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%ApJ in press
%astro-ph/0512288
%http://www.aoc.nrao.edu/~jmacquar/SgrA.pdf
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\shorttitle{Radio variability of Sgr A*}
\shortauthors{Macquart \& Bower}
\begin{document}
\title{Understanding the Radio Variability of Sgr A*}
%% Use \author, \affil, and the \and command to format
%% author and affiliation information.
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\author{Jean-Pierre Macquart\altaffilmark{1}}
87801, U.S.A. \email{jmacquar@nrao.edu}}
\author{Geoffrey C. Bower}
\affil{Astronomy Department and Radio Astronomy Laboratory, University of C=
alifornia, Berkeley, Berkeley, CA 94720, U.S.A. \email{gbower@astron.berke=
ley.edu}}
\begin{abstract}
We determine the characteristics of the 7\,mm to 20\,cm wavelength radio va=
riability in Sgr A* on time scales from days to three decades. The amplitu=
de of the intensity modulation is between 30 and 39\% at all wavelengths. =
Analysis of uniformly sampled data with proper accounting of the sampling e=
rrors associated with the lightcurves shows that Sgr A* exhibits no 57- or =
106-day quasi-periodic oscillations, contrary to previous claims. The caus=
e of the variability is investigated by examining a number of plausible sci=
ntillation models, enabling those variations which could be attributed to i=
nterstellar scintillation to be isolated from those that must be intrinsic =
to the source. Thin-screen scattering models do not account for the variab=
ility amplitude on most time scales. However, models in which the scatteri=
ng region is extended out to a radius of 50-500\,pc from the Galactic Cente=
r account well for the broad characteristics of the variability on $>4$-day=
time scales. The $\sim 10$\% variability on $<4$-day time scales at $0.7-=
3\,$cm appears to be intrinsic to the source.
% Centimeter-wavelength variations on time scales less than $5\,$days appea=
r to be intrinsic to the source.=20
The degree of scintillation variability expected at millimeter wavelengths =
depends sensitively on the intrinsic source size; the variations, if due to=
scintillation, would require an intrinsic source size smaller than that ex=
pected.
\end{abstract}
\keywords{galaxies: active --- Galaxy: center --- scattering}
%--------------------------------------------------------------------------=
------------------
\section{Introduction}
% INTRODUCTION
%--------------------------------------------------------------------------=
------------------
The compact radio source associated with the black hole at the Galactic Cen=
ter, Sgr A*, is known to vary at millimeter and centimeter wavelengths on t=
ime scales from hours to years (Brown \& Lo 1982;=20
%Genzel \et\ ???? ;
Zhao \et\ 1992; Bower \et\ 2002; Herrnstein \et\ 2004). The origin of thes=
e variations remains unclear, with strong arguments for both extrinsic and =
intrinsic mechanisms having been advanced (e.g. Zhao \et\ 1989 compared wit=
h Zhao \et\ 2001).
Interstellar scintillation is the primary mechanism which may cause any ext=
rinsic variability.
The same plasma that is responsible for the scatter-broadening of Sgr A* at=
millimeter and centimeter wavelengths (e.g. Lo \et\ 1998, Bower \et\ 2004)=
is also expected to cause the source to exhibit refractive intensity varia=
tions. It has been argued that much of the monthly to yearly variability i=
n Sgr A* at wavelengths longer than 6\,cm can be explained in terms of refr=
active interstellar scintillation provided that scattering material moves a=
cross our line of sight at a relatively high speed of $\sim 1000-2000\,$km\=
,s$^{-1}$ (Zhao \et\ 1989). However, the variability amplitude is not so =
easily accounted for: a scattering medium modeled as a single thin screen u=
nderpredicts the observed variability amplitude ({\it ibid.}) while extende=
d medium models, which are in principle capable of explaining higher refrac=
tive modulation amplitudes for the same degree of scatter broadening, have =
not been investigated in the context of the Galactic Center.
Recent interpretations favor an intrinsic origin for much of the centimeter=
wavelength variability. These center around claims of 106-day quasi-perio=
dic variations at wavelengths shorter than $\sim 3\,$cm (Zhao \et\ 2001) an=
d of 57-day quasi-periodic behavior at 2.3\,GHz (Falcke 1999). The oscilla=
tions possess only a modest spectral purity, with the highest purity $\nu/\=
Delta \nu=3D2.2 \pm 0.3$ reported at 1.3\,cm. Zhao \et\ (2001) discuss the=
origin of these oscillations in terms of periodic flares from a jet nozzle=
or an instability in the accretion disk triggering, for example, quasi-per=
iodic production of convection bubbles. It is widely supposed that the osc=
illations must reflect a process intrinsic to Sgr A* itself because scintil=
lation is incapable of producing such regular oscillations. Yet observatio=
ns of certain intra-day variable quasars, whose variations are proven to be=
scintillation-induced, invalidate this argument because their fluctuation=
s often exhibit even higher degrees of spectral purity (Kedziora-Chudczer \=
et\ 1997; Rickett, Kedziora-Chudczer \& Jauncey 2002).
Nonetheless there is little dispute that at least some of the variability i=
s intrinsic.
Detections of flares at millimeter, IR and X-ray wavelengths (Wright \& Bac=
ker 1993; Tsuboi, Miyazaki \& Tsutsumi 1999, Eckart \et\ 2004; Baganoff \et=
\ 2001) conclusively demonstrate that the source is intrinsically variable.=
A possible connection between X-ray flaring and unusually large flux dens=
ity excursions at 7\,mm is also reported (Zhao \et\ 2004). However, it is =
difficult to ascertain how much variability observed at centimeter waveleng=
ths could be attributed to flaring since neither the duty cycle nor the ene=
rgy distribution of mm or X-ray flares is well-constrained, much less the p=
hysical connection between centimeter and mm or X-ray behavior.
%The literature is replete with pithy remarks about whether scintillation i=
s a viable explanation for any of the variations observed in Sgr\,A*.
Despite the many recent observational results concerning the properties of =
Sgr A*'s variability, a dearth of corresponding theoretical efforts has fai=
led to place these results in context, leaving us none the wiser as to thei=
r cause. For instance, while it is acknowledged that scintillation variabi=
lity is likely to be important at centimeter wavelengths, no variations hav=
e been specifically attributed to it, and no realistic modeling has been ap=
plied to investigate what contribution it could conceivably make. This pap=
er aims to redress the balance by investigating two outstanding issues: (i)=
what exactly does a model of Sgr A*'s variability need to explain and (ii)=
can one deduce which variations {\it must} be intrinsic to the source by e=
liminating the variations that can be explained by interstellar scintillati=
on? The next section of this paper is devoted to the former question, incl=
uding a critical examination of the $\sim 100\,$day quasi-periodic oscillat=
ions reported in Sgr A* (Zhao \et\ 2001), while \S\ref{Scint} addresses the=
latter question. We compare the models to the observations in \S\ref{Comp=
arison}, and summarize our findings and briefly detail their implications i=
n \S\ref{Conclusions}.
%--------------------------------------------------------------------------=
------------------
\section{Data Analysis} \label{Data}
% DATA ANALYSIS
%--------------------------------------------------------------------------=
------------------
Sgr A* has been the subject of numerous VLA monitoring campaigns since 1975=
=2E The resulting data are published in Zhao \et\ (1992, 2001) and Herrnst=
ein \et\ (2004). The latter lists the results of a three-year effort to me=
asure weekly variations at 7\,mm, 1.3\,cm and 2\,cm.
We combine all these data to form lightcurves at 7\,mm, 1.3\,cm, 2\,cm, 3\,=
cm, 6\,cm and 20\,cm in an attempt to quantify the variability of Sgr A* on=
time scales of a few days to decades.
Additional daily Green Bank Interferometer (GBI) monitoring at 2.3 and 8.3\=
,GHz (Falcke 1999) quantifies variations in Sgr A* on shorter time scales. =
We reanalyze these data here, but do not incorporate them with the VLA flu=
x density measurements because the GBI is highly susceptible to confusion i=
n the Galactic Center region. The GBI is a two-element interferometer whos=
e 2400\,m spacing is insufficient to resolve out much of the extended emiss=
ion near Sgr A* which, if not properly accounted for, can cause hour-angle =
dependent variations in the measured flux density of Sgr A*. Falcke (1999=
) attempted to correct for hour-angle dependent gain variations, and to eli=
minate the contribution of confusion by comparing GBI flux density measurem=
ents with available contemporaneous VLA measurements. But only by comparin=
g the visibilities to a complete synthesis image of the crowded Galactic Ce=
nter region can one be confident in removing the effect of confusion.
The lightcurves from the combined datasets are shown in Figure \ref{lightcu=
rves}. Various parts of the lightcurves have been published elsewhere, and=
their main purpose here is to illustrate exactly which data are used here.
Throughout this paper we adopt the structure function as the chief measure =
of source variability because it is ideally suited to the interpretation of=
data which are highly irregularly sampled in time. The intensity structur=
e function, $D_I(\tau) =3D \langle [I(t'+\tau) - I(t')]^2 \rangle$, is a si=
mple statistic which characterizes the variance between measurements separa=
ted by a time interval $\tau$. Since we wish to be confident that we inter=
pret only those features of the variability that are statistically signific=
ant, this requires a rigorous assessment of the errors associated with our =
measure of variability, and a statistic simple enough that the errors are r=
eadily calculable (see \S\ref{ErrorSection} below). The power spectrum, wh=
ich is related to the structure function by a Fourier transform, is a more =
elegant measure of variability, but we do not employ it here because the ir=
regular time sampling of our datasets complicates the error analysis and th=
us the interpretation of the statistic.
In computing a single structure function to characterize variability over t=
he entire lightcurve we make the implicit assumption that the variability s=
tatistics are wide-sense stationary, which is to say that the statistical p=
roperties of the variations themselves do not vary with time. This approac=
h is not strictly valid, for instance, if the source undergoes various ``ph=
ases'' of variability in which the presence of fast time scale variations i=
s modulated by some underlying long-term process; X-ray binaries, whose be=
havior is characterized by infrequent outbursts, represent an obvious count=
erexample. Although the possibility that variability in Sgr A* changes cha=
racter with time cannot be discounted, there is no strong evidence to suppo=
rt the notion.
