Modeling mm- to X-ray flare emission from SgrA*

Abstract

We report on new modeling results based on the mm- to X-ray emission of the SgrA* counterpart associated with the massive black hole at the Galactic Center. Our modeling is based on simultaneous observations carried out on 07 July, 2004, using the ESO NACO adaptive optics instrument and the ACIS-I instrument aboard the Chandra X-ray Observatory as well as the SMA and the VLA. The observations revealed several flare events in all wavelength domains. Here we show that a combined synchrotron self-Compton (SSC) model followed by an adiabatic expansion of the source components can fully account for the observed flare flux densities and delay times covering the spectral range from the X-ray to the mm-radio domain. The derived physical quantities that describe the flare emission give a blob expansion speed of v{exp}=0.005c, magnetic field of < 60G and spectral indices of 0.8 to 1.4. The derived model parameters suggest that the adiabatic expansion takes place in source components that have a bulk motion larger than v{exp} or the expanding material contributes to a corona or disk, confined to the immediate surroundings of SgrA*.

Full text

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arXiv:0904.2460v1 [astro-ph.CO] 16 Apr 2009
Astronomy & Astrophysics manuscript no. eckart
April 16, 2009
c ? ESO 2009
Modeling mm- to X-ray flare emission from SgrA*
A. Eckart1,2, F. K. Baganoff3, M. R. Morris4, D. Kunneriath1,2, M. Zamaninasab1,2, G. Witzel1, R. Sch¨ odel9, M.
Garc´ ıa-Mar´ ın1, L. Meyer4, G.C. Bower5, D. Marrone6, M.W. Bautz3, W.N. Brandt7, G.P. Garmire7, G.R. Ricker3, C.
Straubmeier1, D.A. Roberts8, K. Muzic1,2, J. Mauerhan4, and A. Zensus2,1
1I.Physikalisches Institut, Universit¨ at zu K¨ oln, Z¨ ulpicher Str.77, 50937 K¨ oln, Germany
e-mail: eckart@ph1.uni-koeln.de
2Max-Planck-Institut f¨ ur Radioastronomie, Auf dem H¨ ugel 69, 53121 Bonn, Germany
3Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
e-mail: fkb@space.mit.edu
4Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095-1562, USA
e-mail: morris@astro.ucla.edu
5Department of Astronomy and Radio Astronomy Laboratory, University of California at Berkeley, Campbell Hall, Berkeley,
CA 94720, USA
e-mail: gbower@astro.berkeley.edu
6Harvard-Smithsonian Center for Astrophysics, Cambridge MA 02138, USA
e-mail: dmarrone.cfa.harvard.edu
7Department of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802-6305, USA
8Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
9Instituto de Astrof´ ısica de Andaluc´ ıa, CSIC, Camino Bajo de Hu´ etor 50, 18008 Granada, Spain e-mail: rainer@iaa.es
Received / Accepted
ABSTRACT
Context. We report on new modeling results based on the mm- to X-ray emission of the SgrA* counterpart associated with the mas-
sive ∼4×106M⊙ black hole at the Galactic Center.
Aims. We investigate the physical processes responsible for the variable emission from SgrA*.
Methods. Our modeling is based on simultaneous observations carried out on 07 July, 2004, using the NACO adaptive optics (AO)
instrument at the European Southern Observatory’s Very Large Telescope⋆and the ACIS-I instrument aboard the Chandra X-ray
Observatory as well as the Submillimeter Array SMA⋆⋆on Mauna Kea, Hawaii, and the Very Large Array⋆⋆⋆in New Mexico.
Results. The observations revealed several flare events in all wavelength domains. Here we show that the flare emission can be de-
scribed withacombination of asynchrotron self-Compton (SSC)model followed byan adiabatic expansion of the source components.
The SSC emission at NIR and X-ray wavelengths involves up-scattered sub-millimeter photons from a compact source component. At
the start of the flare, spectra of these components peak at frequencies between several 100 GHz and 2 THz. The adiabatic expansion
then accounts for the variable emission observed at sub-mm/mm wavelengths. The derived physical quantities that describe the flare
emission give a blob expansion speed of vexp∼ 0.005c, magnetic field of B around 60 G or less and spectral indices of α=0.8 to 1.4,
corresponding to a particle spectral index p∼2.6 to 3.8.
Conclusions. A combined SSC and adiabatic expansion model can fully account for the observed flare flux densities and delay
times covering the spectral range from the X-ray to the mm-radio domain. The derived model parameters suggest that the adiabatic
expansion takes place in source components that have a bulk motion larger than vexpor the expanding material contributes to a corona
or disk, confined to the immediate surroundings of SgrA*.
Key words. black hole physics, X-rays: general, infrared: general, accretion, accretion disks, Galaxy: center, Galaxy: nucleus
1. Introduction
Stellar motion and variable emission allow us to associate
Sagittarius A* (SgrA*) at the center of the Milky Way with a
super-massive black hole (Eckart & Genzel 1996, Genzel et al.
1997, 2000, Ghez et al. 1998, 2000, 2003, 2005, Eckart et al.
2002, Sch¨ odel et al. 2002, 2003, Eisenhauer 2003, 2005).
Recent radio, and near-infrared through X-ray observations
have detected flaring and polarized emission and give detailed
insight into the physical emission mechanisms at work in SgrA*
(e.g. Baganoff et al. 2001, 2002, 2003, Eckart et al. 2003, 2004,
2006ab, 2008ab, Porquet et al. 2003, 2008, Goldwurm et al.
2003, Genzel et al. 2003, Ghez et al. 2004a, and Eisenhauer et
Send offprint requests to: A. Eckart (eckart@ph1.uni-koeln.de)
al. 2005,Hornstein et al. 2007,Yusef-Zadehet al. 2006ab,2007,
2008, Marrone et al. 2008).
Variability at radio through sub-millimeter wavelengths has
been studied extensively, showing that variations occur on
timescalesfromhourstoyears(Wright&Backer1993,Boweret
al. 2002, 2003, 2004, 2005a, 2006, Herrnstein et al. 2004, Zhao
et al. 2003, 2004, Eckart et al. 2006a, Mauerhan et al. 2005,
Yusef-Zadeh et al. 2007, 2008, Miyazaki et al. 2006, Marrone
et al. 2008). Several flares have provided evidence for decay-
ing millimeter and sub-millimeter emission following NIR/X-
ray flares. Simultaneous multi-wavelength observations indicate
the presenceof adiabaticallyexpandingsource componentswith
a delay between the X-ray and sub-mm flares of about 100 min-
utes (Eckart et al. 2006a, Yusef-Zadeh et al. 2008, Marrone et

