The 230 GHz Variability of Numerical Models of Sagittarius A*. I. Parameter Surveys on Varying the Ion-to-electron Temperature Ratio Under Strongly Magnetized Conditions

The 230 GHz lightcurves of Sagittarius A* (Sgr A*) predicted by general relativistic magnetohydrodynamics and general relativistic ray-tracing (GRRT) models by the Event Horizon Telescope Collaboration have higher variability M ΔT compared to observations. In this series of papers, we explore the origin of such large brightness variability. In this first paper, we performed large GRRT parameter surveys that span from the optically thin to the optically thick regimes, covering the ion-to-electron temperature ratio under strongly magnetized conditions, R Low, from 1 to 60. We find that increasing R Low can lead to either an increase or a reduction in M ΔT depending on the other model parameters, making it consistent with the observed variability of Sgr A* in some cases. Our analysis of GRRT image snapshots finds that the major contribution to the large M ΔT for the R Low = 1 models comes from the photon ring. However, secondary contributions from the accretion flow are also visible depending on the spin parameter. Our work demonstrates the importance of the electron temperature used for modeling radiatively inefficient accretion flows and places new constraints on the ion-to-electron temperature ratio. A more in-depth analysis for understanding the dependencies of M ΔT on R Low will be performed in subsequent papers.


