Magnetic reconnection plasmoid model for Sagittarius A* flares

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 with super Keplerian velocity. 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. We consider a spherical large plasmoid ejected along a conical orbit around the black hole. This plasmoid is assumed to be formed by successive mergers of smaller plasmoids produced through magnetic reconnection. Non-thermal electrons are injected in the plasmoid. We compute the evolution of the electron-distribution under the influence of synchrotron cooling. We solve the radiative transfer problem 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. 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 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. These results support the picture of magnetic reconnection as a viable model for Sagittarius~A* infrared flares.


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) × 10 −9 M yr −1 and a bolometric luminosity of (6.8 − 9.2) × 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 ∼ 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 submm but with a time lag of several minutes (Eckart et al. 2008(Eckart et al. , 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. 2008Eisenhauer et al. , 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 ∼ 20 • ) 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(Komissarov , 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;?), 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 γ ∼ 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 100r 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 possi-ble 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.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-σ 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.

Quiescent flow impact on astrometry: shifting and rotating the orbit
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.

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 κ 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 κ 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  2020a) of 1.1 ± 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.
The hot spot model is composed of a plasma sphere of radius 1 r g (fixed) with a uniform but time-dependent κ-distribution for the electrons.The emissivity j ν and absorptivity α ν 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 T e (t) = T hs e exp where t σ 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 σ = B 2 /4πm 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. Hz 1 ] Fig. 2. Left: Spectrum associated to the best-fit of the torus-jet model (see Table 1) for the quiescent state of Sgr A* (χ 2 red = 0.91 with ndof=27).The data are taken from Bower et al. (2015) for ν < 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 µm of the torus-jet model with a field of view of 150 µas seen with an inclination of 20 • and a Position Angle of the Line of Nodes (PALN) of π 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 ∼(0, −2.2). case of a constant-emission hot spot, as well as the varyingemission 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.

Parameter
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-σ significance in the the Gravity Collaboration et al. (2018) data.
We note another non-trivial effect appearing in the varyingemission hot-spot orbit without any quiescent radiation (reddotted 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).

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 ϕ 0 of the hot spot along its orbit, in order to change the dephasing between the time of the maximum intrinsic emission (t = t ref ) which is fixed, and the time

Light Curve
Fig. 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 σ >> 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 σ = 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.Ω 160

Parameter
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.
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 ϕ 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.

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 sec-tion 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 rec /v out with v rec the velocity of matter injected into the reconnection region, and v out the bulk outflow velocity of particles accelerated by the reconnection event.The outflow velocity is of the order of the Alfven speed, v out ≈ v A , which is itself of the order of the speed of light, v A ≈ 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 rec,PIC ≈ 0.1v A ≈ 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*1 .GRRMHD simulations point towards a slower rate of around 1%, so that v rec,MHD ≈ 0.01c (see the discussion in Ripperda et al. 2022) in green and 270 • 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 ϕ 0 as the maximum of emission occurs at different ϕ.The Gaussian modulation which has a typical duration of t σ = 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*. 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 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 ∼ 100r g /c ∼ 30 min and the resulting hot spot orbits for ∼ 500r g /c ∼ 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 Ypoint 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).

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.

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 θ = θ 0 and the initial conditions r 0 , θ 0 , ϕ 0 , v r0 , and v ϕ0 .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 The azimuthal angle is obtained by integrating the previous Eq. 3 ). (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.

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 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 growth corresponds to the lifetime of the reconnection engine, i.e. the duration of magnetic flux dissipation; after t = t 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 100r 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 2r 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 (∼ 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 ∼200r 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 growth of the growth phase of the order of 100r 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.

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.D.7 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 where γ is the Lorentz factor of the electrons, Q inj is the injection rate and N e = dn e /dγ is the electron number density distribution, using the EMBLEM code (Dmytriiev et al. 2021).The term of the right hand side describes the synchrotron cooling of the plasmoid particles, with U B = B 2 /(8π).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 inj (γ) in Eq. 5, we use the following expression, assuming a constant injection rate: 4πN κ e (γ) t growth in the growth phase, 0 in the cooling phase, where N κ e (γ) is the distribution of the injected particles which follows the kappa distribution i.e. a thermal core with a powerlaw tail following: with a normalization factor N = 1/2n e (κ−2)(κ−1)κ −2 Θ −3 e , where n e and Θ e are the density and dimensionless temperature of the injected plasma.The index κ is defined as where following Ball et al. (2018); Werner et al. (2018), where p is the powerlaw index of the non-thermal part of the distribution, σ b 1 is the plasma magnetization of the accelerating site and β b 1 is the ratio of proton thermal pressure to magnetic pressure of the accelerating site.If the magnetization at the accelerating site satisfies σ b ≥ 100 (Crinquand et al. 2021), then κ is in the range of [2.8, 4.4] depending on β b .This implies that the spectral index α (νF ν ∝ ν α ) 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 κ, closer to the low boundary.
The bounds of the electron Lorentz factor are chosen to satisfy γ min = 1 and γ 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) = n max e × t/t growth in the growth phase, n max e in the cooling phase. (11) Such high maximum Lorentz factor is needed to also power Xray flares with only synchrotron.However, Ripperda et al. (2022) suggest a lower maximum Lorentz factor γ max ∼ 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 EM-BLEM.This step is performed by means of the GYOTO 2 raytracing 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 σ p ∼ 10 −2 from the end of the growth phase since neither the density nor the magnetic field evolve.

Importance of evolving the electron distribution
One of the most important aspects of our model is the selfconsistent 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 γ > 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.D.7 and assuming Θ e = γ/3.While this model gives a similar intrinsic light curve as our model, the dimensionless temperature required is twice as high as ours (Θ 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 nonthermal 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) κ-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.

