11institutetext: LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Université Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
11email<EMAIL_ADDRESS>22institutetext: Centre for Space Research,
North-West University, Potchefstroom, 2531, South Africa 33institutetext:
Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 44institutetext:
LUTH, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules
Janssen, 92190 Meudon, France

# Magnetic reconnection plasmoid model for Sagittarius A* flares

N. Aimar 11 A. Dmytriiev 22 F. H. Vincent 11 I. El Mellah 33 T. Paumard 11 G.
Perrin 11 A. Zech 44

(Received September 09, 2022; accepted January 27, 2023)

###### Abstract

Context. Sagittarius A*, the supermassive black hole at the center of our
galaxy, exhibits episodic near-infrared flares. The recent monitoring of three
such events by the GRAVITY instrument has shown that some flares are
associated with orbital motions in the close environment of the black hole.
The GRAVITY data analysis points at super-Keplerian azimuthal velocity, while
(sub-)Keplerian velocity is expected for the hot flow surrounding the black
hole.

Aims.  We develop a semi-analytic model of Sagittarius A* flares based on an
ejected large plasmoid, inspired by recent particle-in-cell global simulations
of black hole magnetospheres. We model the infrared astrometric and
photometric signatures associated to this model.

Methods.  We consider a spherical macroscopic hot plasma region, that we call
a large plasmoid. This structure is ejected along a conical orbit in the
vicinity of the black hole. This plasmoid is assumed to be formed by
successive mergers of smaller plasmoids produced through magnetic reconnection
that we do not model. Non-thermal electrons are injected in the plasmoid. We
compute the evolution of the electron-distribution function under the
influence of synchrotron cooling. We solve the radiative transfer problem
associated to this scenario and transport the radiation along null geodesics
of the Schwarzschild spacetime. We also take into account the quiescent
radiation of the accretion flow, on top of which the flare evolves.

Results.  For the first time, we successfully account for the astrometric and
flux variations of the GRAVITY data with a flare model that incorporates an
explicit modeling of the emission mechanism. We find good agreement between
the prediction of our model and the recent data. In particular, the azimuthal
velocity of the plasmoid is set by the magnetic field line it belongs to,
which is anchored in the inner parts of the accretion flow, hence the super-
Keplerian motion. The astrometric track is also shifted with respect to the
center of mass due to the quiescent radiation, in agreement with the
difference measured with the GRAVITY data.

Conclusions. These results support the picture of magnetic reconnection in a
black hole magnetosphere as a viable model for Sagittarius A* infrared flares.

###### Key Words.:

Accretion, accretion disk - Magnetic reconnection - Black hole physics -
Relativistic processes - Radiative transfer - Radiation mechanisms: non-
thermal

##  1 Introduction

The Galactic Center hosts the compact radio source Sagittarius A* (Sgr A*)
with an estimated mass of 4.297 million solar masses at a distance of only
8.277 kpc (GRAVITY Collaboration et al., 2022). This makes the compact object
associated to Sgr A* the closest supermassive black hole (SMBH) candidate to
Earth. Sgr A* is a low-luminosity accretion flow with an accretion rate of
$(5.2-9.5)\times 10^{-9}M_{\odot}\ yr^{-1}$ and a bolometric luminosity of
$(6.8-9.2)\times 10^{35}$ erg s-1 (Bower et al., 2019; Event Horizon Telescope
Collaboration et al., 2022b) and thus is accreting at a very sub-Eddington
rate. It has been the subject of numerous observing campaigns over the past
two decades in order to test the massive black hole (MBH) paradigm (see
Gravity Collaboration et al. (2020b)) and study the physics of radiatively
inefficient accretion flows (RIAF) around SMBH.

Sgr A* shows a slow and low amplitude variability in radio (Lo et al., 1975;
Backer, 1978; Krichbaum et al., 1998; Falcke, 1999; Bower et al., 2006;
Michail et al., 2021b), in millimetre and submillimetre (Mauerhan et al.,
2005; Macquart et al., 2006; Yusef-Zadeh et al., 2006; Marrone et al., 2008;
Brinkerink et al., 2015; Wielgus et al., 2022a), but also large amplitude and
rapid variability in near infrared (NIR; Genzel et al., 2003; Ghez et al.,
2004; Hornstein et al., 2007; Hora et al., 2014) and in X-rays (Baganoff et
al., 2001; Nowak et al., 2012; Neilsen et al., 2013; Barrière et al., 2014;
Ponti et al., 2015). The flux distribution in the NIR of Sgr A* has been the
subject of numerous studies. Some claim a single state modeled by rednoise
(Witzel et al., 2018; Do et al., 2019) for the variability of Sgr A* while
others claim that there are two states for Sgr A* (Genzel et al., 2003; Dodds-
Eden et al., 2011; Gravity Collaboration et al., 2020a; Witzel et al., 2021):
a continuously low amplitude variable state called ”quiescent state” and the
”flare state” described by short and bright flux with a typical timescale of
30 minutes to 1 hour with a rate of $\sim$ 4 a day. Multi-wavelength studies
show that when an X-ray flare is observed, there is a counterpart in NIR
suggesting a common origin but the reverse is not true (Fazio et al., 2018).
Moreover, the flare can also be observed in sub-mm but with a time lag of
several minutes (Eckart et al., 2008, 2009; Dodds-Eden et al., 2009; Michail
et al., 2021a; Witzel et al., 2021) following a dimming (Wielgus et al.,
2022a; Ripperda et al., 2022).

Recently, the GRAVITY instrument (Gravity Collaboration et al., 2017;
Eisenhauer et al., 2008, 2011; Paumard et al., 2008) was able to resolve the
motion of the NIR centroid during three bright flare events, showing a
clockwise, continuous rotation at low inclination close to face-on ($i\sim
20\degree$) consistent with a region of emission located at a few
gravitational radii $r_{g}=GM/c^{2}$ from the central black hole (Gravity
Collaboration et al., 2018). These flares are thus powered very close to the
event horizon of the black hole. The exploration of a relativistic accretion
region as close to the event horizon with high-precision astrometry and
imaging techniques like GRAVITY and the Event Horizon Telescope (EHT) (Event
Horizon Telescope Collaboration et al., 2022a) promises important information
for physics and astronomy, including new tests of the MBH paradigm.

Significant efforts have been made to explain the flares of Sgr A*: rednoise
(Do et al., 2009), hot spot (Hamaus et al., 2009; Genzel et al., 2003;
Broderick & Loeb, 2006), ejected blob (Vincent et al., 2014), star-disk
interaction (Nayakshin et al., 2004) and disk instability (Tagger & Melia,
2006). The GRAVITY observations in 2018 (Gravity Collaboration et al., 2018)
support the hot spot model. However, the physical origin of such hot spots
remains an open question. Instabilities in black hole accretion disks are a
candidate, for instance the triggering of Rossby Waves Instabilities (RWI;
Tagger & Melia, 2006; Vincent et al., 2014). Alternatively, it could originate
from the dissipation of electromagnetic energy through magnetic reconnection.
This modification of the magnetic field topology results from the inversion of
the magnetic field orientation across a current sheet which eventually breaks
into magnetic islands called plasmoids (Komissarov, 2004, 2005; Komissarov &
McKinney, 2007; Loureiro et al., 2007; Sironi & Spitkovsky, 2014; Parfrey et
al., 2019; Ripperda et al., 2020; Porth et al., 2021). In the past years,
numerical simulations have repeatedly highlighted the ubiquity of magnetic
reconnection in BH magnetospheres, whatever the physical point of view
adopted: global particle-in-cell (PIC) simulations in Kerr metrics (El Mellah
et al., 2022; 2022PhRvL.129t5101C), resistive general-relativistic magneto-
hydrodynamics (GRMHD) simulations (Ripperda et al., 2020; Dexter et al.,
2020a, b) or resistive force-free simulations (Parfrey et al., 2015). PIC
simulations show that magnetic reconnection in the collisionless corona of
spinning BHs can accelerate leptons up to relativistic Lorentz factors of
$\gamma\sim 10^{3...7}$ (El Mellah et al., 2022), sufficiently high to
generate the variable IR (and X-ray) emission (Rowan et al., 2017; Werner et
al., 2018; Ball et al., 2018; Zhang et al., 2021; Scepi et al., 2022).

The GRMHD and PIC frameworks each have different limitations. GRMHD
simulations describe the evolution of the accretion flow over long time
scales, typically of the order of several 100,000 $r_{g}/c$, but they rely on
a fluid representation. Consequently, they cannot self-consistently capture
the kinetic effects which are important to constrain dissipation, particle
acceleration and subsequent non-thermal radiation. On the other hand, PIC
simulations provide an accurate description of the microphysics but at the
cost of simulations which can only span a few 100$r_{g}/c$ in time and with
limited scale separation between global scales and plasma scales.

We develop a semi-analytical model, fed by the knowledge accumulated by recent
GRMHD and GRPIC simulations. The aim is to condense into a reasonably small
set of simple parameters the complex physics of GRMHD and GRPIC models, and
thus allow to probe a large parameter space within a reasonable computing
time. We also want to remain as agnostic as possible regarding the initial
conditions of the flow. In this context, we discuss the interpretation of the
Gravity Collaboration et al. (2018) flare data paying particular attention to
the following diagnostics:

  * •

the marginally detected shift between the astrometric data and the center-of-
mass location;

  * •

the tension between the data and the hot spot model used by Gravity
Collaboration et al. (2018), which assumes a Keplerian orbit;

  * •

the physical origin of the rising and decaying phases of the flare light curve
in the context of magnetic reconnection.

The first point can be discussed in the context of a very simple hot-spot
model and is the main topic of Sect. 2. Sect. 3 is the core of our study and
focuses on the second and third points above. It presents a semi-analytical
large plasmoid model due to magnetic reconnection. It highlights in particular
the impact of considering a self-consistent evolution of the electron
distribution function through kinetic modeling. This section shows that our
plasmoid model is able to reasonably account for the Gravity Collaboration et
al. (2018) flare data. The limitations of our plasmoid model are discussed in
Sect. 4. The conclusions and perspectives are given in Sect. 5.

Figure 1: Scheme of the torus-jet model for the quiescent state in blue and
flares in red. Two trajectories are considered for the flare, which can either
rotate in the torus (hot spot model) or be ejected along the jet sheath
(plasmoid model). The jet is parametrized by the angles $\theta_{1}$ and
$\theta_{2}$ that describe the angular opening of the radiation-emitting
sheath, by the base height $z_{b}$, the constant Lorentz factor $\Gamma_{j}$,
and the temperature power-law index $s_{T}$. The jet is symmetrical with
respect to the equatorial plane, and axisymmetric.

##  2 Quiescent flow impact on astrometry: shifting and rotating the orbit

Gravity Collaboration et al. (2018) used a hot spot model in an equatorial
circular orbit to fit the astrometry of three bright flares. They considered a
constant radiation flux from the emitting region orbiting the black hole to
fit the orbital motion. The effect of out-of-plane motion and orbital shear
have also been studied by Gravity Collaboration et al. (2020c) to model the
flares. However, the impact of the quiescent radiation surrounding the hot
spot was not taken into account. The aim of this section is to show that
taking into account the quiescent radiation can lead to shifting and rotating
the orbit on sky. We note a 1-$\sigma$ difference between the center of the
orbit of the hot spot and the center of mass derived from the orbit of S2 in
Gravity Collaboration et al. (2018) which makes this shift marginal.

In this section, we will use a simplified hot spot model that is sufficient to
highlight the main effects of the quiescent radiation. This simple model will
also allow us to introduce the most important relativistic effects at play,
that were already studied in many previous works (Broderick & Loeb, 2006;
Hamaus et al., 2009). These reminders will be helpful when we turn to a more
complex hot spot model in the Sect. 3, which is the main aim of this paper.

