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\begin{document}

\title{On the orientation of the Sagittarius A* system}

\author{L. Meyer\inst{1}\thanks{Fellow of the International Max Planck Re=
search School (IMPRS) for Radio and Infrared Astronomy at the Universitie=
s of Bonn and Cologne.} \and R. Sch\"odel\inst{1} \and A. Eckart\inst{1,2=
} \and W. J. Duschl\inst{3,4} \and V.~Karas\inst{5} \and M. Dov\v{c}iak\i=
nst{5}  }

\institute{I.Physikalisches Institut, Universit\"at zu K\"oln, Z\"ulpiche=
r Str. 77, 50937 K\"oln, Germany \and Max-Planck-Institut f\"ur Radioastr=
onomie, Auf dem H\"ugel 69, 53121 Bonn, Germany \and Institut f\"ur Theor=
etische Physik und Astrophysik, Universit\"at zu Kiel, 24098 Kiel, German=
y \and Steward Observatory, The University of Arizona, 933 N. Cherry Ave.=
 Tucson, AZ 85721, USA \and Astronomical Institute, Academy of Sciences, =
Bo\v{c}n\'{i} II, CZ-14131 Prague, Czech Republic }

\date{Received  / Accepted }

%%%%%
%
\abstract {The near-infrared emission from the black hole at the Galactic=
 center (Sgr~A*) has unique properties. The most striking feature is a su=
ggestive periodic sub-structure that has been observed in a couple of fla=
res so far.}
{Using near-infrared polarimetric observations and modelling the quasi-pe=
riodicity in terms of an orbiting blob, we try to constrain the three dim=
ensional orientation of the Sgr~A* system.}
{We report on so far unpublished polarimetric data from 2003. They suppor=
t the observations of a roughly constant mean polarization angle of $\sim=
 60\degr \pm 20\degr$ from 2004 -- 2006. Prior investigations of the 2006=
 data are deepened. In particular, the blob model fits are evaluated such=
 that constraints on the position angle of Sgr~A* can be derived.}
{Confidence contours in the position -- inclination angle plane are deriv=
ed. On a $3\sigma$ level the position angle of the equatorial plane norma=
l is in the range $\sim 60\degr - 108\degr$ (east of north) in combinatio=
n with a large inclination angle. This agrees well with recent independen=
t work in which radio spectral/morphological properties of Sgr~A* and X-r=
ay observations, respectively, have been used. However, the quality of th=
e presently available data and the uncertainties in our model bring some =
ambiguity to our conclusions.}
{}

\keywords{black hole physics -- infrared: accretion, accretion disks -- G=
alaxy: center}

\titlerunning{On the orientation of the Sgr A* system}

\maketitle

\section{Introduction}

The near-infrared (NIR) regime plays an outstanding role in Galactic cent=
er research. The proper motion of stars visible in this waveband showed t=
he existence of a supermassive black hole (BH) right in the center of our=
 galaxy named Sagittarius~A* (Sgr~A*), see e.g. Eckart \& Genzel~\cite{ec=
kigenzel}; Ghez et al.~\cite{ghez98}, \cite{gheznature}; Sch\"odel et al.=
~\cite{rainer1, rainer2}; Eisenhauer et al.~\cite{eisenhauer}. In 2003 al=
so NIR emission directly from Sgr~A* has been detected (Genzel et al.~\ci=
te{genzel}; Ghez et al.~\cite{ghez04}), which is the most underluminous B=
H accretion system so far accessible to observations (with a bolometric l=
uminosity which is nine orders of magnitude lower than the Eddington lumi=
nosity). Short periods of increased radiation (called `flares') sometimes=
 seem to be accompanied by a quasi-periodic oscillation (QPO), at least a=
t $\lambda =3D 2.2\,\mu m$ (Genzel et al.~\cite{genzel}; Eckart et al.~\c=
ite{ecki2}; Meyer et al.~\cite{ich2, ich}; see also Belanger et al.~\cite=
{belanger}; Aschenbach et al.~\cite{aschenbach1}; Yusef-Zadeh et al.~\cit=
e{yusef}). However, note that all detections of a periodic sub-structure =
have been reported from observations with NACO/VLT only. While short time=
scale structure has also been reported from Keck observations (e.g. Ghez =
et al.~\cite{ghez05}), an independent confirmation of periodicity is stil=
l lacking (Ghez, priv. comm.). The suggestive QPO manifests itself as sub=
-flares with constant separation that are superimposed on a larger flare.=
 Unfortunately, the exact cause of the QPO is far from clear. While the f=
act that the frequency of QPOs in BH binaries scale with one over the BH =
mass and that Sgr~A* seems to fit in this relation suggests Keplerian mot=
ion as the cause (Aschenbach~\cite{aschenbach2}, \cite{aschenbach3}; Abra=
mowicz~\cite{abramowicz05}), recent magneto-hydrodynamic simulations disa=
gree with that and instead identify pattern rotation as the source for th=
e modulation (Chan et al.~\cite{chan}; Falanga et al.~\cite{falanga}).=20

