text:
\section{Introduction}
\labsecn{sec:introduction}

High resolution soft X-ray spectra from \textit{Chandra} and XMM-Newton show
blueshifted absorption line when observing some stellar mass black holes in binary
systems (black hole binaries, hereafter BHBs). These lines are signatures of
winds from the accretion disk around the black hole. The velocity and
ionization state of the gas interpreted from the absorption lines vary from
object to object and from observation to observation. In most cases, we see
lines only from H- and He-like Fe ions (e.g.  \citealt{lee02},
\citealt{neilsen09} with GRS 1915+105, \citealt{miller04} with GX 339-4,
\citealt{miller06} with H1743-322, and \citealt{king12} with IGR J17091-3624).
Some spectra of some of the objects, however, display lines from a wider range
of ions from O through Fe (e.g. \citealt{ueda09} for GRS 1915+105,
\citealt{miller08, kallman09} for GRO J1655--40). The variations in the wind
properties seem to indicate variations in the temperature, pressure, and
density of the gas, not only from one object to another, but also from one
spectrum of the same object to another spectrum, depending on the accretion
state of the black hole.

The spectral energy distributions (SEDs) of BHBs in the different accretion
states have varying degrees of contribution from the accretion disk and the
non-thermal power-law components. The winds are not present in all the states.
The absorption lines are more prominent in the Softer (accretion disk
dominated) states \citep{miller08, neilsen09, blum10, ponti12}. Some authors
\citep[e.g.][in the case of H1743-322]{miller12} show that the changing
photoionising flux is responsible for the variation in the wind strength;
others conjecture that the underlying outflowing gas changes its physical
properties.

The observable properties of the accretion disk winds are often used to infer
the driving mechanism of the winds \citep{lee02, ueda09, ueda10, neilsen11,
neilsen12}. So, the changes in the wind (or its disappearance) through the
various states of the BHB, has been interpreted as a variation in the driving
mechanism. To elaborate, let us use the case of GRO J1655--40. A well-known
\textit{Chandra} observation of GRO J1655--40 \citep{miller06, miller08,
kallman09} shows a rich absorption line spectrum from OVIII - NiXXVI, which led
the authors to conclude that a magnetic driving mechanism is accelerating the
wind. \citet{neilsen12} have analysed the data from another observation for the
same source taken three weeks later and found absorption lines from only Fe\,
{\sc xxvi}!  It was shown that such a change in the wind was not due to mere
change in photoionization flux. The disappearance of most of the absorption
lines suggest that in the wind in GRO J1655--40, the long term changes may be
driven by the variations in the thermal pressure and/or magnetic fields.  This
wind was hailed to be the most convincing evidence for magnetic driving in
BHBs.  Since the spectrum of GRO J1655-40 with this wide range of ions, is a
unique one among BHBs, it is repeatedly investigated by different authors. More
recently, \citet{neilsen16} and \citet{shidatsu16} have studied the same
spectrum. Both papers use complimentary optical and near-infrared observations
around the Chandra observations and hypothesise the presence of a
Compton-thick, almost completely ionized, gas component which caused scattering
and hence caused reduction in the observed X-ray luminosity. These conjectures
point to the possibility of a near Eddington flow, whence radiation pressure
and Compton heating may be significant contributors to the wind that has
been hailed to be a magnetically driven wind.

In order to have a consolidated picture of these systems, it is necessary to
understand the relation between the accretion states of the BHBs and the
driving mechanisms of the winds.  In \citet[][hereafter called Paper
I]{chakravorty16}, we have investigated the magneto-hydrodynamic (hereafter
MHD) solutions as driving mechanisms for winds from the accretion disks around
BHBs:
cold solutions (with no disk surface heating) from \citet[][hereafter
F97]{ferreira97} and warm solutions (which involve an ad-hoc disk surface
heating term)
from \citet{casse00b} and \citet{ferreira04}. We had concluded that only the
warm class of solutions, which have heating at the disk surface, can have
sufficiently high values of the ejection efficiency $p (\gtrsim 0.1)$ so that
the winds are dense enough to explain the observed absorption lines and the
inferred physical properties.
However, since then it was possible to investigate the variations in the
disk magnetisations more elaborately. \citet{jacquemin19} have derived cold MHD
solutions with $p (\sim 0.1)$, but magnetisation very different from the ones
in Paper I. Further, their analysis showed that $p$ can go as high as $\sim 0.2 - 0.3$ for weakly magnetised solutions.

In Paper I, we published the important physical properties of the gas, for
example, the density and velocity of the flow and specified the regions which
would have the correct ionisation properties to yield FeXXVI absorption lines.
In this paper we are taking the next crucial step - producing synthetic spectra
which can be matched to observations from current (Chandra) and future (XRISM
and Athena) missions.
The goal of this paper is to set up a methodology of looking for the
`correct' MHD outflow models within our suite of solutions, based on the
observable line profiles that they produce. We want to study the variation of
the line profiles as functions of the physical properties like the line-of-sight (LOS)
angle (i, from the surface of the disk), the ejection index (p), the
extent of the disk. We intend this paper to be the 1st among a list of
publications where we shall look at the details of the observables (e.g. the Fe
XXVI K$\alpha$ doublet line profiles) in a X-ray spectra and how they are
related to the MHD outflow model parameters, including some of the disk
parameters like the ejection index and the disk magnetisation. The intention is
to have an uniform, systematic study of a suite of MHD models, and not just a
preferred few to suit specific observed spectrum. This long term project would
be an attempt to build a library of models that can be tested out (against
other non-magnetic driving mechanisms) using the XRISM and Athena spectra
which may be able to tell the difference looking at the line profiles. For this
1st paper, where the methodology of our efforts are being laid out, we
are using the densest MHD models that we have - namely, the warm solutions with
the highest $p$ values, from which we expect to get prominent absorption lines
from various ions (because of the high density of these outflows). Further,
among the disk parameters, we vary only the ejection index ($p$), for this
paper. However, we will follow through in our next publication with the
variation of the disk magnetisation.

Paper I \citep[and also][]{chakravorty13} also discusses that
thermodynamic instability can be one of the causes that absorption lines are
not seen in the hard spectral states in BHBs. This idea has found further
support from \citet{bianchi17}, which shows that accretion state dependent
thermodynamic instability may be at work even for neutron star X-ray binaries.
\citet{petrucci21} goes into rigorous details of the possible evolution of
the stability curves along a BHB outburst. Using the thermodynamic stability
analysis the authors could also argue that from the soft to the hard state, a
mere change of the ionising luminosity cannot explain the changes in the wind
signatures (from presence to absence of absorption lines). They infer that an
associated change in the density of the flow is likely if we follow the
stability curves.

In this paper we will restrict ourselves to deriving spectra for the
Soft accretion state only. To derive spectra for the Hard state we have to
formulate a self-consistent way of addressing the thermodynamic instability,
which is beyond the scope of the current paper, but will be attempted soon in a
following publication.

Our Paper I and the current project underway, is not the 1st attempts to
discuss MHD models for diffused winds in black hole systems.
\citet{fukumura10a,fukumura10b,fukumura14} extensively discuss MHD models for
various wind components in super-massive black hole systems or the Active
Galactic Nuclei (AGN). \citet{fukumura17,fukumura21} extend the same class of
models to explain winds in Black hole X-ray binaries. See
\subsecn{subsec:discuss_pastlit} where we discuss how our MHD models are
different from the ones used in the various Fukumura et al.\  papers. There is
also a quantitative comparison in \fig{fig:MhdSols} pertaining to a particular
model used by \citet{fukumura17}.

The following is the layout of the paper: \secn{sec:Model} talks about the
models we use - for the ionising radiation, as well as the outflowing gas which
is ionised. In the same section we also mention how we use knowledge from
observables and atomic physics processes to constrain which part of the outflow
is relevant for our study. Thus narrowing down our parameter space of
exploration, the theoretical spectra of the light coming from the compact
object and passing through the outflow, are generated in
\secn{sec:SyntheticCloudySpectra} using photoionisation code CLOUDY. These
theoretical spectra need to be convolved with the instrumental response of
X-ray telescopes to give us an idea of what an observer will see. We want to
assess the capabilities of the future telescopes like Athena and XRISM as
discerning instruments for outflow science in BHBs -
\secn{sec:AthenaXrismSpectra}. \dsecn{sec:discussion}{secn:conclusion}
respectively discusses issues and concludes our findings.

\section{The Model}
\labsecn{sec:Model}

\subsection{Spectral energy distribution for the Soft state}
\labsubsecn{subsec:SED}

\begin{figure}
\begin{center}
\includegraphics[scale = 1, width = 9 cm, trim = 0 75 0 20, clip, angle = 0]{fig/Fig01_SedIf.ps}
\caption{{\it Top Panel:} The SED corresponding to the fiducial Soft state of the
	outburst of a black hole of $10 M_{\odot}$. The two important
	components of the SED, namely, the disk spectrum and the power-law have
	been added following the scheme described in \citet{remillard06}. See
	\subsecn{subsec:SED} for the details. {\it Bottom Panel:} The Soft SED
	is used in C08 (version C08.00 of photoionisation code CLOUDY) to generate the
	ionisation fraction of FeXXVI, to show the favourable range of the
	ionisation parameter, for this ion to be present in the gas (see
	\subsecn{subsec:AtomicPhysicsConstraints} for details). The peak of the
	distribution is marked and the corresponding $\log \xi$ value is
	labeled.}
\labfig{fig:SED_If}
\end{center}
\end{figure}

The SED of BHBs essentially has two components \citep{remillard06}. (1) A
thermal component which is radiated by the inner accretion disk around the
black hole, and is conventionally modeled with a multi-temperature blackbody
often showing a characteristic temperature ($T_{in}$) near 1 keV. (2) A
non-thermal power-law component with a photon spectrum $N(E) \propto
E^{-\Gamma}$ which can be radiation due to inverse-Comptonisation of photons
from the disk. During different states of the
outbursts the BHBs show varying degrees of contribution from the
above-mentioned components. The state where the radiation from the inner
accretion disk dominates and contributes more than 75\% of the 2-20 keV flux is
fiducially called the Soft state \citep{remillard06}, whereas in the canonical
Hard state, the contribution of the accretion disk radiation drops below 20\%
in the 2-20 keV flux.

The sum of local blackbodies emitted at different radii, can be used to model
the radiation from a thin accretion disk. The temperature $T_{in}$ of the
innermost annulus (with radius $r_{in}$) of the accretion disk is proportional
to $\left[\dot m_{obs} / (M_{BH} r_{in}^3) \right]^{1/4}$
\citep{peterson97,frank02}, where
\begin{equation}
\dot m_{obs} = L_{rad}/L_{Edd},
\labequn{eqn:mdot_obs}
\end{equation}
$L_{rad}$ being the luminosity in the energy range 0.2 to 20 keV and $L_{Edd}$
being the Eddington luminosity. The package
XSPEC\footnote{http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/}
\citep{arnaud96} includes a standard model for emission from a thin accretion
disk, called $diskbb$ \citep[][]{mitsuda84, makishima86}. $Diskbb$ in version
11.3 of XSPEC is used to generate the disk spectrum $f_{disk}(\nu)$ - with
inputs $T_{in}$ and the normalization which is proportional to $r_{in}^2$.  A
hard power-law with a high energy cut-off (at 100 keV) is added to
$f_{disk}(\nu)$ to yield the full SED -
\begin{equation}
	f(\nu) = f_{disk} (\nu) + [A_{pl} \nu^{-\alpha} \times \exp{^{-\frac{\nu}{\nu_{max}}}}] \times f \times \exp{^{-\frac{\nu_{min}}{\nu}}}.
\labequn{eqn:FullSed}
\end{equation}

The parameters for the fiducial SED to represent the Soft accretion state of a
black hole of $10 M_{\odot}$ (\fig{fig:SED_If} top panel) are chosen according
to the prescription given in \citet{remillard06}. We carefully selected the
input parameters to closely match the SED that they used as a standard for soft
state SED - that of the March 24, 1997 outburst of GROJ1655-40. In the Soft
state the accretion disk extends all the way to $r_{in} = 3R_s = 6r_G$. The
photon index of the power-law is $\Gamma = 2.5$ and $A_{pl}$ in
\equn{eqn:FullSed} is adjusted so that the disk contributes 80\% of the 2-20
keV flux. $\nu_{max}$ = 500 keV. For a $10 M_{\odot}$ black hole, $L_{Edd} =
1.23 \times 10^{39} \rm{erg \, s^{-1}}$. The disk accretion is chosen in such a
way that $T_{in} = 0.56 \unit{keV}$ and our fiducial Soft state SED corresponds to
$\dot m_{obs} = 0.14$, which we round off to 0.1 for the rest of the paper.
Note that two additional factors - a fudge factor $f$ and an exponential lower
energy cut-off $\nu_{min}$ are used to ensure that the powerlaw does not blow
up at low frequencies.  First, $\nu_{min} = 20$ eV (and not higher) was chosen
to ensure that the powerlaw had 100\% flux at 2 keV, because we wanted to leave
the 2-20 keV part of the power-law unadulterated, since this energy range is
used to determine $A_{pl}$. Then an additional fudge factor $1/1.01$ was needed
to ensure that below 2 keV the power-law will never be more than 10\% of the
diskbb flux.

