA~emission detected by \textit{HST}/WFPC2 \citep{kraemer2008a}.\par 

In addition, \citet{kraemer2020a} analyzed \textit{Chandra}/HETG spectra of the X-ray emission-line gas in NGC 4151. The images show extended H- and He-like O and Ne, up to a distance $r \sim 200$ pc from the nucleus. They also measured an average line velocity of $\sim -230$ ${\rm km~s^{-1}}$, which suggests a significant amount of outflow of X-ray gas. They found total mass and mass outflow rates that are similar to those determined using [O~III] (see Section \ref{sec:kraemer_results}), implying that the X-ray gas is a major outflow component. \par 

The use of \textit{Chandra}/HETG to characterise the X-ray outflows in NGC 4151 is an improvement compared to the ACIS imaging data used in \citealt{trindadefalcao2021b} to analyse the X-ray winds in Mrk 34. On the other hand, it is important to note that the data from \textit{Chandra}/HETG is still not enough to to fully characterise the extended emission, as noted above, since it is impossible to directly measure the kinematics profile, as we discuss later on in this study. Furthermore, this analysis can only be done for a few nearby AGNs. \par 



\subsection{The Study of Mass Outflow of the X-ray Emission Line Gas}
\label{sec:kraemer_results}
In this section we summarise the findings of \citet{kraemer2020a} on the mass outflows of the X-ray gas in NGC 4151. By combining the available datasets \citep[see][for more details]{kraemer2020a}, the authors measured the fluxes of emission lines and radiative recombination continua (RRCs) over a local background. They then generated photoionisation models to fit the 1$^{st}$ order HETG spectra, which required three distinct emission-line components. Once a satisfactory fit was achieved, the predicted emission line fluxes from the photoionisation models were compared to their measured fluxes. The result of this phase of the analysis is what is called the “global” fit, i.e., not spatially-resolved \citep[e.e.g,][]{revalski2018a}.\par 

After analysing the 1$^{st}$ order HETG spectra, \citet{kraemer2020a} created emission-line profiles using the 0th order HETG spectra. The emission-line profiles were created for those lines which were used in the Cloudy model fits, as above. The density laws, combined with the spatially-resolved 0th order fluxes, were used to generate photoionisation models for the X-ray emitting gas as a function of radial distance (for detailed information, see \citealt{kraemer2020a}). \par 

To estimate the total mass of ionised gas and the mass outflow rates, \citet{kraemer2020a} applied the model parameters to fit the 0th order emission-line profiles of Ne~IX and Ne~X. Because they mapped Ne~IX-rif (the resonance
line, the intercombination lines, and the forbidden line) and Ne~X $\alpha$, they generated photoionisation models only for the higher ionisation components from their 1$^{st}$ order analysis. In doing so, they assumed that density decreases as $n_{H} \propto r^{−1.65}$, which is the same as for the optical/UV emission-line gas used in \citealt{crenshaw2015a}.\par 

They determined that the total mass of the ionised gas is $\approx$ $5.4 (\pm 1.1)\times10^{5}~{\rm M\textsubscript{\(\odot\)}}$. In addition, by assuming the same kinematic profile as that for the [O~III] gas, derived from the analysis by \citet{kraemer2000b} of \textit{HST}/STIS spectra, the peak X-ray mass outflow rate was $\approx$ 1.8 ${\rm M\textsubscript{\(\odot\)}}$ $yr^{−1}$, at r $\sim$ 150 pc. The total mass and mass outflow rates are similar to those determined using [O~III] \citep[][]{crenshaw2015a}, which implies that the X-ray gas is a major outflow component. \par 

The AGN in NGC 4151 is radiating at a few \% of its Eddington limit \citep{couto2016a}, therefore is too weak by at least an order of magnitude to radiatively accelerate such highly-ionized gas. For this reason, \citet{kraemer2020a} proposed that the X-ray emitting gas arose from expansion of denser [O~III] emitting knots. However, more luminous AGN could be capable of directly driving X-ray emitting outflows \citep[e.g.,][]{trindadefalcao2021b}.\par

In summary, based on their results, the X-ray emission line gas possesses a similar mass and peak mass outflow rate as the optical/UV gas, which makes it an important component in AGN-driven winds. In addition, the fact that it does not appear to be a drop in mass outflow rates for distances greater than 100 pc may indicate that the X-ray component has a greater effect on the host galaxy than the optical/UV gas and that X-ray winds might be a more efficient mechanism for AGN feedback. \par 

