\section{INTRODUCTION}

Under the unification picture of active galactic nuclei (AGN), radio galaxies
and radio-loud quasars are believed to be the same type of
object. Under this unification scheme only viewing angle dictates
whether we see the object optically as a quasar with broad permitted
lines in emission and an unresolved optical core, or as a radio galaxy
with only narrow emission lines and where the host galaxy is resolved
(see e.g. Urry \& Padovani 1995).

Applying the same reasoning to radio-quiet quasars, which outnumber
their radio-loud counterparts by a factor of $\sim 10-100$
(e.g. Goldschmidt et al. 1999), there should be a population of
radio-quiet quasars which are obscured in the same way as radio
galaxies such that their quasar nuclei shine along the plane of the
sky, the so-called type-II quasars. However, such sources are extremely difficult to detect, because
they would appear as normal elliptical galaxies in optical or near-infrared images and only spectroscopy would detect the strong narrow-emission lines.

From an X-ray perspective the hypothesis that there must be a
population of type-II quasars, in which the direct light from the quasar
is obscured and only narrow emission lines are observed, is a
cornerstone assumption for models of the hard-X-ray background
(XRB; e.g. Wilman \& Fabian 1999; Wilman, Fabian \& Nulsen 2000). This requirement for type-II quasars by models of the XRB fits in
well with the orientation based arguments for radio-loud objects.

With the advent of the latest generation of X-ray satellites (CHANDRA
and XMM-Newton), the
majority of the XRB in the
Chandra band ($< 8$~keV) has been resolved (e.g. Alexander et al. 2003). The bulk of which is
produced by Seyferts and quasars with moderate obscuration
(N$_{\rm H} < 2 \times 10^{23}$~cm$^{-2}$), where the Seyferts evolve
quite rapidly to $z=0.8$ and
the quasars evolve somewhat more slowly. But of this population, only a handful
are bona fide type-II quasars (e.g. Gandhi et al. 2004).

The situation at harder energies is quite different. The majority of
the hard sources lie at the faint end of the flux distribution
(e.g. Fig.~2 in Giacconi et al. 2001) and only 50 per cent of the XRB
above 7 keV is resolved (e.g. Worsley et al. 2004). The unresolved flux has the spectrum of a
highly obscured AGN,
suggesting that the range of column density at $z \sim 1$ is similar
to that locally, implying that there is a bias towards highly
obscured, Compton-thick objects. This essentially means that there may
be many obscured AGN which have yet to be discovered in the large,
well-studied fields in the literature.

In addition to the high obscuring column densities causing
the hardening of the X-ray spectra there are other reasons why some
quasars may be obscured and others not. These include factors such as
intrinsic luminosity (e.g. Simpson, Rawlings \& Lacy 1999; Ueda et
al. 2003; Hasinger et al. 2004) and the rate at which the black-hole
accretes mass (e.g. Fabian 1999), which may be needed to derive a
complete unification model for active galactic nuclei (AGN). This is
borne out by the diversity of the hard-X-ray sources when observed in
the optical and near-infrared (e.g. Barger et al. 2002). From these studies it is
apparent that the frequency and distribution of gas and dust plays a
major r\^ole in determining how we see these quasars.

Thus, any alternative method in which
these obscured quasars may be discovered would be extremely important.

In this letter we report the serendipitous discovery of a genuine
type-II quasar from large-volume integral-field observations with VIMOS
on the VLT.

In section~\ref{sec:obs} we briefly summarise the
observing strategy and data reduction procedure and in
section~\ref{sec:results} we provide detailed information of the
type-II quasar including host galaxy properties and an estimate of its
black-hole mass. In section~\ref{sec:conc} we summarise our
conclusions and discuss how the discovery
of this object with integral-field observations may open up a new
window on the obscured AGN population.
We use $\Omega_{\rm M} = 0.3$, $\Omega_{\Lambda}= 0.7$ and $H_{\circ}
= 70$~km~s$^{-1}$~Mpc$^{-1}$ throughout, unless stated otherwise.

\section{Observations and data reduction}\label{sec:obs}

All of our observations were made on the nights of 29th April to the
2nd May 2003 with the Visible Multi-object Spectrograph integral-field
unit (VIMOS-IFU) on UT3 of the very large telescope (VLT). The IFU
consists of an array of $80 \times 80$ fibres coupled to
microlenses. Our observations were carried out in the low-resolution
mode with the LR-Blue grism giving a wavelength coverage of
3500\AA-7000\AA\, at 5.35\AA\, per pixel. We also used the
low-resolution spatial sampling which means that each fibre samples an
area of 0.67~arcsec resulting in a field-of-view per exposure of $54
\times 54$~arcsec$^{2}$.

The observations were centred on the powerful radio galaxy MRC0943-242
at $z = 2.92$ (R\"ottgering et al. 1995). The total exposure time was 9~hours, split into 18
exposures of 30~minutes which were dithered by 10~arcsec around the
central radio galaxy to enable accurate sky removal and cosmic-ray
rejection. All observations were taken in seeing of $< 0.8$~arcsec.

\begin{figure}
{\hbox to 0.48\textwidth{\null\null \epsfxsize=0.48\textwidth\epsfbox{fig1new.ps}}}
{\caption{\label{fig:spec} The 1-D spectrum of the type-II quasar,
    J0945-2428 at $z = 1.65$. The spectrum was extracted over the
    whole galaxy (seven IFU fibres). The shaded regions show the wavelengths affected by sky-line emission. }}
\end{figure}

\begin{figure}
{\hbox to 0.48\textwidth{\null\null \epsfxsize=0.48\textwidth\epsfbox{v0943axis.ps}}}
{\caption{\label{fig:vband} V-band image of J0945-2428, clearly
    showing that the host galaxy is resolved.
}}
\end{figure}

We briefly discuss the data reduction here, although a more complete
description can be found in van Breukelen \&
Jarvis (in prep.).  To reduce the data we used the VIMOS Interactive
Pipeline and Graphical Interface (VIPGI) software (Scodeggio et
al. 2004) which takes the raw images and performs the usual bias
subtraction, flat-fielding, wavelength calibration and combining of
the individual frames.  For the IFU data the final dataset is combined
into a datacube of two-spatial dimensions and a third spectral
dimension. The spectrophotometric calibration is achieved via a script
in VIPGI which provides an accurate solution for the relative
flux-calibration between fibres, however, the absolute
flux-calibration is unreliable. Therefore, to overcome this we used a
calibrated spectrum of the central radio galaxy MRC0943-242
and summed the fibres in our IFU data cube along the slit direction,
and the V-band photometry of the type-II quasar, allowing us to bootstrap the spectrophotometry. The final flux
calibration is good to $\sim 20$~per cent.

The final datacube was used to find emission-line objects within the
volume probed with the IFU, from which we discovered the type-II
quasar discussed in this letter. A detailed investigation of the
number of Lyman-$\alpha$ emitters in the volume is deferred to a
subsequent paper (van Breukelen \& Jarvis in prep.).

\section{Results}\label{sec:results}
\begin{table*}
\begin{center}
\caption{\label{tab:emlines} Emission-line measurements for the type-II quasar J0945-2428. The uncertainties on the line-flux measurements are $\approx$30 per cent.}
\begin{tabular}{cccccc}\hline
Emission & $\lambda$ & $z$ & FWHM & Flux / $10^{-19}$ & $L / 10^{35}$ \\
line & (\AA) &  & (km~s$^{-1}$) & (W~m$^{-2}$) & (W) \\
\hline
CIV$\lambda$1549 & 4112 & 1.654 & $<1376$ & 3.5 & 6.3 \\
HeII$\lambda$1640 & 4355 &  1.655 & $<1127$ & 3.2 & 5.8 \\
CIII]$\lambda$1909 & 5065 & 1.653 & $<1167$ & 2.4 & 4.3 \\
\hline
\end{tabular}
\end{center}
\end{table*}

\subsection{A bona fide type-II quasar?}

Figure~\ref{fig:spec} shows the extracted 1-D spectrum of the type-II
quasar discovered in our datacube at a redshift of $z=1.65$ at
$\alpha=09^h45^m31.74^s$, $\delta=-24^{\circ}28^{\prime}31.3^{\prime\prime}$. Details
of the emission-line characteristics can be found in
table~\ref{tab:emlines}.  The type-II spectrum is very similar to that of
a powerful radio galaxy (see e.g. Jarvis et al. 2001a) with bright
narrow ($v < 1500$~km~s$^{-1}$) emission lines where the the CIV emission lines have luminosities of $10^{35}$~W$ < L_{\rm CIV} < 10^{36.5}$~W.  The host-galaxy
morphology is also resolved in the optical bands (see
figure~\ref{fig:vband}), implying
that the central engine is obscured along the line-of-sight, again similar to radio galaxies.

To confirm this as a genuine type-II quasar
and not just a radio galaxy we determine whether the source has any powerful radio emission. We use the 5~GHz radio map of Carilli et al. (1997) of the
field surrounding MRC0943-242\footnote{We also analysed the 1.4~GHz
  map from the VLA archive, but this was only sensitive to $\sim
  0.7$~mJy (3$\sigma$).}. At the position of the type-II quasar
there is no detectable radio emission down to a 3$\sigma$ sensitivity
of $S_{5 \rm GHz} = 0.15$~mJy. Given the redshift of the type-II quasar and
assuming a typical spectral index for the radio emission from the core
of a radio-quiet quasar of $\alpha = 0.7$ (Kukula et al. 1998),
this corresponds to a radio luminosity of $L_{5 \rm GHz} < 1.6 \times
10^{23}$~W~Hz$^{-1}$~sr$^{-1}$. This is a factor of 10 fainter than the traditional division between
radio-loud and radio-quiet quasars (e.g. Miller, Peacock \& Mead
1990). Thus, the object is radio-quiet.

Further, the emission-line ratios and equivalent widths are very
similar to radio galaxies, as opposed to quasars and Seyfert II
galaxies. The emission-line ratios of McCarthy (1993) show that for a
radio galaxy, the typical ratio of CIV/HeII= 1.1, and HeII/CIII] $\sim
$1.8, in J0945-2428 we find CIV/HeII= 1.1 and HeII/CIII]=1.3.  For
quasars and Seyfert II galaxies the same ratios from McCarthy (1993)
are (CIV/HeII)$_{\rm QSO}$ = 8.6 and (CIV/HeII)$_{\rm SyII}$ = 5.9 and
(HeII/CIII])$_{\rm QSO}$ = 0.26 and (HeII/CIII])$_{\rm SyII}$ =
0.37. Thus, the emission-line ratios are much more consistent with
those of powerful radio galaxies than other types of AGN.

The CIV/HeII ratio in J0945-2428 is however marginally smaller than
other type-II quasars discovered via their X-ray emission in the
literature. Stern et al. (2002) find that CXO~52 has a CIV/HeII
flux-ratio of $\approx 2$ and HeII/CIII] $\sim 0.8$, although the
CIV/HeII emission-line ratio of CDF-S 202 (Norman et al. 2002) is much
higher at CIV/HeII $\sim 3.5$, albeit still far from the ratio of a
typical quasar (e.g. Vanden Berk et al. 2001). This can possibly
explained by the the fact the CIV is a resonant line and as such some
of its flux may be absorbed along the line of sight, thus making it
appear fainter in our low-resolution observations. Another possible
effect is that carbon may deplete onto dust grains
(e.g. Villar-Mart\'\i n, Binette \& Fosbury 1996). Both of these factors
could easily reduce the CIV luminosity whereas the HeII luminosity
would remain roughly constant.

