# Time delays between radio and X-ray and between narrow radio bands of Sgr A*
flares in the shock oscillation model

Toru Okuda1 , Chandra B. Singh2, Ramiz Aktar3
1 Hakodate Campus, Hokkaido University of Education, Hachiman-Cho 1-2,
Hakodate 040-8567, Japan
2 South-Western Institute for Astronomy Research, Yunnan University,
University Town, Chenggong, Kunming 650500,
People’s Republic of China
3 Department of Physics and Institute of Astronomy, National Tsing Hua
University, 30013 Hsinchu, Taiwan
E-mail:bbnbh669<EMAIL_ADDRESS>
(Accepted 2023 April 3. Received 2023 February 28; in original form 2023
January 18 )

###### Abstract

We examine the time delay between radio and X-ray and between narrow radio
frequency flares in Sagittarius A* (Sgr A*), from analyses of the synchrotron,
bremsstrahlung and monochromatic luminosity curves. Using the results of 2D
relativistic radiation magnetohydrodynamic (MHD) simulations based on the
shock oscillation model, we find three types of time delay between the
synchrotron and bremsstrahlung emissions: Type A with a time delay of 2 – 3 h
on the shock descending branch, Type B with no time delay and Type C with an
inverse time delay of 0.5 – 1 h on the shock ascending branch. The time delays
in Types A and C are interpreted as a transit time of Alfvén and acoustic
waves between both emission dominant regions, respectively. The delay times
between 22 and 43 GHz flares and between 8 and 10 GHz flares are $\sim$ 13 –
26 min and 13 min, respectively, while the inverse delay also occurs
dependently on the shock location branch. These time delays between the narrow
radio bands are interpreted as the transit time of the acoustic wave between
the frequency-dependent effective radii $R_{\tau_{\rm\nu=1}}$, at which the
optical depth $\tau_{\rm\nu}$ at the accretion disc surface becomes $\sim$
unity. The shock oscillation model explains well the observed delay times of
0.5 – 5 h between radio and X-ray, 20 – 30 min between 22 and 43 GHz and
$\sim$ 18 min between 8 and 10 GHz in Sgr A*.

###### keywords:

black hole physics – (magnetohydrodynamic) MHD – radiation mechanism: thermal
– shock waves. – Galaxy: centre.

††pagerange: LABEL:0–LABEL:0††pubyear: 2022

##  1 Introduction

Sagittarius A* (Sgr A*), the supermassive black hole at the center of our
galaxy with a mass of $\sim$ 4 $\times$ 10${}^{6}M_{\odot}$ and located at
8.27 kpc away, have revealed very peculiar observations of too low luminosity
and its complicated spectra since its discovery (Genzel et al. 2003, Genzel,
Eisenhauer & Gillessen 2010). The observed luminosity is five orders of
magnitude lower than that predicted by the standard thin disc model (Shakura &
Sunyaev 1973, hereafter, SS73 model) and the spectrum of Sgr A* differs from
the multi-temperature black body spectra obtained from the SS73 model. Various
theoretical models which can explain too low luminosity and characteristic
spectral properties have been proposed during the past three decades. Among
them, advection dominated accretion flow (ADAF) model with high angular
momentum is successful in explaining the observations well (see Narayan &
McClintock 2008; Yuan 2011; Yuan & Narayan 2014, for review).

The latest observations in the range of radio to X-ray and simultaneous
observation campaigns through such energy bands showed flares with a time
scale of $\sim$ hours to days of Sgr A* (Degenaar et al. 2013; Neilson et al.
2013, 2015; Ponti et al. 2015) and also time delays between radio, IR, and
X-ray emissions (Yusef-Zadeh et al. 2006a, 2008, 2009; Rauch et al. 2016;
Ponti et al. 2017; Capellupo et al. 2017). Since the pioneering works of
magnetized discs with shear instability (Balbus & Hawley 1991; Hawley & Balbus
1991), multidimensional magnetohydrodynamic (MHD) simulation works have shown
that the magnetic field and resultant magnetorotational instability (MRI) play
important roles not only in the structure of accretion flow but also the time-
variations of luminosity and spectra around black holes (Machida, Hayashi &
Matsumoto 2000; Machida, Matsumoto & Mineshige 2001; Stone & Pringle 2001;
Igumenshchev, Narayan & Abramowicz 2003; Narayan, Igmenshchev & Abramowicz
2003; Narayan et al. 2012; Yuan, Bu & Wu 2012; Yuan et al. 2015). In this
respect, several MHD simulation works have attempted to address the rapid
flares of Sgr A* ( Proga & Begelman 2003; Chan et al. 2009; Dexter, Agol &
Fragile 2009; Yuan et al. 2009; Dodds-Eden et al. 2010; Ball et al. 2016;
Ressler et al. 2017; Roberts et al. 2017; Li, Yuan & Wang 2017).