\subsection{The MHD Solutions}
\labsubsecn{subsec:MhdSols}

In Paper I we had thoroughly investigated ``cold'' (described in F97) and ``warm'' \citep[described in][]{casse00b, ferreira04} MHD solutions to check if they form feasible models to explain observable winds in BHBs - winds that can be detected via absorption lines of (at least) the FeXXVI and FeXXV ions. While the details of those investigations can be checked out from Paper I and references therein, we mention some of the salient points of the MHD solutions, here, in this section.

We use MHD solutions that describe steady-state, axi-symmetric outflows, from the turbulent disk mid-plane to the ideal MHD jet asymptotic regime. The main assumptions are: \\
(1) The existence of a large-scale magnetic field of bipolar topology threading the whole accretion disk. The strength of the required vertical magnetic field component is obtained as a result of the solution \citep[][F97]{ferreira95}.\\
(2) The disk is fully turbulent, which leads to enhanced (anomalous) transport coefficients, such as viscosity and magnetic diffusivities  allowing the plasma to diffuse through the field lines inside the disk. As turbulence is expected to be active only within the disk, the vertical profiles of these coefficients were assumed to decrease on a disk scale height.

The rigorous mathematical details of how the isothermal MHD solutions for the accretion disk outflow are obtained are given in the above-mentioned papers and summarized in Paper I. Hence,  we refrain from repeating them here. In Paper I we found that the radial exponent, $p$ (referred hereafter as ejection index) in the relation $\dot M_{acc} \propto r^p$ \citep[labeled $\xi$ in F97, ][etc.]{ferreira06, petrucci10} is a key parameter that affect the density $n^+$ [or $\rho^+$ (see \equn{eqn:rho+})] of the outflowing material at a given radius $r$ in the disk.

The larger the exponent $p$, the more massive and slower is the outflow. The wind density is a crucial quantity when studying absorption features. For a disk accretion rate $\dot M_{acc}(r)$ at a given cylindrical radius $r$, the corresponding wind density at the disk surface scales as
\begin{eqnarray}
n^+ = \frac{\rho^+}{m_p} &\propto & \frac{p}{\varepsilon} \frac{\dot M_{acc}}{4 \pi m_p V_K r^2}
\labequn{eqn:rho+}
\end{eqnarray}
where $m_p$ is the proton mass and the superscript "+'' stands for the height where the flow velocity becomes sonic\footnotemark.
\footnotetext{Here, the sonic speed only provides  a convenient scaling for the velocity, especially in isothermal flows. In MHD winds, however, the critical speed that needs to be reached at the disk surface is the slow magnetosonic speed, which is always smaller than the sonic speed.}
Here, $V_K=\Omega_K r=\sqrt{GM_{BH}/r}$ ($G$: gravitational constant) is the Keplerian speed and $\varepsilon= \frac{h}{r}$ is the disk aspect ratio, where $h(r)$ is the vertical scale height at the radius $r$.  In Paper I, our detailed analysis showed that for the detectable wind, the ejection index $p$ plays a much more definitive role in determining the physical parameters of the wind than $\varepsilon$. Hence, for the rest of this paper, we hold $\varepsilon = 0.01$ and study the resulting physical properties and spectrum for variations of $p$.

The asymptotic speed of the magnetically-driven wind $u_\infty$, flowing along a magnetic surface anchored at a radius $r$, can reach a quite large value. Indeed $u_\infty= V_K\sqrt{2\lambda-3}$, where $\lambda\simeq 1 + \frac{1}{2p}$ is the magnetic lever arm parameter (F97), so the smaller $p$ and the faster the wind. However, as shown in Paper I, wind signatures are emitted from the densest regions right above the disk, where the flow velocity is still very low.

We point the readers to note that both the density at the base of the flow and the geometry of the matter stream lines in the flow are intrinsically related to the disk parameters. So whenever such models will be used for fitting observed data in the future, one would always have constraints on some of the disk parameters.

In Paper I, we studied two classes of MHD solutions - "cold'' and "warm''. The outflow is termed "cold" when its enthalpy is negligible when compared to the magnetic energy, which is always verified in near Keplerian accretion disks. The cold MHD solutions cannot achieve high values of $p \gtrsim 0.1$, when the magnetic surfaces are isothermal (F97) or even when the magnetic surfaces are
changed to be adiabatic \citep{casse00a}. On the other hand, if some additional heat is deposited at the disk surface layers (through illumination for instance, or enhanced turbulent dissipation at the base of the corona), it is possible to get warm MHD solutions which can attain larger values of $p \lesssim 0.45$. The reason for emphasis on larger values of $p$ is because, in Paper I we found clear indication that a large value of $p > 0.1$ is needed to have the physical parameters of the wind consistent with observations. Cold MHD solutions have been therefore ruled out as models for winds in accretion disks around BHBs.

Even while considering just the warm solutions, in Paper I we were limited to solutions with $p \leq 0.11$. In order to better explore the effect of the disk ejection efficiency $p$, we played also with the turbulence parameters, allowing to obtain two additional warm solutions with $p = 0.30$ and $0.45$. In Appendix \secn{newsol}, we have presented a brief description on the physics of obtaining these denser warm MHD solutions.

\subsection{Constraints from atomic physics}
\labsubsecn{subsec:AtomicPhysicsConstraints}

The MHD solutions that were described in details in Paper I, can be used to
predict the presence of outflowing material over a wide range of distances
because of the self-similarity criterion - the associated equations can be
found as Equations 5 - 8 of Paper I. As a result, the outflowing material, in
each of these solutions, spans wide ranges in all the physical parameters like,
ionization parameter, density, column density, velocity, and timescales.
However, only part of this outflow can be detected through absorption lines; we
refer to this part as the ``detectable wind''. Literature survey of BHB high
resolution spectra shows that commonly absorption lines from H- and He-like Fe
ions are detected (e.g. \citealt{lee02}, \citealt{neilsen09},
\citealt{miller04}, \citealt{miller06}, and \citealt{king12}). Further,
\citet{ponti12} made a very important compilation of detected winds in BHBs,
also concentrating their discussion around the line from FeXXVI. Hence, we
choose the presence of the ion FeXXVI as a proxy for ``detectable winds''.

It is the ionization parameter, which is the key physical parameter to
determine the region of the ``detectable wind'', because the observable
tracers, in this case, are absorption lines from different ionic species. Since
the absorption lines in question (mainly from FeXXVI and FeXXV) manifest
themselves in X-rays, we use the definition most commonly used by X-ray high
resolution spectroscopists, namely $\xi = L_{ion}/(n_H r^2_{sph})$
\citep{tarter69}, where $L_{ion}$ is the luminosity of the ionizing light in
the energy range 1 - 1000 Rydberg (1 Rydberg = 13.6 eV) and $n_H$ is the
density of the gas located at a distance of $r_{sph}$. There is a simplifying
assumption that gas at any given point within the flow, is illuminated by light
from a central point source. This approach is not a problem unless the wind is
located at distances very close to the black hole ($\leq 100 r_G$). As we have
seen in Paper I that the ''detectable wind'' is much further out (also see
\fig{fig:PhysParams}), we can continue with this reasonable assumption. The SED
for this ionising radiation is the same as discussed above, in
\subsecn{subsec:SED}.

Ion fraction (IF) $I(X^{+i})$ of the $X^{+i}$ ion measures the probability of its
presence.  $$I(X^{+i}) = \frac{N(X^{+i})}{f(X) \, N_{\rm{H}}}$$ gives the
fraction of the total number of atoms of the element $X$ that are in the
$i^{\rm{th}}$ state of ionization; where $N(X^{+i})$ is the column density of
the $X^{+i}$ ion and $f(X) = n(X)/n_{\rm{H}}$ is the ratio of the number
density of the element $X$ to that of hydrogen. The bottom panel of
\fig{fig:SED_If} shows the ion fraction of FeXXVI calculated using version
C08.00 of CLOUDY\footnotemark\ \citep[][; hereafter C08]{ferland98}
\footnotetext{URL: http://www.nublado.org/ }
for Solar metallicity \citep{allendeprieto01, allendeprieto02} gas that is illuminated by the Soft state SED.  The ion fraction is seen to
peak at $\log \xi = 4.86$. However, note that even at $\log \xi = 6.0$, the ion
fraction is 0.1, i.e. 10\% of iron is in FeXXVI.  Thus, $\log
\xi_{max}|_{FeXXVI} = 6.0$ - the conservative upper limit beyond which the ion
fraction for FeXXVI drops below 10\%.  We shall use this limit of $\log
\xi_{max} = 6.0$ to constrain our MHD solutions, throughout this paper (unless otherwise mentioned).

\subsection{Constraining the MHD solution to find the ``detectable wind''}
\labsubsecn{subsec:FindWind}

For locating the ``detectable wind'' region within the MHD outflows, we impose the following physical constraints to remain consistent with the various observational criterion mentioned in the previous sections:
\begin{itemize}
\item[(1)] In order to be defined as an outflow, the material needs to have
	positive velocity along the vertical axis ($z_{cyl}$);
\item[(2)] To ensure a wind which is not Compton thick we impose that the
	integrated column density along the line-of-sight satisfies $N_H <
	10^{24.18} {\rm{cm^{-2}}}$. Through this we get an
	idea about how close to the accretion disk can we probe
	through the absorption lines.
\item[(3)] We impose that $\xi \leq 10^6 \,\, {\rm{erg \, cm}}$ (where FeXXVI
	has IF $\gtrsim 0.1$) for the Soft state, because over-ionized gas
	cannot cause any absorption and hence cannot be detected.
\end{itemize}

\begin{figure}
\begin{center}
\includegraphics[scale = 1.0, width = 9 cm, trim = 0 25 0 25, angle = 0]{fig/Fig02_pVar_Innermost_rsph_imin.ps}
	\caption{Physical parameters for the IP of the wind region detectable through FeXXVI absorption lines. The vertical dotted line shows the extent of $p$ that was used in Paper I. Further, in both the panels, the constraint of $\log \xi_{max}|_{FeXXVI}$ are evident by the use of two different legends. The dotted and dashed, black lines interspersed with the green squares, shows the more stringent cut-off $\log \xi_{max}|_{FeXXVI} = 4.86$ used in Paper I. In the current paper we use $\log \xi_{max}|_{FeXXVI} = 6.0$, whose results are indicated by the solid, red lines interspersed by the blue circles. {\it Top Panel:} It shows how far the absorber is from the central black hole. This is the minimum distance derived from $\log \xi_{max}|_{FeXXVI}$ (= 6.0 or 4.86). The two numbers placed close to some of the points indicate the corresponding values of log$n_H$ and log$v_{obs}$ of the wind at this minimum distance. {\it Bottom Panel:} The lowest line-of-sight along which we can expect to probe the gas, derived from the condition of $N_H < 10^{24.18} {\rm{cm^{-2}}}$.
	}
\labfig{fig:ClosestPointPars}
\end{center}
\end{figure}

