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/div>

{The black hole at the Galactic Center, Sgr A*, is the prototype of =
a

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\begin{docu=
ment}

\title{Jet-lag in Sgr A*: What size and timing =
measurements tell us about the central black hole in the Milky =
Way}

\author{Heino Falcke\inst{1,2} \and Sera Markoff\inst{3} =
\and Geoffrey C. Bower\inst{4}}

\offprints{H.\ Falcke, \email{H.Falcke@astro.ru.nl}=
} \institute{Department of

Astrophysics, Institute =
for Mathematics, Astrophysics and Particle Physics, Radboud =
University,

P.O. Box 9010, 6500 GL Nijmegen, The =
Netherlands \and

ASTRON, =
Oude

Hoogeveensedijk 4, 7991 PD Dwingeloo, The =
Netherlands \and

Astronomical Institute ``Anton =
Pannekoek'', University of

Amsterdam, Kruislaan =
403, 1098SJ Amsterdam, The Netherlands \and

UC Berkeley, =
601 Campbell Hall, Astronomy Department \& Radio Astronomy =
Lab,

Berkeley, CA 94720, =
USA

}

\date{December 25, =
2008}

\abstract

%Context

galactic nucleus at a very low level of =
activity. Its radio through

submm-wave emission is =
known to come from a region close to the

event =
horizon, however, the source of the emission is still =
under

debate. A successful theory explaining the =
emission is

based on a relativistic jet =
model scaled down from =
powerful

quasars.}

%Aims

{ We =
want to test the predictive power of this established jet =
model

against newly available measurements of =
wavelength-dependent time

lags and the =
size-wavelength structure in Sgr A*. }

% =
Methods

{Using all available closure amplitude VLBI data from =
different

groups, we again derived the intrinsic =
wavelength-dependent size of Sgr

A*. This allowed =
us to calculate the expected =
frequency-dependent

time lags of radio flares, =
assuming a range of in- and outflow

velocities. =
Moreover, we calculated the time lags expected =
in

the previously published pressure-driven jet =
model. The predicted

lags are then compared to =
radio monitoring observations at 22, 43,

and 350 =
GHz.}

% Results

{\mybf The combination of time lags =
and size measurements imply a

mildly relativistic =
outflow with bulk outflow speeds =
of

$\gamma\beta\simeq0.5-2$. The newly measured =
time lags are

reproduced well by the jet =
model without any major fine =
tuning.

}

%Conclusions

{\mybf The =
results further strengthen the case for the cm-to-mm =
wave

radio emission in Sgr A* as coming from a =
mildly relativistic jet-like

outflow. The =
combination of radio time lag and VLBI =
closure

amplitude measurements is a powerful new =
tool for assessing the flow

speed and direction in =
Sgr A*. Future VLBI and time =
lag

measurements over a range of wavelengths =
will

reveal more information about Sgr A*, such as =
the existence of a jet

nozzle, and measure the =
detailed velocity structure of a

relativistic jet =
near its launching point for the first =
time.}

\keywords{galaxies: jets =
---

galaxies: active --- galaxies: nuclei --- black hole =
physics ---

Galaxy: center --- radiation mechanisms: =
non-thermal}

\authorrunning{Falcke, Markoff, =
Bower}

\titlerunning{Jet-lag in Sgr =
A*}

\maketitle

\section{=
Introduction}

\label{s:intro}

The Galactic center =
hosts by far the best constrained super-massive

black hole =
candidate: the compact radio source \object{Sgr =
A*}

\citep[see][for a =
review]{MeliaFalcke2001}.

Its mass is believed to be =
around

$4\times10^6M_\odot$ based on stellar proper motion =
measurements

\citep{SchoedelOttGenzel2002,GhezSalimHornstein2005=
}. Linear polarization measurements indicate

that it is =
extremely underfed, with an accretion rate of less =
than

$10^{-7}M_\odot/{\rm =
yr}$

\citep{Agol2000,BowerFalckeWright2005,MacquartBowerWright20=
06,MarroneMoranZhao2007}. The accretion rate and low radio flux put Sgr =
A*

at the tail end of the local luminosity =
function

\citep{NagarFalckeWilson2005} of low-luminosity =
active galactic nuclei

(LLAGN). This makes Sgr A* an ideal =
laboratory to study supermassive

black hole physics in the =
quasi-quiescent state in which most galactic nuclei

exist =
today.

Sgr A* has been detected at radio =
\citep{BalickBrown1974} and now

near-infrared =
\citep{GenzelOttEckart2003} and X-ray =
wavelengths

\citep{BaganoffBautzBrandt2001}. The radio =
spectrum of the source is

variable, slightly inverted, and =
peaking in a submm-bump which

originates close to the event =
horizon

\citep{ZylkaMezgerLesch1992,FalckeGossMatsuo1998,FalckeM=
eliaAgol2000,MeliaFalcke2001,MiyazakiTsutsumiTsuboi2004,EckartBaganoffScho=
edel2006}.

The latter is of particular importance since =
it may eventually allow

imaging of the shadow cast by the =
event =
horizon

\citep{FalckeMeliaAgol2000,HuangCaiShen2007,BroderickLoe=
b2006}. However, until recently

no structural information was =
available for Sgr A*. At wavelengths

shorter than that of the =
submm-bump, the resolution of current telescopes =
is

insufficient and at long wavelengths, where high-resolution =
very long

baseline interferometry (VLBI) techniques can be =
used, the source

structure is blurred by interstellar =
scattering.

This ambiguity has led to a =
longstanding debate about the actual nature of the Sgr

A* =
emission. One class of models suggests that the radio through X-ray =
emission

is caused by accreting hot plasma flowing into the =
black =
hole

\citep{Melia1992a,NarayanMahadevanGrindlay1998,Quatae=
rtGruzinov2000a}.

