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%astro-ph/0901.3723
%http://www.astro.ru.nl/~falcke/publications.html#jetlag
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\begin{document}
\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
{The black hole at the Galactic Center, Sgr A*, is the prototype of a
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,MacquartBowerWright2006,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
,FalckeMeliaAgol2000
,MeliaFalcke2001,MiyazakiTsutsumiTsuboi2004,EckartBaganoffSchoedel2006}.
The latter is of particular importance since it may eventually allow
imaging of the shadow cast by the event horizon
\citep{FalckeMeliaAgol2000,HuangCaiShen2007,BroderickLoeb2006}.
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,QuataertGruzinov2000a}.
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
,FalckeBiermann1999,FalckeMarkoff2000,YuanMarkoffFalcke2002}.
Only recently have measurements of the intrinsic size of Sgr A* become
available
\citep
{BowerFalckeHerrnstein2004,ShenLoLiang2005,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{Size 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,BowerFalckeHerrnstein2004}
the actual source size $\phi_{\rm Sgr\,A*}$ is given by
\begin{equation}
\label{e:quadrature}
\phi_{\rm Sgr\,A*}=\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}, $\phi_{\rm scatt}=(1.31\pm0.02)\,{\rm
mas}\;(\lambda/{\rm cm})^2$. 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{\label{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 $m=1.3$ and $m=1.7$.
\citep{ShenLoLiang2005}, who use only short-wavelength data, find
$m=1.09$.
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-lambda} 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}=(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}}
\caption{\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}
\phi_{\rm Sgr\,A*}=(0.52\pm0.03)\,{\rm mas}\times\left({\lambda/{\rm
cm}}\right)^{1.3\pm0.1}.
\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
$m=1.44-0.19+0.16$, 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}}
\caption{\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
,MiyazakiTsutsumiTsuboi2004
,MauerhanMorrisWalter2005,MarroneBaganoffMorris2008}
with yet larger rms variations and outbursts of a factor of several
over the quiescent level.
%rmsflux = {{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 $s=\phi_{\rm Sgr\,A*}D_{\rm GC}$
of Sgr A* from observations -- $D_{\rm GC}$ is the Galactic center
distance $D=8$ 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
$\phi_1=0.73$ mas and $\phi_2=0.32$ mas or $s_1=8.8\times10^{13}$ cm
and $s_2=3.9\times10^{13}$ 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 $\Delta \tau=20$ min the flow velocity is
$v=(s_1-s_2)/\Delta \tau=1.4 c$. 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 $\Delta\tau=(s_1/v(s_1) - s_2/v(s_2))$.
The easiest model is one with a constant flow speed $v(s)=$const. To
allow for relativistic speeds we write this as $v(s)=\gamma\beta c$,
where $\gamma=\sqrt{1-\beta^2}\,^{-1}$ and $\beta=v/c$. For this
subsection, we will ignore projection effects for the sake of
simplicity. The time lag then is $\Delta \tau=D_{\rm GC}\Delta
\phi_{\rm Sgr A*}/\gamma\beta c$. 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
$\gamma\beta=1$ (red, solid line) or $\gamma\beta=0.5$ 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}
\subsection{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)=-
{\partial\over\partial z}P.
\end{equation}
$\omega=m_{\rm p}nc^2+U_{\rm j}+P_{\rm j}$ is the enthalpy density of
the jet, $U_{\rm j}$ is the internal energy density, $n$ is the
particle density, and $P_{\rm j}=(\Gamma-1)U_{\rm j}$ 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 $\omega=(1+\Gamma)U_{\rm j}$ and ${\omega/n}=(1+\Gamma)m_{\rm p}
c^2=$const at the sonic point $z=z_{0}$. 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}
\resizebox{\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 $\Gamma=4/3$ the solution (see Fig.~\ref{f:sgr-gbplot}) is
given implicitly as
\begin{equation}\label{gb}
\gamma_{\rm j}\beta_{\rm j}=f^\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),
\end{equation}
where $f'(y)=x$ is the inverse function of $f$ such that $f(x)=y$, $x=
\gamma_{\rm j}\beta_{\rm j}$, and
\begin{equation}
f(x)=-28 \ln x+{9\over5}42^{2/3}x^{5/3}+42x^2.