INTRODUCTION * Croucher Scholar
The nature of the centre of our Galaxy, Sagittarius A* (Sgr A*), was a mystery for decades.Although radio (Lo et al. 1993) and X-ray observations (Baganoff et al. 2003), as well as stellar proper motion data (Eckart et al. 2002), suggested that it could be a supermassive black hole, direct evidence was lacking.It was not until the recent progress made by the Event Horizon Telescope (EHT) Collaboration (Akiyama et al. 2022) that the mystery of Sgr A* was unveiled.By directly imaging the black hole accretion disk, it is now confirmed that Sgr A* is a low-luminosity accreting supermassive black hole.
Sgr A* is estimated to be accreting at a sub-Eddington rate of ∼ 10 −5 M ⊙ year −1 (Quataert et al. 1999;Quataert & Gruzinov 2000).The low luminosity of Sgr A* indicates that it is a radiatively inefficient accretion flow (RIAF) -the accretion disc is believed to be geometrically thick and optically thin while transporting energy to the black hole mainly through advection.Under such circumstances, electrons are decoupled from the ions, and the plasma attains a twotemperature state (Shapiro et al. 1976), because i ) the electrons are unable to cool efficiently by radiation and ii ) Coulomb coupling between the ions and the electrons is weak (Begelman 2014).In addition, electrons could attain a non-thermal ( Özel et al. 2000) number density distribution presumably through acceleration by shocks (Sironi & Spitkovsky 2011), magnetic reconnections (Werner et al. 2018), and/or turbulence (Zhdankin et al. 2019), and electrons are unable to be thermalized easily due to weak Coulomb coupling and low cooling efficiency.When modelling the emission from RIAF, accurate electron distributions and precise temperatures are required.However, both are still poorly understood 1 .To the first-order approximation, it is reasonable to parameterize an energy partition between the electrons and ions.We assume the ion temperature T i is a fraction R of the electron temperature T e .The fraction R could depend on the local environment and/or global accretion flow properties.
Earlier work, such as that by Dexter et al. (2009), assumed R was constant over the entire domain, producing reasonable agreement with the millimetre VLBI visibility of Sgr A*.However, it was soon found to overproduce near-infrared and X-ray fluxes (Mościbrodzka et al. 2009).Following this, Chan et al. (2015) proposed a prescription function where R depends on the plasma magnetization β = 2P/|B| 2 , where P is the gas pressure and B is the magnetic field strength.In particular, R is a step-function of β at some threshold β Crit .This step-function prescription helps clearly distinguish emissions coming from the disk and the jet, where β differs drastically.It was motivated by the fact that the electron temperature can drastically differ from the ion's when the density is low, and the magnetic field is strong.Using this prescription function, Chan et al. (2015) obtained parameter spaces where suites of accreting black hole models agree with observations.
The electron and ion temperature partitioning used by Event Horizon Telescope Collaboration et al. (2022) in their theoretical studies of Sgr A* reads: where b = β/β Crit .This prescription was first used by Mościbrodzka et al. (2016), where R High (R Low ) represent the ion-electron temperature ratios in the weakly (strongly) magnetized regime.This function smooths the sharp transition of the step-function prescription by Chan et al. (2015).It was motivated by the results of particle-in-cell simulations, which suggest that collisionless plasma preferentially heat the ions for β > 1 (see Akiyama et al. 2019, and references therein).The power-law dependencies on b guarantee that radiations from strong and weak ion-electron coupling regions are easily distinguishable.
In Event Horizon Telescope Collaboration et al. ( 2022), both β Crit and R Low are fixed at a constant value of 1.This simple description yielded results for accreting Sgr A* images that passed almost all constraints, such as the image size, visibility amplitude morphology, M-rings fits, infrared and near-infrared fluxes, etc., except for the 230 GHz flux variability.The variability is quantified as the ratio M ∆T = σ ∆T /µ ∆T , where σ ∆T is the standard deviation and µ ∆T is the mean of the 230 GHz flux over a time ∆T .For the EHT Collaboration study, ∆T = 530 GM c −2 = 3 hours.The constraint of M ∆T for Sgr A* could not be met by most of the fiducial models in Event Horizon Telescope Collaboration et al. (2022).The reason why the 230 GHz flux of the theoretical models obtained through generalrelativistic magnetohydrodynamics (GRMHD) simulations and general-relativistic ray-tracing (GRRT) postprocessing are more variable than observations remains a mystery.
Here, we investigate whether varying R Low can reduce the variability of the 230 GHz flux of the theoretical models of Sgr A*.The motivation for varying R Low can be understood in two ways.First, it is the only parameter that has not been studied in-depth before, and it is natural to investigate, with a wider parameter space, if it has any effect on M ∆T .Second, we can consider the functional form of Equation 1, which is visualized in Figure 1.We find that R undergoes a sharp transition around β = 10 −1 and 10 1 , for β Crit = 1 and R High = 160.Given that the major contributions to the   1) with increasing RLow.In this case, R High is fixed at 160, and we assume βCrit = 1.The blue-shaded region indicates where R is sensitive to variations in β.
electromagnetic emissions at 230 GHz are mostly plasma located near the horizon (Dexter et al. 2020), where β has a similar order of magnitude, a slight fluctuation in β could lead to a drastic change in R, and thus T e , potentially inducing high variability in the electromagnetic emission.By varying R Low , one can narrow the gap between R High and R Low , making R less sensitive to β around 10 −1 and 10 1 , and potentially solving the variability problem.We note that Event Horizon Telescope Collaboration et al. (2022) increased R Low up to 10 and found no systematic reduction in the 230 GHz variabil-ity.Thus, to extend their work, we will further increase R Low to explore a wider range of parameter spaces.This paper is structured as follows: In Section 2, we describe our methods, namely the GRMHD simulation libraries of Sgr A*, the tools for GRRT parameter surveys, and the parameter spaces that we consider.In Section 3, we present the results of our parameter searches, showing how M ∆T changes as R Low is increased for theoretical models of Sgr A*.In fact, we find a reduction of M ∆T , and we will compare the reduced value with observations.In the same section, we will also unravel why GRMHD models of Sgr A* are more variable than observations.Finally, we conclude our studies in Section 4.

METHODOLOGY
This section describes our methods of obtaining the GRMHD models, performing the GRRT parameter surveys, and constructing the GRRT image libraries of Sgr A* with varying R Low .

