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%astro-ph/0608232

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\shorttitle{Flux and Spectral Index Correlation}
%\shortauthors{Djorgovski et al.}

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\title{Correlation between Flux and Spectral Index  during Flares in
Sagittarius A*}

%% Use \author, \affil, and the \and command to format
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\author{Jonathan M. Bittner,\altaffilmark{1} Siming Liu,\altaffilmark{2}
Christopher L. Fryer,\altaffilmark{2, 3} and
Vah\'e Petrosian\altaffilmark{4}
06520-8120; jonathan.bittner@yale.edu}
liusm@lanl.gov}
\altaffiltext{3}{Physics Department, The University of Arizona, Tucson, AZ
85721; fryer@lanl.gov}
\altaffiltext{4}{Center for Space Science and Astrophysics, Department of
Physics and Applied Physics, Stanford
University, Stanford, CA 94305; vahe@astronomy.stanford.edu}
%\altaffiltext{3}{Physics Department and Steward Observatory, The
%University of Arizona,
%Tucson, AZ 85721; melia@physics.arizona.edu; Sir Thomas Lyle Fellow and
%Miegunyah Fellow.}

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\begin{abstract}
Flares in Sagittarius A* are produced by hot plasmas within a few
Schwarzschild radii of the supermassive black hole at the
Galactic center. The recent detection of a correlation between the
spectral index and flux during a near infrared (NIR) flare
provides a means to conduct detailed investigations of the plasma
heating and radiation processes. We study the evolution of
the electron distribution function under the influence of a turbulent
magnetic field in a hot collisionless plasma.  The
magnetic field, presumably generated through instabilities in the
accretion flow, can both heat the plasma via
resonant wave-particle coupling and cool the electrons via radiation.
The electron distribution can generally be
approximated as relativistic Maxwellian. To account for the observed
correlation, we find that the magnetic field needs
to be anti-correlated with the electron ``temperature''. NIR and X-ray
light curves are produced for a cooling and a heating
phase. The model predicts simultaneous flare activity in the NIR and
X-ray bands, which can be compared with
observations. These results can be applied to MHD simulations to study
the radiative characteristics of collisionless
plasmas, especially accretion flows in low-luminosity AGNs.
\end{abstract}

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\keywords{acceleration of particles --- black hole physics --- Galaxy:
center ---
plasmas --- radiation mechanisms: thermal--- turbulence}
%\object{NGC 6624}, \objectname[M 15]{NGC 7078},
%\object[Cl 1938-341]{Terzan 8})}

%% From the front matter, we move on to the body of the paper.
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%% to the KEY in the \bibitem in the reference list below. We have
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%% the last two numeral of the year of publication as our KEY for
%% each reference.

\section{Introduction}

It is generally accepted that the near-infrared (NIR) and X-ray flares
in Sagittarius A*, the compact radio source associated
with a mass $M\sim 3-4\times 10^6\;M_\odot$ supermassive black hole at
the Galactic Center (Sch\"{o}del et al. 2002; Ghez et
al. 2004; the Schwarzschild radius $r_S =10^{12}(M/3.4\times
10^6\;M_\odot$) cm), are produced by relativistic
electrons within a few Schwarzschild radii of the black hole via
synchrotron and synchrotron self-Comptonization (SSC)
processes, respectively (Baganoff et al. 2001, 2003; Genzel et al.
2003; Belanger et al. 2006; Eckart et al.  2004, 2006;
Yusef-Zadeh et al. 2006a).
Studying the details of the electron acceleration therefore plays an
important role in revealing the properties of the
flaring plasma and its interaction with the surroundings and the
space-time created by the black hole, which may lead to a
measurement of the spin with comprehensive flare observations and
general relativistic MHD and ray-tracing
simulations of the black hole accretion and the corresponding
radiation transfer.

The electrons are likely accelerated by turbulent plasma waves
generated in an accretion torus via instabilities,
which induce the release of gravitational energy of the plasma (Balbus
\& Hawley 1991; Tagger \& Melia 2006). In the simplest
case, in which there is no large scale magnetic field, the turbulent
magnetic field $B$ can play the dual role of both
heating the electrons
(presumably via resonant wave particle interactions) and cooling them
(via SSC radiation), since the heating and cooling rates have
different energy dependencies. Earlier, we showed that the {\it
steady-state} electron spectrum can be
approximated as a relativistic Maxwellian distribution with the
``temperature'' $\gamma_cm_ec^2$ depending on the plasma
density $n$ and the coherent length of the magnetic field, which
should be comparable to the size of the flare region $R$.
Simultaneous spectroscopic observations in the NIR and X-ray bands
(with four measured quantities: fluxes and spectral
indexes in both bands) can then be used to determine the four basic
model parameters, namely $B$, $R$, $\gamma_c m_ec^2$ and
$n$, and to test the model (Liu et al. 2006a, 2006b).

These investigations, however, also indicated that the steady-state
solutions are not always applicable, especially for NIR
flares with very soft spectra. A time-dependent approach to the
evolution of the electron distribution under the influence
of the turbulent magnetic field is required to study the flares in
quantitative details. In light of the recent
discovery of a correlation between the flux
$F_\nu\propto\nu^{\alpha-1}$ and the spectral index $\alpha$ in the
NIR band
(Ghez et al. 2005; Gillessen et al. 2006; Krabbe et al. 2006;
Hornstein et al. 2006), we carry out such an investigation.
We show that relativistic Maxwellian distributions in general give a
good description of the electron spectra with the
corresponding ``temperature'' evolution determined by the initial
temperature, the magnetic field and its coherent length and
the gas density. It is therefore possible to couple these results with
MHD simulations to study the radiative characteristics
of collisionless plasmas self-consistently.