Following conditions 1 and 2, we can find the $\sim$ lowest line-of-sight along
which we can probe the gas - expressed as $i_{min}$ in the bottom panel of
\fig{fig:ClosestPointPars}. Then we impose condition 3 to ensure the presence
of the ion of our interest (in this case FeXXVI). Using the last condition, we
can thus find the closest point ``Innermost Point'' (hereafter IP) to the black
hole which satisfies these conditions. As said at the end of the previous
section, this point will have a ionisation parameter $\log \xi_{max}|_{FeXXVI}
\sim 6.0$.  However, for different MHD solutions, these IPs (shown using solid,
red lines interspersed by the blue circles in \fig{fig:ClosestPointPars}) will
be at different distances from the black hole, and will also have different
values of density and velocity, although all of them will have near constant
ionisation parameter $\log \xi_{max}|_{FeXXVI} \sim 6.0$. The Top Panel of
\fig{fig:ClosestPointPars} gives us a quantitative comparison of the physical
parameters of the IP of the various warm MHD solutions. A careful
consideration of this comparison aids us in choosing the MHD outflow models for
further investigation. Further note that, in Paper 1, we had used a more
stringent constraint on $\log \xi_{max}|_{FeXXVI} \,\, (= 4.86)$. In
\fig{fig:ClosestPointPars}, we have also shown the position (and physical
parameters) of the resultant (Paper I) IPs using a different legend - dotted
and dashed, black lines interspersed with the green squares.

\begin{figure*}
\begin{center}
\includegraphics[scale = 1.0, height = 18 cm, trim = 265 20 140 20, angle = 90]{fig/Fig03_MhdSols/Fig03_MhdSols.ps} \\
\includegraphics[width = 0.6\columnwidth, trim = 0 0 0 0]{fig/Fig03_MhdSols/Solp0.10_ftheta.ps} \includegraphics[width = 0.6\columnwidth, trim = 0 0 0 0]{fig/Fig03_MhdSols/SolAllp_ftheta.ps}
	\caption{{\it Top panels:} The spacial distribution of the MHD solutions is shown - p = 0.10 (left panel), p = 0.30 (middle panel) and p = 0.45 (right panel). The colour gradient shows the distribution of the ionisation parameter (in logarithmic scale). The thin, solid, black lines are the streamlines through which matter flows out following the magnetic field lines. The thick, solid, black line indicates the position of the Alfven surface for each of the solutions. The solid, red line shows the minimum line-of-sight angle (from the mid-plane of the accretion disk) above which it is possible to detect the wind through absorption lines of FeXXVI. The values of these angles are marked close to this line of sight. {\it Bottom panels:} We show a comparison of the density functions, which is the variation of the density against $\theta (= 90^{\circ}) - i$, i.e. the variation of density along the stream line. Comparison of our $p = 0.1$ MHD model with the one that was used by \citet{fukumura17} is presented in the left panel and on the right panel, we show the comparison among the three MHD models that are used in this paper.
	}
\labfig{fig:MhdSols}
\end{center}
\end{figure*}

The density reported for most of the observed BHB winds $\ge 10^{11}
\,\rm{cm^{-3}}$ and the distance estimates place the winds at $\le 10^{10} \,
\rm{cm}$ \citep{schulz02, ueda04, kubota07, miller08, kallman09}. Our analysis
in Paper I had shown that the ``detectable wind'' part of the cold MHD outflows
were too far away from the black hole and hence had lower density ($n_H <
10^{10} \rm{cm^{-3}}$) and lower (line-of-sight) velocity ($v_{obs} < 10^2
\rm{km/s}$) than what is expected from the above mentioned observations. Cold
MHD solutions were thus ruled out as viable models for disk winds in BHBs. We
had further investigated the warm MHD solutions in Paper I and found that they
indeed, can have higher $p$ values which are needed to have dense enough
outflows, so that the IP moves closer to the black hole, so that
the values of the physical parameters of the IP are consistent
with the observations. In Paper I, even for the warm solutions we were limited
to $p = 0.11$. We realised that some observations require the wind to be closer
to the black hole and thus have higher velocity and density than what the
densest $p = 0.11$ warm solution could predict as a model. Thus, we felt the
need for denser warm MHD solutions with higher $p$ values (see Section 6.1.2 of
Paper I).

\fig{fig:ClosestPointPars} shows comparisons among the MHD outflow models and
also comparison between the constraints considered in Paper I and in this
paper. Notice that the $\log \xi_{max}|_{FeXXVI} = 4.86$ lines (black,
dotted-and-dashed) do not extend to p = 0.45 - that is because such low
ionization gas was not found within the Compton thin region of the p = 0.45 MHD
outflow. In comparison, the red, solid curve joining the blue circles show the
result for the `standard' $\log \xi_{max}|_{FeXXVI} = 6.0$ used in this paper.
Although the lowest line-of-sight angle, where we can expect to detect FeXXVI
absorption, does not vary much as a function of $\xi_{max}$, the distance of
the IP to the BH drops by a factor of $\sim 100$ for almost all the MHD
solutions tested, while the density at that point goes up by $\sim 1.5-3$ dex
and velocity by $\sim 0.7 - 1$ dex.  In Paper I, we had predicted this
possibility in Section 6.1.1. The vertical black, dotted line, cutting across
the panels in \fig{fig:ClosestPointPars} shows the other major change that has
happened in our analysis, since Paper I.  As mentioned above, in Paper I, we
were limited to warm MHD solutions with $p \lesssim 1.1$ - the value which has
been marked by the dotted line. For this paper we have been able to investigate
two more solutions which are more ``massive'', with p = 0.3 and 0.45. See
Appendix \secn{newsol}, for a brief description on the physics of obtaining
these denser warm MHD solutions.

We choose to investigate the warm solutions with $p = 0.1$, 0.3 and 0.45 for
further understanding and for deriving spectra. The aim is to see how different
the line-of-sight absorption features are, as a function of $p$, $i$ (or
$\theta$) and the size of the disk.

The top three panels of \fig{fig:MhdSols} gives a visual comparison of the
three solutions. As is evident from the changing gradient of $\log \xi$ values
across the three panels, as p increases, the value of $\xi$ decreases at a
given $r$, because the solutions are denser with increasing $p$. The red solid
line, in each of those panels, near which the value of $i$ is labeled, is the
lowest line-of-sight for which the wind is {\bf not} Compton thick. The wind
will be detected via FeXXVI absorption lines, only from regions above the red,
solid line. Notice that the smallest angle for which the wind can be detected,
moves further away from the accretion disk, as $p$ increases. For $p = 0.30$
and 0.45, the relevant region is also beyond the Alfven surface.  In the next
section, where we derive spectra, we choose representative line-of-sight close
to the value (of minimum $i$) mentioned in \fig{fig:MhdSols}.

In the bottom panels, we have shown comparison between the density functions of
the MHD solutions. The left panel shows a comparison between our $p = 0.1$ MHD
model and the outflow model used in \citet{fukumura17}. On the right, it is the
comparison of $f(\theta)$ of the three MHD models that are being studied in
this paper. It is interesting to note that there is a non-monotonic behaviour
of $f(\theta)$ as function of p. As mentioned before, we were motivated to find
warm MHD models with higher values of $p$, as indicated necessary from Paper I.
However, to obtain MHD solutions with high $p$ values one needs to change other
disc parameters (namely turbulence, heating). This translates into different
outflow geometries, as discussed in \secn{newsol} showing that some solutions
bend and open up much more than others (\fig{fig:figA}). This is the reason of
the non-monotonic behavior of the density function - the link between the
$f(\theta)$ profile and $p$ is not straight forward.

\begin{figure}
\begin{center}
\includegraphics[scale = 1.0, width = 8 cm, trim = 70 20 70 30, angle = 0]{fig/Fig04_PhysicalParametersMhdSols.ps}
	\caption{The three most important observables predicted from C08 simulations for our models and lines of sight. The models and the lines of sight are explicitly labeled in the middle panel which shows the distribution of the column density of the ion FeXXVI along the line of sight. The same line styles and colours are maintained in the top panel (showing distribution of ionisation parameter) and bottom panel (showing distribution of the line of sight velocity).
	}
\labfig{fig:PhysParams}
\end{center}
\end{figure}

\begin{figure*}
\begin{center}
\includegraphics[width = 0.9\columnwidth, trim = 0 0 0 0]{fig/Fig05/1e6Rg_6to8keV_3DenseSolns.ps} \includegraphics[width = 0.9\columnwidth, trim = 0 0 0 0]{fig/Fig05/1e6Rg_1to10keV_3DenseSolns.ps}
\caption{This Figure shows variation of spectrum as the MHD Solutions
	are varied. The left panel just zooms in on the 6-8 keV energy range
	concentrating on the FeXXV and FeXXVI line complexes. On the right
	panel, we show the entire energy range of 1-10 keV relevant for X-ray
	observations of X-ray binaries.}
\labfig{fig:CSp_pvar}
\end{center}
\end{figure*}

\section{Theoretical Spectra of MHD winds using CLOUDY}
\labsecn{sec:SyntheticCloudySpectra}

The 1st step in calculating synthetic spectra using C08, is to determine the
velocity or energy resolution for which we want to derive the spectra. The
maximum energy resolution that C08 can provide corresponds to 75 km/s (1.6 eV)
at $\sim 6.5$ keV. In comparison the projected resolution at XRISM is 300 km/s
(6.4 eV) and that of Athena is $~$150 km/s (3.2 eV) at $~$ 6.5 keV. The
average velocity resolution that Chandra provides at 1 keV is 1500 km/s (32 eV) and
deteriorates for 6.5 keV. Since the resolution of the theoretical spectra
should be better than that for the observed spectra, we choose to accept the
best theoretical resolution possible using C08, at this stage - 75 km/s. Our
theoretical velocity resolution is still better than even what Athena will
provide a decade later.

For any given line-of-sight (e.g. $i = 10^{\circ}$ for p=0.1) the gas is
modeled as a continuous set of slabs/boxes separated by $\Delta v = 75$ km/s.
Thus each slab/box is a `small' unit of the flow itself, and the velocity of
the box is simply the flow velocity at that physical point. We find the `gas
slab' with log$\xi$ = 6 and start C08 calculations from that distance. The 1st
box sees the light coming from the central source, but redshifted by the
velocity of the 1st box, away from the source. Box 2 onwards, any box n `sees'
the SED transmitted to it from box n-1, but blueshifted by the velocity
difference of the two boxes.
For each box, we are calculating the transmission coefficients as a function of
energy, which results in the absorption spectrum. In this paper, we do not
include the emission lines, from any of the boxes.
While calculating the absorption spectrum from each box, we are also recording
the important physical parameters (some of which are plotted in
\fig{fig:PhysParams}), which will help us understand the spectrum along a line
of sight.  Note that while analysing each box, we explicitly check that the box
is not Compton thick, by checking $N_H < 10^{24.18} \,\, \rm{cm}^{-3}$, as well
as the cumulative column along the line-of-sight is $< 10^{24.18} \,\,
\rm{cm}^{-3}$.  The calculation is continued until we reach the end of the
disk, or velocity of the box drops below 75 km/s. The extent of the wind is
determined by the maximum anchoring radius, which is varied as $r_o|_{\rm{max}}
= 10^5 \, \& \, 10^6 r_G$ to test the effects on the spectra. We repeat these
calculations for different lines of sight for the three different warm MHD
solutions with p = 0.10, 0.30, 0.45.

We note that the disk extent of $10^6 r_G$ for a 10 solar mass black hole is
$1.5 \times 10^{12}$ cm. It is on the higher side of what can be expected for
disk sizes for stellar mass black holes. GRS 1915+105, the brightest galactic
black hole is known to have a large disk $\sim 10^{12}$ cm
\citep{done04,remillard06}. Hence we keep our simulations till $r_o|_{\rm{max}}
= 10^6 r_G$. We further checked that there is very little effect of the disk
size on the spectrum or the equivalent widths, as will be mentioned in
subsequent sections as well. Hence we show many of our simulations for
$r_o|_{\rm{max}} = 10^6 r_G$, instead of for $10^5 r_G$.

In \fig{fig:PhysParams}, we see the results of probing p=0.1 along i = 10,
p=0.3 along i = 15, and p = 0.45 is probed along i=25. We can trace the column
density of the FeXXVI as a function of the distance from the black hole. The
figure demonstrates that for denser MHD solutions we can probe material closer
to the black hole, since we get intense FeXXVI absorption closer to the black
hole. Also, note that closer to the black hole means higher velocities. Hence
for denser solutions we have intense FeXXVI absorption lines which are further
blueshifted. However, along a line-of-sight since there is `continuous'
absorption over a range of $\xi$, the resultant absorption will be an
amalgamation of many absorption lines/components spread over a range of $\xi$
and $v$. The lines will be more skewed for the denser MHD solutions, because of
this reason. We will see this effect in
\dfig{fig:CSp_pvar}{fig:CSp_I}.