The only other emission lines that we would expect in our spectrum
would be OIII]$\lambda1663$ and CIII]$\lambda2326$. From our spectrum
we can set a 3$\sigma$ limit on the OIII]$\lambda1663$ emission line
of $< 4 \times 10^{34}$~W, at least a factor of 15 fainter than the
CIV line. This is again in line with observations of high-redshift
radio galaxies. The CIII]$\lambda2326$ line would lie at 6164\AA, and
is thus very close to the atmospheric emission highlighted in
Fig.~\ref{fig:spec}.

\subsection{The host galaxy of J0945-2428}

\begin{table}
\begin{center}
\caption{\label{tab:photo} Photometric properties of the type-II
  quasar. The B-band photometry was measured from a 75~minute exposure of this field with FORS2 on the VLT. The V- and I-band photometry was measured from 93~minute and 40~minute exposures respectively with LRIS on the Keck telescope.   }
\begin{tabular}{cl}\hline
Band & Magnitude \\
\hline
B & 23.00 $\pm 0.02$ \\
V & 22.88 $\pm 0.02$ \\
I & 22.2 $\pm 0.2$ \\
K & 20.5 $\pm 0.2$ \\
\hline
\end{tabular}
\end{center}
\end{table}

The type-II quasar has a continuum magnitude of $V = 22.8$,
which if we assume typical galaxy colours of an evolved elliptical
galaxy at $z = 1.65$ of $V - K \sim 5$ (Poggianti 1997) leads to $K
\sim 17.8$, typical of a radio galaxy at $z = 1.65$ (e.g. Jarvis et
al. 2001b; Willott et al. 2003). However, using archival NTT-SofI data
[programme ID: 70.A-0514(A)] on this field we find that the AGN has $K
= 20.5 \pm 0.2$. The $K-$band light samples the old stellar population
at this redshift and should therefore provide a good estimate of the
stellar mass of the host, but we find that the galaxy is much fainter
than the expected luminosity for a massive host galaxy. Indeed,
$K=20.5$ corresponds to $\sim 0.2~L^{\star}$ at $z = 1.65$ completely
at odds with the radio galaxy $K-z$ relation (e.g. Jarvis et
al. 2001b).

The broad-band colours (Table~\ref{tab:photo}) also show that the host
galaxy is extremely blue, with $B-I = 0.8$ and $I-K=1.7$, indicating
that the host may have a large amount of ongoing star
formation. Coupled with the faintness of the host in the near-infrared
this suggests that the host may be undergoing its first major
star-forming event and sub-millimetre observations would be able to
confirm this.

There are also no obvious signs of merger activity, with the galaxy
profile exhibiting a disk-like morphology with a scale-length of $r =
9.2 \pm 0.7$~kpc, similar to the largest Seyfert galaxies
(e.g. Virani, De Robertis \& VanDalfsen 2000). However, it is worth
pointing out that we are sampling the UV-emitting stellar population
in our V-band (where we have the best signal-to-noise ratio), and that
the longer wavelength data sample the old stellar population
may not have such a morphology or scale-length, and further
observations will be need to investigate this.

\subsection{The mass of the black-hole in J0945-2428}\label{sec:mbh}

\begin{figure}
{\hbox to 0.48\textwidth{\null\null
\epsfxsize=0.48\textwidth\epsfbox{MbhMr.ps}}}
{\caption{\label{fig:MbhMr} The position of J0945-2428 (filled star)
on bulge luminosity versus black-hole mass relation. We use the
back-hole mass derived from the strength of the UV emission lines as
detailed in section~\ref{sec:mbh}, this is a lower limit which assumes
the black hole is accreting at its Eddington limit. $M_{R}$ is
calculated from the fact that $K=20.5$ corresponds to 0.2$L^{\star}$
from the $K-$band luminosity function (Kochanek et al. 2001) and
$M_{R}^{\star} = -22.2$ (Lin et al. 1996). The solid line represents
the best fit relation from McLure \& Dunlop (2002) for inactive and
active galaxies at $z< 0.5$ and the small stars represent the objects
from their data set. All points are for $\Omega_{\rm M} =1$,
$\Omega_{\Lambda}=0$ and $H_{\circ} = 50$~km~s$^{-1}$~Mpc$^{-1}$ to
allow comparison with previous work.}}
\end{figure}

If we use the typical line ratios for radio galaxies then we are able
to estimate the strength of the [O{\footnotesize{II}}] emission lines
and gain an estimate of the bolometric luminosity, $L_{\rm Bol}$ of
the central engine. We use the relation given in Willott et al. (1999)
relating $L_{\rm Bol}$ to the luminosity of the
[O{\footnotesize{II}}]$\lambda3727$ line $L_{\rm [OII]}$, $L_{\rm Bol}
= 5 \times 10^{3} L_{\rm [OII]}$~W, along with the line ratios from
McCarthy (1993). Thus, assuming
C{\footnotesize{IV}}/[O{\footnotesize{II}}] = 1.0 means J0945-2428 has
a bolometric luminosity $L_{\rm Bol} \sim 3.2 \times 10^{39}$~W. If
the central engine is accreting at its Eddington limit, then this
corresponds to a black-hole mass of $\sim 3 \times 10^{8}$\Msolar.

Using the relation linking bolometric luminosity to the monochromatic
luminosity at 2500\AA ($L_{2500}$) from Elvis et al. (1994) and given
the relation between X-ray luminosity at 2~keV ($L_{2\rm keV}$), and
$L_{2500}$ from Risaliti, Elvis \& Gilli (2002) with an optical--X-ray
spectral index of $\alpha_{OX} = 1.6$, this provides a very rough
estimate of the expected X-ray flux at 2~keV. Combining these
relations shows that J0945-2428 would have an X-ray luminosity of
$L_{2\rm keV} \sim 10^{37}$~W. This is also consistent with taking the
typical line ratios found in radio galaxies and linking the X-ray
luminosity to the H$\beta$ luminosity (e.g. Ward et al. 1988).

These properties are typical of a powerful AGN at these redshifts and imply that
it is in the regime of black-hole mass where powerful radio
emission is able to be produced (e.g. McLure \& Jarvis 2004).
However, such a massive black hole should be hosted by a massive
galaxy if the relation between black-hole mass and bulge luminosity
(Magorrian et al. 1998)
holds to high redshifts. From our $K-$band image we find that this is
not the case, which in turn implies that the black-hole mass -- bulge
luminosity correlation may break down (Fig.~\ref{fig:MbhMr}) in the early Universe. Thus
massive black holes may be in place before the host galaxy has fully
built up. However, this is highly tentative and obviously more observations are
needed of this object and other such quasars before such a statement
can be made in earnest.

\subsection{Obscuration toward the nucleus of J0945-2428}

To estimate the amount of obscuration toward the quasar nucleus we
adopt the Bolometric correction of $L_{\rm Bol} = 5.6 \times L_{2500}$ from Elvis et al. (1994), to calculate the continuum flux
expected in our optical spectrum. This leads to a monochromatic
luminosity at rest-frame 2500\AA\, of $L_{2500} = 5.3 \times
10^{38}$~W. From our extracted spectrum (Fig.~\ref{fig:spec}) we find
a monochromatic luminosity of $L_{2500} \sim 10^{34}$~W. This leads to
a visual extinction of A$_{\rm V} \mbox{{\raisebox{-0.4ex}{$\stackrel{>}{{\scriptstyle\sim} 4$~mag toward the quasar
nucleus, which can easily explain the lack of a nuclear point source
in the $K-$band image. Again this is in line with studies of powerful
radio galaxies (e.g. Simpson et al. 1999).

 \subsection{The space density of obscured AGN}

\begin{figure}
{\hbox to 0.48\textwidth{\null\null \epsfxsize=0.48\textwidth\epsfbox{fig4new.ps}}}
{\caption{\label{fig:blqso} The 1-D spectrum of J0945-2429, the $z = 1.79$ unobscured quasar in our IFU volume. The spectrum was extracted over three IFU fibres.
}}
\end{figure}

Obviously with a single object it is very difficult to estimate the
space density of obscured AGN in the high-redshift Universe. However,
it is interesting to note that there are two other AGN in the volume
probed by our IFU observations, the powerful radio galaxy that we
targeted and a traditional broad-line quasar at $z = 1.79$ at
$\alpha=09^{h}45^{m}32.92^{s}$, $\delta=-24^{\circ}29^{\prime}09.1^{\prime\prime}$  (Fig.~\ref{fig:blqso}). Therefore, the best we can say is that obscured AGN are
consistent with being as abundant as their type-I counterparts, but
obviously a larger volume search with IFUs will be able to pin-down
this fraction in a highly efficient and unbiased manner.

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

We have reported the first discovery of a genuine type-II quasar from
optical data alone. This was made possible by the unique capabilities
of the wide-field integral-field unit VIMOS on the VLT.

J0945-2428 has narrow emission lines ($v <1500$~km~s$^{-1}$), a
resolved host galaxy and no detectable radio emission down to a flux-density
limit of $S_{5 \rm GHz} = 0.15$~mJy ($L_{5\rm GHz} = 1.6 \times
10^{23}$~W~Hz$^{-1}$~sr$^{-1}$) at $z = 1.65$, indicative of a bona fide
radio-quiet type-II quasar.

Using typical emission-line ratios of powerful radio galaxies and
assuming that it is accreting at its Eddington limit, we show that
J0945-2428 contains a supermassive black hole of mass $\sim 3 \times
10^{8}$\Msolar. However, from deep $K-$band imaging we find that the
host galaxy has a luminosity of $\sim 0.2~L^{\star}$, placing this
type-II quasar approximately $3\sigma$ away from the relation between
galaxy bulge luminosity and black-hole mass. This may be indicative of
the supermassive black hole in this object being in place before the
host galaxy is fully formed. This is also reinforced by the very blue
colour of the host galaxy, indicative of a large amount of ongoing
star formation activity. Although further observations of this source
and others are needed to confirm these results.

This serendipitous discovery of an obscured quasar highlights the way
in which wide-area integral-field units on large telescope could open up a
unique window on the Universe. VIMOS is currently the only instrument
that has the capability of large spectral coverage coupled with a
$\sim 1$~square arcminute field of view. However, future instruments,
such as the Multi-Unit Spectroscopic Explorer
(MUSE; http://clio.univ-lyon1.fr/MUSE/), will expand the initial work which is taking place with VIMOS.

Furthermore, obscured AGN are not the
only extragalactic sources which may be discovered with volumetric
surveys. Surveys with integral-field units will discover other objects such as bare quasars with unusual optical
spectra which precludes them from being identified as AGN in
colour-colour diagrams, Lyman-$\alpha$ emitters (van Breukelen \&
Jarvis in prep.) and possibly objects yet to be discovered.

\section*{ACKNOWLEDGEMENTS}

We thank Chris Carilli for providing the 5~GHz map of
the field surrounding MRC0943-242, Wil van Breugel for the V-, R- and
I-band images and spectrum of MRC0943-242, Steve Rawlings for looking
at the 1.4~GHz map and Ross McLure for
supplying the data points shown in Fig.~3. We also thank the referee
for useful comments.
MJJ and RJW acknowledge the support of PPARC PDRAs and CVB would like to acknowledge the financial support from the ERASMUS program.