GRMHD Simulations
Our non-radiative GRMHD models of Sagittarius A* are obtained using the GRMHD code kharma, which is the GPU-enabled version of iharm3d (Prather et al. 2021).kharma solves the ideal GRMHD equations (with c = 1) and the divergence-less constraint of magnetic fields using the flux-constrained transport scheme (Tóth 2000).kharma assumes the 'spherical-polar' version of the Kerr-Schild coordinates with the metric tensor given as: where r and θ are spherical distances and polar angles.Σ = r 2 + a 2 cos 2 θ, where a is the black-hole spin.kharma uses the WENO reconstruction method (Jiang & Shu 1996) to reconstruct primitive variables and the Lax-Friedrichs solver to solve the Riemann problem on cell boundaries.kharma evolves the GRMHD equations using the 2nd-order Strong Stability Preserving Runge-Kutta Method (Gottlieb et al. 2011) with a CFL number of 0.7.We refer readers to Wong et al. (2022) to look for the detailed GRMHD equations being solved and their technical details.
The kharma simulation libraries for the EHT campaign include both the Standard and Normal Evolution (SANE) and Magnetic Arrested Disc (MAD) initial conditions.In this work, we consider only the MAD state since the MAD models are favoured in Event Horizon Telescope Collaboration et al. (2022).The accretion disc assumed an ideal gas equation of state with an adiabatic index of 4/3, with the initial conditions being a geometrically thick torus computed using the method by Fishbone & Moncrief (1976).The initial torus has an inner radius at 20 and a pressure maximum at 41, all in units of GM c −2 .To initialize the MAD state, we set the initial magnetic field within the torus using the vector potential (Wong et al. 2022): where ρ max is the maximum density and r 0 is the inner radial boundary of the computational domain.Magnetic fields are computed using ⃗ B = ∇ × ⃗ A. Finally, the simulation library employed the funky modified Kerr-Schild coordinates (Prather 2022) to overcome time step restrictions along the pole.The radial outer boundary is located at 1,000 GM c −3 , the simulation resolution is 288 × 128 × 128, and the simulation duration is 30,000 GM c −2 .

GRRT
We compute the 230 GHz images and fluxes of the GRMHD models using the open-source GRRT code IPOLE (Mościbrodzka & Gammie 2018), which is based on the GRRT code GRTRANS (Dexter 2016).In principle, IPOLE solves the polarized radiative transfer code, solving for the Stokes parameters (I, Q, U , V ).However, the Q, U , and V parameters are out of scope in this study.Thus, to reduce computational time cost, we solve only the unpolarized radiative transfer equation built-in IPOLE.The unpolarized, covariant radiative transfer equation reads: where the subscript ν means the quantity for a given photon frequency, I ν is the intensity, j ν is the emissivity, and α ν is the absorptivity.λ is the affine parameter along the photon geodesic: where k ν is the photon four-vector.We use the transfer coefficients presented in Dexter (2016) to solve Equation 4. Following the work of Event Horizon Telescope Collaboration et al. (2022), we fix the black-hole image's field of view (FOV) to be 200 muas.The resolution of the image is taken as 400 × 400 pixels, which is fine enough to ensure convergence.The distance from Earth to Sgr A* is taken as 8,178 parsec, while the mass of Sgr A* is assumed to be 4.154 × 10 6 M ⊙ (Abuter et al. 2019).To investigate the effect of varying R Low to M ∆T , we perform GRRT parameter surveys on the open science grid (Pordes et al. 2007;Sfiligoi et al. 2009;OSG 2006), which is a set of pools of shared computing and data capacity for distributed high-throughput computing, supported by the National Science Foundation and the Department of Energy.Table 1 shows the parameters we considered in this study, which include the spin of the black of a, R High , and the inclination angle of the observer θ.In particular, we consider the time interval τ = (29,465 -29,995) GM c −2 of snapshots, which covers a duration of 530 GM c −2 ∼ 3 hours, to compute M ∆T .This interval represents the last sets of snapshots of the simulations, in which inflow equilibrium near the horizon, where the 230 GHz emissions are dominated, shall be established.In general, GRMHD simulations of accreting systems are scale-invariant, so there is a free parameter M Unit that scales the gas density of the system.M Unit converts from the code unit to the cgs unit.We consider 10 sets of M Unit ranging from 10 17 to 10 21 (in the log scale), which spans the optically thin to optically thick regime.Additionally, we consider 10 sets of R Low for a given R High , ranging from R Low = 1 to R Low = MIN(60, R High ).We set an upper limit for R Low because the black-hole luminosity drops quickly at large R Low , where the highly magnetized electron becomes cold.Additionally, as we will show in the later section, an increase in R Low accompany an increase in Thus, R Low should not increase further, or the assumption of Sgr A* being optically thin will be broken.Altogether, given a certain set of a, R High , θ, and a particular GRMHD snapshot, we would have 10 × 10 = 100 of GRRT images spanning different R Low and M Unit .