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2 Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*
al. 2008). The adiabatic expansion is also supported by the ex-
pected swing in polarization as indicated by the measurements
ofYusef-Zadehet al.(2008).Frommodelingthemm-radioflares
at individual frequencies Yusef-Zadehet al. (2007,2008) invoke
expansion velocities in the range from vexp=0.003-0.1c, This is
alsosupportedbytheresults ofrecentNIR/sub-mmobservations
inMay2008usingNACOattheVLTandtheLABOCA bolome-
ter at the Atacama Pathfinder Experiment (APEX), at 0.87 mm
wavelength (345 GHz) (Eckart et al. 2008b, Garc´ ıa-Mar´ ın 2008
in prep.). Here we find an expansion speed of 0.005 c. The
speed is well below the asymptotic upper limit of c/√3 obtained
for a system of relativistically interacting particles (e.g. Bowers
1972) expected in the vicinity of the super-massive black hole
(Blandford & McKee 1977). It is also low compared to the ex-
pected orbital velocities that may be of the order of 0.5 c close to
the last stable orbitaroundthe SMBH. Thelow expansionveloc-
ities suggestthattheexpandinggascannotescapefromSgrA*or
must have a large bulk motion (Yusef-Zadeh et al. 2008, Eckart
et al. 2008b).
In order to investigate this question in more detail we revis-
ited the first observations of a flare with simultaneous coverage
in the NIR/X-ray and sub-mm/mmwavelength domainobserved
on July 07, 2004. Eckart et al. (2006a) showed that the observed
amplitudes of the flux density variations are generally consistent
with adiabatic expansion of a synchrotron self-absorbed source
(van der Laan 1966). Here we present a detailed time depen-
dent model of the flare emission from the X-ray to the short cm-
wavelength domain.
For optically thin synchrotron emission we refer throughout
this paper to photon spectral indices (α) using the convention
Sν∝ ν−αand to spectral indices (p) of electron power-law dis-
tributions using N(E) ∝ E−pwith p = (1 + 2α). The assumed
distance to SgrA* is 8 kpc (Reid 1993), consistent with more
recent results (e.g. Ghez et al. 2005, Eisenhauer et al. 2003).
2. Observations and data reduction
In 2004 from July 05 to 08 Sgr A* was observed from the radio
millimeter to the X-ray wavelength domain. On July 07 a strong
simultaneous NIR/X-ray flare event was observed immediately
followed by simultaneous SMA and VLA observations. Putting
emphasis particularly on the NIR/X-ray data the details for the
entire observingrun have been analyzed in Eckart et al. (2006a).
For completeness we give in the following a brief summary of
the data acquisition and reduction for the X-ray, NIR and sub-
mm/mm domain with emphasis on the essentials important for
thepresentedanalysis.Theobservationalresultsare summarized
in Table 1.
2.1. The NACO NIR adaptive optics observations
Near-infrared (NIR) observations of the Galactic Center (GC)
were carriedout with the NIR cameraCONICA andthe adaptive
optics (AO) module NAOS (briefly “NACO”) at the ESO VLT
unit telescope 4 on Paranal, Chile, during the nights between 05
July and 08 July 20041In all observations, the infrared wave-
front sensor of NAOS was used to lock the AO loop on the NIR
bright (K-band magnitude ∼6.5) supergiant IRS 7, located about
5.6′′north of Sgr A*. All exposures were sky subtracted, flat-
fielded, and corrected for dead or bad pixels. In order to enhance
1Based on observations at the Very Large Telescope (VLT) of the
European Southern Observatory (ESO) on Paranal in Chile; Programs:
073.B-0775 July 2004
III IV III
φ3 φ2
X−ray
NIR
φ2/3
φ4
NIR Flux Density in mJy
X−ray Flux Density in nJy
UT in hours
0
50
100
150
200
0001020304
Fig.1. The X-ray and NIR 2.2 µm light curves obtained on July 07,
2004 (Eckart et al. 2006a). Here we plot the data with a UT time axis
and separate flux density axes for the NIR (left) and X-ray (right) data.
In addition to the flare nomenclature introduced in Eckart et al. (2006a)
we labeled the section of the X-ray light curve that corresponds to the
NIR feature II as φ2/3 as it is located between φ2 and φ3.
050100150 200 250300
0
5
10
flux [mJy]
06 July 2004
06 − 07 July 2004
23:19:38 to 04:11:40
050100150200250300
time [min]
0.1
0.2
0.3
0.4
psf fwhm [arcsec]
UT in hours
01020304
00
b)
a)
Fig.2. Results of the re-reduction of the NIR 2.2 µm data from July
07, 2004. In panel a) the flux density scale is given in mJy. The top
(green) data points represent the light curve of the nearby flux star S1
used for flux density calibration and the black data points with the blue
interpolation linerepresent the SgrA*light curve. Thetimeaxisisgiven
in minutes offset from the start time at 23:19:38 UT on 6 July 2004
and in UT hours at the bottom. In the bottom panel b) we show the
FWHM of the PSF in the AO images that reflect the combination of AO
correction and input seeing.
the signal-to-noise ratio of the imaging data, we created median
images comprising 9 single exposures each. Subsequently, PSFs
were extractedfromthese images with StarFinder (Diolaiti et al.
2000). The images were deconvolvedwith the Lucy-Richardson
(LR)andlinear Wienerfilter (LIN)algorithms.Beam restoration
was carried out with a Gaussian beam of FWHM corresponding
to the final resolution at 2.2 µm of 60 milli-arcseconds. The flux
densities of the sources were measured by aperture photometry
with circularaperturesof52 mas radiusand correctedforextinc-
tion, using AK= 2.8. Calibration of the photometry and astrom-

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Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*3
X-ray
flare
ID
φ2
φ2/3
φ3
φ4
NIR
flare
ID
I
II
III
IV
X-ray flux
density
(nJy)
31±27
<20
223±27
37±27
NIR K-band
flux density
(mJy)
≥5.7
∼3.0
6±1.5
5±1.5
Table 1. NIR/X-ray flare flux densities. The peak flux densi-
ties of the flares detected in the individual wavelength bands are
given. The X-ray flares φ2, φ3 and φ4 have been detected simul-
taneouslyin theNIR (Eckartet al.2006a).IndividualAO images
for the NIR event II presented by Eckart et al. (2006a)as well as
the newly reduced light-curve shown here in Fig.2 demonstrate
that Sgr A* clearly was in an “on” state.
etry was done with the known fluxes and positions of 9 sources
within 1.6” of Sgr A*.
In Fig. 1 we show the July 07 NIR and X-ray data in com-
parison. Four NIR flares (I - IV) can be identified. In Fig. 11 of
Eckart et al. (2006a) individual AO images correspond to sepa-
rate points in time and include the flares discussed here. These
images demonstrate that even during the weak NIR flare feature
II Sgr A* clearly was in an “on” state and significantly weaker
before and after. For the flare feature II the NIR flux density ex-
cess is of the order of 3 mJy.
To re-assess the presence of the flare zone II we re-reduced
the NIR data and show the results in Fig. 2. The re-reduction in-
cludes the following additional features: 1) We used sub-pixel
shifting of the data applying the ‘jitter‘-routine in ECLIPSE
(Devillard 1997). 2) We subtracted a constant backgroundbased
on a StarFinder analysis. 3) We rejected low quality images
based on the number of stars detected by StarFinder. 4) We re-
moved low level common trends (≤ 15%) that became apparent
in the reference star data by applying a detrending routine (by
Nicolas Marchili, IMPRS, MPIfR). The generation of the trend
involves binning and splining of the reference star data and is
similar to a procedure described in Villata et al. (2004). The re-
sulting trend then is then removed from the SgrA* NIR light
curve as well. 5) For comparison we also plot the FWHM of the
nearby reference stars as an indication of the combinedinstanta-
neous NIR seeing and the quality of the AO correction.
We find that within the uncertainties the result of the re-
reduction is in very good agreement with the original data re-
duction used in Fig. 1. The improved analysis shows that flare
zone II is a reliably feature in the NIR light curve.
2.2. The Chandra X-ray observations
In parallel to the NIR observations, SgrA* was observed with
ChandrausingtheimagingarrayoftheAdvancedCCD Imaging
Spectrometer (ACIS-I; Weisskopf et al., 2002) for two blocks of
∼50ks on 05–07 July 2004 (UT). We reduced and analyzed the
data using CIAO v2.32software with Chandra CALDB v2.223.
We extracted counts within radii of 0.5′′, 1.0′′, and 1.5′′
around Sgr A* in the 2–8 keV band. Background counts were
extracted from an annulus around Sgr A* with inner and outer
radii of 2′′and 10′′, respectively, excluding regions around dis-
crete sources and bright structures (Baganoff et al. 2003). The
2Chandra
http://cxc.harvard.edu/ciao
3http://cxc.harvard.edu/caldb
InteractiveAnalysisofObservations(CIAO),
4
4
681 01 2
0
1
2
3
4
SMA 340 GHz
810126
1
2
0
3
4
SMA5
SMA4
VLA 43 GHz
VLA1
Flux Density in Jy
UT in hours
Fig.3. The340 GHzSMA and 43 GHzVLA total intensitylight curves
from July 07. The individual data points are connected by straight lines.
The July 07 VLA data represent the excess flux density compared to
the mean of the July 06 and 08 VLA data. The constant flux density of
about 1.4 Jy that has to be added to this excess is indicated by a dashed
line. For further details see text and Eckart et al. (2006a).
mean (total) count rates within the inner radius subdivided into
the peak count rates during a flare and the corresponding inter-
mediate quiescentflux values are listed in Table 4 in Eckart et al.
(2006a).Thebackgroundrateshavebeenscaledtotheareaofthe
source region. The 1.0′′aperture provides the best compromise
between maximizing source signal and rejecting background.In
Fig. 1 we have labeled the section of the X-ray light curve that
correspondstotheNIR featureIIas φ2/3,as itis locatedbetween
φ2 and φ3. For φ2/3 the X-ray flux density excess above the qui-
escent bremsstrahlung component of SgrA* is below 20 nJy.
2.3. The SMA observations
The
Submillimeter Array4(SMA) on Mauna Kea, Hawaii (Ho,
Moran, & Lo 2004). The observations of SgrA* were made at
340 GHz (890 µm wavelength) for three consecutive nights,
05-07 July 2004 (UT), at an angular resolution of 1.′′5×3.′′0.
Nearby quasars were used for phase and gain calibration. On
bothJuly6 and7we obtainedmorethan6hoursofsimultaneous
X-ray/sub-millimeter coverage with 340 GHz zenith opacities
from 0.11 to 0.29 for July 5 to 7, respectively. This is reflected
in the larger time bins and scatter in the later light curves.
The same 5 antennae with the best gain stability were used
to form light curves, resulting in a typical synthesized beam of
1.′′5×3.′′0. The SgrA* data are phase self-calibrated after the ap-
plication of the quasar gains to remove short-timescale phase
variations, then imaged and cleaned. Finally, the flux density is
extracted from a point source fit at the center of the image, with
the error taken from the noise in the residual image. The overall
sub-millimeterobservationsweremadewiththe
4TheSubmillimeter Array isa joint project between the Smithsonian
Astrophysical Observatory and the Academia Sinica Institute of
Astronomy and Astrophysics, and is funded by the Smithsonian
Institution and the Academia Sinica.