Besides the high angular momentum flow like ADAF, the low angular momentum
flow models which exhibit the formation of standing shock near the event
horizon have been developed to explain the quasi-periodic oscillations (QPOs)
around black holes. The studies of the standing shock in an astrophysical
context were pioneered by Fukue (1987) and then Chakrabarti (1989) and further
studies of the low angular momentum flows are refered to Okuda et al. (2019)
and Singh, Okuda & Aktar (2021). Similarly to other MHD simulation works but
based on the low angular momentum flow model, we examined the shock
oscillation model for Sgr A* using 2D MHD and 2D radiation MHD simulations and
showed that the magnetized flows yield large modulations of luminosities with
a time-scale of $\sim$ 5 and 10 d (Okuda et al. 2019; Singh, Okuda & Aktar
2021) and time delay of 1 – 2 h between radio and X-ray flares of Sgr A*
(Okuda, Singh & Aktar 2022). This paper is complementary to the time lag
relation in the previous paper (Okuda, Singh & Aktar 2022), where a simple
two-temperature model with constant ratio of electron to ion temperatures is
used. Here, we examine further time lag relation of flares between radio and
X-ray and between narrow radio frequency bands, using an improved two-
temperature model where ion and electron temperatures are obtained by solving
the radiation energy equilibrium equation.

##  2 Numerical method

###  2.1 Two-temperature model

We use the previous numerical results of 2D relativistic radiation MHD
simulations for the long-term flares of Sgr A* (Okuda, Singh & Aktar 2022)
which were calculated using the RadRMHD module of the public library software
${\sc PLUTO}$ (Melon Fuksman & Mignone 2019). Accordingly, we know already the
time variations of primitive variables of density $\rho$, velocity v, magnetic
field B, gas pressure $p_{\rm g}$, radiation energy density $E_{r}$ and
radiation flux F obtained from a one-temperature model. The computational
domain is $0\leq R\leq 200$ and $-200\leq z\leq 200$ with the resolution of
$410\times 820$ cells in the cylindrical coordinates (R, $\phi$, z), where $R$
and $z$ are expressed in the unit of the Schwarzschild radius $R_{\rm g}$
hereafter. The adiabatic index for studying the flow has been set as 1.6 for
all simulation runs.

In this paper, we use a two-temperature model to evaluate the synchrotron and
bremsstrahlung emissions, solving the radiation energy equilibrium equation
that Coulomb collisions transfer energy rate $q^{\rm ie}$ from ions to
electrons equals to the sum of synchrotron cooling rate $q_{\rm syn}$ and
bremsstrahlung cooling rate $q_{\rm br}$ in electrons. $q^{\rm ie}$ is given
as follows (Stepney & Guilbert 1983).

$\displaystyle q^{\rm ie}$ $\displaystyle=5.61\times 10^{-32}{{n_{\rm e}n_{\rm
i}(T_{\rm i}-T_{\rm e})}\over{K_{2}(1/\theta_{\rm e})K_{2}(1/\theta_{\rm
i})}}$ (1) $\displaystyle\times\left[{{2(\theta_{\rm e}+\theta_{\rm
i})^{2}+1}\over(\theta_{\rm e}+\theta_{\rm i})}K_{1}({{\theta_{\rm
e}+\theta_{\rm i}}\over{\theta_{\rm e}\theta_{\rm i}}})+2K_{0}({{\theta_{\rm
e}+\theta_{\rm i}}\over{\theta_{\rm e}\theta_{\rm i}}})\right]$
$\displaystyle\hskip 108.12054pt{\rm erg\ cm^{-3}\;s^{-1}},$

where $n_{\rm e}$ and $n_{\rm i}$ are the number density of electrons and
ions, $K_{0}$, $K_{1}$ and $K_{2}$ are modified Bessel functions, and the
dimensionless electron and ion temperature are defined by

$\theta_{\rm e}={kT_{\rm e}\over{m_{\rm e}c^{2}}},\;\;\;\theta_{\rm
i}={kT_{\rm i}\over{m_{\rm p}c^{2}}},$ (2)

where $k$, $m_{\rm e}$, $m_{\rm p}$ and $c$ are Boltzmann constant, electron
mass, proton mass and the light velocity.