Nevertheless, the fact that \citet{kraemer2020a} used the [O~III] kinematic profile to track the X-ray gas still remains a problem. There is no direct evidence that the X-ray gas follows the [O~III] kinematics, therefore in this study we present a different way to approach the issue, as mentioned in Section \ref{sec:introdution}.\par 





\begin{figure*}
  \centering
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Al9_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Ca8_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Fe7_medion_er.png}
\end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Fe10_medion_er.png}
 \end{minipage}\qquad 
 
 \caption{Computed radial flux distributions for X-ray footprint lines [Al~IX] 2.04$\mu$m, [Ca~VIII] 2.32$\mu$m, [Fe~VII] 6078\AA, and [Fe~X] 6375\AA. The fluxes are obtained using the analysis described in section \ref{sec:models}. The dust-free models are plotted in red and the 0.5 ISM models are plotted in blue. The models assume $\delta r/r$ $<$ 1 or $\delta r$ fixed at 50pc, where $\delta r$ is the deprojected width for each extraction bin. The uncertainties were obtained based on the S/N of the fluxes in the 0th order image from \citealt{kraemer2020a}.}
\label{fig:fluxes1}
\end{figure*}


\begin{figure*}
  \centering
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Fe11_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Fe14_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Mg7_medion_er.png}
\end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Mg8_medion_er.png}
 \end{minipage}\qquad 

\caption{Computed radial flux distributions for X-ray footprint lines [Fe~XI] 7892\AA, [Fe~XIV] 5303\AA, [Mg~VII] 5.50$\mu$m, and [Mg~VIII] 3.02$\mu$m. The fluxes are obtained using the analysis described in section \ref{sec:models}. The dust-free models are plotted in red and the 0.5 ISM models are plotted in blue. The models assume $\delta r/r$ $<$ 1 or $\delta r$ fixed at 50pc, where $\delta r$ is the deprojected width for each extraction bin. The uncertainties were obtained based on the S/N of the fluxes in the 0th order image from \citealt{kraemer2020a}.}
\label{fig:fluxes2}
\end{figure*}

\begin{figure*}
  \centering
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{S9_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Si7_medion_er.png}
\end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Si9_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Si10_medion_er.png}
 \end{minipage}\qquad 
 
\caption{Computed radial flux distributions for X-ray footprint lines [S~IX] 1.25$\mu$m, [Si~VII] 2.48$\mu$m, [Si~IX] 3.92$\mu$m, and [Si~X] 1.43$\mu$m. The fluxes are obtained using the analysis described in section \ref{sec:models}. The dust-free models are plotted in red and the 0.5 ISM models are plotted in blue. The models assume $\delta r/r$ $<$ 1 or $\delta r$ fixed at 50pc, where $\delta r$ is the deprojected width for each extraction bin. The uncertainties were obtained based on the S/N of the fluxes in the 0th order image from \citealt{kraemer2020a}.}
\label{fig:fluxes3}
\end{figure*}

 

\section{Photionisation Modeling Analysis}
\label{sec:models}

\subsection{Photoionisation Models}
\label{sec:photoionisation}
For our analysis, we generate photoionisation model grids\footnote{The model predictions for the ions fractional abundances present in Figure \ref{fig:cloudy} use the SED of Mrk 34 \citep{trindadefalcao2021a} and parameters as discussed in the caption of the figure.}, computed using Cloudy \citep[version 17.00;][]{ferland2017a}. We assume a continuum source with a SED that can be fitted using a power law of the form $L_{\nu} \propto \nu^{-\alpha}$. For our study we adopt the same SED used by \citet{kraemer2020a} in their study on NGC 4151, i.e. 
\medskip

 $ \alpha$ = 1.0 for $1\times 10^{-4}$ $\leq$ ${ h \nu}$ $\leq$ 13.6 eV;  \par
 $ \alpha$ = 1.3 for 13.6 eV $\leq$ ${ h \nu}$ $\leq$ 500 eV;  \par
 $ \alpha$ = 0.5 for 500 eV $\leq$ ${ h \nu}$ $\leq$ 30 keV; \par
 