% Ds saturates at some sufficiently long time scale.=20
% so we make the assumption from the outset that this is not the case, and =
assume that whatever processes cause the variability always cause variabili=
ty.
% discuss interpretation of peaks, dips, etc.?
\subsection{Assignment of Errors} \label{ErrorSection}
% How we determined the errors assuming that they are drawn randomly=20
% form a wide-sense stationary ensemble.
%--------------------------------------------------------------------------=
------------------
The correct determination of errors associated with any measure of the vari=
ability is crucial in assessing the significance of variability time scales=
or even periodicities in the data, particularly when they are irregularly =
sampled and when=20
the time scales under consideration are comparable to the entire data lengt=
h. It is also crucial when comparing the observed variations to theoretica=
l models, as most theories only predict ensemble-average quantities (i.e. t=
hey predict the climate, not the weather).
The largest contribution to the error arises because our observations only =
sample the stochastic fluctuations over a finite duration. A simple argume=
nt would suggest, for instance, that in a dataset spanning 1000 days a proc=
ess that operates on a time scale of 200\,days contains {\it at most} five=
independent measurements of the variations caused by this process. Forma=
l arguments show that the error in the structure function at delay $\tau$ f=
rom an observation of total duration $T$ is (Jenkins and Watts 1968, see al=
so Appendix B of Rickett, Coles \& Markannen 2000)
\begin{eqnarray}
var[D(\tau)] =3D \frac{4}{(T-|\tau|)^2} \int_{0}^{T-\tau} \left[ \gamma (|r=
-\tau|) +\gamma(|r+\tau|) - 2 \gamma(r) \right]^2 (T-\tau-r) dr,
\end{eqnarray}
where the function $\gamma(\tau)$ is the ensemble-average autocovariance of=
the intensity fluctuations at wavelength $\lambda$. This function is unkn=
own, so we assume that the measured autocovariance is a reasonable represen=
tation of its corresponding ensemble-average counterpart.=20
% NB the result is a factor of 4 lower if we use structure functions inste=
ad of auto-correlations in the above equn.
%Similarly, Bartlett's formula (Jenkins \& Watts 1968) characterizes the er=
ror in the cross-correlation between two datasets, here labeled by suffixes=
1 and 2,
%\begin{eqnarray}
%var[C_{12}(\tau)] =3D \frac{T-\vert \tau \vert}{T^2} \int_{-T+|\tau|}^{T-|=
\tau|} \left[ \gamma_{11}(r) \gamma_{22} (r) + \gamma_{12}(r+\tau) \gamma_{=
12}(-r-\tau) \right] \left(1 - \frac{|r|}{T-|\tau|} \right) dr,
%\end{eqnarray}
%where $T$ is the span of the datasets, assumed identical here, and $\gamma=
_{ij}(\tau),$ is the ensemble-average auto- or cross- correlation between d=
atasets $i$ and $j$. Again, since the ensemble average auto- and cross-cor=
relations are unknown we bootstrap by assuming that the measured auto- and =
cross-correlations are reasonable representations of their corresponding en=
semble-average counterparts.
%%The issue is further complicated if additional variability occurs on long=
er time scales. The long time scale process may be statistically independe=
nt when averaged over the ensemble of all possible variations, but chance c=
orrelations on a sufficiently short dataset can cause power to leak between=
these two nominally independent oscillations.
%--------------------------------------------------------------------------=
------------------
\subsection{Characteristics of the variability} \label{VarCharacteristics}
% Lack of quasi-periodic oscillations.=20
%--------------------------------------------------------------------------=
-------------------
The structure functions derived from the lightcurves are displayed in Figur=
es \ref{StructureFns} and \ref{LinearDs}. Figure \ref{StructureFns} is the=
main result of this section, but the plots in Figure \ref{LinearDs}, shown=
on linear scales, allow closer scrutiny of the variability characteristics=
on time scales shorter than 200 days.
\subsubsection{Intra- to inter-day fluctuations} \label{IntraDay}
Sgr A* is reported to exhibit variations on scales down to less than one da=
y (Brown \& Lo 1982). It is possible to determine whether the present data=
sets show evidence for intra-day ($\leq 4$-day) flickering by examining the=
behavior of the structure functions at small time lags. Short time scale =
flickering is present if the value of the structure function in the smalles=
t time bin differs significantly from zero. A proper assessment of the pre=
sence of flickering depends crucially on the correct determination of the e=
rrors associated with the flux density measurements. We have subtracted th=
e contribution of measurement errors from each structure function using the=
errors quoted in the papers from which the observations are derived\footno=
te{The means of removing measurement errors is obvious when one writes the =
structure function in terms of the autocovariance: $D(\tau) =3D 2 [C(0) - C=
(\tau)],$ where $C(\tau) =3D \langle \Delta I(t+\tau) \Delta I(t) \rangle$.=
Measurement noise is assumed to be uncorrelated between samples and indep=
endent of the true intensity fluctuations, so it only makes an (additive) c=
ontribution to $C(0)$. It is removed by subtracting the variance of the me=
asurement errors from $C(0)$.}. However, if the errors are under-estimated=
the structure function is biased towards high values, leading to an over-e=
stimate of the amplitude of short time scale flickering. Conversely, an o=
ver-estimate of the errors biases the structure function to lower values, t=
he most clear manifestation of which is to cause dips in the structure func=
tion below zero, which is clearly unphysical.
The structure functions in Fig.\,\ref{LinearDs} suggest that Sgr A* undergo=
es appreciable flickering on $<10$-day time scales at wavelengths from 7\,=
mm to 3\,cm. In all cases the root-mean-square fluctuations are approximat=
ely $10$\% of the mean flux density. Inspection of more finely-binned stru=
cture functions reveals that Sgr A* exhibits 10\%, 6\% and 8\% variability =
at 7, 13 and 20\,mm respectively on $<4$-day time scales. However, we cau=
tion that our estimate of the flickering amplitude depends critically on a =
correct assessment of the errors associated with the observations. Such an=
estimate is called into question for the 20\,cm lightcurves, for instance,=
where the estimated contribution of measurement errors leads to negative s=
tructure function values at certain lags.
The amplitude of the intra-day variability must be subtracted when comparin=
g the observed structure functions to models which only apply to variations=
on longer time scales. However, the amplitude of any intra-day variation =
is small compared to total amplitude of the intensity variations, and this =
correction is small compared to the uncertainty in the total variability am=
plitude on long time scales at most wavelengths.
\subsubsection{Quasi-periodic variations}
Figure \ref{LinearDs} can also be used to assess whether any variations exh=
ibited by Sgr A* are quasi-periodic. Such variability is characterized by =
the presence of oscillations in the structure function. The origin of this=
behavior is understood by noting that the structure function is related to=
the power spectrum of the lightcurve, $P(\omega)$, by a Fourier transform:=
$D_I (\tau) \propto \int [1 - \exp(i \omega \tau)] P(\omega) d\omega$. A =
purely sinusoidal signal in the lightcurve would manifest itself as a sharp=
peak in the power spectrum, and the structure function would exhibit a pea=
k at a time scale corresponding to the period, followed by sinusoidal oscil=
lations peaking at multiples of the fundamental period. The amplitude of t=
hese oscillations would decrease with time lag if the variations are spectr=
ally impure.
It is important to distinguish between oscillations in the structure functi=
on from spikes which are devoid of accompanying oscillations at longer time=
scales. A structure function containing sharp, isolated spikes indicates =
that the {\it power spectrum} contains quasi-periodic features. This in tu=
rn indicates the presence of sharp spikes in the corresponding lightcurve. =
The structure function contains spikes (or sharp dips) when at least two f=
lares are present in the lightcurve. For instance, flares at times $t_1$ a=
nd $t_2$ each of duration $\Delta t$ would give rise to a feature in the st=
ructure function at a time lag $|t_2-t_1|$ of width $2 \Delta t$. In the a=
bsence of any quasi-periodic or flaring behavior the structure function is =
expected to increase monotonically with time until it saturates at the long=
est time scale of the variations present in the data. Figure \ref{QPOdiags=
} illustrates how quasi-periodic oscillations and flares are manifested in =
structure functions.
We investigate the form of the structure function on $<150\,$day time scale=
s, motivated by reports of quasi-periodic variations on time scales of $\si=
m 106\,$days between 7\,mm and 3\,cm (Zhao \et\ 2001). The structure funct=
ions derived from the long duration VLA datasets (Fig. \ref{LinearDs}) are =
most useful in assessing the data for the presence of any unusual features.=
A simple test for the significance of any features is obtained by fittin=
g a single line through each structure function at time lags from $\Delta t=
=3D5-150\,$days and computing the reduced $\chi^2$ statistic, as listed in =
Table 1. The departure of the structure function from a line, indicated by=
a high $\chi^2$, signifies the presence of peaked features above the gener=
ally-increasing trend with time. The statistics in Table 1 show that the $=
7\,$mm structure function is well-fit by a single line, and there is no sig=
nificant detection of any quasi-periodic variability. However, the reduced =
$\chi^2$ statistic suggests the presence of significant deviations at all l=
ower wavelengths, as is obvious by inspection of Fig. \ref{LinearDs}. The =
1.3, 2 and 3\,cm structure functions appear to exhibit peaks at lags of $\D=
elta t=3D30-45$, $45-65$ and $45-55$ days respectively. The 2\,cm structur=
e function also exhibits a peak at $\Delta t=3D80-95\,$day which appears to=
coincide with a marginally significant peak at a corresponding time lag in=
the 7\,mm structure function, however the coincidence does not necessarily=
increase the significance of the peaks because the errors bars at the two =
wavelengths are not independent if the two lightcurves are partially correl=
ated. In addition, the 6 and 20\,cm structure functions possess highly sig=
nificant peaks at $\Delta t \approx 140-170 $\,days; the 1.3\,cm structure =
function appears to exhibit a feature at similar time lags, but inspection =
of the error bars in Fig. \ref{StructureFns} suggests the detection is of m=
arginal significance.