\end{equation}
$\gamma_{\rm j}$ and $\beta_{\rm j}c$ are the bulk jet Lorentz
factor and velocity, $z=Z/z_{\rm nozz}$ 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 $z=1$, where
$\gamma_{\rm j}\beta_{\rm j}$ equals the proper sound speed
$\gamma_{\rm s}\beta_{\rm s}=\sqrt{\Gamma(\Gamma-1)(\Gamma+1)^{-1}}$,
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={\Delta \phi D_{\rm GC} \over \sin i \beta_{\rm j}(z) c}
\left(1-\beta_{\rm j} \cos i\right).
\end{equation}
$\Delta \phi=\phi_{\rm Sgr\,A*}(\lambda_0)-\phi_{\rm
Sgr A*}(\lambda)$ and $\lambda_0=1.35$ cm is chosen as the
reference wavelength. This formulation recovers the well-known formula
for apparent superluminal motion ($\beta_{\rm app}=\Delta
\phi D/\Delta\tau$), if the implied flares were observed as moving
blobs.
For the dimensionless length we take $z=\phi_{\rm
Sgr\,A*}(\lambda)/\phi_{\rm Sgr\,A*}(\lambda_{\rm nozz})$ with
$\lambda_{\rm nozz}=0.8$mm. The latter represents the next observing
band above the highest currently available VLBI measurements and
corresponds to a projected size of about $4 R_{\rm g}$ ($R_{\rm
g}=GM_{\bullet}/c^2$), for a black hole mass of
$M_\bullet=3.6\times10^6M_{\odot}$. This is also the typical nozzle
size used in spectral fits \citep[e.g.,
][]{FalckeMarkoff2000,MarkoffBowerFalcke2007}.
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}
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}}
\caption{\label{f:sgr-jet-time-delay-vs-lambda-long}Same as Fig.~
\ref{f:sgr-jet-time-delay-vs-lambda}
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}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{sgr-jet-time-delay-vs-lambda-
diffsize.ps}}
\caption{\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}
%\begin{equation}
%t_{\rm e}= 22\,{\rm h}\,\left({B/{\rm 10\, G}}\right)^{-2}
\left({\gamma_{\rm e}/100}\right)^{-1},
%\end{equation}
\section{Summary and discussion}
\label{s:conclusions}
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}
-------------------------------------------------------------------------------------------------------
Prof. Dr. Heino Falcke
Professor of Astroparticle Physics and Radio Astronomy - Radboud Univ.
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)
Web: www.astro.ru.nl/~falcke
Tel: +31 (0)243652020 Mobile: +49 (0)15123040365 (new!)
Fax: +31 (0)243652807 Secr: +31 (0)243652804
-------------------------------------------------------------------------------------------------------
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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}}
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
{The black hole at the Galactic Center, Sgr A*, is the prototype of =
a 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}.
<=
div>Only recently have measurements of the intrinsic size of Sgr A* =
become
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}
\phi_{\rm Sgr\,A*}=3D(0.52\pm0.03)\,{\rm =
mas}\times\left({\lambda/{\rm =
cm}}\right)^{1.3\pm0.1}.\end{equation}
Again, this is consistent with the previous results and will be =
usedin 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 =
severalover 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),
\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}.
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}}
\caption{\label{f:sgr-jet-time-delay-vs-lambda-long}Same as =
Fig.~\ref{f:sgr-jet-time-delay-vs-lambda} 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}
<=
div>
-------------------------------------------------------=
------------------------------------------------
Prof. Dr. Heino Falcke =
=
=
=
Professor of Astroparticle Physics and =
Radio Astronomy - Radboud Univ.
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)
=
Tel: +31 =
(0)243652020 Mobile: +49 =
(0)15123040365 (new!)
Fax: +31 =
(0)243652807 Secr: +31 =
(0)243652804
---------------------------------------------------=
----------------------------------------------------
=
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