\fig{fig:CSp_pvar} shows how the spectrum changes when we consider different MHD solutions.
The denser solutions with higher value of p have more skewed and deeper lines (left panel).
Notice that the denser solutions also show "satellite lines". Lines that come from lower ions of iron \citep{kallman04, witthoeft11}. The requirement for these lines is that they need low ionisation (much lower than FeXXV).
The 1-10 kev spectra of the same models show that absorption lines of sufficient strength can be observed from low ionisation ions (corresponding to elements with lower atomic number) for the denser (higher $p$) MHD models.

\begin{figure}
\begin{center}
\includegraphics[scale = 1.0, width = 8 cm]{fig/Fig06.ps}
	\caption{This Figure shows variation of spectrum as the line-of-sight angle is varied for the p=0.1 MHD solution.}
\labfig{fig:CSp_I}
\end{center}
\end{figure}

We also generated spectra for different wind extent, while keeping $p$ and
$i (\theta)$ constant. A larger wind extent (for a given solution) will come
from extending the disk further so that the last anchoring radius can be
further out from the black hole.  The effect of a larger size of the accretion
disk is similar as having a denser (higher $p$) MHD model, in one way - we can
have wind further out from the black hole. As a result the ionisation of the
gas which is furthest out will be lower. Hence effects of lower ionisation will
show up for larger disk. However, we found one distinction from the case of the
denser MHD models as well. The lower $\xi$ ions for the larger disks have lower
velocity (and hence narrower line profiles or lesser skewness) than those for denser MHD
solutions.

\fig{fig:CSp_I} shows the variation of spectrum as the angle of line-of-sight
changes.  As we go up from the accretion disk (i increases, $\theta$
decreases), the line strengths decrease. Here we show only the case of p=0.1
where the effect is the most noticeable compared to larger $p$ values.  The p =
0.1 model has a sharper gradient (fall of density with decreasing LOS angle
$\theta$) at $\theta = 80^{\circ} (i = 10^{\circ})$ than what p = 0.3 has at
$\theta = 75^{\circ} (i = 15^{\circ})$ or p = 0.45 has at $\theta \sim
65^{\circ} (i \sim 25^{\circ})$.  Hence, as $\theta$ decreases (and i
increases), the p = 0.10 solution changes more significantly and hence the
lines change significantly.

\begin{figure}
\begin{center}
	\includegraphics[height = \columnwidth,angle=-90]{fig/Fig07.ps}
	\caption{Athena simulation of a spectrum corresponding to a MHD disk wind with an ejection efficiency $p = 0.30$, a radial extension of 10$^6 r_G$ observed along an inclination angle of $15^{\circ}$. The source has a flux of 100 mCrab and the simulation has a 100ks exposure. This corresponds to a total   number of counts of 240$\times 10^3$ in the 6-8 keV energy range. The spectrum has been re-binned in order to have at least 10 counts per bin. {\bf Top:} the solid, green line show the best fit model which combines two power laws for the continuum and 4 Gaussian absorption lines at fixed energies 6.7, 6.96, 7.8 and 7.88 keV. {\bf Bottom:} Ratio data/model.}
\labfig{fig:simuStandard}
\end{center}
\end{figure}

\begin{figure*}
\begin{center}
\includegraphics[width = 0.85\columnwidth]{fig/Fig08/Fig08a.ps}\includegraphics[width = 0.85\columnwidth]{fig/Fig08/Fig08b.ps}
	\caption{EW of the Fe XXVI  line versus the total number of counts in the 6-8 keV range. The simulations correspond to a wind extension of $10^6 r_G$. {\bf Left:} We look for the effect of varying the ejection index $p$ at same or similar inclination angles. Models with $p = 0.1$ (black-square) and $p=0.3$ (blue-pentagon) were studied at line-of-sight angle $i = 20^{\circ} \,\, (\theta = 70^{\circ})$, whereas the $p = 0.45$ (red-circle) model was examined at $i = 23^{\circ}$ (because it is optically thick below this angle). {\bf Right:} We study the effect of varying inclination angle for a fixed ($p = 0.1$) MHD model. The black-pentagon, blue-square and red-circle points respectively correspond to inclination angle of $i =$ 10, 20 and 30 degrees ($\theta$ = 80, 70 and 60 degrees).}
\labfig{fig:EW}
\end{center}
\end{figure*}

This section shows us clearly that there are variations in the theoretical
spectra, and the line shapes, as the MHD flow parameters are changed, or when
we view the flow through different inclination angles. However, this analysis
is not sufficient to tell us if these theoretically apparent variations are
large enough to be distinguished in observations. Hence we dedicate the whole
next section trying to answer that question.

\section{Athena and Xrism Spectra of BHB winds}
\labsecn{sec:AthenaXrismSpectra}

\subsection{Basics of simulating XIFU and XRISM synthetic spectra}

We simulated observations for the high resolution instruments on two planned
	future X-ray missions, the Resolve spectrometer on the X-ray Imaging
	and Spectroscopy Mission \citep[XRISM;][]{makoto18} and the X-ray
	Integral Field Unit (X-IFU) on Athena \citep{barret18}.  Simulations
	for Resolve,  were made based on response matrices, effective area
	curves, and backgrounds for the Hitomi Soft X-ray Spectrometer
	\citep{leutenegger18} which were provided by NASA-GSFC
	(https://heasarc.gsfc.nasa.gov/docs/xrism/proposals/). For Athena, we
	used the X-IFU simulation pipeline xifupipeline, which is part of
	end-to-end simulation package SIXTE \citep{dauser19}. The simulation
	employed here includes the official effective area of Athena's
	telescopes with an effective area of $1.4\,\mathrm{m}^2$. For bright
	source simulations, xifupipeline includes a model for the degradation
	of the energy resolution of the micro-calorimeter array that happens
	when photons hit the same pixel close in time to each other.  For the
	brightest assumed source fluxes, we also performed simulations where a
	Be-filter with a thickness of $100\,\mu$m is put in front of the
	detector array. This approach cuts the flux of the soft X-rays and
	allows X-IFU to perform observations even of bright sources.

\subsection{Single Gaussian Fits to the fake spectra}
\labsubsecn{SingleGaussian}

\subsubsection{General methodology}

We develop an automatic procedure to test the presence of absorption lines in
the simulated spectra. For most of the analysis, we concentrate on the 6-8 keV
range where the most predominant absorption lines reside, especially from Fe
XXV and Fe XXVI. We first fit the continuum with two power laws (to reproduce
any potential curvature in the continuum), ignoring data between 6.5 and 7.2
keV and above 7.7 keV. Then we add, one by one, 4 Gaussian lines in absorption
centred at 6.7, 6.96, 7.8 and 7.88 keV. These energies correspond to FeXXV
(1s$^2$-1s2p), FeXXVI (1s-2p), NiXXVII (1s$^2$-1s2p) and FeXXV (1s$^2$ -1s3p)
respectively. For each Gaussian, the energy was fixed but their width, redshift
(z) and normalisation were left free to vary.

An example of best fit obtained through this procedure is shown at the top of
\fig{fig:simuStandard}, while the ratio data/model is shown at the bottom. This
is an Athena simulation of a spectrum corresponding to a MHD disk wind with an
ejection efficiency $p = 0.30$, a radial extension of 10$^6 r_G$ observed along
an inclination angle of $i = 15$deg ($\theta = 75$deg). The source has a flux
of 100 mCrab and the simulation has a 100ks exposure. This gives a total number
of counts in the 6-8 keV range of about 240$\times 10^3$. We clearly see the
presence of absorption features around 6.7, 7, 7.8 and 7.9 keV.

The procedure correctly finds these four main absorption features but clear
residuals are present around each line - indicating that the line has a more
complex profile than a simple Gaussian one. We also see small absorption
features to the left of the FeXXV line at $\sim$6.7 keV, which are the
'satellite lines' either from the lower ions of iron, or the x, y, z resonance
lines of the FeXXV ion.  This simpler method is used in the immediate next
subsections to get an idea of - i) the number of counts needed to detect the
lines; ii) the detectability of the line asymmetries as the ejection index, and
LOS angle vary; and iii) to study Athena spectra in the 1 - 10 keV energy
range. The single Gaussian fits clearly demonstrate the need to fit the
absorption lines with more complex profiles or multiple Gaussians. The latter
is attempted in \subsecn{MultiGauss}.

\subsubsection{Minimum counts needed}
\labsubsubsecn{subsubsec:MinCount}

We have reported in \fig{fig:EW} the EW of the Fe XXVI K$\alpha$ line, deduced
from our fitting procedure, versus the total counts in the 6-8 keV range for
different sets of parameters.  This figure corresponds to the case of a wind
radial extension of $10^6 r_G$. On the left of \fig{fig:EW}, the inclination is
chosen to be $ ~ 20^{\circ}$ and the wind simulations have different p=0.1, 0.3
and 0.45 for the black squares, blue diamonds and red circles respectively. On
the right of \fig{fig:EW}, the ejection parameter p is fixed to 0.1 and the
wind simulations have different inclination angle of 80, 70 and 60 degrees for
the blue diamonds, black squares and red circles respectively.  As expected,
the absorption lines are stronger (the EW are larger) for larger ejection
parameter p and larger inclination angles.

Interestingly, this figure gives an estimate of the required number of counts,
and consequently the required combination of source flux and exposure, so that
the Fe XXVI K$\alpha$ line can be detected. The counts in the 6-8 keV band has
to be larger than a few thousands to allow good detection. Lines with EW as low
as a few eV should be detectable if the 6-8 keV counts is larger than
$10^4$-$10^5$ for the less favourable simulated cases i.e. low p (=0.1) and
higher inclination angles in terms of i (lower inclination angle in terms of
$\theta$).

\begin{figure}
\includegraphics[width = 0.9\columnwidth]{fig/Fig09.ps}
	\caption{{\bf Top:} 1-10 keV simulated spectra with Athena assuming $p=0.3$, an angle $i=15$deg and a disk extension of $10^6 r_G$. The source has a flux of 100 mCrab and the exposure was 100 ks. {\bf Middle:} A zoom of the 3-4.5 keV energy range. Absorption lines of Ar and Ca are clearly visible. {\bf Bottom:} Same simulation but with XRISM. The spectra have been re-binned in order to have 15$\sigma$ per bin.}
\labfig{fig:spec110}
\end{figure}

\begin{figure*}
\begin{tabular}{cc}
\includegraphics[angle=-90,width = \columnwidth, trim = 50 100 0 50]{fig/Fig10/Fig10a.ps} & \includegraphics[angle=-90,width = \columnwidth, trim = 50 100 0 50]{fig/Fig10/Fig10b.ps}
\end{tabular}
\caption{ EW of ArXVIII and CaXX lines for MHD solutions with different
	ejection indices - p = 0.1, 0.3, 0.45, but of same disk extent
	$r_o|_{max} = 10^6 r_G$. The {\bf left} plot is for Athena simulations
	and the {\bf right} one for XRISM. For the simulations we assume a
	source flux of 100 mCrab and an exposure of 100 ks}
\labfig{fig:lowionsEW}
\end{figure*}

\subsubsection{Athena spectra variation with $p$ and $i$}

Complementary to the analysis in \secn{sec:SyntheticCloudySpectra}, we
conducted Athena simulations (still assuming a source of 100 mCrab flux in 2-10
keV and observed for 100 ks) for spectra which vary with ejection index $p$ and
LOS angle $i$. For the details and Figures we urge the reader to check
\secn{AthenaVarypNi}, in the Appendix. Here, we summarise our finds, based
primarily, on the FeXXVI K$\alpha$ absorption line.

As expected the absorption lines are stronger for larger ejection efficiency
parameter $p$. We found that it is not possible to detect the asymmetry in the
line profile for p = 0.1, even if it can be seen in the theoretical spectra.
Like the strength of the lines, the asymmetry also grows as p increases (0.3 or
higher). Even the single Gaussian fit (even if inaccurate) captures the fact
that the absorption line profile peaks at higher energy (stronger blueshift)
for higher $p$.

Since, we could not detect line asymmetry for $p=0.1$, we chose the $p=0.3$ MHD
solution for studying the variation of Athena spectra as LOS angle changes.
Since we chose a higher $p$ solution, the line remains detectable even viewed
at large inclination from the disk surface - $i=40^{\circ} = (\theta  =
50^{\circ})$. The closer the LOS to the disk (i.e. the smaller the LOS
inclination) stronger the absorption line and more asymmetric the line profile.
This is expected because the wind density increases when approaching the disk
surface. Beyond $i=30^{\circ} = (\theta  = 60^{\circ})$, line asymmetry cannot
be detected.  Here we do not see any significant shift in the blueshift of the
Gaussian fit line, as was the case for $p$ variation.