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4Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*
flux scale is set by observations of Neptune, with an uncertainty
of approximately 25%.
We attribute a flux density value of ∼2.4 Jy as a constant
or only slowly variable part or the light curve that may be due
to more extended source components. Since the final data point
in the July 07 light curve is significantly below the minimum
of ∼2.4 Jy that is usually obtained on SgrA* at 340 GHz (e.g.
Yusef-Zadeh et al. 2008, Marrone et al. 2008) and due to the
steep drop in flux density towards the end of the observations at
low elevations we did not consider this data point in our models
of the light curve.
2.4. The VLA 7 mm observations
The Very Large Array (VLA) observed Sgr A* for ∼ 5 hours on
6,7and8July2004at43GHz(7mmwavelength).Observations
covered roughly the UT time range 04:40 to 09:00, which
is a subset of the Chandra observing time on 6 and 7 July.
Observations on 7 July immediately followed the VLT NIR ob-
servations. The VLA was in D configurationand achieved a res-
olutionof2.5×0.9arcsec at the observingwavelengthof 0.7cm.
Theabsoluteamplitudecalibrationwassetbyobservationsof3C
286. Flux densities were determined for Sgr A* and J1744-312
throughfitting of visibilities at (u,v)distances greaterthan 50 kλ
in order to remove contamination from extended structure in the
Galactic Center.
In Fig. 3 we show the 340 GHz SMA and 43 GHz VLA total
intensity light curves from July 07. Here the July 07 VLA light
curve was calculated as the difference between the mean flux
density data at the same interferometer hour angle obtained on
July 06 and 08 (see Fig. 8 in Eckart et al. 2006a). The minimum
compact flux density of ∼1.4 Jy obtained between 7 and 8 hours
UT has to be added to the resulting excess flux density in order
to derive a complete 43 GHz light curve of SgrA* obtained in
the VLA D configuration at an angular resolution of 2.′′5×0.′′9.
3. Radiation mechanisms
Dueto the shortflare durationthe flare emissionverylikelyorig-
inates from compact source components. The simultaneous X-
ray/NIR flare detections of the SgrA* counterpart implies that
the same population of electrons is responsible for both the IR
and the X-ray emission (e.g. Eckart et al. 2004).The spectral en-
ergy distribution of SgrA* is currently explained by models that
invoke radiatively inefficient accretion flow processes (RIAFs:
Quataert 2003, Yuan et al. 2002, Yuan, Quataert, & Narayan
2003, 2004, including advection dominated accretion flows
(ADAF): Narayan et al. 1995, convection dominated accretion
flows (CDAF): Ball et al. 2001, Quataert & Gruzinov 2000,
Narayan et al. 2002, Igumenshchev 2002, advection-dominated
inflow-outflowsolution (ADIOS): Blandford& Begelman1999;
see also Ballantyne,¨Ozel, Psaltis 2007, Le & Becker 2005), jet
models (Markoffet al. 2001,see also Markoff 2005),and Bondi-
Hoyle models (Melia & Falcke 2001). Also combinations of
models such as an accretion flow plus an outflow in the form
of a jet are considered (e.g. Yuan, Markoff, Falcke 2002).
3.1. Adiabatically expanding source components
To model the sub-mm/mmlight curves we assume an expanding
uniform blob of relativistic electrons with an energy spectrum
n(E) ∝ E−pthreaded by a magnetic field. As the blob expands,
the magnetic field declines with increasing blob radius as R−2,
05 10 1520
0
0.5
1
1.5
2
8
6
2
0
4
05 1015 20
v = 0.016 c
v = 0.008 c
v = 0.004 c
v = 0.002 c
Flux Density in Jy
UT in hours
340 GHz
Fig.4. A comparison of a 340 GHz light curve calculated with adia-
batic expansion velocities that differ by factors of two.
−2024
4
6
0
0.5
1
1.5
2
0.0
1.0
0.5
1.64 THz
8 THz
32 THz
Flux Density in Jy
602 −2
340 GHz
43 GHz
time delay in hours
Fig.5. The adiabatic expansion of a single source component with a
peak fluxdensityat 1.64 THzof 10 Jy, starttimeat 0hours, andconstant
expansion velocity of 0.08 c. The 1.64 THz, 8 THz and 32 THz light
curves have been scaled down by a factor of 5.2.
the energy of relativistic particles as R−1and the density of par-
ticles as R−3(van der Laan 1966).The synchrotronoptical depth
at frequency ν then scales as
?−(p+4)/2?R
R0
and the flux density scales as
?5/2?R
R0
1 − exp(−τ0)
Since the goal is to combine the description of an adiabatically
expandingcloudwitha synchrotronself-Comptonformalismwe
τν= τ0
?ν
ν0
?−(2p+3)
(1)
Sν= S0
?ν
ν0
?31 − exp(−τν)
.
(2)