The synchrotron cooling rate $q_{\rm syn}$ is given by (Narayan & Yi 1995;
Esin et al. 1996)

$\displaystyle q_{\rm syn}$ $\displaystyle=$ $\displaystyle{{2{\rm\pi}kT_{\rm
e}\nu_{\rm c}^{3}}\over{3Hc^{2}}}+6.76\times 10^{-28}{n_{\rm
i}\over{K_{2}(1/\theta_{\rm e})a_{1}^{1/6}}}$ (3) $\displaystyle\times$
$\displaystyle[{1\over a_{4}^{11/2}}\Gamma({11\over 2},a_{4}\nu_{\rm
c}^{1/3})+{a_{2}\over a_{4}^{19/4}}\Gamma({19\over 4},a_{4}\nu_{\rm c}^{1/3})$
$\displaystyle+$ $\displaystyle{a_{3}\over a_{4}^{4}}(a_{4}^{3}\nu_{\rm
c}+3a_{4}^{2}\nu_{\rm c}^{2/3}+6a_{4}\nu_{\rm c}^{1/3}+6){\rm
e}^{-a_{4}\nu_{\rm c}^{1/3}}]$ $\displaystyle\hskip 108.12054pt{\rm erg\
cm^{-3}\ s^{-1}},$

where $H$ is a scale height of the disc,

$\displaystyle a_{1}={2\over{3\nu_{0}\theta_{\rm e}^{2}}},\;a_{2}={0.4\over
a_{1}^{1/4}},\;a_{3}={0.5316\over a_{1}^{1/2}},\;a_{4}={1.8899a_{1}^{1/3}},$
$\displaystyle\Gamma(a,x)=\int_{x}^{\infty}t^{a-1}{\rm
e}^{-t}dt,\nu_{0}={eB\over{2{\rm\pi}m_{\rm e}c}}{\;\rm and}\;\nu_{\rm
c}={3\over 2}\nu_{0}\theta_{\rm e}^{2}x_{\rm M}.$ (4)

Here, $e$ and $B$ are the electron charge and the strength of the magnetic
field and $x_{\rm M}$ is determined from the next equation as

$\displaystyle{\rm exp}(1.8899x_{\rm M}^{1/3})$ $\displaystyle=$
$\displaystyle 2.49\times 10^{-10}{4{\rm\pi}n_{\rm e}R\over
B}{1\over{\theta_{\rm e}^{3}K_{2}(1/\theta_{e})}}$ $\displaystyle\times$
$\displaystyle\left({1\over x_{\rm M}^{7/6}}+{0.40\over x_{\rm
M}^{17/12}}+{0.5316\over{x_{\rm M}^{5/3}}}\right).$

The bremsstrahlung cooling rate $q_{\rm br}$ is given as follows (Stepney &
Guilbert 1983).

$\displaystyle q_{\rm br}=q_{\rm ei}+q_{\rm ee},$ (6)

$\displaystyle q_{\rm ei}=1.48\times 10^{-22}n_{\rm e}^{2}F_{\rm
ei}(\theta_{\rm e})\hskip 28.45274pt{\rm erg\ cm^{-3}\;s^{-1}},$ (7)

$\displaystyle\;\;\;F_{\rm ei}(\theta_{\rm
e})=\left\\{\begin{array}[]{ll}1.02\theta_{\rm e}^{1/2}(1+1.78\theta_{\rm
e}^{1.34})&\mbox{ for $\theta_{\rm e}<1$},\\\ 1.43\theta_{\rm e}[{\rm
ln}(1.12\theta_{\rm e}+0.48)+1.5]&\mbox{ for $\theta_{\rm
e}>1$},\end{array}\right.$ (10)

$\displaystyle q_{\rm ee}=\left\\{\begin{array}[]{llll}2.56\times
10^{-22}n_{\rm e}^{2}\theta_{\rm e}^{1.5}(1+1.10\theta_{\rm e}+\theta_{\rm
e}^{2}-1.25\theta_{\rm e}^{2.5})&\\\ \hskip 85.35826pt{\rm erg\
cm^{-3}\;s^{-1}}\mbox{ for $\theta_{\rm e}<1$},&\\\ 3.40\times 10^{-22}n_{\rm
e}^{2}\theta_{\rm e}[{\rm ln}(1.123\theta_{\rm e})+1.28]&\\\ \hskip
85.35826pt{\rm erg\ cm^{-3}\;s^{-1}}\mbox{ for $\theta_{\rm e}>1$}.&\\\
\end{array}\right.$ (15)

Furthermore, from matching of one-temperature model to two-temperature model,
we have,

$T_{\rm e}+T_{\rm i}=2T,$ (16)

because we set the gas pressure $p_{\rm g}$ = $2nkT$ = $n_{\rm e}kT_{\rm
e}+n_{\rm i}kT_{\rm i}$ and $n=n_{\rm e}=n_{\rm i}$, where $T$ and $n$ are the
temperature and the number density, respectively, in the one-temperature
model. We solve numerically the equations of $q^{\rm ie}=q_{\rm syn}+q_{\rm
br}$ and $T_{\rm e}+T_{\rm i}=2T$ and obtain ion temperature $T_{\rm i}$ and
electron temperature $T_{\rm e}$.

