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\begin{document}
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   \title{Are Ultra Luminous X-Ray Sources Microblazars ?}

   \author{E. K\"ording
          \and
          H. Falcke
          \and
          S. Markoff
          }

   \institute{Max-Plank Institut f\"ur Radioastronomie, 
                Auf dem H\"ugel 69, 53121 Bonn, Germany         
             }

   \abstract{
ROSAT and Chandra-observations have discovered several ultra-luminous X-ray
sources (ULXs)  exceeding luminosities of 
$5 \times  10^{39} \mbox{erg} / \mbox{s}$.
Assuming isotropic emission, these sources should obey the Eddington
limit, requireing the existence of intermediate-mass black holes
of 20-500 $M_{\sun}$.
However, the measured inner disk temperatures are too high for these
masses and there is no convincing creation mechanism known for these objects.
Recently, Markoff, Falcke, Fender (2001)
suggested that jets could be dominantly contributing to the
hard X-ray emission from X-ray binaries (XRB) at least in the Low/Hard State,
meaning that some X-ray sources could be beamed.
A beaming model would reduce the required black hole masses for ULXs
to normal values. To test the hypothesis of beamed emission
we consider a simple population synthesis model for XRBs,
where the X-ray emission is produced by both a jet (beamed) 
and an accretion disk (isotropic).
The model is tested on a combined dataset of X-ray point sources of
nearby galaxies. It can explain the known population of ULXs
with $ M < 15 M_\odot$ and bulk Lorentz factors for jets of
$\gamma_j  \sim 5$.
If this is true, the ULXs would be the stellar-mass
analogues to blazars in the Universe.
}

\maketitle

%
%________________________________________________________________

\section{Introduction}
During the last years X-ray observations have revealed several ultra-luminous
X-ray sources (ULXs) with luminosities 
$L_X \approx 10^{39} - 10^{40} \mbox{erg}/ \mbox{s}$ in nearby galaxies (e.g., La Parola et al. \cite{parola}; Mizuno,
Kubota \& Makishima \cite{mizuno}; Bauer et al. \cite{bauer} ).
Some of the ULXs have shown spectral transitions from a soft spectrum to a
hard power law and are highly variable (e.g., Mizuno \& Kubota \& Makishima \cite{mizuno}; Kubota et al. \cite{kubota}), supporting 
the idea that they can be attributed to accreting objects. But for accretion
powered objects the Eddington limit 
$L_{\mathrm{Edd}} \approx 1.25 \times 10^{38} \frac{M}{M_\odot} \mbox{erg} \ 
\mbox{s}^{-1}$ generally applies, implying that ULXs are super-Eddington for
stellar mass objects. 
Therefore, if the observed X-ray luminosities are created by
isotropically radiating accretion disks, we need to postulate  
a population of intermediate-mass black holes
of $50 - 500 M_\odot$. As discussed in Kubota et al. (\cite{kubota}), however,
the measured inner-disk temperatures of the ULXs 
($T_{\mbox{in}} = 1.0 - 1.8 {\rm keV} $) are too high for theses masses. 
Furthermore, there is no established formation scenario for such 
high mass black holes.
These problems have already been
discussed by King et al. (\cite{king}), where the authors propose 
anisotropic emission as an alternative, 
but this is difficult to achieve via disk models.  
%Analyzing the spectrum of XTE J1118+480  
Markoff, Falcke \& Fender (\cite{markoff}) suggested 
that the spectrum of some X-ray
binaries could be explained by a coupled disk/jet model, where some of
the X-ray emission is produced by synchrotron and inverse-Compton
radiation in the jet. 
The jet emission would naturally be relativistically beamed.
This follows the idea that there may be a unification scheme between AGN and
XRBs, in which --- once one has established the jet model and the geometry ---
one only has to scale the accretion rates.
Thus in analogy to blazars one will expect a population of microblazars.
These are  microquasars with relativistically beamed jets pointed towards the
observer (Mirabel \& Rodriguez \cite{mirabel}), which leads to a high
amplification of the jet emission (one candidate: V4641 Sgr see Orosz \& Kuulkers \cite{orosz}).  
Here, we will first present a population synthesis model for 
disk/jet emission (K\"ording et al. \cite{koerding}) 
and then give possibilities for further tests of the model.

