\section{Introduction}

There is strong evidence that all massive galaxies
(not including dwarfs) with a spheroidal component
(ellipticals, lenticulars, and early-type
spirals with bulges) host supermassive black holes (SMBHs)
in their nuclei (Richstone et al.~1998).
For a few dozen galaxies, there exist direct
determinations of their nuclear masses, either from measurements
of spatially-resolved stellar or gas kinematics (for quiescent
nuclei) or from "reverberation mapping" of their broad-line regions
(for active nuclei) (Gebhardt et al.~2003; Ferrarese et al.~2001; Ho 2002;
Onken et al.~2004; Peterson et al.~2004). The masses of SMBHs correlate
well with fundamental properties of their host galaxies:
bulge mass (Magorrian et al.~1998), central velocity dispersion
(Ferrarese \& Merritt 2000, Gebhardt et al. 2000),
central light concentration (Graham et al.~2001) and
mass of the dark-matter halo (Ferrarese~2002).
However, this is not the case for the accreting luminosities
of SMBHs: while a few are active (AGN), sometimes with
bolometric luminosities close to or above their Eddington limits
(Collin et al.~2002; Wang \& Netzer 2003), the majority are not.
X-ray observations of these ``silent'' SMBHs
have revealed faint sources with X-ray
luminosities as low as $\sim 10^{-8} L_{\rm Edd}$
(Fabbiano \& Juda 1997; Ho 1999; Ho et al.~2001;
Pellegrini et al.~2003a; Pellegrini 2005;
Fabbiano et al.~2003, 2004).

The bolometric luminosity can be expressed as $L_{\rm bol}
\approx \eta \dot{M}
c^2$, where $\dot{M}$ is the mass accretion rate
and $\eta$ is the efficiency of conversion of gravitational energy
into radiation. It is still not clear whether the extremely
low luminosity of most SMBHs is primarily due to
a low accretion rate, or to a low radiative efficiency
(i.e., $\eta \ll 0.1$); see Narayan (2002)
for a review. If it is due to
a low accretion rate, we need to determine whether
this is caused by a lack of gas available in the inner
few pc, or is instead the result of other phenomena
(for example accretion disk winds or jets) that may prevent
the gas from accreting onto the SMBH.

The most likely explanation appears to be that both low $\eta$
and low $\dot{M}$ play a part.
It has become clear that a lack of fuel
cannot be a general explanation: many elliptical
galaxies do have large supplies of cold/warm interstellar gas
at their centers (Caon, Macchetto \& Pastoriza 2000;
Macchetto et al.~1996).
Fabian \& Canizares (1988) pointed out that the accretion
rate of the hot interstellar medium (ISM) expected
from the standard Bondi (1952) model would still be high enough
to produce X-ray luminosities at least two orders of magnitude
higher than observed in the nuclei of
most ellipticals, if accretion occurs efficiently ($\eta \sim 0.1$).
Recent high-resolution {\it Chandra} observations
have led to the measurement of both
the nuclear X-ray emission and of the
physical parameters of the surrounding
hot ISM, greatly strengthening
this conclusion (e.g., Pellegrini 2005 and references
therein).

Advection of energy into the SMBH
(e.g., Narayan \& Yi 1994, 1995) is a plausible
mechanism for low radiative efficiency, and may be
the explanation for many extremely sub-Eddington sources.
However, it cannot be the only explanation
for all sources, either. For example, advective accretion cannot
be invoked to explain some very under-luminous stellar-mass
compact objects in our Galaxy (Perna et al.~2003). Low $\dot{M}$
in addition to low $\eta$ has also been suggested as the major
reason for the overall faintness of Sgr A* (Bower et al.~2003;
Quataert 2003).

Hence, there is now considerable observational
evidence that quiescent or very faint SMBHs
are at least partly due to low radiative efficiency,
and partly to accretion rates
onto the compact object lower than predicted
from the amount of gas available. However,
the physical processes responsible
for this reduced accretion rate are still not well understood,
and various competing models have been proposed.
Hydrodynamical simulations (e.g., Perna et al.~2003,
their Sect. 2.3 and references therein)
have shown that disk winds and outflows
may prevent most of the inflowing gas
from reaching the compact object, so that only a small
fraction of the available mass is accreted.
Another possibility is that convection sets up
a barrier to accretion, by increasing the pressure
and preventing the gas from falling in supersonically
(Narayan, Igumenshchev, \& Abramowicz~2000).
Alternatively, Tan \& Blackman (2005) speculated
that efficient cooling
inside the Bondi radius causes the infalling gas
to condense into stars in a gravitationally unstable disk,
instead of sinking into the SMBH.

Another promising scenario is that most
of the accretion power is not radiated,
but is instead carried away
by a steady, relativistic jet, with bulk
kinetic luminosity $L_{\rm J} \sim 0.1 \dot{M}c^2
\gg \eta \dot{M}c^2$. In some cases
(e.g., M\,87: di Matteo et al.~2003, IC\,4296: Pellegrini et al.~2003b;
IC\,1459: Fabbiano et al.~2003),
it has indeed been shown that including the kinetic
energy of the jet reconciles the observed energy budget
with the expected accretion rate.  If a relativistic
jet is present, synchrotron radio emission is also expected.
The ``fundamental-plane'' relation (Merloni, Heinz, \& Di Matteo 2003)
between the observed radio luminosities,
X-ray luminosities, and BH masses supports the hypothesis
that inactive SMBHs have low accretion rates
as well as low radiative efficiency ($\eta \la 10^{-3}$),
and that most of the accretion power is carried outwards
by a steady jet.
However, in other cases (e.g., NGC\,821: Fabbiano et al.~2004)
it is more likely that the accretion power is released
in intermittent, short-lived outbursts (Binney \& Tabor 1995;
Ciotti \& Ostriker 2001; Jones et al.~2002; Omma et al.~2004);
if that is the case, the relative fraction of SMBHs observed
in an active or quiescent state
would depend on the duration of the duty cycle.
Alternatively, in a few cases the X-ray emission from the SMBH
may be absorbed by dust (often found in the inner $\sim 100$ pc
of elliptical galaxies: Rest et al.~2001) and re-emitted in the infrared (IR).
Finally, it is still possible that a few luminous
elliptical galaxies with partially depleted stellar cores
may not have large gas supplies to feed the SMBH.
Determining which of these physical processes
are at play in quiescent SMBHs will also
help us understand and quantify the SMBH feedback
onto its host galaxy.

For a sample of 18 nearby galaxies with nuclear
X-ray luminosity $L_{\rm X}$ and Bondi mass accretion rate
of hot gas estimated from {\it {Chandra}} data,
Pellegrini (2005) found that $L_{\rm X}$ can vary
by orders of magnitude and shows
a large scatter when plotted against
BH mass or Bondi rate. While in some cases
the observed luminosities are consistent with the prediction
of advective models, for other nuclear sources
the observed luminosities are much lower, and
therefore scenarios including outflows or convection
are needed. A scenario with feedback from accretion
on the surrounding gas was considered the most promising.

\subsection{Objectives and targets of our study}

To test these scenarios, we chose a further
sample of {\it six nearby,
quiescent elliptical galaxies} with direct SMBH mass determinations,
from Gebhardt et al.~(2003) and Kormendy et al.~(1997).
Stringent upper limits on AGN activity exist for all these SMBHs,
from radio, optical (Balmer emission lines),
and previous X-ray studies (Ho 2002), and from
a lack of UV bumps. Our goal is to explore the most extreme cases
of sub-Eddington nuclear activity, so we only focused on galaxies
with SMBH masses $> 10^7 M_{\odot}$ and X-ray luminosity $< 10^{40}$
erg s$^{-1}$. We also excluded galaxies more distant than 30 Mpc.

In five of the six galaxies, the SMBH mass has been accurately
determined with three-integral models
(Gebhardt et al.~2003)\footnote{In three of those five nuclei,
the dynamical sphere of influence of the SMBH
has been resolved by {\it HST}
(Merritt \& Ferrarese 2001; Ho 2002),
making the mass determination more reliable; in the remaining
two cases, NGC\,821 and NGC\,3377, the sphere of influence
is not resolved.}.
The only exception is NGC\,4486B, whose mass estimate
was based on two-integral models (Kormendy et al.~1997)
and is therefore much more uncertain. For example,
a revised error range in Kormendy \& Gebhardt (2001)
allows for masses between $2 \times 10^7$ and $10^9 M_{\odot}$.
A mass $\approx 7 \times 10^8 M_{\odot}$ was
quoted as a preliminary result of an ongoing
study by Gebhardt (2005, priv.~comm.).
As a corollary of our study, we have obtained
an alternative (indirect) SMBH mass
determination for NGC\,4486B
(Soria et al.~2005 = Paper II), using
the $M_{\rm BH}$--velocity dispersion
(Tremaine et al.~2002)
and $M_{\rm BH}$--S\'{e}rsic index
(Graham et al.~2003, 2005 in prep.) relations.
We are aware that the larger mass uncertainty,
and its peculiar morphological type, make NGC\,4486B
somewhat different from the rest of our {\it Chandra} targets,
among which it was included for historical reasons.
Classified as a compact elliptical (cE) galaxy,
NGC\,4486B may have originated
as the bulge of a much larger,
tidally stripped disk galaxy
(Jerjen \& Binggeli 1997; see also Graham 2002).

Our main objective is to {\em determine the energy
and mass budget} for the nuclear regions of
our sample galaxies, and
for a few others for which SMBH mass and luminosity
are available in the literature.
We want to answer three
fundamental questions: how much gas is available for accretion?
how much energy is radiated by the SMBH?
where does the rest of the mass and energy go?
Moreover, this may also help
to explain why most of the
SMBHs in the local universe are very faint or quiescent.

We begin by using {\it Chandra} data to investigate
the spatial and spectral properties of the nuclear
X-ray emission. By comparing the observed X-ray luminosity
of the SMBH with the temperature and density
of the surrounding gas, we can constrain some of
the accretion parameters. In particular, if we
could precisely determine the mass accretion rate
onto the SMBH, we would find the efficiency.
However, a direct measurement of the accretion rate
is still beyond today's observational capabilities,
given the characteristic size of SMBH event
horizons ($\sim 10^{-5}$ pc).
So, we need to turn the problem around.
We shall use model predictions for the radiative
and non-radiative efficiency (e.g., standard disks,
or advective accretion) and compare them with
the observed luminosity and gas density available
inside the SMBH accretion radii (characteristic
size $\sim 10$ pc). From this, we will infer
what fraction of the gas available is really
accreted. In Paper I, we shall use {\it Chandra}
to estimate the hot gas component to the accretion
flow; we shall look for correlations between
the classical Bondi inflow rate from the hot ISM,
and the observed SMBH luminosity.
In Paper II, we shall use optical
data to determine whether additional sources
of fuel (e.g., warm gas from stellar winds)
may have been missed by the X-ray study.
We shall then discuss the correlation between
the total gas accretion rate and the observed
SMBH luminosities.
Finally, we shall discuss how to balance the
mass and energy budget, taking into account
total gas injection, accretion onto the SMBH,
and outflows.

\begin{deluxetable*}{lccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Targets of our study \label{tbl-1}}
\tablewidth{0pt}
\tablehead
{
\colhead{Galaxy} & \colhead{Type} & \colhead{d} & \colhead{$M_B$}
	& \colhead{$M_{\rm BH}$}
	& \colhead{Obs. Date} & \colhead{Obs. ID}
	& \colhead{Exp.~time} \\[2pt]
   	& & \colhead{(Mpc)} & \colhead{(mag)} &
	\colhead{($10^8 M_{\odot}$)} & & & \colhead{(ks)}\\[2pt]
\colhead{(1)}  &  \colhead{(2)}  &  \colhead{(3)}  & \colhead{(4)}  &
                  \colhead{(5)}  &  \colhead{(6)}  &  \colhead{(7)}
		 &  \colhead{(8)}}
\startdata
\, & \, & & & & & & \\[6pt]
NGC\,821
	& E6 & $24.1 \pm 2.0$ & $-20.7$ & $0.85^{+0.35}_{-0.35}$
	& 2002/11/26 & 4408 & 25.3\\
 & & & & & 2002/12/01 & 4006 & 13.7\\[4pt]
NGC\,3377 & E5-6& $11.2 \pm 0.5$ & $-19.2$ & $1.0^{+0.9}_{-0.1}$
	& 2003/01/06 & 2934 & 39.6\\[4pt]
NGC\,4486B & cE0 & $16.9 \pm 1.3$ & $-16.8$ &
	$\left[6.0^{+3.0}_{-2.0}\right]$
	& 2003/11/21 & 4007 & 36.2\\[4pt]
 &&&& $0.5^{+0.5}_{-0.2}$ &&\\[4pt]
NGC\,4564 & E6/S0 & $15.0 \pm 1.2$ & $-19.0$ & $0.56^{+0.03}_{-0.08}$
	& 2003/11/21 & 4008 & 17.9\\[4pt]
NGC\,4697
	& E6 & $11.7 \pm 0.8$ & $-20.3$&$1.7^{+0.2}_{-0.1}$
	& 2000/01/15 & 784 & 39.3\\
 & & & & & 2003/12/26 & 4727 & 39.9\\
 & & & & & 2004/01/06 &  4728 & 35.7\\
 & & & & & 2004/02/12 &  4729 & 38.1\\[4pt]
NGC\,5845 & E3 & $25.9 \pm 2.7$ & $-18.8$& $2.4^{+0.4}_{-1.4}$
	& 2003/01/03 & 4009 & 30.0\\[4pt]
\enddata
\tablecomments{
Col.(1): see Fabbiano et al.~(2004) for a detailed
 study of NGC\,821, and Sarazin, Irwin \& Bregman (2001) for the analysis
of the first observation of NGC\,4697.
Col.(2): from the NASA/IPAC Extragalactic Database (NED).
Col.(3): from Tonry et al.~(2001), except for NGC\,4486B,
	from Neilsen \& Tsvetanov (2000).
Col.(4): from NED, adopting the distances in Col.(3).
Col.(5): from Gebhardt et al.~(2003), except for NGC\,821
        (Richstone et al.~2004), and for NGC\,4486B.
	The latter has two alternative values. The higher mass
	(in square brackets)
	is from Kormendy et al.~(1997), using a two-integral model.
	A similar result ($5.0^{+4.9}_{-4.8} \times 10^8 M_{\odot}$)
	was obtained with the same method by Kormendy \& Gebhardt
	(2001). However, the reliability of this value
	is questionable, both because of the method on which
	it is based, and because it is uniquely discrepant
	from the BH mass inferred from all indirect indicators
	(i.e., the global correlations with stellar velocity
	dispersion, S\'ersic index, bulge mass).
	The alternative, indirect mass determination
	was obtained in our Paper II (Soria et al.~2005), from
	the global correlation between S\'ersic index
	in the optical brightness profile
	and BH mass (Graham et al.\ 2003, 2005 in prep.).
}
\end{deluxetable*}