On the other hand, it has been =
suggested that Sgr A* resembles the

compact radio cores of =
active galactic nuclei (AGN) and therefore most of =
the

emission is associated with a (mildly) relativistic =
outflow or =
jet

\citep{ReynoldsMcKee1980,FalckeMannheimBiermann1993,FalckeBi=
ermann1999,FalckeMarkoff2000,YuanMarkoffFalcke2002}.

available

\citep{BowerFalckeHerrnstein2004,Shen=
LoLiang2005,DoelemanWeintroubRogers2008},

providing crucial =
new input. The new intrinsic size measurements

agree well with =
the predictions of the traditional jet =
model

\citep{BowerFalckeHerrnstein2004,MarkoffBowerFalcke2007}, =
however, a

direct confirmation of an outflow is still =
lacking.

Clearly, additional information is =
required to determine the speed and direction of the flow responsible =
for the emission in Sgr A*. Such additional information has now become =
available with the first reliable time lag measurements of radio =
outbursts at different wavelengths =
\citep{Yusef-ZadehRobertsWardle2006,Yusef-ZadehWardleHeinke2008}.{\mybf =
These observations show that high radio frequencies lead the =
lower radio frequencies by some 20 minutes around 43 GHz. Because =
the radio emission is considered to be optically thick due to its =
flat-to-inverted spectrum, and the synchrotron loss timescale is much =
longer, the radio flux variations are tracing actual adiabatic =
expansion or contraction of the emitting plasma. This scenario is =
in marked contrast to observations in the optically thin part of =
the spectrum at near-infrared (NIR) and X-ray bands, where the cooling =
time scales are faster than adiabatic. At these higher =
frequencies, observations =
\citep{MarroneBaganoffMorris2008,Dodds-EdenBartkoEisenhauer2008} =
show a near simultaneity between NIR and X-ray flares within =
minutes and a delay between X-ray/NIR with respect to the radio =
emission on ther order of hours. The expectation therefore is that =
radio timing observations trace bulk plasma properties, while =
X-ray/NIR variability is dominated by heating and cooling of =
particle energy distributions in the plasma. Which physical =
parameters determine a potential lag between X-rays/NIR and radio/submm =
\citep{MarroneBaganoffMorris2008}, is not immediately =
obvious.}

In this paper we focus on the radio =
time lag data and size

measurements to obtain information on =
the plasma flow speed. To do

this we re-derive the intrinsic =
size of Sgr A* by combining all

existing VLBI data in =
Sec.~\ref{s:vlbisize}, thereby resolving some of

the apparent =
discrepancies between the results of different groups in

the =
literature. We then compute the predicted time lags for =
various

inflow/outflow speeds in Sec.~\ref{s:timedelay} and =
present time lag

predictions of the canonical jet model in =
Sec.~\ref{s:jetmodel}. Here

we also present the only =
analytical velocity profile of a

pressure-driven jet in a =
closed form. The predictions are then

compared with the data =
{\mybf under the assumption that the =
region

causing the variability roughly follows a =
similar size-frequency

relation as seen by VLBI, =
tracing the bulk of the plasma}. Our main

conclusions are then =
summarized and discussed =
in

Sec.~\ref{s:conclusions}.

\section{S=
ize and time lag data in Sgr A*}

\subsection{VLBI size of Sgr =
A*}

\label{s:vlbisize}

The radio size of Sgr A* is =
extremely difficult to determine for several reasons. The radio source =
itself is very compact and hence VLBI techniques have to be used, where =
radio telescopes with separations of several thousand kilometers are =
combined to obtain interferometric information of the source structure. =
However, the major high-frequency VLBI telescopes are in the Northern =
hemisphere, making Sgr A* a low-elevation source which is difficult to =
calibrate. Moreover, the source is located in the Galactic center behind =
a large scattering screen that broadens the intrinsic source size =
significantly at long wavelengths. To escape scattering effects requires =
observing at shorter wavelengths, which are even more difficult to =
calibrate. Hence, the breakthrough for the detection of the =
intrinsic size \citep{BowerFalckeHerrnstein2004} came via the =
introduction of closure amplitude analysis =
\citep{DoelemanShenRogers2001}, a method relatively insensitive to =
common station-based calibration =
errors.

Closure amplitudes provide good means =
to measure the source size with

very high accuracy, especially =
if the source structure is

simple. Since the broadening of the =
source structure by scattering

follows a $\lambda^2$ law =
\citep{DaviesWalshBooth1976,vanLangeveldeFrailCordes1992,LoShenZhao1998,Bo=
werFalckeHerrnstein2004}

the actual source size $\phi_{\rm =
Sgr\,A*}$ is given =
by

\begin{equation}

\label{e:quadrature}

\ph=
i_{\rm Sgr\,A*}=3D\sqrt{\phi_{\rm obs}^2-\phi_{\rm =
scatt}^2},

\end{equation}

where $\phi_{\rm obs}$ and =
$\phi_{\rm scatt}$ are the actually

observed and the expected =
scattering size respectively. $\phi_{\rm

scatt}$ =
can be obtained by measuring the source size at =
long

wavelengths, where the intrinsic size is negligible, and =
extrapolating

with a $\lambda^2$-dependence to shorter =
wavelengths. The validity of

this extrapolation of the =
$\lambda^2$-law has been discussed =
by

\citet{BowerFalckeHerrnstein2004} and demonstrated using =
the measured

Gaussianity of the scattered =
image.

In the following we employ this =
procedure using all currently

available data, and discuss the =
origin of apparently conflicting results for

the =
wavelength-size structure of Sgr A*.