 In addition, the choice of input parameters, specifically, the
radial distances of the emission-line gas with respect to the central source ($r$), number density ($n_{H}$), the ionisation parameter ($U$), and column density ($N_{H}$) of the gas, is that to match the input parameter used by \citet{kraemer2020a}, which we will briefly describe. \par 


\begin{figure*}
  \centering
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{O7_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{O8_medion_er.png}
 \end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Ne9_medion_er.png}
\end{minipage}\qquad 
 \begin{minipage}[b]{0.4\textwidth}
  \includegraphics[width=7cm, height=4.3cm]{Ne10_medion_er.png}
 \end{minipage}\qquad 
 
 
\caption{Computed radial flux distributions for X-ray lines O~VII 22.1\AA, O~VIII 19.97\AA, Ne~IX 13.69\AA, and Ne~X 12.13\AA. The fluxes are obtained using the analysis described in section \ref{sec:models}. The dust-free models are plotted in red and the 0.5 ISM models are plotted in blue. The models assume $\delta r/r$ $<$ 1 or $\delta r$ fixed at 50pc, where $\delta r$ is the deprojected width for each extraction bin. The uncertainties were obtained based on the S/N of the fluxes in the 0th order image from \citealt{kraemer2020a}.}
\label{fig:fluxes4}
\end{figure*}



\citet{kraemer2020a} scaled the ionising luminosity, $Q$, using an average UV flux at 1350 \AA~ observed over the last 7 years, and computed the value for $Q$ using the assumed SED and a distance of 15 Mpc. They obtained a value of $Q = 3\times 10^{53}$ photons ${\rm s^{-1}}$ and a bolometric luminosity L$_{bol}$ = 1.4$\times10^{44}$ ${\rm erg~s^{−1}}$ or 0.03 of the Eddington Luminosity. \par 

In addition, the elemental abundances in the NLR of NGC 4151 were initially determined in \citealt{kraemer2000b}, and based on recent estimates of solar elemental abundances \citep[e.g.,][]{asplund2005a}, they correspond to roughly 1.5x solar. We assume the same values for the present paper, as follows (in logarithm, relative to H, by number):
He=~-$1.00$, C=~-$3.47$, N=~-$3.92$, O=~-$3.17$, Ne=~-$3.96$, Na=~-$5.76$, Mg=~-$4.48$, Al=~-$5.55$, Si=~-$4.51$, P=~-$6.59$, S=~-$4.82$, Ar=~-$5.60$, Ca=~-$5.66$, Fe=~-$4.4$, and Ni=~-$5.78$.\par 

We also generate photoionisation models that account for internal dust, assuming grain abundances of 50\% of those determined for the interstellar medium (ISM) \citep[e.g.,][]{mathis1977a}, with the depletions from gas phase scaled accordingly \citep[e.g.,][]{snow1996a}, i.e., C=~-$3.64$, O=~-$3.29$, Mg=~-$4.78$, Al=~-$6.45$, Si=~-$4.81$, Ca=~-$5.96$, Fe=~-$4.70$, and Ni=~-$6.08$. The grain abundances of 50\% is the same grain abundance adopted by \citet{kraemer2000b} in their photoionisation study of the NLR of NGC 4151.\par

If the X-ray gas was created from the expansion of [O~III] clouds, as discussed in \citealt{trindadefalcao2021b}, we expect it to possess internal dust, since the predicted dust temperatures of the X-ray models are too low to be able to destroy the dust grains \citep[e.g.,][]{barvainis1987a}. On the other hand, the destruction of dust grains is possible as a result of shocks. If the X-ray gas is entraining clouds of [O~III]-emitting gas \citep[][]{trindadefalcao2021b} and effectively "sweeping" them up, the shocks that occur during this process could be enough to shatter the dust grains within the gas \citep[e.g.,][]{jones1997a}. \par 


  \begin{table}
\footnotesize
  \centering
\begin{tabular}[b]{|l|c|c|r|}
\hline
 \textbf{Distance$^{a}$} & \textbf{log$U$}  & \textbf{log $n_{H}^{a}$} & \textbf{log $N_{H}^{b}$}  \\
  & & ${\rm cm^{-3}}$ & ${\rm cm^{-2}}$ \\
\hline
12 pc & 0.05 & 2.71 & 22.28\\
50 pc & -0.17 & 1.69 & 21.88\\
100 pc & -0.27 & 1.19 & 21.38\\
153 pc & -0.34 & 0.89 & 21.06 \\
205 pc & -0.38 & 0.68 & 20.81 \\
257 pc & -0.42 & 0.52 & 20.71 \\
309 pc & -0.45 & 0.39 & 20.58 \\
360 pc & -0.47 & 0.28 & 20.27\\
412 pc & -0.49 & 0.18 & 20.37\\
464 pc & -0.50 & 0.09 & 20.28\\