Figure 1: Time variations of synchrotron luminosity $L_{\rm syn}$ (blue
square), bremsstrahlung luminosity $L_{\rm brem}$ (orange Delta) (erg s-1) and
oscillating shock location $R_{\rm s}$ (green circle) on the equator during
$t=4\times 10^{4}-1.0\times 10^{5}$ in unit of $R_{\rm g}/c$ for model A with
$\lambda$= 1.65, $\epsilon=6.89\times 10^{-3}$ at the outer boundary $R_{\rm
out}$ = 50 and the magnetic field strength $\beta_{\rm out}$ = $10^{4}$. The
scale of $L_{\rm brem}$ is shown inside on the right vertical axis The circle
points on the curves are plotted every time interval of $200R_{\rm g}/c$
($\sim 0.02$) s.  Figure 2: Alfvén velocity $V_{\rm A}$ (solid line) and sound
velocity $V_{\rm s}$ (dashed line) on the equator at $t=4\times 10^{4}(R_{\rm
g}/c)$ for model A. Here, the sound velocity is $\sim 0.1c$ in the region of
$R\leq 20$, while the Alfvén velocity is roughly $\geq$ 0.003$c$.

We confirm here the optical thickness $\Delta\tau=\kappa\rho\Delta R$ across
the mesh size $\Delta R$ for the present model for Sgr A* by (see Okuda, Singh
& Aktar, 2022),

$\displaystyle\Delta\tau\sim 2\times
10^{-28}\left(\frac{\rho}{10^{-16}}\right)^{2}\left(\frac{T}{10^{9}}\right)^{-3.5}\left(\frac{\Delta
R}{0.2}\right)\ll 1,$ (17)

where the opacity $\kappa$ is given by the Kramers approximation corresponding
to the bremsstrahlung emission. Accordingly, the gas is fully optically thin
to the bremsstrahlung emission but not to the synchrotron emission throughout
the region considered here. If we consider the monochromatic radiation in the
radio frequencies where the synchrotron radiation is dominant, the gas may be
optically thick or thin because the opacity becomes large dependently on $\nu$
and the location $R$. In PLUTO code, we use the one-temperature model and only
the radiative energy loss by the fre-free emissions. Then, the total radiative
luminosity $L_{\rm rad}$ corresponds to the free-free emissions and is
evaluated at the outer z-boundary and the outer R-boundary surfaces, as
follows

$\displaystyle L_{\rm rad}=\int{\bf F}{\rm d}{\bf S},$ (18)

where ${\bf F}$ is the radiation energy flux at the boundary surfaces. On the
other hand, the total luminosities $L_{\rm syn}$ and $L_{\rm brem}$ due to the
synchrotron and bremsstrahlung emissions, respectively, are approximately
given

$\displaystyle L_{\rm brem}=\int q_{\rm brem}{\rm d}V,$ (19)

$\displaystyle L_{\rm syn}=\int q_{\rm syn}{\rm d}V,$ (20)

where the volume integration is carried out over all computational zones. The
expression for $L_{\rm syn}$ may be not exact because the synchrotron emission
becomes optically thick at a limited range of the frequencies in the inner
region of the disc as shown in later numerical results (Figs. 10 – 11) but
gives important amounts of radiation energy in the radio bands from the
viewpoint of the time-delay analysis between the radio and X-ray emission
peaks.

###  2.2 Monochromatic radiation energy flux $F_{\rm\nu}$ and luminosity
$L_{\rm\nu}$

To examine time delay between flares at narrow radio frequency band such as
22, 43 and 350 GHz, we calculate monochromatic radiation flux $F_{\rm\nu}$ at
the disc surface and luminosity $L_{\rm\nu}$. Assuming a locally plane-
parallel approximation for the accretion disc flow, the radiation flux
$F_{\rm\nu}$ at a given radius R is given by (see Manmoto, Mineshige &
Kusunose 1997)

$\displaystyle F_{\rm\nu}(R)={{2\pi}\over\sqrt{3}}B_{\rm\nu}\left[1-{\rm
exp}(-2\sqrt{3}{\tau_{\rm\nu}}^{*})\right]\;\;{\rm erg}\;{\rm cm}^{-2}\;{\rm
s}^{-1}{\rm Hz}^{-1},$ (21)

where $B_{\rm\nu}$ is the Planck function,
${\tau_{\rm\nu}}^{*}\equiv(\pi^{1/2}/2)\kappa_{\rm\nu}(0)H$ is the optical
depth at the accretion disc surface and $\kappa_{\rm\nu}(0)$ is the absorption
coefficient on the equatorial plane. Assuming LTE,
$\kappa_{\rm\nu}=\chi_{\rm\nu}/(4\pi B_{\rm\nu})$, where $\chi_{\rm\nu}$ is
the emissivity. Since we are interested in the emissions at radio bands such
as 22, 43 and 350 GHz, the relation (15) is valid for the gas which includes
optically thick disc region. We consider the synchrotron emissivity
$\chi_{\rm\nu,syn}$ by a relativistic Maxwellian distribution of electrons as
follows (Narayan & Yi 1995)