\section{Simple jet/disk model}

Black hole candidate XRBs can be found mostly 
in two distinct states: a high/soft state
where the observed spectrum is soft and thermally-dominated and a
low/hard state dominated by a non-thermal hard power law spectrum
(e.g., Nowak \cite{nowak}). These states seems to be determined mainly by
the accretion rate. One scenario for the evolution of XRBs is that the
inner part of the accretion disk consists of an optically thin,
advection-dominated accretion flow (ADAF) extends up to a transition
radius $r_{\mathrm{t}}$ where the accretion flow turns into a standard
(Shakura \& Sunyaev \cite{ss73}) optically thick disk (e.g., Esin et
al. \cite{esin97}).  The low/hard state appears to be accompanied by
persistent radio jets with optically thick synchrotron emission
extending up to the near-infrared and optical (Fender \cite{fender}). 
The jet could produce soft X-rays by synchrotron and inverse Compton 
emission in the low/hard and the high/soft state.
(Markoff et al. \cite{markoff} , \cite{markoffb} \& 2002 in prep)

To create a simple population synthesis model, including beaming effects of 
the jet emission, we make the following assumptions:

\begin{itemize}
\item We only consider neutron stars of mass $1.4 M_\odot$ and black
holes within a mass range of 5-15 $M_\odot$. 
For simplicity, we use a mass distribution of
black holes given by d$N$/d$M =\mathcal{V} (M)=$const.
\item The ratio of active black holes to active neutron stars 
( $L_{\mbox{X}} > 5 \cdot 10^{36} \mbox{erg} \ \mbox{s}^{-1}$) 
has been fixed to 13 \% (e.g., Tanaka \& Lewin \cite{lewin}).
\item black hole XRBs as well as neutron star XRBs can only be in two distinct spectral states (low/hard and high/soft). In which state a given XRB is depends
only on the accretion rate.
\item The underlying driver of the emission is the accretion rate  $\dot M$
(i.e., not the {\it luminosity} directly).  The probability that a given
XRB has the accretion rate $\dot{M}$ is given by $\mathcal{W}
(\dot{M})$ which we assume as a power law $(\dot{M}^\xi)$ with a
cutoff representing the Eddington limit.
\item  The distribution of accretion rates and jet parameters is
identical for neutron stars and black holes.
\item The soft X-ray emission is produced by an isotropically radiating
disk and a relativistically beamed jet with efficiencies 
as discussed below.
\end{itemize} 

With these assumptions the most important point for our disk/jet model
is the translation of the accretion rate into X-ray luminosity of the
disk and the jet in each state.
To simplify we assume a sharp transition between the
two states, accuring at a critical accretion rate $\dot{M}_C$.
In the high state the disk luminosity increases linearly with 
the accretion rate as expected for a standard accretion disk.
Below $\dot{M}_C$, the disk luminosity increases with $\dot{M}^2$ as 
expected for optically thin ADAFs (Narayan \& Yi \cite{narayan}; 
for a constant $\alpha$-parameter).
Assuming that the jet power scales linearly with
$\dot{M}$, the optically thick jet synchrotron emission will scale
roughly as $L_{\mathrm{x,jet}}\propto\dot{M}^{1.4}$ (Falcke \& Biermann
\cite{falcke95} \& \cite{falcke99}).

At high accretion
rates the jet models for XRBs are not yet well developed, so 
we will discuss the two possibilities that there is a jet in the 
high state or that it does not form at all 
(see Merloni \& Fabian \cite{merloni} ).
In the latter case we assume that the jet breakes down immediately
at the critical accretion rate $\dot{M}_{\mathrm{C}}$.
If there is X-ray emission from the jet the scaling ( $L_{\mathrm{x,jet}}\propto\dot{M}^{1.4}$ ) must 
break down when a significant fraction of the
jet power is radiated away.  In this phase the radiated power can only
increase linearly with jet power. This happens in the high state,
roughly at $\dot M>\dot M_{\mathrm{C}}$, where the jet may be inverse-Compton
cooled (radiating soft X-rays) by the accretion disk.
However, since the jet for this state is not well understood,
we simply fix the luminosity of the jet at $L_{\mbox{jet}} = \eta
L_{\mbox{disk}}$ at $\dot{M}_C$, where $\eta$ is a free parameter.