Several scenarios have been suggested to explain the observed
correlation between flux and spectral index (Gillessen et al.
2006). We find here that the dynamical processes of plasma heating and
cooling are the most likely mechanism and the
observation requires that the magnetic field be anti-correlated with
the electron temperature. Since the evolution of the
magnetic field depends on the turbulence generation mechanism, which
needs to be addressed via MHD simulations, we calculate
the evolution of the electron distribution function and the
corresponding radiation spectrum for a cooling and a heating
phase, where the magnetic field is set to be inversely proportional to
the mean electron energy. The model naturally recovers
the observed correlation between the NIR flux and spectral index and
predicts the X-ray emission characteristics accompanying
the NIR flares. Simultaneous flare observations in the NIR and X-ray
bands can readily test the model and uncover the
underlying physical processes producing these flares.

The kinetic equation of the electron acceleration and its
time-dependent solutions are described in \S\ \ref{acc}. In \S\
\ref{apl} the model is applied to flares in Sagittarius A* by
calculating the radiation from a flaring plasma with
prescribed magnetic field evolutions. The implication and limitation
of these results are discussed in \S\ \ref{dis}.

\section{Time Dependent Solutions of Electron Acceleration by Plasma Waves}
\label{acc}

The theory of electron acceleration by a turbulent magnetic field in a
hot collisionless plasma has been discussed
by Liu et al. (2006b). Regardless the details of resonant
wave-particle coupling, it was shown that the evolution of the
electron distribution function $N(\gamma,t)$ can be described by the
following kinetic equation
\begin{equation}
   {\partial N \over \partial t}
=  {\partial \over \partial \gamma}\left[{\gamma^2\over \tau_{\rm
ac}}{\partial N \over \partial
\gamma} + \left({\gamma^2\over \tau_0}-{2\gamma\over \tau_{\rm
ac}}\right)N\right]\,,
\label{kinetic}
\end{equation}
where the synchrotron cooling and acceleration times are given, respectively, by
\begin{eqnarray}
\tau_{\rm syn}(\gamma) &=&\tau_0/\gamma \equiv {9m_e^3c^5/4 e^4B^2\gamma}\,, \\
\tau_{\rm ac} &\equiv& 2\gamma^2/<\Delta\gamma\Delta\gamma/\Delta t> =
C_1{3R c/v_{\rm A}^{2}}
= {12\pi C_1 n m_p cR/ B^2}\,,
\label{tac}
\end{eqnarray}
the Alfv\'{e}n velocity $v_{\rm A} = B/(4\pi n m_p)^{1/2}$, $C_1$ is a
dimensionless quantity
of order 1, and we have assumed that there is no particle escape and
therefore no return current in association with the
electron acceleration \footnote{These processes are not expected to
affect the results significantly as far as the escape time
scale is much longer than $\tau_{\rm ac}$ (Liu et al. 2006a).}. The
size of the flaring region, the magnetic field and gas
density are given respectively by $R$, $B$ and $n$. The scattering
mean free path of the electrons, $C_1 R$, is comparable to
the source size. $m_e$, $m_p$, $c$, $e$, and $\gamma$ are the electron
mass, proton mass, speed of light, elemental charge
unit and the Lorentz factor of the electron, respectively.

To demonstrate the behavior of the time-dependent solutions, we
consider the simplest case, where $\tau_{\rm ac}$ and
$\tau_0$ remain constant in time, and normalize the distribution
function. Then the steady state is $N(\gamma)= (\gamma^2/
2\gamma_c^3)\exp(-\gamma/\gamma_c)\,, \, {\rm with} \ \gamma_c =
\tau_0/\tau_{\rm ac}=3m_e^3c^4/16\pi C_1 e^4m_p\ n\ R$. Here
the integral over $\gamma$ has been extended from 1 to 0 to simplify
the expression. We note that $\gamma_c$ is independent
of $B$ because both the synchrotron cooling and acceleration rates are
proportional to $B^2$, while $R$ and $n$ control
$\gamma_c$ by affecting the turbulence heat rate via modification of
the coherent length and Alfv\'{e}n velocity $v_{\rm A}$,
respectively. Once one specifies the initial electron distribution
with $\tau_{\rm ac}$ as the time unit, the solution only
depends on $\tau_0$ or the steady-state temperature $\gamma_c$.

In what follows, the energy unit is assumed to be $m_ec^2$. Figure
\ref{fig1.ps} shows the evolution of $N(\gamma, t)$
starting with a relativistic Maxwellian distribution with an initial
temperature $\gamma_i$ until $t=4\tau_{\rm ac}$. The
left panel has $\gamma_i=10$ and $\gamma_c = 200$, which corresponds
to a heating phase of the plasma. We note that the
particle distributions are already close to the steady-state solution
after the time step $t=\tau_{\rm ac}$. The right panel
has $\gamma_i=200$ and $\gamma_c=10$ corresponding to a cooling phase.
Because the synchrotron cooling (with a timescale
$\gamma_c\tau_{\rm ac}/\gamma$) dominates above $\gamma_c$, the
particle distribution evolves faster than in the heating
phase (in unit of $\tau_{\rm ac}$). We also note the pileup of
electrons to a nearly monotonic distribution in the early
section
of this cooling phase, due
to more rapid cooling of higher energy electrons. Later the
distribution is broadened by the diffusion term in the kinetic
equation (the first term on the right hand side of eq. [\ref{kinetic}]).

Figure \ref{fig2.ps} shows the correlations between $N(\gamma, t)$ and
the electron spectral index $\beta\equiv \partial
\ln{N(\gamma, t)}/\partial \ln{\gamma}$ at $\gamma=50$ (left) and
$100$ (right) for the two runs in Figure \ref{fig1.ps}.
Here the solid and dashed lines are for the heating and cooling
phases, respectively (one may consider the time $t$ as a
parametric variable). The rising of $\beta$ to above 2 is due to the
pileup of electrons in the early cooling phase. The
correlations are quite different for the two phases and energies. For
a given magnetic field, these correlations mimic
similar correlations in the synchrotron radiation, which we
investigate in \S\ \ref{apl}.

%that the observed correlation between $F_\nu$ and $\alpha$ in the NIR
band  In principle the two phases can be
%distinguished by observations of these correlations.