There are =
currently only four papers which contain reliable major =
{\it

and} minor axes for Sgr A*: =
\citet{BowerFalckeHerrnstein2004} for

$\lambda$6 cm to =
$\lambda$7 mm data, \citet{BowerGossFalcke2006} =
for

$\lambda$24 cm to $\lambda$17 cm data, and =
\citet{ShenLoLiang2005} and

\citet{Shen2006} for $\lambda$3 mm =
and $\lambda$7 mm data. Sizes at

wavelengths longer than 20 cm =
are from VLBA closure-amplitude

measurements and at longer =
wavelengths high-quality VLA data is

available. There is =
also a closure amplitude size at $\lambda$3 mm

from =
\citet{DoelemanShenRogers2001}, however, that was only =
reliably

obtained for a circular Gaussian source and has been =
superseded by

\citet{Shen2006}, who fit an elliptical =
Gaussian. In addition there

was one older measurement at =
$\lambda$1.3 mm, by

\citet{KrichbaumGrahamWitzel1998}, based =
on a single baseline

detection. {\mybf The latter has now been =
superceded by a more recent

detection by =
\citet{DoelemanWeintroubRogers2008}, based on =
three

baselines and higher signal-to-noise ratio. =
The $\lambda$1.3 mm

observations yield the =
smallest sizes and the largest excursion from

the =
scattering law. We therefore include these data points in =
our

analysis for completeness, even though they do =
not represent a

closure amplitude measurement and =
one cannot distinguish between

major and minor =
axis. Their inclusion, however,

does not change =
our results =
significantly.}

Figure =
\ref{f:sgr-size-vs-lambda} shows the size data of Sgr\,A* =
as

function of wavelength together with the scattering law =
from

\citet{BowerGossFalcke2006}, $\phi_{\rm =
scatt}=3D(1.31\pm0.02)\,{\rm

mas}\;(\lambda/{\rm cm})^2$. One =
can clearly see how the overall size

of Sgr A* follows the =
$\lambda^2$-law closely at long =
wavelengths.

\begin{figure}

\resizebox{=
\hsize}{!}{\includegraphics{sgr-size-vs-lambda.ps}}

\caption{\la=
bel{f:sgr-size-vs-lambda}Measured radio source size (major axis) of Sgr =
A* as function of observing wavelength in =
centimeters.}

\end{figure}

The next =
step is to subtract the scattering law in quadrature from =
the

observed size according to Equation~\ref{e:quadrature}. =
For this, the

exact normalization of the scattering law is =
vitally important. The

normalizations used by =
\citet{BowerGossFalcke2006} and

\citet{ShenLoLiang2005} differ =
only slightly. This has little impact on the

intrinsic source =
size at $\lambda$3 mm and $\lambda$7 mm, but markedly

affects =
the size at longer wavelengths. As

\citet{BowerGossFalcke2006} =
showed, this changes the

size-vs-wavelength relation =
(size

$\propto\lambda^{m}$). \citet{BowerGossFalcke2006} find =
power laws in

the range between $m=3D1.3$ and =
$m=3D1.7$.

\citep{ShenLoLiang2005}, who use only =
short-wavelength data, find $m=3D1.09$.

The=
biggest problem, therefore, is the systematic uncertainty =
introduced

by the inclusion or non-inclusion of =
long-wavelength data sets.

We investigate this =
uncertainty in the following discussion. Note however that the =
difference in

the scattering law primarily affects intrinsic =
sizes at long

wavelengths; short wavelength sizes are largely =
unaffected because the

contribution of the scattering angle to =
the observed size is =
much

less.

\subsection{Robustness of =
the Sgr A* size =
measurements}

Figure~\ref{f:sgr-normsize-vs-lambd=
a} shows the observed sizes divided

by $\lambda^2$. Here we =
have averaged the data for the various

observing bands, in =
order to avoid having the final fit be biased by the number =
of

observations in one band. For the averaging we divided the =
sizes by

$\lambda^2$ to take out the frequency dependence, and =
weighted them by

their error bars. This gives us one data =
point per band. In

particular, all 20 cm data from =
\citet{BowerGossFalcke2006} are

averaged into one point here. =
The error bars we show are the standard

deviations of the =
measurements in one band, where multiple

measurements were =
available. In principle this should be a more robust

measure =
of the error.

The non-homogeneous error =
distribution is problematic, but as it is

a limit of the =
available observational data base, it cannot be overcome. =
The

three data points at $\lambda$7 mm, $\lambda$3.5 cm, and =
$\lambda$20

cm tend to dominate any fitting and a combined =
multi-parameter fit of

scattering-law and intrinsic size does =
not converge. Therefore it is

customary to only fit the =
scattering law to long-wavelength data. The

range of currently =
used scattering laws then depends exclusively on

which data to =
include. Any unknown systematic error at $\lambda$3.5 cm

or =
$\lambda$20 cm would drastically affect the result. To quantify =
the

robustness of the inferred sizes, we performed a series =
of

weighted fits to the data below $\lambda$1 cm, with one =
random data

point dropped. Doing this we find a range of =
possible scattering laws

(Fig.~\ref{f:sgr-normsize-vs-lambda}) =
given by

\begin{equation}

\phi_{\rm =
scatt}=3D(1.36\pm0.02)\,{\rm mas}\times(\lambda/{\rm =
cm})^2.