Page 5

Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*5
use the definition of τ0 as the optical depth corresponding to
the frequency at which the flux density is a maximum (van der
Laan 1966)rather than the definition of τ0as the optical depth at
whichthefluxdensityforanyparticularfrequencypeaks(Yusef-
Zadeh et al. 2006b).Thereforeτ0depends only on p throughthe
condition
eτ0− τ0(p + 4)/5− 1 = 0
andrangesfrom0 to 0.65as p rangesfrom1 to 3. Thusgiventhe
particle energy spectral index p and the peak flux S0in the light
curve at some frequency ν0, this model predicts the variation in
fluxdensityat anyotherfrequencyas a functionoftheexpansion
factor (R/R0).
A modelfor R(t) is requiredto convertthe dependenceon ra-
dius to time: we adopt a simple linear expansion at constant ex-
pansion speed vexp, so that R−R0=vexp(t−t0). Here we assume
that the source componentis decoupledfrom energyinput and is
freely expanding, i.e. neither accelerations nor decelerations of
theexpansionaredominant(seeendofsection4.1.4). Apossible
magnetic confinement of the spot will be described in the model
as a low expansionspeed,i.e. it is containedin the modelof R(t).
For t ≤ t0we have made the assumption that the source has an
optical depth that equals its frequency dependent initial value τν
at R = R0. So in the optically thin part of the source spectrum
the flux initially increases with the source size at a constant τν
and then decreases due to the decreasing optical depth as a con-
sequence of the expansion. For the ∼4×106M⊙ super-massive
black hole at the position of Sgr A*, one Schwarzschild radius
is Rs=2GM/c2∼1010m and the velocity of light corresponds to
about 100 Rs per hour. For t > t0 the decaying flank of the
curve can be shifted towards later times by first increasing the
turnover frequency ν0or the initial source size R0, and second,
by lowering the spectral index αsynchor the peak flux density S0.
Increasing the adiabatic expansion velocity vexpshifts the peak
of the light curve to earlier times. Adiabatic expansion will also
result in a slower decay rate and a longer flare timescale at lower
frequencies.
(3)
3.2. Description and properties of the SSC model
We have employeda simple SSC modelto describethe observed
radio to X-ray propertiesof SgrA* using the nomenclaturegiven
by Gould (1979) and Marscher (1983). Inverse Compton scat-
tering models provide an explanation for both the compact NIR
and X-ray emission by up-scattering sub-mm-wavelength pho-
tons into these spectral domains. Such models are considered
as a possibility in most of the recent modeling approaches and
mayprovideimportantinsightsintosomefundamentalmodelre-
quirements. The models do not explain the entire low frequency
radiospectrumandthebremsstrahlungX-rayemissionthatdom-
inates the IQ state. Also high power X-ray fares (e.g. Porquet
etal. 2003, 2008) may involve additional emission mechanisms.
However,for X-ray flares of up to several 10 times the quiescent
emission the SSC models provide a successful description of the
compact IQ and flare emission originating from the immediate
vicinity of the central black hole. A more detailed explanation is
also given by Eckart et al. (2004).
We assume a synchrotron source of angular extent θ. The
source size is of the order of a few Schwarzschild radii
Rs=2GM/c2with Rs∼1010m for a ∼4×106M⊙ black hole. One
Rsthen corresponds to an angular diameter of ∼8 µas at a dis-
tance to the Galactic Center of 8 kpc (Reid 1993, Eisenhauer et
al. 2003, Ghez et al. 2005). The emitting source becomes opti-
cally thick at a frequency νmwith a flux density Sm, and has an
optically thin spectral index α following the law Sν∝ν−α. This
allows us to calculate the magnetic field strength B and the in-
verse Compton scattered flux density SSSCas a function of the
X-ray photon energy EkeV. The synchrotron self-Compton spec-
trum has the same spectral index as the synchrotron spectrum
that is up-scattered i.e. SSSC∝EkeV−α, and is valid within the
limits Eminand Emaxcorresponding to the wavelengths λmaxand
λmin(see Marscher et al. 1983 for further details). We find that
Lorentz factors γefor the emitting electrons of the order of typ-
ically 103are required to produce a sufficient SSC flux in the
observed X-ray domain. A possible relativistic bulk motion of
the emittingsourceresults ina Dopplerboostingfactorδ=Γ−1(1-
βcosφ)−1. Here φ is the angle of the velocity vector to the line of
sight, β the velocityv in units of the speed of lightc, andLorentz
factor Γ=(1-β2)−1/2for the bulk motion. Relativistic bulk motion
is not a necessity to produce sufficient SSC flux density but we
have used modest values for Γ=1.2-2 and δ ranging between 1.3
and 2.0 (i.e. angles φ between about 10◦and 45◦) since they will
occur in cases of relativistically orbiting gas as well as relativis-
tic outflows - both of which are likely to be relevant to SgrA*.
4. Modeling the light curves
Our primary goal was to generate a model that includes the en-
tire data set on the flare event observed on July 7, 2004 from the
mm- to the X-ray domain. Models like F1 or F2 (Eckart et al.
2006, their Table 9) or the dynamical, multicomponent model
presented by Eckart et al. (2008a) reproduce the NIR/X-Ray
properties of the observed flare φ3/III and φ4/IV very well. We
have repeated this modeling under the premise of achieving fits
with a simultaneous match to the SMA and VLA data.
The six so far reported coordinated SgrA* measurements
that include sub-mm data (Eckart et al. 2006b; Yusef-Zadeh
et al. 2006b; Marrone et al. 2008, Eckart et al. 2008b) have
shown that the observed submillimeter flares follow strong NIR
or X-ray events. If the events were unrelated we would expect
an equal number of submillimeter flares leading and following
the NIR/X-ray events (see detailed discussion in Marrone et al.
2008). We therefore assume that the sub-millimeter flare pre-
sented here is related to the observed IR flare events.
For the July 7 SMA and VLA data Eckart et al. (2006a)have
shown that the observed amplitudes of the flux density varia-
tions are generally consistent with adiabatic expansion of a syn-
chrotron self-absorbed source (van der Laan 1966). Following
the THz peaked NIR/X-ray flare events III/φ3 and IV/φ4 on July
7 (see Fig. 1 and modeling results given by Eckart et al. 2006a)
the radio flux density will first rise and later drop as the source
evolves.
However, a detailed comparison to theoretical light curves
of adiabatically expanding source components shows that a sin-
gle source component cannot give a satisfactory fit to the sub-
mm/mm data. In Fig. 5 we show that when the 340 GHz light
curve is decaying, the 43 GHz curve is still rising (straight bold
face section in the correponding lines). When the 43 GHz curve
is decayingthen the decaying340GHz light curveis at flux den-
sity levels well below the 43 GHz curve (dashed bold face sec-
tion in the correponding lines). Both scenarios are inconsistent
with the observations on July 04. This result is also indepen-
dent of the expansion velocity. Modeling the 2004 July 07 radio
data therefore must involve a minimum of two source compo-
nents. Quite naturally two components can be associated with
the NIR/X-ray flares III/φ3 and IV/φ4.
Our goal is to fit the variable part of the sub-mm/mm light
curves (see section 2.3) with a source model that is also able