\subsubsection{The 1-10 keV spectrum}
\labsubsubsecn{1to10keV}

In the right panel of \fig{fig:CSp_pvar} we have seen theoretical spectra
harbouring absorption lines at energies less that 6 keV. We need to check if
these lines can be detected by Athena and XRISM. It would be interesting to
check if in observed spectra we can trace the change in the line observable as
the MHD flow changes. These lines were not strong enough or asymmetric enough
to warrant multiple Gaussian fits. Hence, single Gaussian fits have been used
to calculate EWs.

We have reported in \fig{fig:spec110} the 1-10 keV simulated spectra expected
with Athena and XRISM with $p=0.3$, an angle $i=15^{\circ}$ and a disk
extension of $10^6 r_G$. Lines are observed in the entire 1-10 keV range and we
clearly see Argon and Calcium lines in Athena and, while less significantly,
also in XRISM.

We have discussed in \secn{sec:SyntheticCloudySpectra} that having a higher $p$
dense MHD model and/or having a larger extent of the accretion disk increases
the chances of `seeing' the absorption lines from the H and He-like ions of
lower atomic number elements like Argon and Calcium. Through the simulations
for Athena and XRISM we can verify the above quantitatively. In
\fig{fig:lowionsEW} we show the EW of ArXVIII and CaXX lines for ATHENA (left
panel) and XRISM (right panel). The variation of the EW is with respect to
changing MHD models (p = 0.1, 0.3 at LOS $i = 20^{\circ}$ and p =0.45 at LOS $i
= 23^{\circ}$), while the disk is assumed to have a constant extent of
$r_o|_{max} = 10^6 r_G$. We had also tried changing the disk extent from
$r_o|_{max} = 10^5$ to $10^6 r_G$ (not shown in the Figure), while keeping the
MHD solution parameters constant at p = 0.3, $i = 15^{\circ}$.

We found that changing the MHD model has a greater impact on the presence and
strength of the absorption lines from Ar and Ca, than extending the accretion
disk further out. While the disk was extended from $r_o|_{max} = 10^5$ to $10^6
r_G$, the EWs of the lines changed only by a factor of a few. However, as $p$
increases from 0.1 through 0.3 to 0.45 the EW change by $\sim 2$ dex.

\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[height = 0.75\columnwidth,angle=-90, trim = 80 75 20 25]{fig/Fig11/fig11a_new.ps} & \includegraphics[height = 0.75\columnwidth,angle=-90, trim = 80 75 20 25]{fig/Fig11/fig11c_new.ps} \\ \\ \\
\includegraphics[height = 0.75\columnwidth,angle=-90, trim = 80 75 20 25]{fig/Fig11/fig11b_new.ps} & \includegraphics[height = 0.75\columnwidth,angle=-90, trim = 80 75 20 25]{fig/Fig11/fig11d_new.ps}
\end{tabular}
	\caption{{\bf Left:} Zoom of the Athena simulation of \fig{fig:simuStandard} around the FeXXV (top) and FeXXVI (bottom) lines. The black line and black residuals correspond to best fits with a single Gaussian. The blue line and blue residuals correspond to best fits with a multiple-Gaussian model. In these examples, 3 Gaussians have been used to fit the FeXXV line + 3 other Gaussians to fit the absorption features below 6.68 keV, and 5 Gaussians to fit the FeXXVI line. {\bf Right:} The same but for XRISM simulations. We need 2 Gaussians to fit the FeXXV line and 3 more Gaussians to fit the satellite lines at $\sim$6.63 and 6.7 keV. And we need 3 Gaussians to fit the FeXXVI line}
\labfig{fig:linezoom}
\end{center}
\end{figure*}

\subsection{Multiple Gaussian fits to the asymmetric absorption lines}
\labsubsecn{MultiGauss}

In most cases, a fit of the absorption lines with a single Gaussian showed
clear residuals around the lines as exemplified in \subsecn{SingleGaussian}.
The reasons are two-fold. First, the expected line profiles are clearly
asymmetric and this asymmetry becomes detectable when the statistics is good
enough. On the other hand, the presence of other weak absorption lines, close
to the FeXXV ones especially, "pollutes" the fit. This is shown in
\fig{fig:linezoom} where we plot zooms around the energy range of the FeXXV and
FeXXVI lines. The black lines represent the fit obtained with a single Gaussian
profile and the corresponding data/model ratio are reported in black at the
bottom of each figures. The blue lines (and the corresponding blue data/model
ratio) have been obtained with a multiple-Gaussians model. There is a clear
improvement of the residuals when we use several Gaussians to fit the main
absorption profile. We find that 3 (resp. 5) Gaussians give a statistically
good fit in the case of FeXXV (resp. FeXXVI) and the addition of another
Gaussian has an F-test probability smaller than 90\%. We also need to add other
Gaussians to fit the weak absorption features in the red part of the FeXXV
line.

The XRISM simulations shown in \fig{fig:linezoom} have been obtained for a 100
mCrab source at line-of-sight angle $i = 15^{\circ}$ observed for 100ks. The
MHD model used was the p = 0.3 outflow extending up to $\rm{r_o|_{max}} = 10^6
r_G$. The fake fit needed 2 Gaussians to fit the FeXXV line and 3 more
Gaussians to fit the satellite lines at $\sim$6.63 and 6.7 keV. And we need 3
Gaussians to fit the FeXXVI line

In this paper we limit ourselves to just demonstrating the power of Athena and
Xrism to detect the asymmetry in the absorption lines. Of course rigorous
quantitative analysis of these fits are warranted, so that more accurate
measures of the EW can be prescribed. However such analysis is beyond the scope
of the current paper. We intend to dedicate a separate publication, in near
future, which will explicitly deal with this analysis.

\section{Discussion}
\labsecn{sec:discussion}

\subsection{Comparison with Chandra}
\labsubsecn{subsec:discuss_chandra}

\begin{figure}
\begin{center}
\includegraphics[width = 0.9\columnwidth,angle=0]{fig/Fig12_Chandra/Fig12a.ps}
\includegraphics[width = 0.9\columnwidth,angle=0]{fig/Fig12_Chandra/Fig12b.ps}
	\caption{Simulated spectra for Chandra: for MHD model with $p = 0.30$, the disk extending to $10^6$ observed at LOS $i = 15^{\circ}$. The central source is assumed to be at 100 mCrab. The exposure times for the fake observations are chosen to be 100 ks (top) and 1 Ms (bottom) }.
\labfig{fig:chandra}
\end{center}
\end{figure}

We show in \fig{fig:chandra} the simulation of p = 0.30, i = 15$^{\circ}$, $r_o|_{max}$
= $10^6 r_G$ MHD solution for a source of 100 mCrab observed by Chandra for 100
ks (top panel) and 1 Ms (bottom panel). The lines, of $\sim 30 - 50$ eV each,
are clearly detected at more than 99.99\% confidence already with 100 ks but
the asymmetry is not. In 1 Ms however we clearly detect the asymmetry at least
for the Fe XXVI line. Given the variability of XrB in outburst, in flux and in spectral
shape, this 1Ms (~11 days) Chandra observation is of course not a realistic
case. It rather shows that we have to rely on the future missions like XRISM
and Athena to bring such studies within practical reach.

\subsection{The FeXXVI K$\alpha$ doublet lines}
\labsubsecn{subsec:discuss_doublet}

\begin{figure*}
\begin{center}
	\includegraphics[height = \columnwidth, trim = 0 0 500 0, angle=-90]{fig/Fig13_Doublet/FeXXVI_Doublet_pVary_4Warm.ps} \includegraphics[height = \columnwidth,angle=-90]{fig/Fig13_Doublet/FeXXVIDoublet_CLOUDYnew.ps} \\
\includegraphics[height = \columnwidth,angle=-90]{fig/Fig13_Doublet/FeXXVIDoublet_XSTARnew.ps} \includegraphics[height = \columnwidth,angle=-90]{fig/Fig13_Doublet/FeXXVIDoublet_XSTAR_2gaunew.ps}
	\caption{{\it Top Left:} Xstar simulations of the FeXXVI K$\alpha$ absorption line, highlighting the doublet feature of the line. {\bf Top Right:} C08 simulation of the Fe XXVI line for $p = 0.30$, $i = 15$, disk extension $10^6$ and 100 mCrab for 100ks. The line is a best fit with a simple Gaussian line with energy fixed to 6.96 keV. {\bf Bottom Left:} Same simulation than the top one but using Xstar. The FeXXVI K$\alpha$ doublet is now included. {\bf Bottom Right:} Fit of the XSTAR simulation using two Gaussians at energy fixed to 6.952 and 6.973 keV.}
\labfig{fig:doublet}
\end{center}
\end{figure*}

In nature, the Lyman $\alpha$ line from the H-like ions of any element is actually split into a doublet. For iron, they are Fe XXVI Ly$\alpha_1$ at 6.973 keV and Fe XXVI Ly$\alpha_2$ at 6.952 keV. It is very difficult to detect these two lines distinctly even in a Chandra-HETG spectrum, which has the best energy spectral resolution available, among current X-ray observatories. \citet{tomaru20} have analysed the Chandra HETG 3rd order spectrum of neutron star low mass X-ray binary GX 13+1 and model these two lines. They have discussed in details the importance of these lines in the upcoming era of X-ray astronomy. Upcoming observatories like XRISM (and subsequently Athena) with significantly improved energy spectral resolution will be able to `easily' find the doublet. Analysing the shape of the lines will shed light on the launching and acceleration mechanisms of the outflowing wind.

The C08 atomic data base records the Fe XXVI Ly$\alpha$ as a single line - a blend of the doublets. It is not a problem for most current observations, because these features can hardly be observed as separate lines. All our above analysis is based on this blended line. To consider the details of the doublets is beyond the scope of this paper, however in this section we show some preliminary results of how the doublet may be seen by Athena. For the analysis in this section, we had to use XSTAR.

In top left panel of \fig{fig:doublet} we show the zoom of the Fe XXVI Ly$\alpha$ line region. Three different warm MHD models have been considered. As $p$ increases the doublets get smeared out. We use the $p = 0.3$ model and its spectrum to further do Athena spectrum (bottom panels of \fig{fig:doublet}). The top right panel of \fig{fig:doublet} show Athena simulations derived out of the C08 calculations. The Fe XXVI Ly$\alpha$ line in the Athena fake data is fit with a single Gaussian with line energy fixed at 6.96 keV. Notice that significant residuals remain. When the same analysis is done, but this time using the XSTAR generated spectrum (bottom left panel of \fig{fig:doublet}), the residuals improve. However, if instead of one Gaussian, if two are used with line centres placed at the natural energies of the doublets, the fit leaves almost no residuals (bottom right panel of \fig{fig:doublet}) - clearly indicating that Athena spectrum can constrain the doublets.

Further detailed quantitative analysis is beyond the scope of this paper. Because in this paper we are dealing with `denser' MHD models, the doublets from such flows are more smeared out. However in future publications we will deal with lower density MHD models and look for their observable signatures. It would be more relevant at that stage to look into further details of the doublet lines and their variations (with the MHD models), and also check the potential of the lines as a deterministic observable.

\subsection{Comparison with past literature}
\labsubsecn{subsec:discuss_pastlit}

While there is a significant amount of literature on theoretical development of MHD outflow models and also their application to Jets' observations, there is only a limited number of attempts in applying MHD models as explanations for the observed lower velocity ($\sim 100$ km/s) diffused winds. \citet{fukumura10a,fukumura10b,fukumura14} is where they were first attempted for explaining the various wind components in super-massive black hole systems or the Active Galactic Nuclei. \citet{fukumura17,fukumura21} show extensions of the same models and their methods to explain winds in Black hole X-ray binaries.

First paper \citet{fukumura10a} gives us the framework of the models and shows different outflow models, mainly varying $\theta$ (the inclination angle), while holding the other parameters (namely $\eta_w$ and $\dot{m}$) constant (see Eqns. 28 and 29 of the paper). The calculations of the paper were done keeping AGN absorbers in mind, however the paper acknowledges that the same models are scalable to different black hole masses. In the 2nd paper \citet{fukumura10b} the authors show that a wide range of physical properties can be made possible within the same outflow, so that ions responsible for Broad Absorption lines, like C IV (with outflow velocity $\lesssim 0.1$c) can co-exist with ions seen as ultrafast outflows Fe XXV ($\lesssim 0.6$c). However this is possible for a special class of AGN SEDs with steep $\alpha_{ox} = -2$.