\section{X-ray Observations and Data analysis}

The six elliptical galaxies that are the
object of our study (Table 1) were
observed with {\it Chandra} (Weisskopf et al.~2000)
ACIS-S, centered on chip 7 (i.e., ACIS-S3).
For one of the galaxies, NGC\,4697, we used all 4 observations
currently available in the public archive.
We analysed the data with the {\footnotesize {CIAO}} v3.2
software\footnote{http://cxc.harvard.edu/ciao/}
for NGC\,4697, and with {\footnotesize {CIAO}} v3.1 for the
other targets; we checked that the difference between
the two {\footnotesize {CIAO}} versions is negligible
for the purposes of this work.
We downloaded the bias files for each observations,
and built new bad-pixel files\footnote{Thread at
http://cxc.harvard.edu/ciao/threads/acishotpixels/}.
We then
built new level=1 and level=2 event files with the {\footnotesize {CIAO}}
script {\tt {acis\_process\_events}}, which includes the application
of a time-dependent gain correction to the data. (The time-dependent
gain adjustment was not applied to the 2000 observation
of NGC\,4697, because it was taken with a focal plane temperature
different from the normal $-120$ C).
We also improved the astrometric accuracy by calculating
and removing the (negligible) aspect offsets,
using the {\footnotesize {CIAO}} Aspect
Calculator\footnote{Thread at
http://cxc.harvard.edu/ciao/threads/arcsec\_correction/}.
No significant background flares were observed in these data,
so no time filtering was necessary. The background level
was generally very low except for the observation of NGC\,4486B.
The energy range was filtered to $0.3$--$10$ keV, with
the {\footnotesize {CIAO}} task {\tt {dmcopy}}.
We identified the point sources with the {\tt {wavdetect}} task,
and for the brightest of them, we extracted their spectra
with {\tt {psextract}} (which also builds response
and auxiliary response files,
with the {\tt {mkrmf/mkarf}} tasks).
After subtracting the discrete sources, we used the
{\footnotesize{CIAO}} script {\tt{acisspec}} to extract
the spectra of diffuse emission from extended regions,
together with their weighted response files and backgrounds.
We then used {\footnotesize {XSPEC}} 11.3.1 (Arnaud 1996)
for the spectral fitting of point-like and extended
sources\footnote{http://heasarc.gsfc.nasa.gov/lheasoft/xanadu/xspec/}.

\section{Results}

We focus in particular on the new results
for NGC\,3377, NGC\,4486B, NGC\,4564 and NGC\,5845.
The remaining two galaxies have already been studied
in detail (NGC\,821: Fabbiano et al.~2004;
NGC\,4697: Sarazin et al.~2001): here we re-analyse
and discuss their nuclear X-ray properties
in the context of our new sample of massive,
quiescent elliptical galaxies. In particular,
for NGC\,4697 we use both the 2000 {\it Chandra}
observation that was studied by Sarazin et al.~(2001),
and three more recent observations that became
public in early 2005.

\begin{deluxetable*}{l@{\ \ \ \ }r@{\ \ }rrr@{\ \ \ \ }r@{\ \ }rrr}
\tabletypesize{\scriptsize}
\tablecaption{ACIS-S net counts and count rates from the nuclear sources\label{tbl-2}}
\tablewidth{0pt}
\tablehead
{
\colhead{Galaxy} & \multicolumn{4}{c}{ACIS-S Net Counts}
	& \multicolumn{4}{c}{ACIS-S Count Rates ($10^{-4}$ s$^{-1}$)}\\
 & $0.3$--$10$ keV & $0.3$--$1$ keV & $1$--$2$ keV & $2$--$10$ keV &
	$0.3$--$10$ keV & $0.3$--$1$ keV & $1$--$2$ keV & $2$--$10$ keV  }
\startdata
\, & \, & & & & & & &\\[-3pt]
NGC\,821 (core) &\multicolumn{8}{c}{not detected}\\
NGC\,821 (``jet'')
	& $137.2\pm26.4$ & $34.3 \pm 7.7$ & $63.5\pm9.5$ & $39.4\pm23.4$
	 &$36.2 \pm 7.0$ &$9.0 \pm 2.0$ & $16.8 \pm 2.5$ &$10.4 \pm 6.2$\\[2pt]
NGC\,3377 & $113.1\pm11.6$ & $41.5 \pm 6.8$ & $45.5 \pm 6.8$ & $26.1\pm 5.4$&
	$29.3 \pm 3.1$ & $10.9 \pm 1.8$ & $11.5 \pm 2.4$ & $6.9 \pm 1.4$\\[2pt]
NGC\,4486B & $38.2\pm8.2$ & $27.7 \pm 6.2$ & $6.8 \pm 7.0$ &$3.7\pm 5.1$
	& $10.6 \pm 2.3$ &$7.7 \pm 1.7$ & $1.9 \pm 1.9$ & $1.0 \pm 1.4$\\[2pt]
NGC\,4564 &$72.3\pm9.3$ & $28.3 \pm 5.9$ & $27.5 \pm 5.6$ &$16.5\pm 4.4$
	& $43.7 \pm 5.6$ &$17.0 \pm 3.5$ & $16.7 \pm 3.4$ &
	$10.0 \pm 2.7$\\[2pt]
NGC\,4697 (1)
	&$128.9\pm11.6$ & $57.3 \pm 7.8$ & $40.4 \pm 6.4$ &$31.4\pm 5.7$
	& $32.8 \pm 3.0$ &$14.6 \pm 2.0$ & $10.3 \pm 1.6$ & $8.0 \pm
1.4$\\
NGC\,4697 (2)
	&$288.0\pm17.3$ & $97.0 \pm 10.2$ & $120.7 \pm 11.1$ &$69.3\pm 8.5$
	& $25.3 \pm 1.5$ &$8.5 \pm 0.9$ & $10.6 \pm 1.0$ & $6.1 \pm 0.7$\\[2pt]
NGC\,5845 &$117.8\pm11.8$ & $36.5 \pm 6.8$ & $56.9 \pm 7.9$ &$24.4\pm 5.3$
	& $39.4 \pm 3.9$ &$12.2 \pm 2.3$ & $19.0 \pm 2.7$ & $8.2 \pm 1.8$ \\
\enddata
\tablecomments{
The values for NGC\,821 are from the combined 2002 Nov/Dec
observations; see Fabbiano et
al.~(2004) for details on the definition of the jet-like
nuclear feature as opposed to a point-like (non-detected) core. The
first line for NGC\,4697 is from the 2000 Jan observation, and the second
is for the combined 2003 Dec--2004 Feb observations; the reduction
in count rate is due to the degraded performance of ACIS-S.}
\end{deluxetable*}

\subsection{Morphology of the nuclear X-ray sources}

In all six galaxies we detect X-ray emission at and around
the nuclear position. In five cases (all except NGC\,821,
see Fabbiano et al.~2004), the centroid
of the nuclear X-ray source coincides with
the optical/IR nucleus identified in the
Two-Micron All Sky Survey (2MASS)
and United States Naval Observatory (USNO)
Catalogues, within the astrometric
accuracy of {\it Chandra} (the $90\%$ uncertainty circle
has a radius of $\la 0\farcs6$).
For NGC\,4697, relative astrometry with an accuracy
better than $\approx 0\farcs3$ can be obtained,
because a few non-nuclear X-ray sources have
point-like optical counterparts (globular clusters).

Firstly, we analyzed the morphology of the nuclear X-ray sources
to determine whether they are consistent
with point-like emission from the SMBH
or are instead extended. From the theoretical ACIS-S3
point spread function and from our fitting of other point
sources detected in the central region of the chip,
we expect a Gaussian FWHM $= 0\farcs89 \pm 0\farcs15$
for a point-like source in the $0.3$--$8$ keV energy band.
The (weak) nuclear source of NGC\,4486B is consistent
with being point-like (FWHM $\approx 0\farcs9$),
implying a size $\la 70$ pc;
it is also well isolated: no other X-ray
sources are detected in that galaxy.
The nuclear source in NGC\,4697 (Figure~1, bottom left)
is also point-like (size $\la 50$ pc) but is located
in a region with high source density,
with at least another 5 point-like sources
detected within a radius of $300$ pc. Their
spatial distribution around the nuclear source is probably
a chance grouping of X-ray binaries in the galactic bulge.

The nuclear source of NGC\,4564 (Figure~1, top right)
has a FWHM $\approx 2\farcs2 \approx 160$ pc
in the $0.3$--$8$ keV band; therefore,
it is probably extended.
In the remaining three galaxies, NGC\,3377, NGC\,5845
and NGC\,821, structure in the nuclear region
is more clearly visible. In NGC\,5845 (Figure~1,
bottom right), the nuclear emission has
an irregular, ``fuzzy'' appearance over
a region $\approx 600$ pc across. In the other
two galaxies, NGC\,821 and NGC\,3377, the emission
has instead an elongated, ``jet-like'' appearance.
In NGC\,3377 (Figure~1, top left), an elongated feature
seems to link the nuclear source
with another, fainter source located $\approx 3\arcsec
\approx 160$ pc to the East. For a discussion
of the elongated substructure in the
nucleus of NGC\,821, see Fabbiano et al.~(2004).
Limiting our imaging study to the $1.5$--$8$ keV band
(Figure 2), we still find some extended structure
in NGC\,5845.

\subsection{X-ray colors and spectra of the nuclear sources}

We extracted X-ray spectra of the nuclear sources,
using the extraction regions plotted in Figure~1
(dashed white circles); the background regions
were chosen as annuli around the nuclear sources,
and we took care of excluding nearby sources.
For NGC\,4697, we extracted separate spectra
for the four observations. For some tasks, it is possible
and convenient to coadd the 2003 Dec--2004 Feb
observations and study them together; however,
the analysis of the first dataset (2000 Jan)
has to be carried out separately because
of the ACIS-S sensitivity degradation, particularly
at lower energies, over those four years.

Firstly, we compared the broad-band X-ray colors (Table 2),
before introducing any spectral modelling.
In five of the six nuclear sources, the X-ray colors
are consistent with typical AGN spectra,
a power-law spectrum of index $\approx 1.5$--$2$ (Figure 3).
The only exception is NGC\,4486B, which has softer
colors, with most of the counts detected below 1 keV,
although the errors are large because
of the relatively high background level and the small
number of counts.