###  2.1 Simple hot spot + quiescent model for the flaring Sgr A*

The quiescent radiation of Sgr A* is modeled by means of the torus-jet model
as derived in Vincent et al. (2019), to which we refer for all details. Figure
1 resumes the main features of the model. The torus emits thermal synchrotron
radiation, while the flux emitted by the jet follows a $\kappa$ distribution
(i.e. a thermal core with a power-law tail). The multi-wavelength spectrum of
the quiescent Sgr A* is well fit with this model. The $\kappa$ distribution
emission from the jets dominates at most wavelength except at the sub-mm bump
where the flux comes mostly from the thermal disk. We summarize the best-fit
parameters in Table 1, and the resulting best-fit quiescent spectrum is given
in Fig. 2. More details on the fitting procedure are given in Appendix A. With
these parameters, the flux of the torus-jet model at $2.2\ \mu m$ is $1.1$
mJy. It is in perfect agreement with the median quiescent dereddened flux
provided by Gravity Collaboration et al. (2020a) of $1.1\pm 0.3$ mJy. At this
wavelength, the torus is optically thin and its emission is negligible
compared to the jet. In the remainder of this paper, where we focus only on
the infrared band, we will thus neglect the torus and consider a pure jet
quiescent model, unless otherwise noted.

The only relevant features of our quiescent model for the rest of this paper
are the location of its infrared centroid and its NIR flux. As depicted in the
right panel of Fig. 2, the centroid of our jet-dominated model lies very close
to the mass center. We have checked that considering a disk-dominated model
only very marginally changes the position of the quiescent centroid at low
inclination (see the blue and green dots in the left panel of Fig. 3). Our
conclusions are thus not biased by our particular choice of a jet-dominated
quiescent model.

Parameter | Symbol | Value
---|---|---
Black Hole |  |
mass [$M_{\odot}$] | $M$ | $4.297\times 10^{6}$
distance [kpc] | $d$ | $8.277$
spin | $a$ | $0$
inclination [$\degree$] | $i$ | $20$
Torus |  |
angular momentum [$r_{g}/c$] | $l$ | $4$
inner radius [$r_{g}$] | $r_{in}$ | $8$
polytropic index | $k$ | $5/3$
central density [cm-3] | $n_{e}^{T}$ | $1.2\times 10^{9}$
central temperature [K] | $T_{e}^{T}$ | $7\times 10^{9}$
magnetization parameter | $\sigma^{T}$ | $0.002$
Jet |  |
inner opening angle [$\degree$] | $\theta_{1}$ | $20$
outer opening angle [$\degree$] | $\theta_{2}$ | $\theta_{1}+3.5$
jet base height [$r_{g}$] | $z_{b}$ | $2$
bulk Lorentz factor | $\Gamma_{j}$ | $1.15$
base number density [cm-3] | $n_{e}^{J}$ | $3.5\times 10^{6}$
base temperature [K] | $T_{e}^{J}$ | $3\times 10^{10}$
temperature slope | $s_{T}$ | $0.21$
$\kappa$ index | $\kappa^{J}$ | $5.5$
magnetization parameter | $\sigma^{J}$ | (fixed) $1$
Table 1: Best fit parameters of the torus+jet quiescent model. We keep the
same geometrical parameters, bulk Lorentz factor and $\kappa$-index as Vincent
et al. (2019) and we fit the base number density, base temperature and
temperature slope of the jet considering the correction (see bellow) and the
new value of the jet magnetization parameter. The parameters of the torus are
unchanged. Figure 2: Left: Spectrum associated to the best-fit of the torus-
jet model (see Table 1) for the quiescent state of Sgr A*
($\chi^{2}_{red}=0.91$ with ndof=27). The data are taken from Bower et al.
(2015) for $\nu<50$ GHz, Brinkerink et al. (2015) for the 2 points around 100
GHz, Liu et al. (2016) for the 492 GHz point, Marrone et al. (2006) for the
690 GHz point, von Fellenberg et al. (2018) for the far infrared upper limits,
Witzel et al. (2018) for the mid infrared data, and Baganoff et al. (2001) for
the X-ray bow-tie. We note that as in Vincent et al. (2019), the X-ray data
are not fitted as we do not take into account bremsstrahlung nor Comptonized
emission. Right: Best-fit image at $2.2$ $\mu m$ of the torus-jet model with a
field of view of $150$ $\mu as$ seen with an inclination of $20\degree$ and a
Position Angle of the Line of Nodes (PALN) of $\pi$ rad. The color bar gives
the values of specific intensity in cgs units in log-scale. The outer region
emission comes from the backward jet’s part while the emission close to the
center comes from the forward part of the jet. The centroid of the jet is
represented by the blue dot at $\sim$($0,-2.2$).

The hot spot model is composed of a plasma sphere of radius 1 $r_{g}$ (fixed)
with a uniform but time-dependent $\kappa$-distribution for the electrons. The
emissivity $j_{\nu}$ and absorptivity $\alpha_{\nu}$ coefficients depend on
the density, temperature, and magnetic field which we considered uniform. We
use the fitting formula of Pandya et al. (2016) to compute these coefficients.
The typical light curve of a flare is characterized by a phase with increasing
flux and one with decreasing flux. We model this behavior by a Gaussian time
modulation on the density and temperature as follows

$n_{e}(t)=n_{e}^{hs}\
exp\left(-0.5\times\left(\frac{t-t_{ref}}{t_{\sigma}}\right)^{2}\right),$ (1)
$T_{e}(t)=T_{e}^{hs}\
exp\left(-0.5\times\left(\frac{t-t_{ref}}{t_{\sigma}}\right)^{2}\right)$ (2)

where $t_{\sigma}$ is the typical duration of the flare. As $n_{e}$ varies
over time (Eq. 1), the magnetic field strength also varies since we set a
constant magnetization $\sigma=B^{2}/4\pi m_{p}c^{2}n_{e}$.

Contrary to Gravity Collaboration et al. (2020c), we keep the circular
equatorial orbit of Gravity Collaboration et al. (2018) as we assume that the
hot spot is formed in the equatorial plane and we do not take into account any
shearing effect and assume a constant spherical geometry of the hot spot. We
summarize all the input parameters of the hot spot in Table 2.

###  2.2 Shifting the orbit on sky

Figure 3 shows the impact of taking into account the quiescent radiation on
the astrometry of the flare, considering the trivial case of a constant-
emission hot spot, as well as the varying-emission hot spot introduced in
section 2.1.

Obviously, whether or not the hot spot intrinsic emission varies, the first
effect of adding a quiescent radiation is to shrink the orbit’s size, because
the overall centroid is moved towards the quiescent radiation’s centroid,
which always lies close to the mass center.

A slightly less obvious effect is that, when the hot spot emission varies in
time, the orbit can shift in the plane of sky, and no longer be centered at
the center-of-mass location. This is clearly apparent on the solid-red orbit
of the left panel of Fig. 3. This is simply due to the time variation of the
intensity ratio between the quiescent and the hot spot radiation. At early and
late times, the hot spot has a weaker emission than the quiescent component,
and the overall centroid coincides with the quiescent centroid. As the hot
spot emission increases and dominates, the overall centroid will be driven
towards it. Such a shift between the astrometric data and the center-of-mass
position is visible at 1-$\sigma$ significance in the the Gravity
Collaboration et al. (2018) data.

We note another non-trivial effect appearing in the varying-emission hot-spot
orbit without any quiescent radiation (red-dotted orbit in Fig. 3). The orbit
is not closing, due to the time delay between the primary and secondary
images. Indeed, at the end of the simulation, the flux from the secondary
image is intrinsically higher than the primary (the emission times of the
primary and secondary are different), and is amplified by the beaming effect.
When the centroid is computed, the secondary image has a larger impact at this
time than before, resulting in a closer centroid position relative to the
black hole. This astrometric impact of the secondary image was already
discussed by Hamaus et al. (2009).

Figure 3: Astrometry (left) and light curves (right) of the hot spot - jet model with two values for the quiescent state corresponding to no quiescent (dashed lines) and the with quiescent state (full lines). In shade of blue, the hot spot has a nearly constant emission ($t_{\sigma}>>t_{orbit}$). The effect of beaming is reflected in the light curves. In shade of red, the hot spot has a Gaussian time emission with $t_{\sigma}=30$ min. The parameters of the hot spot are listed in Table 2. We synchronise the maximum of beaming and the intrinsic maximum of the Gaussian modulation. The black, blue and green dots in the left panels represent the position of Sgr A*, the jet’s centroid and the disk’s centroid respectively. Parameter | Symbol | Value
---|---|---
Hot spot |  |
number density max [cm-3] | $n_{e}^{hs}$ | $1.05\times 10^{7}$
temperature max [K] | $T_{e}^{hs}$ | $9.03\times 10^{10}$
time Gaussian sigma [min] | $t_{\sigma}$ | $30$
magnetization parameter | $\sigma^{hs}$ | $0.01$
$\kappa$-distribution index | $\kappa^{hs}$ | $5$
orbital radius [$r_{g}$] | $R^{hs}$ | $9$
initial azimuth angle [$\degree$] | $\varphi_{0}^{hs}$ | $90$
Position Angle of the Line of Nodes [$\degree$] | $\Omega$ | $160$
Table 2: Summary of parameters of the hot spot model. We note that we used the
maximum number density and temperature of the jet best-fit in Table 1 as
reference and scale them for the hot spot by a factor $3.01$. Figure 4:
Astrometry (left) and light curves (right) of the hot spot - jet model for
four initial azimuthal angle $\varphi_{0}$ of $0\degree$ in blue, $90\degree$
in orange, $180\degree$ in green and $270\degree$ in red. The dashed black
line shows the primary image centroid track with no quiescent jet (clock-
wise). The jet dominates the beginning and the end of the flares. The observed
centroids thus start and end close to the one of the jet. The apparent orbits
rotate around the latter with $\varphi_{0}$ as the maximum of emission occurs
at different $\varphi$. The Gaussian modulation which has a typical duration
of $t_{\sigma}=15$ min (grey dashed line; which is the same for the four the
curves) is affected by relativistic effects. For negative X (right part of the
astrometry), the beaming, combined with relativistic Doppler effect, amplify
the flux from the hot spot while in the positive X (left part of the figure),
they lower it. The black dot in the left panels represents the position of Sgr
A*.

###  2.3 Rotating the orbit on sky

It is not an easy task to disentangle the intrinsic time variability of the
hot spot from the variability due to the relativistic beaming effect. Figure 4
illustrates the impact on astrometry and light curve of playing with the
relative influence between the intrinsic and beaming-related variability.
Here, we simply change the initial azimuthal coordinate $\varphi_{0}$ of the
hot spot along its orbit, in order to change the dephasing between the time of
the maximum intrinsic emission ($t=t_{\mathrm{ref}}$) which is fixed, and the
time of the maximum constructive beaming effect (when the hot spot moves
towards the observer). The orbit rotates around the quiescent centroid
following the variation of $\varphi_{0}$ (left panel of Fig. 4). The light
curve is also strongly affected, reaching much brighter levels when the
intrinsic emission maximum is in phase with the constructive beaming effect.

Here we show that the quiescent state of Sgr A* can have significant impact on
the observed astrometry by shrinking the apparent orbit, creating a shift
between the center of the latter and the position of the mass center. One
should have these effects in mind for the comparison to the flare data at the
end of the following section.

##  3 Plasmoid model from magnetic reconnection

In this section, we develop a semi-analytical hot-spot-like model in order to
interpret the rise and decay of Sgr A* flares, thus going one step further
with respect to the model we used in section 2, where a Gaussian modulation of
the emission is enforced without physical motivation.