GROJ-1655 is an interesting source in the context of BHB diffused winds because this source showed a particularly unique spectrum which has numerous absorption lines similar to the warm absorbers in AGN. In \citet{fukumura17} the authors use the same MHD models, but scaled in mass to stellar mass black holes, to explain the observed spectrum of GROJ-1655. The density normalisation ($n_o$) that they use at the footpoint of each streamline of the outflow is proportional to the mass accretion rate and the ratio of the outflow rate in the wind to accretion rate at $r_{ISCO}$. The same formalism is used in \citet{fukumura21} to explain observed spectra of four prominent BHBs which often show wind signatures during their outbursts.

The principle difference between the models used by the various papers
mentioned above and the MHD models used in this paper is the {\it relation
between the outflow and the accretion disk}.  See the discussion section of
\citet{fukumura21} where they state that their outflow models lack a direct
connection with the accretion process.  While in \citet{fukumura14}, the
authors show the variation of the geometry of the flow as a function of some
input flow parameters like the specific angular momentum (H), and the
fluid-to-magnetic flux ratio (F), note that these parameters are essentially
flow parameters and are not directly related to the accretion disk. Hence using
these models it is not possible to make any statement about the accretion disk
properties. For example, for the models used for our paper, in order to get
higher $p$ values, we had to change disk conditions (heating at the disk
surface layers, turbulence properties). This results in different magnetic
bending and shear at the disk surface (as shown in \tablem{tab:sol}) which lead
to different flow geometries. The magnetisation ($\mu$), fixed in the present
work, is also a crucial parameter in accretion-ejection MHD structures. Hence
we are in the process of understanding which of the disk parameters are the
most important variants in the determining the line shapes that we observe in
the BHB spectra. For example, in this paper, we are concentrating on the
ejection index (p) and in our next work we shall analyse the role of the disk
magnetisation ($\mu$) in details.

We show a comparison between our p = 0.1 MHD model and the one used by
\citet{fukumura17} in \fig{fig:MhdSols} (bottom left panel), where we compare
the density variation as a function of $\theta$ along the streamline.

\section{Conclusion}
\labsecn{secn:conclusion}

We have used the magnetohydrodynamic accretion-ejection dense and warm solutions developed by \citet{casse00b, ferreira04} and \citet{jacquemin19} to predict possible observable spectra from such diffused outflows (winds) in black hole binary systems. While explaining Jets in black hole systems might have been the original motivation for developing these models, they are analytical solutions and self similarity conditions can be used to make these physical models span large (up to $\sim 10^6 \rm{r_G}$) distances (and hence wide range of other physical parameters) from the black hole. At any given radius the density at the base of the outflow is explicitly dependent on the accretion rate of the disk at that radius. Thus the outflow cannot be treated as independent of the inflow in these models, unlike some of the other models used in the literature (see \subsecn{subsec:discuss_pastlit}).

We have found that the ejection index $p$ is one of the key MHD parameters that decides if the flow has the right physical quantities so that absorption lines can manifest in the high resolution X-ray spectra. In Paper I we found that $p >\sim 0.1$ is a staple requirement for dense enough (to be detected) winds. In this paper, we choose the densest warm solutions derived by \citet{casse00b, ferreira04} and \citet{jacquemin19} and develop our methodology to predict high resolution spectra concentrating our efforts on the 1-8 keV energy range. We convolved our theoretical spectra with the currently available response functions of Athena and XRISM to predict what these new age instruments will observe seeing through MHD outflows of our kind.

Along with the MHD parameter, the ejection index $p$ which was varied from 0.1-0.45, we also varied two other external parameters, namely, the extent of the disk ($\rm{r_o|_{max}}$ from $10^5 - 10^6 \,\, \rm{r_G}$) and the angle of the line of sight $i \sim 10 - 25$ (or $\theta \sim 65 - 80$). From our investigations we can list some of the trends
\begin{itemize}
	\item[$\bullet$] The denser the MHD (higher $p$), the broader the line profiles. This happens because higher density ensures that the ionization parameters are low enough at even high velocities (closer to the black hole) to generate enough H and He-like Fe ions for them to absorb the X-ray continuum. For the same reason, the denser the MHD model, more the chances that one can detect the H and He-like ions of species of lower atomic number, like Calcium and Argon. Note that to detect these ions, the gas needs to have a lower (than that required for the Fe ions) ionisation parameter.
	\item[$\bullet$] External parameters like larger extent of $\rm{r_o|_{max}}$ can also result in high density of the absorbing gas, increasing possibility of detection of the lines. However, note that in this case the gas would be absorbing at lower velocities than the case of high $p$ MHD models. This is because, here the density increases because of the density profiles that (typically) rises with distance (albeit with different slopes for different $p$ models). Larger distance translates to lower velocity.
	\item[$\bullet$] Varying the line of sight angle also affects the density. For all the MHD models, the density drops as one moves away from the surface of the disk. Thus, as $i$ increases (i.e. $\theta$ decreases), the absorption lines become relatively weaker. However, the rate of this drop in density is not same for all the models. For example we find that for the $p = 0.1$ model the lines drop in strength and vanishes by $i = 40^{\circ}$, whereas for the $p = 0.3$, outflow, the line strength drops, but the line is still detectable at $i = 40^{\circ}$.
	\item[$\bullet$] In an observed spectrum, the above mentioned variations in the lines, manifest as, in addition to strength, asymmetry in the profiles. Thus the two most important tests that we conducted with the simulated Athena and XRISM spectra from our models, were a) if the lines will be detected and b) if the asymmetry in the lines are detected. For both the tests, we found that Athena can detect the lines and their asymmetries for a standard 100 ksec observation of a 100 mCrab. The counts in the 6-8 keV band has to be larger than a few thousands to
allow good detection. Lines with EW as low as a few eV should be
detectable if the 6-8 keV counts are larger than $10^4 - 10^5$ for the less
favourable simulated cases i.e. low $p ( \lesssim 0.1)$ and higher inclination
angles in terms of $i$ (lower inclination angle in terms of $\theta$).
\end{itemize}

Even if our MHD solutions have intrinsic differences from the MHD models of Fukumura et.al. papers, the trends that we get generally agree with their predictions. Our aim is not to stop at this stage after deriving these trends. Our goal at this stage is to thoroughly examine spectra for the entire suit of outflow emitting disk models available from the above mentioned papers, using the uniform methodology that we have introduced in this paper. While we have rigorously tested the effect of variation of $p$, we feel that another MHD parameter, namely the disk magnetisation ($\mu$) will also be an important parameter in defining the nature of the outflows. As shown by \citet{jacquemin21}, the accretion-ejection structure indeed, strongly depends on $\mu$. A radial evolution of $\mu$ in the accretion flow could also well explain the spectral behavior of X-ray binaries in outbursts \citep[e.g.][]{marcel18a, marcel18b, marcel19, marcel20} highlighting the importance of this parameter. In our next papers we shall examine spectra and line (and their asymmetries) detectability as a function of the disk magnetisations. These series of papers (the current draft is Paper II) will let us take the step to form xspec models that can fit the observed spectra from XRISM and Athena {\it for the disk parameters} - a unique possibility that will be proffered by only, the outflow emitting disk kind of models.

\section*{Data Availability}
We have not used any observational data in this paper. All other analysis techniques and tools have been duly acknowledged.

\section*{Acknowledgments}
The authors acknowledge funding support from the CNES and the High Energy Programme (PNHE) of the CNRS/INSU.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
From its inception, the Schwarzschild solution of Einstein's equations
has been the subject of controversy over the possibility of
compressing matter within the gravitational radius $2GM$.
Given the accumulated evidence for supermassive compact objects  ranging from
a few $10^{6}M_{\odot}$ to a few $10^{9} \, M_{\odot}$,
the existence of black-hole-like objects is beyond doubt \cite{melia,korm}
 What does
remain an issue is whether the Schwarzschild metric correctly describes
the physics of the interior.
Alternatives to classical black holes have been proposed
with no singularities in the interior
\cite{viol5,viol6,chap1,chap2,mazu3,dym,mazu4,math,lob3,ben,bro}.

The simplest model proposed  for supermassive compact objects at the
galactic centers is a self-gravitating
degenerate fermion gas composed of, e.g.,
 heavy sterile neutrinos
 \cite{viol5,viol6,bil7,bil8,mun9}
   However, this scenario
cannot cover the whole mass range of supermassive black-hole
candidates with a single sterile neutrino
species \cite{bil1}.

Recently, Chapline et al \cite{chap1,chap2}  put forth an
interesting proposal based on analogies to condensed matter
systems where the effective general relativity was an emergent phenomenon.
Specifically, they suggested that the sphere
 where the lapse function vanished marked a quantum phase
transition, the lapse function increasing again at $r < 2GM$.
As this required negative pressure, the authors of \cite{chap2}
assumed the interior vacuum condensate to be described by de Sitter space
with the equation of state $p=-\rho$.

Subsequently, the idea of gravitational vacuum condensate, or `gravastar',
 was taken up by Mazur and Mottola \cite{mazu3,mazu4},
replacing the horizon with a shell of stiff matter astride
the surface at $r=2GM$.
Visser and Wiltshire \cite{viss4} and recently Carter \cite{car} also examined the stability
of the gravastar using the Israel thin shell formalism \cite{isra5}.
Despite the fact that general relativity is an emergent phenomenon
in string theory \cite{gree6}, the gravastar has met with a cool reception.
Certainly, the assumption of a de Sitter interior presents a quandary:
on the one hand, the quantum phase transition would suggest that
the associated cosmological constant is a fundamental parameter;
on the other hand, to accommodate the mass range of supermassive black hole
candidates, it must vary over some six orders of magnitude.
Thus, it seems prudent to explore other possibilities for the gravastar interior.

One obvious extension of the de Sitter equation of state
$p+\rho=0$ would be an equation  that satisfies
$p+\rho \leq 0$,
  thus violating the dominant energy condition.
The fluid of which the equation of state violates the dominant energy condition
has been
dubbed {\em phantom energy} \cite{cald1,cald2}
and has recently become a popular alternative
to quintessence and to the
cosmological constant \cite{phantom}.
The motivation for introducing the phantom energy is that
the equation of state $w\equiv p/\rho < -1$ produces a superaccelerated
cosmological expansion  which seems to be favored by the combined analysis of
the CMB and super nova type I data \cite{super}.
However,  a superaccelerated expansion may also be obtained
without violating the dominant energy condition
in scalar-tensor theories of gravity \cite{polar}
and in models with variable gravitational constant \cite{stef}
or variable cosmological constant \cite{sola}.
Some  astrophysical aspects of the phantom have recently been discussed,
such as phantom energy accretion \cite{bab},
phantom energy wormholes \cite{lob2},
 and
a possible relation of the phantom tachyon model to
supermassive black holes \cite{gonz}.

In this paper we consider a gravastar interior consisting of a self-gravitating scalar field described by a  Born-Infeld type Lagrangian which
 yields the Chaplygin gas equation of state \cite{kamen7,bil2}.
Hence, we look for static solutions of the self-gravitating Chaplygin gas.
In particular, we consider static Chaplygin gas configurations in the phantom regime,
i.e., when $p+\rho<0$, and we show that these configurations could
provide an alternative scenario for compact massive objects at galactic
centers.

The paper is organized as follows: In section \ref{model} we introduce the
basics of the model. In section \ref{static} we investigate static
solutions using Tolman-Oppenheimer-Volkoff equations.
In section \ref{galactic} we discuss a possible interpretation of
galactic centers as Born-Infeld phantom gravastars.
A stability analysis is given in section \ref{stability} and we conclude the paper
by section \ref{conclude}

\section{The model}
\label{model}
Consider the equation of state
\begin{equation}
p = - \; \frac{A}{\rho}
\label{eq006}
\end{equation}
in the phantom regime, i.e., when
\begin{equation}
\rho < \sqrt{A} \, .
\label{eq106}
\end{equation}
 Equation (\ref{eq006}) describes the Chaplygin gas which,
for $\rho \geq \sqrt{A}$, has attracted some attention as a
dark energy candidate \cite{kamen7,bil2}.
Astrophysical objects made of the so-called {\em generalized Chaplygin gas} \cite{bento}
have recently been discussed \cite{bert}.
The generalized Chaplygin gas has also been exploited in the phantom
regime \cite{hao2}.
As we shall shortly demonstrate, static solutions
to Einstein's equations with matter described by (\ref{eq006}) with (\ref{eq106})
cover the range of masses and radii required to fit
the phenomenology of supermassive dark compact objects
at the galactic centers. Moreover, Eq. (\ref{eq006})
yields the de Sitter gravastar solution
in the limit when the central density of the static solution
approaches the value $\sqrt{A}$.