In this research note we focus on polarimetric measurements of Sgr~A* in =
the NIR (for polarization properties of Sgr~A* in the mm-regime see Marro=
ne et al.~\cite{marrone2, marrone1}) and their interpretation in terms of=
 the orbiting spot model, i.e. Keplerian motion in strong gravity is adop=
ted as the cause for the sub-flares. Note that the orbiting blob model ca=
n be tested (and perhaps rejected), although the task cannot be achieved =
now, given the insufficient quality of data
available at present. In principle, constraints can be imposed on the
model by tracking all four Stokes parameters and comparing their time
evolution against the model. It is known that general relativity should
imprint specific features in the time evolution of a polarized signal
when a blob orbits near to a black hole (which is what we suggest here); =
the
direction of the polarization vector should wobble within a range
determined by the distance of the blob from the hole and the viewing angl=
e
of the observer (e.g., Connors et al.~\cite{connors80}). Here, we discuss=
 in particular the constraints that this modelling sets on the position a=
ngle of the normal to the equatorial plane of the spinning BH.=20

In the next two sections we first report so far unpublished polarimetric =
data of Sgr~A* from 2003 that show that the mean polarization angle fluct=
uated only slightly for at least four years. Afterwards, this preferred d=
irection is interpreted within the blob model.

\section{The data and their reduction}

The data we present here are polarimetric observations of Sgr~A* at $2.2\=
, \mu$m from October 2003\footnote{They are freely available on the ESO a=
rchive, program 072.B-0285(A)}. They have not been published before and a=
re important to identify a favored orientation of the Sgr~A* system. They=
 have been taken with the near-infrared camera and adaptive optics system=
 NACO at ESO's Very Large Telescope (VLT) in combination with a wire-grid=
=2E The observations have been conducted in such a way that after $\sim 5=
$ min the wire-grid has been rotated. While it is now clear (Genzel et al=
=2E~\cite{genzel}; Eckart et al.~\cite{ecki2}) that this time resolution =
is too low due to the high variability of Sgr~A* in the NIR, nevertheless=
 a mean polarization angle can be inferred. The data are of very high qua=
lity and show an exceptionally bright flare.

We carried out standard reduction techniques, i.e. sky subtraction, flat =
fielding and bad pixel correction. The point spread function was extracte=
d on each individual image (Diolaiti et al.~\cite{diolaiti}) and then use=
d for a Lucy-Richard deconvolution. After restoration with a Gaussian bea=
m, aperture photometry on the diffraction limited images for individual s=
ources with known flux and Sgr~A* was done.  For the extinction correctio=
n we assumed
$A_K=3D2.8$\,mag. Estimates of uncertainties were obtained from the
standard deviation of fluxes from nearby constant sources. The
calibration was performed using the overall interstellar polarization
of all sources in the field, which is 4\% at $25\degr$
(Eckart et al.~\cite{ecki95}; Ott et al.~\cite{ott}).

The dereddened flux of Sgr~A* and of a nearby constant star is shown in F=
igure~\ref{flux}. The flux was calibrated such that each angle seperately=
 matched the total flux of known sources. That means that actually Figure=
~\ref{flux} shows approximately twice the flux for each angle. The first =
gap between $\sim 25-50$\,min is due to sky observations, the reason for =
the second gap is not traceable. The observations started exactly at the =
base of the peak.
=20

\section{The mean polarization angle}

Figure~\ref{flux} shows the high variability of Sgr~A* with a very short =
rise and fall timescale consistent with previous observations. From these=
 observations (Genzel et al.~\cite{genzel}; Eckart et al.~\cite{ecki2}; M=
eyer et al.~\cite{ich2}; Trippe et al.~\cite{trippe}) the following pheno=
menology of K-band flares has emerged: the first component is a broad und=
erlying flare that lasts 50-120\,min. The second component is sub-flares =
that are superimposed on the broad flare and show a constant seperation o=
f $17\pm 3$\,min. Having this context in mind and regarding the incomplet=
eness of the data here, the single peak seen in the lightcurve in Figure~=
\ref{flux} may be interpreted as one sub-flare superimposed on an underly=
ing flare. Note that although only one possible sub-flare can be seen, it=
s duration is $\sim$20\,min and therefore exactly what is expected from p=
revious observations that showed suggestive QPO activity.  =20