Black hole magnetospheres naturally lead to the development of equatorial
current sheets corresponding to a strong spatial gradient of the magnetic
field which changes sign at the equator (Komissarov, 2004; Komissarov &
McKinney, 2007; Parfrey et al., 2019; Ripperda et al., 2020). Such a
configuration results in magnetic reconnection, i.e. a change of the topology
of the field lines forming X points (Komissarov, 2005; Loureiro et al., 2007;
Sironi & Spitkovsky, 2014). This process is intrinsically non-ideal and thus
can only be captured either by resistive MHD or kinetic simulations. For
suitable values of the magnetic diffusivity, the reconnecting current sheet
can break into chains of plasmoids, i.e. magnetic islands separated by X
points (Loureiro et al., 2007; Parfrey et al., 2019; Ripperda et al., 2020;
Porth et al., 2021).

The reconnection rate (i.e. the typical rate at which magnetic energy is
dissipated into particle kinetic energy) is equal to the ratio
$v_{\mathrm{rec}}/v_{\mathrm{out}}$ with $v_{\mathrm{rec}}$ the velocity of
matter injected into the reconnection region, and $v_{\mathrm{out}}$ the bulk
outflow velocity of particles accelerated by the reconnection event. The
outflow velocity is of the order of the Alfven speed, $v_{\mathrm{out}}\approx
v_{A}$, which is itself of the order of the speed of light, $v_{A}\approx c$,
for strongly magnetized environments. The reconnection rate has been shown to
be rather independent of the details of the chosen parameters. For PIC
simulations, it lies around $10\%$, i.e. $v_{\mathrm{rec,PIC}}\approx
0.1v_{A}\approx 0.1c$, for magnetized collisionless plasmas (Sironi &
Spitkovsky, 2014; Werner et al., 2018; Guo et al., 2015), which are the
typical conditions in the inner flow surrounding Sgr A*111It is likely that
the accretion flow surrounding Sgr A* is in a Magnetically Arrested Disk (MAD,
see Narayan et al., 2003) regime, i.e. with strong poloidal magnetic fields in
the inner regions (Gravity Collaboration et al., 2018; Dexter et al., 2020b)..
GRRMHD simulations point towards a slower rate of around $1\%$, so that
$v_{\mathrm{rec,MHD}}\approx 0.01c$ (see the discussion in Ripperda et al.
2022), but this applies to collisional environments, thus less similar to Sgr
A* vicinity.

Fresh plasma flows into the current sheet at the reconnection rate
$v_{\mathrm{rec}}$, is accelerated by the electric field generated in the
current sheet, usually giving rise to power-law energy distributions of
electrons (Sironi & Spitkovsky, 2014; Werner et al., 2018). Inside the current
sheet, the particles get trapped in the plasmoids which act as particle
reservoirs (Sironi & Spitkovsky, 2014) which can merge in a macroscopic
magnetic island, that is, a large plasmoid. In Ripperda et al. (2022),
magnetic flux dissipation through reconnection last for $\sim 100r_{g}/c$
$\sim 30$ min and the resulting hot spot orbits for $\sim 500r_{g}/c$ $\sim
150$ min before it disappears by losing its coherence through interaction with
the surrounding flow.

In the global PIC simulation of El Mellah et al. (2022), the authors study
magnetic reconnection in the sheath of relativistic jet working with magnetic
field loops coupling the BH to the accretion disk. The resulting plasmoids
evolve off-plane, propagate away from the BH and are prone to merge with each
other to form macroscopic plasmoids susceptible to radiate high amounts of
energy under the form of non-thermal radiation. The underlying mechanism,
first described by Uzdensky (2005) and de Gouveia dal Pino & Lazarian (2005),
relies on the accretion of poloidal magnetic field loops onto a spinning BH.
Once the inner footpoint of the loop reaches the BH ergosphere, the magnetic
field line experiences strong torques due to the frame dragging effect while
its other footpoint on the disk rotates at the local Keplerian speed. Thus,
the toroidal component of the magnetic field quickly grows in the innermost
regions, propagates upstream along the field line and leads to the opening of
the magnetic loop above a certain magnetic loop size. On the outermost closed
magnetic field line (called the separatrix), a Y-point appears where plasmoids
form and flow away along an inclined current sheet above the disk (Fig. 5). In
the PIC simulations of El Mellah et al. (2022), a cone-shaped reconnecting
current sheet forms where vivid particle acceleration takes place. Electrons
and positrons pile up into outflowing plasmoids where they cool through
synchrotron radiation. This topological configuration where some magnetic
field lines anchored in the disk close within the event horizon is coherent
with what is seen in resistive GRMHD simulations during the short episodes of
flux repulsion which separates different accretion regimes. During
approximately 100 rg/c, these simulations show an essentially force-free
funnel surrounded by merging plasmoids formed in the jet sheath and in the
equatorial plane Ripperda et al. (2022); Chashkina et al. (2021).

###  3.1 Plasmoid model from magnetic reconnection

The aim of this section is to develop a semi-analytic large plasmoid model
(which will be named plasmoid model), inspired from the reconnection
literature reviewed above. The interest of such a model, compared to state-of-
the-art GRMHD or GRPIC modeling, is twofold:

  * •

it allows to remain as agnostic as possible regarding the physical conditions
close to Sgr A* and encapsulate into a single model a large parameter space;

  * •

it allows to perform simulations within a limited computing time, allowing to
explore the large parameter space and compare to astrometric and photometric
data.

Our hope is that such a model can not only be fed with the results of more
elaborated simulations, but also bring constraints to these simulations by
determining what features of the modeling are important in order to explain
the data.

The main features of our model are illustrated in Fig. 5. inspired by the
recent GRPIC results of El Mellah et al. (2022). Here, we consider a single
plasmoid, which is modeled as a sphere of hot plasma with a constant radius.
This macroscopic plasmoid is understood as the end product of a sequence of
microscopic plasmoid mergers. The spherical geometry is chosen only for
simplicity, given that current data are certainly unable to make a difference
between various geometries.

####  3.1.1 Plasmoid motion

We consider that the magnetic reconnection event occurs close to the black
hole, and the resulting plasmoid is ejected along the jet sheath (Ripperda et
al., 2020; El Mellah et al., 2022). Thus, we define a conical motion (as in
Ball et al. (2021)) defined by a constant polar angle $\theta=\theta_{0}$ and
the initial conditions $r_{0}$, $\theta_{0}$, $\varphi_{0}$, $v_{r0}$, and
$v_{\varphi 0}$ . The subscript $0$ reflects the initial value of a given
parameter in Boyer-Lindquist coordinate. As in Ball et al. (2021), we set a
constant radial velocity $v_{r}=v_{r0}$ and the azimuthal velocity is defined
through the conservation of the Newtonian angular momentum

$v_{\varphi}(t)=v_{\varphi 0}\frac{r_{0}^{2}}{r(t)^{2}}.$ (3)

The azimuthal angle is obtained by integrating the previous Eq. 3

$\varphi(t)=\varphi_{0}+r_{0}^{2}\frac{v_{\varphi
0}}{v_{r}}(\frac{1}{r_{0}}-\frac{1}{r(t)}).$ (4)

In the GRPIC simulations performed by El Mellah et al. (2022), the plasmoids
are formed in the vicinity of the black hole at the Y-point and are ejected
into the black hole magnetosphere, we thus restrict our study to $v_{r}>0$.

An important feature of our model is the fact that the initial azimuthal
velocity of the plasmoid is naturally super-Keplerian. Indeed, the Y-point,
from which the plasmoid is generated, is anchored to the equatorial plane of
the accretion flow through the separatrix field line. So it will typically
rotate at the Keplerian speed corresponding to the foot point of the line,
thus at a velocity higher than the Keplerian velocity corresponding to the
initial cylindrical radius of the plasmoid.

####  3.1.2 Growth and cooling phases

We consider two phases in the lifetime of the plasmoid that aim at modeling
the ascending and descending phases of the observed flare light curves:

  * •

during the growth phase, which lasts a total time $t_{\mathrm{growth}}$, the
plasmoid continuously receives fresh accelerated particles at a constant rate
resulting from the merging of microscopic plasmoids from reconnection into our
large plasmoid which mix with ”old” electrons cooled by synchrotron radiation.
The growth time $t_{\mathrm{growth}}$ corresponds to the lifetime of the
reconnection engine, i.e. the duration of magnetic flux dissipation;

  * •

after $t=t_{\mathrm{growth}}$, the plasmoid enters the cooling phase: we
assume that magnetic reconnection is quenched and plasmoids no longer merge so
injection of fresh plasma stops and the plasmoid cools by emitting synchrotron
radiation. We neglect particle escape and adiabatic losses.

The duration of the growth phase is set both by the reconnection rate and the
speed at which magnetic flux is advected by the accretion flow into the
current sheet. In Parfrey et al. (2015), the accretion of successive magnetic
loops of opposite polarity activates this process, with typical duration of
transition of the order of 100$r_{g}/c$. This duration is representative of
the dissipation of the magnetic flux of one magnetic loop which is set by both
the size of the loop and the accretion speed, that the authors fix to 2$r_{g}$
and $c/200$ respectively, and the reconnection rate, fixed by the prescribed
resistivity. Resistive GRMHD simulations of magnetically arrested disks
(Narayan et al., 2003) bring support to these values (e.g. Ripperda et al.,
2020) but fail at reaching reconnection rates realistically high (Bransgrove
et al., 2021a). In the more ab initio PIC simulations of El Mellah et al.
(2022), the reconnection rate is more realistic ($\sim 10\%$, Sironi &
Spitkovsky, 2014; Werner et al., 2018) but due to the high computational cost
of the simulations, they did not work over duration long enough to model the
inward drift of the magnetic footpoints on the disk. As a consequence, the
reconnection rate is accurate but the fueling magnetic flux is artificially
steady and act as an infinite reservoir over the $\sim$200$r_{g}/c$ covered by
the simulation. A coupling between GRMHD, force-free and PIC simulations to
jointly describe the disk, the corona and the current sheet respectively is
still missing. In this context, we considered duration $t_{\mathrm{growth}}$
of the growth phase of the order of 100$r_{g}/c$, corresponding to a typical
episode of magnetic flux dissipation set by the two rates at which magnetic
flux is advected into the current sheet and dissipated by magnetic
reconnection.

Figure 5: Sketch of magnetic reconnection in the black hole magnetosphere as
shown by El Mellah et al. (2022) on which our plasmoid model is built. There
are three types of magnetic field lines: the ones threading the event horizon
which goes to infinity, the ones anchored in the disk which also go to
infinity, and the separatrix which link the disk and the black hole event
horizon. The latter form a Y-point and a current sheet where chain of
plasmoids are formed. Here we model a single plasmoid as the result of
multiple mergers.