We then estimated the emitted fluxes and luminosities
by assuming an absorption column density equal
to the Galactic line-of-sight (Table 3; from
Dickey \& Lockman 1990). The color-color
diagram (Figure 3) and the X-ray spectra (at least for the nuclear
sources with enough counts) justify this assumption,
suggesting that the total column density
is not much higher than the Galactic value;
hence, we are only slightly underestimating the emitted
luminosities. We fitted the (ungrouped) spectra
with an absorbed power-law (Table 3), using the Cash statistic
(Cash 1979)\footnote{This statistic
can be used in the same way as the $\chi^2$ statistic
to find confidence intervals. One finds the parameter
values that give C $=$ C$_{\rm min}$ $+$ N, where N
is the same number that gives the required
confidence for the number of interesting parameters
as for the $\chi^2$ case (Arnaud 1996).
See the XSPEC manual for details.}.
We also grouped the same spectra to achieve
a minimum signal-to-noise ratio of $3$ in each spectral
bin, and fitted them with the $\chi^2$ statistics
(Table 3). For sources with a very limited number of counts,
we consider the results based on the
Cash statistic to be more reliable.
For NGC\,4697, we increased the signal-to-noise ratio
by coadding the spectra from 2000 and
2003--2004, using the algorithm of
Page, Davis \& Salvi (2003), which generates
a count-weighted average of their response matrices
(Figure~4). From a power-law fit (Table 3)
we calculate an emitted luminosity
$L_{\rm x} = (3.9\pm0.2) \times 10^{38}$ erg s$^{-1}$
in the $0.3$--$10$ keV band; we also checked that
there are no significant spectral or luminosity changes
between the two epochs.
The emitted luminosities of the nuclear sources
in the other five galaxies are between $\approx 3 \times 10^{38}$
and $\approx 3 \times 10^{39}$ erg s$^{-1}$,
in the same band (Table 3).

For NGC\,4486B, we have also tried fitting
the nuclear source with a thermal-plasma
model, to account for the possibility
that it is a very compact patch of thermal
emission from the ISM.
We obtain a best-fit $kT = 3.7^{+8.0}_{-1.6}$
keV, but the fit is statistically worse
(C-statistics $=54/297$) than
for the power-law model (C-statistics $=46/297$).
Henceforth, we shall assume for this work
that most of the X-ray emission comes from the SMBH.

\begin{figure*}[t]
\epsscale{1.0}
\begin{center}
\includegraphics[angle=0,scale=0.37]{f1a.ps}
\includegraphics[angle=0,scale=0.37]{f1b.ps}\\
\includegraphics[angle=0,scale=0.37]{f1c.ps}
\includegraphics[angle=0,scale=0.365]{f1d.ps}
\end{center}
\caption{ACIS-S image ($0.3$--$8$ keV band)
of the nuclear regions in four
of the galaxies showing extended structures.  (For the other
two galaxies in our sample: see Fabbiano et al.~(2004) for
NGC\,821; the nuclear source is point-like in NGC\,4486B).
In each panel, the dashed white circle is the source region
used to determine the nuclear luminosity, spectrum
and colors (Tables 2, 3 and Figure 3). The green circles
(radius of $0\farcs6$) are the 2MASS positions
of the galactic nuclei, except for NGC\,4697, where,
for consistency with previous work, the green circle
marks the average between the 2MASS and the USNO B1.0
Catalogue positions (the difference between
the two is $0\farcs4$). North is up, East to the left.
\label{fig1}}
\end{figure*}

\begin{figure*}[t]
\epsscale{1.0}
\begin{center}
\includegraphics[angle=0,scale=0.37]{f2a.ps}
\includegraphics[angle=0,scale=0.37]{f2b.ps}\\
\includegraphics[angle=0,scale=0.37]{f2c.ps}
\includegraphics[angle=0,scale=0.37]{f2d.ps}
\end{center}
\caption{As in Figure~1, but limited to the $1.5$--$8$ keV band.
The green circles (radius of $0\farcs6$) mark the best
available optical/IR position for the nuclei. North is up,
East to the left.
\label{fig1b}}
\end{figure*}

\begin{figure}
\includegraphics[angle=270,scale=0.36]{f3.eps}
\caption{X-ray color-color diagram for the nuclear emission
in the six galaxies of our sample. We overplotted the expected
colors for an optically-thin thermal plasma model
and for power-law models of different photon indices,
with varying line-of-sight column density ($n_{\rm H}$
running from $2 \times 10^{20}$ to $5 \times 10^{21}$ cm$^{-2}$
along each curve, from the bottom to the top).
The galaxies are labelled from 1 to 6 as in Table 1.
For NGC\,821, for which no point-like source
is detected at the nuclear position, we plotted the average colors
of the elongated, jet-like feature.
For NGC\,4697, we only used the 2003--2004 data,
to minimize the effect of color changes due to the
sensitivity degradation. The model curves are the colors
expected for an observation in mid-2003.
\label{fig2}}
\end{figure}

\begin{figure}
\includegraphics[angle=270,scale=0.35]{f4.eps}
\caption{Spectrum of the nuclear source in NGC\,4697,
with best-fit model and $\chi^2$ residuals. The data
have been fitted with an absorbed power law, of photon
index $\Gamma = 1.5 \pm 0.2$ (Table 3).
\label{fig3}}
\end{figure}

\begin{table*}
\begin{center}
\caption{Spectral parameters and emitted luminosities from the nuclear
sources. All spectra have been fitted with an absorbed power law,
and column density fixed to the line-of-sight galactic value. For each
source, the first fit is based on the Cash statistics, the second (when
applicable) on the $\chi^2$ statistics. Errors are 90\%
confidence levels for one interesting parameter ($\Delta \chi^2 = 2.7$)
\label{tbl-3}}
\begin{tabular}{l@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }r}
\tableline\tableline
Galaxy & Galactic $n_{\rm H}$\tablenotemark{a}
&  Photon index $\Gamma$ & Normalization
at 1 keV & $\chi^2$/dof
& $L_{0.3-10 {\rm keV}}$\tablenotemark{b}\\
 & ($10^{20}$ cm$^{-2}$) & & ($10^{-6}$ ph. keV$^{-1}$ cm$^{-2}$ s$^{-1}$)
 & &($10^{38}$ erg s$^{-1}$)\\
\tableline\\[-3pt]
NGC\,821 (``jet'') & $6.4$ & $1.53_{-0.31}^{+0.32}$& $5.3_{-1.0}^{+1.1}$
	&& $25.5 \pm 2.3$\\[4pt]
	& &$1.50_{-0.29}^{+0.31}$ & $4.9 \pm1.0$ & $7.5/6$
	& $24.4 \pm 2.2$\\[6pt]
NGC\,3377 & $2.9$ & $1.65^{+0.32}_{-0.25}$ & $3.3^{+0.6}_{-0.6}$
	&& $3.6^{+0.7}_{-0.8}$\\[4pt]
	& &$1.85^{+1.6}_{-1.2}$ & $3.9^{+1.8}_{-1.8}$ & $1.6/7$
	& $3.4^{+2.6}_{-1.7}$ \\[6pt]
NGC\,4486B\tablenotemark{c}
	& $2.6$ & $2.4^{+0.5}_{-0.5}$ & $1.6^{+0.5}_{-0.5}$ &
	& $2.6^{+1.3}_{-1.0}$ \\[4pt]
NGC\,4564 & $2.4$ & $2.4^{+0.4}_{-0.4}$ & $4.5^{+1.1}_{-1.2}$ &
	& $5.9^{+1.2}_{-1.8}$\\[4pt]
	&  & $1.6^{+1.5}_{-2.1}$ & $5.3^{+3.7}_{-3.7}$ & $0.76/4$
	& $10.5^{+25.3}_{-5.9}$ \\[6pt]
NGC\,4697\tablenotemark{d} & $2.1$ & $1.53^{+0.23}_{-0.22}$ & $2.9^{+0.5}_{-0.5}$ &$6.6/22$
	& $3.9^{+0.2}_{-0.2}$\\[6pt]
NGC\,5845 &$4.3$ & $2.03^{+0.24}_{-0.29}$ & $5.4^{+0.5}_{-1.6}$ &
	& $24.0^{+2.5}_{-2.5}$ \\[4pt]
	& & $1.4^{+0.8}_{-0.8}$ & $5.1^{+2.1}_{-2.2}$ & $1.3/7$
	& $39^{+19}_{-25}$\\[5pt]
\tableline
\end{tabular}
\tablenotetext{a}{from Dickey \& Lockman 1990.}
\tablenotetext{b}{Emitted luminosity inferred from the spectral fit.}
\tablenotetext{c}{Fitted with Cash statistics only.}
\tablenotetext{d}{Fitted with the $\chi^2$ statistics.}
\end{center}
\end{table*}

\subsection{Scattering and obscuration of the nuclear SMBH?}

The nuclear X-ray source in NGC\,5845 appears extended
on the scale of a few hundred pc, larger
than the FWHM of the {\it Chandra}/ACIS PSF for
a point source (Section 3.1). The count rate
is insufficient to map the X-ray colors or
the spectral properties across the nuclear region.
However, we can at least say that the ``fuzzy''
emission around the nuclear position is not
significantly softer than a typical AGN-like
power-law spectrum: the emission is extended
even in the hard band ($2$--$10$ keV).
The simplest explanation is that the emission is due
to a few faint, unresolved LMXBs in the inner region.
If that was the case, the true luminosity
of the SMBH in NGC\,5845 would be
a few times lower than estimated
in Table 3.

Alternatively, we note that the X-ray morphology is reminiscent
of the extended X-ray emission associated with
obscured Seyfert nuclei (for NGC\,4151:
Elvis, Briel \& Henry~1983; Elvis et al.~1990;
Ogle et al.~2000; for NGC\,1068: Ogle et al.~2003),
which is understood as scattered nuclear emission.
Dust is known to be present in the nuclear
region of most ellipticals, especially
those with disky isophotes (van den Bosch et al.~1994;
Kormendy et al.\ 2005 and references therein).
The nuclear region of NGC\,5845 contains a dusty disk of radius
$\approx 100$ pc (Quillen, Bower \& Stritzinger 2000),
i.e. an order of magnitude larger than the Bondi
accretion radius.
In fact, NGC\,5845 is one of only two galaxies
known for containing both a dust disk
and an associated, nuclear stellar disk
(the other one is NGC\,4486A). It has been
suggested (Kormendy et al.\ 2005; Kormendy et al.\ 1994)
that, as cold gas and dust settle
towards the center, they form stars and build
a stellar disk.

In NGC\,5845, the disk is seen almost edge-on (Figure 5).
Thus, it is possible
that most of the direct X-ray emission from the SMBH
along our line of sight is obscured by dust,
and most of the observed, extended emission comes
from the surrounding photo-ionized plasma which is
scattering the nuclear continuum.
If this was the case, the true SMBH luminosity
in NGC\,5845 could instead be at least an order of magnitude
higher than estimated in Table 3.
Given the apparent thin-disk geometry
of the dusty feature (Figure~5), we may expect
that the disk/torus can intercept
and re-radiate in the far-infrared
a fraction $\sim 10$--$20\%$ of the bolometric luminosity.
If the far-infrared nuclear luminosity
$\ga 10^{41}$ erg s$^{-1}$, it is potentially
detectable by {\it Spitzer} (flux density $\sim 10$ mJy
in the {\it Spitzer}/MIPS bands). This will
be investigated in a follow-up work, for which {\it Spitzer}
time has already been obtained.
For the present paper, we assume that
the observed X-ray luminosity of the SMBH
in NGC\,5845 is a good approximation
of the emitted luminosity.

A dusty disk or torus of outer radius $\approx 200$ pc
is also apparent in the nuclear region of NGC\,4697 (Figure 6).
However, obscuration or scattering of the central X-ray
source may not be as significant as in NGC\,5845
because our viewing angle is $\approx 75^{\circ}$.
A more detailed investigation of the physical
processes at work in a dusty nuclear disk is beyond
the scope of this paper.

\begin{figure}[t]
\includegraphics[angle=0,scale=0.45]{f5.eps}
\caption{Greyscale {\it HST}/WFPC2 image (F555W filter)
of the inner region of NGC\,5845, with X-ray contours
of the nuclear emission overplotted (cf. Figure 1,
bottom right panel). North is up, East to the left.
The nucleus contains an
$\approx 100$-pc-radius dusty and stellar disk,
which may contribute to obscuring or scattering
the central X-ray source. In an alternative scenario,
the dusty disk may reduce the accretion rate onto the BH
by storing cold gas and triggering star formation.
\label{fig13}}
\end{figure}

\begin{figure}[t]
\includegraphics[angle=0,scale=0.45]{f6a.ps}\\
\includegraphics[angle=0,scale=0.45]{f6b.ps}
\caption{Greyscale {\it HST}/ACS image (F475W filter)
of the inner region of NGC\,4697 (top panel),
with a close-up view of the nuclear region (bottom panel).
In both panels, North is up, East to the left.
The X-ray contours are on a square-root scale with arbitrary
zeropoint.
We improved the relative astrometry of the {\it HST}
and {\it Chandra} images
by taking advantage of a few X-ray sources
with an optical counterpart (possibly globular clusters
in NGC\,4697). Four of them are visible here
(in addition to the nuclear source), labelled
as ``1'' through ``4''. The optical nucleus
is point-like in {\it HST} (size $\la 0\farcs1 \approx 5$ pc)
and is probably a stellar cluster.
\label{fig14}}
\end{figure}

\subsection{Jet-like features}

The morphologies of the X-ray emission
in the nuclear regions suggest possible
jet-like features in two of our target galaxies,
NGC\,821 and NGC\,3377. The former was discussed
in Fabbiano et al.~(2004), who suggest
that the X-ray flux is consistent with synchrotron
emission in a jet, or with hot thermal plasma
shocked by intermittent nuclear outbursts.
For NGC\,3377, we compared the {\it Chandra} image
with an archival {\it HST}/WFPC2 image
in the F555W filter,
to determine whether the elongated feature
in the X-ray image has an optical counterpart; however,
we do not see any obvious distortions in the optical
isophotes (Figure 7, top panel).