\section{Introduction}

Mrk 501 is among the brightest and closest ($z=0.034$) extragalactic X-ray
sources in the sky. It is classified as a BL Lacertae object and is located
in the elliptical galaxy UGC 10599 (Stickel et al. 1993). BL Lac objects
belong to a more general class of radio-loud active galactic nuclei known
as blazars, which are characterized by rapid variability and non-thermal
emission at nearly all wavelengths. The emission from a blazar is generally
thought to be dominated by radiation from a relativistic jet that is
directed roughly along the line of sight (Urry \& Padovani 1995). The
spectral energy distribution (SED) of blazars invariably shows two
characteristic ``humps'' in the $\nu F_{\nu}$ representation, with one
located at optical--X-ray energies and the other at GeV--TeV energies
(Fossati et al. 1998). Mrk 501 is among a small number of blazars that
have been detected at TeV energies (e.g., Quinn et al. 1996; Bradbury et
al. 1997).

For TeV blazars, there is a general correlation between fluxes at X-ray
and TeV energies (where the SED peaks), although exceptions have recently
been noted (Krawczynski et al. 2004; Cui et al. 2004). While the origin of
gamma rays is still being debated, there is a general consensus that
X-rays originate in the synchrotron radiation from highly relativistic
electrons in the jet. Two classes of models have been proposed to explain
gamma-ray emission from blazars. In the leptonic models, the TeV emission
is attributed to the synchrotron self-Compton (SSC) process (e.g., Marscher
\& Gear 1985; Maraschi et al. 1992). The SSC models are attractive for
their conceptual simplicity. They can quite naturally account for the
observed X-ray--TeV correlation, and have also enjoyed some success in
fitting the observed SEDs (e.g., Kataoka et al. 1999; Sambruna et al. 2000;
Petry et al. 2000; Krawczynski et al. 2002; Konopelko et al. 2003). However,
such models still face challenges, such as the presence of ``orphan TeV
flares'' (i.e., those TeV flares with no apparent counterparts in X-rays;
Krawczynski et al. 2004; Cui et al. 2004). In the hadronic models, the TeV
emission is probably due to synchrotron radiation from relativistic protons
in the jet (Aharonian 2000; M\"ucke et al. 2003), although other hadronic
processes might also contribute. Although the models are being pushed to
the limit by the observed rapid variability of TeV blzars, they are, by no
means, ruled out yet. It remains to be seen whether they can account for
the X-ray--TeV correlation in a quantitative manner.

TeV blazars are known to undergo flaring activities both at X-ray and TeV
energies. The flares seem to occur on all timescales but vary greatly in
magnitude, as illustrated vividly by the X-ray flaring hierarchy in Mrk 421
(Cui 2004). The hierarchy strongly implies the scale-invariant nature of
flaring processes in the source, prompting comparison to solar flares and
seemingly similar rapid flares in stellar-mass black hole systems. The
flares from Mrk 501 have been well observed at X-ray and TeV energies (e.g.,
Quinn et al. 1996; Bradbury et al. 1997; Catanese et al. 1997; Hayashida
et al. 1998; Kataoka et al. 1999; Sambruna et al. 2000; Petry et al. 2000).
In fact, the detection of a very rapid X-ray flare from the source was
reported (Catanese \& Sambruna 2000), which represented the first case of
sub-hour X-ray flares in any TeV blazar. The short duration of the flare
would lead to severe constraints on the properties of the flaring region,
as in the case of Mrk 421 (Cui 2004).

Motivated by the recent finding of hierarchical X-ray flaring phenomenon
in Mrk 421 (Cui 2004), we conducted a systematic search for X-ray flares
in Mrk 501 over a broad range of timescales, making use of the rich
{\em RXTE} archival database. We describe the data and data reduction
procedure in \S~2. The results are presented in \S~3, along with a
critical examination of the reported sub-hour X-ray flare. Finally,
in \S~4, we discuss the results and their implications on the proposed
models.

\section{Data and Data Reduction}

Mrk 501 has been observed frequently with {\em RXTE}. For this work, we
obtained publicly available data from the archival databases,
including data both from the All-Sky Monitor (ASM) and the Proportional
Counter Array (PCA). We obtained the ASM light curves of Mrk~501 from the
MIT archive.\footnote{See http://xte.mit.edu/asmlc/srcs/mkn501.html\#data}
We chose to filter out data points with error bars (on the raw count rates
in the summed band) greater than 2.0 counts/s, which constitute about 5\%
of the data. We then weighted the rates by $1/\sigma^2$ and rebinned
them to produce light curves with 21-day time bins. The light curves are
available in three energy bands: 1.5--3, 3--5, and 5--12 keV, providing
crude spectral information.

The PCA is a narrow-field, pointing instrument. It consists of five nearly
identical proportional counter units (PCUs) and covers a nominal energy
range of 2--60 keV. Due to operational
constraints, some of the PCUs were often switched off. Only PCU 0 and PCU 2
are nearly always in operation. Data analysis is further complicated by
the loss of the front veto layer in PCU 0 (in 2000), because the data from
PCU 0 is now more prone to contamination by events caused by low-energy
electrons entering the detector. This is particularly relevant to the study
of variability of weak sources like Mrk~501. Fortunately, it only affects a
about 15\% of the data used in this work. We excluded from timing analyses
PCU 0 data collected in 2000. Most PCA observations were made in ``snap-shot''
modes, with typical effective exposure times ranging from half a kilosecond
to a few kiloseconds. The PCA has numerous data modes and multiple modes
are usually employed in an observation. For this work, however, we only
used the {\em Standard 2} data, which have a time resolution of 16 s.

We followed Cui (2004) closely in reducing and analyzing the PCA data.
Briefly, the data were reduced with {\em FTOOLS 5.2}. For an observation,
we first filtered data by following the standard procedure for faint
sources (see the online {\em RXTE} Cook Book),\footnote{See
http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook\_book.html.} which
resulted in a list of good time intervals (GTIs). We then simulated
background events for the observation with the latest background model
that is appropriate for faint sources (pca\_bkgd\_cmfaintl7\_eMv20031123.mdl).
Using the GTIs, we proceeded to extract a light curve from the data (by
combining all active PCUs) in each of the following energy bands: 2.0--5.7,
5.7--11, 11--60, and 2.0--60 keV. Note that the boundaries of each band are
matched up as closely as possible across different PCA epochs but are only
approximate (up to $\pm$0.2 keV). We repeated the steps to construct the
corresponding background light curves from the simulated
events. Finally, we obtained light curves of the source by subtracting
off the background. Following a similar procedure, we constructed an X-ray
spectrum and the associated background spectrum for each observation. We
note that for spectral analysis we only used data from the first xenon layer
of each PCU (which is most accurately calibrated). For the purpose of
detailed spectral modeling (see \S~3.4), we included a 1\% systematic
uncertainty uniformly across the entire energy range of interest, to
take into account residual calibration uncertainties. Note that we did
include PCU 0 data from the 2000 observations in the spectral analysis
because the background model for the detector seems to work fine in all
cases.

\section{Results}

\subsection{Long X-ray Flares: Months to $>$ Year}

Figure 1 shows the ASM light curve of Mrk 501 for the 1.5--12 keV band over
roughly an eight-year period. The source was very active in X-rays in 1997.
The activity continued, at a much lower level, in 1998 and also the early
part of 1999. Mrk 501 has been relatively quiet ever since. At least two
major flares are easily identifiable from the light curve,
which lasted for months to over a year. The flares show substantial
sub-structures, indicating variability on shorter timescales, perhaps in
the form of unresolved weaker and shorter flares.

Though crude, hardness ratios make it possible to study, in a
model-independent manner, spectral variability of a source. Here, we
computed the ratio between the ASM count rate in the 3--12 keV band to
that in the 1.5--3 keV band to examine spectral evolution of Mrk 501
during the 1997 giant flare. The results are shown in Figure 2. For
clarity, we have averaged the raw ASM count rates with variable time bins
(6-week, 2-week, and 1-week), depending on the brightness of the source.
The figure clearly shows a hardening trend of the X-ray spectrum as the
source approaches the peak of the flare. This is consistent with
findings of previous works (e.g., Pian et al. 1998). Such a
hardness-intensity correlation seems to hold in general for X-ray bright
blazars (Giommi et al. 1990).

To investigate the phenomenon of spectral hysteresis, which is often
observed in blazars (e.g., Takahashi et al. 1996, Kataoka et al. 2000,
Zhang 2002, Giebels et al. 2002, Falcone et al. 2004, Cui 2004)
but may be too subtle to be seen in the ASM data, we made a similar
hardness-intensity diagram from the PCA data. There are three observing
periods, with two covering the rising portion of the 1997 flare and one the
decaying portion, as shown in Figure~3 (top panel). Unfortunately, the
coverage of the flare with the PCA is quite sparse, especially during the
decaying phase. While there appears to be indication of spectral
hysteresis during the rising phase (perhaps associated with smaller
flares) and between the rising and decaying phases, it is difficult to
draw any definitive conclusions from these results. On the other hand,
the trend of spectral hardening with increasing fluxes is also apparent
in the PCA data.

\subsection{Intermediate X-ray Flares: Days to Weeks}

The PCA data span a period of 4 years, which covers weaker flaring activities
in 1998, 1999, and 2000, besides the giant flare in 1997 (see Fig.~1). The
light curves reveal the presence of X-ray flares with shorter durations, from
days to weeks, nearly all the time, as shown in Figure~4. Although the ASM
light curve shows that the source seems to be very quiet during the PCA
observations in 1999 and 2000, the more sensitive PCA light curves reveal
flaring activities during these time periods. Therefore, like Mrk 421, there
is no apparent ``quiescent'' state for Mrk 501 either. Unlike Mrk 421,
however, the flux of Mrk 501 does not seem to vanish, implying the presence
of steady-state emission. The amplitude of these intermediate flares can also
be very large, up to 4--5 times the steady-state flux. We attempted to
investigate spectral evolution and hysteresis of the source across individual
flares but found that the results were ambiguous, mainly due to the lack of
quality of the data in this case.

\subsection{Rapid X-ray Flares: $<$ Hour}

Catanese \& Sambruna (2000) reported the first case of a sub-hour X-ray
flare from Mrk 501. Surprisingly, however, we initially failed to find it in
the PCA light curves constructed from the same observation. Examining
the light curves more carefully, we realized that there was actually a
data gap almost exactly at the time of the flare. We suspected that the
data had probably been filtered out during the data cleaning process in our
case. We then relaxed the data filtering criteria one at a time and in
the end found that it was the high ``ELECTRON2'' values that deemed the
time interval (of the reported flare) ``bad''. In general, the ``ELECTRON''
parameter provides a measure of contamination of the data by events
induced by electrons entering the detector. Some of such events (due to
low energy electrons) may be registered as good X-ray events because
they fail to trigger any of the vetoing logics. Since the PCU 2 is always
on, we used ``ELECTRON2'' in the data cleaning procedure (as recommended
in the {\em RXTE} Cookbook).

To show possible electron contamination of the data, we plotted the X-ray
count rates and the values of the ``ELECTRON2'' parameter for the
observation in Figure~5 (left panels). The figure clearly shows that there
is a strong electron flare nearly at the same time as the reported X-ray
flare, which raises the possibility of the latter being an artifact.
However, we note that the X-ray flare seems to have started about 200 s
before the electron flare and that the profile of the X-ray flare appears
asymmetric, unlike that of the electron flare. We cannot think of an
obvious explanation for either ``anomaly''. On the other hand, we found
another rapid X-ray flare (which has not been reported) from a different
observation (obsid: 30249-01-01-01), which also has an
asymmetric profile, is accompanied by an electron flare, and leads the
electron flare by about the same amount of time ($\sim$200 s), as also
shown in Fig.~5 (right panels) for a direct comparison. The presence of
two such flares raised serious doubts in our mind about them being
physically associated with Mrk 501.