####  3.1.3 Evolution of the electron distribution

Next, we prescribe the emission/absorption mechanism in the plasmoid. Most
studies use chosen electron distributions, with analytical prescriptions for
their evolution at best. Ball et al. (2021) use a fixed thermal distribution
with a linear increase of the number density for the rising part of the light
curve and a decrease of the temperature following Eq. 28 for the cooling.
Scepi et al. (2022) use a kappa distribution with an exponential cutoff and a
synchrotron cooling break for the plasma emission generating X-ray flares.
While their evolution of the plasma parameters (number density, temperature,
magnetic field) is more elaborate than in our model, their approximation is
valid only while injection and cooling are balanced. When the injection stops,
the shape of the electron distribution changes rapidly (see Fig. 6). Here, we
choose a different approach by evolving the electron distribution in the
plasmoid by solving the kinetic equation

$\dfrac{\partial N_{\text{e}}(\gamma,t)}{\partial
t}=\dfrac{\partial}{\partial\gamma}\left(-\dot{\gamma}_{\mathrm{syn}}\,N_{\text{e}}(\gamma,t)\right)+Q_{\text{inj}}(\gamma),$
(5)

where $\gamma$ is the Lorentz factor of the electrons, $Q_{\text{inj}}$ is the
injection rate and $N_{e}=\mathrm{d}n_{e}/\mathrm{d}\gamma$ is the electron
number density distribution, using the EMBLEM code (Dmytriiev et al., 2021).
The term

$-\dot{\gamma}_{\mathrm{syn}}N_{\text{e}}=\frac{4\sigma_{T}U_{B}}{3m_{e}c}(\gamma^{2}-1)N_{e}$
(6)

of the right hand side describes the synchrotron cooling of the plasmoid
particles, with $U_{B}=B^{2}/(8\pi)$. In our approach, we do not model the
details of the magnetic reconnection process but instead describe the supply
of freshly accelerated particles to the plasmoid by magnetic reconnection.
Therefore, for the injection rate $Q_{\text{inj}}(\gamma)$ in Eq. 5, we use
the following expression, assuming a constant injection rate:

$Q_{\text{inj}}(\gamma)=\left\\{\begin{array}[]{ll}\dfrac{4\pi
N_{e}^{\kappa}(\gamma)}{t_{\mathrm{growth}}}&\mbox{in \>the\>growth \>phase,
}\\\ 0&\mbox{in \>the\>cooling \>phase,}\end{array}\right.$ (7)

where $N_{e}^{\kappa}(\gamma)$ is the distribution of the injected particles
which follows the kappa distribution i.e. a thermal core with a power-law tail
following:

$N_{e}^{\kappa}(\gamma)=\dfrac{N}{4\pi}\gamma(\gamma^{2}-1)^{1/2}\,\left(1+\dfrac{\gamma-1}{\kappa\Theta_{e}}\right)^{-(\kappa+1)}$
(8)

with a normalization factor
$N=1/2n_{e}(\kappa-2)(\kappa-1)\kappa^{-2}\Theta_{e}^{-3}$, where $n_{e}$ and
$\Theta_{e}$ are the density and dimensionless temperature of the injected
plasma. The index $\kappa$ is defined as

$\kappa=p+1=A_{p}+B_{p}\tanh{(C_{p}\ \beta_{b})}+1$ (9)

where

$A_{p}=1.8+0.7/\sqrt{\sigma_{b}},B_{p}=3.7\ \sigma_{b}^{-0.19},C_{p}=23.4\
\sigma_{b}^{0.26},$ (10)

following Ball et al. (2018); Werner et al. (2018), where $p$ is the powerlaw
index of the non-thermal part of the distribution,
$\sigma_{b}\leavevmode\nobreak\ \gg\leavevmode\nobreak\ 1$ is the plasma
magnetization of the accelerating site and $\beta_{b}\ll 1$ is the ratio of
proton thermal pressure to magnetic pressure of the accelerating site. If the
magnetization at the accelerating site satisfies $\sigma_{b}\geq 100$
(Crinquand et al., 2021), then $\kappa$ is in the range of [$2.8,4.4$]
depending on $\beta_{b}$. This implies that the spectral index $\alpha$ ($\nu
F_{\nu}\propto\nu^{\alpha}$) is between $-0.5$ and $0.5$ which is in perfect
agreement with the measured indices for flares (Fig. 32 in Genzel et al.,
2010). We note that realistic values for the magnetization in the funnel
region of Sgr A* can be orders of magnitude higher than $100$ (see Ripperda et
al., 2022) which result in a smaller parameter space for $\kappa$, closer to
the low boundary.

The bounds of the electron Lorentz factor are chosen to satisfy
$\gamma_{\mathrm{min}}=1$ and $\gamma_{\mathrm{max}}=10^{6}$ (Eq.3 of Ripperda
et al., 2022). When solving the kinetic Eq. 5, we assume that the density of
the plasmoid particles follows

$n_{e}(t)=\left\\{\begin{array}[]{ll}n_{e}^{\mathrm{max}}\times
t/t_{\mathrm{growth}}&\mbox{in \>the\>growth \>phase, }\\\
n_{e}^{\mathrm{max}}&\mbox{in \>the\>cooling \>phase.}\end{array}\right.$ (11)

Such high maximum Lorentz factor is needed to also power X-ray flares with
only synchrotron. However, Ripperda et al. (2022) suggest a lower maximum
Lorentz factor $\gamma_{\mathrm{max}}\sim 10^{4}$ in the plasmoid as electrons
cool during their travel time between the acceleration site and the later.
Such lower value results in a marginally lower flux in NIR, as most of the
emission at this wavelength comes from lower energy electrons, which can be
compensated with a slightly higher maximum number density.

The temperature of the injected particles remains fixed in the growth phase,
and we define a uniform and time-independent tangled magnetic field in the
plasmoid. This is also a simplifying assumption, and we intend to consider in
future work the impact of the magnetic field geometry on the polarized
observables.

The EMBLEM code does not only solve for the evolution of the electron
distribution, it also provides the associated synchrotron emission and
absorption coefficients of the plasmoid particles. We can thus compute an
image of our plasmoid scenario by backwards integrating null geodesics in the
Schwarzschild spacetime from a distant observer screen, and integrate the
radiative transfer equation through the plasmoid by reading the tabulated
emission and absorption provided by EMBLEM. This step is performed by means of
the GYOTO222https://gyoto.obspm.fr ray-tracing code (see Appendix B for
details; Vincent et al., 2011; Paumard et al., 2019). The input parameters
that we used for the code are listed in Table 3. With these values of density
and magnetic field strength, we obtain a magnetization inside the plasmoid of
$\sigma_{p}\sim 10^{-2}$ from the end of the growth phase since neither the
density nor the magnetic field evolve.

###  3.2 Importance of evolving the electron distribution

One of the most important aspects of our model is the self-consistent
evolution of the electron distribution function, and corresponding radiative
transfer, in the plasmoid. Here, we illustrate the importance of taking into
account the evolution of the electron distribution by comparing our model with
another reconnection plasmoid model inspired by Ball et al. (2021).

We show in the top-left panel of Fig. 6 the evolution of the electron
distribution in our plasmoid model, for the parameters listed in Tables 3 and
4 (see Sect. 3.3 for details) and the associated spectral energy density (SED)
in Fig. 7. During the growth phase ($t_{obs}<10$ min), for $\gamma>10^{3}$ the
distribution is stationary as the injection is balanced by the cooling. After
the end of the growth phase, the shape of the distribution changes rapidly as
there is only cooling left. We show in the right panel of Fig. 6 the light
curve obtained with our model (in red) and with a model inspired by Ball et
al. (2021) who do not take into account the non-thermal electrons, i.e. using
a thermal distribution, with a linear increase of the number density with a
fixed temperature during the growth phase and an analytical prescription for
the temperature decrease during the cooling phase using Eq. 28 and assuming
$\Theta_{e}=\gamma/3$. While this model gives a similar intrinsic light curve
as our model, the dimensionless temperature required is twice as high as ours
($\Theta_{e}=109$) with a magnetic field of $B=20$ G to cool faster lower
energy electrons. The evolution of the distribution with this model is shown
in the bottom left panel of Fig. 6. We do not need such high a temperature as
most of the emission comes from high-energy electrons which are non-thermal in
our model as suggested by PIC simulations (Rowan et al., 2017; Werner et al.,
2018; Ball et al., 2018; Zhang et al., 2021). Our temperature could be even
lower with a harder (i.e. lower) $\kappa$-index. The cooling of the electron
distribution through synchrotron radiation is difficult to model properly and
needs a kinetic approach as we do in our plasmoid model.

Figure 6: (Top-left) Evolution of the electron distribution in our model with
EMBLEM at each observing time of the July 22, 2018 flare. The black dotted
line correspond to the injected $\kappa$ electron distribution composed of a
thermal core with a power law tail. The parameters used are listed in Table 3.
(Bottom-left) Evolution of the electron distribution in the Thermal model
inspired by Ball et al. (2021) at each observing time of the July 22, 2018
flare. The parameters used for this distribution are the same as in our model
(listed in Table 3) but with the dimensionless temperature of $\Theta_{e}=109$
and the magnetic field of $B=20$ G. (Right) Full intrinsic light curves of the
two models. Note that in this panel we plot the light curve from the beginning
of the growth phase while in the left panels we plot the distribution at the
observed time of Fig. 8 ($t_{obs}=t_{intrinsic}-29.6$ min). Figure 7:
Intrinsic Spectral Energy Density (SED) evolution from radio to X-rays of our
plasmoid model with the parameters listed in Table 3. Color code the time as
in Fig. 6. Grey lines are the SED out of the observing time. The SED peak
occurs in NIR with this set of parameters, but with a lower $\kappa$-index,
i.e. a harder power law tail, the peak could reach X-rays. Here, X-rays flux
drops quickly compare to the typical timescale of X-ray flares as we consider
only synchrotron cooling and not Synchrotron-Self Compton (SSC). We note that
the flux emitted by our plasmoid at 230 GHz is $\sim 0.1$ Jy which is the good
order of magnitude of the variability associated to flares in sub-mm (Wielgus
et al., 2022a).

###  3.3 Comparing GRAVITY 2018 flare data with our plasmoid model

This paper aims at checking whether we can reproduce with our plasmoid model
the general features of the observed light curve and astrometry of the July
22, 2018 flare reported by Gravity Collaboration et al. (2018). We show in
Fig. 8 a comparison between the July 22, 2018 flare data observed by GRAVITY
(in black) and our plasmoid model (red line) with the parameters listed in
Tables 3 and 4. For comparison, we show the intrinsic light curve (dashed
line) obtained by removing all the relativistic effects (Doppler effect,
beaming, secondary image).

This comparison is not the result of a fit and was obtained by estimating the
relevant parameters using simple physical arguments:

  * •

The rise time and slope of the light curve are mainly monitored by (i) the
growth time, (ii) our choice of linear evolution of the electron density
(which enters the injection function), (iii) the relativistic beaming effect,
and thus (iv) the initial azimuthal position of the plasmoid, $\varphi_{0}$,
which has a strong impact on beaming as illustrated in the right panel of Fig.
4.

  * •

The decaying part of the light curve is monitored by the synchrotron cooling
time, thus by the magnetic field strength, and by the beaming effect.

  * •

The maximum of the light curve can be estimated by means of an analytical
formula that we derive in Appendix C.2. This maximum depends mainly on the
maximum number density $n_{e,max}$, as well as on the temperature and
$\kappa$-index of the distribution. These parameters are degenerate and thus
not constrained with only the NIR flare data. Nevertheless, GRMHD (Dexter et
al., 2020b; Ripperda et al., 2022; Scepi et al., 2022) and GRPIC simulations
(El Mellah et al., 2022) of magnetic reconnection suggest that the density in
the plasmoid is higher than its close environment in the current sheet, of the
order of the density at the base of the jet, close to the event horizon, but
lower than in the disk. Still, the two remaining parameters ($\Theta_{e}$,
$\kappa$) which describe the shape of the distribution are fully degenerate.

  * •

The initial position and velocity of the plasmoid have a strong impact on the
astrometric trace on sky. We guess the initial azimuthal velocity based on the
following reasoning. The Keplerian velocity of the plasmoid at its initial
cylindrical radius is $v_{Kep}\sim 0.31c$ (for our choice of initial
cylindrical radius given in Table 4, $r_{cyl}=10.6\leavevmode\nobreak\
r_{g}$). However, as discussed in Sect. 3.1, our model naturally leads to a
super-Keplerian initial velocity to the plasmoid. Indeed, the plasmoid initial
azimuthal velocity is that of the footpoint of the separatrix (see Fig. 5).
Based on Fig. 8 of El Mellah et al. (2022), we can determine the radius of the
footpoint, $r_{fp}$, of a separatrix giving rise to a Y point located at a
cylindrical radius of $\approx 10\,r_{g}$. We find $r_{fp}=(4.7\pm
0.5)\,r_{g}$, which translates in an orbital velocity $v_{\varphi,0}$ between
$0.41c$ and $0.45c$. The upper bound of this interval compares well with the
July 22 flare data. We note that this estimate of the initial azimuthal
velocity is anchored in the model of El Mellah et al. (2022) and thus depends
on their choice of initial condition, in particular on the initial profile of
their magnetic field.