The Chaplygin gas equation of state (\ref{eq006}) with the condition
$\rho < \sqrt{A}$ may be derived from
the Dirac-Born-Infeld type Lagrangian
\begin{equation}
{\cal{L}}_{\rm DBI} = - \sqrt{A}\, \sqrt{ 1 + X}\, ,
\label{eq015}
\end{equation}
where
\begin{equation}
X= g^{\mu \nu}  \vartheta_{,\mu}
 \vartheta_{,\nu} \: .
\label{eq115}
\end{equation}
Clearly,  in the limit $X\rightarrow0$
this Lagrangian
becomes a free scalar field Lagrangian with a ``wrong sign"
kinetic term, hence a phantom\footnote{It has recently been shown that quantum effects could lead to an effective
 dark-energy equation of state violating the dominant energy condition on
cosmological scales even  for a scalar field Lagrangian having
the correct sign of the kinetic energy \cite{onemli}.}.
A  phantom Lagrangian of the type (\ref{eq015}) has been proposed in the
context of the
superaccelerated expansion \cite{hao}.
Note that the Lagrangian (\ref{eq015})
leads to a perfect fluid with  the four-velocity
\begin{eqnarray}
u_{\alpha} = \frac{\vartheta_{, \alpha}}{\sqrt{ X}} \: ,
\label{eq016}
\end{eqnarray}
the pressure $p={\cal{L}}_{\rm DBI}$, and
 the density
\begin{eqnarray}
\rho = \frac{\sqrt{A}}{\sqrt{ 1 + X}}
 \leq  \sqrt{A}
 \: .
\label{eq017}
\end{eqnarray}
Clearly, the pressure and the density
obey Eq. (\ref{eq006}) and $\rho\leq  \sqrt{A}$.
It is worth noting that
the Chaplygin gas cosmological model \cite{kamen7}, in which
the condition
$\rho>\sqrt{A}$ holds, is described by
the  Lagrangian (\ref{eq015})
with a ($-$) sign in front of $X$.
It is as though instead of the lapse function
changing its sign, as it does in the Schwarzschild case for $r < 2 G M$,
the scalar kinetic energy changed its sign to become a phantom.

Equivalent to ${\cal{L}}_{\rm DBI}$ of Eq. (\ref{eq015}) is
\begin{equation}
{\cal{L}} = - \frac{\phi^{2}}{2} \, X -
\frac{1}{2} \; \left( \phi^{2} + \frac{A}{\phi^{2}} \right) ,
\label{eq019}
\end{equation}
as may be seen by eliminating $\phi^{2}$ through its equation of motion.
Now recall that in four dimensions, a three-form field strength is dual to a scalar:
\begin{equation}
H^{\mu \nu \alpha} = \phi^{2} \; \epsilon^{\mu \nu \alpha \beta} \; \vartheta_{, \beta} \; \; .
\label{eq020}
\end{equation}
Then the content of the model is
\begin{equation}
{\cal{L}} =  \frac{1}{12 \phi^{2}} \; H_{\mu \nu \alpha} \;
H^{\mu \nu \alpha} -
\frac{1}{2} \; \left( \phi^{2} + \frac{A}{\phi^{2}}  \right) \; \; .
\label{eq021}
\end{equation}
Notably, this is also the content of low-energy string theory
\cite{gree6} in the Einstein frame where
$H_{\mu \nu \alpha}$ is the Kalb-Ramond field, $\phi^{2}$
corresponds to the string coupling and the
dilaton kinetic term is neglected in lieu of a potential
assumed to arise from non-perturbative effects\footnote{Strictly,
 the exterior should have
 $H_{\mu \nu \alpha} \; H^{\mu \nu \alpha} = 6A$ and $\phi^{2} = 0$ to be
Schwarzschild.}.
A potential similar to that in (\ref{eq019}) and (\ref{eq021})
has been considered in the context of polymer scaling and black holes
\cite{rama}.

\section{Static solutions}
\label{static}
Next, we proceed to solve Einstein's equations for a
static, spherically symmetric configuration with an interior
described by (\ref{eq006}) and with a Schwarzschild exterior.
Our approach is similar to that of Armendariz-Picon and Lim \cite{arm}.
However, in contrast to us, they consider
the classes of Lagrangian with strictly space-like gradient of $\phi$,
i.e. with $X<0$, so that in their case the quantity (\ref{eq016})  cannot be
interpreted as a four velocity. As a consequence, their energy-momentum tensor
does not have a standard perfect fluid form and the pressure is not isotropic.
Here, we consider strictly time-like gradient of $\phi$ with $X>0$, but we
allow  phantom-like Lagrangians which violate the dominant energy condition.

For the static, spherically symmetric line element
\begin{equation}
ds^{2} = \xi^{2}(r) \; dt^{2} - \frac{dr^{2}}{ 1 - 2G{\cal{M}}(r)/r} - r^{2} \;
\left( d\vartheta^2 + \sin^{2} \vartheta \; d \varphi^{2} \right)
\label{eq002}
\end{equation}
with
$T_{\mu}^{\nu} = \mbox{diag} \;
( \rho  ,  -p  , - p  , - p)$,
 Einstein's equations become
\cite{oppen1}
\begin{equation}
{\cal{M}}' = 4 \pi r^{2} \; \rho ,
\label{eq003}
\end{equation}
\begin{equation}
\xi' = G \; \xi \;
\frac{ {\cal{M}} + 4 \pi \; r^{3} \; p}{r (r - 2 G \; {\cal{M}})} \, ,
\label{eq004}
\end{equation}
while $T_{\mu \; ; \nu}^{\nu} = 0$ gives
\begin{equation}
p' = - ( \rho + p ) \; \frac{\xi'}{\xi} \, .
\label{eq005}
\end{equation}
We focus on the equation of state
(\ref{eq006})
to close the system
and  we require the solution to (\ref{eq003}) and (\ref{eq004}) to be regular
 at $r=0$.
 Since by rescaling $t$
one may set $\xi (0) = 1$, Eqs. (\ref{eq006}) and (\ref{eq005}) yield
\begin{equation}
\xi (r) = \frac{\rho}{\rho_{0}} \,
\sqrt{ \frac{A - \rho_{0}^{2}}{A - \rho^{2}} } \, .
\label{eq007}
\end{equation}
Combining Eqs.
(\ref{eq006}),
(\ref{eq004}),
and (\ref{eq005}),
 one has
\begin{equation}
\rho' = G \left( 1 - \frac{\rho^{2}}{A} \right)
\left( \frac{ \rho {\cal{M}} - 4 \pi A r^{3} }{r (r - 2G {\cal{M}} ) }  \right) \, .
\label{eq008}
\end{equation}

In Figs. \ref{fig1}, \ref{fig2}, and \ref{fig3}, respectively, we exhibit
the resulting
$\rho (r) / \rho_{0}$, $\xi (r)$, and $2 G {\cal{M}} (r) / r$ for
selected values of
$\rho_{0} / \sqrt{A}$.
The solutions depend essentially
on the magnitude of $\rho_0$ relative to
 $\sqrt{A}$. In the following we summarize the properties of three classes of
 solutions corresponding to whether $\rho_0$ is larger, smaller, or
 equal to $\sqrt{A}$.

{\bf i)} For $\rho_0>\sqrt{A}$, the density
$\rho$ increases and the lapse function $\xi$ decreases with $r$
starting  from the origin
up to the black-hole horizon radius
 $R_{\rm bh}$,
    where $2G{\cal{M}}(R_{\rm bh})=R_{\rm bh}$.
In the limit
$\rho_0 \rightarrow \infty$,
a limiting
solution exists with a singular behavior
\begin{equation}
    \rho(r) \simeq (\frac{7 A}{18\pi G r^2})^{1/3}
\label{eq100}
\end{equation}
  near the origin.
\begin{figure}
\begin{center}
\includegraphics[width=.5\textwidth,trim= 0 2cm 0 2cm]{fig1.ps}
\caption{
 Density profile of the Chaplygin star for
 $\rho_0/\sqrt{A}= 1.2$ (short dashed), 0.98 (dotted),
 0.9 (long dashed).
The limiting singular solution with
 $r^{-2/3}$ behavior at small $r$ is represented by the dot-dashed
 and the de Sitter gravastar by the solid line.
}
\label{fig1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.5\textwidth,trim= 0 2cm 0 2cm]{fig2.ps}
\caption{
Lapse function $\xi/\xi_0$ for various solutions  as in Fig.\ \ref{fig1}.
}
\label{fig2}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=.5\textwidth,trim= 0 2cm 0 2cm]{fig3.ps}
\caption{
Enclosed mass divided by the radius of the star for
various solutions as in Fig.\ \ref{fig1}.
}
\label{fig3}
\end{center}
\end{figure}

{\bf ii)} For $\rho_0<\sqrt{A}$,
both
  $\rho$ and $\xi$ decrease with $r$ up to the radius $R_0$ where they vanish.
  At that point the pressure $p$ blows up to $-\infty$
     owing to (\ref{eq006}).
    The enclosed mass
${\cal M}$ is always less than $r/(2G)$,  never reaching the black-hole horizon,
i.e., the radius where $2G{\cal M}(r)=r$.

{\bf iii)} For $\rho_0=\sqrt{A}$,
the density $\rho$ remains
 constant equal to $\sqrt{A}$ up to  the de Sitter radius
  $R_{\rm dS}=2GM=8 \pi G R_{\rm dS}^3\rho_0/3$.
Hence, the interior is de Sitter, precisely as in Chapline et al.\ \cite{chap2}.
The lapse function is given by
\begin{equation}
\xi=\left(1-\frac{r^2}{R_{\rm dS}^2}\right)^{1/2}
\label{eq101}
\end{equation}
with
\begin{equation}
    R_{\rm dS} = \sqrt{\frac{3}{8\pi G}} A^{-1/4}.
\label{eq102}
\end{equation}

As $\rho_0 \rightarrow \sqrt{A} $  from above or from below,
solutions {\bf i)} or
{\bf ii)}, respectively, converge   to
{\bf iii)}
except at the endpoint.
The lapse function in {\bf iii)}
 joins the Schwarzschild solution outside
\begin{equation}
\xi(r)=\left(1-\frac{2 GM}{r}\right)^{1/2},
\label{eq103}
\end{equation}
continuously,
whereas in {\bf i)} and {\bf ii)} it happens discontinuously.

As in the case of a de Sitter gravastar, in order to
join
our interior solutions to a
Schwarzschild exterior at a spherical boundary of radius $R$,
 it is necessary to put a thin spherical shell\footnote{It has  recently
 been demonstrated that the joining can be made continuous
without the presence of a thin shell
for a gravastar made of the fluid with an
anisotropic pressure \cite{cat}.}
at the boundary with a surface density
and a surface tension
satisfying Israel's junction conditions \cite{isra5}.
In
all the solutions discussed above the pressure
is isotropic  and does not vanish
at the boundary.
Hence, the pressure at the boundary
must be  compensated by a negative surface tension of the membrane.
We postpone this issue for section \ref{stability} where we discuss
the stability of the solution.