We then systematically searched for X-ray flares that were accompanied by
electron flares, by visually inspecting all of the light curves, and found
a total of 21 cases (although it is entirely possilble than some weaker
ones might have escaped our attention), including the two just discussed.
Interestingly, in all other 19 cases, we found no apparent timing offsets
between X-ray and electron flares. Figure~6 shows some representative
examples of those. It is clear, from the figure, that not all electron
flares are registered as X-ray signals and that the profile of an electron
flare can be complex, e.g., with multiple peaks. Moreover, we found that
the spectrum of the X-ray flares varies greatly. In extreme cases, a single
X-ray flare is seen in one energy band that coincides with one of the peaks
of an electron flare, but another flare in a different energy band that
coincides with a different peak of the same electron flare. Therefore, the
phenomenology is complex.

Finally, we derived the latitude and longitude of the {\em RXTE} satellite
at the time of the peak of each of the 21 electron flares. Figure~7 shows
a map that summarizes the results. The two cases shown in Fig.~5 are
highlighted with different symbols. It should be noted that the peaks of
these two electrons flares are separated by almost exactly one satellite
orbit ($\approx$1.6 hours). In general, the electron flares seem to cluster
in the northeast Pacific region, which confirms what is generally known
about the occurrence of such events (K.~Jahoda and C.~Markwardt of the PCA
team, private communication). However, particle fluxes in the region are
usually over two orders of magnitude smaller than those in the well-known
South Atlantic Anomaly (SAA) region, which is why the region is not
screened in the same manner as the SAA. Putting
together all the evidences, we think that the reported rapid flare (and
also the one found in this work) is most likely an artifact caused by
intense flaring in soft electron fluxes. Admittedly, outstanding issues
remain regarding the observed timing offsets or asymmetric profiles (see
Fig.~5). Neither can we rule out the possibility that real flares from
Mrk 501 might sometimes include a very soft second delayed component,
which increases the ``ELECTRON'' parameter through extra counts absorbed
in the propane layer, although the scenario seems a bit contrived.

In our search, we discovered yet another sub-hour X-ray flare. It occurred
during an observation (obsid: 30249-01-01-02) when there was hardly any
enhancement in the soft electron fluxes. Figure~8 shows the flare, along
with the ``ELECTRON2'' parameter for the same time period. The flare is
quite weak (compared with the other two), reaching a peak amplitude of
about 30\% of the steady-state flux. It lasted
for about 800 s and shows sub-structures on even shorter timescales. The
profile shown might represent the composite of two flares or a sudden
decrease and recovery of X-ray fluxes in the middle of a flare. Remarkably,
the rising or decaying time associated with the sub-structures is as short
as about 20--30 seconds! The flare occurred roughly one satellite orbit
after the reported one, which might be a bit worrisome. However, it would
appear about 50\arcdeg\ in longitude further west (as shown in Fig.~7).
Moreover, Mrk 501 was monitored very intensely during this time period,
with data taken in 18 nearly consecutive satellite orbits. Most of these
observations did indeed span periods when the satellite was in the general
area of the ``Northeast Pacific Region'', but no X-ray/electron flare
(other than the ones already mentioned) was detected, which rules out
any systematic effects associated with the region. Finally, if the flare
shown in Fig. 8 were to be false, it would
have to be caused by a surge in particle-induced events that were not
vetoed {\em and} not manifested in the increase of the ``ELECTRON'' factor;
we are not aware of any examples of such background flaring activities.
Therefore, we believe that we have detected a sub-hour flare from Mrk 501.
The rare detection can be attributed, to a large extent, to the much
intensified coverage of the source.

\subsection{Spectral Modeling}

We carried out detailed spectral modeling to investigate, in a more
quantitative manner, spectral variability of Mrk 501. Here, we eliminated
all observations with effective exposure times less than 200 s, because
of the lack of statistics. In subsequent analyses, we fixed the hydrogen
column density at $1.8\times 10^{20}\mbox{ }cm^{-2}$ (Dickey \& Lockman
1990). We experimented with the following empirical models: power law,
power law with an exponential rollover, and broken power law, to fit the
observed X-ray spectra. We found that when the source was relatively weak
(the 2-20 keV flux less than
$2\times 10^{-10}\mbox{ }ergs\mbox{ }cm^{-2}\mbox{ }s^{-1}$), the power-law
model fits the X-ray spectra well. However, the simple model fails
statistically (in terms of $\chi^2$ values) at higher fluxes. The data
taken when the source was brighter (in 1997 and parts of 1998) clearly
favor the broken power-law model. Formally, the resultant fits are
satisfactory (with the reduced $\chi^2$ values all around unity).

Mrk 501 was most active in 1997, with the 2--20 keV flux varying roughly
between $3$--$9.5 \times 10^{-10}\mbox{ }ergs\mbox{ }cm^{-2}\mbox{ }s^{-1}$.
The best-fit broken power-law parameters are summarized in Figure 9. The
first photon index shows an initial decreasing (or spectral hardening) trend
and then levels off toward high fluxes; so does the second photon index,
although the trend is less apparent. In general, the SED of Mrk 501 is
quite flat in the X-ray band during this time period. Spectral variability
seems to be dominated by changes in the break energy (roughly between 5 and
16 keV). In the subsequent years (1998--2000), the source became much weaker
in X-rays, with the flux ranging between
$0.4$--$4 \times 10^{-10}\mbox{ }ergs\mbox{ }cm^{-2}\mbox{ }s^{-1}$. The
broken power-law model is still required for some of the 1998 observations,
as already mentioned, and the results are also shown in Fig.~9 for
comparison. The photon indices show large variations. In some cases, the
X-ray spectrum is significantly steeper than that observed in 1997. On the
other hand, the values of the break energy seem comparable to those from
the 1997 observations.

At lower fluxes
($< 2\times 10^{-10}\mbox{ }ergs\mbox{ }cm^{-2}\mbox{ }s^{-1}$), the data
can no longer effectively constrain the broken power-law model, mainly
due to the lack of statistics (and also to the fact that the photon indices
above and below the break do not differ significantly; see Fig.~9).
Figure 10 shows the results from fits with a simple power law. The trend
of spectral hardening toward high fluxes is more apparent in this case.
The photon index goes roughly from 2.6 down to 1.8, clearly indicating a
shift in the synchrotron SED peak toward higher energies at higher fluxes.
Also apparent from the figure is spectral hysteresis associated with the
1998 observations. As shown in Fig.~1, these observations sampled two
distinct periods: the very end of the decaying phase of the 1997 giant
flare and the rising phase of a subsequent major flare (of much lower
magnitude). To examine spectral hysteresis in more detail, we marked
the data points with different symbols for the two periods in Figures 9
and 10. For the most part, the points seem to bifurcate, with those for
each period clustered around one of the branches. However, the phenomenon
is clearly more complex --- spectral hysteresis also seems to be present
within each period, perhaps associated with smaller flares that are not
resolved.

\section{Discussion}

Like Mrk 421, Mrk 501 also produces X-ray flares on a wide range of
timescales. Unlike in Mrk 421, however, the flares in Mrk 501 do not seem
to occur nearly as frequently, which appears to be the case on all
timescales. This would be consistent with the scale invariance of X-ray
flaring phenomenon, if it is common among TeV blazars. Fig.~11 illustrates
the flaring hierarchy observed in Mrk 501.
In other words, there might not be any fundamental differences among long
flares, intermediate flares, or rapid flares, because they could all be
caused by the same physical process operating on all physical scales.
Though not fully understood, the flares in blazars are often thought to
be related to internal shocks in the jet (e.g., Rees 1978; Spada et al.
2001), or to major ejection events of new components of relativistic
plasma into the jet (e.g., B\"ottcher et al. 1997; Mastichiadis \& Kirk
1997). In these scenarios, the flares of different durations might simply
be the observational manifestation of a hierarchy of inhomogeneities in
the jet, which are energized to produce flares by the shocks. Alternatively,
the flares might be associated with magnetic reconnection events in a
magnetically dominated jet (Lyutikov 2003), perhaps similar to solar
flares in this regard. The flaring hierarchy might be related to
reconnection and subsequent avalanche processes. We should note that the
proton-synchrotron model actually requires a magnetically dominated jet
to account for the observed TeV emission from blazars (Aharonian 2000;
M\"{u}cke et al. 2003).

The X-ray spectrum of Mrk 501 evolved significantly during the 1997 and
1998 major flares. It shows a general trend of hardening toward the
peak of a flare, which is consistent with results from previous
works on X-ray bright blazars (Giommi et al. 1990), perhaps implying a
hardening in the spectral distribution of the emitting electrons during
the major flares. Moreover, there are indications of hysteresis associated
with the spectral evolution of the source during major flares in 1997
and 1998. Though common among blazars, spectral hysteresis is still not
fully understood. Kirk \& Mastichiadis
(1999) showed, using an internal shock model, that complex hysteresis
behavior could arise from the interplay of three characteristic timescales
associated with synchrotron cooling ($\tau_{syn}$), particle acceleration
($\tau_{acc}$), and intrinsic variability ($\tau_{var}$), respectively.
The model can qualitatively describe the observed spectral hysteresis
associated with rapid flares, but would have difficulty in explaining the
phenomenon associated with long-duration flares in Mrk 501, because
$\tau_{var}$ almost certainly dominate over $\tau_{syn}$ and $\tau_{acc}$
in the latter cases.

We critically examined the reported sub-hour X-ray flare from Mrk 501
(Catanese \& Sambruna 2000), in light of intense background flaring
activities during the observation. We found that it was likely associated
with flaring activities in the flux of soft electrons entering the
detectors, as opposed to those of Mrk 501. A similar rapid X-ray flare
was detected in this work, which was also accompanied with a strong
electron flare. On the other hand, we also found a rapid X-ray flare that
occurred during a time that seems to be free of any background
flaring activities. The flare lasted only for about 800 s; it might in
fact be composed of two flares on even shorter timescales. These
variability timescales pose severe
constraints on the physical properties of the flaring region. First of
all, the (jet-frame) size of the region is constrained by
$l \lesssim c t_{flare} \delta/(1+z) = 2.4\times 10^{14} \delta_1\mbox{ }cm$,
where $t_{flare}$ is the duration of the flare ($=800$ s), $\delta$ is the
Doppler factor of the jet ($\delta = 10\delta_1$), and $z$ is the redshift
of Mrk 501 ($= 0.034$). The upper limit is already comparable to the
gravitational radius ($r_g \equiv G M/c^2$) for a $1.3\times 10^9 M_{\odot}$
black hole, which is believed to exist in Mrk 501 (Barth et al. 2003),
if the Doppler factor of the jet is not too much larger than 10 (which
is probably the case; see discussion below). Since the peak flux of the
flare is a significant fraction of the steady-state flux, the size of
the flaring region is probably comparable to the lateral extent of the jet.
If the jet originates from accretion flows, as is often thought to be the
case, the result would also represent an upper limit on inner boundary of
the flows (either in the form of a geometrically thin, optically thick
cold disk or of a geometrically thick, optically thin hot torus).