\hline
\end{tabular}
\caption{\textbf{Cloudy Model Parameters.} The models follow the same parameters as the medium ionisation component from \citealt{kraemer2020a}\\
\small $^{a}$ Based on density law $n_{H}$ $\propto$ $r^{-1.65}$\\
$^{b}$ Assuming $\delta r/r$ $<$ 1 or $\delta r$ fixed at 50 pc, where $\delta r$ is the deprojected width for each extraction bin; for a detailed explanation, see \citealt{kraemer2020a}. }
\label{tab:models}
\end{table}



\subsection{Radial Flux Distributions}

In their study, \citet{kraemer2020a} calculated the emitting area for the X-ray emitting gas. They were able to do this calculation only for Ne~IX 13.69\AA~ and Ne~X 12.13\AA, since the loss of soft-Xray sensitivity for ACIS made it hard to accurately measure the O~VII 22.1\AA~and O~VIII 18.97\AA~fluxes. To construct the radial flux profiles associated with each emission-line, we redo their analysis, and include in our study other X-ray line profiles, e.g., O~VII 22.1\AA~and O~VIII 18.97\AA. In addition, in our study, we only generate photoionisation models for the medium ionisation component from \citealt{kraemer2020a}. The reason is that the highest ionisation models did not predict significant fluxes for the footprint lines. The complete set of parameters used to generate our models is listed in Table \ref{tab:models}.  \par

Using the Cloudy predictions for the X-ray footprint lines, and the emitting areas calculated in \citealt{kraemer2020a}, we also construct radial flux profiles for the footprint lines, such as [Fe~X] 6375\AA~and [Si~X] 1.43$\mu$m. Our results are shown in Figure \ref{fig:fluxes1}, \ref{fig:fluxes2}, \ref{fig:fluxes3}, for the optical/IR lines, and in Figure \ref{fig:fluxes4} for the X-ray lines. In addition, the resulting spatially-integrated fluxes for all the footprint and X-ray lines are listed in Table \ref{tab:fluxes}. \par 



\section{Footprints of X-ray Outflows}
\label{sec:footprints}

By analysing the archival STIS long-slit spectra, we are able to obtain the spatial [Fe~X] and [O~III] distribution in NGC 4151, as shown in Figure \ref{fig:spatial_dist}. Specifically, since the [Fe~X] $\lambda$6375 emission-line is contaminated by [O~I] $\lambda$6364, we obtain an intrinsic [Fe~X] brightness profile by subtracting 1/3 of the [O~I] $\lambda$6300 flux at each point along the slit. In addition, in order to account for the mass of [Fe~X] emitting gas outside of the area sampled by the STIS slit, we use an archival HST/WFPC2 [O~III] image of the NLR for the target \citep[][]{kraemer2020a}. We measure the [O~III] fluxes in regions of 0$\arcsec$.5x3$\arcsec$.0, centered at the nucleus, along a position angle of 140\degree. \par 

We then measure the ratio [Fe~X]/[O~III] inside the STIS slit, which allow us to obtain the full [Fe~X] flux distribution. In Figure \ref{fig:comparison}, we compare the results of our predicted [Fe~X] flux profiles (both for dust-free models and 50\% dust models) and the results of the full [Fe~X] flux distribution (inside and outside the slit). The total spatially-integrated [Fe~X] flux, i.e., the sum of the fluxes inside and outside the STIS slit, is $1.9^{2.5}_{1.7}\times10^{-13}~{\rm erg~s^{-1}~cm^{-2}}$.\par 

In addition, based on the analysis of the STIS long-slit spectra, we are able to calculate an upper limit, integrated over 11 pixels (0.1$\arcsec$ $\times$ 0.55$\arcsec$), for the flux of the [Fe~XIV] 5303\AA~line $<$ $7.5\times 10^{-14}~{\rm erg~s^{-1}~cm^{-2}}$, which agrees with our predicted flux of $1.3\times 10^{-14}~{\rm erg~s^{-1}~cm^{-2}}$ (see Table \ref{tab:fluxes}). Considering that the brightness profile for [Fe~XIV] follows the same brightness profile as [Fe~X], we calculate an upper limit in the central bin 0.1$\arcsec$ $\times$ 0.05$\arcsec$ $<$ $2.6\times10^{-14}~{\rm erg~s^{-1}~cm^{-2}}$, which also agrees with our predicted value of $1.0\times10^{-14}~{\rm erg~s^{-1}~cm^{-2}}$. It is important to note here that the other optical footprint lines are below the detection limit of STIS. \par 