\end{equation}

This includes the best-fit =
scattering laws used by

\citet{BowerGossFalcke2006} =
and

\citet{ShenLoLiang2005} within 3 $\sigma$ limits, which =
have

scaling factors of $1.31\pm0.02$ and $1.39\pm0.02$ =
respectively.

\begin{figure}

\resizebox=
{\hsize}{!}{\includegraphics{sgr-normsize-vs-lambda.ps}}

\captio=
n{\label{f:sgr-normsize-vs-lambda}Measured radio source size (major =
axis) of Sgr A* divided by $\lambda^2$ as function of observing =
wavelength in cm. The solid line represents the average scattering law =
used here, where the dashed lines indicated the 3$\sigma$ limits found =
by randomly dropping one data =
point.}

\end{figure}

Subtraction of =
this scattering law in quadrature yields a slightly

revised =
intrinsic size as shown =
in

Figure~\ref{f:sgr-intrinsicsize-vs-lambda}. The sizes =
at $\lambda$2

cm and $\lambda$3.5 cm are relatively sensitive =
to the scattering law

and therefore ``negative'' source sizes =
are possible within the

errors. Negative sizes are treated as =
lower limits around zero with

the respective error bars. We =
fit the error-weighted intrinsic source

size with a powerlaw =
function, =
yielding:

\begin{equation}\label{sgrsize}

\end{equation}

Again, this is consistent with the previous results and will be =
used

in the following analysis. We have further verified =
this result by

running a Monte Carlo simulation, excluding the =
1.3 mm data, by

randomly varying the observed data and the =
scattering law within the

errors quoted here. To each of these =
trials we then fitted the

intrinsic size law and determined =
the slope parameter $m$. We find

that the distribution of $m$ =
is non-Gaussian, with a more extended

tail towards smaller =
values. The median, however, is again at

$m=3D1.44-0.19+0.16$, =
where the errors are the 25\% and 75\% =
quantiles,

respectively. These may be the more realistic error =
estimates than the

ones from the simple analytic =
fitting.

To improve on this result in the =
future, more and better

closure-amplitude size measurements =
need to be obtained at longer

wavelengths, especially at =
$\lambda$2 and $\lambda$6 cm.

In any case, the =
combined set of currently available data and the

error =
analysis confirm previous conclusions that there is =
a

wavelength-dependent photosphere in Sgr A* from a =
stratified

medium. As expected for optically thick synchrotron =
radiation, the

optical depth is indeed frequency dependent. =
This means that

observations of Sgr A* at two different radio =
wavelengths provide

information about two different spatial =
scales where the emission =
originates.

\begin{figure}

\resizebox{\=
hsize}{!}{\includegraphics{sgr-intrinsicsize-vs-lambda.ps}}

\cap=
tion{\label{f:sgr-intrinsicsize-vs-lambda}Intrinsic radio source size =
(major axis) of Sgr A* obtained by subtracting the scattering diameter =
in quadrature. The solid line represents the best fit powerlaw. The =
upper and lower dashed lines are the intrinsic sizes fitted by Bower et =
al. (2006) and Shen et al. (2005) =
respectively.}

\end{figure}

\subsection=
{Variability and time lags}

\label{s:timedelay}

In =
addition to the size measurement, we now have another new crucial =
parameter: the time lags between different wavelengths during =
small-scale variability outbursts. In the absence of direct imaging of =
source substructure, this provides the only means to determine flow or =
signal speeds in Sgr A*.

The overall =
variability of Sgr A* has been established for a long

time. =
The most comprehensive data sets stem from long-term =
monitoring

programs with the Green Bank Interferometer =
\citep{Falcke1999a} and

the VLA =
\citep{HerrnsteinZhaoBower2004} at cm wavelengths. =
The

reported rms variations of the radio spectrum are 2.5\%, =
6\%, 16\%,

17\%, and 21\% at wavelengths of 13, 3.6, 2, 1.3, =
and 0.7 cm

respectively. \citet{MacquartBower2006} argue that =
most of the

variation at longer timescales (several days) and =
at long wavelengths

is due to interstellar scintillation. =
However, for time scales less

than four days the variations =
may be intrinsic with an rms of $\sim10\%$

for wavelengths =
0.7-3 cm. Variability is also seen at mm and

sub-mm =
wavelengths =
\citep{ZylkaMezgerWard-Thompson1995,ZhaoYoungHerrnstein2003,MiyazakiTsutsu=
miTsuboi2004,MauerhanMorrisWalter2005,MarroneBaganoffMorris2008}

with yet larger rms variations and outbursts of a factor of =
several

over the quiescent =
level.

%rmsflux =3D {{cnv[c/(2.3 GHz)/cm], =
2.5/100.}, {cnv[c/(8.3 GHz)/cm],6/100.}, {2, 0.13/0.834}, {1.3, =
0.16/0.926}, {0.7, 0.21/1.001}};

%{{13.0345, 0.025}, {3.61196, =
0.06}, {2, 0.155875}, {1.3, 0.172786}, {0.7, =
0.20979}}

In most cases, where multiple =
wavelengths were observed, the time

coverage was not dense =
enough to find a reliable time lag between

two wavelengths, =
despite several attempts. =
Recently,

\citet{Yusef-ZadehRobertsWardle2006} and =
\citet{Yusef-ZadehWardleHeinke2008} published data obtained with =
the

VLA in fast switching mode allowing quasi-simultaneous =
high-time

resolution measurements of time variability in Sgr =
A* at two different

wavelengths. {\mybf They find a lag =
between $\lambda$1.3 and 0.7 cm on

the order of 20 minutes. =
Taking the weighted average of Table 1 =
in

\citet{Yusef-ZadehWardleHeinke2008} one finds a lag of =
$21\pm3$ minutes.