The polarimetric observing technique that was chosen for these observatio=
ns here is certainly unsuitable as is known by now. The high variability =
demands the simultaneous measurement of all four position angles of the e=
lectric field vector.=20
However, a shape of the sub-flare can be assumed and fitted to the data t=
o allow a statement on the polarization angle and degree. Here, we approx=
imate the sub-flare by a Lorentz profile of the form
\[
f(x)=3D\frac{s}{s^2+(x-t)^2}, \qquad s>0, \quad -\infty<t<\infty.
\]=20
The choice of a Lorentzian to fit this part of the lightcurve is of cours=
e not unambigous. This clearly brings some uncertainty to the inferred po=
larization properties. Figure~\ref{fit} shows fits of a Lorentzian to eac=
h polarization angle.=20

The polarimetric observations of Sgr~A* from 2004 -- 2006 (Eckart et al.~=
\cite{ecki2}; Meyer et al.~\cite{ich2}; Trippe et al.~\cite{trippe})  hav=
e shown that the degree of linear polarization and the polarization angle=
 vary during a flare. The angle wobbles around a mean value during the hi=
gh flux phase and then goes to different values in the decaying part of t=
he flare (it is important to keep in mind that Sgr~A* is at $2.2 \mu m$ o=
nly detectable in its "flaring state" so that nothing is known about the =
polarization properties in its low flux phase).  The small insert in the =
lower right corner of Figure~\ref{fit} shows that the Lorentz profile fit=
s reproduce this behavior from previous observations qualitatively. The m=
ean angle can be read off to be $\sim 40\degr$ (east of north).=20

Eckart et al.~(\cite{ecki2}) report a mean angle of $60\degr \pm 20\degr$=
 for observations in 2004 and 2005. While this is within the range inferr=
ed here, Meyer et al.~(\cite{ich2}) and Trippe et al.~(\cite{trippe}) arr=
ive at a mean angle of $80\degr \pm 10 \degr$ and $80\degr \pm 25 \degr$,=
 respectively, for the 2006 observing run. Concerning the uncertaintiy of=
 the procedure adopted here, a conclusion that suggests itself is that th=
e mean polarization angle of Sgr~A* changed only slightly during the past=
 four years. This is especially true if one takes into account that a str=
ict stability of the mean polarisation appears to be rather unphysical. S=
mall fluctuations in the magnetic field configuration and/or a precession=
 of the inner accretion disk seem likely. In this regard, our conclusion =
that the roughly constant mean polarisation points to a preferred positio=
n of the Sgr~A* system seems reasonable.     =20
=20

\section{The three dimensional orientation of the Sgr~A* system}

The existence of a favored polarization angle allows to investigate the o=
rientation of the Sgr~A* system on the sky.
Here, we want to study it within the orbiting blob model. Meyer et al.~(\=
cite{ich2,ich}) calculated polarimetric lightcurves from an orbiting spot=
 (here we use spot and blob interchangeably) around Sgr~A* and compared t=
hem to observations. This showed that this simple model, in which general=
 relativistic effects on the radiation of a somehow confined, locally hea=
ted region plays the major role, leads to very good fits of the measureme=
nts. More precisely, in this model the sub-flares are due to a blob on a =
relativistic orbit
around the MBH, while an underlying ring accounts for the broad overall
flare. Relativistic effects like beaming, lensing, and change of
polarization angle imprint on the emitted intrinsic radiation
(e.g. Dovciak et al.~\cite{dovciak}; Connors \& Stark~\cite{connors}; Hol=
lywood \& Melia~\cite{hollywood2}; Bromley et al.~\cite{bromley}; Falcke =
et al.~\cite{falcke}; Broderick \& Loeb~\cite{broderick1, broderick2}; Sc=
hnittman~\cite{schnittman}). In our model we assume that the variability
in the polarization angle and the polarization degree are only due to the=