Model Parameters
After having the black-hole flux at 230 GHz for a particular set of a, R High , θ, and in our considered time interval τ , we can then compute the time-domain average, variance, and hence, the variability for the 230 GHz flux.We could then infer how R Low should vary against M Unit for black-hole models that satisfy the constraint  Note, however, that we assumed the vertical axis of the camera aligns with the vertical axis of the spinning black hole.

RESULTS AND DISCUSSION
This section presents and discusses the results of our GRRT parameter surveys, the dependencies of M ∆T on R Low and M Unit , and the origin of the high variability for the R Low = 1 images.Note that we refer to prograding (retrograding) black holes as black holes with positive spin a > 0 (negative spin a < 0) and that non-rotating black holes are referred to as non-spinning a = 0 black holes.Could M ∆T be lowered with varying R Low ?Here, we include results for the GRMHD model of a = +0.94 in Figure 2 as a reference.We also include some representative plots for other black-hole spins in Figure 3.At the same time, the remaining full results are shown in Figure Set 1.The light-blue markers indicate the models we obtained from GRRT.The black-dotted lines in the contour plots of Figure 2 represent models consistent with the Sgr A* 230 GHz flux.The green lines indicate regions of the parameter spaces where M ∆T ≤ 0.1.This is also the upper limit of the historical distribution of M ∆T for Sgr A*.Depending on the black-hole spins, the parameter space where M ∆T ≤ 0.1 could be consistent with the Sgr A* flux constraint.For instance, black holes with a = +0.94 and a = +0.5 (for θ = 90 Deg) show a limited sequence of Sgr A* models that intersect with the region bounded by the green line (M ∆T ≤ 0.1), which are obtained by increasing R Low beyond 1.Nonspinning black-hole, except for R High = 10, contains sequences of M ∆T ≤ 0.1 that is consistent with the Sgr A* flux.We also note that all R High = 10 Sgr A* models with a = +0.94 and +0.5 could not attain M ∆T ≤ 0.1 irrespective of R Low assumed.

Parameter Surveys
However, black holes with a < 0 differ from those with a ≥ 0. Except for R High = 120, 180 viewed at θ = 0.1 Deg, black-hole with a = −0.5 that have M ∆T ≤ 0.1 are inconsistent with the Sgr A* flux constraint.In fact, we find that no parameter spaces with M ∆T ≤ 0.1 that are consistent with the observed M ∆T for Sgr A* with a = −0.94.This suggests a substantial reduction in M ∆T with varying R Low is modeldependent and sensitive to the black-hole parameters.This also suggests that the effects of varying R Low to the 230 GHz image are different between a ≥ 0 and a < 0 black holes, and we will compare and explain the differences in more detail in our next papers.
In the contour sub-plots, we find an almost loglinear relationship between R Low and M Unit for suites of Sgr A* models for all of the θ and R High parameter values we considered.This could be interpreted qualitatively: as R Low increases, the thermal energy partitioned to the electrons at low β becomes smaller, making the electrons cooler.Note that the major contribution to the 230 GHz flux is synchrotron radiation, for which its emissivity is an increasing function of T e and ρ.Given the 2.4 Jy constraint, the fluid density needs to increase to compensate for a reduction in T e .This explains the observed trend for M Unit .We note a similar pattern between R Low and M Unit for other black-hole spins.