Millimeter and submm-millimeter wave =
timing observations by the same

group are less significant, =
but seem to go in the same direction, with

a lag between 22 =
and 350 GHz of $65\pm+10-23$ =
minutes

\citep{Yusef-ZadehWardleHeinke2008}. =
}

The sign of the lag between 43 and 22 GHz =
($\lambda\lambda$0.7 \& 1.3 cm) is such that the shorter wavelengths =
lead the longer ones. Keeping in mind that shorter wavelength emission =
originates at smaller size regions, this immediately implies that bursts =
propagate outwards from small to larger scales. =

Given that we know the projected size =
$s=3D\phi_{\rm Sgr\,A*}D_{\rm GC}$

of Sgr A* from observations =
-- $D_{\rm GC}$ is the Galactic center

distance $D=3D8$ kpc =
\citep{EisenhauerSchoedelGenzel2003} -- {\mybf the

time lag =
provides a straightforward estimate for the flow speed}. Using =
equation

Eq.~\ref{sgrsize} we find that the intrinsic size of =
Sgr~A* is

$\phi_1=3D0.73$ mas and $\phi_2=3D0.32$ mas or =
$s_1=3D8.8\times10^{13}$ cm

and $s_2=3D3.9\times10^{13}$ cm at =
$\lambda1.3$ cm and $\lambda7$ mm

respectively. Hence, $\Delta =
s$ is $\sim 27$ light minutes. Given a

time lag on the order =
of $\Delta \tau=3D20$ min the flow velocity =
is

$v=3D(s_1-s_2)/\Delta \tau=3D1.4 c$. Therefore, Sgr A* =
needs to a harbor

at least a mildly relativistic outflow, even =
if one allows for an

error of $\sim$50\%. Projection effects =
would tend to increase this

value even =
further.

Alternatively, if one has a model for =
a flow speed $v(s)$ one can

predict the time lags with =
$\Delta\tau=3D(s_1/v(s_1) - s_2/v(s_2))$.

The easiest model is =
one with a constant flow speed $v(s)=3D$const. To

allow for =
relativistic speeds we write this as $v(s)=3D\gamma\beta =
c$,

where $\gamma=3D\sqrt{1-\beta^2}\,^{-1}$ and $\beta=3Dv/c$. =
For this

subsection, we will ignore projection effects for the =
sake of

simplicity. The time lag then is $\Delta \tau=3DD_{\rm =
GC}\Delta

\phi_{\rm Sgr A*}/\gamma\beta c$. =
Figure

\ref{f:sgr-time-delay-vs-lambda} shows the time lags =
for proper speeds

$\gamma\beta$ in the range 0.5-2 c for the =
measured size-wavelength

relation.

We =
note that the source size in Sgr A* is close to linear =
with

wavelength, hence, for constant velocity one expects a =
linear increase

of the time lags with decreasing wavelength =
relative to a fixed

reference wavelength. For comparison, we =
also show the time lags if

the outflow would propagate always =
with the (Newtonian) escape speed

($\sqrt{2 GM/r}$) for a =
$3.6\times10^6M_\odot$ black hole. Here the

time lags would =
become longer and grow nonlinearly towards =
shorter

wavelengths, since the escape speed is significantly =
slower than the

speed of light. Figure =
\ref{f:sgr-time-delay-vs-lambda} shows that a

gravitationally =
bound flow would predict much longer time lags than

what is =
actually =
observed.

\begin{figure}

\resizebox{\hsize}{!}{\includegraphics{sgr-time-delay-vs-lambda.ps}}

\caption{\label{f:sgr-time-delay-vs-lambda}Expected time lag of =
Sgr A* versus wavelength relative to $\lambda$1.3 cm (22 GHz) for the =
observed size-wavelength relation and a proper flow or signal speed of =
$\gamma\beta=3D1$ (red, solid line) or $\gamma\beta=3D0.5$ and 2 =
(orange, dashed). The data points are measured time lags from =
Yusef-Zadeh et al. (2008). The top black solid line shows the Newtonian =
time lag for an outflow just at the escape speed. Long lags above that =
line would correspond to gravitationally bound =
outflows.

}

\end{figure}

\sub=
section{Time lags in the jet =
model}

\label{s:jetmodel}

Given that the time lags =
suggest a mildly relativistic outflow, it

seems appropriate to =
investigate what an actual jet model

predicts. The basic jet =
model for Sgr =
A*

\citep{FalckeMannheimBiermann1993,FalckeMarkoff2000} =
naturally fits

the spectrum, properly predicted the low =
accretion rate of Sgr A* now

inferred from polarization, and =
also was able to explain the VLBI =
size

\citep{MarkoffBowerFalcke2007}. The only major property =
that could not

be tested so far is in fact the flow =
speed.

So far, the underlying assumption for =
the jet model has been that

Sgr A* is not a strongly =
relativistic outflow. Energetically

this is an optimal =
solution in terms of the ratio between total jet

power and =
emitted synchrotron =
radiation

\citep{FalckeMannheimBiermann1993}. On the =
other hand, the sound

speed for a relativistic plasma as well =
as the escape speed from the

black hole are on the order of =
$\sim0.5c$, which sets a lower bound for a

supersonic jet in =
Sgr A*.

{\mybf In the standard =
\citet{BlandfordKonigl1979} model for the

flat-spectrum radio =
emission of compact jets, a constant velocity is

assumed and =
introduced as a free parameter. \citet{Falcke1996a}

pointed =
out that this is in principle inconsistent, since =
the

longitudinal pressure gradient would inevitably lead to =
some acceleration

of a modestly relativistic jet. As a =
first-order assumption the

velocity was then assumed to be =
simply given by a purely

pressure-driven wind in the =
supersonic regime. This approach had the

advantage that the =
actual acceleration mechanism of the jet, which is

likely =
magnetohydrodynamic in origin, could be treated as a black =
box.

Simulations of the actual acceleration =
process are actually very difficult and time consuming \citep[e.g., =
][]{MeierKoidaUchida2001,DeVilliersHawleyKrolik2005}. However, they all =
start with some initial magnetohydrodynamic collimation regime (here =
referred to as the ``nozzle''). After passing through the fast =
magnetosonic point, the

flow eventually becomes over-pressured =
in a phase where the jet expands more or less freely into the ambient =
medium. The latter situation is mainly addressed by simulations of =
pc-scale jets observed with VLBI =
\citep[e.g.,][]{2008arXiv0811.1143M}.

