The unusual X-ray morphology of NGC~4636 revealed by deep Chandra observations: cavities and shocks created by past AGN outbursts
A. Baldi
We present Chandra ACIS-I and ACIS-S observations ($\sim$200~ks in total) of the X-ray luminous elliptical galaxy NGC 4636, located in the outskirts of the Virgo cluster. A soft band (0.5-2 keV) image shows the presence of a bright core in the center surrounded by an extended X-ray corona and two pronounced quasi-symmetric, 8 kpc long, arm-like features. Each of this features defines the rim of an ellipsoidal bubble. An additional bubble-like feature, whose northern rim is located $\sim2$~kpc south of the north-eastern arm, is detected as well. We present surface brightness and temperature profiles across the rims of the bubbles, showing that their edges are sharp and characterized by temperature jumps of about 20-25\%. Through a comparison of the observed profiles with theoretical shock models, we demonstrate that a scenario where the bubbles were produced by shocks, probably driven by energy deposited off-center by jets, is the most viable explanation to the X-ray morphology observed in the central part of NGC 4636.
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\begin{document}

\title{The unusual X-ray morphology of NGC~4636 revealed by deep Chandra observations: cavities and shocks created
by past AGN outbursts}

\classification{95.85.Nv -- 98.52.Lp -- 98.54.Cm}
\keywords      {}

\author{A. Baldi}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA
  02138}
  ,altaddress={now at INAF-Osservatorio Astronomico di Bologna, Via Ranzani 1, 
  I-40127 Bologna, Italy} % additional visiting address
}

\author{W. Forman}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138}
}

\author{C. Jones}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138}
}

\author{R. Kraft}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138}
}

\author{P. Nulsen}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138}
}

\author{L. David}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138}
}

\author{S. Giacintucci}{
  address={Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138}
}

\author{E. Churazov}{
  address={Max Planck Institut f\"ur Astrophysik, Karl-Schwarzschild-Str. 1, 85741
  Garching, Germany}
}


\begin{abstract}
We present Chandra ACIS-I and ACIS-S observations ($\sim$200~ks in total) of the X-ray luminous 
elliptical 
galaxy NGC 4636, located in the outskirts of the Virgo cluster.
A soft band (0.5-2 keV) image shows the presence of a bright core in the center surrounded by an 
extended 
X-ray corona and two pronounced quasi-symmetric, 8 kpc long, arm-like features. Each of this 
features defines the rim
of an ellipsoidal bubble. An additional bubble-like feature, whose northern rim is located 
$\sim2$~kpc south of the 
north-eastern arm, is detected as well.
We present surface brightness and temperature profiles across the rims of the bubbles, showing 
that their edges are sharp 
and characterized by temperature jumps of about 20-25\%. 
Through a comparison of the observed profiles with theoretical shock models, we demonstrate that 
a scenario 
where the bubbles were produced by shocks, probably driven by energy deposited off-center by jets, 
is the most 
viable explanation to the X-ray morphology observed in the central part of NGC 4636.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MAINMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