Secondly, the observed decaying time of the flare
sets a firm upper limit on the synchrotron cooling time of the emitting
electrons. The cooling time is given by Rybicki \& Lightman (1979),
$\tau_{syn} \approx 6 \pi m_e c/\sigma_T \gamma_p B^2$, where
$m_e$ is the electron rest mass, $\sigma_T$ is the Thomson cross
section, $B$ is the strength of the magnetic field in the region, and
$\gamma_p$ is the characteristic Lorentz factor of those
electrons that contribute to the bulk of the observed X-ray emission
(at $E_{p}\sim 10$ keV, where the synchrotron peak of the SED lies).
From $\tau_{syn}/\delta < t_d$, where $t_d$ is the decaying time of the
flare ($\approx 400 s$), we can derive a lower limit on the magnetic field
strength, $B > 1.4 \delta_1^{-1/2} \gamma_{p,5}^{-1/2} \mbox{ }G$, where
$\gamma_p = 10^5\gamma_{p,5}$. This is already larger than typical values
of $B$ inferred from fitting the SED with the SSC models (e.g., Kataoka
et al. 1999; Sambruna et al. 2000; Petry et al. 2000; Krawczynski et al.
2002; Konopelko et al. 2003). Finally, for synchrotron photons to reach
X-ray energies (at $\sim E_p$), the Lorentz factor of the emitting
electrons must be sufficiently high,
$E_{p}=\delta h\nu_c \equiv (3 e h/4 \pi m_e c) \delta \gamma_{p}^2 B$
(Rybicki \& Lightman 1979).
Using the derived lower limit on $B$, we can derive an {\em upper} limit
on the Doppler factor, $\delta < 6.4 E_{p,1}^2 (\gamma_{p,5}/3)^{-3}$,
which seems quite low compared to typical values inferred from the SSC
models.

\acknowledgments

We thank Keith Jahoda and Craig Markwardt for insightful discussions on
the issue of electron contamination and the referee for his/her very
detailed and constructive comments. This work was supported in part by
grants from the US Department of Energy and National Aeronautics and
Space Administration. We made use of data obtained through the High
Energy Astrophysics Science Archive Research Center Online Service,
provided by the NASA/Goddard Space Flight Center.


\section{Surfaces of decagonal quasicrystals
\label{sec:decagonal}}

Decagonal Al$_{65}$Cu$_{15}$Co$_{20}$ (d-AlCuCo) is
periodic in $z$-direction~\cite{Kor90},
with periodicity t=4.13~{\AA}. The  $z$-direction is
orthogonal to the quasiperiodic decagonal $x-y$ plane.
The phase crystallizes in a shape of long thin
decagonal prisms, with the 2fold surfaces considerably
larger than the 10-fold surfaces,
see Fig.~\ref{fig:deca}(a).
\begin{figure}[]
\includegraphics[width=0.27\textwidth]
{PK-Fig1a.eps}
\includegraphics*[width=0.30\textwidth]
{PK-Fig1b.eps}
\includegraphics*[width=0.40\textwidth]
{PK-Fig1c.eps}
 \caption{
 (a)~(top) Long, thin decagonal prisms~\cite{K0}
           of the d-AlCuCo;
 (b)~(middle) Estimated positions of 2fold surfaces
        on an STM image of the 10fold terrace-like
         surface of d-AlCuCo;
 (c)~(bottom) 2fold bulk terminations in ${\cal M}({\cal T}^{*(A_4)})$\
         marked on a 10fold one~\cite{Kram95}.
      Note the $1.23$~{\AA} gaps on both sides of
      a termination.
\label{fig:deca}
}
\end{figure}
The size of the surface area is evidently a parameter
of it's stability.
In Fig.~\ref{fig:deca}(b), over the image of the 10fold
surface \cite{Kor90}, we determine the positions of
possible 2fold surfaces, orthogonal to the 10fold
surface.
In the model ${\cal M}({\cal T}^{*(A_4)})$\ (t=4.18~{\AA})~\cite{Bur93} we
investigate the densities of the ``thin'', 2fold layers
containing 2 atomic planes.
Among these, the most dense one is a $0.47$~{\AA}-layer
of the density $\rho_{2}^{0.47A}=0.124$~{\AA}$^{-2}$.
Comparing it to the much smaller 10fold surface,
which is of the density $\rho_{10}=0.146$~{\AA}$^{-2}$,
we conclude that the $0.47$~{\AA}-layer can not
represent the 2fold termination. But, on some positions
in the bulk these layers appear in
pairs, $0.29$~{\AA} apart. Such, a rather ``thick''
$1.23$~{\AA}-layer of 4 planes on mutual
small distances (see Fig.~\ref{fig:deca}(c)) is a
candidate for a terminating layer. These layers appear
on distances mutually scaled by the factor
$\tau=(1+\sqrt{5})/2$ ($12.3$~{\AA}, $19.9$~{\AA},
$32.2$~{\AA}, $52.1$~{\AA}\ldots,
as in Fig.~\ref{fig:deca}(c)), in excellent agreement
with those found on the STM image ($21$~{\AA},
$31$~{\AA}, $54$~{\AA}, see Fig.~\ref{fig:deca}(b)).

\section{ Surfaces of icosahedral quasicrystals
\label{sec:ico}}

A feature of the surface of \mbox{i-AlPdMn}, not
accounted for by the ``thin'' layer analysis \cite{PP2},
is that not all types of maximally dense layers appear
as surfaces:
for example, $(q,b)$ layers~\cite{classes},
$0.48$~{\AA} apart, are seen in 5fold
surfaces but equally dense $(b,q)$ layers,
also $0.48$~{\AA} apart,
are not. If both kind of layers were possible
terminations, the sequence of much
shorter terrace heights, than observed, could appear.
\begin{figure}[]
\includegraphics*[width=0.48\textwidth]
{PK-Fig2a.eps}
\includegraphics*[width=0.48\textwidth]
{PK-Fig2b.eps}
\caption{
(a)~(top)
Density graph $\rho_{5f}(z_{\perp})$ of the ``thick''
5fold layers $(q,b,b,q)$ (full line). The plateau
(maximum density) of the graph defines the terminations.
It is compared to the density graph of the $0.48$~{\AA}
thin, plane-like layer $(q,b)$ (dotted line).
The symbol \ding{196}\ is the standard distance along
a 5fold axis $z_{\perp}$ in the coding space $\mathbb{E}_\perp$.
(b)~(bottom) Density graph of the
5fold layers $(b,a,q,b,q,a,b)$, with spacings as in the
image.
The bottom of the
cavity
defines a sequence of the
{\em minimum density} layers in ${\cal M}({\cal T}^{*(2F)})$, situated above
a {\em subsequence} of the
 5fold terminations.
\label{fig:5f-thick-term}
}
\end{figure}
If one chooses to define a
termination incorporating the neighboring planes too,
as we did in decagonal case
(in Section~\ref{sec:decagonal}),
one could introduce a ``thick'' layer as a bundle of
high density planes (or thin, plane-like layers).
A 5fold termination can be considered to
be a ``thick'' layer consisting of a
$(q,b)$  layer and a $(b,q)$ layer,
each with the spacing $0.48$~{\AA}.
Such a layer contains 4 planes with
spacings:
$q$-plane, $0.48$~{\AA}, $b$-plane, $1.56$~{\AA},
$b$-plane, $0.48$~{\AA}, $q$-plane.
For a bundle we define an effective
(averaged) density of within contained ``thin''
layers/planes

\vspace{0.3cm}
\noindent
{ \small
$\rho_{5f}(z_{\perp})=({\rho}_{q_1}(z_{\perp})+
{\rho}_{b_1}(z_{\perp}))/2 +
({\rho}_{q_2}(z_{\perp})+
{\rho}_{b_2}(z_{\perp}))/2$
}.

\vspace{0.3cm}
\noindent
As we see in Fig.~\ref{fig:5f-thick-term}(a),
whereas for the thin-layer concept the width of the
support of the plateau is approximately
$\frac{2{\tau}^2 }{\tau+2}$\ding{196}\ broad,
and consequently~\cite{PP2} encodes the
Fibonacci sequence of terrace heights $S=4.08$~{\AA}
and $L=\tau S=6.60$~{\AA}
($\tau=(1+\sqrt{5})/2$),
in the thick-layer concept the width is
exactly
$\frac{2{\tau} }{\tau+2}$\ding{196}\
(see  Fig.~\ref{fig:5f-thick-term}(a))
and encodes the Fibonacci sequence of by factor $\tau$
larger terrace heights, i.e. $L=6.60$~{\AA} and
$L+ S=\tau L=10.68$~{\AA}.
Whereas on the clean surfaces, obtained at lower
annealing temperature, even the terrace height
$\tau^{-1}S=2.52$~{\AA} appears~\cite{Shen99},
on the
surfaces obtained at the highest
annealing temperature, on the contrary,
a terrace height $L+ S$ appears
(see Fig.~1 in Ref.~\cite{KPKB}).

The height of the plateau  of the graph
$\rho_{5f}$ (see Fig.~\ref{fig:5f-thick-term}(a))
defines the densities of the ``thick'' layer
terminations to be $0.134$~{\AA}$^{-2}$,
see table~\ref{tab:densities}.
The 5fold layers intertwining the terminations are
of densities not higher than
$0.072$~{\AA}$^{-2}(\ll 0.134$~{\AA}$^{-2}$).
It is also a fact, that the density graphs of the
``thin'' and the ``thick'' layers have a strong overlap,
see Fig.~\ref{fig:5f-thick-term}(a).
Hence, almost any  $(q,b)$ ``thin'' layer termination
occurs within such a $(q,b,b,q)$ ``thick'' layer
termination.
The same holds true for any icosahedral quasicrystal
described by the ${\cal M}({\cal T}^{*(2F)})$\ model.

Above each termination, there is a $2.04$~{\AA} gap,
if we dare to neglect an $a$-plane of a density
smaller than $0.013$~{\AA}$^{-2}$.
But, if each gap of $2.04$~{\AA}
in the model ${\cal M}({\cal T}^{*(2F)})$\ would be declared as
a criterion of a termination to appear below it,
as in Ref.~\cite{Y04}, the $2.04$~{\AA} terrace heights,
that were not observed, should appear as well.
However in the model ${\cal M}({\cal T}^{*(2F)})$\ there is a
low density 5fold layer $(b,a,q,b,q,a,b)$
(see Fig.~\ref{fig:5f-thick-term}(b)),
$4.08$~{\AA} broad.
The width of the
cavity
on the density graph of these layers,
$W=(2/(\tau +2))$\ding{196}, encodes a Fibonacci
sequence with the intervals
$\tau L=10.68$~{\AA} and  $\tau^2 L=17.28$~{\AA}.
These are the {\em minimum density} layers
of equal, $0.041$~{\AA}$^{-2}$ density,
placed in the model over a {\em subsequence} of
the terminations.
Hence, the minimum density layer sequence
alone can not define
the terminations, because
it does not reproduce
the pairs of large terraces $6.60$~{\AA}
apart, which were frequently observed.

\begin{figure}[]
\includegraphics*[width=0.48\textwidth]
{PK-Fig3a.eps}
\includegraphics*[width=0.48\textwidth]
{PK-Fig3b.eps}
  \caption{
(a)~(top)
Density graph $\rho_{2f}(z_{\perp})$
of the ``thick'' 2fold
layers (full line). The plateau (maximum density)
of the graph defines the terminations. It is compared
to the ``thin'' layer termination (dotted line),
which is a single, $abq$-plane termination.
The symbol \ding{193}\ is the standard distance
along a 2fold axis $z_{\perp}$ in the coding space
$\mathbb{E}_\perp$.
(b)~(bottom)
Density graph of the
2fold layer $(bq,bq)$. The cavity
defines a sequence of the {\em  minimum density} layers
in ${\cal M}({\cal T}^{*(2F)})$,
of which a {\em subsequence} is situated above
{\em all} the 2fold terminations.
\label{fig:2f-thick-term}
}
\end{figure}
In the case of 2fold surfaces, we may replace a single
dense 2fold terminating atomic
$abq$-plane~\cite{PP2} by a layer of 4 atomic planes
with spacings:
$abq$-plane, $1.48$~{\AA}, $bq$-plane, $0.92$~{\AA},
$bq$-plane, $1.48$~{\AA}, $abq$-plane.
For a bundle we define an effective
(averaged) density of planes