We note that the fiducial values proposed in Tables 3 and 4 represent a set of
parameters with values that are inspired by numerical simulations of
reconnection which reproduce the key observational features of the July 22
flare data. This setup is not unique and is not the result of a fit. We let
the exploration of the full parameters space (freeing some fixed/constrained
parameters like maximum number density, growth time, inclination) to a future
work. Nevertheless, our model disfavor low growth time
($t_{\mathrm{growth}}<50r_{g}/c$) for this particular flare.

Overall, our plasmoid model jointly describes the astrometry and the flux
variation of the 22 July 2018 flare measured by (Gravity Collaboration et al.,
2018) for the first time, considering a model with a specific emission
prescription. Magnetic reconnection is thus a viable scenario to explain Sgr
A* flares.

Parameter | Symbol | Value
---|---|---
Plasmoid |  |
magnetic field [G] | $B_{\text{p}}$ | 15
plasmoid radius [$r_{g}$] | $R_{\text{p}}$ | $1$
minimal Lorentz factor | $\gamma_{\text{min}}$ | 1
maximum Lorentz factor | $\gamma_{\text{max}}$ | $10^{6}$
kappa distribution index | $\kappa$ | 4.0
kappa distribution temperature | $\Theta_{e}$ | 50
maximum electron number density [cm-3] | $n_{\text{e,max}}$ | $5\times 10^{6}$
growth timescale [$r_{g}/c$] | $t_{\mathrm{growth}}$ | 120
Table 3: Input parameters of the EMBLEM code for the simulation of the electron distribution evolution. These parameters are used for the July 22 flare of Gravity Collaboration et al. (2018). Parameter | Symbol | July 22
---|---|---
Plasmoid |  |
time in EMBLEM at zero observing time [min] | $t_{obs,0}^{emblem}$ | $29.6$
initial cylindrical radius [$r_{g}$] | $r_{cyl,0}$ | $10.6$
polar angle [$\degree$] | $\theta$ | $135$
initial azimuthal angle [$\degree$] | $\varphi_{0}$ | $280$
initial radial velocity [$c$] | $v_{r,0}$ | $0.01$
initial azimuthal velocity [$c$] | $v_{\varphi,0}$ | $0.45$
X position of Sgr A* [$\mu as$] | $x_{0}$ | $0$
Y position of Sgr A* [$\mu as$] | $y_{0}$ | $0$
PALN [$\degree$] | $\Omega$ | $160$
Table 4: Orbital parameters of the plasmoid model following a conical motion
used for the comparison of the July 22, 2018 flares observed by Gravity
Collaboration et al. (2018). Figure 8: Data and plasmoid models of the flares
from July 22, 2018. The left panels shows the astrometry of the flare while
the right panel shows the observed (full line) and intrinsic (dashed line)
light curves. The parameters of the model are listed in Tables 3 and 4. Note
that this is not the result of a fit. The black dot in the left panels
represents the position of Sgr A* in GYOTO and the orange cross represent the
position of Sgr A* measured through the orbit of S2.

##  4 Limitations of our plasmoid model

Our plasmoid model is vastly simplified with respect to the complexity of
realistic magnetic reconnection events in the environment of black holes. We
review here its main limitations:

  * i.

We consider a single plasmoid while the instability of thin current sheets
gives rise to a dynamic flow of merging magnetic islands. Our argument for
this simplification is that the merging process is certainly very dependent on
the unknown initial conditions, and that the final, bigger and brighter
product of the cascade is likely to dominate the observed signal;

  * ii.

The initial condition on the plasmoid’s velocity is simply imposed for the
radial motion, and based on a particular GRPIC model as regards the azimuthal
motion;

  * iii.

The evolution of the plasma parameters (density, temperature, magnetic field)
are chosen to be either constant or linear, so very simplified compared to a
realistic scenario. However, we consider that these evolutions are very likely
to be strongly dependent on the initial conditions of the flow, so that they
are weakly constrained;

  * iv.

The values of almost all the parameters except mass and distance of Sgr A* are
poorly constrained. We choose a set of values which are reasonable according
to simulations. Future work is needed to investigate the details of the
parameter space.

  * v.

We model the plasmoid by a homogeneous sphere for simplicity from the circle
plasmoid seen in 2D GRMHD (Nathanail et al., 2020; Ripperda et al., 2020;
Porth et al., 2021) and PIC simulations (Rowan et al., 2017; Ball et al.,
2018; Werner et al., 2018). The 3D aspect of such plasmoid is cylindrical
(flux ropes) both in GRMHD (Bransgrove et al., 2021b; Nathanail et al., 2022;
Ripperda et al., 2022) and PIC (Nättilä & Beloborodov, 2021; Zhang et al.,
2021) simulations. Thus, a realistic geometry of the flare source is likely
more complex than in our model. We note that the exact geometry of the flare
is not relevant as we only track the centroid position, as much the flare
source is not too extended, and we consider tangled magnetic field. However,
the coherence time of the structure might be shorter in 3D and might have an
impact on the rise time of the light curve. Further 3D simulations studies are
needed to better model the shape of the flux ropes and their evolution;

  * vi.

We neglect any shearing of the plasmoid and consider that it remains identical
to itself throughout the simulation. Differential rotation is however likely
to stretch the plasmoid over its orbit and destroy its coherence (Hamaus et
al., 2009; Gravity Collaboration et al., 2020c);

  * vii.

We consider a tangled magnetic field in the plasmoid and thus do not consider
the impact of the magnetic field geometry on the observable. The magnetic
field geometry of the quiescent flow is likely to be ordered and vertical if
Sgr A* is strongly magnetized. The magnetic field in the plasmoid, which is
our interest here, could be either helical (plasmoids) or vertical for large
flux tubes (Ripperda et al., 2022).

  * viii.

During a flare, the quiescent state can change in a non axisymmetrical way
(Ripperda et al., 2022). This will push the centroid position of the quiescent
further away from the center-of-mass location which will affect the offset. We
choose to use a static and axisymmetric quiescent model during flare to avoid
adding more degrees of freedom which would lead to higher degeneracies, rather
than clearer constraints.

  * ix.

We choose a high maximum Lorentz factor $\gamma_{\mathrm{max}}=10^{6}$ to be
able to power X-ray flares (but without any constrain for this study).
However, high energy photons lead to pair-production and thus increase the
number density in the plasmoid which we do not take into account.

Despite these many limitations, we consider that our model is very interesting
for fitting flare data, because it allows to cover a much broader set of
physical scenarios than more elaborate simulations that strongly depend on
their assumptions regarding the relevant physics and the initial conditions.

##  5 Conclusion and perspectives

This paper is mainly focused at developing a new plasmoid model for Sgr A*
flares, inspired by magnetic reconnection in black hole environments. Our
semi-analytic model allows to study a broad parameter space within a
reasonable computing time, thus being well suited for data analysis.

Our model considers non-thermal electrons accelerated by magnetic reconnection
and injected into a spherical large plasmoid. We evolve the electron
distribution through a kinetic equation taking into account synchrotron
cooling and particle injection at a constant rate. We show in Appendix C.1
(Fig. 11) the importance of taking into account the cooling of the electrons
already in plasmoid during the growth phase. Our model also naturally accounts
for a super-Keplerian velocity of the flare source, through the dynamical
coupling between the plasmoid and the inner regions of the accretion flow
through magnetic field lines. One of the main results of this paper is that
for the first time we model the astrometry and lightcurve of the flares
measured by (Gravity Collaboration et al., 2018) by explicit modeling of the
emission zone.

Our conclusions regarding the three main points raised in the introduction are
the following:

  * •

the marginally detected shift between the astrometric track of Gravity
Collaboration et al. (2018) and the center of mass might be due to the impact
of the quiescent radiation of the background accretion flow;

  * •

a dynamical coupling between the plasmoid and the inner accretion flow through
closed magnetic field lines might naturally account for the super-Keplerian
speed obtained by Gravity Collaboration et al. (2018);

  * •

in general, a large plasmoid due to magnetic reconnection in a thin current
sheet in the black hole magnetosphere is a reasonable model to account for the
main features of the Gravity Collaboration et al. (2018) observables.

Sect. 3.3 shows that the temperature, density, and $\kappa$ parameters of the
plasmoid are degenerate. This degeneracy might be removed by simultaneous
observations of NIR and X-ray flares. Moreover, synchrotron cooling leads to a
translation of the electrons from the NIR-emitting band to the millimeter-
emitting band, which could explain the sub-mm flare and its time lag with
respect to NIR. We thus intend to consider the multi-wavelength properties of
our plasmoid model in future work, in order to better assess its ability to
account for the complete flare data set of Sgr A*.

A crucial recent observable of Sgr A* flares are the polarization QU loops
(Gravity Collaboration et al., 2018, 2020d; Wielgus et al., 2022b). We also
intend to study the polarized properties of our plasmoid model and compare it
to these recent constraints.

###### Acknowledgements.

NA and FHV are very grateful to B. Cerutti, B. Crinquand, S. von Fellenberg,
S. Gillessen, S. Masson, B. Ripperda, N. Scepi, and M. Wielgus for fruitful
discussions. This work was granted access to the HPC resources of MesoPSL
financed by the Region Ile de France and the project Equip@Meso (reference
ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the
Agence Nationale pour la Recherche.

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##  Appendix A Torus - Jet model for the quiescent state

We used the Torus-jet model of Vincent et al. (2019). Their jet model is
restricted to an emitting sheath with an empty funnel in agreement with GRMHD
simulations (e.g. Mościbrodzka & Falcke 2013; Ressler et al. 2017; Davelaar et
al. 2018; Porth et al. 2019). In their model, Vincent et al. (2019) define an
opening and closing angle $\theta_{1}$ and $\theta_{2}$ respectively and a
base height $z_{b}$ to define the geometry of the jet. The number density and
the temperature are defined by their values at the base height of the jet
($n_{e}^{J}$ and $T_{e}^{J}$ respectively) and their profiles along the jet.
The profile of the number density is fixed ($\propto r_{cyl}^{-2}$ with
$r_{cyl}$ the projected radius in the equatorial plane) and the one of the
temperature is set by the temperatures slope $s_{T}$ ($\propto z^{-s_{T}}$
with $z$ the height along the vertical/spin axis). The jet emits synchrotron
radiation from a $\kappa$ electron distribution.

The torus is defined by its central density and temperature ($n_{e}^{T}$ and
$T_{e}^{T}$ respectively). The profiles of these two quantities in the torus
are governed by the polytropic index $k$ and its geometry. The latter is
defined by the inner radius $r_{in}$ and the angular momentum $l$ but also on
the metric (see Vincent et al. (2019) for more details). Contrary to the jet,
we consider that the electron distribution of the torus is purely thermal.

We use the same algorithm as in Vincent et al. (2019) after the correction of
a small technical issue leading to an overestimation of the number density and
temperature. However, we change the choice of the magnetization parameter in
the jet sheath. As illustrated e.g. by Porth et al. (2019), the jet sheath,
which corresponds to the dominating emission region of the jet, coincides with
the transition between the highly-magnetized ($\sigma\gg 1$) funnel and the
less-magnetized ($\sigma\ll 1$) main disk body. Consequently, we fix the
magnetization to $\sigma=1$ in the emitting jet sheath, while Vincent et al.
(2019) used a low magnetization both in the jet and in the torus. Our choice
leads to a smaller density in the jet sheath compared to Vincent et al.
(2019). We found a best-fit with a $\chi^{2}_{red}=0.91$ using the same data
points as Vincent et al. (2019). The values are reported in Table 1 and Fig. 2
shows the associated spectrum and the image at $2.2\ \mu m$. We obtain a
magnetic field strength of $257\ \mathrm{G}$ for the jet and $212\ \mathrm{G}$
at the center of the torus.These values are higher than in Bower et al. (2019)
who considers a full thermal electron population with a higher temperature but
are of the same order as in Scepi et al. (2022).