It is well known that active galactic nuclei (AGN) play an important role in the
evolution of the hot gas in both individual galaxies and clusters of galaxies
\citep[e.g. ][]{churazov00,bruggen02,cavaliere02,hoeft04,forman05,mcnamara07}.
The so-called `AGN feedback' can cause re-heating of the central cooling regions
of a cluster and balance the cooling due to the X-ray emission \citep[see][for a review]{mcnamara07}.
AGN are also effective in shaping the morphology of the hot gas halos around individual
galaxies and clusters, e.g. giving rise to cavities and bubbles \citep{birzan04}.\\
While several examples of subsonic bubble inflation due to AGN outbursts in individual galaxies are present
in the literature \citep[e.g. NGC~507, M~84; ][]{kraft04,finoguenov08}, supersonic bubble expansion,
giving rise to shocks in the hot X-ray emitting gas, has only been observed in a handful of cases
\citep[e.g. M~87, NGC~4552; ][]{forman05,machachek06}.
Another possible case of supersonic bubble expansion could be that of NGC 4636, 
the dominant galaxy of a group on the outskirts of the Virgo Cluster 
\citep[10$^\circ$ or 2.6 Mpc on the sky to the south of M87, at a distance to NGC~4636 of 15~Mpc;][]{tonry01}.
%The optical surface brightness profile of NGC 4636 flattens in the inner regions
%as expected for a luminous, slowly rotating elliptical galaxy. As observed by Hubble Space
%Telescope exposures, the central region has low eccentricity and this galaxy is thus 
%classified as an E0, although, at low surface brightness, 
%the galaxy is flattened \citep[E4;][]{sandage61}. The H$\alpha$ luminosity is measured to be 
%$2.3\times10^{38}$~ergs s$^{-1}$ \citep{ho97}.
%This galaxy has been observed extensively 
%by every major X-ray imaging observatory to date. 
%Spectral imaging with Einstein showed the galaxy 
%to be surrounded by an extensive halo of hot gas with a temperature 
%of 0.78~keV \citep{forman85}. 
%Further studies using ROSAT and ASCA found gradients in both the temperature and abundance 
%profiles, along with a very extended halo
%\citep{awaki94,trinchieri94,matsushita97,finoguenov00,buote00,osullivan03}. 
%The metallicity is about solar in the central 9~kpc ($2^\prime$) region, decreasing to 
%0.2-0.3 solar at $\sim45$~kpc ($10^\prime$) from the galaxy center \citep{matsushita97}.
%Although the galaxy is located in the outskirts of the Virgo Cluster, X-ray 
%emission associated with Virgo has not been observed around NGC 4636.
%More recently, NGC~4636 has been 
%observed by both Chandra and XMM-Newton. The initial Chandra ACIS-S 
%observation showed an unusual morphology in the core of the galaxy, 
%most notably quasi-symmetric spiral arm-like features and a possible ring of 
%high surface brightness emission, thought to be the product of shocks 
%driven by a previous AGN outburst 
%\citep{jones02}.
%As a confirmation to the outburst driven shock scenario, a change 
%of the gas temperature across one of the arms was 
%observed as well. 
%From the same Chandra data, \citet{ohto03} performed a detailed spectral 
%analysis, suggesting 
%the presence of a cavity in the X-ray halo on the west side of the core, with
%AGN activity being the most likely cause also in this case. In this scenario, plasma from 
%the AGN jets may be able to displace the gas of the X-ray halo, leaving 
%an apparent void as currently observed in the X-ray morphology.
%Using data from XMM-Newton and Chandra, \citet{osullivan05} mapped the 
%temperature and abundance structure of the inner halo of NGC 4636, confirming the 
%detection of a cavity to the west of the galaxy and revealing the presence of 
%a plume of cool metal-rich gas extending beyond the cavity to the southwest. 
%Both cavity and plume appear to be the product of past AGN activity. Moreover, their
%abundance determination shows a depression in the central few kpc ($\sim0.2$ solar)
%in disagreement with the results of \citet{matsushita97}.   
%An upper limit of $2.7\times10^{38}$~ergs s$^{-1}$ is 
%estimated for the present X-ray emission of the AGN \citep{loewenstein01}.\\
%
%In this paper we take advantage of the long Chandra ACIS-I
%observations ($\sim150$~ks) combined with the previously observed ACIS-S 
%exposure ($\sim$40~ks) to analyze in more detail the features revealed by the previous
%Chandra and XMM-Newton observations. 
%In particular we are able to study in more detail the morphology and
%temperature structure in the observed cavities and look for gradients in temperature
%and abundance to test the shock scenario invoked for their formation
%\citep{jones02,ohto03,osullivan05}. Moreover we will investigate with unprecedented detail 
%the morphology and the properties of the hot gas in the galaxy core.

%At the distance of NGC~4636 \citep[15~Mpc;][]{tonry01},
%$1^{\prime\prime}\approx70$~pc. Throughout the paper, 
%all the uncertainties are quoted at 1$\sigma$ (68\%) for one
%interesting parameter, unless otherwise stated.
%The abundance estimates are relative to the
%cosmic values given in \citet{anders89}.