\noindent
\vspace{0.3cm}
\noindent
{\small$\rho_{2f}(z_{\perp})=
(1/4)[{\rho}_{abq_1}(z_{\perp})+
 {\rho}_{bq_1}(z_{\perp})+
 {\rho}_{bq_2}(z_{\perp})+
 {\rho}_{abq_2}(z_{\perp})$]},

\vspace{0.3cm}
\noindent
For this ``thick'' 2fold layer
the peak of $\rho_{2f}(z_{\perp})$ is a perfectly
flat plateau, see Fig.~\ref{fig:2f-thick-term}(a).
The height of the plateau defines the effective
densities of terminations to be $0.086$~{\AA}$^{-2}$,
see Table~\ref{tab:densities}.
The support of the width of the plateau equals
$W=(1/2)$\ding{193}\ and encodes the
Fibonacci sequence of 2fold terminations with the
terrace heights $S=6.3$~{\AA} and
$L=\tau S=10.2$~{\AA}.
The standard distances along 5, 2 and 3fold axes
in icosahedral structures are
\ding{196}~/$\sqrt{\tau +2}={}$~\ding{193}~/2$(=1/\sqrt{2(\tau +2)})
={}$~\ding{194}~$/\sqrt{3}$, where $\tau=(1+\sqrt{5})/2$.
The heights of the larger 2fold terraces were
measured to be
$S=
6.2$~{\AA} and $L=9.5$~{\AA}
(see Fig.~3(a) and 3(c) in Ref.~\cite{PP2}),
in good agreement with the predicted values.
The 2fold layers intertwining the terminations are
of densities not higher than
$0.079$~{\AA}$^{-2}(<0.086$~{\AA}$^{-2}$).
The small 2fold terraces, or rather the
pits within the big terraces
(see Fig.~3(a) and 3(c) in Ref. \cite{PP2})
may be explained by the comparatively large distances
between the atomic planes inside of the ``thick''
terminating layer ($1.5$~{\AA}, $2.4$~{\AA} and
$3.9$~{\AA}).
These excellently reproduce the measured values
($2.4$~{\AA} and $3.6$~{\AA}),
see Fig.3(a) and 3(c) in Ref. \cite{PP2}.

In the model ${\cal M}({\cal T}^{*(2F)})$\ there is a low density
2fold layer
$(bq,bq)$:
$0.92$~{\AA} gap, $bq$-plane, $0.57$~{\AA},
$bq$-plane, $0.92$~{\AA} gap
(see Fig.~\ref{fig:2f-thick-term}(b)).
The width of the
cavity
on the graph of these layers,
$W=(\tau/2)$\ding{193}, encodes a Fibonacci
sequence with the intervals
$\tau^{-1} S=3.9$~{\AA} and $S=6.3$~{\AA}.
These are the {\em minimum density} layers
of almost equal density,
somewhat above $0.063$~{\AA}$^{-2}$.
A member of a {\em subsequence} of these layers
is placed over  {\em each} 2fold termination.
Nevertheless, in the 2fold case the
minimum density layer sequence {\em alone}
can not define the terminations, because it predicts
by $\tau^{-1}$ shorter terrace heights between the
large 2fold terminations, than observed.

The 3fold terminations could also be modeled
as ``thick'' layers of atomic planes in ${\cal M}({\cal T}^{*(2F)})$,
see table~\ref{tab:densities}.
But, inspecting the intertwining 3fold layers,
we see that these are of the densities {\em comparable}
to the ``terminating'' ones.
We also know that the 3fold surfaces facet readily
\cite{Shen00},
and some correlated STM measurements,
(as those in Fig.3(a) and 3(c) of Ref. \cite{PP2})
for the 3fold surfaces do not exist so far.
\begin{table}[h]
\begin{center}
\caption{
Relative and absolute densities of the planes
and layer terminations orthogonal to 5, 2 and 3fold
symmetry axes in ${\cal M}({\cal T}^{*(2F)})$\  of \mbox{i-AlPdMn}.
The corresponding data for \mbox{i-AlCuFe} are similar.
\label{tab:densities}
}
\begin{tabular}{l|lll}
\hline
   & 5fold & 2fold & 3fold\\
\hline
Densest planes (abs.) & 0.086\,{\AA}$^{-2}$&
0.101\,{\AA}$^{-2}$&0.066\,{\AA}$^{-2}$ \\%[1mm]
Densest ``thin'' layers (abs.) &0.133\,{\AA}$^{-2}$&
0.101\,{\AA}$^{-2}$&0.066\,{\AA}$^{-2}$\\
Densest ``thin'' layers (rel.) & 1 & 0.76 & 0.50\\
Densest ``thick'' layers (abs.) &0.134\,{\AA}$^{-2}$&
0.086\,{\AA}$^{-2}$&0.058\,{\AA}$^{-2}$\\
Densest ``thick'' layers (rel.) & 1 & 0.64 & 0.44\\
\hline
\end{tabular}
\end{center}
\end{table}

On the STM measurements it is in general
hard to judge whether ``thin'' layer or ``thick''
layer terminations best model the physical surfaces.
However, in the 5fold case, the ``thick'' layer concept
removes the contradiction with respect to the
Bravais' rule, that some, equally dense layers
do not appear on the surfaces, see Ref.~\cite{PP2}.
In the 2fold case the ``thick'' layer concept is
evidently better, it treats differently
the large terraces compared to the small pits inside.
That the effective densities of the 2fold terminations
are somewhat lower than the densities of some single
2fold planes is not contradictory, because these are
included in the most dense layers.

Fig.~\ref{fig:compSEI}(a)
shows the secondary-electron pattern
\cite{B04}
obtained from the clean pentagonal surface of a
quasicrystalline
Al$_{70}$Pd$_{20}$Mn$_{10}$
sample~\cite{K0}.
\begin{figure}
\includegraphics[width=0.465\textwidth]
{PK-Fig4.eps}
  \caption{
   (a)~(left) Secondary-electron pattern obtained from
       the pentagonal surface of a single icosahedral
       Al$_{70}$Pd$_{20}$Mn$_{10}$ quasicrystal.
       The center of the pattern is obscured by the
       shadow of the electron gun used for the
       excitation. The edge of the screen corresponds
       to~\cite{B04} $\theta=52^\circ$.
(b)~(right) Calculated secondary-electron pattern based
    on the single scattering approximation of electrons
    using model~\cite{E03} ${\cal M}({\cal T}^{*(2F)})$.
\label{fig:compSEI}
}
\end{figure}
Secondary-electron images
(SEI) represent an orthogonal
projection to the sphere of the symmetry directions
below a near-surface region
of the sample.
Apart from the icosahedral symmetry of the pattern,
some groups of bright patches are seen to lie
within bands, similar to Kikuchi bands
\cite{B04} connecting the
2fold-, 3fold-, and 5fold-symmetry directions.
In case of crystals these
bands are a direct consequence of well-defined dense
{\it planes}
of atoms.
The bands observed by a quasicrystal
should be also a consequence of dense planes or
plane-like layers within the quasicrystal, which,
according to the secondary-electron pattern,
lie  perpendicular to the principal directions of the
icosahedron.
Hence, the pattern in Fig.~\ref{fig:compSEI}(a)
carries an
information of the long-range order.
We note that there are bands
perpendicular to 2fold- and 5fold-symmetry directions,
but not to 3fold-symmetry directions.

A quantum mechanical single-site scattering calculation
\cite{B04} is a faithful representation of the SEI
pattern because it accounts for the wave nature of the
secondary electrons.
Fig.~\ref{fig:compSEI}(b) illustrates the results of
the calculation~\cite{E03} using the coordinate set
of the model ${\cal M}({\cal T}^{*(2F)})$. This approximation overestimates
the scattering intensity along chains of atoms
\cite{B04}
but suffices for
purposes, since our interest is
mainly in the presence or absence of bands on the
screen.
As in case of crystals \cite{B04} the band width
is inversely proportional to the spacing of
crystallographic planes,
and the band width is related to the distance
of interatomic planes
\cite{B04}.
The observed band widths reveal that the
inter-planar distances of the highly dense
2fold planes are broader than these of the
5fold planes/plane-like layers
by a factor
1.6.
In 5fold case, from the model we predict the distance
$d_5=\tau/(\tau+2)$\ding{196}$=2.04$~{\AA}, which is the
distance between the highly dense $(q,b)$ and the
$(b,q)$ plane-like layers in the terminating
thick layer $(q,b,b,q)$:
$2.04$~{\AA}$=0.48$~{\AA}$+1.56$~{\AA},
see also Ref.~\cite{Z04}.
In the 2fold case, between the highly dense planes
in the above defined ``thick'' layer, appears
once the distance of 0.92~{\AA} and twice the distance
of 1.48~{\AA}.
Hence, an average distance between the highly dense
planes is
$d_2=\frac{\tau}{3\sqrt{\tau+2}}$\ding{193}$=1.29$~{\AA}.
And their ratio is $d_5/d_2=3/\sqrt{\tau+2}
\approx 1.6$.

The SEI method does not determine the bulk termination,
(it is testing the bulk circa 30~{\AA} below
the surface). However,
in the case of the ordinary
crystals, the Kikuchi bands are related to the most
dense atomic planes, and we show that the same holds
true in the case of quasicrystals as well.
Hence we may claim that SEI images are supporting the
thick atomic layers of the high effective
density to be the bulk terminations, if the Bravais'
rule should be valid in quasicrystals as well.
The not existing 3fold Kikuchi bands  are also
supporting
the model ${\cal M}({\cal T}^{*(2F)})$, in which we find that the
atoms collected by the 3fold  planes are almost
uniformly distributed among these, without the
notable repetitive layers of higher densities.


\section{Introduction}

 Press-Schechter theory has been written into textbooks, but two
 important parameters $\delta_c$ and $\Delta_c$ of it extracted from the studying of
 spherical collapse models are not determined well enough. \cite{GunnGott, LaceyCole}
 are two early works studying the spherical models. They considered
 this model in a background universe without dark energies.
 \cite{Barrow, ECF, WangSteinhardt1, LokasHoffman} are four later works studying this
 model in the flat $\Lambda$CDM or open cosmologies.
 Our purpose here is to provide a simple and direct method
 for the calculation of
 $\delta_c(\Omega_{m0},a_c)$ and $\Delta_c(\Omega_{m0},a_c)$. Our
 background is flat $\Lambda$CDM cosmology but the method can be easily
 generalized to the QCDM \cite{WangSteinhardt1} as well as
 non-flat cosmologies. Comparing with the existed results on this
 model, our results are analytical and exact.

 \section{Press-Schechter Theory}

 Press-Schechter Theory \cite{PressSchechter} predicts that the
 fraction of volume which has collapsed at a certain red-shift $z$ is
 \beq{}
 f_{coll}(M(R),z)=\frac{2}{\sqrt{2\pi}\sigma(R,z)}
 \int_{\delta_c}^{\infty}d\delta e^{-\delta^2/2\sigma^2(R,z)}.
 \nonumber\\
 \label{Press-Schechter}
 \end{eqnarray}
 Here, $R$ is the radius over which the density field has been
 smoothed, $\sigma(R,z)$ is
 the rms of the smoothed density
 field \cite{Dodelson}. $\delta_c$ is the threshold of density contrast at time
 $z$ (count stopping time)
 beyond which objects collapse.