We also consider the lightcurves measured with the GBI. Our reanalysis of =
the GBI dataset reproduces the structure functions reported by Falcke (1999=
) on whose basis a 57-day quasi-periodic variability cycle is claimed at 2.=
3\,GHz. These variations were suggested to be quasi-periodic because the s=
tructure function subsequently oscillates weakly after peaking at $\sim 57$=
\,days. However, our estimate of the errors associated with these structu=
re functions casts doubt on the significance of any claim of quasi-periodic=
behavior. At both 3.6 and 13\,cm even the 1-$\sigma$ error troughs are c=
onsistent with structure functions that increase monotonically and subseque=
ntly saturate without undergoing any oscillatory behavior. The insignifica=
nce of the quasi-periodic behavior is not affected by changes in the tempor=
al binning of the structure function. The error troughs reflect only the =
error incurred by trying to infer the ensemble average behavior of the var=
iations from a dataset of finite duration.
In summary, all of the particular features identified in the VLA structure =
functions are single, isolated peaks. None of these can be attributed to o=
scillatory behavior. As a further, more sensitive test for quasi-periodic v=
ariability we present in Fig.\,\ref{LombFig} Lomb periodograms for 2 cm, 1.=
3 cm and 0.7 cm lightcurves based on well-sampled data from 2000 to 2003. =
We compare the power spectral density for the data against the 99th percent=
ile expectation of uniform noise (dot-dashed curve) and noise with a red sp=
ectrum (dashed line). The red spectrum is calculated using 300 Monte Carlo
simulations that use the sampling function of the data sets. We see clearl=
y that there are no significant periods in the data. In particular, there =
are no peaks in the vicinity of the 106-day period reported by Zhao \et\ (2=
001). The spikes in the 2\,cm and 1.3\,cm data at $\sim 0.003 {\rm\ days}^=
{-1}$ have a significance of only $\sim 90$\% for the uniform noise case. =
Moreover, these periods have only been sampled $\sim 3$ times in this data =
set and they are not apparent in PSDs from longer (but less well-sampled) d=
ata sets. Against the red noise case, these peaks have minimal significanc=
e. Results are similar to the uniform noise case when one estimates the PS=
D significance through Monte Carlo simulations in which the lightcurves are=
generated through reordering of the data. We conclude, then, that there is=
no evidence for periodic or quasi-periodic oscillations in the radio light=
curves of Sgr A*.
\subsubsection{Essential characteristics of the variability}
The structure functions presented in Fig. \ref{StructureFns} are the key ob=
servable that any variability theory must reproduce. Broadly speaking, eac=
h structure function may be characterized by a monotonically increasing por=
tion until it saturates at a certain amplitude and time scale. A viable e=
xplanation of the variability should explain the shape of the structure fun=
ction over an appreciable range of time lags. Even if the model does not e=
xplain every significant peak and wobble in the structure function, it is s=
till viable if it explains the (i) amplitude and (ii) time scale at which t=
he structure function saturates and (iii) the slope of the structure functi=
on over an appreciable range in time lags. In the following sections we ga=
uge the success of our model by its ability to reproduce these three charac=
teristics.
As remarked above, many of the structure functions exhibit spikes. These ar=
e caused by large, rapid and isolated flux density excursions in the lightc=
urves. We shall not attempt to explain these features in this paper. We m=
erely remark that they likely represent either flaring activity intrinsic t=
o Sgr A* itself, or a manifestation of Extreme Scattering Events (Fiedler \=
et\ 1987) which appear to occur when the Earth traverses a caustic surface =
of certain lens-like discrete objects in the interstellar medium (Romani \e=
t\ 1987; Walker \& Wardle 1998).
The prevalence of these flares in most of the structure functions indicates=
that the mechanism responsible operates over a factor of ten range in wave=
length.
%We shall see below that these features cannot be reproduced by scintillati=
on models considered below, and must be produced by a qualitatively differe=
nt phenomenon.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\tableline
$\lambda$ & saturation & saturation & index of power law increase & reduce=
d $\chi^2$ from linear fit \\
(cm) & amplitude & time scale (days) & before saturation & to $D_I$ betwee=
n 5 \& 150 days \\
% & reduced $\chi^2$ (fourth-order polynomial) \\
%& probability of getting $\chi^2$ as high as this by chance \\
\tableline
0.7 & 0.05 & 6 & $1.2 \pm 0.5$ & 1.92 \\=20
% & 1.36 & 0.17\\
1.3 & 0.05 & 30 & $0.7 \pm 0.2$ & 4.33 \\
%& 2.25 & 0.008 \\
2 & 0.05 & 50 & $0.8 \pm 0.15 $ & 4.2 \\
% & 1.87 & 0.03 \\
3 & 0.1 & 40 & $1.0 \pm 0.2$ &15.6 \\
%& 14.6 & 0 \\
6 & 0.1 & 200 & $1.0 \pm 0.2$ & 12.5 \\
% & 8.1 & 0\\
20 & 0.1 & 100-1000? & $1.6 \pm 0.6$ & 34.4 \\=20
% & 22.6 & 0 \\
\tableline
\end{tabular}
% nb last column computed from Chi-sq distn with 14 d.o.f. e.g.
% f[x_]=3D PDF[ChiSquareDistribution[14],x]=20
% prob =3D 1- 14* Integrate[PDF[14 x],{x,0,1.2}]
\caption{Characteristics of the structure functions in Figures \ref{Structu=
reFns} \& \ref{LinearDs}. The index of the power law in the increasing par=
t of the structure function is obtained from a fit of the function $A+ B \t=
au^\alpha$ to the data between days 5 and 40 in the logarithmically-binned =
structure functions. The constants $B$ and $\alpha$ are allowed to vary, b=
ut the offset $A$ is chosen to remove any contribution that reflects intra-=
day variations whose contribution to the structure function does not appear=
to be associated with the variations on time scales longer than 5\,days (i=
=2Ee. intra-day variability); this parameter is nonzero at 7, 13 and 20\,mm=
only (see the discussion in \S \ref{IntraDay}).=20
The last column lists reduced $\chi^2$ values resulting from attempts to fi=
t each structure function at lags between $5$ and $150$\,days by a straight=
line.}
\end{table}
%--------------------------------------------------------------------------=
------------------
\section{Scintillation Variability} \label{Scint}
% Use a refractive scintillation model and see what it explains so we know
% what must be attributed to intrinsic variability
%
% How much of these variations could be due to scintillation?=20
%--------------------------------------------------------------------------=
------------------
In this section we attempt to distinguish between variations intrinsic to S=
gr A* itself and those due to refractive interstellar scintillation. Sever=
al detailed scintillation models that span the range of possible scattering=
conditions are constructed in order to isolate those variations which cann=
ot be explained under any plausible scattering conditions, and must therefo=
re be attributed to intrinsic source variability. The distinction between i=
ntrinsic and extrinsic variability is made on the basis of a scintillation =
model because the physics of any centimeter-wavelength intrinsic variabilit=
y in Sgr A* is ill-constrained; indeed even the fraction of the radio emiss=
ion originating in the jet and accretion disk is disputed (e.g. Falcke \& M=
arkoff 2000; Quataert \& Narayan 1999; Yuan, Markoff, Falcke 2002). On the=
other hand, the basic physics of interstellar scintillation is well-unders=
tood and makes robust predictions that can be compared directly to the obse=
rved intensity variations.
The shortcoming of this approach lies in the uncertainty of the exact distr=
ibution of scattering material along the line of sight (Lazio \& Cordes 199=
8; Yusef-Zadeh \et\ 1994), which in turn affects the amplitude and time sca=
le of the predicted variations. To encompass the range of variations possi=
ble we consider a model in which the scattering material is entirely locate=
d in a single thin screen, either at a distance of $50$ or $500$\,pc from S=
gr A*, and one in which the material is distributed in an extended medium n=
ear Sgr A*, again with a scale length of either $\Delta z =3D50$ or $500\,$=
pc.=20
% more comments on extremely strong scattering in GC region?
Only variations caused by refractive interstellar scintillation are investi=
gated here. Fluctuations caused by diffractive scintillation are possible =
in principle, but the extremely strong scattering observed toward Sgr A* re=
nders the time scale of such scintillation of order seconds at centimeter w=
avelengths. Such variability is expected to be strongly quenched, given th=
at recent estimates suggest the intrinsic source size far exceeds the angul=
ar scales probed by this phenomenon (Bower \et\ 2004).
%(Caveat: their measurement of an overall source size does not preclude the=
existence of smaller internal substructure.)
%--------------------------------------------------------------------------=
------------------
\subsection{The scattering model}
% Introduce the extended medium and thin screen models we will use
%--------------------------------------------------------------------------=
------------------
The distribution of scattering material along the line of sight to Sgr A* i=
s described in terms of the power spectrum of electron density fluctuations=
=2E This is modeled in the following standard form:
\begin{eqnarray}
\Phi_{N_e} ({\bf q},z) =3D C_N^2 (z)\, \left( \frac{q_x^2}{R} + R q_y^2 \ri=
ght) ^{-\beta/2} \exp \left[ - \left( \frac{q l_0}{2} \right)^2 \right], \l=
abel{PhiNe}
\end{eqnarray}
where the amplitude of the power spectrum is written as a function of the s=
patial wavenumber, ${\bf q}$, and the distance, $z$, from the source. The =
quantity $l_0$ is the inner scale of the fluctuation spectrum, and is usual=
ly identified with the turbulent dissipation scale.