\section{Chandra Data Preparation and X-ray Morphology} \label{dataprep}

NGC~4636 was observed twice by ACIS-S and twice by ACIS-I.
In our analysis, we use the longer ACIS-S observation (performed on 2000, January 26; ObsID: 323) 
and both the ACIS-I observations, performed one immediately after the other 
on 2003, February 14 (ObsID: 3926) and on 2003, February 15 (ObsiD: 4415).
Details on data preparation and analysis can be found in \citet{baldi09}.
%\section{X-ray morphology and surface brightness}\label{radanal}
Fig.~\ref{xrayimage}a shows the merged 0.3-2~keV Chandra image of all three observations.
The galaxy presents a very bright central core of radius $\sim1~kpc$ (15$^{\prime\prime}$) centered on
the nucleus.
A lower surface brightness region surrounds the nucleus extending out to $\sim6$~kpc ($80^{\prime\prime}$).
Two quasi-symmetric arm-like features are embedded in this lower surface brightness emission. These features were 
previously reported by \citet{jones02} who analyzed the shorter ACIS-S observation. However, combining
these data with the two ACIS-I observations we are able to observe that the south-western arm is clearly
part of an X-ray cavity extending as far as $\sim9$~kpc ($2^{\prime}$) from the center. 
The hint of a similar structure can 
be observed in coincidence with the north-eastern arm-like feature. It is not clear from the X-ray
image whether there is a cavity in this case.
The cavities are more evident if we remove the contribution from the general diffuse
emission of the galaxy. Fig.~\ref{xrayimage}b shows the merged Chandra image in the 0.3-2 keV band where
a $\beta$-model fitted to the galaxy diffuse emission was subtracted. This processing allowed to highlight
fainter structures.
\begin{figure}
%\epsscale{.80}
\includegraphics[width=6cm]{f6}
\includegraphics[width=6cm]{f1b}
\caption{{\it Left:} Chandra ACIS-I+ACIS-S image of NGC~4636 in the 0.3-2 keV band. The three bubble-like features
detected are labelled and identified by cyan ellipses. The yellow arrows are pointing toward 
the detected rims of the bubbles.
{\it Right:} Chandra ACIS-I+ACIS-S image after the subtraction of a $\beta$-model fitted to
the general diffuse X-ray emission. 
\label{xrayimage}}
\end{figure}
%
%%Indeed, both the south-western and the north-eastern cavities are clearly visible in this Figure.
%Moreover two additional quasi-symmetric cavities can be seen, just to the East of the nucleus and to
%the North-West of it, with the rims of the former being better defined than the latter.\\ 
%The radio data overlaid on the X-ray image (Fig.~\ref{xrayradio}) shows a correlation with the cavities.
%However, the radio lobes do not fill the SW cavity, or break through the cavity boundaries
%as in other galaxies presenting cavities \citep[e.g. M~84,][]{finoguenov08}. However, the lobes
%detected at 610~MHz (Giacintucci et al. in prep.) with the Giant Metrewave Radio Telescope (GMRT) extend 
%further than those 
%observed at 1.4~GHz and also the NE cavity seems to be
%%filled by plasma detected at 235~MHz with the GMRT (Giacintucci et al. in prep.). The
%radio lobes are weak, having a luminosity at 1.4~GHz of $\sim1.4\times10^{38}$ erg s$^{-1}$.
%However the plasma creating
%the cavities remains partially undetected by the radio observations of NGC~4636 performed so far.

%Figure~\ref{hardsoft} shows Chandra images of NGC~4636 in two different bands: 0.3-0.9 keV and 0.9-1.3 keV.
%A $2^{\prime\prime}$ width gaussian smoothing was applied to them to enhance the features visible in the two
%images.
%The morphology is quite different between them, with the X-ray arm-like features becoming prominent only at 
%energies larger then 0.9 keV, a possible indication that the temperature in these features is higher than
%in the surrounding hot interstellar medium of the galaxy. 