##  Appendix B Ray-tracing setup

We consider a Kerr black hole with dimensionless spin parameter $a=0$,
described in Boyer-Lindquist ($t,r,\theta,\varphi$) coordinates. We work in
units where the gravitational constant and the speed of light are equal to 1,
$G=c=1$. Radii are thus expressed in units of the black hole mass $M$.

We use the backward ray-tracing code GYOTO333https://gyoto.obspm.fr (Vincent
et al. 2011; Paumard et al. 2019) to compute images of our models at different
epochs. Each pixel of our image corresponds to a direction on sky. For each
pixel of the image (i.e. each direction), a null geodesic is integrated
backwards in time from the observer towards the black hole, integrating along
this path the radiative transfer equation

$\frac{dI_{\nu}}{ds}=-\alpha_{\nu}I_{\nu}+j_{\nu}$ (12)

using the synchrotron emissivity $j_{\nu}$ and absorptivity $\alpha_{\nu}$
coefficients, considering various electron distribution functions. This allows
us to determine the flux centroid for each epoch and trace its motion. In
addition to astrometry we also determine the total flux emitted as the sum of
the intensity weighted by the element of solid angle subtended by each pixel,
again, for each epoch which allows us to plot the light curve.

The images produced are 1000x1000 pixels with a field of view of $300\ \mu as$
vertically and horizontally which makes a resolution $<0.1\ \mu as^{2}$/pixel.
This high resolution is needed to resolve properly the secondary image which
has a very important role in both astrometry and light curve (see Sect. 2).

We model the quiescent state of Sgr A* at $2.2\ \mu m$ with a jet (see Sect.
2.1). However, computing an image of the jet is $\sim 200$ times longer than
an image of the flare source (i.e. the hot spot or the plasmoid, Sect. 2 and 3
respectively) because the jet is much more extended, and integrating the
radiative transfer equation is thus much longer. The absorption in the jet is
negligible thus the flux emitted by the flare which crosses the jet is fully
transmitted. We can compute a single image of the jet that we add to each
images of the hot spot a posteriori. We then calculate the total flux by
summing the jet flux with the one of the flare at a given time. The final
centroid position is calculated by a simple barycenter of the two centroids
(jet and flare).

##  Appendix C Intrinsic emission of the Plasmoid

###  C.1 Tests on the kinetic simulations

In our model, we follow the evolution of the electron distribution taking into
account the injection of accelerated electrons by the merging of small
plasmoids into our large plasmoid and their cooling through synchrotron
radiation. The emissivity $j_{\nu}$ and absorptivity $\alpha_{\nu}$
coefficients, needed to integrate the radiative transfer Eq. 12, are computed
through the formula of Chiaberge & Ghisellini (1999); Rybicki & Lightman
(1986) (with our notation)

$j_{\nu}(t)=\frac{1}{4\pi}\int_{\gamma_{min}}^{\gamma_{max}}d\gamma
N_{e}(\gamma,t)P_{s}(\nu,\gamma),$ (13) $\alpha_{\nu}(t)=-\frac{1}{8\pi
m_{e}\nu^{2}}\int_{\gamma_{min}}^{\gamma_{max}}\frac{N_{e}(\gamma,t)}{\gamma
l}\frac{d}{d\gamma}[\gamma lP_{s}(\nu,\gamma)]$ (14)

with

$P_{s}(\nu,\gamma)=\frac{3\sqrt{3}}{\pi}\frac{\sigma_{T}cU_{B}}{\nu_{B}}x^{2}\\\
\left\\{K_{4/3}(x)K_{1/3}(x)-\frac{3}{5}x[K_{4/3}^{2}(x)-K_{1/3}^{2}(x)]\right\\}$
(15)

where $l=(\gamma^{2}-1)^{1/2}$ is the electron momentum in units of $m_{e}c$,
$x=\nu/(3\gamma^{2}\nu_{B})$, $\nu_{B}=eB/(2\pi m_{e}c)$ and $K_{a}(t)$ is the
modified Bessel function of order $a$. We note that the Eq. 15, is already
averaged over pitch angle. For standard distributions as thermal, power-law
and $\kappa$-distributions, these formulae are equivalent to the fits of
Pandya et al. (2016) (see Appendix D) that we used for computing the quiescent
synchrotron flux.

As electrons start to cool as soon as they are injected in the plasmoid, the
full distribution is no more a $\kappa$-distribution. However, turning off the
cooling during the growth phase allows us to compare the results of EMBLEM to
the fitting formulae of Pandya et al. (2016). As we inject electrons following
their definition of the $\kappa$-distribution with a linear increase of the
number density, the two approach show similar results (see Appendix D). In our
cases, the absorption is very low, thus we neglect the absorption in these
tests. We can derive an analytical formula for the specific intensity from the
high frequency limit emissivity Eq. 24 depending on the number density
$n_{e}$, the electron temperature $\Theta_{e}$ and the magnetic field $B$ in
case the cooling is switched off during growth. We find in the case without
cooling that, keeping $\kappa$ constant

$I_{\nu,max}\propto n_{e,max}\ \Theta^{\ \kappa-2}\ B^{\ \kappa/2}\text{, if
}X_{\kappa}>2000.$ (16)

We show the relative error of the maximum specific intensity between the
EMBLEM code (red dots) and the formulae of Pandya et al. (2016) (black curve)
depending on the electron temperature and the magnetic field in Fig. 9 and 10.
We fix the others parameters to $n_{e}=5\times 10^{6}$ cm-3, $\kappa=4$,
$t_{\mathrm{growth}}=75$ $r_{g}/c$. The values of EMBLEM are in good agreement
with the previous analytical expression (Eq. 16) for low values of
$\Theta_{e}$ and $B$. For high values, we are beyond the validity of our
approximations (in the intermediate frequency regime of the fitting formula,
see Appendix D). Comparing the results of EMBLEM (without cooling) with the
full fitting formula of Pandya et al. (2016) (black curves) results in an
error lower than $5\%$ showing the good agreement between the two approaches.

Figure 9: Specific intensity at the end of the growth phase
($t=t_{\mathrm{growth}}=75$ $r_{g}/c$) of a $\kappa$-distribution with
$n_{e}=5\times 10^{6}$ cm-3, $B=10$ G, $\kappa=4$ for a range of $\Theta_{e}$
computed from the full fitting formulae of Pandya et al. (2016) (black curve),
by the EMBLEM code (red dots) and with the high-frequency limit analytical
expression (dashed blue curve). We overplot in light blue the range of
$\Theta_{e}$ where $X_{\kappa}>2000$ i.e. where the relative error between the
high frequency limit and the full formula is lower than $20\%$.

Figure 10: Same as in Fig.9 for a range of $B$ and with $\Theta_{e}=10$.

###  C.2 Analytical estimate of the intrinsic light curve

Next, we compute the light curve emitted by the plasmoid which will be
affected by the relativistic effects. To reproduce a given light curve, we can
estimate the values of the parameters through characteristic scales. The
growth time, which is a ”free” parameter of the model, can be estimated from
the light curve taking into account the beaming effect and thus, depends on
the orbital parameters. The synchrotron cooling time of an electron with
Lorentz factor $\gamma$ in a magnetic field $B$ reads

$t_{cool}=\frac{3}{4}\frac{m_{e}c}{\sigma_{T}U_{B}\gamma}$ (17)

with $\sigma_{T}$ the electron Thomson cross section and $U_{B}$ the magnetic
energy density. In a Dirac spectrum approximation, the Lorentz factor of an
electron emitting an IR photon at $2.2\ \mu m$ is (Rybicki & Lightman 1979)

$\bar{\gamma}=\left(\dfrac{\nu m_{e}c}{\eta eB}\right)^{1/2}$ (18)

with $\eta=(0.29\times 3)/(2\pi)$ a dimensionless numerical factor (see
Appendix E.2). One can thus constrain the magnetic field from the synchrotron
cooling time as

$t_{cool}=19\times\left(\frac{B}{20G}\right)^{-1.5}\
\left(\frac{\lambda}{2.2\mu m}\right)^{0.5}\text{min.}$ (19)

Taking into account the cooling of the electrons during the growth phase leads
to a lower flux than what we estimate from Eq. 16. Indeed, as electrons start
to cool directly after being injected, the integral of the distribution in Eq.
13 and so the emissivity will always be lower than without cooling. The key
parameter of synchrotron cooling is the cooling time scale (Eq. 17), which
depends on the magnetic field strength and the initial energy of the
electrons. It has to be compared to the growth time. Indeed, with a low growth
time, only high-energy electrons have the time to cool. Increasing the growth
time will allow lower energy electrons to cool and so decrease even more the
maximum flux of the light curve.

With some approximations (see Appendix E for the details), one can estimate
the flux with cooling at $t=t_{\mathrm{growth}}$

$\nu
F_{\nu}^{\mathrm{syn}}(\nu,t)=\dfrac{n_{e}R_{b}^{3}\,\bar{\gamma}m_{e}c^{2}}{12D^{2}t_{\mathrm{growth}}\,\kappa\theta^{2}}\begin{cases}\left[\Psi(\bar{\gamma})-\Psi(\xi(\bar{\gamma},t))\right],&\text{for
}\nu<\tilde{\nu}(t)\\\ \Psi(\bar{\gamma}),&\text{for
}\nu\geq\tilde{\nu}(t)\end{cases}$ (20)

where $\tilde{\nu}(t)=(\eta eB)/(m_{e}cb_{c}^{2}t^{2})$ is the frequency
corresponding to the condition $\bar{\gamma}=1/(b_{c}t)$ and

$\Psi(x)=\left(1+\dfrac{x-1}{\kappa\theta}\right)^{-\kappa}\,\left[x^{2}(\kappa-1)+2x(\kappa\theta-1)+2\theta(\kappa\theta-2)\right].$
(21)

We plot the maximum light curve evolution relative to the magnetic field with
EMBLEM with (blue crosses) and without (red crosses) cooling during the growth
phase and the previous analytical expression in Fig. 11 (black line). As
expected, the cooling becomes more significant with a strong magnetic field
until the maximum flux starts to decrease for very high values ($B>100$ G).
The two regimes of Eq. 20 are clearly visible in Fig. 11 with a turning point
at $B=16.2$ G. This approximation has a maximum relative error lower than
$30\%$ compared to the results of EMBLEM in the stationary regime and below
$7\%$ for the non stationary regime making it a good approximation estimate
the peak light curve flux.

Figure 11: Evolution of the maximum flux $\nu F_{\nu}(t_{\mathrm{growth}})$
(at the end of the growth phase $t_{\mathrm{growth}}=75$ $r_{g}/c$) as a
function of the magnetic field. We show the results of EMBLEM without cooling
(red crosses) as in Fig. 10. Allowing the cooling during the growth phase
results in a lower maximum flux (blue crosses). One can estimates the maximum
flux with cooling through Eq. 20 (see Appendix E for details). This equation
is divided in two regimes, the equilibrium regime where the magnetic field is
strong enough to compensate the injection and creates a stationary state
($B\geq 16.2$ G) and non stationary regime where not all electrons has cooled
at $t_{\mathrm{growth}}$ ($B<16.2$ G). The relative error between the
analytical formula and the results of EMBLEM (with cooling) is below 30$\%$ in
the whole domain and below 7$\%$ in the non stationary regime.