\section{Black holes at galactic centers}
\label{galactic}
Case {\bf ii)}, together with {\bf iii)}, is of particular
interest as we would like to interpret the supermassive
compact
dark
objects at the galactic centers in terms of
phantom energy rather than in terms of a classical black hole.
It is natural to
assume that the most massive such object is described by
the de Sitter gravastar, i.e., solution {\bf iii)}
(depicted by the solid line in Figs. 1, 2, and  3).
If we identify the most massive black-hole candidate
observed at
the center of M87, with mass
$M_{\rm max}=
3\times 10^9 M_{\odot}$, with the de Sitter gravastar,
then $A^{1/8}=9.7 {\rm keV}^4$, to be contrasted with
the $10^{-3} {\rm eV}$ values wanted for  cosmology
\cite{bil2}.
The radius of this object is $R_{\rm dS}$ equal to the Schwarzschild radius
 $2GM_{\rm max}$.
Clearly,  solutions belonging to class {\bf ii)}, can fit
all masses $M<M_{\rm max}$.
However, for the phenomenology  of supermassive
galactic centers
it is important to find, at least approximately,
the mass-radius relationship for these solutions.
This may be done in the low central density approximation, i.e.,
$\rho_0 \ll \sqrt{A}$,
which is similar to the Newtonian approximation but,
in contrast to the  Newtonian approximation, one cannot neglect the pressure  term
in Eq. (\ref{eq004}). Moreover,
 as may be easily shown, in this approximation
${\cal{M}} \ll r^3 p$, so that the pressure term becomes dominant.
Next, neglecting $2G \cal M$ with respect to $r$, as in the usual
Newtonian approximation,
 Eq. (\ref{eq008})
simplifies to
$
\rho' =
 4 \pi G A r $,
\label{eq201}
with the solution
\begin{equation}
\rho = \rho_0 \left(
1-\frac{r^2}{R_0^2}\right) \, ;
\;\;\;\;\;
R_0^2= \frac{\rho_0}{2 \pi G A} \, ,
\label{eq202}
\end{equation}
which gives a mass-radius relation
\begin{equation}
\frac{M}{R_0^5}= \frac{16\pi^2}{15}GA={\rm constant}.
\label{eq205}
\end{equation}

The mass-radius relationship
$M \propto R_0^{5}$ which phantom gravastars obey, offers the prospect
of unifying the description of all supermassive compact dark objects that have been
observed at the galactic centers, as Born-Infeld phantom gravastars
with masses ranging from
$M_{\rm min} = 10^{6}  M_{\odot}$ to
$M_{\rm max} = 3 \times 10^{9}  M_{\odot}$.
Indeed, assuming that the most massive compact dark object, observed at
the center of M87, is a Born-Infeld phantom gravastar near the
black-hole limit, with
$R_{\rm max} = 2GM_{\rm max} = 8.86 \times 10^{9} \mbox{km} = 8.21 \; \mbox{lhr}$,
the compact dark object at the center of our Galaxy, with mass
$M_{\rm GC} =  3 \times 10^{6} M_{\odot}$, would have a radius $R_{\rm GC}$ = 2.06 lhr
if the scaling law (\ref{eq205}) holds. This radius is well below the distances
of the closest approach to Sgr A$^{*}$ which the stars SO-2
($R_{\rm min}$ = 17 lhr = 123 AU, \cite{eck}) and SO-16 ($R_{\rm min}$ = 8.32 lhr = 60 AU \cite{ghez})
recently had and
beyond which the Keplerian nature of the gravitational potential of Sgr A$^{*}$
is well established.

\section{Dynamical stability}
\label{stability}
None of the solutions discussed in
the preceding section
will be stable unless there is a membrane,
e.g. in the form of a thin shell,
placed at the boundary with surface density
$\sigma$ and  surface tension $\theta$
satisfying Israel's junction conditions.
Following Visser and Wiltshire \cite{viss4}
we consider a dynamical thin shell
allowed to move radially
at the boundary of the phantom gravastar
  and discuss
under which conditions will a gravastar configuration will be stable.
A dynamical thin shell connecting
two general static
spherically symmetric spacetimes
have also  recently been considered \cite{lob1} in a slightly
different context.

Israel's junction conditions
read \cite{isra5}
\begin{equation}
\left[ \left[ K_{a}^{b} - \delta_{c}^{b}  K_{c}^{c} \right] \right]
= 8 \pi G  S_{a}^{b}\, ,
\label{eq009}
\end{equation}
where $S_{a}^{b}$ = diag ($\sigma,  \theta,  \theta$) is the
surface stress energy,
and $[[f]]$ denotes the discontinuity in $f$ across the boundary,
i.e.,
\begin{equation}
\left[ \left[ f (r) \right] \right] =
\lim_{\epsilon \rightarrow 0}  \,
\left( f (R + \epsilon) - f (R - \epsilon) \right)\, .
\end{equation}
 The tensor
  $K_{ab}$ is the extrinsic curvature
defined by
\begin{equation}
K_{ab}=h_{a}^{c}h_{b}^{d} n_{d;c}
\label{eq109}
\end{equation}
where $n_a$ is a spacelike unit vector orthogonal to the timelike boundary and $h_{ab}$ is
the induced metric on the shell.

The angular components of the extrinsic curvature
may be easily calculated from (\ref{eq109})  yielding
\begin{equation}
K_{\vartheta}^{\vartheta} = K_{\varphi}^{\varphi} = \frac{1}{r} \,
(\Delta+\dot R)^{1/2}  \, ,
\label{eq210}
\end{equation}
precisely as in \cite{viss4},
where the dot denotes the derivative with respect to the proper time
and
where
\begin{equation}
\Delta=
 1 - \frac{2 G {\cal{M}}}{r} \, .
\label{eq110}
\end{equation}
The calculation of the time-time component $K_t^t$ is
slightly more involved and is most easily done
following Israel \cite{isra5}.
By making use of the Gauss normal coordinates and orthogonality
from (\ref{eq109}) it follows
\begin{equation}
K_{\tau}^{\tau} =  u^au^b K_{ab} = -n_a u^b {u^a}_{;b}
\label{eq211}
\end{equation}
where $u^a$ is the four-velocity
of a point on the shell. Its non-vanishing components
 for the metric
(\ref{eq002}) are
\begin{equation}
 u^t = \frac{1}{\xi}\left(
 1+\frac{\dot{R}}{\Delta}
  \right)^{1/2};
  \;\;\;\;\;
  u^r= \dot{R}\, .
\label{eq212}
\end{equation}
Similarly, the non-vanishing components of $n_a$ are
\begin{equation}
n_t = \frac{\xi \dot{R}}{\Delta^{1/2}}\, ;
    \;\;\;\;\;
  n_r=\frac{(\Delta+ \dot{R}^2)^{1/2}}{\Delta}\, .
\label{eq213}
\end{equation}
Using these expressions it may be shown that the four-acceleration
$u^b {u^a}_{;b}$
satisfies \cite{isra5}
\begin{equation}
n_a u^b {u^a}_{;b}=
 -\frac{1}{(\Delta+ \dot{R}^2)^{1/2}}
(\ddot{R} +\Gamma^r_{ab} u^a u^b) .
\label{eq214}
\end{equation}
A straightforward calculation yields
\begin{equation}
K_{\tau}^{\tau} =
\frac{1}{(\Delta+ \dot{R}^2)^{1/2}}
\left[ \ddot{R}+\frac{\xi'}{\xi} (\Delta+ \dot{R}^2 )
+\left(\frac{\cal{M}}{r}\right)'\frac{\dot{R}^2}{\Delta}
\right].
\label{eq216}
\end{equation}

Then, from (\ref{eq009}) using (\ref{eq003}) and (\ref{eq004}) we find
\begin{equation}
4\pi\sigma =-\frac{1}{G}\left[\left[ \frac{\sqrt{\Delta+\dot{R}^2}}{R}
\right]\right] ,
 \label{eq301}
\end{equation}
\begin{equation}
4\pi (\sigma-2\theta)=
\left[\left[\frac{R^2\ddot{R}
+G{\cal{M}}-4\pi G R^3\rho
}{ G R^2\sqrt{\Delta+\dot{R}^2}}
+\frac{4\pi  R\sqrt{\Delta+\dot{R}^2}}{\Delta}\,(\rho+p)
\right]\right] .
\label{eq302}
\end{equation}
To derive the stability condition a la Visser and Wiltshire \cite{viss4},
it is now sufficient to replace $\dot{R}^2$
by $-2 V(R)$ and $\ddot{R}$ by $-V'(R)$ in equations (\ref{eq301}) and
(\ref{eq302}), where $V(R)$ is a potential.
The shell will be stable against small radial perturbation
if  there exists an equilibrium position $\bar{R}$ such that
 \begin{equation}
V(\bar{R})=0;
\;\;\;\;
V'(\bar{R})=0;
\;\;\;\;
V''(\bar{R})>0 .
\label{eq303}
\end{equation}
Then, by choosing a suitable potential $V(R)$, equations (\ref{eq301}) and
(\ref{eq302}) define a parametric
 equation of state $\theta=\theta(\sigma)$ for the shell.

For the static shell and the metric (\ref{eq002}) we find

\begin{equation}
\left( 1 - \frac{2 G M}{R} \right)^{1/2} -
\left( 1 - \frac{2 G {\cal{M}} (R)}{R}\right)^{1/2}
= - 4 \pi G  R  \sigma ,
\label{eq013}
\end{equation}
and with the help of Eq. (\ref{eq004}) we obtain
\begin{eqnarray}
M
\left(1 - \frac{2 G M}{R }\right)^{-1/2}
-
 \left({\cal{M}} (R)
-
\frac{4 \pi R^{3} A}{ \rho (R)}\right) \left( 1 -
\frac{2 G {\cal{M}}(R)}{R}\right)^{-1/2}
& &     \nonumber \\
=
4 \pi R^{2} ( \sigma - 2 \theta) ,
& &
\label{eq014}
\end{eqnarray}
where $M$ is the total mass.
Note that if the joining is affected
at the point $R_0$
where
$\xi = 0$ and $p$ is infinite
(the point of naked singularity \cite{cat}), the surface density $\sigma$
is finite, whereas the surface tension
$\theta \rightarrow - \infty$. Hence, to avoid the singularity
the shell must be placed at some $R < R_0$.

For the Newtonian gravastars discussed in the preceding section,
equations (\ref{eq301}) and
(\ref{eq302}) may be considerably simplified.
Using (\ref{eq202}),
 the approximation
$2G{\cal{M}}/R \ll 1$, and assuming that the boundary radius $R$ is close to
  $R_0$, i.e.,
\begin{equation}
y\equiv \frac{R_0-R}{R_0} \ll 1.
\label{eq00}
\end{equation}
 we find
\begin{equation}
\sigma=2\pi G A R_0^3 \frac{y^2}{\sqrt{1-2V}}
  \, ,
\label{eq05}
\end{equation}
\begin{equation}
\theta = -\frac{1}{8\pi G R_0}\frac{\sqrt{1-2V}}{y}
-4\pi G A R_0^3\frac{yV}{\sqrt{1-2V}}
+\pi G A R_0^4\frac{y^2 V'}{1-2V}
  \, .
\label{eq06}
\end{equation}
In general, the potential $V$ is  a function of both $R_0$ and $y$,
depending on the chosen equation of state $\theta=\theta(\sigma)$.
The dynamical stability is achieved if for any $R_0$, $V$ has a minimum
at some point $\bar{y}=1-\bar{R}/R$ which should not depend strongly on $R_0$
and at which $V(\bar{y})=0$.

Instead of postulating $\theta=\theta(\sigma)$ we can
choose a desirable potential and determine the equation of state
{\em a posteriori}.
Static stability (indifferent balance) is obtained by setting $V\equiv 0$.
Depending on the potential $V(R_0,y)$,  this may still be a  complicated parametric
equation of state. However,
for the static shell, i.e. $V\equiv 0$, the equation of state simplifies to
\begin{equation}
\theta =-\left(\frac{A R_0}{32\pi G\sigma}\right)^{1/2} .
\label{eq11}
\end{equation}
This equation  describes a 2-dimensional generalized anti-Chaplygin gas
\cite{gor}
with the equation of state of the form
$p=-\theta= A'/\sigma^{\alpha}$, with $\alpha=1/2$.

\section{Conclusions}
\label{conclude}
In conclusion, we have shown that replacing the de Sitter interior of the gravastar by a Born-Infeld phantom allows a wide range of gravastar mass and radii related to the central density, or equivalently to the velocity of the phantom scalar. We have demonstrated that if the constant $A$, as the only free parameter of the model, is fixed by assuming
that the most massive galactic center object
is the maximal (de Sitter) gravastar, then the model is able to explain all
 supermassive compact dark objects at the center of the galaxies.
Furthermore, as demonstrated above, the phantom gravastar model can lay claim to a connection with low-energy string theory.

\subsection*{Acknowledgment}
 The work of NB   was supported by
 the Ministry of Science and Technology of the
 Republic of Croatia under Contract
 No. 0098002 and partially supported through the
 Agreement between the Astrophysical sector, SISSA, and the
 Particle Physics and Cosmology Group, RBI.
Two of us (RDV and GBT) acknowledge grants from the South
African National Research Foundation (NRF GUN-2053794),
the Research Committee of the University of Cape Town,
and the Foundation for Fundamental Research (FFR PHY-99-01241).

\section*{References}