The $\zeta \approx 2$ axial ratio observed in the scatter-broadened image o=
f Sgr A* (Lo \et\ 1998) indicates that the amplitude of the power spectrum =
varies with direction on the sky, presumably reflecting the orientation of =
the local magnetic field. The observed anisotropy is the result of either =
a change of the inner scale or the amplitude of the power spectrum as a fu=
nction of orientation on the sky. We concentrate on the latter case becaus=
e such anisotropy is expected for MHD turbulence (Goldreich \& Sridhar 1995=
).=20
In eq. (\ref{PhiNe}) the anisotropy is characterized by means of the parame=
ter $R$, and is oriented so that the major axis of the scattering disk is a=
long the $x$-axis. The anisotropy of a scattered image, $\zeta$, is equal=
to the anisotropy parameter $R$ when the length scales probed by angular b=
roadening are much larger than the turbulent dissipation scale. On smaller=
scales the two measures are approximately equivalent (see \S\ref{CN2determ=
ine} below).=20
% When the diffractive scale is much larger than the inner scale, the ratio=
of $r_{\rm diff}$ measured along the $y$-axis to that measured along the $=
x$-axis is exactly $R$.=20
%
% NB it would seem that $C_N^2$ is greater along the y-axis and thus that t=
he scattering disk is along the
% y-axis. But numerical integrations show it's the other way around. To s=
ee this, just write the spectrum:
% propto 1/ (q_x^2/R + R q_y^2)^\beta. So for q_y =3D0 we get an R on the =
numerator associated with q_x.
% The case in which only the inner scale varies is uninteresting because re=
fractive scintillation is sensitive to the low-wavenumber end of the turbul=
ence power spectrum, where the inner scale is irrelevant. In this case the =
scintillations would resemble those from isotropic turbulence.
The spectral index of the turbulence is assumed to be independent of $z$, b=
ut its amplitude is allowed to vary through the quantity $C_N^2(z)$. For a=
thin screen located at a distance $z_0$ from Sgr A* one writes,
\begin{eqnarray}
C_N^2 (z) =3D \left\{ \begin{array}{ll}=20
C_N^2, & |z-z_0| < \Delta L/2\\
0, & \hbox{ otherwise}, \end{array} \right.
\end{eqnarray}
where the thickness of the medium, $\Delta L$, is assumed to be far less th=
an the source-observer distance, $L$. In the thin screen model the scatter=
ing measure ${\rm SM}=3DC_N^2 \Delta L$ and the screen distance are the fre=
e parameters.
When the scattering measure is large, as it is in the Galactic Center envir=
onment (see \S\ref{CN2determine} below), the refractive modulations from a =
thin screen can be small.
To this end, we also investigate a model in which the turbulent fluctuation=
s are extended along the line of sight. Extended media are capable of prod=
ucing larger refractive modulations relative to thin screens (Romani, Naray=
an \& Blandford 1986; Coles \et\ 1987). We consider the specific distribut=
ion in which the amplitude of the turbulence declines as a Gaussian from th=
e Galactic Center:
\begin{eqnarray}
C_N^2 (z) =3D C_N^2 (0) \exp \left( - \frac{z^2}{\Delta z^2} \right).=20
\end{eqnarray}
The free parameters of this model are the scale length of the distribution,=
$\Delta z$, and a normalization constant that sets the overall amplitude o=
f the turbulence, $C_N^2(0)$. The effective scattering measure of the medi=
um in this model is ${\rm SM}=3DC_N^2(0) \pi^{1/2} \Delta z /2$ when $\Del=
ta z$ is much smaller than the distance to the source.
Unfortunately, the additional complexity inherent to the treatment of fluct=
uations in an extended medium forces us to abandon the explicit inclusion o=
f anisotropy for this model. We take $R=3D1$ and normalize the model to a =
scattering strength intermediate to that implied by the scatter-broadening =
along the major and minor axis of Sgr A*. We feel that our failure to take=
anisotropy into account in a self-consistent manner in this model is a min=
or point compared to other uncertainties, such as the true distribution and=
strength of scattering material along the line of sight. In any case, the=
correct manner in which to incorporate anisotropy in an extended scatterin=
g medium model is highly uncertain. Goldreich \& Sridhar (1995) point out =
that that the local value of $R$ in a thin plane of MHD turbulence, where `=
`thin'' is less than the outer scale of magnetic field fluctuations, is exp=
ected to be $\ga 10^3$, and the value of anisotropic image broadening that =
one actually observes is much lower only because it represents an average o=
ver many different orientations of the magnetic field along the line of sig=
ht\footnote{The Goldreich \& Sridhar (1995) theory of Kolmogorov MHD turbul=
ence suggests that the local anisotropic ratio is $\sim (k_\perp L_{\rm out=
})^{1/3}$, where $L_{\rm out}$ is the outer scale of the turbulent magnetic=
field fluctuations and $k_\perp$ is the wavenumber on which the power spec=
trum is probed.}.
% see section 6.2 of Goldreich and Sridhar 1995.
\subsection{Determination of $C_N^2$ appropriate to the models} \label{CN2d=
etermine}
The angular size of the scatter-broadened image of Sgr A* provides a direct=
means to determine the scattering strength appropriate for our scattering =
models. For a source of unit intensity whose intrinsic angular size is muc=
h smaller than the angular broadening size, the visibility\footnote{We deno=
te this observed visibility by $\Gamma$ to distinguish it from the intrinsi=
c source visibility, $V$ used below.} observed on a baseline ${\bf s}$ is
\begin{eqnarray}
\Gamma({\bf s}) =3D \exp \left[ -\frac{1}{2} \int_0^L D_\phi'({\bf s} z/L,z=
) \, dz \right]. \label{MutCoher}
\end{eqnarray}
The quantity $D_\phi'$ is the derivative of the phase structure function wi=
th respect to $z$ and is given by
\begin{eqnarray}
D_\phi'({\bf s},z) =3D 4 \pi r_e^2 \lambda^2 \int_{-\infty}^{\infty} \Phi_{=
N_e}({\bf q},z)\, (1-e^{i {\bf q} \cdot {\bf s}} ) \, d^2{\bf q}. \label{Dp=
hiPrime}
\end{eqnarray}=20
The baseline ${\bf s}_0$ at which the visibility falls to $1/e$ of its maxi=
mum value at wavenumber $k=3D2 \pi/\lambda$ is related by $\btheta_0=3D2/k =
{\bf s_0}$ to the angular radius at which the brightness distribution falls=
to $1/e$ of its maximum value, $\theta_0$.
% \footnote{This is easily seen when the visibility has the form $V(s) =3D =
\exp(-s^2/2 a^2)$. The $1/e$ point is at $s_0=3D\sqrt{2} a$.=20
%The image brightness is then $I(\theta) \propto \exp(- k^2 \theta^2 a^2/2)=
$. So the $1/e$ point of the brightness distribution occurs at $\theta =3D=
\sqrt{2} /k a =3D \sqrt{2}/(k s_0/\sqrt{2}) =3D 2/k s_0$.}.=20
One therefore solves $\int_0^L D_\phi'(2z/k \btheta_0 L,z)dz=3D2$ to determ=
ine the scattering strength.
% NB 1/e point of visibility at s_0 is related to \theta_0 by=20
%\theta=3D 1/k s_0 (where the brightness of the scattered image falls to $e=
^{-1}$ of its maximum value).=20
% It's easy to see this: the visibility goes as V(s) =3D exp[-s^2/2 a^2]. =
1/e point is at s_0=3Dsqrt{2} a.=20
% FTing this gives I(theta) propto exp[- k^2 \theta^2 a^2/2]. So the 1/e p=
oint of the brightness distn is
% \theta =3D \sqrt{2} /k a =3D theta_0 =3D \sqrt{2}/(k s_0/sqrt{2}) =3D 2/k=
s_0.
%The scattering models are normalized with the condition
%\begin{eqnarray}
%2=3D \int_0^L D_\phi'({\bf s}_0 z/ L,z) \, dz =3D 4 \pi r_e^2 \lambda^2 \i=
nt_0^L dz \int \Phi_{N_e}({\bf q},z)\, (1-e^{i z\,{\bf q} \cdot {\bf s}_0/=
L}) \, d^2{\bf q} ,
%\end{eqnarray}
% NB \Gamma(s) =3D 1/e =3D exp[-D(s)/2]. So this requires us to solve D(s)=
=3D 2=20
% which is equivalent to D(1/k\theta) =3D 2.
Measurements of angular broadening probe the structure function on a length=
scale $\sim s z/L=3D2 z/k \theta_0 L$, which is of order kilometers for th=
e case of Sgr A*. Consider, for instance, the scale probed by angular-broa=
dening at 6\,cm, where the angular diameter of Sgr A* is $49.6\,$mas along =
its major axis. Assuming the scattering occurs at $z=3D100\,$pc and taking=
the distance to the Galactic center as $L=3D8.5\,$kpc, one finds that angu=
lar broadening is sensitive to structure on scales of only $2 \times 10^3\,=
$m.
This scale is much smaller than the expected inner dissipation scale. Span=
gler \& Gwinn (1990) argue that the inner scale of the turbulent cascade is=
plausibly identified with the larger of the ion inertial length=20
\begin{eqnarray}
l_i=3D \frac{v_A}{\Omega_i} =3D 228 \left( \frac{\rho}{1\,{\rm cm}^{-3}}\ri=
ght)^{-1/2} \, {\rm km},
\end{eqnarray}
or the ion Larmor radius=20
\begin{eqnarray}
r_i =3D \frac{v_{\rm th}}{\Omega_i}=3D 300 \left( \frac{T_i}{10^3\,{\rm K}}=
\right)^{1/2} \left( \frac{B}{1\,\mu{\rm G}} \right)^{-1} \, {\rm km},
\end{eqnarray}
where $T_i$ is the ion temperature. The dissipation scale is larger than s=
everal kilometers for the range of plausible densities, temperatures and ma=
gnetic fields in the Galactic Center.
The length scale on which angular broadening probes the scattering medium i=
s important in determining the scattering strength because the character of=
the phase structure function changes when its argument falls below the tur=
bulent dissipation scale. The structure function scales as $s^{\beta-2}$ a=
bove this point and as $s^2$ below it. Defining $s_{\rm diff}$ as the leng=
th scale over which the rms phase change is one radian, one sees that $s_{\=
rm diff} \propto \theta_0^{-1}$. One has $s_{\rm diff} \propto \lambda^{-2=
/(\beta-2)}$ for $s_{\rm diff} > l_0$ and $s_{\rm diff} \propto \lambda^{-2=
}$ for $s_{\rm diff} < l_0$.