relativistic effects. As the emitted radiation of Sgr~A* is
synchrotron radiation (emitted in the disk corona), we assumed two differ=
ent magnetic field
configurations to fit the light curves with our model. The first is
analogous to a sunspot and results in a constant $E$-vector
perpendicular to the disk. The second configuration is a global
azimuthal magnetic field that leads to a rotation of the $E$-vector
along the orbit. With this model at hand, observed polarimetric lightcurv=
es can be fitted to investigate the parameters of the Sgr~A* system. The =
inclination angle, the dimensionless spin parameter $a_\star$, the bright=
ness excess of the spot with respect to the disk, the initial phase of th=
e spot on the orbit, the orientation of the equatorial plane on the sky, =
and the polarization degree of the disk and the spot are the free paramet=
ers.  In their discussion, Meyer et al.~(\cite{ich2,ich}) focused on the =
viewing angle and the spin parameter of Sgr~A*. In this note we extend th=
e discussion to the position angle of the system, i.e. the angle of the e=
quatorial plane normal.


Figure~\ref{conf} shows the $1\sigma-$ and $3\sigma-$confidence contours =
in the position angle ($\theta$) -- inclination angle ($i$) plane. The co=
ntours are results of the fits to the 2006 data presented in Meyer et al.=
~(\cite{ich2}). The magnetic field configuration corresponds to the sunsp=
ot case. The contours have been calculated such that on each point in the=
 $\theta$ -- $i$ plane the $\chi^2$-minimum with respect to the other fre=
e parameters has been taken. The only exception is the dimensionless spin=
 parameter $a_\star$ which has been fixed to $a_\star =3D 0.6$ throughout=
 the calculations. The overall $\chi^2$-minimum lies then at $\theta =3D =
105\degr$, $i=3D55\degr$ (marked with the black dot). As the contours sho=
w, this minimum is not unambigous, but it is nevertheless noteworthy that=
 the formal $\chi^2$-minimum coincides exactly with the finding of Markof=
f et al.~(\cite{sera}). In their paper, they fitted spectral and morpholo=
gical properties of Sgr~A* within the jet model. They also arrive at a po=
sition angle of 105\degr (east of north) for the jet axis together with a=
n almost edge on viewing angle. The inclination angle in our work (Figure=
~\ref{conf}) may not be strict edge-on, but it has the trend to be. Also =
note that our approach can only cover the region $i\lessapprox 70\degr$. =
Furthermore, Muno et al.~(\cite{muno}) and Eckart et al.~(\cite{ecki2}) p=
resent X-ray and NIR data of one long, narrow feature of synchrotron-emit=
ting particles that point toward Sgr~A* and may be identified as a jet. T=
he angle of this jet-like feature on the sky is close to the $105\degr$ i=
nferred here and thereby supports our finding.

It is interesting that different models and methods seem to converge to a=
 viewing angle which is close to edge-on. Not only Markoff et al.~(\cite{=
sera}) and Meyer et al.~(\cite{ich2}) but also Falanga et al.~(\cite{fala=
nga}) and Huang et al.~(\cite{huang}) find such a configuration. Using th=
e Rossby wave instability model (Tagger \& Melia~\cite{tagger}), Falanga =
et al.~(\cite{falanga}) arrive at an inclination angle of $i \approx 77\d=
egr$. Huang et al.~(\cite{huang}) noted that large inclination angles are=
 preferred when a radiatively inefficient accretion flow model is coupled=
 to a ray tracing algorithm and compared to recent VLBI size measurements=
=2E Taking the high inclination for granted, Figure~\ref{conf} shows that=
 our model predicts a position angle of $\sim60\degr - 108\degr$. This qu=
ite broad range includes the result from Markoff et al.~(\cite{sera}) and=
 Muno et al.~(\cite{muno}) as well as -- at least in the $3\sigma$ limit =
-- Muzic et al.~(\cite{kora}). The latter work investigated the proper mo=
tion of thin filaments at the Galactic center, which may be driven by som=
e kind of collimated outflow. Sgr~A* is identified as one possible source=
 for such an outflow and if this is true, a preferred ejection direction =
of $\sim 60\degr$ is possible.