We then searched for possible fainter
optical signatures in the nuclear region of NGC\,3377.
We determined the mean 2-D surface brightness distribution
of the central region of NGC\,3377 with the help of a boxcar
pixel-smoothing technique and an isophotal fitting routine
written in {\footnotesize IRAF}. Both techniques were recently employed
to uncover spiral and bar structures in seemingly normal dwarf
elliptical galaxies in the Virgo (Jerjen, Kalnajs \& Binggeli~2000;
Barazza, Binggeli \& Jerjen~2002) and Coma clusters (Graham,
Jerjen \& Guzm\'{a}n~2003).
The reconstructed model galaxy is subtracted from
the original, leaving just the small-scale substructures
in the residual image. The result for NGC\,3377 derived in
this manner is shown in the bottom panel of Figure 7:
we find a small, brighter stellar disk
but no feature corresponding to the X-ray elongation,
neither parallel not perpendicular to the disk plane.

None of the six galaxies in our sample have
significant radio detections; however, five have
at least reliable upper limits to their radio-core emission,
from VLA observations.
In NGC\,821, the radio-core flux at 5 GHz is $< 0.3$ mJy
(Wrobel \& Heeschen 1991); in NGC\,3377, NGC\,4697 and NGC\,5845,
the radio flux at 8.4 GHz is $< 0.1$ mJy
(Krajnovic \& Jaffe 2002). In NGC\,4564, the upper
limit is $0.52$ mJy at 8.4 GHz (Wrobel \& Herrnstein 2000);
however, the core is marginally detected
at the level of $\approx 2$ mJy at 1.4 GHz,
in the NRAO/VLA Sky Survey. A comparison of radio
and X-ray nuclear fluxes is discussed in Paper II.

\begin{figure}
\includegraphics[angle=0,scale=0.45]{f7a.ps}\\
\includegraphics[angle=0,scale=0.45]{f7b.ps}
\caption{Comparison between X-ray ({\it Chandra}/ACIS,
$0.3$--$8$ keV band)
and optical ({\it HST}/WFPC2, in the F555W filter)
images of the inner region of NGC\,3377.
Top panel: the elongated X-ray emission feature
at the nuclear position does not have any correspondence
in the optical isophotes.
Bottom panel: X-ray emission contours overplotted
onto a greyscale residual optical image.
The residual image was obtained by subtracting
the mean 2-D surface brightness from the original
optical image (see text for details). A bright
stellar disk is found in the nuclear region, but has
no apparent association with the elongated X-ray feature.
In both panels, North is up, East to the left.
\label{fig12}}
\end{figure}

\section{X-ray emitting, diffuse gas}

\subsection{X-ray brightness profiles}

Unresolved X-ray emission in elliptical galaxies
is a combination of faint low-mass X-ray binaries (LMXBs)
and hot gas, with $kT \sim 0.3$--$0.7$ keV
(Trinchieri \& Fabbiano 1985; Blanton, Sarazin \& Irwin 2001;
Sivakoff, Sarazin \& Irwin 2003; Randall, Sarazin \& Irwin 2004).
In this work, we focus on the nuclear BH activity
and do not set out to study the X-ray source population
and large-scale structure of the galaxies. However,
we still need to investigate the diffuse hot gas component,
because it provides a constraint to the energy budget
of the nuclear BH. From the surface brightness
and temperature of the diffuse emission, we can in principle
estimate the density of the hot gas in the nuclear region;
hence, we can obtain a lower limit to the gas density
available for accretion onto the BH, which can be
compared to the observed luminosity, to constrain
the accretion efficiency.

To obtain the surface brightness of the unresolved
emission (and, in particular, the thermal gas component),
firstly we filtered the event files
to remove the point sources detected with {\tt {wavdetect}}.
In the case of NGC\,4697, we combined all four observations
to reach a deeper detection limit for the point sources;
then we analysed the unresolved emission separately
for the 2000 and 2003--2004 datasets.
For each galaxy, we created a stack of concentric annular regions
extending up to $120\arcsec$ from the nucleus, and a background
annulus outside this region. We used the {\footnotesize {CIAO}}
task {\tt {dmextract}} to obtain the background-subtracted
radial profiles. Since we are only interested here
in the diffuse gas contribution, we restricted the energy
range to $0.3$--$1.5$ keV, where the hot thermal plasma
component is relatively stronger, compared with
the unresolved LMXB contributions. Above
$\approx 1.5$ keV, the hot gas component is, instead,
negligible, and the unresolved emission is dominated by the
power-law LMXB component.

To determine the physical properties of the extended
X-ray emission, we need to measure the radial profiles
of the gas surface brightness $S(r)$, and the central
surface brightness $S_0$ (a distance-invariant quantity).
The surface brightness $S(r)$ does not come directly
from the observations: what we have instead is
the surface density $S'(r)$ of the count rates
in the $0.3$-$1.5$ kev band, for the total
unresolved emission, including contributions
from both the interstellar gas and faint LMXBs.
We show $S'(r)$ for five of the six galaxies
in our sample (Figures~8, 9). The strongest
emission is found in NGC\,4697 (Figure~10; see also
Sarazin et al.~2001). On the other end,
no unresolved emission is detected in NGC\,4486B,
though the relatively higher background level of that observation
makes it more difficult to obtain a tight upper limit.
To calculate the density of the hot gas,
we need to convert the observed count rates into
physical flux units, and to estimate what fraction
of the unresolved emission is due to diffuse gas.
This requires a spectral analysis.

Only two galaxies (NGC\,4697 and NGC\,3377) have enough detected
counts in the unresolved X-ray emission to allow
for spectral fitting. After excluding the discrete
sources found by {\tt{wavdetect}}, we used the
{\footnotesize{CIAO}} script {\tt{acisspec}} to extract
the spectrum of the unresolved emission from a $90\arcsec$
circle, taking an annulus between $120\arcsec$ and
$150\arcsec$ as background region. For NGC\,4697,
we extracted separate spectra (with their respective
response files) for the 2000 and 2003--2004 observations;
we then combined the two spectra using the algorithm
of Page et al.~(2003).
We fitted the spectra in {\footnotesize{XSPEC}},
assuming an absorbed power-law plus a single-temperature
thermal plasma model, {\tt{wabs*(vapec + po)}}.
First, we fixed the absorption column density to the Galactic value,
the metal abundance of all elements to the solar value,
and the photon index of the power-law component
to $1.7$ (average value for LMXBs), to reduce
the number of free parameters (Tables 4, 5).
This already produces a good fit ($\chi^2 = 43.0/51$)
As previously noted ({\it ROSAT} study: Pellegrini
\& Fabbiano 1994; {\it Chandra} study: Sarazin et al.~2001),
the interstellar gas in NGC\,4697 is cooler ($kT \approx 0.3$ keV)
than is usually found in X-ray bright ellipticals;
this may be the consequence of a shallow gravitational potential.
We further improved the fit by varying the metal abundances
of the main elements: we find that an overabundance
of C and N over the $\alpha$ elements
by a factor of $\sim 5$--$10$ significantly
reduces the $\chi^2$ below 1 keV. No significant Mg or Si lines
are detected (Figure 11), an indication that the X-ray emitting gas
may not be strongly metal-enriched from Type-II SNe.
The gas in NGC\,3377 is consistent with a higher temperature
($kT \approx 0.6$ keV),
although the uncertainty is larger, due to the limited
number of detected counts (only $\approx 130$; Figure 12).
There are not enough counts to vary the metal abundances.

By fitting a spectral model to the unresolved X-ray emission,
we can estimate the conversion factors between
the observed count-rate density $S'(r)$
(units of counts s$^{-1}$ arcsec$^{-2}$
in a specific band), and the bolometric surface brightness $S(r)$
of the diffuse hot gas.
For the spectral model fitted
to NGC\,3377, we obtain a conversion factor of
$1$ count s$^{-1}$ arcsec$^{-2}$ in the $0.3$--$1.5$ keV band
$= (2.5 \pm 0.5)$ erg cm$^{-2}$ s$^{-1}$ arcsec$^{-2}$;
$\approx 50\%$ of the unresolved counts in the $0.3$--$1.5$ keV band
are from truly diffuse emission. This is an average value
over the whole extraction region; in fact,
the diffuse emission may be more centrally peaked
(Kim \& Fabbiano 2003).
For the 2000 observation of NGC\,4697,
$1$ count s$^{-1}$ arcsec$^{-2}$ $= (4.8\pm 0.4)$ erg cm$^{-2}$
s$^{-1}$ arcsec$^{-2}$; in 2003--2004, $1$ count s$^{-1}$
arcsec$^{-2}$ $= (6.7\pm 0.5)$ erg cm$^{-2}$
s$^{-1}$ arcsec$^{-2}$ (this difference is due to the
decrease in sensitivity of the detector);
$\approx 80\%$ of the unresolved counts are from hot gas.
The total emitted luminosity from the hot gas is $\approx 10^{38}$
erg s$^{-1}$ in NGC\,3377 and $\approx 10^{39}$
erg s$^{-1}$ in NGC\,4697.

We can use these values to estimate the conversion
factors for the other galaxies in our sample,
for which we cannot do any spectral analysis.
We can safely assume that the temperature
of the hot ISM is in the range $\sim 0.3$--$0.7$ keV.
The relative contribution of hot gas and LMXB emission
varies from galaxy to galaxy (see a discussion
in Kim \& Fabbiano 2003); our choice of a soft
energy band ($0.3$--$1.5$ keV) should reduce
the uncertainty caused by LMXB contamination.
The conversion factor between unresolved count rate
and flux of the truly diffuse emission
depends strongly on the detection threshold
of the point sources: we used NGC\,4697 to estimate this effect.
When the point-source detection limit is $1 \times 10^{37}$ erg s$^{-1}$,
we find that $\approx 80\%$ of the unresolved counts
in the $0.3$--$1.5$ keV band are
from the ISM, as mentioned earlier;
$\approx 50\%$ for a detection limit of $1 \times 10^{38}$
erg s$^{-1}$; and only $\approx 35\%$ for a detection
limit of $3 \times 10^{38}$ erg s$^{-1}$.
In conclusion, taking into account these arguments,
we adopt a conversion factor between count-rate surface
density and bolometric surface brightness of
$1$ count s$^{-1}$ arcsec$^{-2}$ in the $0.3$--$1.5$ keV band
$= (3.0 \pm 1.5)$ erg cm$^{-2}$ s$^{-1}$ arcsec$^{-2}$
from the hot gas, for NGC\,4564 and NGC\,5845.

\begin{figure}
\includegraphics[angle=270,scale=0.36]{f8.eps}
\caption{Count-rate surface density profiles of the unresolved emission
in the $0.3$--$1.5$ keV band (units of ACIS-S counts
s$^{-1}$ arcsec$^{-2}$),
plotted as a function of angular distance on the sky.
From the brightest to the faintest, symbols (and
colors for the online version of the paper)
are as follows: filled circles for NGC\,4697 (green
for the 2000 observations, magenta
for the 2003--2004 observations); open squares
(black) for NGC\,3377; open circles (blue)
for NGC\,5845; open triangles (red)
for NGC\,4564; crosses (cyan) for NGC\,4486B.\label{fig4}}
\end{figure}

\begin{figure}
\includegraphics[angle=270,scale=0.36]{f9.eps}
\caption{Count-rate surface density profiles of the unresolved emission
in the $0.3$--$1.5$ keV band (units of ACIS-S counts
s$^{-1}$ arcsec$^{-2}$),
plotted as a function of projected physical distance from the nucleus.
Symbols and colors are as in Figure 8.\label{fig5}}
\end{figure}

\begin{figure*}
\epsscale{1.15}
\includegraphics[angle=0,scale=0.48]{f10a.ps}
\includegraphics[angle=0,scale=0.79]{f10b.eps}
\caption{Left panel: adaptively-smoothed ACIS-S image
of NGC\,4697, in the $0.3$--$8$ keV band. The ellipse
marks the $D_{25}$ isophote; the box (dashed white line)
is shown in greater details on the right. Right panel:
true-color X-ray image of the inner region of
NGC\,4697. Red $= 0.3$--$1$ keV;
green $= 1$--$2$ keV; blue $= 2$--$8$ keV.
In both panels, North is up, East to the left.\label{fig6}}
\end{figure*}

\begin{figure}
\includegraphics[angle=270,scale=0.35]{f11.eps}
\caption{Spectrum of the unresolved emission from NGC\,4697,
with best-fit model and $\chi^2$ residuals. The data have been
fitted with a variable-abundance thermal plasma model
({\tt{vapec}} in {\footnotesize{XSPEC}}) plus a power-law. The fit parameters
are listed in Table 4.\label{fig7}}
\end{figure}