Since for =
our simple Sgr A* jet model only the part after the sonic =
point

was considered, the only main parameter is then the =
location of the

sonic point and the sound speed. For powerful, =
relativistic jets the

sonic point is expected to be up to =
thousands of Schwarzschild radii

away from the central engine =
\citep{MarscherJorstadDArcangelo2008}

while for Sgr~A* a =
relatively small value, of a few

Schwarzschild radii, seems =
required by the data

\citep{MarkoffBowerFalcke2007}. The =
magnetized plasma is here treated

as a single-component fluid =
with adiabatic index 4/3 -- i.e., in the

relativistic limit of =
a ``photon gas''. The supersonic jet evolution is =
then

calculated from the modified, relativistic Euler equation =
for a freely

expanding jet propagating along the $z$ axis in a =
cylindrical

coordinate system, which we reproduce here from =
\citet{Falcke1996a}: =
}

\begin{equation}\label{euler1}

\gb =
n {\partial\over\partial z}\left(\gb{\omega\over =
n}\right)=3D-{\partial\over\partial =
z}P.

\end{equation}

$\omega=3Dm_{\rm p}nc^2+U_{\rm =
j}+P_{\rm j}$ is the enthalpy density of

the jet, $U_{\rm j}$ =
is the internal energy density, $n$ is the

particle density, =
and $P_{\rm j}=3D(\Gamma-1)U_{\rm j}$ is the pressure

in the =
jet (all in the local rest frame). With a =
``total

equipartition'' assumption one gets $U_{\rm j}\simeq =
m_{\rm p}nc^2$,

hence $\omega=3D(1+\Gamma)U_{\rm j}$ and =
${\omega/n}=3D(1+\Gamma)m_{\rm p}

c^2=3D$const at the sonic =
point $z=3Dz_{0}$. In the free jet with conical

shape =
the energy density evolves as $U_{\rm =
j}\propto

\left(\gb\right)^{-\Gamma}z^{-2}$. The final =
equation is then given by

Eq. 2 in =
\citet{Falcke1996a}.

{\mybf Note that for =
simplicity this equation lacks a =
gravitational

term. This term becomes negligible =
quickly at larger distances and

for unbound flows =
corresponding to typical radio frequencies =
for

Sgr~A*. This is clearly a deficiency when =
discussing the nozzle

region in detail. In the =
following we will subsume this effect in

the =
nozzle size as a free parameter. =
}

\begin{figure}

\resize=
box{\hsize}{!}{\includegraphics{sgr-gbplot.ps}}

\caption{\label{=
f:sgr-gbplot}

The proper flow speed (Eq.~\ref{gb}) of a =
pressure driven jet plotted

versus the logarithm of the =
distance, in units of the nozzle size

$z_{\rm nozz}$, along =
the z-axis.}

\end{figure}

Previously =
the solution of the equation was only available numerically

in =
the code. In in the following we present it in a closed form =
that

allows testing it against time lag observations. For an =
adiabatic

index of $\Gamma=3D4/3$ the solution (see =
Fig.~\ref{f:sgr-gbplot}) is

given implicitly =
as

\begin{equation}\label{gb}

\gamma_{\=
rm j}\beta_{\rm j}=3Df^\prime\left(8+12\left({4\times3^5\over 5^6\times =
7}\right)^{1/6}-28\ln\left({2\over\sqrt{21}}\right)+42\ln{(z)}\right),

<= /div>

<=
div>

\end{equation}

where $f'(y)=3Dx$ is the inverse =
function of $f$ such that $f(x)=3Dy$, $x=3D\gamma_{\rm j}\beta_{\rm j}$, =
and

\begin{equation}

f(x)=3D-28 \ln =
x+{9\over5}42^{2/3}x^{5/3}+42x^2.

\end{equation}

$\gam=
ma_{\rm j}$ and $\beta_{\rm j}c$ are the bulk jet =
Lorentz

factor and velocity, $z=3DZ/z_{\rm nozz}$ is the =
dimensionless length

along the jet axis ($Z$), and $z_{\rm =
nozz}$ marks the location of the

jet nozzle. The equation thus =
has a critical point at $z=3D1$, where

$\gamma_{\rm =
j}\beta_{\rm j}$ equals the proper sound speed

$\gamma_{\rm =
s}\beta_{\rm s}=3D\sqrt{\Gamma(\Gamma-1)(\Gamma+1)^{-1}}$,

and =
is only valid in the supersonic regime =
$z>1$.

This relatively simple quasi-analytical =
description had first been developed for \object{M81*} and was then =
integrated into the Sgr A* jet papers thereafter. While naturally overly =
simplified, we retain it here, treating it as a published prediction. =
{\mybf However, one should not consider this as the only possible =
solution, but rather as representative of a broader class of =
models for modestly relativistic accelerating =
jets.}

Using this description we now calculate =
the time lags based on the

assumption that any flare is =
essentially due to an increase in the

accretion power. This =
increased accretion will turn into an increased

outflow rate, =
based on the ``jet-disk symbiosis''-ansatz of a =
linear

coupling between inflow and outflow =
rate

\citep{FalckeBiermann1995}. The increased power and mass =
flow will

then propagate along the jet essentially with the =
local flow

speed. Here we ignore the slightly increased =
acceleration due to the

increased longitudinal pressure =
gradient in an overdense region, which

would be a second order =
effect.

{\mybf We note that earlier we have =
argued that the X-ray flares in

Sgr A* are not due =
to a similar increase in accretion, but rather

due =
to additional heating or acceleration of the internal =
particle

distributions =
\citep{MarkoffFalckeYuan2001}. This is =
entirely

consistent with our approach here, since =
in the same paper we showed

that such heating =
processes only marginally affect the radio flux =
in

the optically thick region. Hence simultaneous =
radio-X-ray flare are

not necessarily required. =
Radio flares, however, required an actual

increase =
in accretion rate as also argued here. Of course, it is =
not

inconceivable that a sudden increase in =
accretion rate also leads to

additional heating =
and particle accreleration in the inner region =
of

disk and =
jet.