Centimeter-wavelength observations show that the apparent size of Sgr A* sc=
ales as $\lambda^{2.01 \pm 0.03}$ for $\lambda \geq 2\,$cm (e.g. Lo \et\ 19=
98, Bower \et\ 2004), which indicates either $\beta \approx 4$ or $ s z /L =
< l_0$. However, in view of the foregoing arguments, only the latter exp=
lanation is viable. Scatter-broadening measurements at centimeter waveleng=
ths yield no information on the spectral index of the electron density fluc=
tuations.
We normalize each scattering model by finding the appropriate $C_N^2 \Delta=
L$ or $C_N^2(0)$ to reproduce the observed angular radius of the image of =
Sgr A*. In the thin screen model the phase structure function at $rz. \\
\end{array} \right.=20
\end{eqnarray}
The assumption that the source is point-like is somewhat more problematic w=
hen the scattering medium extends close to the source, since clearly here =
the apparent angular source size must be appreciable relative to the local =
size of the scattering disk. Moreover, it is possible for the source size =
to affect the amplitude of intensity fluctuations without exerting a strong=
influence on the shape of the scatter-broadened image. This is because sc=
attering material closer to the observer influences the angular broadening =
more strongly; eq. (\ref{MutCoher}) shows that one requires a smaller basel=
ine ${\bf s}$ for a larger value of $z$ to satisfy $s z/L=3D s_{\rm diff}$.
The effect of source size is investigated here by calculating the structure=
functions for a variety of intrinsic source sizes, from $1\,\mu$as to $10\=
,$mas. The source is modeled as a circular disk whose visibility falls to =
zero on baselines longer than $1/k \theta_{\rm src}$ so that the term $V({\=
bf q} (L-z)/k)$ appearing in the integrand above is approximated as $H[ 1/(=
L-z) \theta_{\rm src} - q]$, where $H$ is the unit step function.
% i.e. H(x)=3D0 when x < 0 and =3D1 when x>1. We want a function that is o=
ne only when q< 1/(L-z) \theta.
We also make the simplifying assumption that the scattering is always stron=
g so that, even for very small $z$, the exponential in eq. (\ref{Crs}) cuts=
off the integrand before the sine-squared term begins to oscillate (i.e. =
$r_{\rm diff}(z) < \sqrt{z/k}$). We therefore expand the sine function fo=
r small argument and obtain the following expression for the intensity auto=
covariance from an extended scattering medium
\begin{eqnarray}
C_I (\tau) &=3D& \frac{r_e^2 \lambda^4 C_N^2(0) }{L^2} \int_0^L dz \, z^2 (=
L-z)^2=20
e^{-z^2/\Delta z^2} \int_0^\infty dq \, q^{5-\beta} \, {\rm J}_0 \left( \fr=
ac{q \, v_{\rm ISS} \tau \, z}{L} \right)=20
H \left[ \frac{1}{(L-z) \theta_{\rm src}} - q \right] \nonumber \\
&\null& \qquad \qquad \qquad \qquad \qquad \times
\exp\left[ - \left( \frac{q l_0}{2}\right)^2 -\int_0^L D_\phi' \left(\fra=
c{q h(t,z)}{k} ,t \right)\,dt \right]. \label{CIintermediate}
\end{eqnarray}
The integrand over $q$ cuts-off sharply once the source size becomes import=
ant, at $(L-z)^{-1} \theta_{\rm src}^{-1}$, or when either of the two argum=
ents of the exponential exceed unity, at $2 l_0^{-1}$, or $q_\phi$ which is=
defined by the implicit equation=20
\begin{eqnarray}
1 &=3D& \frac{8 \pi^2}{\beta-2} C_N^2(0) \Gamma \left(2 - \frac{\beta}{2} \=
right) r_e^2 \lambda^2 \left(\frac{l_0}{2} \right)^{\beta-2} \left\{
L \int_0^{z/L} e^{-t^2 {\cal L}^2} \null_1 F_1 \left[1-\frac{\beta}{2};1;- =
\left( \frac{q_\phi t (L-z)}{l_0 k} \right)^2 \right] dt \right. \nonumber =
\\ &\null& \left. \qquad \qquad \qquad \qquad +=20
L \int_{z/L}^1 e^{-t^2 {\cal L}^2} \null_1 F_1 \left[ 1-\frac{\beta}{2};1;-=
\left( \frac{q_\phi z (1-t)}{l_0 k} \right)^2 \right] dt
- \frac{\sqrt{\pi}\, {\rm erf}({\cal L})}{2} \right\},
\end{eqnarray}
where we write ${\cal L} =3D L/\Delta z$. Noting that the scintillation ve=
locity is well-approximated by $|{\bf v}_{\rm srcreen,o} - {\bf v}_{\rm sr=
c,o}|L /z$ for the values of $L/z \gg 1$ under consideration here, the argu=
ment of the Bessel function in eq. (\ref{CIintermediate}) becomes $q v_{\rm=
eff} \Delta t$, where $v_{\rm eff} =3D | {\bf v}_{\rm screen,o}-{\bf v}_{=
\rm src,o}|$. The intensity autocovariance thus reduces to=20
%\begin{mathletters}
\begin{eqnarray}
C_I(\tau) &=3D& \frac{r_e^2 \lambda^4 C_N^2(0)\, L^3}{6-\beta} \int_0^1 dz=
\, z^2 (1-z)^2=20
e^{-z^2 {\cal L}^2} q_{\rm max}^{6-\beta} \null_1 F_2 \left( 3-\frac{\beta}=
{2}; 1, 4- \frac{\beta}{2} ; - \frac{q_{\rm max}^2 v_{\rm eff}^2 \tau^2}{4}=
\right) , \nonumber \\
&\null&
\qquad \qquad \qquad \qquad \qquad q_{\rm max} =3D \min[(L-z)^{-1} \theta_=
{\rm src}^{-1},2 l_0^{-1}, q_\phi], \quad \beta < 6. \label{CrsIntegrate}
\end{eqnarray}
Below we also use eq. (\ref{CrsIntegrate}) to predict the amplitude of fluc=
tuations at millimeter wavelengths. Since the scattering is weaker at shor=
ter wavelengths, an additional wavenumber cut-off, at the inverse of the Fr=
esnel scale, $q_{\rm max} =3D \sqrt{4 \pi L/\lambda z (L-z)}$, is introduce=
d so that the amplitude of the variability is not overestimated.
%\begin{eqnarray}
%q_{\rm max} &=3D& 2 k \left[ \frac{\alpha\, \Gamma(1+\alpha/2) }{8 \pi^2 \=
Gamma(1-\alpha/2) C_N^2(0)
% r_e^2 \lambda^2 L^{1+\alpha} g(z) }\right]^{1/\alpha}, \\
%g(z) &=3D& \frac{1}{2} (1 -z)^\alpha {\cal L}^{-1-\alpha}=20
%\left[ \Gamma \left( \frac{1+\alpha}{2}\right) - \Gamma \left(\frac{1+\alp=
ha}{2},z^2 {\cal L}^2 \right) \right] + z^\alpha \int_z^1 (1-t)^\alpha \,=
e^{-t^2 {\cal L}^2} \,dt . \label{Crsgz}
%\end{eqnarray} \end{mathletters}
% NB q_{\rm max} solution comes from assuming a negligible inner scale. Wh=
ich may or may=20
% not be correct under these circumstances.
Equation (\ref{CrsIntegrate}) is integrated numerically to derive the inten=
sity structure function expected due to refractive scintillation in an exte=
nded medium. These functions are shown in Figs. \ref{ExtCIFig1} and \ref{E=
xtCIFig2}.
The time scale at which the structure functions saturate can be understood =
in terms of the time required for the turbulent medium to traverse the scat=
tering disk, of order $L \theta_0/2 v_{\rm ISS}$. A more rigorous estimate=
of the time scale is obtained by comparing the spatial wavenumber of the c=
ut-off in eq. (\ref{Crs}) with the form of eq. (\ref{MutCoher}). The power=
spectrum of intensity fluctuations cuts-off when the term $\int_0^L D_\phi=
' (q h(t,z)/k,t) dt$ reaches unity. The smallest wavenumber cut-off occurs=
at the outer boundary of the scattering medium, when $z=3D\Delta L$ and $h=
(t,z) \approx t$. Eq. (\ref{MutCoher}), which describes angular broadening=
due to the scattering medium, involves an integral of similar form. By eq=
uating arguments in the two integrals one sees that the power spectrum cuts=
-off at $q=3D2/\theta_0 L$, where $\theta_0$ is the scatter-broadened size.=
This corresponds to a time scale $\tau =3D L \theta_0/2 v_{\rm screen}$.=
=20
% so one equates the arguments of the two integrals $1=3D\int_0^L D_\phi'(s=
z/L)dz =3D \int_0^L D_\phi'(q z/k)dz$ to show that $q=3Ds_0 k/L$.
The intensity variations potentially yield information on the source size o=
n scales well below the size of the scatter-broadened image. For instance,=
observations at 1\,cm could distinguish between a source size of $1\,\mu$a=
s and $10\,\mu$as simply on the basis of its variability properties. This =
might seem surprising because such small angular diameters represent only 0=
=2E1 and 1\% respectively of the angular diameter of the scattering disk at=
this wavelength. This sensitivity to detail well below the scatter-broade=
ned size is possible because phase fluctuations along the path of propagati=
on are weighted differently between observations of scatter-broadening, whi=
ch measures a second-order moment of the wavefield, and intensity variabili=
ty, which represents a fourth-order moment.