Page 6

6 Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*
to describe the observed NIR/X-ray properties. We calculated
model light curves at 340 GHz and 43 GHz for the model with
a smallest (4) and larger (6) number of source components.
Smaller numbers of components cannot account for all essen-
tial features of the sub-mm/NIT/X-ray light curves. A detailed
explanation is given in the following.
In Fig. 6 we show the decomposition of the overall light
curves into the contribution of individual source components. In
Figs. 7 and 8 we show the model light curves for comparison to
the measured 340 GHz and 43 GHz data.
SSC modeling with adiabatically expanding source compo-
nents: We iterated between the SSC modeling of the NIR/X-
ray data and the modeling of the sub-mm/mm data as adiabat-
ically expanding sources using the same component parameters
as givenin Table3. Thesourceexpansionis also motivatedbyan
indicationofahotspotevolutionwithinapossibleaccretiondisk
based on a May 2007 flare event. Eckart et al. (2008a) have de-
scribed the July 2004 flare using a multi-component disk model
allowing for a source size increase of (at least) 30% over about
40 minutes in order to explain the strong decrease of the X-ray
flux density between φ3 and φ4.
In Table 3 we summarize the propertiesof 6 differentmodels
that we considered to represent the 2004 July 7 radio, NIR,and
X-ray data. For each model the ’x’ symbols in column 2 and 3
indicate which of the source components have been considered
for the models. Correspondingly they are labeled A1, A2, B1,
B2, C1 and C2. The 340 GHz flare SMA5 is accounted for by
either a double (γ,δ) or a single (η) component in A1, B1, C1
and A2, B2, C2, respectively, using the source labels marked in
columns 1, 2 and 3. The decaying 43 GHz flux density com-
ponent VLA1 is accounted for by simple flux density offsets in
models A1 and A2 (see caption of Table 3) or by 2 components
(ǫ,ζ)inmodels(B1,B2,C1, C2).Incolumn4 theindividualadi-
abatically expanding source components are labeled and identi-
fied with the flares detected in the different wavelength regimes
using the nomenclature by Eckart et al. (2006a). The modeling
was done with constant expansionvelocities of 0.005 c for mod-
els (A, B) and 0.006 c for models (C). Results of the modeling
are discussed in section 4.1.
The predictions from the SSC-modeling and of the optically
thin NIR flux density from the sub-mm data are especially sen-
sitive to variations of the model parameters. The uncertainties
of the model parameters given in the first row of Table 3 were
derived from a comparison of observed and predicted NIR and
X-ray model flux densities and from reduced χ2values calcu-
lated by comparing the SMA and VLA data with the adiabatic
expansion models.
A global variation of a single parameter by the value listed
in the corresponding column results in an increase of ∆χ = 1.
Here global variation means: adding for a single model parame-
ter but for all source components the 1σ uncertainty, such that a
maximum positive or negative flux density deviation is reached.
Alternatively, a variation by the listed uncertainty for only a
single source component results in a variation of the model pre-
dicted NIR and X-ray flux density by more than 30%. Judging
from the ∆χ based on the sub-mm data only the global uncer-
tainties for Smax,obs, αsynchand R0could be doubled.
The minimum number of source components is 5. In the
reduced χ2fit we used 5 times 4 (Smax, αsynch, R0, νmax) plus
one common expansion velocity vexp and time offset (leaving
the time differences between the components fixed), i.e. 22 de-
grees of freedom. The model parameters can unfortunately not
all be considered as being independent, e.g. the width and peak
of a light curve signature depends to a varying extend on all 4
parameters Smax, αsynch, R0, and νmax. Therefore we stayed for
all models with the minimum number of source components (5)
to estimate the degrees of freedom. Leaving all parameters free
(∆t, vexp, Smax, αsynch, R0, and νmax) would double the number
of degrees of freedomand reduce the χ2values correspondingly.
Since the VLA data consists of 5 to 6 times the number of SMA
data points we weighted the squared SMA flux deviations and
number of data points by factor of 6. The χ2test was then car-
ried out using the sum of the squared flux deviations and data
points of the VLA and SMA datasets.
Of course the models have been set up under the constraint
of minimizing the number of free parameters (and to maximize
the description of significant flare features in the observed light
curves). The flux excursions labeled (VLA1,φ2/3) could easily
be explained by 3 or 4 components rather than 2 (ǫ and ζ) in the
same way as SMA5 can be explained by 2 (γ and δ) rather than
a single component (η).
In the following we comment on the detailed modeling of
different sections of the light curves.
Modeling of individual portions of the light curves:
Significant NIR and X-ray flux density is only produced
from the initial THz peaked source components α and β. Using
a linear expansion at constant speed vexpwe then calculate light
curves in the selected sub-mm/mm bands. For an expansion
speed of ∼0.005 c we find that the radio model component β
(due to the III/φ3 NIR/X-ray flare event) can account for a ma-
jor portion of the decreasing part of the 340 GHz light curve -
SMA4 in Fig. 3 - between 06 and 12 UT on July 2004.
Components α and β also account for a portion of the final
section (after about 6 hours UT - VLA1 in Fig. 3) of the de-
creasing 43 GHz light curve. For models A and B the spectral
indices of components α and β of αsynch=0.8 are consistent with
the value of 0.6±0.2 found by Hornstein et al. (2007; see discus-
sion in section 4.1.1). While the NIR flux density in models (A,
B) is provided by direct synchrotron emission, the flux density
in model (C) is produced via the low frequency section of the
scattered SSC specrum. In this case the spectral index must be
steep (αsynch=1.2) and is close to the value derived for the over-
all NIR/X-ray spectral index of α and β of αNIR/X−ray=1.2±0.2
(Eckart et al. 2006a). This is consistent with the fact that the
optically thin synchrotronspectral index (sub-mm to NIR) is ex-
pected to equal the broad band spectral index of the SSC spec-
trum.
Components α and β cannot fully account for the 43 GHz
radio flux density. This is especially true for the the initial sec-
tion (before 6 hours UT) of the decreasing 43 GHz light curve.
The remaining flux density of the VLA1 flare event then has
to be explained by source components that provide an almost
flat light curve between 5 and 9 hours UT. Alternatively, source
components that can be identified with the the NIR flare compo-
nent II/φ2/3 (see Fig. 1) are required to deliver this flux density
contribution. To model the light curve in this region a minimum
of 2 model components (ζ and ǫ) is needed. The observed flux
densities (or limits) and implied time differences require rather
steep spectral indicesof αsynch=1.3forthese sourcecomponents.
Especially for models C1 and C2 the predicted peak flux densi-
ties for the 340 GHz and 43 GHz bands lie well within the range
of observed flux density values (e.g. Marrone et al. 2008, Yusef-
Zadeh et al. 2008, Eckart et al. 2006a).
The 340 GHz flare event SMA5 requires at least one (η)
or two model components (γ, δ). For two source components
a steep spectral index with αsynch=0.9-1.4 results in a weak
NIR flare event with an X-ray flux that would have been non-
detectable for Chandra in the presence of the quiescent X-ray

Page 7

Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*7
0
0
1 0
10
2 0
0
1
2
3
4
−55 15
340 GHz
Flux Density in Jy
UT in hours
3
1
4
2
0
α
β
ζ
ε
γδ
0
0
1 0
10
2 0
0
0.2
0.4
0.6
−55 15
UT in hours
43 GHz
0.0
Flux Density in Jy
0.6
0.2
0.4
δ
γ
β
α
ζ
ε
Fig.6. Decomposition of the 340 GHz and 43 GHz light curve for
model B1 (see Table 3) into the contributions of the different source
components (α to ζ). The data are shown with the offsets discussed in
sections 2.3 and 2.4 removed.
bremsstrahlung component of SgrA*. For a single component a
flat spectral index of αsynch=0.2 (corresponding to p=0.6) is re-
quired. This results in a significant NIR/X-ray flare about 4±1
hours after the III/φ3 flare event. Fig.6 in Eckart et al. (2006a)
shows that no X-ray event was detected by Chandra over this
period of time. Therefore a flat spectral component η can be ex-
cluded.
4.1. Discussion of the modeling results
Our modeling shows that the observed mm through X-ray data
of the July 7, 2004 flare event can be successfully modeled by a
combinationofanSSC andanadiabaticexpansionmodel.While
our analysis cannotfully supportor even provethe model of adi-
abatic expansion for the flare emission of SgrA* by itself, it is
−5
−5
0
0
5
5
1 0
10
1 5
15
0
1
2
3
4
340 GHz
43 GHz
Flux Density in Jy
UT in hours
0
1
4
2
3
−5
−5
0
0
5
5
1 0
10
1 5
15
0
1
2
3
4
340 GHz
43 GHz
Flux Density in Jy
UT in hours
0
4
1
2
3
−5
−5
0
0
5
5
1 0
10
1 5
15
0
1
2
3
340 GHz
43 GHz
Flux Density in Jy
UT in hours
1
0
2
3
C1
B1
A1
Fig.7. The observed 43 GHz and 340 GHz light curves shown in com-
parison to model A1, B1, and C1. The data is shown with the offsets
discussed in sections 2.3 and 2.4 removed.