$\displaystyle\chi_{\nu,syn}=4.43\times 10^{-30}{{4\pi n_{\rm
e}\nu}\over{K_{2}(1/\theta_{\rm e})}}I(x)\;\;{\rm erg}\;{\rm cm}^{-3}\;{\rm
s}^{-1}{\rm Hz}^{-1},$ (22)

where $x\equiv{4\pi m_{\rm e}c\nu}/({3eB{\theta_{\rm e}}^{2}})$,

$\displaystyle I(x)={4.0505\over x^{1/6}}\left(1+{0.40\over
x^{1/4}}+{0.5316\over x^{1/2}}\right){\rm exp}(-1.8899x^{1/3}).$ (23)

The monochromatic luminosity $L_{\rm\nu}$ is given by

$\displaystyle L_{\rm\nu}=\int 2\pi RF_{\rm\nu}(R)dR,$ (24)

where the integration is carried out up to the outer radial boundary.

##  3 Numerical results

###  3.1 Application to a typical oscillating shock model under weak magnetic
field

In the previous paper (Okuda et al. 2019), to find general effects of the
magnetic field on 2D hydrodynamical flow with standing shock around a black
hole of mass 10 $M_{\odot}$, we examined a 2D magnetohydrodynamical flow
(hereafter model A) with a typical set of flow parameters of the specific
angular momentum $\lambda$= 1.65, the specific energy $\epsilon=6.89\times
10^{-3}$ at the outer boundary $R_{\rm out}$ = 50. The result shows steady
standing shocks under a very weak magnetic field with $\beta_{\rm out}$=
$10^{5}$ and $10^{9}$, a periodic shock oscillation between $R$ = 22 – 25
under a weak magnetic field with $\beta_{\rm out}=10^{4}$, and a chaotic
variation of the shock under a strong magnetic field with $\beta_{\rm
out}=4\times 10^{3}$, where $\beta_{\rm out}$ is the ratio of gas pressure to
magnetic pressure at the outer R-boundary $R_{\rm out}$. Applying the present
two-temperature model to the above second case with $\beta=10^{4}$, we
examined how the time delay between the synchrotron and bremsstrahlung
emissions occurs, because this is a simple example to understand the time lag
relation. Fig. 1 shows the time variations of synchrotron luminosity $L_{\rm
syn}$ (blue square), bremsstrahlung luminosity $L_{\rm brem}$ (orange Delta)
(erg s-1) and standing shock location $R_{\rm s}$ (green circle) on the
equator for model A, where the curves are plotted every time interval of 200
$R_{\rm g}/c$ ($\sim 0.02$ s). Hereafter, we term the plotted time interval as
one point time interval. The bremsstrahlung luminosity begins to increase
after the maximum shock location and becomes maximum after the shock contracts
minimally because the temperature, density and magnetic field strength behind
the shock are mostly enhanced in the innermost region. In Fig. 1, the
synchrotron luminosity peak delays the bremsstrahlung one by $\sim$ 8 –10
points time interval. From examining the radial dependence of the synchrotron
and bremsstrahlung volume emissions, we find that most of the synchrotron
luminosity is emitted in the inner region of $R\leq R_{\rm syn}$ (= 5) at all
phases. While the bremsstrahlung emission dominant region is confined at
$R\sim R_{\rm brem}$ (=10). Therefore, even after the bremsstrahlung
luminosity is attained to the maximum, the perturbations of acoustic and
Alfvén waves at $R\sim R_{\rm brem}$ influence the downstream flow. As a
result, the synchrotron luminosity peak delays the bremsstrahlung one by a
transit time of Alfvén wave between $R_{\rm syn}$ and $R_{\rm brem}$, because
the synchrotron emission depends strongly on the magnetic field.

Fig. 2 shows Alfvén velocity $V_{\rm A}$ (solid line) and sound velocity
$V_{\rm s}$ (dashed line) on the equator at $t=4\times 10^{4}R_{\rm g}/c$ near
the bremsstrahlung luminosity peak in model A, where $V_{\rm s}\sim 0.1c$ in
the inner region of $R\leq 20$, while $V_{\rm A}\geq$ 0.003$c$. The delay time
of the synchrotron luminosity peak to the bremsstrahlung one is estimated as
$(R_{\rm brem}-R_{\rm syn})/V_{\rm A}=1.67\times 10^{3}R_{\rm g}/c$ ($\sim$ 8
points time interval), which agrees well with the above time delay of 8 – 10
points time interval between the synchrotron and bremsstrahlung luminosity
peaks. Also, the synchrotron luminosity dip delays the bremsstrahlung one
nearly by $\sim$ 8 points time interval, because $R_{\rm brem}$ and $R_{\rm
syn}$ are not so variable. It should be noted that, in the accretion flow, the
gas always flows inwards near the equator and the outflow appears only along
the rotational axis. This shows that the time delay occurs without any
expanding hot blob which is discussed later with relevance to the plasmon
model.