Indeed, recent observations =
\cite{MarroneBaganoffMorris2008} seem to

show that =
there is a rather long lag (on the order of hours) =
between

X-ray/IR-flares and radio flares. =
This time scale is much longer

than =
free-fall or rotational time scales and consistent with =
viscous

processes in the accretion flow linking =
the two types of flares.}

The predicted radio =
time lags in the jet model are then calculated =
as

\begin{equation}

\Delta\tau=3D{\Delt=
a \phi D_{\rm GC} \over \sin i \beta_{\rm j}(z) c}\left(1-\beta_{\rm j} =
\cos i\right).

\end{equation}

$\Delta \phi=3D\phi_{\rm=
Sgr\,A*}(\lambda_0)-\phi_{\rm

Sgr A*}(\lambda)$ and =
$\lambda_0=3D1.35$ cm is chosen as the

reference wavelength. =
This formulation recovers the well-known formula

for apparent =
superluminal motion ($\beta_{\rm app}=3D\Delta

\phi =
D/\Delta\tau$), if the implied flares were observed as moving =
blobs.

For the dimensionless length we =
take $z=3D\phi_{\rm

Sgr\,A*}(\lambda)/\phi_{\rm =
Sgr\,A*}(\lambda_{\rm nozz})$ with

$\lambda_{\rm nozz}=3D0.8$mm.=
The latter represents the next observing

band above the =
highest currently available VLBI measurements and

corresponds =
to a projected size of about $4 R_{\rm g}$ =
($R_{\rm

g}=3DGM_{\bullet}/c^2$), for a black hole mass =
of

$M_\bullet=3D3.6\times10^6M_{\odot}$. This is also the =
typical nozzle

size used in spectral fits =
\citep[e.g.,

][]{FalckeMarkoff2000,MarkoffBowerFalcke2007}.

\caption{\label{f:sgr-jet-time-delay-vs-lambda-long}Same as =
Fig.~\ref{f:sgr-jet-time-delay-vs-lambda}

Figure~\ref{f:sgr-jet-time-delay-vs-lambda} shows =
the expected time

lag for the measured size and the velocity =
field of the

pressure-driven jet. The prediction is consistent =
with the 21 min time

lag found between $\lambda$7 mm and =
$\lambda$1.3 cm. We stress

that this is based solely on the =
combination of the observed sizes and

the previously published =
velocity field for the jet =
model.

\begin{figure}

\resizebox{\hsize=
}{!}{\includegraphics{sgr-jet-time-delay-vs-lambda.ps}}

\caption=
{\label{f:sgr-jet-time-delay-vs-lambda}

Expected time lag of =
Sgr A* versus wavelength relative to $\lambda$1.3

cm (22 GHz) =
for the observed size-wavelength relation and a flow =
speed

according to the jet model for three inclination angles. =
The data

points are the same as in =
Fig.~\ref{f:sgr-time-delay-vs-lambda}.}

\end{figure}

<=
br>

Quite noticeable is the quick rise of the time lag, =
relative to

$\lambda$1.3 cm, towards shorter wavelengths. The =
rise comes from the

fact that the jet first needs to =
accelerate beyond the nozzle, which

yields initially slower =
flow velocities and accordingly longer time

lags. This should =
be a characteristic signal of a nozzle, which future

time lag =
measurements could help to identify. Of course, one has =
to

bear in mind that the model is overly simplistic and in =
reality

this feature may look less drastic. In particular, =
general relativistic

effects will start to play an important =
role. Also, the exact location

of this kink is very sensitive =
to the size of the nozzle, which

is a free parameter within a =
factor of two or so and therefore

difficult to predict. On the =
other hand, an exact localization of the

kink would =
effectively constrain the nozzle size.

For =
future use, we also extrapolate the predicted time lag to =
longer

wavelengths =
(Fig.~\ref{f:sgr-jet-time-delay-vs-lambda-long}). One can

see =
that the lag becomes on the order of a day at cm wavelengths. This =
may

therefore be difficult to observe, given the limited =
observability of Sgr A*

in the Northern hemisphere, but would =
certainly provide crucial

information on the large-scale =
structure of Sgr A* that is otherwise

impossible to obtain due =
to the strong scatter broadening.

In addition =
we consider the effect of the range of =
possible

size-wavelength relations for Sgr A* =
in

Figure~\ref{f:sgr-jet-time-delay-vs-lambda-diffsize}. Not =
surprisingly

the time lags do not show much of a difference at =
short wavelengths,

but differ markedly at long =
wavelengths.

\begin{figure}

\resizebox{=
\hsize}{!}{\includegraphics{sgr-jet-time-delay-vs-lambda-long.ps}}

but extended =
to longer wavelengths. For reference we also show the

time =
lags for a marginally bound outflow as in =
Fig.~\ref{f:sgr-time-delay-vs-lambda}.}

\end{figure}

<=
br>

\begin{figure}

\resizebox{\hsize}{!=
}{\includegraphics{sgr-jet-time-delay-vs-lambda-diffsize.ps}}

\c=
aption{\label{f:sgr-jet-time-delay-vs-lambda-diffsize}Same as =
Fig.~\ref{f:sgr-jet-time-delay-vs-lambda-long}

but now =
we show the time lags for a jet model inclined by =
$60^\circ$

for the different size-wavelength laws considered. =
This has a

significant effect at long cm =
waves.}

\end{figure}

%\b=
egin{equation}

%t_{\rm e}=3D 22\,{\rm h}\,\left({B/{\rm 10\, =
G}}\right)^{-2} \left({\gamma_{\rm =
e}/100}\right)^{-1},

%\end{equation}

\s=
ection{Summary and =
discussion}

\label{s:conclusions}

<= /div>

By now Sgr A* is probably the best studied supermassive black =
hole

with imaging and timing information on scales very close =
to the event

horizon over a wide range of frequencies. New =
VLBI measurements, which

we have here revisited, have =
confirmed theoretical predictions that

Sgr A* has a =
frequency-dependent photosphere at radio wavelengths,

with =
sizes scaling roughly as =
$\lambda^{1.3\pm0.1}$.