\begin{figure}
\includegraphics[angle=270,scale=0.35]{f12.eps}
\caption{Spectrum of the unresolved emission from NGC\,3377,
with best-fit model and $\chi^2$ residuals. The data have been
fitted with a thermal plasma model
({\tt{apec}} in {\footnotesize{XSPEC}},
with fixed solar abundance) plus a power-law.
The fit parameters are listed in Table 5.\label{fig8}}
\end{figure}

\begin{table}
\begin{center}
\caption{Best-fit parameters for the spectrum of the unresolved
X-ray emission in NGC\,4697. Spectral model:
{\tt wabs$_{\rm Gal}$*wabs*(vapec+po)}. Values in brackets were kept
fixed. Errors are 90\%
confidence levels for one interesting parameter ($\Delta \chi^2 =
2.7$).
\label{tbl-5}}
\begin{tabular}{l@{\ \ \ \ \ }c@{\ \ \ \ \ }r}
\tableline\tableline\\[-6pt]
Parameter & Model 1 & Model 2\\
 & Value & Value\\
\tableline\\[-3pt]
$n_{\rm {H,Gal}}$\tablenotemark{a} & $(2.1)$ & $(2.1)$\\[6pt]
$n_{\rm {H}}$ & $(0.0)$ & $2.2^{+6.6}_{-2.2}$\\[6pt]
$\Gamma$\tablenotemark{b}  & $(1.7)$ & $(1.7)$\\[6pt]
$N_{\rm {pl}}$ at 1 keV\tablenotemark{c} &  $8.4^{+2.4}_{-2.2} \times 10^{-6}$
	& $8.1^{+2.3}_{-2.2} \times 10^{-6}$\\[6pt]
$kT_{\rm{gas}}$ (keV) &$0.34^{+0.02}_{-0.02}$ & $0.32^{+0.03}_{-0.03}$ \\[6pt]
$Z_{\rm C,N} (Z_{\odot})$ & $(1.0)$ & $9.1^{+7.3}_{-5.1}$\\[6pt]
$Z_{\rm 0-Ca} (Z_{\odot})$ & $(1.0)$ & $(1.0)$\\[6pt]
$Z_{\rm Fe,Ni} (Z_{\odot})$ & $(1.0)$ & $1.6^{+1.0}_{-0.6}$\\[6pt]
$K_{\rm{vapec}}$\tablenotemark{d} &$3.2^{+0.3}_{-0.4} \times 10^{-5}$
	 & $2.7^{+1.3}_{-0.9} \times 10^{-5}$\\[6pt]
$\chi^2$/dof & $43.0/51$ & $30.7/48$\\[6pt]
\tableline\\[-3pt]
$L_{\rm x, 0.3-10}$\tablenotemark{e} &$2.0^{+0.3}_{-0.3} \times 10^{39}$
	& $2.3^{+0.3}_{-0.3} \times 10^{39}$ \\[6pt]
$L_{\rm x, 0.3-1.5}$\tablenotemark{f} & $1.3^{+0.1}_{-0.1} \times 10^{39}$
	& $1.6^{+0.1}_{-0.1} \times 10^{39}$\\[6pt]
$L_{\rm x, th}$\tablenotemark{g} &$1.0^{+0.1}_{-0.1} \times 10^{39}$
	& $1.3^{+0.1}_{-0.1} \times 10^{39}$\\[6pt]
\tableline
\end{tabular}
\tablenotetext{a}{From Dickey \& Lockman (1990).}
\tablenotetext{b}{Fixed to the typical LMXB value of 1.7.}
\tablenotetext{c}{Units of ph. keV$^{-1}$ cm$^{-2}$ s$^{-1}$.}
\tablenotetext{d}{Apec/vapec model normalization $K_{\rm vapec} =
10^{-14}/\{4\pi\,[d_A\,(1+z)]^2\}\,
\int n_e n_H {\rm{d}}V$, where $d_A$ is
          the angular size distance to the source (cm), $n_e$ is the electron
          density (cm$^{-3}$), and $n_H$ is the hydrogen density (cm$^{-3}$).}
\tablenotetext{e}{Total unabsorbed luminosity of the unresolved emission
in the $0.3$--$10$ keV band; units of erg s$^{-1}$.}
\tablenotetext{f}{Total unabsorbed luminosity of the unresolved emission
in the $0.3$--$1.5$ keV band; units of erg s$^{-1}$.}
\tablenotetext{g}{Unabsorbed luminosity of the diffuse thermal-plasma
component in the $0.3$--$1.5$ keV band; units of erg s$^{-1}$.}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Best-fit parameters for the spectrum of the unresolved
X-ray emission in NGC\,3377. Spectral model: {\tt wabs*(apec+po)}.
Values in brackets were kept fixed. Errors are 90\%
confidence levels for one interesting parameter ($\Delta \chi^2 =
2.7$). Parameters and units defined as in Table 4.
\label{tbl-6}}
\begin{tabular}{l@{\ \ \ \ \ }r}
\tableline\tableline\\[-6pt]
Parameter & Value\\
\tableline\\[-3pt]
$n_{\rm H}$\tablenotemark{a} & $(2.9)$ \\[6pt]
$\Gamma$  & $(1.7)$ \\[6pt]
$N_{\rm {pl}}$ at 1 keV
	& $3.1^{+2.7}_{-2.5}\times 10^{-6}$ \\[6pt]
$kT_{\rm{gas}}$ (keV) & $0.57^{+0.19}_{-0.27}$ \\[6pt]
$Z (Z_{\odot})$ & $(1.0)$ \\[6pt]
$K_{\rm{apec}}$ &  $2.5^{+1.7}_{-1.5} \times 10^{-5}$\\[6pt]
$\chi^2$/dof & $1.4/5$ \\[6pt]
\tableline\\[-3pt]
$L_{\rm x, 0.3-10}$
	& $4.2^{+3.2}_{-2.2} \times 10^{38}$ \\[6pt]
$L_{\rm x, 0.3-1.5}$
	& $2.0^{+1.0}_{-0.8} \times 10^{38}$ \\[6pt]
$L_{\rm x, th}$
	& $1.0^{+1.4}_{-0.5} \times 10^{38}$ \\[6pt]
\tableline
\end{tabular}
\tablenotetext{a}{fixed to the Galactic line-of-sight value, units of
$10^{20}$ cm$^{-2}$; from Dickey \& Lockman (1990).}
\end{center}
\end{table}

\begin{deluxetable*}{lccc@{\hspace{-0.2cm} }ccccr}
\tabletypesize{\scriptsize}
\tablecaption{Fitting parameters of the isothermal $\beta$-model
for the extended soft X-ray emission, and central gas
densities. Errors are 90\%
confidence levels for one interesting parameter ($\Delta \chi^2 = 2.7$)
\label{tbl-4}}
\tablewidth{0pt}
\tablehead{
\colhead{} &&&&&&&&\\[6pt]
\colhead{Galaxy} & \colhead{Det. limit} & \colhead{Counts}
	& \colhead{$S'_0$} & \colhead{$S_0$}
	& \multicolumn{2}{c}{$r_c$} & \colhead{$\beta$} &
	\colhead{$n_0$}\\
 & \colhead{(erg s$^{-1}$)} & \colhead{($0.3$--$1.5$ keV)} &
	\colhead{(ct s$^{-1}$ arcsec$^{-2}$)}
	& \colhead{(erg cm$^{-2}$ s$^{-1}$ arcsec$^{-2}$)}
	& \colhead{(arcsec)} & \colhead{(pc)} &
	& \colhead{(cm$^{-3}$)}}
\startdata
\, & \, & & & & & & & \\[6pt]
N821\tablenotemark{a} & $2 \times 10^{38}$
	& \multicolumn{6}{c}{Too faint for spatial analysis}
	& $0.03^{+0.07}_{-0.02}$\\[6pt]
N3377 & $4 \times 10^{37}$ & $152 \pm 18$
	& \hspace{-0.2cm}$(15.9 \pm 3.1) 10^{-6}$ & $(3.8\pm0.8) 10^{-17}$ &
	$4.0\pm0.5$ & $215\pm25$ & $0.50^{+0.03}_{-0.02}$
	& $0.022\pm0.004$\\[4pt]
N4486B & $1 \times 10^{38}$
& \multicolumn{6}{c}{No significant detection of unresolved emission}
	& $ < 0.01$\\[4pt]
N4564 &$2 \times 10^{38}$ & $25\pm8$& \hspace{-0.2cm}$(7.7\pm4.2) 10^{-6}$ &
	$(2.1\pm1.3) 10^{-17}$ & $8.7^{+2.3}_{-2.8}$ & $630^{+170}_{-200}$
	&$0.76^{+0.36}_{-0.14}$ & $0.011\pm0.006$ \\[4pt]
N4697\tablenotemark{b}
	& $(1 \times 10^{37})$\tablenotemark{c}
	& $1038\pm62$ & \hspace{-0.2cm}$(19.2\pm1.7) 10^{-6}$&
	$(9.3\pm1.2) 10^{-17}$ & $6.1\pm1.7$ & $344\pm100$
	& $0.40^{+0.01}_{-0.01}$
	& $0.017\pm0.003$\\[4pt]
N4697\tablenotemark{d}
	& $(1 \times 10^{37})$\tablenotemark{c}
	& $2167\pm92$ & \hspace{-0.2cm}$(14.0\pm1.0) 10^{-6}$&
	$(9.4\pm1.2) 10^{-17}$ & $4.6\pm1.7$ & $260\pm100$
	& $0.37^{+0.01}_{-0.01}$
	& $0.019\pm0.003$\\[4pt]
N5845 & $3 \times 10^{38}$
	& $55\pm12$& \hspace{-0.2cm}$(10.7\pm3.6) 10^{-6}$&
	$(3.3\pm1.0) 10^{-17}$&$5.5\pm1.1$&$690 \pm150$&$0.59^{+0.10}_{-0.06}$
	& $0.011\pm0.004$\\
\enddata
\tablenotetext{a}{See Fabbiano et al.~2004.}
\tablenotetext{b}{2000 Jan observation.}
\tablenotetext{c}{Detection limit for the point sources from all datasets
combined.}
\tablenotetext{d}{Combined 2003--2004 observations.}
\end{deluxetable*}

\subsection{Density of the hot ISM}

The gas surface brightness $S(r)$ is the projection on the sky
of the plasma emissivity $\epsilon \approx n_e^2 \Lambda(T_{\rm {gas}})$
where $n_e$ is the gas electron density and $\Lambda(T_{\rm {gas}})$
the cooling function at a gas temperature $T_{\rm {gas}}$.
We can obtain the gas
density $n_e$ by de-projecting the surface brightness,
if we know the radial dependence
of $T_{\rm {gas}}$. The problem can be simplified
by assuming that the temperature is constant,
and that the radial profile
of the surface brightness can be parameterized
with isothermal $\beta$-models (Cavaliere \& Fusco-Femiano
1976, 1978) of the form:
\begin{equation}
S(r) = S_0 \left[1+ \left(r/r_c\right)^2\right]^{0.5-3\beta},
\end{equation}
where the fitting parameters are $S_0$ (central surface
brightness), $r_c$ (core radius) and $\beta$.
Physically, it can be shown that
$\beta \sim (\sigma^2/T_{\rm {gas}})$
where $\sigma$ is the stellar velocity
dispersion (e.g., Ettori 2000 and references therein).
In this approximation,
one can derive an analytical expression for the density profile
(Ettori 2000):
\begin{equation}
n_e(r) = n_0 \left[1+ \left(r/r_c\right)^2\right]^{-3\beta/2},
\end{equation}
where
\begin{equation}
n_0^2 = E_0\,\pi^{-0.5}\,\frac{\Gamma(3\beta)}{\Gamma(3\beta -
0.5)}\,\frac{1}{r_c\,\Lambda(T_{\rm {gas}})}.
\end{equation}
In equation~3, $\Gamma(x)$ is the gamma function, and
the central surface emissivity $E_0$ (luminosity
per unit surface area of the emitter, units of
erg s$^{-1}$ cm$^{-2}$) is simply related to the central
surface brightness $S_0$ (received flux from a unit
angular element on the sky, units of
erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$) by a constant
geometrical factor\footnote{To obtain
this conversion factor, recall that the area of an element of sky
seen under a solid angle of $1$ arcsec$^2$ corresponds
to a physical size of $(48.5 {\rm{pc}})^2 \times
(d/10{\rm{Mpc}})^2 = 2.35 \times 10^{-11} (d/{\rm{cm}})^2$ cm$^2$,
where $d$ is the distance to the source. The flux $S$
received from a unit angular element of sky
is $1/(4\pi d^2)$ times the luminosity emitted from the
same unit angular element, or
$(2.35 \times 10^{-11}d^2)/(4\pi d^2)$ times the luminosity $E$
emitted from a unit surface area of the source.
Inverting this relation gives the numerical factor
between surface emissivity and surface brightness
used in the text.},
$E_0 = 5.34 \times 10^{11} S_0$.