---
Figure 3: (Left-hand panel): 2D contours of ratio $T_{\rm e}/T_{\rm i}$ of
electron temperature $T_{\rm e}$ to ion temperature $T_{\rm i}$ at
$t=1.98\times 10^{7}$ s for model Rad1. (Right-hand panel): 1D profiles of
$T_{\rm e}$ (small dash line), $T_{\rm i}$ (dash-dot line) and ratio $T_{\rm
e}/T_{\rm i}$ (solid line) on the equator. The shock location on the equator
is found at $R\sim 50$.
---
Figure 4: Time variations of synchrotron luminosity $L_{\rm syn}$ (blue
square), bremsstrahlung luminosity $L_{\rm brem}$ (orange Delta) (erg s-1) and
oscillating shock location $R_{\rm s}$ (green circle) on the equator during
$t=1.9-2.1\times 10^{7}$ s in model Rad1. The circle points on the lines are
plotted every time interval of 50 $R_{\rm g}/c$ ($\sim$ 0.55 h). The arrow
shows the local maximum points on the synchrotron luminosity curve and the
marks of A, B and C show delay types: Type A with a positive time delay of the
synchrotron to bremsstrahlung flare, Type B with no time delay and Type C with
a negative time delay at which the bremsstrahlung flare delays the synchrotron
one.

###  3.2 Application to Sgr A*

In this section, we apply the present two-temperature model to the numerical
results of 2D relativistic radiation MHD simulations for the long-term flares
of Sgr A*. We consider the mass of Sgr A* as $M=4\times 10^{6}M_{\odot}$.
Here, the unit of distance $R_{\rm g}$ is $2GM/c^{2}\sim 1.19\times 10^{12}$
cm.

####  3.2.1 Time delay between radio and X-ray flares

To examine the time delay between the synchrotron and bremsstrahlung emissions
from the radiation MHD simulations, we select time evolutions during 1.90 -
2.1 $\times 10^{7}$ s at model Rad1 in the previous paper (Okuda, Singh &
Aktar 2022) and recalculated the simulations using a smaller time step of
50$R_{\rm g}/c$ ( $\sim$ 0.55 h) in order to get better time resolution for
the time delay analysis. First, we show the electron and ion temperatures
obtained from the two-temperature model and compare them with other results in
the advection-dominated two-temperature model. Fig. 3 shows 2D contours of
$T_{\rm e}/T_{\rm i}$ (left-hand panel) and 1D profiles of $T_{\rm e}$ (small
dash line), $T_{\rm i}$ (dash-dot line) and $T_{\rm e}/T_{\rm i}$ (solid line)
on the equator (right-hand panel) at $t=1.98\times 10^{7}$ s for Rad1 case.
Here, the shock location on the equator is found at $R\sim 50$. The left panel
shows that the electron temperatures are not so different from the ion
temperatures in far distant region from the center but the ratio $T_{\rm
e}/T_{\rm i}$ becomes small as 0.01 – 0.1 in the inner region of $R\leq$ 20
and along the rotational axis. In the right panel, the temperature
distributions of $T_{\rm e}$ and $T_{\rm i}$ are qualitatively similar to
those in the steady solutions of advection-dominated accretion flow without
shock by Manmoto, Mineshige & Kusunose (1997) and also of advective flows with
shocks by Mandal & Chakrabarti (2005) and Dihingia, Das & Mandal (2018),
except that the difference between $T_{\rm e}$ and $T_{\rm i}$ becomes very
large across the shock in our case. Then, we may regard the two-temperature
model to be reasonable.

The radiative luminosity $L_{\rm rad}$ is evaluated at the vertical outer
z-boundaries and the outer R-boundaries, while the synchrotron and
bremsstrahlung emissions are effectively emitted near the equatorial plane.
Accordingly, the former ($L_{\rm rad}$) is expected to delay the latters
($L_{\rm syn}$ and $L_{\rm brem}$) by a transit time of the light from the
equator to the outer z-boundary if there is no time lag between the
synchrotron and bremsstrahlung emissions on the equator. The transit time is
200$R_{\rm g}/c$ ($7.9\times 10^{3}$ s $\sim$ 2 h). Actually, we confirm that
$L_{\rm rad}$ quite often delays $L_{\rm brem}$ by $\sim$ 2 h at local maximum
and minimum points of the luminosity curves but that the delay is different
for $L_{\rm syn}$ because the synchrotron emission delays the bremsstrahlung
emission in most flares. In our radiation MHD simulations using the RadRMHD
module of the ${\sc PLUTO}$ code, it should be noted that the radiative
cooling is taken into account only the free-free emission between the ions and
electrons because the one-temperature model is used in the ${\sc PLUTO}$ code.
Then the radiative luminosity is underestimated compared with the
bremsstrahlung luminosity due to the two-temperature model, since in the
latter the electron-electron bremsstrahlung is also considered in addition to
the electron-proton bremsstrahlung. However, it will not affect our time delay
analysis in this paper.

