During synchrotron self-Compton emission, X-ray photons are produced by the
              scattering of the synchrotron radiation from radio to NIR on their own electron
              population. If we compare the fluxes produced by the synchrotron and SSC emissions,
              the variation of the X-ray/NIR ratio constrains the size evolution of the flaring
              source. Let us consider a spherical source of radius R with a power law
              energy distribution of relativistic electrons. Following Van der Laan (1966), the radiative transfer for the synchrotron
              radiation can be computed as (2)\begin{equation} S_\mathrm{SYN}=\int_{0}^{R}{\frac{\epsilon_\nu}{\kappa_\nu}\,\left(1-{\rm e}^{-\tau_\nu(r)}\right)\,2\pi r\,{\rm d}r} , \end{equation}


S
SYN

=

∫
0
R





ϵ
ν


κ
ν



 

(
1
−

e

−

τ
ν

(
r
)


)

 
2
πr
 
d
r

,

with κν ∝
                  B(p +
                    2)/2ν−(p +
                4)/2 the absorption coefficient, ϵν ∝
                  B(p +
                    1)/2ν−(p −
                1)/2 the emission coefficient, B the magnetic field
                (Lang 1999) and τν(r)
              the optical depth, which can be computed at each distance r from the sphere
              center as: (3)\begin{equation} \tau_\nu(r)=\int_{0}^{2\sqrt{R^2-r^2}}{\kappa_\nu {\rm d}l} . \end{equation}


τ
ν

(
r
)
=

∫
0

2





R

2


−

r

2










κ
ν

d
l

.

Assuming that we are in the optically
              thin regime (i.e., τν(r) ≪
                1), we utilize formula 3 of Marrone
                et al. (2008): SSYN ∝ B(p +
                  1)/2ν−(p −
                  1)/2R3. For synchrotron
              radiation, we have \hbox{$B \propto R^4\,\nu_\mathrm{m}^{5}\,S_\mathrm{m}^{-2}$}
B
∝

R
4

 

ν
m
5

 

S
m
-2

 with Sm the
              maximum flux density of the spectral energy distribution occurring at frequency
                νm (Marscher 1983). Finally, the synchrotron radiation can be expressed using
                p = 2α
                + 1 as (4)\begin{equation} S_\mathrm{SYN} \propto R^{4\alpha+7}\,\nu_\mathrm{m}^{5(\alpha+1)}\,S_\mathrm{m}^{-2(\alpha+1)}\,\nu^{-\alpha}\, . \end{equation}


S
SYN

∝

R

4
α
+
7


 

ν
m

5
(
α
+
1
)


 

S
m

−
2
(
α
+
1
)


 

ν

−
α


 
.

The SSC radiation of X-ray photons is
              (formula 4 of Marscher 1983):
                  (5)\begin{equation} S_\mathrm{SSC} \propto R^{-2(2\alpha+3)}\,\nu_\mathrm{m}^{-(3\alpha+5)}\,S_\mathrm{m}^{2(\alpha+2)}\,\ln \left(\frac{\nu_2}{\nu_\mathrm{m}}\right)\,\nu^{-\alpha} . \end{equation}


S
SSC

∝

R

−
2
(
2
α
+
3
)


 

ν
m

−
(
3
α
+
5
)


 

S
m

2
(
α
+
2
)


 
ln




ν
2


ν
m




 

ν

−
α


.

The natural logarithm in this equation
              could be approximated by c1
                  (ν2/νm)c2
              with c1 =
                1.8 and c2 = 0.201 (Eckart et al. 2012b). The synchrotron-to-SSC flux ratio is
                  (6)\begin{equation} \frac{S_\mathrm{SSC}}{S_\mathrm{SYN}} \propto R^{-(8\alpha+13)}\,\nu_\mathrm{m}^{-(8\alpha+10+c_2)}\,S_\mathrm{m}^{4\alpha+6}\, . \end{equation}




S
SSC


S
SYN



∝

R

−
(
8
α
+
13
)


 

ν
m

−
(
8
α
+
10
+

c
2

)


 

S
m

4
α
+
6


 
.

We therefore have three parameters that
              may vary during the flare to explain the increased ratio of X-ray and NIR flux (Fig.
                15). Considering the plasmon model, for which
              a spherical source of relativistic electrons expands and cools adiabatically, we have
                (Van der Laan 1966): νm ∝
                    R−(8α + 10)/(2α +
                  5) and Sm ∝ R−(14α +
                    10)/(2α + 5). Thus, SSSC/SSYN ∝
                  R−β with
                β ≡
                  (8α2 + (30−8
                  c2)α + 25−10
                  c2)/(2α + 5). We
              first consider the adiabatic expansion. For our observation, the ratio between the
              X-ray and the NIR flux increases during the 2014 Mar. 10 flare, implying that
                R−β must increase
              as the radius R increases. This condition is satisfied if the
              exponent β is negative and thus if the α value is lower than
                −2.5 or is between
                −2.3 and −1.25, which is inconsistent because
                α must
              be positive. The expansion case is thus likely to be rejected under the hypothesis of
              an optically thin plasmon that expands adiabatically. We can also consider the case
              where the plasmon is compressed during its motion through a bottle-neck configuration
              of the magnetic field. We can still use the equations of Van der Laan (1966), since the conservation of the magnetic flux is
              explicitly taken into account. The compression case is thus preferred, because it
              allows positive values of α for β> 5.4. Thus, for
              the SYN-SSC process, the plasmon must be adiabatically compressed with at least
                SSSC/SSYN ∝
                  R-5.4. Therefore, the observed
              increase of the X-ray-to-NIR flux ratio by a factor of 10 in 1.2 h implies a decrease
              of the radius by a factor of about 0.6. The average compression velocity is estimated
              as Vcomp =
                  ΔR/ Δt, leading to
                | Vcomp |
                  /c 0.0034
                R/Rs with
                Rs the Schwarzschild radius
                (Rs = 1.2 ×
                  1012 cm for Sgr A*, which corresponds to 0.08 au). For
              comparison, the expansion velocities computed with this model in the literature range
              between 0.0028 and 0.15c (Yusef-Zadeh et
                al. 2006a, 2009; Eckart et al. 2008), which is of the same order as the compression
              velocity computed here. Thus, the model of an adiabatic compression of a plasmon is
              the likely hypothesis to explain the variation of the ratio between X-ray and NIR
              flux, in the context of the SYN-SSC process.
Citation: Marscher 1983
Citation function: Uses; Uses
Citation: Marscher 1983
Citation function: Uses