To characterize the evolution of $N(\gamma, t)$ in general, we next
consider the evolution of the mean energy $<\gamma>(t) =
\int \gamma N(\gamma, t) \d \gamma/\int N(\gamma, t)\d \gamma$.
Because the heating of electrons by turbulence depends on the
derivative of $N(\gamma, t)$ with respect to $\gamma$ via the
diffusion term, the evolution of $<\gamma>$ can be very complicated.
Fortunately for any smooth function $N(\gamma)$
one may approximate this derivative term with a heating term, i.e. the
total energy change rate
\begin{equation}
\dot{<\gamma>} \simeq -{<\gamma>^2\over \tau_{\rm
ac}\gamma_c}+{(2+a)<\gamma>\over \tau_{\rm ac}}\,,
\label{heq}
\end{equation}
where a ``\ $\dot{}$\ '' indicates a derivative with respect to time
and $a$ is a parameter to be determined by the steady
state solution. Then we have
\begin{equation}
<\gamma>_t={(2+a)\gamma_c\gamma_i\over
\gamma_i-(\gamma_i-\gamma_c)\exp[-(2+a)t/\tau_{\rm ac}]}\,.
\label{mgamt}
\end{equation}
In the steady state, $<\gamma>_t=3\gamma_c$, therefore $a=1$.

However the above approximation is not good enough for the intermediate
steps of the heating phase when the particles have a broad distribution
(Fig. \ref{fig1.ps}) and the heating by the diffusion term is more
efficient. To fit the numerical results we set the ``$a$'' in the
numerator of equation (\ref{mgamt}) equal to one to be consistent with the
steady state solution and leave the ``$a$'' in the denominator as a free
parameter:
\begin{equation}
<\gamma>_t={3\gamma_c\gamma_i\over
\gamma_i-(\gamma_i-\gamma_c)\exp[-(2+a)t/\tau_{\rm ac}]}\,.
\label{mgamtf}
\end{equation}
We find that $a=1.6$ and $1.0$ give good fits to the $<\gamma>$ evolutions
of the heating and cooling phases, respectively.
Figure \ref{fig3.ps} shows the evolution of $<\gamma>$ and the
corresponding fits for the heating
(left) and cooling (right) phases. Figure \ref{fig4.ps} gives the
ratio of $<\gamma>$ to the fit values
for several initial and final temperatures. The relative error is
within $20\%$ for the heating phase
and within $10\%$ for the cooling. The error also increases with the
increase of the dynamical range,
which is the ratio of the initial temperature to steady state
temperature for the cooling phase and vice versa
for the heating phase.
%The discontinuties at $t=0.01\tau_{\rm ac}$ for the cooling phase are
due to numerical
%error\footnote{We have two runs for each curve, one from 0 to
$0.01\tau_{\rm ac}$ and another from 0 to $4\tau_{\rm ac}$}.
Equation (\ref{mgamtf}) therefore gives a good description of the
evolution of $<\gamma>$ under the
influence of a turbulent magnetic field.

We note that equation (\ref{mgamtf}) can be generalized to address the
heating and cooling of collisionless plasma by
turbulent plasma waves. The heating time $\tau_{\rm ac}$ (see
eq.[\ref{tac}]) usually is a function of time.  Taking into
account the effects discussed in the previous paragraph, equation
(\ref{mgamt}) leads to
\begin{equation}
<\gamma>_t={3\gamma_c\gamma_i\over
\gamma_i-(\gamma_i-\gamma_c)\exp[-(2+a)\int\d t/\tau_{\rm ac}(t)]}\,.
\label{mgamtfg}
\end{equation}
MHD simulations can give the time evolution of $B$, its coherent
length, and $n$. Equation (\ref{mgamtfg}) can then be
used in these simulations to address the heating and cooling of
electrons. In \S\ \ref{apl} we study the
SSC emission of these electrons during flares in Sagittarius A*.

\section{Synchrotron Emission and SSC}
\label{apl}

We consider the case where the particle distribution is isotropic with
respect to the turbulent magnetic field.
Then the synchrotron flux density and emission coefficient at
frequency $\nu$ are
given, respectively, by (Pacholczyk 1970)
\begin{equation}
F_{\nu}(\nu)={4\pi R^3\over 3 D^2}{\cal E}_\nu\,,\ \ \ \
{\cal E}_\nu(\nu)={\sqrt{3} e^3\over 4\pi m_e c^2} B\,n\, \int_0^\infty\d \gamma
\int_0^1 \d \mu (1-\mu^2)^{1/2} N(\gamma) F(x)
\,,
\end{equation}
where
\begin{eqnarray}
x&=&{\nu\over \nu_c}\equiv {4\pi m_e c\ \nu\over 3eB(1-\mu^2)^{1/2}\gamma^2}\,,
\\
F(x) &=& x\int_x^\infty K_{5/3}(z)\d z \simeq {2.1495
x^{1/3}e^{-2x}}+1.348 x^{1/2}e^{-x-0.249x^{-1.63}}\,,
\end{eqnarray}
$D$, $\mu$, and $K_{5/3}$ are the distance to the Galactic Center, the
cosine of the angle between the magnetic field and line
of sight, and the corresponding Bessel function, respectively, and the
approximation for $F(x)$ is accurate within $8\%$. We
therefore have the spectral index in a given narrow frequency range
$\alpha \equiv {{\rm d}\ln(\nu F_\nu)/ {\rm d}\ln\nu}$.