The ability to distinguish between such small source sizes arises in any sc=
attering medium that extends very close to the source. Refractive modulati=
ons are strongest when the scattering strength is near unity. The scatteri=
ng strength can be expressed as the ratio of the Fresnel scale $r_{\rm F} =
=3D \sqrt{z (1-z/L)/k}$ to the transverse scale in the scattering medium ov=
er which the phase changes by one radian (i.e. the length scale, $r_{\rm di=
ff}$, at which $ D_\phi(r_{\rm diff},z) =3D1$). The scattering strength i=
ncreases with distance from the source as both the Fresnel scale and the am=
ount of scattering material encountered along the ray path increase. Thus =
material close to the source can contribute greatly to the intensity variat=
ions provided that the source is sufficiently small that it does not substa=
ntially quench this contribution.
%The corresponding solution for a thin screen of scattering material is
%\begin{mathletters}
%\begin{eqnarray}
%C({\bf s}) &=3D& \frac{r_e^2 \lambda^4 C_N^2 \Delta L}{(6-\beta) L^2} z_0=
^2 (L-z_0)^2 q_{\rm max}^{6-\beta} \null_1 F_2 \left(3 - \frac{\beta}{2}; 1=
, 4 -\frac{\beta}{2}; - \frac{q_{\rm max}^2 s^2 z_0^2}{4 L^2 } \right),
%\end{eqnarray}
%where $q_{\rm max}$ is defined implicitly by
%\begin{eqnarray}
%1 =3D \frac{8 \pi^2}{\alpha} \Gamma \left(1 - \frac{\alpha}{2} \right) C_N=
^2 \Delta L r_e^2 \lambda^2=20
%\left( \frac{l_0}{2} \right)^{\alpha} \left[ \null_1 F_1\left(-\frac{\alph=
a}{2}; 1; - \frac{q_{\rm max}^2 z_0^2 (1- z_0/L)^2}{k^2 L^2} \right) - 1 \r=
ight].
%\end{eqnarray}
%\end{mathletters}
%\subsection{Refractive Variations from Two Thin Screens} \label{TwoThinScr=
een}
%Here we investigate the intensity fluctuations resulting from scattering a=
t two thin screens.=20
%Equation (\ref{Crs}) specifies the autocovariance of the intensity fluctua=
tions as a function of displacement $s$ on the observer's plane:
%\begin{eqnarray}
%C_I({\bf s}) &=3D& 16 \pi^2 r_e^2 \lambda^2 \left\{ {\rm SM}_1 \int_0^\inf=
ty dq\,q^{1-\beta} \sin^2 \left[\frac{q^2 z_1 (L-z_1)}{2 k L} \right] e^{-(=
q l_0/2)^2} {\rm J}_0 \left( \frac{s q z_1}{L} \right) \right. \nonumber \=
\
%&\null& \qquad \qquad \qquad \left. \times \exp \left[ -D_{\phi_1} \left(=
\frac{q z_1 \left(\frac{z_1}{L}-1 \right)}{k} \right) - D_{\phi_2} \left( =
\frac{q z_1 \left(\frac{z_2}{L}-1 \right)}{k} \right) \right] \right. \no=
number \\
%&\null& \left. \qquad + {\rm SM}_2 \int_0^\infty dq \, q^{1-\beta} \sin^2=
\left[\frac{q^2 z_2 (L-z_2)}{2 k L} \right] e^{-(q l_0/2)^2} {\rm J}_0 \le=
ft( \frac{s q z_2}{L} \right) \right. \nonumber \\
%&\null& \qquad \qquad \qquad \left. \times \exp \left[ -D_{\phi_1} \left(=
\frac{q z_1 \left(\frac{z_2}{L}-1 \right)}{k} \right) - D_{\phi_2} \left( =
\frac{q z_2 \left(\frac{z_2}{L}-1 \right)}{k} \right) \right] \right\}, \=
label{TwoScreenCI}
%\end{eqnarray}
%where the phase structure function $D_{\phi_i}$ specifies the phase fluctu=
ations on the the $i$th screen.
%The time scales of the intensity fluctuations are determined by the appare=
nt velocities of the scattering screens. Equations (\ref{TwoScreenCI}) is =
converted into a temporal autocovariance by replacing the arguments of the =
Bessel functions by ${\bf v}_{\rm ISS_1} \Delta t q z_1/L$ and ${\bf v}_{\r=
m ISS_1} \Delta t q z_2/L$ respectively, where the scintillation pattern sp=
eeds are given by eq. (\ref{viss}).
%The pattern velocities of the screens may be quite different, but for the =
sake of simplicity the velocities of the scattering screens are here assume=
d equal, so the arguments of the Bessel functions both reduce to $q v_{\rm =
eff} \Delta t$ assuming $L \gg z_{2} > z_1$.
%Figure \ref{TwoScreenMp} shows how the peak modulation index changes as a =
function of the relative scattering measure between the two screens, $f=3D{=
\rm SM}_2/{\rm SM}_1$ when the screen locations are $z_1=3D50\,$pc and $z_2=
=3D500\,$pc.
%--------------------------------------------------------------------------=
------------------
\section{The distinction between Scintillation-Induced and Intrinsic variat=
ions} \label{Comparison}
% \section{Il cimento dell=D5 armonia e dell=D5 inventione}
% The contest between Harmony and Invention
% Vivaldi op. 8.
% Having discussed a scintillation model, what definitely must be intrinsic=
to the source?
%--------------------------------------------------------------------------=
------------------
We now consider to what extent scintillation can account for the observed v=
ariability. To qualify as a viable explanation of the variability on any g=
iven time scale it must account for a large fraction of the modulation ampl=
itude at that time scale.
By this criterion no thin-screen model constitutes a viable explanation of =
the intensity variations in Sgr A* on any time scale. Larger variability a=
mplitudes are predicted in this model when the scattering material is place=
d further from the source, but even when the screen is placed 500\,pc from =
Sgr A* the variability amplitude is more than a factor of two below that ob=
served.
%These models are thus also unable to account for any the intensity fluctua=
tions on time scales shorter than the saturation time scale.=20
Another shortcoming of the thin screen model is its failure to account for =
the slopes of the observed structure functions at small time lags. None of=
the observed structure functions rise more steeply than $\tau^{1}$, wherea=
s the models predict that they should rise as $\tau^{\beta-2}$. Moreover, =
even if a thin-screen model were to reproduce the amplitudes and time scale=
s at which the structure functions saturate it would still not explain any =
variations on shorter time scales. There does not appear to be any range o=
f time scales for which any of the structure functions rise as steeply as t=
he thin-screen model predicts. We thus conclude that thin-screen models co=
nstitute a poor explanation for the intensity variations observed on any ti=
me scale.
Extended medium models fare significantly better at explaining the amplitud=
e of the observed intensity variations. As can be seen from Fig.\,\ref{Ext=
CIFig1}, models with shallow electron density power spectra ($\beta \leq 11=
/3$) still fail to account for most of the intensity variations, particular=
ly at long wavelengths. However, the steep $\beta=3D3.9$ power spectrum =
model reproduces both the saturation amplitudes $\approx 0.1$ of the observ=
ed structure functions and their weak dependence on wavelength.
These structure functions also saturate at roughly the time scale observed =
in the data.=20
The saturation time scales in the models can be understood as the time requ=
ired by the scattering medium to traverse the spatial extent of the scatter=
ing disk. This rule of thumb estimates the time scale correct to within a =
factor of two relative to the time scales indicated by the structure functi=
ons in Figs. \ref{ExtCIFig1} and \ref{ExtCIFig2}.=20
%Assuming the measured scatter-broadened size of 50\,mas at 6\,cm and that =
angular broadening measurements probe phase fluctuations within the inner s=
cale of the turbulence,=20
The predicted time scale is=20
\begin{eqnarray}
\tau_0 \approx=20
%\frac{L \theta_0}{2 v_{\rm screen}} =3D=20
368 \left( \frac{L}{8.5\,{\rm kpc}} \right)=20
\left( \frac{\lambda}{ 6\,{\rm cm}} \right)^2=20
\left(\frac{v_{\rm screen}}{1000\,{\rm km\,s}^{-1}} \right)^{-1} \quad {\rm=
days}.
\end{eqnarray}
Certain extended-medium models also reproduce the generally shallow slope o=
f the observed structure functions at small time lags. This behavior depen=
ds on the source size relative to the angular scales probed by the scintill=
ation. Shallow slopes are present in some model structure functions over a=
large range of time scales, spanning up to two orders of magnitude in time=
scale before the structure function saturates.
To illustrate the distinction between thin and extended-medium models, Fig.=
\,\ref{DsComparison} plots the observed 1.3 and 2\,cm structure functions a=
gainst several models. It is interesting to note that the $\beta=3D3.9$ ex=
tended-medium model with an intrinsic source size of $\sim 300\,\mu$as woul=
d appear to closely match the observed structure functions. This size is c=
omparable to that recently deduced by Bower \et\ (2004) using VLBI.
%Why does the structure function rise more slowly in certain portions?
Our conclusion is that extended medium models with steep $\beta \ga 3.9$ po=
wer spectra are capable of reproducing the gross features of the variabilit=
y at centimeter wavelengths. They do not explain {\it all} the detailed fe=
atures of the structure functions. In particular, they fail to account for=
any of the various peaks apparent in the structure functions in the range =
$50-100$\,days which, as discussed in \S\ref{VarCharacteristics}, which are=
probably due to flaring. Although it is hard to see how these features =
could be reproduced by a simple scintillation model, it is nonetheless pert=
inent to consider how the assumptions used in the models bear on the predic=
ted scintillation properties.
In all extended medium models it is assumed that all layers of the scatteri=
ng medium move with identical peculiar velocities. When this is not the ca=
se different layers of the medium may cause intensity fluctuations on diffe=
rent time scales. The extent to which this could occur depends on the rela=
tive contributions that layers at different distances make to the intensity=
fluctuations. Significant variations will only be observed on different t=
ime scales if two layers which both contribute substantially to the scatter=
ing each possess different transverse velocities. It is possible to see wh=
ich layers contribute most to the scattering by comparing the amplitude of =
the structure functions for medium scale lengths of 50 and 500\,pc. The in=
tensity fluctuations are dominated by scattering layers within $50$\,pc fro=
m Sgr A* if the amplitude on a given time scale is identical in the two mod=
els. Conversely, when they differ, most of the intensity fluctuations orig=
inate in the scattering medium beyond $50\,$pc from Sgr A*.