%\subsection{Global X-ray properties}\label{global}
%Although the morphology of the X-ray emission around the nucleus of NGC~4636 is extremely complex, it could
%be useful to study its average radial properties.
%The surface brightness profile of NGC~4636 is shown in Figure~\ref{sbprof}. The emission does not show a central
%peak and is quite flat at the center, then it starts to decline monotonically at $\sim5^{\prime\prime}$ ($0.35$~kpc) from
%the center, showing
%another `plateau' at $25^{\prime\prime}<r<60^{\prime\prime}$ ($1.75$~kpc$<r<4.2$~kpc). Both these features can be noticed 
%looking at the
%Chandra image (Fig.~\ref{xrayimage}a).
%To study the radial spectral properties, we subdivided the emission from the galaxy into 25 annuli centered on the
%X-ray peak.
%We extracted a spectrum from each annulus (excluding the point sources) using
%$specextract$, which generates source and background spectra building appropriate RMFs and ARFs. 
%The background is taken from the re-normalized blank field observations using the same region as the 
%source.
%The spectra were analyzed using XSPEC v12 \citep{arnaud96} and fitted with a 
%single-temperature {\tt APEC} model \citep{smith01}, combined
%with the XSPEC {\tt projct} deprojection model.
% in which the 
%ration
%between the elements is fixed to the solar value as in Anders \& Grevesse (1989). 
%However, as explained in \S1, these
%values for the solar metallicities have more recently been superseeded by the new values by
%Grevesse \& Sauval (1998) and A05. For clarity and completeness, we 
%also performed the fits using solar abundances by A05.
%The free parameters in the model are 
%the gas temperature $kT$, the gas metallicity $Z$ and the normalization. 
%The spectral band considered is 0.5-4 keV. 
%We choose not to consider the data below 0.5 keV because of 
%uncertainties
%in the ACIS calibration below that energy.
%All the spectra were rebinned to have at least 20 counts per bin.
% and to be 
%able to use the $\chi^2$ statistics. 
%Although because of our low energy cut at 0.6 keV we become less sensitive in measuring the
%$N_H$, we avoid to introduce systematics due to the uncertainties in the calibration 
%of the instrument at soft energies. 
%We have measured $N_H$ from the X-ray data, finding it consistent (within 1$\sigma$) 
%with the Galactic value
%along the line of sight, as derived from radio data 
%\citep[$1.9\times10^{20}$ cm$^{-2}$;][]{stark92}.
%The de-projected temperature profiles of NGC~4636 
%are shown in Figure~\ref{ktz}a. No abundance profile trend was detected, we therefore decided to fix the
%abundance at $Z=0.5$ times solar. This is in agreement with the findings of \citet{osullivan05}, who
%also found no trend in $Z(r)$ for the azimuthally-averaged data. Interestingly, they did however find 
%a strong
%azimuthal abundance variation.  The temperature shows clear evidence for a decline in the
%center where $kT\sim0.5$~keV, and increases toward the outskirts of the galaxy ($kT\ge 0.8$~keV). 
%Several substructures, due to the
%complex X-ray morphology of the galaxy, are observed.
%The electron density profile derived from the deprojection (Fig.~\ref{ktz}b) shows instead a strong central
%peak and a `plateau' coincident with the one observed in the surface brightness profile located at 
%2~kpc$\le r\le$4.5~kpc from the center.

%The radial profile includes also the arm-like structures observed
%right outside the core of the galaxy. In principle, this could influence the general trend in the X-ray properties
%observed introducing a possible bias.
%To avoid this problem we performed a deprojection of the X-ray emission SE to the galaxy center
%($185^\circ$-$235^\circ$), where no X-ray structure
%is present, thus avoiding the complications introduced by the presence of the cavities.
%In this deprojection analysis, we follow the approach used in \citet{churazov03} and described
%in more detail in \citet{churazov08}. Spherical symmetry is assumed, but no specific 
%assumption about the form of the underlying gravitational potential is made. 
%The procedure differs slightly from the often sed onion peeling algorithm or 
%\verb#projct# in XSPEC \citep{arnaud96}. It yields however consistent results and
%is particularly useful in the case a large number of annuli is used.
%The resulting spectra were fitted in XSPEC v12 \citep{arnaud96} with a
%single-temperature APEC model \citep{smith01} and absorption fixed at
%the Galactic value $N_H=1.9 \times 10^{20}~{\rm
%cm}^{-2}$ \citep{stark92}. The gas temperature and
%normalization were free parameters of the model, while the metal abundance
%was fixed at 0.5 solar. 
%The electron density was derived from the normalization of the
%spectra, fixing the proton to electron ratio to $0.83$ and taking into
%account the distance to NGC~4636. The resulting temperature and electron density
%are plotted in Figure~\ref{deprojSE} as a function of distance from the center of
%NGC~4636. Red and blue points correspond to the deprojected and projected values,
%respectively. The deprojected temperature shows a central value of $kT=0.55$~keV,
%showing a decrease at $r>0.3^\prime$ ($\sim1.2$~kpc), down to a value of $\sim0.35$~keV
%at $0.7^\prime$ ($\sim2.9$~kpc) from the center. At $r>0.8^\prime$ ($\sim3.4$~kpc)
%the temperature grows monotonically reaching a value of 0.9~keV at $6.5^\prime$ 
%($\sim27$~kpc) from the center.
%The deprojected electron density has a behaviour similar to the one observed for the azimuthally averaged
%radial analysis, 
%showing a `plateau' at $0.5^{\prime}<r<1.4^{\prime}$ ($2.1$~kpc$<r<5.9$~kpc), shifted at larger
%radii than the one observed in the azimuthally averaged profile. 