 In existing literatures, there are two conventions for the
 definition of $\sigma(R,z)$ and $\delta_c$. In the first
 convention \cite{Dodelson, WangSteinhardt1, VianaLiddle},
 $\sigma(R,z)$ contains growth factor,
 while $\delta_c$ only depends on the partition of
 the cosmological component today and the count stopping epoch $a_c$.
 In the second convention \cite{ECF}, $\sigma(R)$ does not contain the growth factor,
 but $\delta_c$ contains both the cosmological component partition
 effects and the growth factor effects. We will take the first
 convention in this paper. Our $\delta_c$ here will be
 corresponded to the $\delta_c(0)$ of \cite{ECF}, which was
 plotted in the upper panel of figure.1 of it.

 Operationally, $\delta_c$ is obtained by extrapolating the primordial
 perturbation to the collapse epoch using the growth law of
 linear perturbation theory, i. e.,
 \beq{}
 \delta_c=\left[(\frac{\rho_{mc}(a,a_c)}{\rho_{mb}(a)}-1)\frac{1}{D_1(a)}\right]_{a\rightarrow0}D_1(a_c)
 \label{deltacDefinition}
 \end{eqnarray}
 Where $\rho_{mc}$ and $\rho_{mb}$ are the mass density of the
 clusters and the background respectively; while $D_1(a)$ is the
 growth function of linear perturbation theory \cite{Dodelson},
 \beq{}
 D_1(a)=\frac{5\Omega_{m0}H^2_0}{2}H(a)\int_0^ada^{\prime}[a^{\prime}H(a^{\prime})]^{-3}.
 \end{eqnarray}
 It can be shown that $D_1(a)_{a\rightarrow0}\rightarrow a$. Using the method of
 \cite{LaceyCole} we can show that
 \beq{}
 \left[\frac{r}{a}\right]_{a\rightarrow0}\propto(1-\alpha\cdot a)
 \label{roaagoesto0}
 \end{eqnarray}
 So,
 \beq{}
 \delta_c=3\alpha\cdot D_1(a_c).
 \label{deltacOperationDefinition}
 \end{eqnarray}
 In a given cosmological model, $D_1(a)$ are known
 \cite{Dodelson}, the purpose of studying spherical collapse
 model is to determine $\alpha$.

 It should be noted that besides the partition of the
 cosmological components and the observational epoch, the value
 of $\delta_c$ is also quite dependent on the choice of
 smoothing window used to obtain the dispersion
 $\sigma(R,z)$ \cite{LaceyCole}. We will not consider
 this effect in this paper.

 \section{Spherical Collapse Model}

 To calculate parameter $\alpha$ of eq(\ref{deltacOperationDefinition}),
 let us start with the Friedman equations for
 both the over-dense region and back ground cosmology:
 \beq{}
 (\frac{\dot{r}}{r})^2&&\hspace{-3mm}=\frac{8\pi G}{3}[\frac{\rho_{mc,ta}r^3_{ta}}{r^3}+\rho_{\Lambda 0}]-\frac{\kappa}{r^2}
 \label{FriedmanC} \\
 (\frac{\dot{a}}{a})^2&&\hspace{-3mm}=\frac{8\pi G}{3}[\frac{\rho_{mb,ta}a^3_{ta}}{a^3}+\rho_{\Lambda 0}]
 \label{FriedmanB}
 \end{eqnarray}
 , where $r$ and $a$ denote the radius of the over-dense region
 and scale factor of background universe respectively; $\kappa$ is
 a constant; while subscript $_{ta}$ and $_c$ denote
 the turn around and collapse point time. Because at the turn around time $\dot{r}=0$,
 \beq{}
 \frac{\kappa}{r^2_{ta}}=
 \frac{8\pi G}{3}[\rho_{mc,ta}+\rho_{\Lambda 0}].
 \label{kappa}
 \end{eqnarray}
 Dividing eq(\ref{FriedmanC}) by (\ref{FriedmanB}) and using (\ref{kappa}) we get
 \beq{}
 \frac{\dot{r}^2r^{-2}_{ta}}{\dot{a}^2a^{-2}_{ta}}
 =\frac{\rho_{mc,ta}r^{-1}r_{ta}+\rho_{\Lambda 0}r^2r^{-2}_{ta}-(\rho_{mc,ta}+\rho_{\Lambda 0})}{\rho_{mb,ta}a^{-1}a_{ta}+\rho_{\Lambda
 0}a^2a^{-2}_{ta}}
 \label{FriedmannV}
 \end{eqnarray}
 After letting
 \beq{}
 x=\frac{a}{a_{ta}},y=\frac{r}{r_{ta}},\zeta=\frac{\rho_{mc,ta}}{\rho_{mb,ta}},\nu=\frac{\rho_{\Lambda0}}{\rho_{mb,ta}}
 \label{x-y-zeta-nu-definition}
 \end{eqnarray}
 eq(\ref{FriedmannV}) becomes
 \beq{}
 (\frac{dy}{dx})^2=\frac{\zeta y^{-1}+\nu y^2-(\zeta+\nu)}{x^{-1}+\nu
 x^2}.
 \label{dydxMainEq}
 \end{eqnarray}
 We will not try to solve this equation, we derive it here
 because we need to use its asymptotic behavior to calculate
 $\delta_c$.

 In the $\Lambda$CDM cosmologies, the quantity $\zeta$ and
 $\nu$ in eq(\ref{dydxMainEq}) are only functions of $\Omega_{m0}$ and $a_c$.
 They do not depends on $x$ or $y$. Using eq(\ref{roaagoesto0}),
 we can write
 \beq{}
 \left[\frac{y}{x}\right]_{x\rightarrow0}=
 \left[\frac{rr^{-1}_{ta}}{aa^{-1}_{ta}}\right]_{a\rightarrow0}=\zeta^{\frac{1}{3}}(1-\alpha\cdot
 a_{ta}x).
 \label{yoverx}
 \end{eqnarray}
 So,
 \beq{}
 \frac{d}{dx}[y_{x\rightarrow0}]=\zeta^{\frac{1}{3}}(1-2\alpha\cdot a_{ta}x).
 \label{dydx}
 \end{eqnarray}
 Substitute eqs(\ref{yoverx}) and (\ref{dydx}) into
 eq(\ref{dydxMainEq}), expand it and keep only the linear
 term, we get
 \beq{}
 \alpha=\frac{1}{5}a^{-1}_{ta}[\zeta^{\frac{1}{3}}+\nu\cdot
 a_{ta}\zeta^{-\frac{2}{3}}].
 \label{alpha-result}
 \end{eqnarray}
 Substitute this result into eq(\ref{deltacOperationDefinition}),
 we get:
 \beq{}
 \delta_c(\Omega_{m0},a_c)=\frac{3}{5}a^{-1}_{ta}[\zeta^{\frac{1}{3}}+\nu\cdot
 a_{ta}\zeta^{-\frac{2}{3}}]D_1(a_c)
 \label{deltacAnalytical}
 \end{eqnarray}
 This is the first important result of this paper. We will
 derive the differential equation for determining $\zeta(\Omega_{m0},a_c)$ and
 the fitting formula eq(\ref{zetaLCDMold}) for it as well as the analytical solution for
 $\nu(\Omega_{m0},a_c)$ eq(\ref{LCDMtcAndtta}) (and the notes there) in the technique details
 section. Comparing with the method of \cite{ECF, WangSteinhardt1, LokasHoffman}
 our method here is more easier to operate and the result is more
 simply-looking.

 \begin{figure}
 \includegraphics[scale=0.4]{deltac3Dp.eps}
 \caption{$\delta_c^{\prime}$s
 dependence on the cosmological models characterized by
 $\Omega_{m0}$ and the collapse epoch $a_c$.
 }\label{deltac3D}
 \end{figure}
 \begin{figure}
 \includegraphics[scale=0.4]{deltacECF3Dp.eps}
 \caption{
 $\delta_c^{\prime}$s
 dependence on $\Omega_{m0}$ and
 the collapse epoch matter percentage $\Omega_{mb,c}$.
 Points on the same space curves have the same
 collapse epoch $a_c$. The outmost curve has $a_c=1$,
 the next one has $a_c=0.9$ and so on. The deeper a
 curve lies in the paper, the smaller $a_c$ it has.
 }\label{ECFd}
 \end{figure}

 In FIG.\ref{deltac3D} we plotted
 $\delta_c^{\prime}$s dependence on $\Omega_{m0}$ and $a_c$.
 From the figure we see that if $\Omega_{m0}$ is fixed, then
 $\delta_c$ is a decreasing function of $a_c$. Physically this is
 because, in the given cosmological model, $\Omega_{m0}$ fixed,
 the earlier an objects collapse, the more denser it is required
 to assure collapse occur successfully. On the other hand, if
 $a_c$ is fixed, then $\delta_c$ increases as $\Omega_{m0}$
 increases. Superficially this means that in those lower matter (thus high dark energy)
 percentage universes, collapse occurs more easily! This is not
 the fact. Because in the lower matter percentage universes,
 density perturbation grows more slowly. As the result of linear
 extrapolation of the primary perturbations
 eq(\ref{deltacDefinition}), the more smaller growth factor comes from
 the lower matter percentage suppresses the primary perturbations
 more strongly so gives more smaller $\delta_c$.

 To compare our results with that of \cite{ECF}, we plotted
 $\delta_c^\prime$s dependence on $\Omega_{m0}$ and
 $\Omega_{mb,c}$ in FIG.\ref{ECFd}. From this figure we see that
 what was shown in the upper panel of figure.1 of \cite{ECF} is
 the projection of our results in the $\delta_c$-$\Omega_{mb,c}$
 plane.