Parameter Dependence
How would M ∆T vary as R Low for suites of Sgr A* models?We extract the values of M ∆T and R Low along the black-dotted lines in Figure 2 and present them in Figure 4. To extract the values of M ∆T , we perform Bivariate spline approximations with default parameters provided by the package Scipy, and we did it for all the spins we considered.In sum, we find three different patterns for the M ∆T versus R Low curves: i ) M ∆T increases as R Low , ii ) M ∆T decreases as R Low , and iii ) M ∆T first decreases to a local minimum and then increases.We observe that except for the case of R Low = 10, the shape of the M ∆T versus R Low curves is less sensitive to the variations of R Low , but is more sensitive to the changes in θ.
Almost every a ≥ 0 black-hole model, with R High > 10, shows pattern iii as R Low first increases beyond 1.Whether or not M ∆T reduces for a < 0 black-hole models depends on a and θ.Also, the M ∆T versus R Low curves for a < 0 black holes differ from those of a > 0 black holes.For instance, none of the a < 0 black-hole curves shows strong dips in the M ∆T versus R Low relations, and they vary more slowly and smoothly than those of the a > 0 black holes.This also proves that the GRRT images of changing R Low for the a < 0 black holes fundamentally differ from those for the a > 0 black holes.
We also find that some suites of Sgr A* models with R Low > 10 show a reduction in M ∆T as R Low increases.For instance, it is possible to achieve M ∆T ≲ 0.1 for almost every model with spin a = +0.94 and a = 0.This result contrasts those presented in Event Horizon Telescope Collaboration et al. ( 2022) and is discovered by considering a more comprehensive range of parameter spaces.Thus, we successfully show that it is possible to reduce M ∆T by varying R Low .We note, however, that M ∆T remains larger than 0.1 for almost every model for the remaining spins.Thus, whether or not a substantial reduction in M ∆T occurs when R Low is varied along suites of Sgr A* models highly depends on the underlying model parameters.

Large M ∆T Uncovered -Image Domain
What are the origins of large M ∆T for the GRRT images of Sgr A* at R Low = 1 (the value assumed in Event Horizon Telescope Collaboration et al. 2022)?To understand this, we computed the time domain, pixel-wise i ) averaged and ii ) standard deviation of image luminosity across all a, and θ.We show the time-averaged, pixel-wise luminosity in Figure 5 for the a = +0.94GRMHD models.We also include representative plots for other black-hole spin in Figure 6   disk is optically thin.Hence, for θ = 0.1, 179.9 Deg, the GRRT images show a thin photon ring.As θ progresses towards 90 Deg, most of the emission is concentrated to the left of the black-hole shadow due to the Doppler beaming effect.We find that for all the R Low = 1 images, the emissions are contributed from a thin photon ring.This is true even for other spins a.
Having all emissions contributed from a small area indicates that a slight variation in the photon ring lumi-nosity would induce large variability in the 230 GHz flux.Indeed, we show the time-domain, pixel-wise standard deviation of the 230 GHz luminosity for the a = +0.94GRMHD models in Figure 7.We also include representative plots for other black-hole spin in Figure 8. Results for the remaining a are shown in Figure Set 3. We find that the photon rings contribute most to the image variability.However, some minor contributions from the accretion flow could also be seen.This is true also for the a = +0.5 and 0 models.
On the other hand, the situation for a < 0 black holes is different from that of a ≥ 0 black holes.In addition to the photon-ring variability, the contribution to the flux variability from the accretion flow is more significant for the a < 0 black-hole models, and their contribution is the largest for the a = −0.94model.This could be due to the misalignment between the black-hole and torus angular momentum, creating highly irregular accretion flow dynamics.
We note that the image asymmetry due to the Doppler beaming effect is reduced as the black hole is spinning more negatively.Such an effect is first mentioned in Medeiros et al. (2022), where they consider image asymmetry up until a = 0. Here, we briefly extend their work to consider also a < 0 black holes.The image becomes more symmetric for a = −0.5 and a = −0.94.This is probably due to the stronger retrograde motion black hole, which drags the rotating plasma in the opposite direction to the plasma angular velocity, reducing the angular velocity differences between the plasma on the  In particular, we consider R High = 200, 220, and 240.We performed GRRT with varying R Low given these R High and to look for models that satisfy the 2.4 Jy constraints.We then extract the corresponding minimum of M ∆T .We use two methods to achieve this.The first is to directly look for the minimum of M ∆T along the curves generated by the Bivariate spline approximations (c.f. Figure 4).The second is to look for the value of R Low (and M Unit ) that produces such a M ∆T , and we further perform GRRT to compute the corresponding M ∆T .In addition, we extract ALMA and SMA observations of Sgr A* at 230 GHz and compute the mean and standard deviation of the observed M ∆T .To do this, we perform windowed sampling with a window size of 3 hours, and and that the starting point of this window moves forward by 0.5 hours for each sampling.We also note that the ALMA and SMA observations are performed with the High and Low bands.We compute the M ∆T separately.
These results are shown in Figure 9.We find that all R High = 10 models are inconsistent with the M ∆T constraint even if we vary R Low .R High ≥ 50 models could attain M ∆T consistent with the observations.As R High increases beyond 180, the minimum of M ∆T could be as low as ∼ 0.03.Also, we find that the minimum of M ∆T decreases for a larger R High , but the reduction rate also decreases.Indeed, the minimum of M ∆T for R High = 240 is close to that of R High = 180.This might They are shown as scatter points.This is for the particular black-hole parameter with a = +0.94 and θ = 0.1 Deg, which shows exceptionally small M∆T .In each subplot, the upper right box shows the telescope data and the observational band filter that we use to compute the 3 hours variability.The solid red horizontal lines and the red shaded represent the observed mean and standard deviation of M∆T , respectively.We use Bivariate spline approximations to obtain the blue-square scatter points while using direct GRRT to compute the green-diamond scatter points.
suggest a threshold value of M ∆T , where M ∆T could not be lowered than such value even if R Low is considered.More comprehensive statistical studies will help reveal whether such value exists.