%\clearpage

%\clearpage

\section{The X-ray Bubbles}
The most prominent features observed in the Chandra image are the two quasi symmetric 8~kpc
long X-ray arm-like structures, already observed by \citet{jones02} and by \citet{osullivan05}. The SW arm (SW1) 
is however part of a cavity
which extends at least $\sim5$~kpc in radius to the North.
We performed a spectral analysis in a strip perpendicular to the SW arm-like feature, dividing
the strip into rectangles $\sim0.5^\prime$ wide, 
to look for variations in temperature or metal abundance. 
Although the metallicity does not vary significantly
across the bubble rim, a sharp variation in the temperature was detected coincident with SW1 
(Fig.~\ref{SWshock}) 
showing a temperature decrease from $kT\sim0.75$~keV to $kT\sim0.64$~keV, well above
the measurement errors ($\sim0.01$~keV). A temperature jump was not observed across SW2 most likely because
of the complicated geometry of the X-ray emission. Indeed, SW2 seems to be partially embedded in another
bubble-like feature located just North of it.\\
The symmetry of the X-ray arm-like features is highly suggestive of the presence of a symmetric bubble NE of the nucleus
of NGC~4636. However the southern rim of the bubble is not clearly visible and it looks instead to be embedded
in another round shaped bubble located to the East of the nucleus. If we examine the surface brightness profile of the
NE cavity we find a shape which is very similar to the SW bubble. Performing a spectral analysis across the
northern cavity rim NE1, 
we also observe a 20\% temperature jump across the rim. However,
the scenario in this part of the galaxy is more complex because of the presence of an additional feature just East of the
nucleus with the shape of another cavity. This cavity looks less elongated than the other two cavities observed.
However, the surface brightness profile has a shape similar to the one across the NE cavity, while the temperature profile shows
a temperature jump similar to the one observed in the other two cavities.

%\begin{figure}
%%\epsscale{.80}
%\plottwo{kTprof_SE_deproj.ps}{neprof_SE_deproj.ps}
%\caption{{\it Left:} Deprojected temperature radial profile along a wedge located SE of the nucleus of NGC~4636. 
%The direction of the wedge was chosen to avoid any X-ray feature such as cavities or bubbles. {\it Right:} 
%Density radial profile along the SE wedge.\label{SEradial}}
%\end{figure}
\begin{figure}
%\epsscale{.80}
\includegraphics[width=6cm]{f8a}
\includegraphics[width=6cm]{f8b}
\caption{{\it Left:} Surface brightness profile perpendicular to the shock front for the SW bubble. The three
colored solid lines represents the prediction from a numerical hydro-dynamical shock model at different
Mach numbers for the shock. The best fit shock model to the observed data has $M=1.72$. {\it Right:} 
Temperature profile across the southern rim of the SW bubble. The temperature jump is consistent with
the predictions of a shock model with $M=1.72$.\label{SWshock}}
\end{figure}

\subsection{The origin of the cavities: a simple shock model}
The most likely scenario for the origin of the cavities observed in the X-ray morphology of NGC~4636
is that they were the result of successive outbursts of the central AGN.
% similarly to what happened in the 
%case of the elliptical galaxy M~84 \citep{finoguenov08}.
In this scenario the jet propagated rapidly from the center, creating a long thin cavity which then inflated in
all directions. Perpendicular to the axis of the jet, the expansion has an approximate cylindrical symmetry.
The expanding cavities drive shocks into the surrounding gas.

A 1-dimensional, cylindrically symmetric, time-dependent hydrodynamic
model was used to investigate the properties of the shocks for the
SW bubble.
In this model the sound speed in the relativistic gas that fills the cavity (the piston that drives the shock) 
is assumed very high, keeping the pressure in the piston nearly uniform. The ratio of the pre-shock
pressure to the post-shock pressure determines the strength of the shock. As a result, the shock is weakest 
(slowest) in the region close to the AGN and fastest in the region farthest from the AGN. 
Further details on the shock model can be found in \citet{baldi09}.
%The
%pressure required to drive the shock observed at the middle of the cavity now is lower than the ambient 
%pressure close to the AGN, so the piston can no longer drive the shock towards the AGN. Indeed,
%the end of the cavity close to the AGN is being pushed outward, as denser gas falls inward and displaces it.