As shown by Liu et al. (2006b), most of the synchrotron radiation is
emitted in the optically thin region for flares in
Sagittarius A*. For a uniform spherically symmetric source, the
self-Comptonization flux density is then given by (Blumenthal
\& Gould 1970)
\begin{eqnarray}
F_X(\nu) \simeq {\pi e^4\nu n R\over 6 m_e^2 c^4}\int_0^\infty
\d\gamma
{N(\gamma)\over \gamma^4}
\int_{\nu\over 4\gamma^2}^\infty
\d \nu^\prime
{F_\nu(\nu^\prime)\over \nu^{\prime3}}
\left(2\nu\ln{\nu\over
4\gamma^2\nu^\prime}+\nu+4\gamma^2\nu^\prime-{\nu^2\over
2\gamma^2\nu^\prime}\right)
%{\nu\over 4\nu_c\gamma_c^2}\int_{\gamma_c^{-1}}^\infty {\rm d}
%x\int_0^1{\rm d}y\;
%\exp(-x) I\left({\nu\over
%4\nu_c\gamma_c^2}{1\over x^2 y}\right)\left(2\ln{y}+1-2y+{1\over
%y}\right)\nonumber\\
%= 2.121\ n_7^2\ B_1\ R_{12}^4\ D_8^{-2}
%G({\nu/4\nu_c\gamma_c^2})
%\mu{\rm Jy}
\,,
\end{eqnarray}
Similarly one can define the X-ray spectral index $\alpha_X \equiv
{{\rm d}\ln(\nu F_X)/ {\rm d}\ln\nu}\,.$

\subsection{Flux and Spectral Index Correlations with a Constant Magnetic Field}
\label{conb}

To demonstrate how the heating and cooling processes affect the
evolution of the radiation spectrum, we first consider the
cases where the magnetic field $B$ and the total number of electrons
$4\pi nR^3/3$ remain constant. Figure \ref{fig5.ps}
shows the evolution of the normalized synchrotron flux density spectrum
\begin{equation}
\epsilon_\nu(\nu, t) \equiv {F_\nu(\nu, t) \sqrt{3} m_e c^2 D^2\over
e^3 B n R^3}=\int_0^\infty\d \gamma
\int_0^1 \d \mu (1-\mu^2)^{1/2} N(\gamma, t) F(x)
\label{epss}
\end{equation}
for the two runs in Figure \ref{fig1.ps} with $B=100$ G. Compared to
the evolution of $N(\gamma)$, the evolution of
$\epsilon_\nu(\nu)$ is less dramatic because the synchrotron spectrum
is dominated by emission from the more energetic
electrons, and the radiation spectra are very similar to thermal
synchrotron spectra. However, the difference between the
heating and cooling phases is obvious.

Figure \ref{fig6.ps} shows the correlations between $\alpha$ and
$\epsilon_\nu$ at $\nu=1.4\times 10^{14}$ Hz (left) and
$1.4\times 10^{13}$ Hz (right). The dotted lines indicate the observed
correlations with different background
subtraction methods (Gillessen et al. 2006). The cooling phase fits
the observations marginally, while the heating phase
predicts a correlation much weaker than is observed. We note that for
constant $B$ and $nR^3$ the observed flux
density is proportional to $\epsilon_\nu$. The correlation between
$\epsilon_\nu$ and $\alpha$ can be compared with
observations directly by adjusting the normalization factor $BnR^3$.
We also note that for given $N(\gamma, t)$ the
correlation between $\epsilon_\nu$ and $\alpha$ only depends on
$\nu/B$. Therefore for the same runs the correlations at
$1.4\times10^{14}$ Hz with $B=1000$ G will be the same as those shown
in the right panel.

In \S\ \ref{acc} we showed that the evolution of $N(\gamma, t)$ is
very similar in the heating and cooling phases for
different initial and final temperatures, and that adjusting these
temperatures changes the starting and ending points of the
correlation in the $\epsilon_\nu$-$\alpha$ plane. The shape of the
correlation, however, does not change dramatically. We
therefore conclude that for a constant (and uniform) magnetic field
the observed correlation between flux and spectral index
in the NIR band can marginally be attributed to the cooling of a
plasma heated up instantaneously at the onset of the
flare.

Adiabatic expansion or compression and Doppler boosting have also been
suggested to explain the correlations between spectral
index and flux during flares (Yusef-Zadeh et al. 2006b; Gillessen et
al. 2006). To produce NIR flares with very soft spectra
the electrons producing the NIR emission need to have a sharp high
energy cutoff. As shown by Liu et al. (2006a, 2006b) and
discussed in \S\ \ref{acc}, a relativistic Maxwellian distribution
gives the most natural approximation to the electron
spectrum. For a temperature of $\gamma_0$ (Liu et al. 2006b) we have,
\begin{equation}
\epsilon_\nu(\nu) = 0.5 x_M I(x_M)\,,
\end{equation}
where
\begin{eqnarray}
I(x_M)&=& 4.0505x_M^{-1/6}(1+0.40x_M^{-1/4} +
0.5316x_M^{-1/2})\exp(-1.8899\,x_M^{1/3})\,,
\label{Im}
\\
x_M &=& {\nu/\nu_0}
\equiv {4\pi m_e c\nu/3 e B \gamma_0^2}
%= 1412\ C_1^2\ \nu_{14} R_{12}^2\ n_7^2\ B_1^{-1}
\,,
\label{xm}
\end{eqnarray}
and
\begin{equation}
\alpha = 1.833 - 0.6300
x_M^{1/3}-{0.1000x_M^{1/4}+0.2658\over
x_M^{1/2}+0.4000x_M^{1/4}+0.5316}\,.
\label{alpha}
\end{equation}

The left panel of Figure \ref{fig7.ps} shows the correlation between
$\epsilon_\nu$ and $\alpha$. As mentioned above, when
the magnetic field is constant, this can be compared with observations
directly. Such a correlation can be produced when a
plasma is expanding or compressed adiabatically in a uniform magnetic
field so that the temperature $\gamma_0$ changes.
The theoretical prediction is clearly inconsistent with the observed
results. A very unusual electron spectrum is needed to
make the model of adiabatic expansion in a uniform magnetic field in
line with observations.

Because ${\cal E}_\nu/\nu^2$ is a Lorentz invariant, $F_\nu/\nu^2$ is
approximately a constant under the Lorentz transform.
The correlation between $F_\nu$ and $\alpha$ due to Doppler boosting
effects is therefore given by the correlation between
$\epsilon_\nu/x_M^2$ and $\alpha$, which is much flatter than the
correlation between $\epsilon_\nu$ and $\alpha$  and
therefore can not explain the observations.

We conclude that in a constant and uniform magnetic field only SSC
cooling of a hot plasma heated instantaneously is
marginally in agreement with the observed spectral index and flux
correlation in the NIR band. Turbulent heating, adiabatic
processes and Doppler effects produce a correlation which is much
flatter than the observed results.