CONCLUSION
In this paper, we performed a comprehensive set of GRRT parameter surveys using the Open Science Grid to study the reason for the unexpectedly high variability M ∆T of Sgr A* predicted by theoretical models in Event Horizon Telescope Collaboration et al. (2022).Our parameter surveys vary the R Low parameter in the electron temperature prescription function used by Event Horizon Telescope Collaboration et al. ( 2022), which spans the optically thin to the optically thick regimes, and covers R Low from 1 to 60.Our results contrast to those in Event Horizon Telescope Collaboration et al. (2022).We find that it is possible to reduce M ∆T to a value lower than 0.1, the threshold values of the historical distribution of M ∆T for Sgr A*, by varying R Low , for specific sets of black-hole spin a, inclination angle θ, and R High .
We find that models with a = +0.94,θ = 0.1 Deg, and R High ≥ 50 could produce a low-level of variability consistent with the observed M ∆T of Sgr A*.We also note that the minimum of M ∆T reduces for larger R High for this particular set of a and θ, and M ∆T ∼ 0.03 for R High ≥ 180.Although this might be inconsistent with the statistical average of M ∆T , we note that the minimum of M ∆T across the historical observational distribution could be as low as 0.03.Thus, such a result sheds light on whether R Low could be an important parameter to consider if one intends to perform GRRT on suites of Sgr A* simulation libraries.
We also analyze the GRRT images for the R Low = 1 models, which are also the values assumed in Event Horizon Telescope Collaboration et al. (2022), across different a and θ.By computing the time domain, pixel-wise, averaged, and standard deviation of the image luminosity, we find that all R Low = 1 models have their luminosity contributions to be dominated by a thin photon ring.We also find that the thin photon ring contributes most to the observed variability for all a ≥ 0 models.For a < 0 black-hole models, secondary contributions of variability from the accretion flow are more visible due to the irregular flow dynamics induced by misalignment between the black-hole and torus angular momentum.Our findings on the origin of high M ∆T suggest that a more serious study on the electron temperature used to model RIAF should be adopted.These results, together with the findings that the behaviour of M ∆T versus R Low relations are different between the a < 0 and a ≥ 0 black holes, also lay the foundations for our successive papers, where we would analyze why a variation of R Low could reduce/increase M ∆T for different black-hole parameters.
Note that there are still several caveats in this preliminary study.We assumed only R Low ≤ 60 in our study.This might raise concerns about the lack of details on the behaviour of M ∆T against R Low at a larger R Low .However, we have already shown that M ∆T would decrease in the range of parameter spaces we are interested in.Thus, there seems to be no immediate need to consider higher R Low .Also, we show an almost log-linear relationship between R Low and M Unit for suites of Sgr A* models.Thus, a further increase in R Low would make the accretion disk more optically thick.This would violate our assumption of Sgr A* being a low-luminosity, geometrically thick, and optically thin accretion flow.
Computing the effects of R Low on M ∆T for one particular given time interval τ = (29,465 -29,995) GM c −2 only might induce concerns about whether our results are statistically significant.Also, coarse parameter spaces that we considered in our work might pose worries about whether the minimum of M ∆T is due to interpolation errors.However, we show in Figure 9 that the minimum of M ∆T computed by the Bivariate spline approximations and direct GRRT are close.Nonetheless, these limitations are due to the limited computational storage and time constraints.We plan to perform more comprehensive studies to understand better the role of R Low in a more serious statistical framework on the increase/decrease of M ∆T .