It is only useful to show the fits to the 2006 data here. This is for two=
 reasons. First, the confidence contours of the 2005 and 2006 data are ve=
ry similar and second, the 2006 data are of far better quality than the 2=
005 data. But it is of major importance to note the shortcomings of our a=
pproach. First of all, the magnetic field geometry of the blob is complet=
ely unknown. As described above we used two simple approximations to desc=
ribe the spot's field: (i) a sunspot like geometry in which the resulting=
 $\vec{E}$-vector is always perpendicular to the equatorial plane, (ii) a=
n azimuthal magnetic field, which is suggested by MHD simulations of magn=
etized accretion disks that often develop a toroidal
component of the magnetic field. The former leads to better fits, that is=
 why we have discussed it at length. But the latter case can not be disca=
rded. The corresponding confidence contours are shown in Figure~\ref{toro=
}. Unfortunately, its $\chi^2$-minimum lies at rather small position angl=
es, so the above conclusions get weakened. However, this minimum lies $\D=
elta \chi^2 \approx 5$ higher than in the sunspot scenario ($\chi^2 \appr=
ox 3$ and $\chi^2\approx 8$, respectively; all values are reduced-$\chi^2=
$ values). Although this is not a $3\sigma$ exclusion, it makes the sunsp=
ot scenario more likely. Another caveat concerns the unknown foreground p=
olarization. Its value at the position of Sgr~A* is important in the dete=
rmination of the position angle. We had to assume that it is equal to the=
 average value in the field. With these restrictions in mind, our work at=
 least shows the consistency of previous independent findings with our mo=
del.



\section{Summary}

The purpose of this research note is to show that the mean polarization a=
ngle in the near-infrared was almost constant from 2003-2006, and to infe=
r the position angle of Sgr~A* from this preferred direction in terms of =
the orbiting spot model. For a high inclination angle the $3\sigma$ range=
 for the position angle is derived to be $\sim 60\degr - 108\degr$, there=
by supporting the findings of Markoff et al.~(\cite{sera}), and -- less l=
ikely -- Muzic et al.~(\cite{kora}) who arrived at position angles of $\s=
im105\degr$ (see also Muno et al.~\cite{muno}) and $\sim60\degr$, respect=
ively, with complementary methods. However, there are severe caveats. The=
 unknown magnetic field structure, the uncertain foreground polarization,=
 and the lack of a clear understanding of the detailed hydrodynamics of t=
he blob challenge our conclusions.


\begin{figure}[p]
\centering
\resizebox{10cm}{!}{\includegraphics[angle=3D270]{8009fig1.jpg}}
\caption{Dereddened flux of Sgr~A* and the constant reference star S2 (sh=
ifted 15\,mJy upward for better comparison). Note that the duration of th=
e single sub-flare is $\sim 20$\,min. The flux for each channel is calibr=
ated such that it matches the total flux, i.e. each Stokes parameter is i=
ndividually calibrated to the total flux of the references sources and no=
t the sum of two orthogonal angles. }
\label{flux}
\end{figure}

\begin{figure}[p]
\centering
\resizebox{10cm}{!}{\includegraphics[angle=3D0.3]{8009fig2.jpg}}=20
\caption{Fits of a Lorentz profile to each channel of figure~\ref{flux}. =
With this approximate behavior, the polarization angle and degree of line=
ar polarization can be inferred. They are shown in the insert in the lowe=
r right corner. The polarization angle is represented by the black line, =
the polarization degree is the red line.}
\label{fit}
\end{figure}

\begin{figure}[p]
\centering
\resizebox{8cm}{!}{\includegraphics[angle=3D0]{8009fig3.jpg}}
\caption{$1\sigma-$ and $3\sigma-$confidence contours calculated from fit=
s on the 2006 data reported by Meyer et al.~\cite{ich2}. The vertical axi=
s shows the position angle of the normal to the equatorial plane of the b=
lack hole on the sky (east of north). The horizontal axis shows the viewi=
ng angle (0\degr: face-on). The $\chi^2$-minimum is represented by the bl=
ack dot. The assumed magnetic field configuration corresponds to the sun =
spot scenario (see text for details). }
\label{conf}
\end{figure}

\begin{figure}[p]
\centering
\resizebox{8cm}{!}{\includegraphics[angle=3D0]{8009fig4.jpg}}
\caption{$1\sigma-$ and $3\sigma-$confidence contours calculated from fit=
s on the 2006 data reported by Meyer et al.~\cite{ich2}. The vertical axi=
s shows the position angle of the normal to the equatorial plane of the b=
lack hole on the sky (east of north). The horizontal axis shows the viewi=
ng angle (0\degr: face-on). The $\chi^2$-minimum is represented by the bl=
ack dot. An azimuthal magnetic field has been assumed. }
\label{toro}
\end{figure}



\begin{acknowledgements}
We are grateful to Sera Markoff and Andrea Ghez for fruitful discussions =
and comments. This work was supported in part by the Deutsche Forschungsg=
emeinschaft (DFG) via grant SFB 494 and the Center for Theoretical Astrop=
hysics in Prague.=20
\end{acknowledgements}

\clearpage

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