Fig. 4 shows the time variations of the synchrotron luminosity $L_{\rm syn}$
(blue square), bremsstrahlung luminosity $L_{\rm brem}$ (orange Delta) and
oscillating shock location $R_{\rm s}$ (green circle on the equator during
$t=1.9-2.1\times 10^{7}$ s in model Rad1. The curves are plotted every
$50R_{\rm g}/c$ time interval (1.98$\times 10^{3}$ s $\sim$ 0.55 h). The
attached “arrow” indicates local peaks on the synchrotron luminosity curve and
the marks of A, B and C expresses delay types; Type A with a positive time
delay of the synchrotron to bremsstrahlung flare, Type B with no time delay
and Type C with a negative time delay at which the bremsstrahlung flare delays
the synchrotron one. From the figure, we can find roughly the distribution of
delay-type of the flares.

---
Figure 5: Details of the luminosity curves of $L_{\rm rad}$ (red diamond),
$L_{\rm syn}$ (blue square), $L_{\rm brem}$ (orange Delta) and oscillating
shock location $R_{\rm s}$ (green circle) on the equator during $t$= 1.9237 –
1.9445 $\times 10^{7}$ s in model Rad1. The symbols “arrow”,
“$\bigtriangledown$” and “$\bigtriangleup$” show the corresponding peaks in
the curves of $L_{\rm syn}$, $L_{\rm brem}$ and $L_{\rm rad}$, respectively.
The letters ${\textbf{S}i}$, ${\textbf{B}i}$ and ${\textbf{R}i}$ express the i
th flare event on the synchrotron, bremsstrahlung and radiative luminosity
curves, respectively. The number within the bracket of the synchrotron symbol
S shows the delay point number relative to the bremsstrahlung emission. For
example, ${\textbf{S}1(-4)}$ shows that the 1 th synchrotron flare event
delays the bremsstrahlung emission by 4 points time interval ($\sim$ 2 h). The
scale of $L_{\rm rad}$ is shown inside on the right vertical axis.
---
Figure 6: Same as Fig. 5 but during $t$= 1.965 – 1.990 $\times 10^{7}$ s.

In order to show details of the time delay, we plot typical two cases of the
luminosity curves $L_{\rm rad}$ (red diamond), $L_{\rm syn}$ (blue square) and
$L_{\rm brem}$ (orange Delta) over a shock oscillation cycle in Figs. 5 and 6.
The luminosity curve $L_{\rm rad}$ is useful to distinguish the peak and dip
points in the bremsstrahlung luminosity curve because $L_{\rm rad}$ delays
nearly always $L_{\rm brem}$ by 200$R_{\rm g}/c$, that is, 4 points time
interval. To be able to distinguish the time delay between $L_{\rm rad}$,
$L_{\rm syn}$ and $L_{\rm brem}$ clearly, the scale of $L_{\rm rad}$ is shown
inside on the right vertical axis. In the figures, the symbols “arrow”,
“$\bigtriangledown$” and “$\bigtriangleup$” show the corresponding peaks in
the curves of $L_{\rm syn}$, $L_{\rm brem}$ and $L_{\rm rad}$, respectively,
and the letters ${\bf Si}$, ${\bf Bi}$ and ${\bf Ri}$ express the ${\bf i}$ th
flare event in the synchrotron, bremsstrahlung and radiative luminosity
curves, respectively and the number within the bracket of the synchrotron
symbol shows the delay point number relative to the bremsstrahlung emission.
For example, ${\bf S1(-4)}$ shows that the 1 th synchrotron flare event delays
the bremsstrahlung emission by 4 points time interval ($\sim$ 2 h). Type A
flares (events 1, 2 and 5) occur on the shock descending branch and have a
delay of 4 – 6 points time interval (2 – 3 h). Type B is the intermediate case
between Type A and Type C and occurs only when the shock is far away from the
center. Type C flares (events 3, 4, 6 and 7) with a delay time of 1 – 2 points
time interval (0.5 – 1 h) occurs when the oscillating shock attains the
minimum location and expands.