\subsection{Predicting the Mass of the Extended Gas}
\label{sec:mass_prediction}

In order to recompute the mass of the extended X-ray gas in NGC 4151 based on our X-ray footprint analysis, we use our predicted [Fe~X] radial flux profile. It is important to note here that the methodology used in this study to estimate the mass of the extended X-ray gas in the target is different from the methodology used by \citet{kraemer2020a}. As discussed in Section 
\ref{sec:kraemer_results}, \citet{kraemer2020a} mapped the Ne~IX rif and Ne~X $\alpha$, instead of [Fe~X], and applied the model parameters to fit the 0th order emission-line profiles of these lines.\par 

We calculate the masses in each bin by using the line luminosities and the Cloudy-predicted fluxes for the [Fe~X], as follows:

\begin{equation}
    M_{bin} = N_{H}\mu m_{p} \left(\frac{L_{[Fe X]}}{F_{{[Fe X]}_{c}}}\right)
\end{equation}

\noindent where $N_{H}$ is the hydrogen column density of each photoionisation model generated by \citet{kraemer2020a}, assumed to be the same as the column density modelled by Cloudy, $\mu$ is the mean mass per proton, which we assume to be 1.4 (consistent with roughly solar abundances), and $m_{p}$ is the mass of a proton. In order to get the mass of [Fe~X] the column density needs to be multiplied by an effective area. The term in parentheses gives the effective surface area of the emitting gas as seen by the observer. $F_{{[Fe X]}_{c}}$ is the [Fe~X] luminosity per ${\rm cm^{2}}$,
predicted by Cloudy, and $L_{[Fe X]}$ is the observed luminosity determined using the total flux. Specifically,

\begin{equation}
   L_{[Fe X]} = 4\pi D^{2} F_{[Fe X]_{total}}
\end{equation}

\noindent where $D$ is the distance to NGC 4151, and $F_{[Fe X]_{total}}$ is the total [Fe~X] flux.
Physically, the equation for $M_{slit}$ determines the area of the emitting clouds through the ratio of the luminosity and flux, and then multiplies this by the column density to yield the total number of particles, which, when multiplied by the mean mass per particle, gives the total ionised mass. The total mass of X-ray-emitting gas is $7.8(\pm 2.1)\times 10^{5}~M_{\odot}$, with the uncertainties being added in quadrature for each bin, and the mass profile for the target is shown in Figure \ref{fig:comparison_masses} (purple points). We compare our results to the results of \citealt{kraemer2020a}, as shown in Figure \ref{fig:comparison_masses} (orange points), which found that the total mass of X-ray gas within the outflow regions for NGC 4151 is $5.4(\pm 1.1)\times 10^{5}~M_{\odot}$. \par 

Even though our results for the total mass of X-ray-emitting gas are very promising, due to the poor resolution of the STIS grating used for this observation, along with a low S/N, we are not able to obtain any accurate kinematic information for [Fe~X]. This means that we cannot compute the outflow rates for NGC 4151, and a comparison to the values for $\dot{M}$ found by \citet{kraemer2020a}, i.e., $\dot{M}$ = 1.8M$_{\odot}$ ${\rm yr^{-1}}$, is not possible given the available STIS low-dispersion spectrum.