##  Appendix D Computation of the synchrotron coefficients for the Plasmoid

###  D.1 Fitting formulae of Pandya et al. (2016)

In the hot spot model and for the test of EMBLEM, we used the fitting formula
of Pandya et al. (2016) to compute the emissivity $j_{\nu}$ and absorptivity
$\alpha_{\nu}$ considering a well defined $\kappa$-distribution. This
distribution has two regimes, the low and high frequency regimes.

In the low frequency limit, the emissivity is

$j_{\nu,low}=\frac{n_{e}e^{2}\nu_{B}}{c}\ X_{\kappa}^{1/3}\ sin(\theta)\
\frac{4\pi\Gamma(\kappa-4/3)}{3^{7/3}\Gamma(\kappa-2)}\\\ $ (22)

and the absorption coefficient is

$\alpha_{\nu,low}=\frac{n_{e}e^{2}}{\nu m_{e}c}\
X_{\kappa}^{-2/3}3^{1/6}\frac{10}{41}\ \frac{2\pi}{(\Theta_{e}\
\kappa)^{10/3-\kappa}}\frac{(\kappa-2)(\kappa-1)\kappa}{3\kappa-1}\\\
\times\Gamma\left(\frac{5}{3}\right)_{2}F_{1}\left(\kappa-\frac{1}{3},\kappa+1,\kappa+\frac{2}{3},-\Theta_{e}\
\kappa\right)$ (23)

where ${}_{2}F_{1}$ is the hypergeometric function.

In the high-frequency limit, the emissivity is

$j_{\nu,high}=\frac{n_{e}e^{2}\nu_{B}}{c}\ X_{\kappa}^{-(\kappa-2)/2}\
sin(\theta)\ 3^{(\kappa-1)/2}\\\ \times\frac{(\kappa-2)(\kappa-1)}{4}\
\Gamma\left(\frac{\kappa}{4}-\frac{1}{3}\right)\Gamma\left(\frac{\kappa}{4}+\frac{4}{3}\right)$
(24)

and the absorption coefficient is

$\alpha_{\nu,high}=\frac{n_{e}e^{2}}{\nu m_{e}c}\ X_{\kappa}^{-(1+\kappa)/2}\
\frac{\pi^{3/2}}{3}\ \frac{(\kappa-2)(\kappa-1)\kappa}{(\Theta_{e}\
\kappa)^{3}}\\\ \times\left(\frac{2\Gamma(2+\kappa/2)}{2+\kappa}-1\right)\
\left(\left(\frac{3}{\kappa}\right)^{19/4}+\frac{3}{5}\right).$ (25)

The final approximations for the emissivity and absorption coefficient are

$j_{\nu}=\left(j_{\nu,low}^{-x_{j}}+j_{\nu,high}^{-x_{j}}\right)^{-1/x_{j}}$
(26)
$\alpha_{\nu}=\left(\alpha_{\nu,low}^{-x_{\alpha}}+\alpha_{\nu,high}^{-x_{\alpha}}\right)^{-1/x_{\alpha}}$
(27)

with $x_{j}=3\kappa^{-3/2}$ and
$x_{\alpha}=\left(-\frac{7}{4}+\frac{8}{5}\kappa\right)^{-43/50}$.

The two frequency limits do not have the same dependence on the parameters.
The frequency regime is defined by the dimensionless parameter
$X_{\kappa}=\nu/\nu_{\kappa}$, with
$\nu_{\kappa}=\nu_{B}(\Theta_{e}\kappa)^{2}$. Fig. 12 shows the relative error
of the two regimes (the low frequency in red and the high frequency in blue)
compared to the final emission coefficient. While at very high (respectively
very low) $X_{\kappa}$, the high frequency (resp. low frequency) fitting
formulae work very well, there is a large frequency regime ($10^{-2}\lesssim
X_{\kappa}\lesssim 10^{3}$), hereafter intermediate regime, where both limits
are needed. At $2.2$ $\mu m$, $X_{\kappa}>1$, while $\Theta_{e}\
\kappa\lesssim 10^{3}$ which correspond to our typical set of parameters. This
is why we used the high frequency regime for our test our EMBLEM.

Figure 12: Relative error between the low frequency regime (in red) - resp.
the high frequency regime (in blue) - fit formulae $J_{s,lo}$ (resp.
$J_{s,hi}$) of Pandya et al. (2016) and the full fit formula of the emission
coefficient $J_{s}$ in function of $X_{\kappa}=\frac{\nu}{\nu_{\kappa}}$ with
$\nu_{\kappa}=\nu_{B}\ (\Theta_{e}\kappa)^{2}$.

###  D.2 Chiaberge & Ghisellini (1999) approximation

Modeling the synchrotron cooling of the electrons with a thermal, power-law or
$\kappa$ distribution is not trivial. Indeed, the evolution of the energy of
an electron which emits synchrotron radiation is (e.g., Rybicki & Lightman
(1986))

$\gamma(t)=\gamma_{0}(1+A\gamma_{0}t)^{-1}$ (28) $\text{with
}A=\frac{4}{3}\frac{\sigma_{T}B^{2}}{8\pi m_{e}c},$ (29)

$\gamma$ the Lorentz factor of the electron at time $t$, and $\gamma_{0}$ the
initial Lorentz factor. The energy evolution strongly depends on the initial
energy. The higher the initial energy of the electron, the faster it will
cool. Thus, the initial distribution we could impose will quickly be deformed
(see top-left panel of Fig. 6) and could not be modeled by one (or more) of
the three distribution of Pandya et al. (2016) (thermal, power-law and/or
$\kappa$).

In order to properly model the cooling of electrons, we simulate the evolution
of the electron distribution with injection and synchrotron cooling (Sect. 3).
These simulations give us the electron distribution $N_{e}(\gamma,t)$ at
different times. We compute the emissivity $j_{\nu}$ and the absorptivity
$\alpha_{\nu}$ associated for a range of frequencies from $10^{6}$ to
$10^{21}$ Hz following the formula of Chiaberge & Ghisellini (1999) (with our
notation)

$j_{\nu}(t)=\frac{1}{4\pi}\int_{\gamma_{min}}^{\gamma_{max}}d\gamma
N_{e}(\gamma,t)P_{s}(\nu,\gamma)$ (30)

and the absorption coefficient follows

$\alpha_{\nu}(t)=-\frac{1}{8\pi
m_{e}\nu^{2}}\int_{\gamma_{min}}^{\gamma_{max}}\frac{N_{e}(\gamma,t)}{\gamma
p}\frac{d}{d\gamma}[\gamma pP_{s}(\nu,\gamma)],$ (31)

where $p=(\gamma^{2}-1)^{1/2}$ is the electron momentum in units of $m_{e}c$
and $P_{s}$ is the emissivity of a single electron (see 15).

In order to obtain the emissivity and absorption coefficient at any time and
any frequency (to account the relativistic Doppler effect for example), we
made a bilinear interpolation.

##  Appendix E Analytical approximation for Sgr A* flare peak flux

Here we derive an analytical expression to compute the time-dependent flux
from Sgr A* flares during the growth phase, and obtain an analytical formula
for the peak flare flux. For that, we first obtain the approximate analytical
form of the varying electron spectrum during the growth phase by solving the
kinetic equation, and then compute the approximate synchrotron SED associated
to the time-dependent electron spectrum.

###  E.1 Deriving time-dependent electron spectrum during the growth phase

The kinetic equation describing the evolution of the electron spectrum during
the growth phase is given by Eq. 5:

$\dfrac{\partial N_{e}(\gamma,t)}{\partial
t}=\dfrac{\partial}{\partial\gamma}\left(b_{c}\gamma^{2}N_{e}(\gamma,t)\right)\,+\,Q_{\mathrm{inj}}(\gamma,n_{e},\theta,\kappa)$
(32)

with the injection term $Q_{\mathrm{inj}}(\gamma)$ given by Eq. 7 and Eq. 8,
and synchrotron cooling term
$\dot{\gamma}_{\mathrm{syn}}=-b_{c}(\gamma^{2}-1)$ (see Eq. 6), where
$b_{c}=(4\sigma_{T}U_{B})/(3m_{e}c)$. We use here an approximation
$\dot{\gamma}_{\mathrm{syn}}\approx-b_{c}\gamma^{2}$, as the bulk of the
electrons producing the flare emission are highly relativistic.

We use the method of characteristics to solve the kinetic equation. We search
for characteristic curves in the $\gamma$-t space, along which our
differential equation in partial derivatives becomes an ordinary differential
equation. Let us rewrite the kinetic equation in the following form, expanding
the derivative on the Lorentz factor:

$\dfrac{\partial N_{e}(\gamma,t)}{\partial
t}\,+\,(-1)b_{c}\gamma^{2}\,\dfrac{\partial
N_{e}(\gamma,t)}{\partial\gamma}=Q_{\mathrm{inj}}(\gamma)+2\gamma
b_{c}N_{e}(\gamma,t)$ (33)

When restricting our equation to the characteristic curve ($\gamma(t)$,$t$),
the full derivative of the electron spectrum over time, by the chain rule, is:

$\dfrac{dN_{e}(\gamma,t)}{dt}\,=\,\dfrac{\partial N_{e}(\gamma,t)}{\partial
t}\,+\,\dfrac{d\gamma}{dt}\,\dfrac{\partial N_{e}(\gamma,t)}{\partial\gamma}$
(34)

Comparing this to Eq. 33, we identify
$(-1)b_{c}\gamma^{2}=\dfrac{d\gamma}{dt}$, and therefore along the chosen
characteristic curve, our equation is split into a system of two ordinary
differential equations:

$\begin{cases}d\gamma/dt=-b_{c}\gamma^{2}\\\
dN_{e}(\gamma,t)/dt=Q_{\mathrm{inj}}(\gamma)+2\gamma
b_{c}N_{e}(\gamma,t)\end{cases}$ (35)

The solution of the first equation is (applying the initial condition that
$\gamma(t=0)=\xi$):

$\gamma(t)=\dfrac{1}{b_{c}t\,+\,1/\xi}$ (36)

This equation defines a characteristic curve in the $\gamma$-t space. We have
chosen the initial point of the characteristic curve as $(\xi,0)$. The
physical meaning of $\xi$ is the initial value of the Lorentz factor of an
electron before it starts undergoing the cooling process. Eq. 36 is equivalent
to Eq. 28, and describes how the Lorentz factor of an individual electron
evolves in time due to synchrotron cooling. From this equation, the initial
Lorentz factor $\xi$ is:

$\xi=\xi(\gamma,t)=\dfrac{1}{1/\gamma-b_{c}t}$ (37)

This formula defines the initial Lorentz factor of the characteristic curve
that passes through a point ($\gamma$,t). We denote the function
$N_{e}(\gamma_{\xi}(t),t)=u(t)$ (electron spectrum along the characteristic
curve), and solve the second equation in the system:

$du/dt-2b_{c}\gamma(t)u=Q_{\mathrm{inj}}(\gamma(t))$ (38)

The generic solution of this linear differential equation is:

$u(t)=\dfrac{1}{\mu(t)}\,\int_{0}^{t}\,\mu(t^{\prime})\,Q_{\mathrm{inj}}(\gamma(t^{\prime}))\,dt^{\prime}\,+\,\dfrac{C}{\mu(t)}$
(39)

with $C$ being the integration constant, and the function $\mu(t)$ being the
integration factor, which is equal to:

$\mu(t)=\mathrm{exp}\left(\int-2b_{c}\gamma(t)dt\right)\,=\,\dfrac{1}{(b_{c}t+1/\xi)^{2}}$
(40)

As the electron spectrum at $t=0$ is zero, we set the initial condition
$u(t=0)=0$, which results in $C=0$. Therefore, the solution for $u(t)$ is:

$u(t)\,=\,(b_{c}t+1/\xi)^{2}\,\int_{0}^{t}\,(b_{c}t^{\prime}+1/\xi)^{-2}\,Q_{\mathrm{inj}}(\gamma(t^{\prime}))\,dt^{\prime}$
(41)

Now we have to return back from $u(t)$ to $N_{e}(\gamma,t)$, which is achieved
by substitution of the equation for $\xi=\xi(\gamma,t)$ (Eq. 37) to the
expression for $u(t)$. After doing that, we obtain an expression for the
electron spectrum at a moment of time $t$:

$N_{e}(\gamma,t)=\dfrac{1}{\gamma^{2}}\int_{0}^{t}\Gamma^{2}\,Q_{\mathrm{inj}}(\Gamma)\,dt^{\prime}$
(42)

with
$\Gamma=\Gamma(\gamma,t,t^{\prime})=\left[1/\gamma+b_{c}(t^{\prime}-t)\right]^{-1}$.
We use here an approximation for $Q_{\mathrm{inj}}(\Gamma)$, and more
specifically, for the kappa distribution, to enable analytical integration. As
we are in the relativistic regime, and the peak of the injection spectrum in
our case typically occurs at Lorentz factors $\gamma\gg 1$, we can substitute
$\gamma(\gamma^{2}-1)^{1/2}$ with $\gamma^{2}$ in the Eq. 8. This leads to
some inaccuracies only at very low Lorentz factors, which virtually do not
contribute to the integral value, and do not contribute to the light curve
flux. We therefore use for the injected spectrum:

$Q_{\mathrm{inj}}(\gamma,n_{e},\theta,\kappa)\approx\frac{N}{t_{\mathrm{growth}}}\gamma^{2}\left(1+\frac{\gamma-1}{\kappa\theta}\right)^{-(\kappa+1)}$
(43)

Now we can perform the analytical integration. We use the variable
substitution from $t^{\prime}$ to $\Gamma(\gamma,t,t^{\prime})$. In this case,
the differential $dt^{\prime}=-b_{c}^{-1}\Gamma^{-2}d\Gamma$. Our integral
(Eq. 42) then becomes:

$N_{e}(\gamma,t)=\dfrac{N}{\gamma^{2}t_{\mathrm{growth}}}\int_{0}^{t}\Gamma^{4}\left(1+\dfrac{\Gamma-1}{\kappa\theta}\right)^{-(\kappa+1)}\,dt^{\prime}=\\\
=-\dfrac{N}{b_{c}\gamma^{2}t_{\mathrm{growth}}}\,\int_{0}^{t}\Gamma^{2}\left(1+\dfrac{\Gamma-1}{\kappa\theta}\right)^{-(\kappa+1)}\,d\Gamma$
(44)

To solve the integral, we perform integration by parts, and we obtain:

$\int_{0}^{t}\Gamma^{2}\left(1+\dfrac{\Gamma-1}{\kappa\theta}\right)^{-(\kappa+1)}\,d\Gamma=-\dfrac{\theta\kappa}{(\kappa-2)(\kappa-1)}\Psi(\Gamma)\,\bigg{|}_{0}^{t}$
(45)

with

$\Psi(x)=\left(1+\dfrac{x-1}{\kappa\theta}\right)^{-\kappa}\,\left[x^{2}(\kappa-1)+2x(\kappa\theta-1)+2\theta(\kappa\theta-2)\right]$
(46)

We substitute the variable back from $\Gamma$ to $t^{\prime}$, with
$\Gamma(t^{\prime}=0)=(1/\gamma-b_{c}t)^{-1}=\xi(\gamma,t)$ and
$\Gamma(t^{\prime}=t)=\gamma$, as well as substitute the expression for the
injection spectrum normalization,
$N=(1/2)n_{e}(\kappa-2)(\kappa-1)\kappa^{-2}\theta^{-3}$ (see Eq. 8), and
obtain:

$N_{e}(\gamma,t)\,=\,\dfrac{n_{e}}{2\kappa\theta^{2}b_{c}\gamma^{2}t_{\mathrm{growth}}}\left[\Psi(\gamma)-\Psi(\xi(\gamma,t))\right]$
(47)

One has to consider separately a special case when the $b_{c}t\geq 1/\gamma$,
as this leads to either $\xi\rightarrow\infty$ or $\xi<0$. Obviously, the
latter situation is non-physical, as the Lorentz factor cannot be less than
unity. Qualitatively, $b_{c}t\geq 1/\gamma\,\rightarrow\,t\geq
1/(b_{c}\gamma)$ means that the evolution time of an electron is longer than
its cooling time-scale, and in this regime the equilibrium between the
injection and cooling is already reached. Therefore, one can easily see that
the time-dependent electron spectrum in the Lorentz factor domain $\gamma\geq
1/(b_{c}t)$ will be “frozen” at the steady-state one. A steady-state solution
corresponds to $\xi\rightarrow\infty$, which results in $\Psi(\xi)\rightarrow
0$ (in case $\kappa>2$). Therefore, the final solution for the time-dependent
electron spectrum during the growth phase, is:

$N_{e}(\gamma,t)\,=\dfrac{n_{e}}{2\kappa\theta^{2}b_{c}\gamma^{2}t_{\mathrm{growth}}}\begin{cases}\left[\Psi(\gamma)-\Psi(\xi(\gamma,t))\right],&\text{for
}\gamma<(b_{c}t)^{-1}\\\ \Psi(\gamma),&\text{for
}\gamma\geq(b_{c}t)^{-1}\end{cases}$ (48)

It is worth to note, that one can obtain the same steady-state solution (the
case $\gamma\geq 1/(b_{c}t)$) by directly solving the kinetic equation (Eq.
32) with $\frac{\partial N_{e}}{\partial t}=0$. To find the electron spectrum
at the peak of the flare, i.e. at the moment when the injection is stopped,
one simply calculates $N_{e}(\gamma,t=t_{\mathrm{growth}})$.

###  E.2 Deriving time-dependent synchrotron SED during the growth phase

Now let us proceed to the SED and light curve computation. We use the so-
called $\delta$-approximation for the electron synchrotron emissivity
coefficient. This approximation assumes that a single electron with a Lorentz
factor $\gamma$ emits at a single frequency, rather than a broad spectrum
(Rybicki & Lightman 1979):

$\omega_{\mathrm{peak}}\simeq 0.29\omega_{c}$ (49)

with $\omega_{c}=3\gamma^{2}eB/(m_{e}c)$ (averaged over pitch angles), $e$
being the electron charge, and $B$ being the magnetic field (CGS units). From
this expression, one obtains:

$\nu_{\mathrm{peak}}=\dfrac{\eta e\gamma^{2}B}{m_{e}c}$ (50)

where $\eta=(0.29\times 3)/(2\pi)\approx 0.14$ is a dimensionless numerical
factor. For a distribution of electrons, the synchrotron SED in
$\delta$-approximation, is given by (Dermer & Schlickeiser 2002):

$\nu F_{\nu}^{\mathrm{syn}}(\lambda)=\dfrac{4}{3}\pi
R_{b}^{3}\dfrac{c\sigma_{T}U_{B}}{6\pi
D^{2}}\bar{\gamma}^{3}N_{e}(\bar{\gamma})$ (51)

where $R_{b}$ is the radius of the emitting region, $D$ is the distance
between the observer and the source, and $\bar{\gamma}$ is the Lorentz factor
of electrons emitting synchrotron photons with the frequency $\nu$. We obtain
this Lorentz factor by expressing it from Eq. 50:

$\bar{\gamma}=\left(\dfrac{m_{e}c\nu}{\eta eB}\right)^{1/2}$ (52)

Substituting the expression for $N_{e}(\gamma,t)$ (Eq. 48), and the expression
$b_{c}=(4\sigma_{T}U_{B})/(3m_{e}c)$, into the Eq. 51, we finally obtain the
time-dependent SED during the growth phase in $\delta$-approximation:

$\nu
F_{\nu}^{\mathrm{syn}}(\nu,t)=\dfrac{n_{e}R_{b}^{3}\,\bar{\gamma}m_{e}c^{2}}{12D^{2}t_{\mathrm{growth}}\,\kappa\theta^{2}}\begin{cases}\left[\Psi(\bar{\gamma})-\Psi(\xi(\bar{\gamma},t))\right],&\text{for
}\nu<\tilde{\nu}(t)\\\ \Psi(\bar{\gamma}),&\text{for
}\nu\geq\tilde{\nu}(t)\end{cases}$ (53)

where $\tilde{\nu}(t)=(\eta eB)/(m_{e}cb_{c}^{2}t^{2})$ is the frequency
corresponding to the condition $\bar{\gamma}=1/(b_{c}t)$.

Figure 13: Time evolution of the electron distribution with EMBLEM (full
lines) from $t=0$ to $t=t_{\mathrm{growth}}=75$ $r_{g}/c$ injecting a
$\kappa$-distribution with $\Theta_{e}=10$ and $\kappa=4$ and
$\dot{n_{e}}=5.10^{6}/t_{\mathrm{growth}}$. The magnetic field strength is set
to 30 Gauss resulting in a stationary regime for $\gamma>10^{4}$ from the very
beginning. This regime extends to lower $\gamma$ values as time growths. For
the estimation of the peak flux, we approximate the whole distribution (at
$t=t_{\mathrm{growth}}$) by a simple Dirac at $\bar{\gamma}$ represented by
the dashed grey line.

###  E.3 Evaluating the peak light curve flux

To obtain a light curve during the growth phase at a specific frequency of
interest $\nu_{*}$, one has to compute $\nu
F_{\nu}^{\mathrm{syn}}(\nu=\nu_{*},t)$. To compute the peak light curve flux,
one evaluates the quantity $\nu
F_{\nu}^{\mathrm{syn}}(\nu=\nu_{*},t=t_{\mathrm{growth}})$.

##  Appendix F Additional Setup for July 22 flare

We also find another setup which reproduce well the July 22 flare data. In
such scenario, the magnetic reconnection and so the plasmoid growth phase
occurs way before the observing time and the flare is due to the beaming
effect combined to the slow decrease of the cooling phase. The peak due to the
growth phase occurs during the negative beaming part of the orbit resulting in
a low flux comparable to the quiescent state.

Parameter | Symbol | July 22 bis
---|---|---
Plasmoid |  |
time in EMBLEM at zero observing time [min] | $t_{obs,0}^{emblem}$ | $-53$
initial orbital radius [$GM/c^{2}$] | $r_{0}$ | $15$
polar angle [$\degree$] | $\theta$ | $135$
initial azimuthal angle [$\degree$] | $\varphi_{0}$ | $240$
initial radial velocity [$c$] | $v_{r,0}$ | $0.01$
initial azimuthal velocity [$c$] | $v_{\varphi,0}$ | $0.5$
X position of Sgr A* [$\mu as$] | $x_{0}$ | $0$
Y position of Sgr A* [$\mu as$] | $y_{0}$ | $0$
PALN [$\degree$] | $\Omega$ | $160$
Table 5: Second orbital parameters of the plasmoid model following a conical
motion used for the comparison of the July 22 flares observed by Gravity
Collaboration et al. (2018). Figure 14: Data and plasmoid models of the flares
from July 22, 2018. The left panels shows the astrometry of the flare while
the right panel shows the light curves. Note that this is not the result of a
fit. Contrary to the setup for Fig. 8, the growth time is shorter
$t_{\mathrm{growth}}=50r_{g}/c$ resulting into a two peak light curve with the
first one occurring at $t=-22$ min but being mitigate by the negative beaming
effect. The secondary peak which match the observed flare data shown here is
due to the positive beaming during the cooling phase (as shown by the
intrinsic light curve). The parameter set for this model is similar to the set
of July 22 and is listed in Table 5 with the same physical parameter as in
Table 3 but with $t_{\mathrm{growth}}=50r_{g}/c$, $\Theta_{e}=72$ and $B=10$
G.