In the case of hot gas in elliptical galaxies,
the gas temperature varies at most by a factor $\approx 2$
over the extent of the diffuse emission
(e.g., Sivakoff, Sarazin \& Carlin 2004;
Sivakoff et al.~2003), while the gas
density can vary by a few orders of magnitude.
Hence, taking into account the limited number
of detected counts, we can accept the isothermal model
as a good approximation for our case.
We have now all the elements
to determine the radial density profile $n(r)$
and central density $n_0$ from equations 1--3,
in the isothermal $\beta$-model approximation.
For each galaxy, we use the surface brightness
profile inferred from the count rates of the unresolved
emission, as described earlier, together with the cooling
function $\Lambda(T)$ from Sutherland \& Dopita (1993).
The results are given in Table 6,
and the inferred radial profiles of the electron density
(equation 2) are plotted in Figure 13.

\begin{figure}[t]
\includegraphics[angle=270,scale=0.36]{f13.eps}
\caption{Density profiles of the X-ray emitting gas,
as a function of projected physical distance from the central BH.
Dotted (black) line: NGC\,3377; dash-dotted (red) line:
NGC\,4564; dashed (green) line: NGC\,4697; solid (blue)
line: NGC\,5845. The four curves have been obtained
by deconvolving the best-fit $\beta$-model profiles.
In addition, for three of the four galaxies,
we overplot the density directly estimated
from the X-ray emission in three concentric shells,
using an onion-skin approach, with no model assumptions.
$\beta$-models and empirical onion-skin approach
give a consistent estimate of the hot-gas density
at distances $\ga 150$ pc; inside that radius,
the degree of flattening in a core is somewhat
model-dependent.\label{fig9}}
\end{figure}
The above method for the deprojection of the surface
brightness is based on the assumption of a specific
analytic form for the radial profile ($\beta$-model).
It is also possible to obtain an estimate of the
density profile without assuming any models,
using an onion-skin approach\footnote{An onion-skin
approach consists in dividing
the observed region into concentric shells,
projected as concentric annuli on the sky.
The problem is solved from outside in.
From the measured average temperature
and luminosity of the gas in the outermost
annulus, one calculates the average gas density
in the outermost shell. The observed emission
from the second annulus is given by emission
from the second shell, plus some emission
from the fraction of the outermost shell
projected in front of it; the latter component
is by now known, and can be subtracted.
And so on, until the innermost annulus,
whose emission comes from the innermost
shell plus all other shells projected
in front of it.} together with
the best-fit emission measure from the fitted
spectral model in {\footnotesize{XSPEC}}
(recalling that $K_{\rm{apec}} \sim \int n_e^2 {\rm{d}}V$).
We did so for three galaxies in our
sample\footnote{in NGC\,4697, the diffuse
emission extends to much larger distances; hence this
onion-skin method would give meaningful results
only by using a larger number of shells.
A more detailed analysis of the radial profile
of the diffuse emission in NGC\,4697
is beyond the scope of this work, which
focuses on the central gas density; see also
Sarazin et al.~(2001).}: we
defined three concentric shells,
including the whole surface where diffuse emission
is detected (i.e., to a radial distance of $\approx 60\arcsec$),
and calculated the average count rate and luminosity
from each one. From the best-fit emission measure,
we then inferred the average gas density in the
three shells. The results are in agreement
with the $\beta$-model estimates (Figure 13),
implying gas densities $\sim 1$--$2 \times 10^{-2}$
cm$^{-3}$ at $\sim 100$ pc from the SMBH.

Estimating the hot-gas density in the two remaining
galaxies (NGC\,821 and NGC\,4486B) is more difficult,
because of the limited number of photons.
For NGC\,821, an average density
$n_e = 4.1_{-1.5}^{+10.9} \times 10^{-3}$ cm$^{-3}$
was determined in Fabbiano et al.~(2004).
However, this value was estimated
over a region of radius $20\arcsec$, excluding the point
sources. Therefore, it underestimates the central density.
By scaling this value in analogy with the density profiles
inferred for the other four galaxies,
we estimate that it corresponds to central
densities $n_e = 0.03_{-0.02}^{+0.07}$ cm$^{-3}$
(i.e., $\log n_e = -1.5 \pm 0.5$).
For NGC\,4486B, a central density $\approx 10^{-2}$
cm$^{-3}$ is the upper limit.

For the typical values of ISM temperature and density
in our target galaxies,
the characteristic radiative cooling timescale inside
$r_{\rm acc}$ is $\sim 10^8$ yr,
in the absence of external heating.
In fact, both Type-Ia SNe and feedback
from the SMBH are possible heating sources,
as we shall discuss in Section 6.2.
The radiative cooling timescale is much longer
that the accretion timescale ($\sim$
free-fall timescale), which is $\la 10^5$ yr
within the sphere of influence of the SMBH
(defined as in Section 4.3).

\subsection{Bondi inflow rate of hot gas}

From the hot-gas density (determined in Section 4.2),
we can estimate the accretion rate
onto a point-like mass within a uniform gas,
in the standard model
of steady, spherically symmetric accretion (Bondi 1952;
Frank, King \& Raine 2002; Ruffert \& Arnett 1994):
\begin{equation}
\dot{M}_{\rm B} = \pi G^2 M^2_{\rm BH}
\frac{\rho(\infty)}{c^3_{\rm s}(\infty)}
\left(\frac{2}{5-3\gamma}\right)^{(5-3\gamma)/(2\gamma -2)},
\end{equation}
where $\gamma$ is the polytropic index ($1 \le \gamma \le 5/3$),
$c_{\rm s} = (\gamma kT/\mu m_{\rm p})^{1/2}$ is the sound
speed of the gas, $m_{\rm p}$ is the proton mass,
$\mu m_{\rm p} $ is the mean mass per particle of gas.
We take $\mu = 0.62$ (solar composition)
and $\gamma = 1.33$ for consistency with Pellegrini (2005).
For the gas temperature and density ``at infinity''
we take their values at the Bondi accretion radius
$r_{\rm acc} = 2GM_{\rm BH}/c^2_{\rm s}(\infty)$,
which defines the region inside which the gravitational
pull of the SMBH dominates over the internal thermal
energy of the gas. Henceforth, we shall refer
to this region simply as the SMBH ``sphere of influence''
\footnote{It is also possible to define a dynamical sphere
of influence, with a radius $\sim GM_{\rm BH}/\sigma^2$,
where $\sigma$ is the stellar velocity dispersion. In practice,
the two definitions are very similar for this class of systems,
because the stellar velocity dispersion $\approx$
thermal velocity of virialized hot gas.}.

A spherically-symmetric model is probably
not applicable to the behaviour of the
accretion flow very close to the SMBH
(e.g., Frank et al.\ 2002; Krumholz, McKee \& Klein~2005, 2006);
however, we can still take the Bondi rate from equation~(4) as
a first-order approximation of the rate
at which the hot ISM is captured by the sphere of influence
of the BH, at $r =r_{\rm acc}$. As we shall
discuss later, not all of the inflowing, hot gas
reaches the BH, and, conversely, there are
other sources of gas inside $r_{\rm acc}$.

We can more usefully express the accretion radius
and Bondi rate as:
\begin{equation}
r_{\rm acc}  = 8.4\,\left(\frac{M_{\rm BH}}{10^8 M_{\odot}}\right)\,
\left(\frac{0.5 {\rm ~keV}}{kT}\right)\ {\rm pc},
\end{equation}
\begin{eqnarray}
\dot{M}_{\rm B} &=& 1.56 \times 10^{-5}
	\left(\frac{M_{\rm BH}}{10^8 M_{\odot}}\right)^2\,
	\left(\frac{0.5 {\rm ~keV}}{kT}\right)^{3/2}\nonumber\\
	&&\times \left(\frac{n_0}{0.01 {\rm ~cm}^{-3}}\right) \ \
	M_{\odot}\ {\rm yr}^{-1}.
\end{eqnarray}
The numerical values for our target galaxies
are listed in Table 7, Cols.~3,4,5
For the hot gas phase, we used a temperature
of $0.46$ keV for NGC\,821 (Fabbiano et
al.\ 2004), $0.57$ keV for NGC\,3377,
$0.33$ keV for NGC\,4697, and $0.5$ keV for the other two galaxies
with a detected hot-gas component.

\subsection{Is the central gas density underestimated?}

The hot-gas densities inferred from the deprojection
of the diffuse X-ray emission in our target galaxies
are comparable (though, on the lower end) with
those found in Pellegrini's (2005) sample
($n_e \sim 0.01$--$0.5$ cm$^{-3}$)
and with the hot-gas density recently
estimated for NGC\,3379 ($n_e \sim 0.01$ cm$^{-3}$:
David et al.\ 2005).
There are only two nearby galaxies
for which the accretion radius has been
clearly resolved from {\it Chandra} images.
For the Milky Way ($r_{\rm acc} = 1\farcs8$),
the hot gas density $n_e(r_{\rm acc}) \sim 10^2$ cm$^{-3}$
(Baganoff et al.~2003; Quataert 2003; Cuadra et al.~2005).
For M\,87 ($r_{\rm acc} \approx 1\farcs9$),
the central density is $\approx 0.17$
cm$^{-3}$ (Di Matteo et al.~2003).
Four more galaxies have accretion radii
$r_{\rm acc} \approx 1\arcsec$:
in M\,31, a central density $n_e(r<200~{\rm pc}) \sim 0.1$
cm$^{-3}$ has been estimated (Garcia et al.~2005; Dosaj
et al.~2002); $n_e \approx 0.04$ cm$^{-3}$ in the center
of NGC\,5128 (Evans et al.~2004; Kraft et al.~2003);
$n_e \sim 0.1$ cm$^{-3}$ for the Sombrero galaxy
(Pellegrini et al.~2003a); and
$n_e \sim 0.5$ cm$^{-3}$ for NGC\,4649
(Soldatenkov, Vikhlinin \& Pavlinsky ~2003).

Our target galaxies are slightly more distant
and/or less massive, and, as a result, their accretion radii
are not resolved. The observed radial profiles
provide a good estimate of the gas distribution
at distances $\ga 150$ pc;
however, we do not have any direct measurements
at smaller radii (Figures 9, 13), because
of a lack of spatial resolution.
If we want to determine the amount
of gas available for accretion inside the sphere of influence
of the SMBH, we need to estimate the density
at radii $\sim 10$ pc.
If $\beta$ models are a good approximation,
the presence of a core implies that the density
stays approximately constant inside the inner
$r \sim 100$ pc, making this task easier.
But we cannot {\it a priori} rule out
the possibility that the gas density distribution
has a cusp in the nuclear region,
for example in correspondence of a massive star
cluster, or because cooling gas has accumulated
there. Although we cannot resolve
the nuclear gas emission in the {\it Chandra} images,
we know that it does not dominate the point-like
nuclear sources, which have harder, AGN-like spectra (see, e.g.,
the X-ray colors in Figure 3).
A conservative constraint is that the thermal emission
inside $\sim 10$ pc has an X-ray luminosity
$\la 10^{38}$ erg s$^{-1}$. For typical emissivities
corresponding to temperatures $\sim 0.3$--$0.6$ keV,
this gives an order-of-magnitude constraint on the
``true'' central density $n_e \la 10$ cm$^{-3}$ for
the five target galaxies in which hot gas is found.

Moreover, the hot, X-ray emitting ISM may represent
only a small fraction of the gas in the inner region.
That can be the case in systems where gas can cool efficiently
(cooling timescale $<$ accretion timescale),
or, vice versa, if gas is injected into the inner regions
in a cool or warm phase (for example,
through stellar winds) and is accreted
before it has time to virialize.
Cool and warm gas would of course elude an X-ray
investigation.
Therefore, the Bondi inflow rate of the hot-ISM component
into the sphere of influence of the SMBH represents
a firm lower limit to the total gas mass available for accretion.
We shall compare the SMBH luminosity expected
for gas accretion at the Bondi rate, in the case
of a radiatively-efficient inflow (standard disk,
Shakura \& Sunyaev 1973)
and for radiatively inefficient solutions.
Conversely, we shall also estimate the accretion rate
(expressed as a fraction of the Bondi rate)
required to explain the observed luminosities, if
we assume a radiatively-inefficient scenario.
In Paper II, we shall provide a complementary
estimate of the gas injection rate into
the SMBH sphere of influence, based
on optical studies of the stellar population
inside the accretion radius, following the example
of Fabbiano et al.~(2004).