  \begin{table*}
\footnotesize
  \centering
\begin{tabular}[b]{|l|c|c| c|c|c|}
\hline
 \textbf{Ion} & \textbf{Wavelength}  & \textbf{IP} & \textbf{Predicted Flux df$^{a}$} & \textbf{Predicted Flux d$^{b}$} & \textbf{Observed Flux} \\
  & & (eV) & $({\rm 10^{-16}~erg~s^{-1}~cm^{-2}})$ & $({\rm 10^{-16}~erg~s^{-1}~cm^{-2}})$ & $({\rm 10^{-16}~erg~s^{-1}~cm^{-2}})$\\
\hline
\boldsymbol{${\rm[Al~IX]}$} & 2.04$\mu$m & 284 & 51.8 & 6.9 & 23 ($\pm$16)$^{c}$ \\
${\rm[Ca~VIII]}$ & 2.32$\mu$m & 128 & 3.9 & 5.3 & 151 ($\pm$23)$^{c}$ \\
${\rm[Fe~VII]}$ & 6087\AA & 103 & 15.4 & 26.5 & -\\
\boldsymbol{${\rm[Fe~X]}$} & 6374\AA & 235 & 2360 & 1050 & 1900$^{d}$\\
\boldsymbol{${\rm[Fe~XI]}$} & 7892\AA & 262 & 921 & 302 & -\\
\boldsymbol{${\rm[Fe~XIV]}$} & 5303\AA & 355 & 130 & 34.0 & $<$750$^{d}$\\
\boldsymbol{${\rm[Mg~VII]}$} & 5.50$\mu$m & 186 & 35.20 & 37.1 & -\\
\boldsymbol{${\rm[Mg~VIII]}$} & 3.02$\mu$m & 225 & 251 & 174 & -\\
\boldsymbol{${\rm[S~IX]}$} & 1.25$\mu$m & 328 & 22.0 & 31.1 & 397 ($\pm$23)$^{c}$\\
\boldsymbol{${\rm[Si~VII]}$} & 2.48$\mu$m & 205 & 14.6 & 16.9 & -\\
\boldsymbol{${\rm[Si~IX]}$} & 3.55$\mu$m & 303 & 319 & 195 & -\\
\boldsymbol{${\rm[Si~X]}$} & 1.43$\mu$m & 351 & 559 & 214 & 377 ($\pm$30)$^{c}$\\
\textbf{O~VII f} & 22.1\AA & 138 & 3940 & 3230 & 3400($\pm$769)$^{e}$\\
\textbf{O~VIII} \boldsymbol{$\alpha$} & 18.97\AA & 739 & 1610 & 880 & 1089($\pm$192)$^{e}$\\
\textbf{Ne~IX i} & 13.69\AA & 239 & 851 & 747 & 1041($\pm$96)$^{e}$\\
\textbf{Ne~X} \boldsymbol{$\alpha$} & 12.13\AA & 1195 & 204$^{*}$ & 166 & 480($\pm$48)$^{e}$\\


\hline
\end{tabular}
\caption{\textbf{Predicted Spatially-Integrated Fluxes.} In boldface are the emission-lines that we define as footprint lines, i.e., lines from ions with IP $\geq$ 138 eV.\\
\small $^{a}$ Dust-free models\\
$^{b}$ 50\% dust models\\
$^{c}$ Observed fluxes from \citealt{rodriguez-ardila2011a}\\
$^{d}$ Observed fluxes from this study\\
$^{e}$ Observed fluxes from \citealt{kraemer2020a}\\
$^{*}$ Most of the X-ray emission accounted for this line comes from the high ionisation components modeled by \citet{kraemer2020a}. However, since these models do not significantly contribute to the footprint lines emission we decide not to include them in this study.}
\label{tab:fluxes}
\end{table*}


\begin{figure}
  \centering
  \includegraphics[width=0.5\textwidth]{Fe10_and_O3_flux_profiles_STIS.png}
 \caption{Spatial distribution for [Fe~X] (in black) and [O~III] (in red) in NGC 4151. The values were obtained from the analysis of the STIS G750L spectrum. The fluxes are per cross-dispersion pixel (0.1$\arcsec$ $\times$ 0.05$\arcsec$).}
\label{fig:spatial_dist}
\end{figure}


\begin{figure}
  \centering
  \includegraphics[width=0.5\textwidth]{Fe10_comparison_models.png}
\caption{A comparison between the results of our predicted radial flux profiles for the [Fe~X]-emitting gas (the blue points represent the dust-free model values and the red points represent the 50\% dust model values) and the results of the full [Fe~X] flux distribution (inside and outside the STIS slit). The uncertanties were obtained based on the S/N of the fluxes in the 0th order image from \citet{kraemer2020a}}
\label{fig:comparison}
\end{figure}


\begin{figure}
  \centering
  \includegraphics[width=0.5\textwidth]{predicted_mass_2.png}
\caption{A comparison between the results of our predicted masses for the X-ray-emitting gas (purple points) and the results of \citealt{kraemer2020a} (orange points). The uncertainties were obtained based on the S/N of the fluxes in the 0th order image from \citet{kraemer2020a}.}
\label{fig:comparison_masses}
\end{figure}