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, ϕ 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 κ-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 (Θ e , κ) 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 ∼ 0.31c (for our choice of initial cylindrical radius given in 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 f p , of a separatrix giving rise to a Y point located at a cylindrical radius of ≈ 10 r g .We find r f p = (4.7 ± 0.5) r g , which translates in an orbital velocity v ϕ,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 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.3) but with the dimensionless temperature of Θ 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 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 κ-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 ∼ 0.1 Jy which is the good order of magnitude of the variability associated to flares in sub-mm (Wielgus et al. 2022a).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.

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 γ max = 10 6 to be able to power X-ray flares (but without any constrain for this study).However, high energy photons lead to pairproduction 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.

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. C.3) 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 κ 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.using the synchrotron emissivity j ν and absorptivity α ν 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 µas vertically and horizontally which makes a resolution < 0.1 µas 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 µm with a jet (see Sect. 2.1).However, computing an image of the jet is ∼ 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
Appendix 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 ν and absorptivity α ν coefficients, needed to integrate the radiative transfer Eq.B.1, are computed through the formula of Chiaberge & Ghisellini (1999); Rybicki & Lightman (1986) (with our notation) where l = (γ 2 − 1) 1/2 is the electron momentum in units of m e c, x = ν/(3γ 2 ν B ), ν B = eB/(2πm e c) and K a (t) is the modified Bessel function of order a.We note that the Eq.C.3, is already averaged over pitch angle.For standard distributions as thermal, power-law and κ-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 κ-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 (2016) (black curve), by the EMBLEM code (red dots) and with the highfrequency limit analytical expression (dashed blue curve).We overplot in light blue the range of Θ e where X κ > 2000 i.e.where the relative error between the high frequency limit and the full formula is lower than 20%.
the κ-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.D.3 depending on the number density n e , the electron temperature Θ e and the magnetic field B in case the cooling is switched off during growth.We find in the case without cooling that, keeping κ constant 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 (C.7) Taking into account the cooling of the electrons during the growth phase leads to a lower flux than what we estimate from Eq. C.4.Indeed, as electrons start to cool directly after being injected, the integral of the distribution in Eq.C.1 and so the emissivity will always be lower than without cooling.The key parameter of synchrotron cooling is the cooling time scale (Eq.C.5), 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 growth (C.9) We plot the maximum light curve evolution relative to the magnetic field with EMBLEM with (blue crosses) and without (red   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 ν and absorptivity α ν considering a well defined κ-distribution.This distribution has two regimes, the low and high frequency regimes.
In the low frequency limit, the emissivity is where 2 F 1 is the hypergeometric function.In the high-frequency limit, the emissivity is .The two frequency limits do not have the same dependence on the parameters.The frequency regime is defined by the dimensionless parameter X κ = ν/ν κ , with ν κ = ν B (Θ e κ) 2 .Fig. D.1 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 κ , the high frequency (resp.low frequency) fitting formulae work very well, there is a large frequency regime (10 −2 X κ 10 3 ), hereafter intermediate regime, where both limits are needed.At 2.2 µm, X κ > 1, while Θ e κ 10 3 which correspond to our typical set of parameters.This is why we used the high frequency regime for our test our EMBLEM.

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.

Fig. 1 .
Fig. 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 θ 1 and θ 2 that describe the angular opening of the radiation-emitting sheath, by the base height z b , the constant Lorentz factor Γ j , and the temperature power-law index s T .The jet is symmetrical with respect to the equatorial plane, and axisymmetric.

Figure 3
Figure3shows the impact of taking into account the quiescent radiation on the astrometry of the flare, considering the trivial the Line of Nodes [ • ] Fig.4.Astrometry (left) and light curves (right) of the hot spot -jet model for four initial azimuthal angle ϕ 0 of 0 • in blue, 90 • in orange, 180 • in green and 270 • 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 ϕ 0 as the maximum of emission occurs at different ϕ.The Gaussian modulation which has a typical duration of t σ = 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*.

Fig. 5 .
Fig. 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.

Fig. 6 .
Fig. 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 κ 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 Table3) but with the dimensionless temperature of Θ 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).

Fig. 8 .
Fig. 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 Tables3 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.
Fig. C.1.Specific intensity at the end of the growth phase (t = t growth = 75 r g /c) of a κ-distribution with n e = 5 × 10 6 cm −3 , B = 10 G, κ = 4 for a range of Θ e computed from the full fitting formulae of Pandya et al. (2016) (black curve), by the EMBLEM code (red dots) and with the highfrequency limit analytical expression (dashed blue curve).We overplot in light blue the range of Θ e where X κ > 2000 i.e.where the relative error between the high frequency limit and the full formula is lower than 20%.
Fig. C.2. Same as in Fig.C.1 for a range of B and with Θ e = 10.
Fig. C.3.Evolution of the maximum flux νF ν (t growth ) (at the end of the growth phase t 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. C.2.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.C.8 (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 ≥ 16.2 G) and non stationary regime where not all electrons has cooled at t 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.
crosses) cooling during the growth phase and the previous analytical expression in Fig. C.3 (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.C.8 are clearly visible in Fig. C.3 with a turning point at B = 16.2G.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.Appendix D: Computation of the synchrotron coefficients for the Plasmoid Appendix D.1: Fitting formulae of Pandya et al. (2016) Fig. D.1.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 κ = ν νκ with ν κ = ν B (Θ e κ) 2 .

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).

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).
).One can estimates the maximum flux with cooling through Eq.C.8 (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 ≥ 16.2 G) and non stationary regime where not all electrons has cooled at t 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.