The time scale may also vary from the wavelength scaling predicted if diff=
erent layers of the medium move at different velocities. This is possible =
because the layer that contributes most to the intensity fluctuations chang=
es with wavelength (see Fig.\,\ref{ExtCIFig2}).
Another assumption inherent to the scattering model lies in the simplicity =
of the source structure. Our models assume the simplest possible structure=
: a source comprised of one single circularly-symmetric component. More co=
mplicated structure could cause qualitatively different variability. To be=
specific, eqs. (\ref{Cthin}) and (\ref{Crs}) establish how the power spect=
rum of the intrinsic source brightness distribution, $|V({\bf r})|^2$ alter=
s the power spectrum of scintillation-induced intensity fluctuations.
Consider, for instance, how a source comprised of two compact components (e=
=2Eg. a jet and a counterjet) would alter the scintillation characteristics=
=2E Whereas the visibility of a single point source is constant and indepe=
ndent of baseline length, the visibility amplitude of a double source with=
angular separation $\Delta \theta$ pointing along the direction $\hat{\bth=
eta}=3D\Delta \btheta/\Delta \theta$ oscillates on a scale of length ${\bf =
r}_{\rm osc}=3D\hat{\btheta}/k \Delta \theta$. This oscillation enhances t=
he scintillation fluctuations at certain time scales relative to others. =
The first peak of the oscillation emphasizes the power spectrum of intensit=
y fluctuations at a fundamental spatial wavenumber ${\bf q}=3D\hat{\btheta}=
\, [\Delta \theta \, (L-z)]^{-1}$, for a scattering layer a distance $z$ f=
rom Sgr A*. This corresponds to fluctuations on time scales $\tau_{0} \app=
rox \Delta \theta (L-z)/v_{\rm screen}$. The amplitudes of visibility osci=
llations at higher harmonics depend on the size of the components constitut=
ing the double source. The visibility amplitude of a double source compris=
ed of sufficiently compact components contains peaks comparable to the ampl=
itude of the fundamental peak at integer multiples of ${\bf r}_{\rm osc}$.=
=20
This in turn would enhance the scintillation fluctuations at shorter time s=
cales, with the $n$th harmonic enhancing fluctuations on a time scale $\tau=
\approx \tau_{0}/n$ relative those on surrounding time scales.=20
%This behavior is caused by oscillations in the visibility amplitude which =
boost scintillation power at multiples of the fundamental wavenumber.
It is possible in principle for source structure to explain additional feat=
ures of the fluctuations observed toward Sgr A*. One could appeal to a dou=
ble source structure to explain peaks in the observed structure functions. =
However, this model cannot not explain the peaks at time lags between 30 a=
nd 150\,days that are discussed in \S\ref{VarCharacteristics}. A double so=
urce would give rise to a number of regularly-spaced peaks in the structure=
function, not a single, isolated peak. Moreover, the source separation re=
quired to explain the location of the peaks is sufficiently large that it w=
ould have been observed. A peak on a time scale of $\tau =3D 50\,$days wou=
ld require a double source of separation $3\,$mas for a scattering medium m=
oving at 1000\,km\,s$^{-1}$.
We conclude that the interpretation of the isolated peaks present in the st=
ructure functions discussed in \S\ref{VarCharacteristics} as flares intrins=
ic to Sgr A* is robust to the assumptions made in the scintillation models =
considered here.
\subsection{The predicted role of scintillation at millimeter wavelengths}
Given the success of the $\beta=3D3.9$ extended medium model in explaining =
the broad characteristics of the centimeter variability, we have applied it=
to the predict the variations at millimeter wavelengths. Sgr A* flux moni=
toring is already being carried out at the Sub-Millimeter Array (Zhao et al=
=2E 2003), and similar monitoring will soon be possible using CARMA and, ev=
entually, ALMA.
The predictions at 1 and 3\,mm from the $\Delta L=3D500\,$pc, $\beta=3D3.9$=
extended-medium model are shown in Fig. \ref{ExtCIPredict}. This model im=
plies that a $1\,\mu$as ($10\,\mu$as) source should exhibit 25\% (22\%) roo=
t-mean-square fluctuations on a time scale of $1.4 \, (v_{\rm screen}/1000\=
, {\rm km\,s}^{-1})$\,hours at 3mm. A $100\,\mu$as would exhibit only 2.6\=
% variations, and on a time scale approximately three times longer. Given =
recent measurements of the intrinsic size of Sgr A* at 7\,mm of $\approx 0.=
22$mas (Bower \et\ 2004) and assuming a $\nu^{-1}$ size-dependence it is re=
asonable to expect scintillation variations of order $3$\% at a wavelength =
of 3\,mm.
% NB Bower quotes a size of 24 Rs. 1 Rs for Sgr A* =3D 1.18 x 10^10m. <=
=3D> which is 222.6 muas at
% 8.5 kpc. For reference
The scintillation characteristics are even more sensitive to source size at=
1\,mm. Only a source size of $\sim 1\,\mu$as is sufficiently small to exh=
ibit 26\% fluctuations. Sources of 10 and $100\,\mu$as would exhibit r.m.s=
=2E~fluctuations of 3.3 and 0.29\% respectively. Again, the predicted vari=
ations occur on intra-day time scales of 4 and 12 hours respectively assumi=
ng a scintillation velocity of 1000\,km\,s$^{-1}$.
It is possible to already compare these predictions with a number of observ=
ations. Both Wright \& Backer (1993) and Tsuboi \et\ (1999) report variati=
ons of order 1\,Jy amplitude at a wavelength of 3\,mm. Such large flux de=
nsity excursions are difficult to explain in terms of the present scintilla=
tion model, suggesting that intrinsic source activity is responsible for mo=
st of the variability.
Mauerhan \et\ (2005) have recently claimed the Sgr A* undergoes only $\sim =
20$\% intra-day variations at 3\,mm. These variations, if real, would requ=
ire an intrinsic source size of $10-30\,\mu$as at this wavelength to be con=
sistent with scintillation.
%--------------------------------------------------------------------------=
------------------
%\section{Discussion} \label{Implications}
% DISCUSSION
%--------------------------------------------------------------------------=
------------------
%--------------------------------------------------------------------------=
------------------
\section{Discussion \& Conclusions} \label{Conclusions}
% CONCLUSIONS
%--------------------------------------------------------------------------=
------------------
Our analysis of multi-frequency monitoring data presented by Zhao \et\ (199=
2, 2001), Herrnstein=20
\et\ (2004) and Falcke (1999) indicates that Sgr A* exhibits no quasi-perio=
dic oscillatory behavior on any time scale between one week and 200 days. =
The variability amplitudes are remarkably constant with frequency, varying=
between 30 and 39\%, but the time scale on which they saturate increases w=
ith wavelength.
Several structure functions show evidence for variability on multiple time =
scales. If the errors associated with the flux density measurements are co=
rrect, the data of Herrnstein \et\ (2004) indicate that the source exhibits=
unresolved 6-10\% inter-day ($<4$-day) variations between 7\,mm and 2\,cm.
No structure functions exhibit, within the errors, any evidence for appreci=
able variability with
time scales longer than 1000\,days. The long term stability of the radio f=
lux implies there is very little long-term variation in the accretion rate.=
=20
% Short-term fluctuations can be driven by excitation of high energy electr=
ons rather than accretion rate changes.
We sought to explain the general features of the variability by reproducing=
the shape and amplitudes of the observed structure functions using several=
scintillation models. Both thin-screen and extended-medium models were co=
nsidered. No thin screen model accounts for the properties of the variatio=
ns. The structure functions of the observed lightcurves rise less steeply =
with time lag and saturate at higher amplitudes than predicted. They under=
predict the amplitude of the variability by at least a factor of two at the=
saturation time scale and often by a more than an order of magnitude on sh=
orter time scales. If the medium responsible for the scattering of Sgr A* =
lies on a thin screen all of the observed flux variability must be intrinsi=
c to the source itself.
Certain extended-medium models, on the other hand, do explain the amplitude=
of the fluctuations over a large range of time scales. Models in which th=
e electron density fluctuations follow a Kolmogorov power spectrum, corresp=
onding to $\beta=3D11/3$, and a slightly steeper, $\beta=3D3.9$, power spec=
trum were investigated. Of the two models, only those with a $\beta=3D3.9$=
spectrum account for the amplitude of the fluctuations at all wavelengths.=
If scintillation is to precisely predict the amplitude of the flux densit=
y variability of Sgr A*, this suggests that the power spectrum of the Galac=
tic Center turbulence is slightly less steep, with an index lying in the ra=
nge $3.8 \la \beta < 3.9$.
The most successful extended-medium model examined was used to predict the =
maximum contribution that scintillation could make to future observations o=
f Sgr A* at millimeter wavelengths. The expected variability amplitude dep=
ends strongly on the intrinsic source size. A $1-10\,\mu$as object at 3\,m=
m would undergo fractional root-mean-square fluctuations of $\sim 25$\%, bu=
t a $100\,\mu$as source would exhibit only 3\% variations. Scintillation is=
even more sensitive to source size at 1\,mm, with a $1\,\mu$as source expe=
cted to display 26\% variations, but a $10\,\mu$as source would display onl=
y 3\% variability.
% [From Geoff: We should discuss or at least present physical conditions ap=
propriate for our favored scintillating medium: density, or integrated colu=
mn density. Is there a known medium that we can associate this with? The =
HIM? What does it imply that we favor $\beta=3D3.9$?
With what physical structures can we associate the extended scattering
medium? Given only the approximate match between our model structure funct=
ions
and the actual structure functions, we do not expect that the extended
scattering medium must in fact be parameterized as discussed in the text.
In fact, the extended scattering medium might consist of only a few=20
thin scattering media distributed over a range of distances from Sgr A*.