%The undisturbed gas was assumed to be isothermal, with electron
%density $n_{\rm e,gas} \propto r^{-\eta}$, where $r$ is the
%distance to the AGN.  Corresponding to this, the surface brightness
%profile of the undisturbed gas has the form, ${\rm SB} \propto
%\rho^{1 - 2 \eta}$, where $\rho$ is the projected distance from the
%AGN.  The power law index $\eta$ was determined by fitting the SB
%profile in the region adjacent to the SW arm, outside the shock,
%giving $\eta = 1.3$.  Taking a cut perpendicular to the radius at a
%distance, $d$, from the AGN, gives the electron density profile
%$n_{\rm e,gas} \propto (d^2 + \varpi^2)^{- \eta/2}$, where $\varpi$
%is the cylindrical radius measured from the axis of the lobe.
%Using the distance, $d$, corresponding to $1.2'$, this was the form
%adopt for the density profile of the undisturbed gas in the
%cylindrical flow model.  The initial gas distribution was made
%hydrostatic by imposing a static gravitational field on the model.
%Shocks were initiated by depositing a large amount of energy on the
%symmetry axis and their evolution was followed using the
%cylindrically symmetric hydrodynamic code until the shock had
%expanded to the observed size. Surface brightness profiles for
%the model shocks were determined by projecting the axially symmetric
%models onto the sky, using the density and temperature dependent 0.3
%-- 1.8 keV ACIS-I response.  The temperature scale was
%determined by assuming a preshock temperature of 0.65 keV.  The left
%panel of Fig.~\ref{SWshock} shows the surface brightness profile of
%the SW shock, perpendicular to the radius, in the region shown in
%Fig.~\ref{xrayspecreg}, together with model surface brightness
%profiles for Mach numbers of 1.42, 1.72 and 2.04 (computed
%assuming $\gamma = 5/3$ in the gas). Based on its
%surface brightness profile in the vicinity of the shocks, the Mach
%1.72 shock model provides the best fit.

%The temperature jump expected for a Mach 1.72 shock is a factor of
%1.74.  For a preshock temperature of 0.65 keV, that would make the
%postshock temperature $\simeq 1.13$ keV.  However, as observed, gas
%behind the cylindrical shock is projected onto the surrounding
%unshocked gas.  The shocked gas also expands adiabatically behind the
%shock, reducing its average temperature.  As a result, the observed
%temperature profile shows a considerably lower temperature jump than
%expected for a Mach 1.72 shock.  Complicating matters further, fitted
%temperatures for multiphase gas are not readily determined from the
%emission measure distribution \citep{vikhlinin06}.  In view of this,
%projected emission measure distributions were determined for the Mach
%1.72 model, assuming a preshock temperature of 0.65 keV.  These were
%used to construct multiphase spectral models in XSPEC and the
%resulting models were fitted to a single temperature model 
%to determine the observed temperature
%profile corresponding to the Mach 1.72 shock model.  The results are
%plotted as stars together with the observed temperature profile in the
%right panel of Fig.~\ref{SWshock}.  Within the errors, both the
%temperature and density profiles of this region are consistent with
%the Mach $\simeq1.7$ shock model.