\section{Discussion}

\subsection{Accretion rates and efficiencies}

Let us define a total accretion power
$P_{\rm acc} \equiv \eta' \dot{M} c^2$,
where $\eta'$ is the total accretion
efficiency (radiative plus mechanical), and
$\dot{M}$ is the rate at which matter actually
accretes onto the BH.
One immediate objective of our analysis
is to determine whether $\dot{M}_{\rm B}$
derived from X-ray observations of hot gas
is a good tracer of $\dot{M}$.
The efficiency $\eta' \la 0.3$, with its exact
value depending on the physics of the accretion flow
as well as on the BH spin parameter;
more generally, we shall talk of ``efficient''
accretion when $\eta' \sim 0.1$.
We then define a bolometric luminosity
$L_{\rm bol} = \eta \dot{M} c^2$, where
$\eta$ is the radiative efficiency. In general,
it is  $\eta \equiv f_{\rm r}\eta' \la \eta'$
where $f_{\rm r}$ is the fraction
of the accretion power released as radiation.
In standard accretion models, e.g., for
an accretion disk extended down
to the innermost stable orbit around the BH,
$\eta \approx \eta' \sim 0.1$; however,
it is possible that $\eta \ll \eta'$
if most of the accretion power is released,
for example, in a jet.
Finally, only a fraction $f_{\rm X} < 1$
of the bolometric luminosity
is emitted in the {\it Chandra} X-ray band,
$0.3$--$10$ keV; for accreting SMBHs in typical AGN,
$f_{\rm X} \sim 10\%$ (Elvis et al.~1994; Ho 1999).
We shall also define a dimensionless accretion rate
$\dot{m}\equiv 0.1 \dot{M} c^2/L_{\rm{Edd}}$,
so that
\begin{equation}
L_{\rm bol}/L_{\rm{Edd}} \equiv (\eta/0.1) \dot{m}.
\end{equation}
Similarly, we define
$\dot{m}_{\rm B} \equiv 0.1 \dot{M}_{\rm B} c^2/L_{\rm{Edd}}$.

From the {\it Chandra} observations, the X-ray
luminosity of the nuclear sources in our target galaxies
is extremely sub-Eddington:
$L_{\rm X}/L_{\rm{Edd}}  = f_{\rm X} L_{\rm bol}/L_{\rm{Edd}}
\sim 10^{-8}$--$10^{-7}$ (Table 7).
From the estimated densities of the hot ISM, we infer
dimensionless Bondi accretion rates
$\dot{m}_{\rm B}  \sim 10^{-5}$
(Table 7).

Followig the example of previous studies (e.g., Pellegrini 2005),
we can compare the observed X-ray luminosities
with the inferred Bondi inflow rates.
For the nuclei of our target galaxies,
the X-ray observations imply the following constraint:
\begin{equation}
\frac{L_{\rm{X}}}{0.1\dot{M}_{\rm{B}}c^2}
	= f_{\rm X} f_{\rm r} \left(\frac{\eta'}{0.1}\right)
	\left(\frac{\dot{m}}{\dot{m}_{\rm B}}\right)
	\sim 5 \times 10^{-4}{\rm \,-\,}10^{-2}
\end{equation}
(Table 7, Col.~8).
Theoretical arguments and observational evidence in a variety
of accreting galactic nuclei suggests that $f_{\rm X}$
varies at most within a factor of $\approx 2$--$3$.
The crucial problem is to constrain the
other three parameters
($f_{\rm r}$, $\eta'$ and $\dot{m}/\dot{m}_{\rm B}$)
in equation~(8), to make them consistent with the observed
SMBH luminosities. This will be a test for the predictions
of different accretion scenarios.
We shall take into account both our new sample
of six galaxies, and another eighteen galaxies
for which Bondi rates and SMBH luminosities
have been measured or constrained
(Pellegrini 2005; Garcia et al.~2005;
David et al.~2005).

\begin{deluxetable*}{lccccccccc}
\tablewidth{0pt}
\tablecaption{Physical properties of the nuclear BHs ($L_{\rm X}$ is
the unaborbed luminosity in the $0.3$--$10$ keV band)\label{tbl-7}}
\tablehead
{
\colhead{\,}&\colhead{\ }&\colhead{}&\colhead{}
	&\colhead{}&\colhead{}&&&&\\[8pt]
\colhead{Galaxy} & \colhead{$M_{\rm{BH}}$} & \colhead{$r_{\rm acc}$}
        & \colhead{$\dot{M}_{\rm{B}}$} &
	\colhead{$\log
	\left(\frac{\dot{M}_{\rm{B}}}{\dot{M}_{\rm{Edd}}}\right)$}
	& \colhead{$\log L_{\rm{X}}$}
        & \colhead{$\log
	\left(\frac{L_{\rm{X}}}{L_{\rm{Edd}}}\right)$}
        & \colhead{$L_{\rm{X}}/(0.1\dot{M}_{\rm{B}}c^2)$}
	& \colhead{$\log L_{\rm X,ADAF}$}
	& \colhead{$\dot{M}/\dot{M}_{\rm B}$}	\\[2pt]
  &  \colhead{($10^8 M_{\odot}$)} & \colhead{(arcsec/pc)}
	& \colhead{($M_{\odot}$ yr$^{-1}$)} &
  & \colhead{(erg s$^{-1}$)} & &
\colhead{$= f_{\rm X}
	\left(\frac{\eta}{0.1}\right)
	\left(\frac{\dot{m}}{\dot{m}_{\rm B}}\right)$}
	& \colhead{($\dot{M} = 0.1\dot{M}_{\rm B}$)}
	&\colhead{}\\[4pt]
\colhead{(1)}  &  \colhead{(2)}  &  \colhead{(3)}  & \colhead{(4)}  &
                  \colhead{(5)}  &  \colhead{(6)}  &  \colhead{(7)}
		 &  \colhead{(8)} &  \colhead{(9)} &  \colhead{(10)}}
\startdata
\, & \, & & & & & & & &\\[6pt]
N821 & $0.85^{+0.35}_{-0.35}$ &$0\farcs07/7.8$ & $3.8 \times 10^{-5}$&
$-4.70^{+1.05}_{-0.43}$  & $<38.7$
        & $< -7.3$  &   $<6 \times 10^{-3}$  & $36.4$ & $< 2$\\[4pt]
N3377 & $1.0^{+0.9}_{-0.1}$ & $0\farcs14/7.4$ & $2.8\times10^{-5}$
 & $-4.92^{+0.50}_{-0.24}$ & $38.5$
        & $-7.57^{+0.10}_{-0.39}$  &  $2.3\times10^{-3}$ & $36.0$ &
	$1.3$  \\[4pt]
N4486B
 & $\left[6.0^{+3.0}_{-2.0}\right.$ & $0\farcs61/50.4$ &
 $<1\times10^{-3}$ & $< -4.1$ & $38.4$
        &  $-8.5^{+0.2}_{-0.3}$ &  $> 4\times10^{-5}$  & $<38.2$
	&$\left.>0.1\right]$\\[3pt]
	 & $0.5^{+0.5}_{-0.2}$ & $0\farcs05/4.2$ &
$<1\times10^{-5}$ & $< -5.0$ & $38.4$
        &  $-7.44^{+0.28}_{-0.39}$ &  $> 4\times10^{-3}$  & $<35.2$ &
	$> 2$ \\[4pt]
N4564 & $0.56^{+0.03}_{-0.08}$ & $0\farcs06/4.7$ &
$5.4\times10^{-6}$
  &$-5.38^{+0.40}_{-0.40}$ & $38.9$
        & $-6.96^{+0.23}_{-0.22}$  &  $0.03$ & $34.3$ & $6$\\[4pt]
N4697 & $1.7^{+0.2}_{-0.1}$ & $0\farcs38/21.6$ &
$1.5\times10^{-4}$
 & $-4.41^{+0.11}_{-0.10}$ & $38.6$
        & $-7.74^{+0.03}_{-0.06}$  &  $4.6\times10^{-4}$ & $37.5$
	& $0.3$ \\[4pt]
N5845 & $2.4^{+0.4}_{-1.4}$ & $0\farcs16/20.2$ &
	$9.9\times10^{-5}$
	& $-4.74^{+0.37}_{-0.44}$& $39.4$
        & $-7.00^{+0.26}_{-0.08}$  &  $5.6\times10^{-3}$
	& $36.8$ & $1.4$\\[4pt]
\enddata
\tablecomments{
Col.(1): galaxy ID. NGC\,4486B is listed twice:
the first line (in square brackets) assumes
the mass of Kormendy et al.~(1997); the second line
assumes the lower mass indirectly derived in our Paper II.
Col.(2): SMBH mass, see Table 1.
Col.(3): accretion radius, from equation~5.
Col.(4),(5): Bondi inflow rates estimated from the density
and temperature of the diffuse hot ISM.
Col.(6),(7): unabsorbed X-ray luminosity of the nuclear source,
in the $0.3$--$10$ keV band.
Col.(8): comparison between observationally-determined
X-ray luminosity and bolometric luminosity expected from standard,
radiatively-efficient accretion.
Col.(9):
prediction for the $0.3$--$10$ keV luminosity for
an ADAF accretion flow, with a viscosity parameter $\alpha = 0.1$;
values were suitably rescaled from the fit of Merloni et al.~(2003),
using an accretion rate $\dot{m} \approx \alpha \dot{m}_{\rm B}$,
as expected for an ADAF scenario.
Col.(10): accretion rates required to satisfy the observed
(unabsorbed) X-ray luminosities, as a fraction
of the Bondi inflow rate of hot gas.
}
\end{deluxetable*}

\subsection{Advective solutions (ADAF)}

Standard disk accretion predicts
$f_{\rm r} \sim 1$, $\eta' \sim 0.1$
and $\dot{m}/\dot{m}_{\rm B} \sim 1$.
This is ruled out by the observations: if the gas available
were efficiently accreted at the Bondi rate, we would expect
X-ray luminosities $\sim 10$--$100$ times higher
than observed in our target galaxies.
Moreover, there is a theoretical reason
why we expect standard accretion to be ruled out:
in the limit of low accretion rates
($\dot{m} \la 0.01$), the accretion flow cannot cool
efficiently within the infall timescale; therefore,
most of the gravitational energy is carried by the ions
(protons and nuclei)
and advected into the BH, rather than being
transferred to the electrons and radiated away.
This regime is known as Advection-Dominated Accretion Flow (ADAF,
Narayan \& Yi 1994; Ichimaru 1977).

In the simplest, self-similar ADAF model,
the luminosity
$L_{\rm ADAF} \approx 0.1\dot{M}c^2(\dot{m}/\alpha^2)
\approx (\dot{m}/\alpha)^2 L_{\rm Edd}$
(Narayan \& Yi 1994),
where $\alpha$ is the dimensionless viscosity
parameter, with $0.1 \la \alpha < 1$; a value $\alpha \approx 0.1$
is typically assumed for galactic SMBHs.
This corresponds to a radiative efficiency
$\eta \sim 0.1 \dot{m}/\alpha^2 \sim 10 \dot{m} < 0.1$.
A more rigorous calculation (Merloni et al.~2003),
taking into account synchrotron and inverse-Compton
contributions, indicates that
the X-ray luminosity $L_{\rm X,\,ADAF}
\propto \dot{m}^{2.3} L^{0.97}_{\rm Edd}$. In the parameter range
of our study, the two estimates
agree within a factor of 3.
Moreover, in the ADAF scenario, the radial inflow velocity
inside the Bondi radius is only $\sim \alpha c_{\rm s}$.
For a fixed hot-gas density and temperature
at infinity, this implies that a more accurate
estimate of the accretion rate
is $\dot{m} \sim \alpha \dot{m}_{\rm B} \sim 0.1
\dot{m}_{\rm B}$ (Narayan 2002).