\subsection{Flux and Spectral Index Correlations with a Variable Magnetic Field}
\label{vmag}

To improve the model fitting to the observed correlations, we next
consider the scenario where the magnetic field is
variable. The correlation between flux and spectral index is depicted
by the correlation between $B\epsilon_\nu$ and
$\alpha$. For a relativistic Maxwellian distribution with a
temperature $\gamma_0$, if $B\gamma_0$ is a constant in time,
$B\epsilon_\nu$ is proportional to $x_M\epsilon_\nu$. As shown in the
right panel of Figure \ref{fig7.ps}, the correlation of
the solid line gives a better fit to the observations than all models
discussed above. We note, however, that for $\alpha$
increasing from -4 to 1, the temperature of the plasma has to increase
by about three orders of magnitude (Liu et al. 2006b).

In the more general cases where $B\gamma_0^p$ (with $p$ as a constant)
does not change with time, $B\epsilon_\nu$ is
proportional to $x_M^{p/(2-p)}\epsilon_\nu$. For $p=0$, we recover the
result with a constant magnetic field. It is clear
that $p$ has to be positive and less than 2 to make the flux and
spectral index correlation steeper, implying an
anti-correlation between the magnetic field and the mean electron
energy during the flare evolution. The dashed line in
Figure \ref{fig7.ps} (right) corresponds to $p=4/3$, which fits the
correlation in the soft- and dim- state but predicts a
spectrum, which is softer than the observed ones when the flux is
high. On the other hand, the recent re-analyses of the Keck
observations of flares from Sagittarius A* appear to favor such a
scenario (Krabbe et al. 2006).

To take into account the heating and cooling effects, we choose a
magnetic field inversely proportional to $<\gamma>$ during
the evolution of the flare, which makes the heating and cooling rates
depend on the electron distribution. Compared with the
heating and cooling processes studied in \S\ \ref{acc}, one more
parameter is needed to obtain the solutions: the initial magnetic
field. This parameter also sets the characteristic timescale of the
system, which is related to the
initial value of $\tau_{\rm ac}$. Using an explicit number scheme with
the time steps scaled as $\tau_{\rm ac}(t)$, we
calculate the evolution of the synchrotron spectrum for a cooling and
heating phase.

The dashed line in Figure \ref{fig8.ps} (left) shows the correlation
between $\epsilon_\nu B$ and $\alpha$ for the cooling
phase at $\nu = 1.4\times 10^{14}$ Hz, which is consistent with the
observations. The lower magnetic field, the initial and
final temperatures, are $4$ Gauss, 2000 and 20, respectively. Then we
have $n R =2.1\times10^{19} C_1^{-1}$ cm$^{-2}$ and
$\tau_{\rm ac} = 6.5 \times 10^{3} (B/10\ {\rm G})^{-2}$ mins. Here a
relatively large temperature change is chosen to produce
significant variation of the synchrotron spectral index. The dashed
line in the right panel gives the evolution of the
magnetic field. Although the initial magnetic field is relatively low,
the steady state magnetic field is $400$ Gauss, which
is much higher than the typical value for the quiescent state.

The dashed lines in Figure \ref{fig9.ps} show the light curves of the
flux (left) and spectral index (right) for the cooling phase. We note
that
the correlation between the flux and spectral index mostly occurs
between 130 to 160 minutes. Before 130 minutes, the system
evolves slowly due to the relatively lower magnetic field ($< 20$
Gauss). Because the magnetic field in the accretion torus
in Sagittarius A* is about a few tens of Gauss in the quiescent state,
the evolution before 100 minutes is likely irrelevant
to flares in Sagittarius A* since the magnetic field is below 10 Gauss
during this period. From 100 to 170 minutes the model
predicted correlation is quite in line with observations. The model
not only produces the correct correlation slope, but also
recovers the typical variation timescale of a few tens of minutes.
After 170 minutes, the system reaches a steady state.
The dips near $160$ mins in the flux and spectral index light curves
are related to the turn around of the spectral index
and flux correlation before reaching the steady-state. These are
caused by the sharp cutoff (sharper than an exponential
cutoff) of the electron spectra in the
cooling phase, which makes the spectra softer than that of a
relativistic Maxwellian distribution.
To produce a flare state with $\alpha = -0.5$ and $F_\nu = 5$ mJy, the
total number of electrons
involved in the flare $4\pi n R^3/3 = 2.6 \times 10^{41}$, giving rise
to $R = 5.5 \times 10^{10} C_1^{1/2}$ cm and $n=3.7
\times 10^{8} C_1^{-3/2}$ cm$^{-3}$. These values agree with results
of previous studies (Liu et al. 2006a, 2006b).

The heating phase fits the observed correlation marginally (the solid
line in the left panel of Figure \ref{fig8.ps}). The
initial magnetic field, the initial and final temperatures are $200$
Gauss, 10 and 1000, respectively, giving rise to $n R
=4.1\times10^{17} C_1^{-1}$ cm$^{-2}$ and $\tau_{\rm ac} = 1.3 \times
10^{2} (B/10\ {\rm G})^{-2}$ mins. The solid lines in
the right panel of Figure \ref{fig8.ps} and in Figure \ref{fig9.ps}
depict the evolution of the magnetic field and the flux
and spectral index, respectively. The heating phase proceeds much
faster than the cooling phase due to the relatively high
initial magnetic field and final temperature. The flux reaches the
peak value within $\sim 6$ minutes, after which the
spectral index saturates near 0.2, and both the magnetic field and
flux decrease slowly. To produce even harder spectra, one
has to increase the value of $B \gamma_c^2$ at the steady state
dramatically (Liu et al. 2006b), making the heating phase
proceed on an even shorter timescale ($\propto 1/B^2\gamma_c$). A very
weak initial magnetic field ($<1$ Gauss) and a
very high temperature ($\gg1000$) are required to produce hard NIR
spectra and a flux rising time of a few minutes.
To produce a flare state with $\alpha = -0.5$ and $F_\nu = 5$ mJy, the
total number of electrons involved in the flare $4\pi
n R^3/3 = 4.7 \times 10^{42}$, which is more than ten times larger
than that in the cooling phase.  This leads to $R = 1.6
\times 10^{12} C_1^{1/2}$ cm and $n=2.5 \times 10^{5} C_1^{-3/2}$ cm$^{-3}$.