The calculations of the physical parameters derived from the model were performed for the SW bubble.
The three cavities present similar temperature jumps and surface brightness profiles, so similar
physical parameters are expected.
%The shock strength was determined by matching the observed surface brightness perpendiculary to the cavity rims with the projected
%model (Fig.~\ref{SWshock}a). A Mach number $M\sim1.7$ is found to be the the best match for the observed surface brightness.
%The prediction of a shock model having the aforementioned Mach number is consistent also with the temperature jump 
%observed across the southern boundary (Fig.~\ref{SWshock}b).
From the hydrodynamic model, the age of the shock, $t\sim2\times10^{6}$~yrs, is the time it takes
to expand to its observed size. The age is only sensitive
to the shock strength and the temperature of the unshocked gas. This
age is slightly shorter than the ratio of the shock radius to the
present shock speed, because in the model the shock strength decreases
with time (i.e. the shock expanded faster when it was younger).
The total energy which produced the shock was $10^{56}$~ergs, roughly equal to the enthalpy, 
$H=4pV=7\times10^{55}$~ergs, calculated in the assumption that the bubble is predominantly relativistic 
($\gamma=4/3$). The average mechanical power required to produce the bubble equals to 
$P_{mech}\sim1.6\times10^{42}$~erg s$^{-1}$.
%The age of the cavity is shorter than what is usually observed in the so-called 'ghost' cavities
%where no evident 1.4~GHz radio emission is observed to fill them. However, the cavities in NGC~4636 are not
%really 'ghost' cavities and a weak radio luminosity is indeed expected from the mechanical power derived above.
%\citet{bir08} found indeed a correlation (although with a large scatter) between the cavity (jet) power $P_{jet}$ and 
%the radio synchrotron power $L_{radio}$, of the form $P_{jet}\sim L_{radio}^{\beta}$.
%%, where $0.35\leq \beta\leq 0.70$. 
%The cavities observed in NGC~4636 fall in the lower-left corner of the diagram in Figure 5 of \citet{bir08}, with
%a ratio $P_{jet}/ L_{radio}\sim10^4$, similar to a few objects in the plot presenting a radio luminosity
%four order of magnitudes brighter than NGC~4636. 
%There is a large scatter in the correlation, that is not well understood. 
%However, the low radio luminosity observed in NGC~4636 is consistent with expectations based on this sample.
%If we compare the mechanical power obtained for the cavities with the radio luminosity of the jets creating the cavities
%($L_{1.4\,GHz}\sim1.4\times10^{38}$ erg s$^{-1}$), we observe that the weakness of the radio lobes is just due to a
%scaling factor respect to larger objects such as clusters and groups. In Figure 5 of \citet{bir08} the cavities of NGC~4636
%occupies the lower-left end of the diagram, ho
%\begin{figure}
%\epsscale{.80}
%\plotone{xrayhard.ps}
%\caption{Hard band (2-7 keV) Chandra ACIS-I+ACIS-S image of the core region of NGC~4636. The position of the central 
%point-like radio source is indicated as a white circle. 
%Several point sources are detected and one of the point sources is coincident with the radio position.
%\label{xrayhard}}
%\end{figure}

%\subsection{The nuclear point source}%
%
%The X-ray morphology of NGC~4636 is characterized by the presence of a dense core having a radius 
%of $\sim1$~kpc.
%A hard band image of the core (2-7 keV band) reveals the presence of 
%several point sources. An X-ray point source is
%coincident with the radio nucleus (within 0.5$^{\prime\prime}$) and may be X-ray emission from the
%supermassive black hole (SMBH) at the center of the galaxy. We extracted a spectrum of this point source
%fitting a power-law model to it in XSPEC. The spectrum has $\sim230$ counts (ACIS-S plus ACIS-I) in total
%and the best fit model gives a power-law index $\Gamma=2.5_{-0.5}^{+0.4}$ and
%a flux of 2.3$\pm0.4\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$ and of 6.0$\pm1.0\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$ in the
%2-10 keV and in the 0.5-10 keV bands, respectively. This corresponds to a total luminosity of 
%1.6$\pm0.3\times10^{38}$ erg s$^{-1}$ in the 0.5-10 keV band. This could be the current luminosity of the AGN at the
%center of NGC~4636. However, given the low luminosity, the X-ray emission
%could be also due to a low mass X-ray binary coincident with the radio nucleus. In the latter case, 
%this value can be considered 
%as a more stringent upper limit to the X-ray luminosity of the AGN at the center of the galaxy than
%the one set by \citet{loewenstein01}.%

%To derive an estimate of the mass supply rate for accretion on the SMBH for NGC~4636, we can
%apply the \citet{bondi52} theory of steady, spherical and adiabatic accretion. This requires $T$
%and $n$ at ``infinity'', in practice near the accretion radius (that
%in the case of NGC~4636 is $r_A=GM_{BH}/c_s^2\sim 8$ pc, where $c_s$ is
%the speed of sound).  Unfortunately the spatial resolution does not allow us to measure the 
%temperature and the density at such a small radius. However, we can estimate
%the accretion rate, using the temperature and the density profiles measured in
%in our radial analysis (\S~\ref{global}).
%The accretion rate is given by the formula
%$$
%\dot{M}_{Bondi} = 8.4 \times 10^{21} \;
%M_8^2\;\;T_{0.5}^{-3/2} \;n_1 \; \hbox{$\thinspace 
%g\; s^{-1}$}, 
%$$
%where $M_8$ is the SMBH mass in units of $10^8\; M_{\odot}$ \citep[for
%NGC~4636 $M_{BH}\simeq7.9\times10^7\; M_{\odot}$,][]{merritt01}, $T_{0.5}$ is the
%temperature in units of 0.5 keV and $n_1$ is the density in units
%of 1 cm$^{-3}$.
%% [see, e.g., eq.  (3) of \citet{dim03} ].
%Assuming that the density $n\propto r^{-1}$, from the thermal component normalization
%of the spectrum in the inner
%5$^{\prime\prime}$ ($\sim0.35$~kpc), we obtain that the density at the accretion radius is
%$n\sim5.5$~cm$^{-3}$. The temperature profile was instead fitted with a second order
%%Using this values of $n$ and $kT$, we find that
%$\dot{M}_{Bondi}\simeq 4.7\times10^{-4} M_{\odot}$ yr$^{-1}$. 
%If at very small radius this gas joins an accretion
%disc with a standard radiative efficiency $\eta \sim 0.1$, as in
%brighter AGNs, it should produce a luminosity:
%$$
%L_{acc}=\eta \dot{M}_{Bondi}
%c^2 \simeq 2.7 \times 10^{42} \hbox { erg s$^{-1}$},
%$$
%at least four orders of magnitude higher than the
%observed X-ray luminosity, pointing toward a highly inefficient accretion scenario. \\%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The bibliography can be prepared using the BibTeX program or
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%%
%% The code below assumes that BibTeX is used.  If the bibliography is
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%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{alesbib.bib}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Just a reminder that you may have to run bibtex
%% All of it up to \end{document} can be removed
%% if you don't like the warning.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\IfFileExists{\jobname.bbl}{}
 {\typeout{}
  \typeout{******************************************}
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 }