In summary, we use the following parameterization for the soft X-ray
luminosity of accretion disk and jet:

% (see also Fig.~\ref{LxMd}):

\begin{eqnarray}
L_{\mbox{disk}} & =  & \left\{ 
\begin{array}{c}
\epsilon \left(\frac{\dot{M}}{\dot{M}_C}\right) \dot{M} c^2 \ \ \mbox{if} \ \ \dot{M} < \dot{M}_C  \\
\epsilon \dot{M} c^2  \ \ \mbox{if}  \ \ \dot{M}_C < \dot{M} <\dot{M}_{\mbox{Edd}} 
\end{array}
\right. \qquad \nonumber \\
L_{\mbox{jet}} & = &  \left\{ 
\begin{array}{c}
\eta \epsilon \left(\frac{\dot{M}}{\dot{M}_C}\right)^{0.4} \dot{M} c^2 \ \ \mbox{if} \ \ \dot{M} < \dot{M}_C  \\
\eta \epsilon \dot{M} c^2  \ \ \mbox{if} \ \ \dot{M}_C< \dot{M} <\dot{M}_{\mbox{Edd}} \ \mbox{or} \ \ 0 
\end{array}
\right.  \label{emitlaw}
\end{eqnarray}
Where $\epsilon$ denotes the efficiency of the standard accretion disk 
which we set to $\epsilon = 0.1$ in the following discussion.
For a given mass $M$ the parameter $\dot{M}_{\mbox{Edd}}$ has been chosen
such that the luminosity of the disk and the jet integrated over all
angles is equal to the Eddington luminosity of $1.25 \cdot 10^{38} 
\frac{M}{M_\odot} \mbox{erg} \ \mbox{s}^{-1} $.  

While the disk emission is isotropic, the jet emission depends on the
angle to the line of sight (if $\gamma_{\mathrm{j}}>1$). 
The emission of a continuous jet is given by Lind \& Blandford (\cite{lind}).
The Doppler factor is 
$\delta = \frac{1}{\gamma_{\mathrm{j}} (1-\beta \cos \Theta )}$.  If the
emission in the rest frame of the jet follows a power-law with
spectral index $\alpha$, the observed emission is proportional to
$\delta^{2+\alpha}$.  The probability of seeing an object with an
emission exceeding $L$ when in the rest frame the jet emits
$L_{\mbox{loc}}$ is:
\begin{equation}
P(L,L_{\mbox{loc}}) = \frac{1-\beta}{\beta}\left( 
\left( \frac{L_{\mbox{max}}}{L} \right)^{\frac{1}{2+\alpha}} -1 \right),
\label{eq:prop}
\end{equation}
where $L_{\mbox{max}} = \delta^{2+\alpha}(\Theta = 0) L_{\mbox{loc}}$
is the maximal emission.  To derive this we only consider the jet
component pointing towards us and then integrate over randomly
distributed inclination angles. Since we only discuss jets with
$\gamma_{\mathrm{j}} >2$, the emission of the counter-jet is largely
negligible.
In the high state the jet emits a factor $\eta$ less radiation than the
disk, but due to relativistic beaming the jet will dominate the radiation
at small inclination angles. For example $\gamma_{\mathrm{j}} = 5$ will
boost a fraction of 2\% of the binaries by a factor of $77$, more
than making up for the lower efficiency of the jet.

The parameters $\dot{M}_C$ and $\eta$ of a single population of
XRBs at a given mass and accretion rate are well constrained by jet models:
\begin{itemize}
\item Critical accretion rate: $\dot{M}_C\sim 0.1 \dot{M}_{\mathrm{Edd}} $  
    (see Narayan \& Yi \cite{narayan})  
\item  Jet efficiency: $\eta\la0.3$  
(see Falcke \& Biermann \cite{falcke95} \& \cite{falcke99}) 
\end{itemize}
The Lorentz factor for the jet is not that well constrained. While there
are some reports of very low jet speeds others report higher lorentz factors 
around $\gamma_{\mathrm{j}} \approx 20$. Overall it seems that 
$\gamma_{\mathrm{j}} \simeq 2-5$ is consistent with
the observations. For discussions see Mirabel \& Rodriguez (\cite{mirabel}), 
Fender et al. (\cite{fender99}), Gallo et al. (2002) in prep. 

With this model and the parameters described above we are able to calculate
the luminosity function. The emission in the rest frame of the
jet and the disk is given by  Eq. (\ref{emitlaw}). This
yields together with Eq. (\ref{eq:prop})  for the estimated number of 
XRBs with a X-ray luminosity greater than $L$:

\begin{eqnarray}
N(L) & = & \sum_{i=N,B} 
\mathcal{N}_i \int dM \int d\dot{M} \mathcal{V}_i(M) \mathcal{W}_i(\dot{M})
 \cdot \nonumber \\
& & \ \ \ \ \ P(L-L_{\mbox{disk}}(\dot{M}),L_{\mbox{jet}}(\dot{M})) \label{eqn1}
\end{eqnarray}
where the sum runs over the two populations.  