\section{Discussion}
\label{sec:discussion_1}

In Figure \ref{fig:ratios} we present the Cloudy-predicted [Fe~X]/H${\beta}$ (in red), O~VII~(22.1\AA)/H${\beta}$ (in blue), and O~VII~(21.6\AA)/H${\beta}$ (in green) ratios for NGC 4151, for the 50\% dust fraction models. It is possible to see that when densities are very high, i.e.,  $\gtrsim$ 10$^{8.0}$ ${\rm cm^{-3}}$, the [Fe~X]/H${\beta}$ ratio starts to drop, which means that the [Fe~X] 6375\AA~line starts to get collisionally suppressed. In this scenario the method of predicting the mass of X-ray gas using footprint lines might not be as accurate as when used for lower density gas, since the O~VII~(22.1\AA)/H${\beta}$ ratio only starts to drop at $\approx$ 10$^{9.0}$ ${\rm cm^{-3}}$. For NGC 4151, if $Q \approx$ $3\times 10^{53}~{\rm photons~s^{-1}}$ \citep[e.g.,][]{kraemer2020a}, and log$U$ = 0.0, gas with densities of $\approx$ 10$^{9}~{\rm cm^{-3}}$ is at $r~ \approx$ $2.8\times10^{16}$ cm from the AGN, which is just at the outer edge of the BLR (about 10 light-days) \citep[e.g.,][]{bentz2013a}. However, the fact that our predicted values for the mass of X-ray-emitting gas in NGC 4151 agree very well with the measured values by \citet{kraemer2020a} (see Figure \ref{fig:comparison_masses}) tells us that the contribution from X-ray gas that is located in the BLR of this AGN is negligible. This result is consistent with the findings of \citet{crenshaw2007a}, which shows that the mass outflows from the dense inner region of the AGN do not contribute much to the total gas in the NLR of NGC 4151, and the gas located in this region, i.e., in the NLR, is likely accelerated in-situ.  \par 


As discussed in Section \ref{sec:models}, internal dust can have a significant effect on how we interpret our results. As shown in Figures \ref{fig:fluxes1}, \ref{fig:fluxes2}, and \ref{fig:fluxes3}, the presence of dust within the outflowing gas can have a more pronounced effect in some lines, e.g., [Fe~X] 6375\AA, compared to others, e.g., [S~IX] 1.25$\mu$m. The fact that the [S~IX] fluxes do not seem to be heavily affected by the presence of dust within the gas, as shown in Figure \ref{fig:fluxes3}, tells us that this line might be a good choice of footprint line when the percentage of dust within the gas is unknown (but see below). On the other hand, for lines like [Fe~X] 6375\AA, and [Mg~VIII] 3.02$\mu$m it is important to have a good estimate of the amount of dust present in the gas, since their fluxes are strongly affected by depletion of these element onto dust grains. For instance, if one uses only the [Fe~X] 6375\AA~line to predict the emission of X-ray gas, but underestimates the amount of dust within the gas, the estimated X-ray mass could be overestimated, since the dusty models predict lower fluxes, as shown in Figures \ref{fig:fluxes1}, \ref{fig:fluxes2}, and \ref{fig:fluxes3}. Furthermore, a few of our predicted flux profiles, such as [Ca~VIII], [Fe~VII], [Mg~VII], and [Mg~VIII] (see Figures \ref{fig:fluxes1}, \ref{fig:fluxes2}, and \ref{fig:fluxes3}) show a higher predicted flux for the dusty models in the center bins. This is because, for these emission-lines, we are dealing only with traces of these ions, since, as shown in Figure \ref{fig:cloudy}, their fractional abundaces peak at much lower ionisation parameters. In this case, the presence of dust within the gas can enhance the abundance of these ions at small distances from the SMBH. \par 

In addition, \citet{rodriguez-ardila2011a} measured the fluxes of IR footprint lines for NGC 4151 in their study of near-infrared coronal line spectrum of nearby AGN. As shown in Table \ref{tab:fluxes}, our predicted fluxes for [Al~IX] 2.04$\mu$m and [Si~X] 1.43$\mu$m agree well with the results of \citet{rodriguez-ardila2011a}. The under-prediction of [Ca~VIII] 2.32$\mu$m is consistent with the fact that Ca~VIII peaks at much lower ionisation than our X-ray models (see Figure \ref{fig:cloudy}). Regarding the discrepancy between our measured flux for the [S~IX] 1.25$\mu$m and the observed flux from \citealt{rodriguez-ardila2011a}, we believe that our underpredicted value could be due to collision strength of this transition \citep[e.g.,][]{zhang2002a}, or due to an unmodeled process, such as photo-excitation/fluorescence.\par 