Figure 1 .
Figure 1.Visualization of Equation (1) with increasing RLow.In this case, R High is fixed at 160, and we assume βCrit = 1.The blue-shaded region indicates where R is sensitive to variations in β.

Figure 2 .
Figure 2. Example contour representations of M∆T against RLow and MUnit obtained through GRRT parameter surveys.This is for the GRMHD models with a = +0.94.The upper right box in each sub-grid shows the θ and R High assumed for a particular set of GRRT.Also, the black-dotted contour lines represent models of Sgr A* where the time-averaged flux is 2.4 Jy, which is the observed averaged fluxes of Sgr A* at 230 GHz.For R High = 10 and 50, the slashed region represent the parameter spaces where R High < R low , which is out of our interest.Light-blue markers indicate the point of models we obtained through GRRT.We use a diverging colormap that centers around M∆T = 0.1, the upper limit of the historical distribution of M∆T for Sgr A*.Thus, regions with red (blue) colours indicate models with high (low) M∆T .We also overlay green contour lines to represent the boundary of the parameter space where M∆T ≤ 0.1.The contour plots for other spins are included in Figure Set 1.

Figure 3 .
Figure 3. Same as Figure 2, but for the representative plots of each black hole spin a.We labelled a and θ under the description of each subplot.

Figure 4 .
Figure 4. M∆T against RLow for the black-hole models that satisfy the 2.4 Jy constraint for all the black-hole spin we considered.Each sub-figure represents results for different a.The sub-grid plot in each sub-figure shows the variations of M∆T against RLow for a given θ across different R High .Except for R High = 10, the shape of the M∆T versus RLow curves are less sensitive across different R High than θ.The values of M∆T and RLow are obtained from the parameter search results using Bivariate spline approximation provided by Scipy in the log-log scale.

Figure 5 .
Figure 5. Example of time-averaged, pixel-wise luminosity plots.These images are computed with the GRMHD snapshots within the time interval of τ = (29,465 -29,995) GM c −2 , while spanning different R High and θ.Here, a = +0.94,and we assume RLow = 1.The upper right box in each sub-grid shows the θ and R High assumed.The colour map is scaled down to [0, 1] in each sub-plot.The luminosity plots for other spins are included in Figure Set 2.

Figure 6 .
Figure 6.Same as Figure 5, but for the representative plots of each black hole spin a.We labelled a under the description of each subplot.All images are taken at θ = 45 Deg.

Figure 7 .
Figure 7. Same as Figure 5, but for the pixel-wise, time-domain standard deviation of the luminosity.The standard deviation plots for other spins are included in Figure Set 3.

Fig.
Fig. Set 3. Pixel-wise, time-domain standard deviation of the luminosity across different R High and θ (with RLow = 1) for a = +0.5, 0, −0.5, and −0.94.left and right-hand side of the event horizon and thus weakening the Doppler beaming effect.

Figure 8 .
Figure 8. Same as Figure 7, but for the representative plots of each black hole spin a.We labelled a under the description of each subplot.All images are taken at θ = 45 Deg.

Figure 9 .
Figure9.The variation of the minimum of M∆T , obtained through varying RLow from 1 to 60, against a given R High .They are shown as scatter points.This is for the particular black-hole parameter with a = +0.94 and θ = 0.1 Deg, which shows exceptionally small M∆T .In each subplot, the upper right box shows the telescope data and the observational band filter that we use to compute the 3 hours variability.The solid red horizontal lines and the red shaded represent the observed mean and standard deviation of M∆T , respectively.We use Bivariate spline approximations to obtain the blue-square scatter points while using direct GRRT to compute the green-diamond scatter points.

Table 1 .
List of parameters of the GRMHD models and the GRRT parameter surveys we considered.