These thin media would have characteristics similar to the thin medium
discussed by Lazio \& Cordes (1998), with densities $>10^2 {\rm\ cm^{-3}}$.
The details of the structure functions are not sufficient to constrain
this result further. For comparison purposes, we do compute the
mean density associated with the extended scattering medium as presented.
In this case, the mean density is $\sim 10\,$cm$\,^{-3}$, assuming the out=
er scale of the turbulence to be 1\,pc. This density is substantially gre=
ater than the density of the diffuse hot ionized gas ($T_e \sim 10^7\,$K,=
=20
$n_e \sim 0.05 {\rm\ cm^{-3}}$) detected in the Galactic Center. As Lazio =
\& Cordes (1998) discuss,=20
the scattering medium may be the interface between this hot medium
and molecular clouds in the central 100 pc. Our results are consistent wit=
h the picture reached by Lazio \& Cordes (1998) for the scattering medium w=
ith the modification that the scattering may take place at a range of dista=
nces from Sgr A*.
\acknowledgments
Facilities: \facility{VLA, Green Bank Interferometer}
% \facility{HST(STIS)}, \facility{CXO(ASIS)}.
%% Appendix material should be preceded with a single \appendix command.
%% There should be a \section command for each appendix. Mark appendix
%% subsections with the same markup you use in the main body of the paper.
%% Each Appendix (indicated with \section) will be lettered A, B, C, etc.
%% The equation counter will reset when it encounters the \appendix
%% command and will number appendix equations (A1), (A2), etc.
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\clearpage
\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=3D0.6]{f1a.eps} &
\includegraphics[scale=3D0.6]{f1b.eps} \\
\includegraphics[scale=3D0.6]{f1c.eps} &
\includegraphics[scale=3D0.6]{f1d.eps} \\
\includegraphics[scale=3D0.6]{f1e.eps} &
\includegraphics[scale=3D0.6]{f1f.eps} \\
\includegraphics[scale=3D0.6]{f1g.eps}
=09& \includegraphics[scale=3D0.6]{f1h.eps} \\
\end{tabular}=20
\caption{Lightcurves of the variations in Sgr A* from 7\,mm to 20\,cm. Flu=
x density errors were not included in the GBI data reduced by Falcke (1999)=
, but the spread of the data points at each epoch is a reasonable indicator=
of the uncertainty. } \label{lightcurves}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[angle=3D0,scale=3D0.63]{f2a.eps} & \includegraphics[angle=
=3D0,scale=3D0.63]{f2b.eps} \\
\includegraphics[angle=3D0,scale=3D0.63]{f2c.eps} & \includegraphics[angle=
=3D0,scale=3D0.63]{f2d.eps} \\
\includegraphics[angle=3D0,scale=3D0.63]{f2e.eps} & \includegraphics[angle=
=3D0,scale=3D0.63]{f2f.eps} \\
\includegraphics[angle=3D0,scale=3D0.63]{f2g.eps} & \includegraphics[angle=
=3D0,scale=3D0.63]{f2h.eps} \\
\end{tabular}
\caption{Structure functions of the variations in Sgr A* from wavelengths o=
f 7\,mm to 20\,cm. The continuous and dashed error lines indicate the 1- a=
nd 2-$\sigma$ confidence limits of the measured structure functions due to =
the finite duration of the lightcurves (see \S\ref{ErrorSection}). The tic=
ks associated with individual points display the error caused by the finite=
number of observations that contributed to the measurement at that particu=
lar time lag.} \label{StructureFns}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[angle=3D0,scale=3D0.65]{f3a.eps} & \includegraphics[angle=
=3D0,scale=3D0.65]{f3b.eps} \\
\includegraphics[angle=3D0,scale=3D0.65]{f3c.eps} & \includegraphics[angle=
=3D0,scale=3D0.65]{f3d.eps} \\
\includegraphics[angle=3D0,scale=3D0.65]{f3e.eps} & \includegraphics[angle=
=3D0,scale=3D0.65]{f3f.eps} \\
\end{tabular}
\caption{Structure functions of the variations in Sgr A* from wavelengths o=
f 7mm to 20cm. For small lags $\Delta t < 200\,$days the errors are domina=
ted by the finite number of observations that contributed to the measuremen=
t at that particular time lag. The contribution of measurement errors is e=
stimated using the errors quoted from the observations, and has been subtra=
cted from each structure function. At 6cm and 20cm these clearly overestim=
ate the true error, as these structure functions are negative at certain ti=
me lags.} \label{LinearDs}
\end{figure}
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[angle=3D0,scale=3D0.5]{f4a.eps} &\includegraphics[angle=3D=
0,scale=3D0.5]{f4b.eps} \includegraphics[angle=3D0,scale=3D0.5]{f4c.eps}& \=
\
\includegraphics[angle=3D0,scale=3D0.5]{f4d.eps} &\includegraphics[angle=3D=
0,scale=3D0.5]{f4e.eps} \includegraphics[angle=3D0,scale=3D0.5]{f4f.eps}& \=
\
\end{tabular}
\caption{A simple illustration of the effect of quasi-periodic oscillations=
(top) and flaring (bottom) on the behavior of the structure function. The=
left panels show the sample lightcurve, the middle panel its power spectru=
m and the right panels the resulting structure function. The quasi-periodi=
c oscillations possess a relative bandwidth $\Delta \nu/\nu$ of 0.5.} \labe=
l{QPOdiags}
\end{figure}
\begin{figure}
\includegraphics[angle=3D0,scale=3D1.0]{f5.eps}
\caption{Lomb periodograms of the variations between 2000 and 2003 at 20, 1=
3 and 7\,mm.
The power spectral density for the data (solid line) is plotted with the 99=
th percentile expectation of uniform noise (dot-dashed line) and noise with=
a red spectrum (dashed line).=20
} \label{LombFig}
\end{figure}
\begin{figure}
\includegraphics[scale=3D0.7]{f6a.eps}
\includegraphics[scale=3D0.72]{f6b.eps} \\
\includegraphics[scale=3D0.7]{f6c.eps}
\includegraphics[scale=3D0.78]{f6d.eps}
% NB don't use DsThinxxx100 or 1000.eps because they were incorrectly norma=
lized to Sgr A*'s
% scattering size.
\caption{Intensity structure functions from a thin-screen of scattering mat=
erial with a turbulent power spectrum with index $\beta=3D11/3$ or $\beta=
=3D3.9$, located a distance of 50\,pc or 500\,pc from Sgr A*. The strength =
of the scattering is set by requiring the scattering to reproduce the obser=
ved degree of angular broadening of Sgr A*.=20
%The colour coding represents the fluctuations at 1cm (blue), 3cm (light bl=
ue), 6cm (red) and 20cm (green).=20
Because the turbulence is anisotropic the form of the intensity structure f=
unction depends on the orientation of the anisotropy axis with the scintill=
ation velocity. The two limiting cases are shown for each wavelength: the=
curve which reaches the highest amplitude results when the velocity is ori=
ented parallel to the major axis of the scattering disk, while the lower cu=
rve results when the velocity is oriented along the minor axis. Equation (=
\ref{viss}) should be used to convert the values on the abscissae to units =
of time.} \label{ThinCIFig100}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=3D0.5]{f7a.eps} &=20
\includegraphics[scale=3D0.5]{f7b.eps} \\
\includegraphics[scale=3D0.5]{f7c.eps} &=20
\includegraphics[scale=3D0.5]{f7d.eps} \\
\includegraphics[scale=3D0.5]{f7e.eps} &=20
\includegraphics[scale=3D0.5]{f7f.eps} \\
\includegraphics[scale=3D0.5]{f7g.eps} &=20
\includegraphics[scale=3D0.5]{f7h.eps} \\
\end{tabular}
\caption{Theoretical structure functions as a function of wavelength for th=
e extended medium model for a Kolmogorov spectrum of turbulence ($\beta=3D1=
1/3$) and for medium depths of $50$ and $500$\,pc.
The effect of finite size is illustrated by the various curves in each figu=
re: red, brown, dark green, light green and blue dots represent the structu=
re functions for a source of intrinsic angular size $10, 1, 0.1, 0.01$ and =
$0.001\,$mas respectively. A finite turbulent dissipation scale of $10^5\,=
$m is assumed in the calculation. } \label{ExtCIFig1}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=3D0.5]{f8a.eps} &=20
\includegraphics[scale=3D0.5]{f8b.eps} \\
\includegraphics[scale=3D0.5]{f8c.eps} &=20
\includegraphics[scale=3D0.5]{f8d.eps} \\
\includegraphics[scale=3D0.5]{f8e.eps} &=20
\includegraphics[scale=3D0.5]{f8f.eps} \\
\includegraphics[scale=3D0.5]{f8g.eps} &=20
\includegraphics[scale=3D0.5]{f8h.eps} \\
\end{tabular}
\caption{The same as Figure \ref{ExtCIFig1} but a steeper, $\beta=3D3.9$ el=
ectron density fluctuation power spectrum. } \label{ExtCIFig2}
\end{figure}
\begin{figure}
\includegraphics[scale=3D0.75]{f9a.eps}=20
\includegraphics[scale=3D0.75]{f9b.eps} \\
\caption{A comparison between the observed structure functions and those pr=
edicted by the extended-medium $\Delta =3D500\,$pc, $\beta=3D3.9$ model for=
a plausible range of source sizes, and the $\beta=3D3.9$, $z=3D500$ thin-s=
creen model. A scintillation speed $v_{\rm ISS}=3D1000\,$km\,s$^{-1}$ is a=
ssumed in this plot.} \label{DsComparison}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=3D0.5]{f10a.eps} &=20
\includegraphics[scale=3D0.5]{f10b.eps} \\
\end{tabular}
\caption{The structure function of the predicted variations for the best-fi=
tting, $\Delta L=3D500\,$pc, $\beta=3D3.9$, scintillation model at centimet=
er wavelengths applied at 1 and 3\,mm.} \label{ExtCIPredict}
\end{figure}
\clearpage
%% The following command ends your manuscript. LaTeX will ignore any text
%% that appears after it.
\end{document}
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