Page 8

8 Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*
−5
−5
0
0
5
5
1 0
10
1 5
15
0
1
2
3
4
340 GHz
43 GHz
Flux Density in Jy
UT in hours
0
1
4
2
3
C2
Fig.8. The observed 43 GHz and 340 GHz light curves shown in com-
parison to model C2. The data is shown with the offsets discussed in
sections 2.3 and 2.4 removed.
NIR
X−ray
NIR Flux Density in mJy
X−ray Flux Density in nJy
UT in hours
0
50
100
150
200
00 0102 03 04
α
β
αζ
ε
Fig.9. A comparison between the X-ray and NIR 2.2 µm light curves
shown in Fig. 1 and the modeling results for the A, B (both in green),
and C (red) models. The contributions of individual model components
as listed in Table 3 are labeled.
in full agreement with model results that have been obtained by
analyzingpreviousobservations(e.g.Eckartet al. 2006a,2008b,
Yusef-Zadeh et al. 2007, 2008, Marrone et al. 2008). The mod-
eling also suggests the presence of two flare phases in the si-
multaneous NIR/X-ray events. The fact that a single component
adiabatic expansion model cannot account for the observed sub-
mm/mm light curves, the considerable spread in source parame-
ters (brightness,size, spectal index,sub-mmturnoverfrequency)
as well as the presence of a non rapidly variable (on timescale of
more than one or a few days) flux density component indicates
low level activity in addition to the presence of repeated individ-
ual bright flares. The flare event II/φ2/3 probably is an example
of such low level activity.
Our modeling also requires the low adiabatic expansion ve-
locity that describes other flares of SgrA* (Yusef-Zadeh et al.
2007,2008,Eckartet al. 2008b).Our valuesof ∼0.005c fits well
into the range of velocities of vexp=0.003-0.1c given by Yusef-
Zadehetal.(2008).Itiscurrentlyunclearhowtointerprettheex-
pansion velocity. The velocities are lower than the orbital veloc-
ities for flare components introduced from orbiting spot models
and accretion disk models (Eckart et al. 2006b, 2008ab, Meyer
et al. 2006ab,2007,2008,Trippe et al. 2007).The observedsub-
mm/mm flare emission therefore can be explained as an expan-
sion within the occasionaly present accretion disk around SgrA*
(as shown in Fig. 10) or a flaring of the disk corona.
We can now derivethe delay times to the sub-mm/mmbands
relative to the NIR/X-ray flare times from our modeling. A sum-
mary is given in Table 2. The delay times of the compact (α,β)
components with cutoff frequencies in the THz domain are 1-
2 hours in the 340 GHz and 3-9 hours in the 43 GHz band.
The delay times of the more extended source components (γ
to η) with cutoff frequencies at several 100 GHz are about 20
minutes in 340 GHz and 4-7 hours in 43 GHz band. In general
larger source components (>1.5 Rs) with low peak flux densi-
ties (Smax,obs<3 Jy) and low cutoff frequencies (νmax,obs<a few
100 GHz) show little or no X-ray emission and have shorter de-
lay times towards the 345 GHz band (see also Marrone et al.
2008).
In Fig. 10 we show how a double or extended spot model
can be interpreted in an evolutionary framework for disk struc-
ture. This model is based on work by Hawley & Balbus (1991,
see also Balbus & Hawley 1998 and Balbus 2003) and is mo-
tivated by the fact that accretion disks show magneto-rotational
instabilities in which magnetic field lines provide a coupling be-
tween disk sections at different radii resulting in an efficient out-
ward transport of angular momentum.In a differentially rotating
disk the inner disk portions that lose angular momentum will
slide into lower lying orbits, and rotate more rapidly. The cen-
ter image in Fig. 10 represents the expanded disk component
that may be the source of a flaring corona or inhanced jet ac-
tivity since it is located at the footpoint of a possible short jet
that may be associated with SgrA*. The occasional presence of
a thin disk in which the Hawley & Balbus mechanism is oper-
ating effectively may imply an accretion rate and corresponding
luminosity higher than that expected for an accretion flow that
is otherwise assumed to be radiatively inefficient. This may sup-
port that the actual accretion rate is indeed a strong function of
the radius and much below the rate expected from the mass loss
of the surrounding He-stars (see references in introduction).
Alternatively, the expanding source components may also
have a bulk velocity that can be substantially larger than the ob-
served expansion velocity. As the jet or wind expands, different
regions of it will dominate the emission at successively lower
frequencies(see Falcke & Markoff2000,2001,Yuan et al. 2002,
as well as Fig.12 by Eckart et al. 2008a).
4.1.1. Matching the NIR flux densities
We have re-modeled the July 7, 2004 flare to obtain a smaller
discrepancy between the measured and predicted NIR flux den-
sities. Our models A and B for the light curve features φ3/IIIand
φ4/IVoverestimatetheNIR flarefluxbyafactorof4(Table3)to
7 (Eckart et al. 2006a). Since direct synchrotron emission from
γe∼103electrons delivers the dominant portion of the NIR flux
density this discrepancy becomes larger if we use flatter spectral
indices like α∼0.4 (Model F2 by Eckart et al. 2006a). This ef-
fect cannotbe compensatedforusinga lower turnoverfrequency
νmor flux density Smsince then the X-ray emission cannot be
matched any longer.
The NIR flare emission shows observed spectral indices of
α∼0.6 (Ghez et al. 2005, Hornstein et al. 2006) or even steeper

Page 9

Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*9
component NIR/
X-ray
3.3
3.8
8.3
10.4
1.6
0.7
∼9.0
340 GHz43 GHzdelay
340 GHz
0.7
1.7
0.7
0.6
0.4
0.3
1.0
delay
43 GHz
2.7
7.7
4.7
5.6
7.4
4.3
6.0
α
β
γ
δ
ǫ
ζ
η
46
5.5
9
11
2
1
10
11.5
13
16
9
5
15
Table 2. Columns2to4 list theUT timesin hoursoftheNIR/X-
ray, 340 GHz and 43 GHz flux desity peaks for source compo-
nents of models A,B, and C with properties listed in Table 3
Columns 5 and 6 list the delay times in hours from the time of
birth to the 340 GHz and 43 GHz frequency bands. The time
of origin of the components at their peak frequency are syn-
chronous with the NIR/X-ray flare times. The uncertainties are
of the order of one hour.
(Eisenhauer et al. 2005, Gillessen et al. 2006, Krabbe et al.
2007). Eckart et al. (2006a) assume that steep NIR spectral in-
dices may be a consequence of synchrotron losses represented
by an exponential cutoff in the energy spectrum of the relativis-
tic electrons (see also Liu et al. 2006). This will appear as a
modulation of the intrinsically flat spectra with an exponential
cutoff proportional to exp[−(λ0/λ)0.5] (see e.g. Bregman 1985,
and Bogdan & Schlickeiser 1985) and a cutoff wavelength λ0in
the infrared. If λ0lies in the 4-8µm wavelength range, then the
variation in the spectral index is of the order of ∆λ=0.6-1.0 and
can easily explain the factors of 4 to 10 between the observed
and modeled NIR flux density. This is the case for models A
and B. Small variations in such an exponential damping of the
radiation provide variable infrared spectra with spectral indices
that may be consistent with the observedvalues. Such a scenario
explains NIR flux densities that fall below the values predicted
by the SSC model. In model C the flux density is produced via
the low frequency section of the scattered SSC spectrum with a
steep spectral index (αsynch=1.2). This solution implies no NIR
polarization of source component α but provides a satisfactory
fit to the NIR and sub-mm/mm flux densities.
In comparison to the data plotted in Fig. 1 we show in Fig. 9
the NIR- and X-ray light curves that correspond to the models
listed in Table 3. The SSC-modeling was done referring solely
to the peakvaluesof theindividualflare featuresin the measured
light curves. While in models A and B the components ǫ and ζ
have not been considered (to minimize the number of free pa-
rameters) or result in very small contributions to the NIR and X-
ray flux densities, model C reproduces the overall shape of the
light curves quite satisfactorily. We have not plotted the NIR-
and X-ray contributions of components γ, δ, or η that contribute
to the sub-mm flare SMA5, since there are no NIR observations
available (see comment on SMA5 at the end of the opening of
section 4). Therefore, in Fig. 1 it is also not necessary to dis-
tinguish between models with different model labels (1 or 2) as
listed in Table 3.
4.1.2. Matching the sub-mm/mm flux density offsets
Models A1 and A2 fail to fit the VLA data at 43 GHz. In par-
ticular the slope and the first part of the data are not matched.
Higher expansion velocities help to match the 43 GHz data but
fail to match the first decaying flank of the 340 GHz SMA data.
In addition it is required that a considerable part of the variable
43 GHz flux density has to be modeledas a constant offset in ad-
dition to the 1.4 Jy as the lowest visibility flux density (see sec-
tion 2.4). Therefore, also given the poor match of the NIR flux
densities of light curve components αand β, models A1 and A2
- although requiring the smallest number of source components
- are not very likely as an explanation of the observed mm/sub-
mm light curves.
With S(43 GHz)∼1.7 Jy and S(340 GHz)∼2.4 Jy the spec-
tral index is α43/340∼ −0.17 indicative of an inverted spectrum
that must be related to a compact source component. This is in-
consistent with the assumption that these flux density contribu-
tions are due to a non- or only slowly variable source compo-
nent. Therefore, non-expandingsynchrotronsource components
are not a satisfying explanationforthe constantflux density con-
tributions.
We may, however, assume that the fluxes of the non-
variable part of the light curve are due to a more extended
component. If that component is also self-absorbed and adi-
abatically expanding then the peak flux densities will be
S(ν2) = S(ν1)[ν2/ν1](7p+3)/(4p+6)
S(43 GHz)∼1.4 Jy and S(340 GHz)≤2.4 Jy we find α∼0.26 or
p=1+2α=1.52which is between the value of p=2.2 expected for
a spectral index of 0.6 and a value of p=-0.1 found prior to the
flare peak on July 17, 2006, reported by Marrone et al. (2008).
We conclude that even if we use the smallest number of
source componentsto describe the 2004 flare emission the resid-
ual flux densities that are not taken into account by the model
at both frequencies are consistent with adiabatic expansion of
source components.
(van der Laan 1966). With
4.1.3. The magnetic field
The magnetic field strengths between 5 and 70 Gauss (Eckart et
al. 2006a, 2008a, Yusef-Zadeh et al. 2008) are consistent with
sub-mm/mm variability timescales of synchrotron components
with THz peaked spectra and the assumption that these source
components have an upper frequency cutoff ν2in the NIR, i.e.
that they contribute significantly to the observed NIR flare flux
density.Here the upperfrequencycutoffto thesynchrotronspec-
trum is assumed to be at ν2 = 2.8 × 106Bγ2in Hertz, with the
magnetic field strength in Gauss. The Lorentz factor γ2corre-
sponds to the energy γ2mc2at the upper edge of the electron
powerspectrum.Forγ2∼103andB around60G, thesynchrotron
cutoff falls into the NIR.
A comparison between typical flare timescales and syn-
chrotron cooling timescales can also be used to derive estimates
of the required magnetic field strengths. The flux density vari-
ations of Sgr A* can be explained in a disk or jet model (see
e.g. discussion in Eckart et al. 2006ab, 2008a), or they could be
seen as a consequenceof anunderlyingphysicalprocessthat can
mathematicallybe describedas red-noise(Do et al. 2008,Meyer
et al. 2008).
In the orbiting spot model this timescale will reflect the flux
modulation by the relativistic orbital motion of the spots. While
the overall flare length is of the order of 2 hours (Eckart et
al. 2006a) shorter timescales of about 20 minutes can be at-
tributedtothesub-flares(Genzeletal.2003,Eckartetal.2006b).
However, the spot lifetime is likely of the order of the orbital
timescale or even shorter (Schnittman 2005, Schnittman et al.
2006, Eckart et al. 2008a). In order to match the overall typical
flare timescale of about 2 hours and given a minimum turnover
frequencyaround300GHztheminimumrequiredmagneticfield
strength is of the order of 5 Gauss. This is required as a mini-
mum value to have the cooling time of the overall flare less than
the duration of the flare (Yuan, Quataert, Narayan 2003, 2004,