\section{Data}
A single galaxy has only marginal statistics in the high luminosity 
regime. To test our model we therefore used  
a combination of  properly scaled Chandra data from the galaxies M101, 
M31 and M82 (Pence et al. \cite{pence}; Di
Stefano et al.  \cite{stefano}; Griffiths et al. \cite{griffiths})
in the lower luminosity regime $(\leq 5 \cdot 10^{38} \mbox{erg}/\mbox{s})$  .
These are three close ($D<10$ Mpc) galaxies with good
Chandra data published. 
For the higher luminosities we used data of 49 spiral galaxies 
from the XHFS-sample by Roberts and Warwick (\cite{roberts}).
To derive the higher luminosities the authors of the
papers used an absorbed power
law model with $N_H \approx 10^{21} /\mbox{cm}^2$ and a photon index
 $\alpha \approx 1.7$, but the values are not always directly fitted,
introducing a small additional error ($\approx 10$ \%) 
in the luminosity estimate.
As we are only interested in the slope of the luminosity function 
in a log$N$ - log $S$ plot, these errors
will not play an important role. We corrected the luminosities so that they
all refer to the 2-8 keV band.

The overall scaling of the data is arbitrary, 
because the number of XRBs strongly depends on
the history of star formation, but the slope of the luminosity
function should be more general.
As a reference galaxy we take M101, to
which we scale the populations of the other galaxies in the
overlapping luminosity regime.
For the original data we use standard counting errors 
and normal error propagation. Because we are showing a
cumulative distribution, the errors for each point are not
independent. 

\section{Results}
To compare our simple model with the data we evaluate the integral in
Eq. (\ref{eqn1})  numerically. 
First we discuss the case that there is significant X-ray emission coming
from the jet in the high state.
The free parameter $\xi$ and  absolute normalization have been fitted to 
the data at $L_{\mathrm{x}}\le10^{37} \mbox{erg} \ \mbox{s}^{-1}$. 
We obtain a best-fit
value of the accretion rate index $\xi = 1.4$. 
(note that the luminosity does not scales linearly in this regime). 

 \begin{figure}
   \centering
   %%%\includegraphics[width=3cm]{empty.eps}
   \vspace{180pt}
   \special{psfile=EKoerding_plot2.eps hscale=70 vscale=70 hoffset=-40 voffset=-30}
\caption{Comparison of our model of the luminosity function with the data.
The parameters are $\gamma_{\mathrm{j}} = 5$ , $\eta = 0.3$, $\dot{M}_C = 0.1$, $\xi =1.4$.
Also shown are the individual contributions of the disk and the jet. 
  \label{gamma5}
}
   \end{figure}


Fig.~\ref{gamma5} shows the result for our best-fit model with
$\gamma_{\mathrm{j}} = 5$ and $\eta = 0.3$, together with the combined data
set discussed above.  In this plot we also show the individual
contributions of the disks and jets from neutrons stars and black
holes to the overall distribution.  The Eddington limit clearly shows
up as breaks at the respective luminosities but with $\gamma_{\mathrm{j}} =
5$ the beaming produces emission up to $10^{40} \mbox{erg} \ \
\mbox{s}^{-1}$.

The high luminosity domain depends linearly on $\eta$ while its
dependence on $\gamma_{\mathrm{j}}$ goes as $\gamma_{\mathrm{j}}^{2.7}$, a
slight decrease of $\gamma_{\mathrm{j}}$ can be compensated by an increase
of $\eta$ and vice versa. For $\gamma_{\mathrm{j}}=5.8 $ or $\gamma_{\mathrm{j}}=7.5$ we can find $\eta=0.2$ or $\eta=0.1$, but the fit gets
progressively worse at higher Lorentz factors. Reasonably demanding
$\eta\la0.3$ for the radiative efficiency of the jet sets a lower
limit for $\gamma_{\mathrm{j}}\ga5$.
The model is stable for changes in the other parameters 
(critical accretion rate, power law indices of $L_{\mbox{disk}}(\dot{M})$
and $L_{\mbox{jet}}(\dot{M})$, black hole mass distribution).