In general the spectral index $\alpha$ of the above models is always
less than $4/3$. To produce even harder spectra, one
has to introduce self-absorption effects. For an electron temperature
$\gamma_0\simeq 200$, the size of the emission
region is then given by $$R\simeq (F_\nu /2\pi \gamma_0 m_e)^{1/2}
D/\nu = 3.7\times 10^7 (D/8 {\rm kpc})(F_\nu/5\ {\rm
mJy})^{1/2}(\gamma_0/200)^{-1/2} (\nu/1.4\times 10^{14}{\rm Hz})^{-1}
{\rm cm}\,,$$ which is much smaller than any relevant
length scales. We therefore do not expect NIR flares with spectral
indexes larger than $4/3$. Any solid detection of very
hard NIR spectra will rule out this model and may be in conflict with
any synchrotron models.

\subsection{Flux and Spectral Index Correlation in the X-ray Band}

The spectrum of the Comptonization component and its evolution should
be quite different for the two phases, and may be used
to distinguish between them. Figure \ref{fig10.ps} shows the evolution
of the radiation spectrum during the cooling (left) and
heating (right) phases. Besides indicating the initial and
steady-state spectra, we highlight the spectral evolution when the
correlation between NIR flux and spectral index are prominent,
corresponding to the shaded periods in Figure \ref{fig9.ps} and Figure
\ref{fig11.ps},
where the spectra are indicated by filled circles. For the model
parameters chosen in \S \ref{vmag}, because the electron
column depth $nR$ in the cooling phase is 50 times that in the heating
phase while their synchrotron luminosities are
comparable, much more high energy emission is produced via SSC in the
cooling phase.

In analogy to equation (\ref{epss}), one can define a dimensionless
SSC flux density spectrum
\begin{equation}
\epsilon_X(\nu)\equiv {6\sqrt{3}m_e^3 c^6 D^2F_X(\nu)\over \pi e^7 n^2 R^4 B}
 \simeq {\nu}\int_0^\infty
\d\gamma {N(\gamma)\over \gamma^4}
\int_{\nu\over 4\gamma^2}^\infty
\d \nu^\prime
{\epsilon_\nu(\nu^\prime)\over \nu^{\prime3}}
\left(2\nu\ln{\nu\over
4\gamma^2\nu^\prime}+\nu+4\gamma^2\nu^\prime-{\nu^2\over
2\gamma^2\nu^\prime}\right)\,.
\end{equation}
The left panel of Figure \ref{fig11.ps} shows the evolution of
$B\epsilon_X$ at $0.5\times10^{18}$ (thin lines) and
$2\times10^{18}$ Hz (thick lines) for the cooling (dashed lines) and
heating (solid lines) phases. As expected, higher
energy emission precedes (lags) lower energy emission slightly during
the cooling (heating) phase because they are produced
by more energetic electrons. However the difference in the peak times
is within 10 minutes and may not be distinguished with
the capacities of current instruments.  The time delay between the NIR
and X-ray peaks is also within 30 minutes. We
therefore expect a good correlation between the NIR and X-ray flux
densities. The right panel shows the corresponding
correlation between $\epsilon_X$ and the X-ray spectral index
$\alpha_X$. This correlation gives a correlation between
the X-ray flux density and spectral index, which can be tested with
observations.
Interestingly the X-ray flux density peaks near $\alpha_x\simeq
0.6-0.7$, which explains the hardness of X-ray spectra.

\subsection{Implications on the Source Structure}

The above studies clearly show that if the plasma producing the flares
does not have complicated structure, the observed
correlation between the spectral index and flux in the NIR band can
only be explained by the cooling model with a magnetic
field inversely proportional to the mean energy of the electrons.
This is especially true if  the total number of electrons
involved does not change and the electron energy distribution is
roughly uniform throughout the flare region. Other models
either fit the observed correlation marginally or predict very weak
dependence of the spectral index on the flux in the hard
spectral states, which is inconsistent with observations.

On the other hand, the flare region may have complicated source
structure. The amount of electrons involved in producing the
soft state emission can be quite different from that producing the
hard state emission. If the electron distribution can be
approximated as a Maxwellian spectrum, as is likely true, the observed
correlation can then be used to determine the source
structure. As shown in section \S\ \ref{conb}, $\alpha$ only depends
on $B\gamma_0^2$, and $F_\nu$ also depends on $B n
R^3$. For a given magnetic field, the observed correlation can be used
to give a relation between the temperature $\gamma_0$
and the total number of electrons at such a temperature, which may
shed light on the instabilities triggering the flares.
One may also assume that the sum of the electron and magnetic field
pressures is a constant in the flaring region. The
observed correlation can also lead to a measurement of the amount of
plasma at different temperatures.

It is probably more natural to use MHD simulations to produce some
flaring regions. The formalism developed in this paper can
then be used to calculate the radiation spectrum. By comparing with
the observed correlation, one can then determine what
kind of active regions are likely responsible for the flares. It is
clear that the SSC spectra are quite different for all
these scenarios and simultaneous spectroscopic observations in the NIR
and X-ray bands can be used to distinguish these
models. Current observations do not justify such a comprehensive
investigation, which is also beyond the scope of the paper.

\section{Conclusion and Discussion}
\label{dis}

Flares from the direction of the supermassive black hole in the
Galactic center are produced within a few Schwarzschild
radii. Studying the details of the flare properties plays an important
role in understanding the physics of black hole
accretion. To infer the underlying physical processes with
simultaneous spectroscopic observations over a broad frequency
range, the plasma process of electron acceleration has to be
addressed. Given the closeness of the flare variation timescale,
the dynamical time at the last stable orbit of the black hole, and the
synchrotron cooling time of electrons producing NIR
emission during the flares, a time dependent study of the particle
acceleration is necessary. Although there are already
attempts to carry out this study analytically (Becker et al. 2006;
Schlickeiser 1984), numerical solutions are required
when specific events are considered.