Page 10

10 Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*
model
lable
12 source flare
lable
∆tvexp
in c
Smax,obs
[Jy]
αsynch
R0
νmax,obs
[GHz]
BSNIR,synch
[mJy]
SNIR,SSC
[mJy]
SX−ray,SSC
[nJy] hours[G]
1 σ →±1.0
±0.001
±0.1
±0.1
±0.1
±250
±10
±1.0
±1.0
±20
A B
A B
A B
A B
x
x
x
x
x
x
-
x
x
-
-
x
x
x
α
β
γ
δ
ǫ
ζ
η
φ3
SMA4, φ4
SMA5
SMA5
VLA1, φ2/3
VLA1, φ2/3
SMA5
0.0 0.006
0.006
0.006
0.006
0.006
0.006
0.006
9.0
11.2
2.4
1.2
3.0
3.9
4.0
0.8
0.8
0.9
0.9
0.9
1.4
0.2
0.9
1.8
1.5
1.8
2.9
2.9
1.2
1750
1230
680
525
470
223
1350
50
63
65
63
18
1
85
(39)
(34)
4.5
1.5
0.0
0.0
600
<1.0
<1.0
<1.0
<1.0
<1.0
<1.0
<1.0
230
<10
<10
<10
<10
<10
390
+0.5
+5.0
+7.1
-1.7
-2.6
+4.9
B
B
A B
Cx
x
x
x
x
x
-
x
x
-
-
x
x
x
α
β
γ
δ
ǫ
ζ
η
φ3
SMA4, φ4
SMA5
SMA5
VLA1, φ2/3
VLA1, φ2/3
SMA5
0.0 0.005
0.005
0.005
0.005
0.005
0.005
0.005
8.2
8.5
2.4
1.2
3.5
3.0
2.9
1.2
1.2
0.9
0.9
0.9
1.4
0.2
0.6
1.6
1.5
1.8
2.9
2.9
1.5
1310
1310
680
525
380
220
1060
23
68
50
57
16
4
55
0.0
5.5
5.1
1.6
4.0
0.1
390
7.8
0.5
190
<10
<10
<10
<10
<10
270
+0.5
+5.0
+7.1
-1.7
-2.6
+3.2
<1.0
<1.0
<1.0
<1.0
<1.0
Table 3. Source component parameters for the combined SSC and adiabatic expansion model of the 7 July, 2004 flare. The offset
times ∆t are given with respect to the peak of the brighter NIR flares φ3 (synchronous with the brightest X-ray flare III) at about
7 July, 204, 03:15:00 UT. The adiabatic expansion of the individual source components occurs at a constant velocity of 0.006 c
for model A and B and of 0.005 c for model C. In column 4 we identify the observed flares feartures that are represented by the
corresponding model component, using the nomenclature introduced in Eckart et al. (2006a) and here (see Fig. 1). The different
source component models A, B, C discussed in the text are labeled in column 1. The flare components required for models A1, B1,
C1 and A2, B2, C2 are marked with an ’x’ in column 2 and 3, respectively. Flux density offsets of the VLA and SMA data used for
models A1/A2 are 0.26 Jy and 1.9 Jy. Flux density offsets of the VLA and SMA data used for models B1/B2/C1/C2 and are 0.0 Jy
and 1.8 Jy, respectively. The ∆χ values in comparison to the mm/sub-mmdata that we obtained for the various models are: A1: 2.9;
A2: 3.1; B1: 1.71; B2: 1.81; C1: 1.70; C2: 1.80.
Quataert 2003).Similarly, as the flare expandsit will cool on the
synchrotroncooling timescale ts∼ 3×107ν−0.5
in seconds, B is in Gauss, ν9is frequency in GHz (Blandford &
K¨ onigl 1979). Here B is the magnetic field of the synchrotron
component with turnover frequency νm, turnover flux density
Sm and angular source size θ. Adopting a typical NIR flare
timescale of 20 minutes and ν9∼1000 we get B∼80 G. For much
shorter spot timescales as theoretically indicated (Schnittman
2005, Schnittman et al. 2006) the field strengths may be even
higher. As we show in the present paper, the description of
NIR/X-ray flares as SSC source components with THz peaked
spectra is also compatible with the assumption of adiabatic ex-
pansion of the same component that then can explain the ob-
served sub-mm/mm flare emission.
The assumption of variable synchrotroncomponentsthat be-
come optically thick in the MIR or even NIR is problematic.For
the magnetic field we find B ∼ θ4ν5
turnoverfrequency(∼150THz ratherthan1THz) andlowerflux
densities (mJy rather than Jy) requires source sizes much less
than a Schwarzschild radius in order to allow for magnetic field
strengths of the order of 60 G or even below. In addition we find
that with the low values for Smthe adiabatic expansion of this
source component cannot explain the sub-mm/mm flux densi-
ties.
Fortheouter,largercomponentsǫ andζ whicharealsomuch
less constrained by the NIR and X-ray data, we find values for
the magnetic field well below 60 G. For models (A,B) this is
- to first order - consistent with an expected decay of the field
strength in an adiabatic expansion model (van der Laan 1966)
proportionaltoR−2.Thesecomponentsalsorequiresteeperspec-
9
B−3/2, where tsis
mS−2
m. Given the higher NIR
tral indices as expected - if there is a correlation between NIR
flare strength and spectral index (Ghez et al. 2005, Gillessen
et al. 2006, Krabbe et al. 2006, Hornstein et al. 2006, but see
Hornsteinet al.2007)as is also discussedin Eckartet al. (2006a)
and Bittner et al. (2007). Larger sizes and steeper spectra for
these source components are required in models B1 and B2 to
match the early section of the 43 GHz VLA data and to fulfill
the flux density measures and limits in the NIR and X-ray do-
main.
4.1.4. Bulk motion and detectability of structure
VLBI experimentsat mm-wavelengthshaverevealeda size limit
for SgrA* of about 0.5 AU corresponding to ∼5 RS(Doeleman
et al. 2008, see also Doeleman et al. 2001, Shen 2006, Huang et
al. 2007). If the expansion speed of ∼0.01 c is taken as a bulk
motion then the time to cross 0.5 AU is less than 7 hours which
is well above the flare timescale. The flare flux due to the mov-
ing source component will then not lead to a detectable struc-
ture, extended on scales that can currently be probed with VLBI
techniques. Assuming a flare timescale of 2 hours this also re-
sults in an upper limit of the bulk motion of 0.07 c to lead to
a detectable structure if the VLBI size measurements are taken
during a flare at mm-wavelengths. At this speed, however, only
plasma at distances of ≥11 Rscan leave the ∼4×106M⊙black
hole. Eventually, the relativistic electrons will contribute to a
possible overall jet or outflow with a low surface brightness.
Falcke & Markoff (2000, 2001) and Yuan et al. (2002) propose
that the emission from Sgr A* arises primarily in a jet. In this
picture, a small fraction of the accretion flow is ejected near the