 To this point in this section, we imagined the evolution of an
 over-dense region as: as long as it is over-dense enough,
 it will grow from an $r=0$ point, to the maximum
 radius, then collapse to $r=0$ point. Factually, before the
 region collapse to the $r=0$ point, pressures from the random
 moving of the materials inside the region will balance their
 self-gravity and the system will enter virialization status.
 At this point, the second important parameter
 $\Delta_c:=\frac{\rho_{mc,vir}}{\rho_{mb,vir}}$ in the application of
 Press-Schechter theory \cite{ECF, VianaLiddle, WangSteinhardt1} appears.
 Assuming that at the collapse point, the system has virialized fully, we can write:
 \beq{}
 \Delta_c=
 \frac{\rho_{mc}(a_{ta})r^3_{ta}r^{-3}_c}{\rho_{mb}(a_{ta})a^3_{ta}a^{-3}_c}
 =\zeta\frac{r^3_{ta}}{r^3_c}\frac{a_c^3}{a_{ta}^3}.
 \label{DeltacDefinition}
 \end{eqnarray}
 According to virial theorem and energy conservation law \cite{WangSteinhardt1},
 \beq{}
 &&\hspace{-5mm}E_{kinetic}=-\frac{1}{2}U_{G}+U_{\Lambda}\nonumber\\
 &&\hspace{-5mm}\frac{1}{2}U_{G,c}+2U_{\Lambda,c}=U_{G,ta}+U_{\Lambda,ta}
 \end{eqnarray}
 we have
 \beq{}
 &&\hspace{-5mm}-\frac{1}{2}\frac{3GM^2}{5r_c}-2\cdot\frac{4\pi GM\rho_{\Lambda,c} r^2_c}{5}
 \nonumber\\
 &&\hspace{18mm}=-\frac{3GM^2}{5r_{ta}}-\frac{4\pi GM\rho_{\Lambda,ta} r^2_{ta}}{5}
 \nonumber\\
 &&\Rightarrow\nonumber\\
 &&\hspace{-5mm}\frac{r_{ta}}{r_c}=
 \frac{2(\rho_{mc,ta}+\rho_{\Lambda,ta})\rho^{-1}_{mc,ta}}{(\rho_{mc,c}+4\rho_{\Lambda,c})\rho^{-1}_{mc,c}}
 \nonumber\\
 &&\hspace{5mm}=\frac{2(1+\frac{\Omega_{\Lambda,ta}}{\Omega_{mb,ta}}\frac{1}{\zeta})}
 {1+4\frac{\Omega_{\Lambda,c}}{\Omega_{mb,c}}[\zeta\frac{r^3_{ta}}{r^3_c}\frac{a_c^3}{a^3_{ta}}]^{-1}}
 \label{RtaORc}
 \end{eqnarray}
 Looking as an equation for $\frac{r_{ta}}{r_c}$, eq(\ref{RtaORc})
 can be solved analytically. Substituting the solution into
 eq(\ref{DeltacDefinition}), we get
 \beq{}
 &&\hspace{-3mm}\Delta_c(\Omega_{m0},a_c)=\zeta\cdot\left[\frac{a_c}{a_{ta}}\right]^3\nonumber\\
 &&\hspace{3mm}\times\left\{\frac{2}{3}[\frac{(\mu_t+1)^2}{f(\mu_t,\mu_c)}+(\mu_t+1)+f(\mu_t,\mu_c)]\right\}^{3}
 \label{NLDeltacAnalytical}\\
 &&\hspace{-3mm}f(\mu_t,\mu_c)=[(\mu_t+1)^3-\frac{27}{4}\mu_c
 \nonumber\\
 &&\hspace{15mm}+\frac{\sqrt{27}}{4}\sqrt{\mu_c(27\mu_c-8(\mu_t+1)^3)}]^{1/3}
 \label{fdefinition}\\
 &&\mu_t=\frac{\Omega_{\Lambda,ta}}{\Omega_{mb,ta}}\frac{1}{\zeta},\
 \mu_c=\frac{\Omega_{\Lambda,c}}{\Omega_{mb,c}}\frac{1}{\zeta}\frac{a^3_{ta}}{a^3_c}
 \label{mutmucdefinition}
 \end{eqnarray}
 Eq(\ref{NLDeltacAnalytical}) is the second important result of this
 paper. In $\Lambda$CDM cosmology, all the quantities on the right
 hand side of eq(\ref{NLDeltacAnalytical}) are known functions of
 $\Omega_{m0}$ and $a_c$. We will give the relevant formulaes
 eq(\ref{OmegaLCDM}), (\ref{zetaLCDMold}) and (\ref{LCDMtcAndtta})
 in the technique section. Comparing with \cite{WangSteinhardt1},
 our results here is exact instead of approximated.

 To see the physical meaning of
 eq(\ref{NLDeltacAnalytical}) clearly, we plotted
 $\Delta_c^{\prime}$s dependence on $\Omega_{m0}$ and $a_c$
 in FIG.\ref{NLD3D}.
 From the figure, we see that $\Delta_c$ is an increasing function of
 $a_c$, i. e., the latter a region collapses, the more denser
 should it be. This can be understood physically. Because at more later
 times, dark energy's percentage will be more larger. To
 cancel its counter-collapse effects, more denser
 matter is required to assure collapse.
 It is worth noting that
 \cite{ECF} used a different definition of
 $\Delta_{c(ECF)}:=\frac{\rho_{mc,c}}{\rho_{b,totoal,c}}$ (so different
 mass-temperature relations, please comparing eq(2.2) of \cite{ECF}
 and eq(4) of \cite{WangSteinhardt1}), we
 reproduce the results there in FIG.\ref{ECFD}. From the right
 panel of this figure we see that, what was plotted in the lower
 panel of figure.1 of \cite{ECF} is the projection of our results
 on the $\Delta_c$-$\Omega_{mb,c}$ plane.

 \begin{figure}
 \includegraphics[scale=0.22]{NLD3D1p.eps}
 \includegraphics[scale=0.22]{NLD3D2p.eps}
 \caption{Left panel: $\Delta_c^{\prime}$s
 dependence on the collapse epoch $a_c$, different
 curves denote different cosmological models indexed by $\Omega_{m0}$.
 Right panel: $\Delta_c^{\prime}$s
 dependence on the cosmological models, different
 curves denote different collapse epoch indexed by $a_c$.
 }\label{NLD3D}
 \end{figure}

 \begin{figure}
 \includegraphics[scale=0.22]{NLDECFp.eps}
 \includegraphics[scale=0.22]{NLDECF3Dp.eps}
 \caption{Right panel: $\Delta_{c(ECF)}^{\prime}$s
 dependence on $\Omega_{m0}$ and the collapse point $\Omega_{mb,c}$,
 the same curve has the same collapse epoch.
 Right panel: projection of three space curves from the right
 panel. }\label{ECFD}
 \end{figure}

 \section{Technique Details}
 To calculate parameter $\zeta$ appearing in eqs(\ref{deltacAnalytical}) and
 (\ref{NLDeltacAnalytical}), we write down
 Friedman equations for the back ground cosmology and the space-space
 component of Einstein equation for the radius of the over-dense
 region,
 \beq{}
 (\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}(\rho_{mb}+\rho_\Lambda).
 \label{Friedmann_clusterFormation}
 \end{eqnarray}
 \beq{}
 \frac{\ddot{r}}{r}=-\frac{4\pi G}{3}
 (3p_{\Lambda}+\rho_\Lambda+\rho_{mc}).
 \label{Einstein_clusterFormation}
 \end{eqnarray}
 Using the notations introduced in
 eq(\ref{x-y-zeta-nu-definition}),
 eqs(\ref{Einstein_clusterFormation}) and
 (\ref{Friedmann_clusterFormation}) can be rewritten as
 \beq{}
 (\frac{\dot{x}}{x})^2&&\hspace{-3mm}=\frac{8\pi G\rho_{mb,ta}}{3}\frac{1}{x^3\Omega_{mb}(a)}
 \label{Friedmann_clusterFormation2}\\
 \frac{\ddot{y}}{y}&&\hspace{-3mm}=-\frac{4\pi G\rho_{mb,ta}}{3}
 [(1+3w)\frac{1-\Omega_{mb}(a)}{x^3\Omega_{mb}(a)}+\frac{\zeta}{y^3}]
 \label{Einstein_clusterFormation2}
 \end{eqnarray}
 ,where we preserved the equation of state coefficient $w$ as a
 general parameter. It can even be a function of $a$ or $x$. The
 above two equations can be translated into:
 \beq{}
 &&\hspace{-3mm}\frac{d^2y}{dx^2}
 -\frac{dy}{dx}\frac{1}{2}[\frac{1}{x}+\frac{d\Omega_{mb}}{dx}\frac{1}{\Omega_{mb}}]
 \nonumber\\
 &&\hspace{7mm}+\frac{1}{2}[(1+3w)\frac{y(1-\Omega_{mb})}{x^2}+\frac{\zeta x\Omega_{mb}}{y^2}]
 =0
 \label{zetaDeterminingEq}
 \end{eqnarray}
 Eq(\ref{zetaDeterminingEq}) is a second ordinary differential equation,
 it contains a free parameter $\zeta$, but satisfies three boundary
 condition:
 \beq{}
 &&[\frac{y}{x}]_{x\rightarrow0}=[\frac{rr^{-1}_{ta}}{aa^{-1}_{ta}}]_{a\rightarrow0}=1_{-}\cdot\zeta^{\frac{1}{3}}
 \nonumber\\
 &&y|_{x=1}=1,\ y^{\prime}|_{x=1}=0.
 \label{BC-zeta-eigenValueEq}
 \end{eqnarray}
 This is a two point boundary condition problem. It can also
 be looked as an eigen-value problem and solved numerically by the method
 described in \S17.4 of \cite{PressNR}.

 Noting
 \beq{}
 \Omega_{mb}(a)
 =\frac{\Omega_{m0}a^{-3}}{\Omega_{m0}a^{-3}+(1-\Omega_{m0})}
 =\frac{1}{1+\frac{1-\Omega_{mb,ta}}{\Omega_{mb,ta}}x^3},
 \label{OmegaLCDM}
 \end{eqnarray}
 we see that $\zeta$ solved from eq(\ref{zetaDeterminingEq})
 only depends on $\Omega_{mb,ta}$. So
 we can choose $a_{ta}=1$ and solve eq(\ref{zetaDeterminingEq})
 for different values of $\Omega_{m0}$
 to get the appropriate $\zeta$ and fit
 the result as $\zeta(\Omega_{m0})$. Then change the
 function
 into $\zeta(\Omega_{mb,ta})$ to get $\zeta^\prime$s dependence on
 $\Omega_{m0}$ and $a_c$. We do so by software $Mathematica$ and
 confirmed the results by the method of \cite{PressNR} and get
 \beq{}
 \zeta(\Omega_{mb,ta})=(\frac{3\pi}{4})^2\Omega_{mb}^{-0.7384+0.2451\Omega_{mb}}|_{ta}.
 \label{zetaLCDMold}
 \end{eqnarray}

 Now the final question we need to answer is to express the
 scale factor of turn around point with that of the collapse
 point.
 Note that, in the unperturbed spherical cases, clusters'
 formation process is symmetrical about the turn around
 time $t_{ta}$, so
 \beq{}
 t_c=2t_{ta}\label{tcAndtta}.
 \end{eqnarray}
 According to Friedmann
 equation $H^2=H^2_0\Omega_{m0}/a^3\Omega_m$, the time-scale-factor
 relation is
 \beq{}
 t&&\hspace{-3mm}\propto\int_0^{a}da^{\prime}\sqrt{\Omega_m(a^{\prime})a^{\prime}}
 \nonumber\\
 &&=\int_0^{a}da^{\prime}\sqrt{\frac{a^{\prime}}{1+\nu_0 a^{\prime 3}}}
 \nonumber\\
 &&\propto\textrm{ln}[\sqrt{\nu_0 a^3}+\sqrt{\nu_0 a^3+1}].
 \label{time-a-LCDM}
 \end{eqnarray}
 Substitute eq(\ref{time-a-LCDM}) into (\ref{tcAndtta}) and
 solve it analytically, we get
 \beq{}
 a_{ta}=\left[\frac{\sqrt{1+\nu_0a^3_c}-1}{2\nu_0}\right]^{1/3},
 \label{LCDMtcAndtta}
 \end{eqnarray}
 where $\nu_0$ is today's ratio of dark-energy/mass density and
 $\nu=\nu_0a^3_{ta}$. \cite{WangSteinhardt1} proposes that when
 eqs(\ref{Friedmann_clusterFormation2}) and (\ref{Einstein_clusterFormation2}) solved,
 using eq(A12) and (A13) of it to calculate $\delta_c$ directly.
 This method has difficulty to get high precision.

 \section{Conclusions}

 We studied the spherical collapse model in the flat $\Lambda$CDM
 cosmology and provided exact and analytical formulaes
 eq(\ref{deltacAnalytical}) and (\ref{NLDeltacAnalytical}) for
 the calculation of the two important parameters
 $\delta_c(\Omega_{m0},a_c)$  and $\Delta_c(\Omega_{m0},a_c)$
 in terms of the media variable $\zeta$ eq(\ref{zetaLCDMold}).
 We reproduced the existing results in the literature, but our method is
 more easier to operate and the result is more simply-looking.
 Our result will be useful in
 the studying of galaxy clusters evolution and the application of Press-Schechter
 theory.

 \section{Acknowledgements}
 We thank very much to profess V. R. Eke for his patience on
 explaining \cite{ECF} to us.