In this paper we study the time dependent solutions of electron
heating and cooling by a turbulent magnetic field. When the
amplitudes of large scale magnetic fields are much smaller than the
turbulent component, there are four basic model
parameters, namely the magnetic field and its coherent length, the gas
density and the initial distribution of electrons.
Electrons gain energy from the plasma waves and loss energy via
radiation, and the energy dependences
of the heating and cooling rates are quite different. The evolution of
the electron distribution is determined by these
parameters.
We show that for the optically thin collisionless plasma in
Sagittarius A*, synchrotron cooling dominates and
relativistic Maxwellian distributions usually give a good description
of the electron spectra. A formula is given to describe
the dependence of the temperature evolution on the model parameters,
which can be coupled with MHD simulations to study the
radiation properties of the plasma.

We demonstrate that it is not trivial to produce the observed
correlation between the spectral index and flux in the NIR
band. Doppler effects and adiabatic processes suggested previously for
the correlation (Gillessen et al. 2006; Yusef-Zadeh et
al. 2006b) can not reproduce the observed results unless one has a
very unusual electron spectrum. If the magnetic
field does not change dramatically during the flare, only the cooling
model of electrons heated instantaneously at the flare
onset fits the observed correlation marginally. All other models
predict very weak dependence of the spectral index on the
flux density in the relatively hard spectral states.

We argue that the dynamical processes of electrons heating and cooling
give the most reasonable explanation for the
observations. To fit the correlation, the magnetic field needs to be
anti-correlated with the electron temperature to reduce
the flux density in the hard spectral states, bringing the above
theoretical predictions more in line with the observations.
It is not clear what mechanism may be responsible for such an
anti-correlation. One possibility is that the dependence of the
heating rate on the magnetic field is not as strong as that of the
cooling rate. For example, the heating rate may be
determined by the sound velocity, which can be much higher than the
Alfv\'{e}n velocity. Moreover, the cooling due to inverse
Comptonization will dominate when the electron temperature and column
depth are very high, which is certainly the case in
the early cooling phase\footnote{We note that the X-ray luminosity in
the early cooling phase is higher than the synchrotron
luminosity, suggesting a Compton catastrophe. This is due to the
choice of the model parameters to embrace a large dynamical
range. Including the cooling effect due to inverse Comptonization will
make the electron temperature decrease rapidly to a
value, where the Comptonization luminosity becomes lower than the
synchrotron luminosity.}. All these effects can give rise
to the desired anti-correlation.

The required anti-correlation between the magnetic field intensity and
the electron mean energy suggests
efficient energy exchanges between the field and electrons. While
magnetic reconnection can cause transfer of magnetic field
energy into electrons (corresponding to the heating phase), it is not
clear what mechanism are responsible for the increase of
magnetic field in the cooling phase. If the flare region is in
pressure equilibrium with a relatively stable background, the
magnetic field pressure may increase as the electron pressure
decreases due to cooling.

We note that the magnetic field pressure increases from 1.27 to $1.27
\times 10^4$ ergs cm$^{-3}$ and the electron
pressure decreases from $6.1\times 10^5$ to $6.1\times 10^3$ ergs
cm$^{-3}$ in the cooling phase of the flare discussed in
\S\ \ref{vmag}. For the heating phase, the magnetic field pressure
decreases from $3.2\times 10^3$ to 0.32 ergs cm$^{-3}$,
while the electron pressure increases from 2.1 to $2.1\times 10^2$
ergs cm$^{-3}$. Although the pressures are far from
equipartition in the initial and final states, the observed
correlation is produced when the two are evolving toward an
equipartition. These suggest that the flares are triggered by a
process, which drives the energy densities of the magnetic
field and electrons far from equipartition. The relaxation of the
system toward energy equipartition produces the observed
correlation. It is therefore critical to produce and/or identify
features far from energy equipartition in MHD simulations to
uncover the physical processes driving the flares.

It is also possible that the flare region has complicated structure.
The results of the electron spectral evolution
presented here, which are also applicable to the relatively
quiescent-state of the accretion flow,
then need to be coupled with MHD simulations to study the radiation
characteristics of the accretion flow.
The observed correlation between spectral index and flux suggests
that, if the magnetic field does not change dramatically
during the flare and through the flare region, more electrons are
involved in producing the softer spectral
state flux. Regions with small hot cores flashing in a extended lower
temperature active envelope in MHD simulations may be
identified with the flares.

\acknowledgments

This work was carried out under the auspices of the
National Nuclear Security Administration of the
U.S. Department of Energy at Los Alamos National
Laboratory under Contract No. DE-AC52-06NA25396.
This research was partially supported by NSF grant ATM-0312344, NASA
grants NAG5-12111, NAG5 11918-1 (at Stanford),
%, NSF grant AST-0402502 (at Arizona),
and NSF grant PHY99-07949 (at KITP at UCSB).
SL would like to thank Justin R. Pelzer for developing part of the code.
%FM is very grateful to the University of Melbourne for its support
(through a Miegunyah Fellowship).