The time lag of =
individual bursts seen at different wavelengths provides a powerful new =
tool to constrain the physics at work in Sgr A*. Combining this =
timing data with the increasingly better intrinsic size measurements =
obtained with VLBI can significantly constrain flow speeds. The latter =
is otherwise not measurable with direct imaging due to the extreme =
scatter broadening in the Galactic center.

The =
well-established lag of $\sim 20$ min between

$\lambda7$mm and =
$\lambda1.3$ cm found =
by

\citet{Yusef-ZadehWardleRoberts2006b}, together with the =
intrinsic size

difference of $\sim27$ light minutes at these =
wavelengths, already suggests

that the radio emitting plasma =
is unbound and flows out with mildly

relativistic =
speeds.

This is in contrast with the =
conclusions by

\citet{Yusef-ZadehWardleRoberts2006b} who =
derive a subsonic and

sub-relativistic outflow from the same =
time lag data. However, the

authors base their conclusions =
solely on a fit of their light curve to

a simple =
\citet{vanderLaan1966} model without ever considering the

VLBI =
size measurements. As mentioned by these authors, the van =
der

Laan model describes the adiabatic expansion of a =
spherical plasma

blob and cannot describe bulk outflow, which =
we now know is a major

factor in the extragalactic radio =
sources it was developed for. The

van der Laan model predicts =
source sizes of 8 Schwarzschild radii

\citep[e.g., Fig.~3 =
in][]{Yusef-ZadehWardleRoberts2006b} at 22 GHz

while the =
measured radio size at 22 GHz is 80 Schwarzschild radii. =
The

variations of $\ga20\%$ of the total flux would thus have =
to be

produced by a region that is only $\sim$0.1\% of the =
total volume

compared to the bulk of the plasma. We find this =
scenario unlikely.

The pressure-driven jet =
model \citep{FalckeMarkoff2000}, that has been

successfully =
used to fit size and spectrum of Sgr A* already, is =
quite

consistent with the combined size and time lag data. No =
particular

adjusting of parameters is necessary with respect =
to published jet

models. The main free parameters are the =
nozzle size and the

inclination angle, for which we have used =
canonical

values. Coincidentally, both parameters do not =
affect the $\lambda$7

mm-$\lambda1.3$ cm lag very much. =
However, as one can see in the

figures, these parameters will =
become important once

time lags at other wavelengths are =
available.

The sensitivity to parameters can be =
turned around to state that

measurements at other wavelengths =
in the future will provide

invaluable information about the =
structure of Sgr A*. If time lags can be

found at $\lambda$2 =
cm or $\lambda$3.5 cm, this would constrain

inclination angle =
and the size-wavelength relation much better and

even more =
clearly distinguish between models.

Moreover, =
time lags at $\lambda$3 mm and $\lambda$1 mm could start =
to

show evidence for the acceleration region of the outflow =
(``the

nozzle''), which would be a unique diagnostic for jet =
and accretion

physics. However, here one has to caution =
that our simple analytic

treatment naturally breaks down near =
the nozzle region and in the

vicinity of the black hole. More =
sophisticated numerical

magnetohydrodynamic calculations =
\citep[e.g.,][]{HawleyKrolik2006} are

clearly needed to =
understand the submm-wave emission. Also, since the

jet likely =
originates from an inflow somewhere close to the =
nozzle

region, this inflow could also contribute to the =
submm-bump emission

\citep[e.g.,][]{YuanMarkoffFalcke2002}. =
These additional emission

components might eventually decrease =
the coherence of outbursts across

wavelength in the submm =
\& mm-wave region.

So far we only have a =
single reliable time lag from $\lambda7$ mm to $\lambda$1.3 cm on which =
to base conclusions, with corroborating evidence from a tentative =
$\lambda$ 0.8 mm to $\lambda$ 1.3 cm lag. Therefore further time =
lag measurements at these and other wavelengths and VLBI closure =
amplitude measurements at long wavelengths are highly desirable. The =
former is challenging since the expected time scales at longer =
wavelengths are on the order of a typical observing run. The latter is =
challenging because typical VLBI arrays resolve out Sgr A* at long =
wavelengths and long baselines. Nonetheless, these observations =
are certainly worth the effort. They promise a wealth of detailed =
information about how plasma behaves very close to the event horizon of =
a supermassive black hole. {\mybf Such coordinated multi-wavelength =
studies will eventually allow a range of more sophisticated =
models to be tested in great =
detail.}

\bibliographystyle{aa} =
\bibliography{hfrefs}

\end{document}

-------------------------------------------------------=
------------------------------------------------

=

=
--Apple-Mail-36-1022269342--

Prof. Dr. Heino Falcke =
=
=
=

Professor of Astroparticle Physics and =
Radio Astronomy - Radboud Univ.

LOFAR International Project = Scientist - ASTRON, Dwingeloo

LOFAR International Project = Scientist - ASTRON, Dwingeloo

=

Department of Astrophysics, IMAPP, Radboud =
Universiteit

P.O. Box 9010, 6500 GL Nijmegen, The =
Netherlands

(Visitors: Heyendaalseweg 135, 6525 AJ Nijmegen, The = Netherlands)

(Visitors: Heyendaalseweg 135, 6525 AJ Nijmegen, The = Netherlands)

Tel: +31 =
(0)243652020 Mobile: +49 =
(0)15123040365 (new!)

Fax: +31 = (0)243652807 Secr: +31 = (0)243652804

Fax: +31 = (0)243652807 Secr: +31 = (0)243652804

---------------------------------------------------=
----------------------------------------------------