The second possibility is that in the high state the jet is not forming 
at all. As the total power from the jet in its rest frame is at
most $ \eta \frac{\dot{M}_C}{\dot{M}_{\mathrm{Edd}}} L_{\mathrm{Edd}}$,  
beaming with Lorentz factors of $\gamma_j = 5$ (yields a factor of
$\sim 10^2$) can not explain the existence of ULXs. But with
higher Lorentz-factors around $\gamma_j \approx 15$ the population can
be reproduced, as shown in Fig.~\ref{nohighstate}. The other
parameters besides $\gamma_j$ are chosen as before.
In this figure we also show the contribution from the XRBs in low state.
In this case the low state XRBs dominate the luminosity function at
low luminosities (below $10^{36.5}$ erg/s) and at high luminosities again
due to boosting.   

  \begin{figure}
   \centering
   %%%\includegraphics[width=3cm]{empty.eps}
   \vspace{180pt}
   \special{psfile=EKoerding_Gamma15.eps hscale=70 vscale=70 hoffset=-40 voffset=-30}
    \caption{Model without jet emission in the high state --- 
Lorentz factor $\gamma_j  =15$
\label{nohighstate}} 
   \end{figure}

It should be mentioned that the introduction of a population of
intermediate-mass black holes could explain the population of ULXs as
well as the jet/disk model. We tested that a population of black 
holes with masses of $20-1000 M_{\sun}$ distributed as a power law
with index of roughly $-2$ can fit the luminosity function within the
given errors.

\section{Possible distinctions of the Models}
The spectrum of an XRB in the low/hard state is characterized by a 
hard power law with possibly a black body component, 
while the high state has a softer spectrum. 
Unfortunately the shape of the spectrum of a beamed jet in the high state
is unknown. It could well be that
it is peaked, and due to boosting we look flat part of it,
resulting in a hard spectrum.
Even if the distinction between the
low and the high state is difficult, it is possible to 
make log $N$ - log $S$ plots for hard and soft spectra separately,
giving a hint to the luminosity function of XRBs in the low and the high state.
The prediction of the disk/jet model with emission from the jet in the 
high state is shown in Fig.~\ref{fig:LowHigh}
and could be compared with observations if sufficient data became available.
The plot for the model with no jet in the high state can be found
in Fig.~\ref{nohighstate}.

 \begin{figure}
   \centering
   %%%\includegraphics[width=3cm]{empty.eps}
   \vspace{180pt}
   \special{psfile=EKoerding_LowHighNew.eps hscale=70 vscale=70 hoffset=-40 voffset=-30}
      \caption{ Luminosity functions of XRB in the Low/Hard State and the 
High/Soft State.
         \label{fig:LowHigh}
         }
   \end{figure}

A second possibility to check if non-thermal emission from 
jets plays a role for ULXs is to search for radio emission. 
Boosting a 10 mJy Galactic XRB with a Lorentz factor of $\gamma \sim 5$
(yields a factor of $\sim 10^2$) and placing it at $D \sim 3$ Mpc 
would yield only  a faint 10 $\mu$Jy source. 
Furthermore the emission of the accretion disk will quench the 
radio emission, reducing the flux even more and 
making radio detections difficult.


\section{Summary and Discussion}

To investigate whether beamed emission of jets could explain the
existence of ULXs, we calculated the luminosity distribution of X-ray 
point sources using a coupled disk/jet model. 
We assumed that the emission of accretion disks 
is isotropic and scales as an ADAF below a critical accretion rate
($ 0.1 \dot M_{\mathrm{Edd}}$), while scaling as a standard disk above.  
The soft X-ray emission from the jet is subject to relativistic beaming
and scales with the accretion rate according to the jet models of
Falcke \& Biermann (\cite{falcke99}) and Markoff et
al. (\cite{markoff} \& \cite{markoffb}) in the low state.
For the high state we considered two cases. First that there
is significant X-ray emission from the jet in the high state and 
second that the jet does not form at all.

The calculated luminosity functions are compared with a 
combined data set of three close galaxies and the XHFS sample.
Within the statistical errors both models agree with the
measured data and can fully account for the
population of ULXs as well as the introduction of a new
population of intermediate-mass black holes.

If one assumes the existence of a jet in the high state, 
it is necessary to have moderately high  
Lorentz factors around $\gamma\sim5$ and fairly 
high jet efficiencies ($\eta=0.1-0.3\%$)
to fit the observed luminosity function.
Otherwise, if the jet does not form in the high state, one needs
very high Lorentz factors around $\gamma_j \approx 15$ to create
a sufficient amount of ULXs. Clearly, more intense modeling of jets
in the high state and comparison to X-ray data is needed.

With the current statistics it is not possible to distinguish
between the different models. But using log $N$ - log $S$ plots
for the different spectral states, radio detections or variability 
it could be possible to give a final answer on the nature
of ULXs.


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\end{document}