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\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f1a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f1b.eps}
\end{center}
\caption{
{\it Left:} Evolution of the electron distribution function
$N(\gamma)$. The initial and final temperatures (in unit of
$m_ec^2$) are $10$ and $200$, respectively. The time steps are
$4(i/14)^2\tau_{\rm ac}$, where $i$ is an integer ranging from
0 to 14. The arrow indicates the direction of the evolution. The thick
line is for $i=7$ corresponding to the time step
$\tau_{\rm ac}$. We note that the electron distribution is already
very similar to the steady-state spectrum after $\tau_{\rm
ac}$.
{\it Right:} Same as the left panel except the initial and final
temperatures exchanged. Note the pileup of electrons during
the first few time steps due to the SSC cooling.}
\label{fig1.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f2a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f2b.eps}
\end{center}
\caption{
{\it Left:} Correlation between $N$ and $\beta=\d
\ln{N}/\d\ln{\gamma}$ at $\gamma=50$. The solid line is for the
heating
phase. The dashed line is for the cooling phase. The rising of the
spectral index to above 2 is caused by the pileup of
electrons in the early cooling phase. The correlation is quite
different for the two phases.
{\it Right:} Same as the left panel but for $\gamma=100$.
}
\label{fig2.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f3a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f3b.eps}
\end{center}
\caption{
{\it Left:} Evolution of the mean energy of the electrons in the
heating phase. The dashed line indicates the theoretical
result given by equation (\ref{mgamtf}) with $a=1.6$.
{\it Right:} Same as the left panel but for the cooling phase with $a=1.0$.
}
\label{fig3.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f4a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f4b.eps}
\end{center}
\caption{
{\it Left:} The ratio of the mean energy of the electrons to the
theoretical value as a function of time for several initial
and final temperatures in the heating phase. The solid lines have an
initial temperature of $10$ and final temperatures of
200, 500, 1000. The dashed lines have a final temperature of 200 and
initial temperatures of 20 50, and 100. The relative
error (the difference between the numerical and theoretical results
divided by the theoretical values) increases with the
increase of the ratio of the final to initial temperature (the
dynamical range).
{\it Right:} Same as the left panel but for the cooling phase. The
line with the greatest relative error corresponds to a
cooling from  1000 to 10. The other lines have an initial temperature
of 200 and final temperatures of 10, 20, 50, and 100.
The relative error also increases with the dynamical range.
%The discontinuities at $0.01\ \tau_{\rm ac}$ indicate the
%numerical error. We have two runs for each cooling phase: one from 0
to $0.01\ \tau_{\rm ac}$ and another from 0 to
%$4\ \tau_{\rm ac}$.
}
\label{fig4.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f5a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f5b.eps}
\end{center}
\caption{
Evolution of the normalized synchrotron flux density spectrum
$\epsilon_\nu$ for the two runs shown in Figure
\ref{fig1.ps}. The time steps are the same as in Figure \ref{fig1.ps}
and the magnetic field $B=100$ G. The left and
right
panels correspond to the heating and cooling phases, respectively. The
evolution of $\epsilon_\nu$ is less dramatic than that
of $N(\gamma)$ due to the dominance of synchrotron emission by more
energetic electrons. The spectra are very similar to
thermal synchrotron spectra.
}
\label{fig5.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f6a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f6b.eps}
\end{center}
\caption{Correlation between the normalized flux density
$\epsilon_\nu$ and the spectral index $\alpha$ at
$\nu=1.4\times10^{14}$ Hz (left) and $1.4\times10^{13}$ Hz (right) for
the two runs in Figure \ref{fig5.ps}.
The solid and dashed lines are for the heating and cooling phases,
respectively. The dotted lines give the observational
results. The cooling phase correlation fits the observations
marginally, while the heating phase correlation is too flat
compared with observations. See text for details.
}
\label{fig6.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f7a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f7b.eps}
\end{center}
\caption{
{\it Left:} Correlation between $\epsilon_\nu$ and $\alpha$ for
electrons with a relativistic Maxwellian distribution. The
dotted lines indicate the observed results, which clearly lie below
the theoretical curve.
{\it Right:} Correlations between $x_M\epsilon_\nu$ (solid line),
$x_M^2\epsilon_\nu/50$ (dashed line) and $\alpha$, which
are more consistent with observations. See text for details.
}
\label{fig7.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f8a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f8b.eps}
\end{center}
\caption{
{\it Left:}
Correlation between the flux $\epsilon_\nu B$ and spectral index
$\alpha$ at $1.4\times 10^{14}$ Hz for the heating (solid
line) and cooling (dashed line) phases, where the magnetic field $B$
is chosen to be inversely proportional to the mean energy
of the electrons $<\gamma>$. The initial magnetic field, the initial
and final temperatures are $4$ Gauss, 2000 and 20 and
200 Gauss, 10 and 1000 for the cooling and heating phases,
respectively. The dotted lines indicate the observational results.
The scale of the heating phase flux is indicated on the upper axis.
{\it Right:}
Time evolution of the magnetic field for the heating (solid line;
upper scale) and cooling (dashed line; lower scale) phases.
The regions most relevant to flare (0 to 3 minutes for the heating
phase and 133 to 163 minutes for the cooling phase)
observations are shaded.
}
\label{fig8.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f9a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f9b.eps}
\end{center}
\caption{
Same as the right panel of Figure \ref{fig8.ps} but for the flux
(left) and spectral index (right) at $1.4\times 10^{14}$ Hz.
The scale of the heating
phase flux is indicated on the right axis.  Solid circles correspond
to the NIR spectra in Figure \ref{fig10.ps}. See text for details.}
\label{fig9.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f10a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f10b.eps}
\end{center}
\caption{
Evolution of the SSC spectrum during the cooling (left) and heating
(right) phases shown in Figure \ref{fig8.ps}. Besides the
initial and steady-state spectra, we show the spectral evolution for
the cooling and heating phases when the correlation is
produced. These spectra are indicated by the solid circles in Figure
\ref{fig9.ps} and Figure \ref{fig11.ps}.
%from 130 to 160 mins and from 0 to 5 mins
%respectively. The time steps are uniform between these intervals
}
\label{fig10.ps}
\end{figure}

\begin{figure}[bht]
\begin{center}
\includegraphics[height=8.4cm]{f11a.eps}
\hspace{-0.6cm}
\includegraphics[height=8.4cm]{f11b.eps}
\end{center}
\caption{
{\it Left:}
Light curve of the flux and spectral index at $0.5\times 10^{18}$ Hz
(thin lines, corresponding to $\sim 2.1$ keV) and
$2.0\times 10^{18}$ Hz (thick lines) during the cooling (dashed lines)
and heating (solid lines) phases. The scales for the
heating phase are indicated on the right and top axes.
{\it Right:} The corresponding correlation between the X-ray flux and
spectral index. The scale of the heating phase is
indicated on the top axis.  The solid circles correspond to the SSC
spectra in Figure \ref{fig10.ps}.
}
\label{fig11.ps}
\end{figure}

\end{document}