It is important to note that, in this study, we use [Fe~X] as a tracer for X-ray outflows based on the characteristics of NGC 4151, i.e., the fact that this target is not a very powerful AGN. For AGNs with more highly ionised gas in their NLR, other lines would be a better choice, such as [Si~X] and [Fe~XIV], as shown in Figure \ref{fig:cloudy}. For instance, based on our predicted flux and observed flux for [Fe~XIV] 5303\AA (see Table \ref{tab:fluxes}), there is not a large amount of [Fe~XIV] in NGC 4151 and this is because there is only a small amount of high ionisation gas in the NLR of this target. For more powerful targets, e.g., quasars, the use of [Fe~XIV] as a tracer might be preferable, since these objects are more luminous and powerful than NGC 4151. \par 






\begin{figure}
  \centering
  \includegraphics[width=0.5\textwidth]{line_Hbeta_NGC4151.png}
\caption{Predicted [Fe~X]/H${\beta}$ (in red), O~VII -f/H${\beta}$ (in blue), and O~VII -r/H${\beta}$ (in green) ratios for NGC 4151. The model assumptions are: abundances (1.4x solar), SED from \citealt{kraemer2020a}, constant column density N$_{H}$ = 10$^{21.5}$ ${\rm cm^{-2}}$, and ionisation parameter log$U$ = 0.0.}
\label{fig:ratios}
\end{figure}



\section{Conclusions}
\label{sec:discussion}

Based on our analysis derived from Cloudy photoionisation models, we present a new method that can be used to model X-ray outflows. Our conclusions are as follows:\par 

1. There are ions that peak over the same range in ionisation parameter as the H and He-like ions and can produce detectable optical/IR emission lines, which we call "footprint lines". These lines can be detected and spatially-resolved, e.g., via \textit{HST} or \textit{JWST} observations and, therefore, potentially be used to accurately measure the kinematics and the
dynamics of the extended X-ray gas.\par 

2. We show that the X-ray models present in \citealt{kraemer2020a} produce strong footprint emission lines. We also develop flux profiles to characterise such lines. \par 

3. By analysing the STIS long-slit spectrum and the [O~III] image for NGC 4151, we are able to derive the measured flux profile for [Fe~X] 6375\AA~within similar extraction bins. \par 

4. We compare our model predictions to the observed flux profile for [Fe~X] obtained from the analysis of the STIS G750L spectrum. Our values are in very good agreement with the measured values, which confirm the reliability of our approach. We also compare our model predictions for the IR lines to the observed fluxes from \citealt{rodriguez-ardila2011a}, which show very good agreement for the [Al~IX] 2.04$\mu$m and [Si~X] 1.43$\mu$m (see Section \ref{sec:discussion_1} for caveat on predicted IR footprint lines).\par 

5. We confirm our methodology by determining the mass of X-ray gas using our models predictions and the observed flux profile for the [Fe~X]. As shown in Figure \ref{fig:comparison_masses}, our values are in very good agreement with the values found by \citet{kraemer2020a}, which confirms that our method can accurately produce mass profiles for the X-ray gas. \par 

\medskip
 
Even though we are able to produce accurate mass profiles for the X-ray gas using our method of "footprint lines", we are not able to get accurate kinematic information for these lines from the STIS data. The poor resolution of the STIS G750L grating, combined with low S/N, does not allow us to obtain the necessary kinematic information to compute the values for outflow rates of this target. It would be interesting to obtain observations with higher spectral resolution for NGC 4151, which would allow us to compute the predicted mass outflow rates and again test our proposed methodology. Another possibility is to apply this method on other AGN, such as NGC 1068, for which the combination of X-ray spectra and medium resolution STIS long-slit spectra exist. In addition, it is important to note that, even though we do not need the kinematic profiles derived from X-ray observations to apply our method, we still need X-ray spectra in order to be able to fully characterise these outflows, since we need to be able to constrain the physical characteristics of the gas for our photoionisation modeling analysis.






\section*{Acknowledgements}

The material is based upon work supported by NASA under award number 80GSFC21M0002. Basic research at the Naval Research Laboratory is funded by 6.1 base funding.\par 
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. This paper used the photoionisation code Cloudy, which can be obtained from http://www.nublado.org. We thank Gary Ferland and associates, for the maintenance and development of Cloudy. 

\section*{Data Availability}

 
Based on observations made with the NASA/ESA Hubble Space Telescope, and available from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESAC/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA).







\bibliographystyle{mnras}