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The following lines show an example how to produce a bibliography
%% without the help of the BibTeX program. This could be used instead
%% of the above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{9}

\bibitem{Brown2000}
M.~P. Brown,  and K.~Austin, \emph{The New Physique}, Publisher Name,
  Publisher City, 2000, pp. 212--213.

\bibitem{BrownAustin:2000}
M.~P. Brown,  and K.~Austin, \emph{Appl. Phys. Letters} \textbf{85},
  2503--2504 (2000).

\bibitem{Wang}
R.~Wang, ``Title of Chapter,'' in \emph{Classic Physiques}, edited by
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\bibitem{SJ:1999}
C.~D.~Smith and E.~F.~Jones,  ``Load-Cycling in Cubic Press,'' in
  \emph{Shock Compression of Condensed Matter-1999}, edited by M.~D.~F. et~al.,
  AIP Conference Proceedings 505, American Institute of Physics, New York,
  1999, pp. 651--654.

\end{thebibliography}

\endinput
%%
%% End of file `template-6s.tex'.

baldi_a.bbl
\begin{thebibliography}{14}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\providecommand{\enquote}[1]{``#1''}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{Churazov} et~al.(2000)]{churazov00}
E.~{Churazov}, W.~{Forman}, C.~{Jones}, and H.~{B{\"o}hringer}, \emph{A\&A}
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\bibitem[{Br{\"u}ggen} and {Kaiser}(2002)]{bruggen02}
M.~{Br{\"u}ggen}, and C.~R. {Kaiser}, \emph{Nature} \textbf{418}, 301--303
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\bibitem[{Cavaliere} et~al.(2002)]{cavaliere02}
A.~{Cavaliere}, A.~{Lapi}, and N.~{Menci}, \emph{ApJl} \textbf{581}, L1--L4
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\bibitem[{Hoeft} and {Br{\"u}ggen}(2004)]{hoeft04}
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\bibitem[{Forman} et~al.(2005)]{forman05}
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\bibitem[{McNamara} and {Nulsen}(2007)]{mcnamara07}
B.~R. {McNamara}, and P.~E.~J. {Nulsen}, \emph{ARA\&A} \textbf{45}, 117--175
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\bibitem[{B{\^i}rzan} et~al.(2004)]{birzan04}
L.~{B{\^i}rzan}, D.~A. {Rafferty}, B.~R. {McNamara}, M.~W. {Wise}, and P.~E.~J.
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\bibitem[{Kraft} et~al.(2004)]{kraft04}
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\bibitem[{Finoguenov} et~al.(2008)]{finoguenov08}
A.~{Finoguenov}, M.~{Ruszkowski}, C.~{Jones}, M.~{Brueggen}, A.~{Vikhlinin},
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\bibitem[{Machacek} et~al.(2006)]{machachek06}
M.~{Machacek}, P.~E.~J. {Nulsen}, C.~{Jones}, and W.~R. {Forman}, \emph{ApJ}
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\bibitem[{Tonry} et~al.(2001)]{tonry01}
J.~L. {Tonry}, A.~{Dressler}, J.~P. {Blakeslee}, E.~A. {Ajhar}, A.~B.
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\end{thebibliography}