Page 11

Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*11
Fig.10. Sketch of an expanding hot spot within an inclined tempo-
rary accretion disk of SgrA* based on Hawley & Balbus (1991, see
also Balbus & Hawley 1998 and Balbus 2003). The black center in-
dicates the event horizon of the massive black hole, the solid line the
outer edge of the accretion disk (see e.g. Meyer et al. 2005), The long
dashed line marks the inner last stable orbit. The dotted line represents
a random reference orbit to show the effect of differential rotation of
an extended emission region. The red solid line across the grey shaded
extended spots depicts the magnetic field lines that through magneto-
hydrodynamical instabilities provide a coupling between disk sections
at different radii.
black hole as a short, luminous jet. A source structure in which
an accretion disk is associated with a short jet may explain most
of the observed properties of SgrA*. Jet structures are associ-
ated with almost all galactic nuclei. For the case of SgrA* such
a configuration is sketched in Fig.9 of Eckart et al. (2008a).
It is therefore possible that the emergent spectrum of Sgr A*
is the sum of the emission from a jet and an underlyingaccretion
process. With increasing distance from Sgr A* the plasma in the
jet becomes optically thin at ever longer wavelengths. Hence,
radiation at different radio wavelengths probes different sections
of the jet and results in a correlation between the emission at
different wavelengths. Emission at sub-millimeter wavelengths
arises at the smallest scales, at the foot of the jet at distances of
a few Schwarzschild radii from the black hole. In the immediate
vicinity of the black hole it is hard to distinguish between emis-
sion from an accretion flow and from the foot of a jet. Details of
expected low surface brightness jet geometries are discussed by
Markoff, Bower & Falcke (2007) (see also Markoff, Nowak &
Wilms 2005).
Monitoring of the SgrA* centroid position has the poten-
tial to place significant constraints upon the existence and mor-
phology of inhomogeneities in a super-massive black hole ac-
cretion flow. Reid et al. (2008) present measurements with the
VLBA of the variability in the centroid position of SgrA* rel-
ative to a background quasar at 7 mm wavelength. They find
an average centroid wander of 71 ± 45µas for timescales be-
tween 50 and 100 min and 113 ± 50µas for timescales between
100 and 200 min, with no secular trend. For these particular ob-
servations highly asymmetric flux density distributions can be
ruled out - as they would result from a hot spot with orbital radii
above 15GMSgrA∗/c2=7.5 Rsand a >30% contributionto the to-
tal 7 mm flux. Structural variations at smaller radii or lower flux
density levels remain unconstrained.
The velocity estimates are derived under the assumption of
tangledfields (vanderLaan1966)in the expandingcomponents.
If the expansion, however, takes place in a partially aligned field
- e.g. along an outflow - then the expansion is hampered in di-
rectionsperpendiculartothe fieldlines. Thecomponentwill stay
more confined and lower values for the expansion velocity will
be derived. An expansion of source components through shear-
ing due to differential rotation within the accretion disk may
explain the low expansion velocities. The recent theoretical ap-
proach of hot spot evolution due to shearing is highlighted in
Eckart et al. (2008a) and Zamaninasab et al. (2008; see also
Pech´ aˇ cek et al., 2008). A model that explicitly solves for the
relativistic hydrodynamics and includes low expansion speeds,
with reference to SgrA* and to the work by Hawley & Balbus
(1991, 1998) was recently suggested by Yuan et al. (2008). In
their model expansion velocities of less than 0.01c close to the
accretion disk are explained.
From the 340 GHz light curves (see also Eckart et al. 2006a)
it appears that highly accelerated expansion of the source com-
ponents is unlikely. Highly accelerated or decelerated expansion
would result in sharp drops or rises in the light curves.For Fig. 4
we can deduce that a low velocity expansion would first result
in high flare flux levels for a few hours after the initial event.
A change to a significantly higher speed would then result in a
sudden drop. Similarly for strongly decelerated flares we would
first expect a fast drop to low flux density values followed be a
longer lasting decay.
5. Summary and discussion
We have presented new model results for the July 7, 2004 flare.
We show that the data can successfully be explainedby a combi-
nationof a SSC and an adiabaticexpansionmodel.Based onthis
interpretationwe haveobservedthe emission of the synchronous
NIR/X-ray flares of the compact components (α,β) with cut-
off frequencies in the THz domain delayed by 1-2 hours in the
340 GHz and by 3-9 hours in the 43 GHz bands.The delaytimes
of the more extended source components with lower cutoff fre-
quencies are less than an hour to the 340 GHz and 4-7 hours to
43 GHz bands. We can therefore identify this flare as one with
thebroadestcoverageacrosstheelectromagneticspectrum:from
the X-ray,throughthe NIR, sub-mmto mm-wavelengthdomain.
Otherflare events with a broadfrequencycoveragehavebeenre-
portedby(Yusef-Zadehet al. 2006b;Marroneet al. 2008,Eckart
et al. 2008b).
Themodelingsuggeststhepresenceoftwo separateflaresφ3
and φ4, that are needed to explain the variable flux at 340 GHz
and 43 GHz. Source component β is responsible for fitting the
decaying flank of the 340 GHz data and component α accounts
in combination with ζ and ǫ for the overall slope and the first

Page 12

12 Eckart, Baganoff, Morris, Kunneriath et al.: Modeling flare emission from SgrA*
part of the VLA data. Our modeling also requires the low adia-
batic expansion velocity that describes other flares from SgrA*
(Yusef-Zadeh et al. 2007, 2008, Marrone et al. 2008, Eckart et
al. 200b).
Given that there is considerable structure in the NIR/X-ray
light curves that can be linked to flare activity at sub-mm/mm
wavelengths, it appears difficult to derive an estimate of the ex-
pansion velocity and power-law index p of the relativistic elec-
tron distribution from an analysis of the flare profiles alone.
Modelingtheflares throughanSSC formalismcoupledwithadi-
abatic expansion shows the importance of simultaneous NIR/X-
ray measurements that preceed the radio measurements.
The expansion velocities are lower than the orbital veloci-
ties for flare components introduced from orbiting spot models
and accretion disk models and will not allow material to leave
the immediate vicinity of the massive black hole at the position
of SgrA*. The observed sub-mm/mm flare emission therefore
can be explained as an expansion within the occasionally exist-
ing accretion disk around SgrA* or a flaring of the disk corona.
Alternatively, the expanding source components may also have
a bulk velocitythat can be substantially higherthan the observed
expansion velocity. From the small and not strongly variable
VLBI source sizes as well as the typical flare length of 2 hours,
we find an upper limit on the bulk velocity of 0.07 c. The non-
detection of a bright jet or variable source sizes or positions in-
dicates that an expansion into a short jet (Eckart et al. 2006b,
2008ab) or an occasional accretion disk (Eckart et al. 2004) is
the most likely explanation of the observed sub-mm/mm light
curves.
Acknowledgements. This work was supported in part by the Deutsche
Forschungsgemeinschaft (DFG) via grant SFB 494, the Max Planck Society
through the International Max Planck Research School, as well as special
funds through the University of Cologne. Chandra research is supported by
NASA grants NAS8-00128, NAS8-38252, GO2-3115B, and G05-6093X. We
are grateful to all members of the NAOS/CONICA and the ESO PARANAL
team. Macarena Garc´ ıa-Mar´ ın is supported by the German federal department
for education and research (BMBF) under the project numbers: 50OS0502 &
50OS0801. M. Zamaninasab, D. Kunneriath, are members of the International
Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the
MPIfRand the Universities ofBonn and Cologne. R.Sch¨ odel acknowledges sup-
port by the Ram´ on y Cajal program by the Ministerio de Ciencia e Innovaci´ on
of the government of Spain.
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