Using the ADAF values of efficiency and accretion rate
(from suitable rescalings of
the Merloni et al.~2003 model fit),
we estimated the $0.3$--$10$ keV luminosity
predicted by the advective model for our target galaxies
(Table 7, Col.~9).
Conversely, we also determined the accretion rates
that would be required to reproduce the observed
luminosities, at an assumed ADAF-like efficiency
(Table 7, Col.~10).
We find (Table 7 and Figure 14) that, at an accretion rate
$\dot{m} \approx 0.1 \dot{m}_{\rm B}$
(as predicted by the ADAF model),
the SMBH luminosities are underestimated by
a factor of 10 (for NGC\,4697) and by three to four
orders of magnitude for the other five galaxies.
Conversely, we find that accretion rates
$\dot{m} \approx 0.6$--$6 \dot{m}_{\rm B}$
are required to reproduce the observed luminosities,
at an ADAF-like efficiency.
In particular, an accretion rate $\approx 0.3 \dot{m}_{\rm B}$
suffices to explain the X-ray luminosity of NGC\,4697,
but rates $\dot{m} \ga \dot{m}_{\rm B}$ are required
for the other five galaxies.
If the hot-ISM inflow was the only
source of gas, such a high accretion rate
would be difficult to reconcile with the ADAF prediction.
We take this finding as a strong
indication that, for our target galaxies,
the gas available for accretion has been underestimated:
there are other sources of fuel
for the SMBH, such as stellar winds,
that could be more significant
than the Bondi inflow;
this would explain why we infer $\dot{m} \ga \dot{m}_{\rm B}$.
We shall discuss this scenario in detail
in Paper II (see also Fabbiano et al.~2004; Pellegrini 2005).

\begin{figure}
\includegraphics[angle=270,scale=0.36]{f14a.eps}\\
\includegraphics[angle=270,scale=0.36]{f14b.eps}
\caption{Top panel: relation between normalized X-ray luminosity
$L_{\rm X}/L_{\rm Edd}$ and normalized Bondi accretion
rate $\dot{m}_{\rm B} = \dot{M}_{\rm B}/\dot{M}_{\rm Edd}$.
Bottom panel: zoomed-in view of the same diagram,
to allow a better identification of the
six galaxies of the present study
(datapoints plotted in red in the online version).
The datapoints
in black have been collected by Pellegrini (2005)
(see references therein for the original measurements;
a slightly revised mass has been adopted for NGC\,4636,
from Wang \& Zhang 2003).
The large filled circles (in blue, in the online version)
represent Sag A$^{\ast}$ (Baganoff et al.~2003)
and M\,31$^{\ast}$ (Garcia et al.~2005). The datapoint
plotted in green is NGC\,3379 (David et al.~2005).
We have compared the observed X-ray luminosities with
those predicted, as a function of Bondi accretion rate,
by: a) standard radiatively-efficient
accretion ($\eta = 0.1$, $\dot{m} = \dot{m}_{\rm B}$:
solid line); b) standard ADAF solution ($\eta = 10\dot{m}$,
$\dot{m} = 0.1 \dot{m}_{\rm B}$:
dashed-dotted line); c) a scenario
with ADAF-like efficiency ($\eta = 10\dot{m}$)
but higher accretion rate ($\dot{m} = \dot{m}_{\rm B}$:
dashed line).
SMBHs with X-ray luminosities falling below the standard
ADAF line are likely to require an additional mechanism
to reduce the true mass accretion rate further
below the Bondi value (e.g., ADIOS or CDAF solutions).
Conversely, for the six target galaxies
studied in this paper, the Bondi rate is too low
to be consistent with the observed X-ray luminosities,
if the accretion process has ADAF-like radiative
efficiencies. \label{fig10}}
\end{figure}

More generally, there are $\approx 20$ quiescent
or very faint SMBHs for which the SMBH mass
and X-ray luminosity, and the density of
the surrounding hot ISM,
have been estimated or constrained (Pellegrini 2005;
Garcia et al.~2005; David et al.~2005).
When plotted in an X-ray-luminosity
versus Bondi-rate diagram (Figure 14),
the distribution of the datapoints
does not provide any obvious correlations
between these two parameters,
or between the luminosities observed and
predicted by the ADAF model.
(The standard disk model fares much worse,
and is clearly ruled out for all but one galaxy).
Some SMBHs, including those in the six
galaxies of our sample,
tend to be more X-ray luminous than predicted
by the standard ADAF model (i.e., they lie
above the predicted $\dot{m} \approx 0.1 \dot{m}_{\rm B}$
ADAF line in Figure 14). This seems to be the case
for most galaxies with extremely low
hot-ISM densities in the nuclear region ($n_e \la 0.1$ cm$^{-3}$),
and therefore very low Bondi accretion rates
($\dot{m}_{\rm B} \la 10^{-4}$).
On the other hand, SMBHs in galaxies with estimated
Bondi accretion rates
$\dot{m}_{\rm B} \ga 10^{-4}$ tend to lie below
the ADAF line (Figure 14); that is, the ADAF model
overpredicts their observed luminosities, in some cases
by more than two orders of magnitude.

In the framework of inefficient accretion,
we can explain the former group of galaxies by considering
additional sources of gas which boost the accretion
rate above the classical Bondi value, as
we shall discuss in Paper II.
Conversely, an additional mechanism must be at work
in the latter group of galaxies,
reducing the true accretion rate to values
$\dot{m} \la 10^{-2} \dot{m}_{\rm B}$.
As an aside, we note that
the lack of SMBHs in the bottom left of the diagram
(i.e., with $\dot{m} \la 10^{-5}
\dot{m}_{\rm B}$ and $L_{\rm X} \la 10^{-8} L_{\rm Edd}$)
may be partly due to a selection effect, because
both the nuclear point-like source and the diffuse
hot-ISM emission would be close to or
below the {\it Chandra} detection
limit in those galaxies.

\subsection{Convection and outflows}

The original ADAF solution has subsequently been developed
by taking into account two physical effects that
are likely to further reduce the accretion rate, so that
$\dot{m} \ll \dot{m}_{\rm B}$ (see Narayan 2002
and references therein, for a review). Firstly,
the inflowing gas is likely to become convectively
unstable due to its inefficient cooling (Convection
Dominated Accretion Flow, CDAF:
Narayan et al.~2000); convective instabilities
are also likely to occur when a magnetic field
is frozen into the infalling gas
(Igumenshchev \& Narayan 2002); or when
the inflowing gas has a large angular
momentum (Proga \& Begelman 2003)
or vorticity (Krumholz, McKee \& Klein~2005, 2006). Secondly,
inefficient cooling causes the gas to have positive
Bernoulli parameter, thus triggering large-scale winds
and outflows which reduce the amount of gas reaching
the BH (Advection Dominated Inflow-Outflow Solutions,
ADIOS: Blandford \& Begelman 1999).

Low, ADAF-like efficiencies {\em and}
a reduced accretion rate with respect to the Bondi rate
($\dot{m}/\dot{m}_{\rm B} \sim 10^{-3}$--$10^{-2}$),
via CDAF and/or ADIOS solutions, is a possible explanation
for those galaxies in which hot-gas densities
are $\sim 0.1$--$10$ cm$^{-3}$,
and correspondingly, $10^{-4} \la \dot{m}_{\rm B}
\la 10^{-2}$. They lie below the standard ADAF line
in Figure 14. Among them, the SMBH in the Milky Way, for which
$\dot{m}_{\rm B} \approx 2 \times 10^{-4}$;
an accretion rate $\dot{m}/\dot{m}_{\rm B} \la 10^{-3}$
has been suggested (Quataert 2003).
The SMBH in the nucleus of M\,31 also
belongs to this group (Garcia et al.~2005).
At higher central gas densities ($\ga 10$ cm$^{-3}$,
corresponding to
$\dot{m}_{\rm B} \la 10^{-2}$), theoretical and observational
arguments show that accretion will become radiatively
efficient, with the formation of a standard, optically-thick,
geometrically-thin accretion disk. SMBHs in this regime
will appear as fully-fledged AGN.

\subsection{Intermittent accretion}

We have assumed so far that
the accretion rate can be expressed as a constant
fraction of the gas available.
We also need to take into account
the possibility that accretion is intermittent, so that
the system may go through phases when
the SMBH accretion rate is much lower than the Bondi
inflow rate, and subsequent phases in which it can be
much higher. The nuclear region would alternatively
build up or be depleted of gas.

In one version of this scenario, it is the feedback from the nuclear
X-ray source that drives and self-regulates the activity cycles:
when the nuclear source is active, the surrounding ISM
is radiatively heated (Ciotti \& Ostriker 2001;
Ostriker \& Ciotti 2005),
or, in the case of radiatively-inefficient
accretion, is heated by the bulk kinetic
energy of jets, outflows or convective motions
(Binney \& Tabor 1995;
Omma et al.~2004), until the radial inflow is stopped
and the nucleus switches off. Thus, the Bondi inflow
estimated from the gas density at large radii
(at the limit of the {\it Chandra}
resolution) may not provide a good indication
for the time-dependent accretion rate and SMBH
luminosity. This is a possible explanation for
the observed lack of any relation
between $L_{\rm X}$, $M_{\rm BH}$ and $\dot{M}_{\rm B}$
(Pellegrini 2005).

Alternatively, accretion may cycle
between radiatively inefficient flows (e.g., CDAF)
in which gas accumulates inside the accretion radius,
and a standard, radiatively-efficient accretion disk
in which the stored gas is quickly accreted into the BH.
However, this transition should only happen
at accretion rates $\dot{m} \sim 10^{-2}$,
much higher than estimated in our target galaxies.

Intermittent accretion may also be due to
thermal-viscous ionization instabilities, triggered
by an ionization front propagating inwards or outwards
across an accretion disk
(Siemiginowska, Czerny \& Kostyunin 1996); a change
of the viscosity parameter by a factor of $4$
between the two states may lead to changes
in luminosity by four order of magnitudes.
Again, this scenario requires that the inflowing gas
can cool efficiently to form an accretion disk.

It was suggested (Fabbiano et al.~2004)
that intermittent ejections may be responsible
for the knotty jet-like feature in the nuclear
region of NGC\,821.
If this emission is connected with nuclear activity,
the most likely explanation would be that of thermal
emission from shocks in the hot ISM
resulting from a past nuclear outburst.

\section{Conclusions}

We have used {\it Chandra}, supported by archival
{\it HST} and ground-based optical images,
to study the nature, accretion power
and mass budget of X-ray-faint SMBHs in elliptical/S0 galaxies.
For this purpose, we have chosen a sample of six nearby galaxies
with quiescent SMBHs, five of which with
accurate BH mass determinations (from stellar kinematics).
From the X-ray images, we found different morphologies
for the X-ray emission in the nuclear region.
In NGC\,821, a jet-like feature has been previously
suggested (Fabbiano et al.~2004). In NGC\,4697,
the nuclear emission is resolved into a few point-like sources
(one of which is coincident with the nucleus). In other cases,
(NGC\,3377, NGC\,4564, NGC\,5845) the nuclear emission
is extended on scales of a few hundred pc
but its morphology remains unclear.
Obscuration of the nuclear source may play
a role in some cases; for example, in at least one galaxy
(NGC\,5845), the SMBH may be partly obscured by an edge-on
disk of dust and stars, and the observed extended X-ray
emission may be due to scattering in the surrounding plasma
above and below the disk plane.

In five of the six galaxies, the X-ray colors
of the nuclear emission are hard, consistent with
an AGN spectrum. Hence, it is likely that most of
the nuclear emission comes from the accreting SMBH, rather than from
a central concentration of hot gas.
In addition, softer X-ray emission
from diffuse hot gas is detected
around the nuclear region, out to $\sim 1$ kpc.
We have fitted the soft X-ray brightness profiles
with $\beta$ models to estimate central hot gas densities
(typical values are $\sim 0.02 \pm 0.01$ cm$^{-3}$).
For two of our galaxies, the flux from the diffuse
component is high enough to allow
also for spectral fitting.
NGC\,4486B is the only case where the central,
point-like X-ray source is soft but there is
no evidence of diffuse thermal emission around it.

The Bondi accretion rate gives an estimate
of the rate at which diffuse hot gas flows
into the sphere of influence of the SMBH (characteristic
radius $r_{\rm acc} \sim 10$ pc $\sim 0\farcs1$).
Typical Bondi
inflow rates vary from $\sim$ a few $\times 10^{-6}$
to $\sim 10^{-4} M_{\odot}$ yr$^{-1}$ in our target galaxies
This is a lower limit to the gas injection rate into the sphere
of influence, because it does not take into account
additional gas sources directly inside $r_{\rm acc}$
(for example, stellar winds or gas losses
from Type-Ia supernovae).

Based on the X-ray data, we tried to answer the following
(inter-related) questions:
whether the observed X-ray SMBH luminosities
are consistent with standard or radiatively-inefficient
accretion; whether the X-ray-inferred Bondi rate
is a good estimate of the true accretion rate
onto the SMBH; and whether there is a correlation
between Bondi rate and X-ray luminosity of the SMBH.
We found that standard accretion is ruled out
for all galaxies, as it would overestimate the SMBH
luminosities. On the other hand, our quiescent sample of SMBHs
are also brighter than what is predicted by
radiatively-inefficient solutions (ADAF, ADIOS, CDAF);
we interpret this as evidence
that the Bondi inflow rate of the hot ISM
underestimates the total gas injection rate
into the sphere of influence of the SMBH.
However, when we consider the other X-ray faint SMBHs
for which the nuclear luminosity and the Bondi
rate are known from the literature, we see that
most of them are consistent with, or fainter than
predicted by the ADAF model (as noted in Pellegrini 2005).
In those cases, it appears that the Bondi inflow rate of hot gas
is an over-estimate of the true accretion rate.
Overall, there seems to be no correlation
between the Bondi rate and the X-ray luminosity of the SMBH.
A possible explanation suggested (Pellegrini 2005)
for this lack of correlation
is that accretion is intermittent, self-regulated
by feedback from the BH: it may switch
between a state in which $\dot{m} \ll \dot{m}_{\rm B}$
(when gas builds up in the inner few pc),
and one in which $\dot{m} \gg \dot{m}_{\rm B}$
(as it is depleted).

In Paper II we will use optical observations
to investigate additional sources of fuel
for the SMBH; this will provide a better
estimate of the total accretion rate,
and will allow us to re-analyze its correlation
with the X-ray luminosity. We shall then
be able to estimate the energy and mass budget,
discuss the conditions for a steady-state, and
distinguish between different varieties
of radiatively-inefficient models.

\acknowledgments

We thank Andreas Zezas for his suggestions
on the X-ray data analysis and interpretation,
and Hermine Landt for her detailed comments on the
accretion rate issue. RS acknowleges
support from University College London
via a Marie Curie Fellowship. AB acknowledges partial support
from NASA contract NAS8-39073 and
from NASA grants GO1-2115X and GO2-3135X.