In the same way as the previous paper (Okuda, Singh & Aktar 2022), to examine
the effective emitting regions of $L_{\rm syn}$ and $L_{\rm brem}$, we
calculate local synchrotron and bremsstrahlung luminosities $L_{\rm syn}(R)$
and $L_{\rm brem}(R)$ which are emitted within a sphere of radius $R$. As a
result, we confirmed that most of the synchrotron emission is emitted within a
confined small region of $R\leq$ 5 and 60 – 80 percent of the bremsstrahlung
emission comes from the inner region of $10\leq R\leq 20$ at phases of the
minimum shock location. Then, we regard the synchrotron and bremsstrahlung
emission dominant radii $R_{\rm syn}$ and $R_{\rm brem}$ as 5 and 10,
respectively. The delay time between the synchrotron and bremsstrahlung
emissions in Type A is estimated as the transit time of the Alfvén wave from
$R_{\rm brem}$ to $R_{\rm syn}$. The acoustic wave also contributes to the
synchrotron radiation but is not effective as the Alfvén wave because the
synchrotron emission strongly depends on the magnetic field. Fig. 7 shows
profiles of radial velocity $V_{\rm r}$ (solid line), sound velocity $V_{\rm
s}$ (dashed line) and Alfvén velocity $V_{\rm A}$ (dash-dot-dot line) on the
equator at $t=1.936\times 10^{7}$ s (left-hand panel) and $t=1.980\times
10^{7}$ (right-hand panel). The oscillating shock locations are found at
$R\sim$ 48 on the shock descending branch (left-hand panel) and $\sim$ 50 on
the shock ascending branch (right-hand panel). Here, $V_{\rm s}\geq 0.1c$ and
$V_{\rm A}\geq 0.02c$ at $R\leq 20$ on the shock descending branch. Then, the
transit time ($R_{\rm brem}-R_{\rm syn})/V_{\rm A}$ in Type A is estimated as
$\leq 250R_{\rm g}/c$ = 2.5 h which corresponds well to the above delay time 2
– 3 h.

Type B flare with no time-lag appearing at the far distant shock location is
naturally understood from the above consideration. Near the maximum shock
location, the synchrotron and bremsstrahlung dominant regions are as $R_{\rm
syn}\sim 5$ and $R_{\rm brem}\geq 50$, respectively, the Alfvén velocity is
small as $\sim 10^{-3}$ in the distant region and the shock strength is weak.
Then the transit time of the Alfvén wave in Type B is too long compared with
the above transit time in Type A. Thus, in Type B, there is no interaction of
the perturbed Alfvén wave between the synchrotron and bremsstrahlung dominant
regions, that is, there is no time lag between the synchrotron and
bremsstrahlung emissions on the equator.

Type C flare appears in a special state as follows. During the time evolution
of the magnetized flow under MRI, the magnetic field intermittently increases
and decreases near the event horizon, and the magnetic pressure gradient force
begins to dominate the gas pressure gradient force, the gravitational and
centrifugal forces along the rotational axis and sometimes in the equatorial
direction. This leads to an intermittent high-velocity jet along the
rotational axis and an outflow even in the equatorial direction. The outflow
above and below the equator is at one time faded into the strong accreting
flow, at other times leads to turbulent flow and grows in the outer turbulent
flow as an expanding hot and dense blob as is explained in Figs. 7, 8 and 12
in the previous paper (Okuda, Singh & Aktar 2022). The expanding hot blob
sometimes interacts with the contracting oscillating shock and is incorporated
into the shock, a strong flare occurs as Type C. The time delay of Type C is
estimated as the transit time of the perturbed acoustic wave from $R_{\rm
syn}$ to $R_{\rm brem}$ because the powerful synchrotron emission influences
the outer bremsstrahlung dominant region through the acoustic wave, contrary
to Type A, because the bremsstrahlung emission depends only on the density and
the temperature. The delay time is $\sim(R_{\rm brem}-R_{\rm syn})/V_{\rm
s}\leq 50R_{\rm g}/c$ (0.55 h), using $V_{\rm s}\geq 0.1c$, which is very
small compared with that in Type A.

---
Figure 7: Profiles of radial velocity $V_{\rm r}$ (solid line), sound velocity
$V_{\rm s}$ (dashed line) and Alfvén velocity $V_{\rm A}$ (dash-dot-dot line)
on the equator at $t=1.936\times 10^{7}$ s (left-hand panel) and
$t=1.980\times 10^{7}$ (right-hand panel). The oscillating shock locations are
found at $R\sim$ 48 on the shock descending branch (left-hand panel) and
$\sim$ 50 on the shock ascending branch (right-hand panel). In the latter
case, the outflow triggered by the strong magnetic field near the event
horizon is expanding as a hot blob at $R\sim 40$ and the dip at $R\sim$ 30 of
the Alfvén velocity is due to the weak spot of magnetic field behind the
expanding hot blob.

####  3.2.2 Time delay between 22, 43 and 350 GHz flares

Time delay of $\sim$ 20 min between flares at 22 and 43 GHz have been detected
with simultaneous X-ray observations for Sgr A*. To examine the time delay
between the radio bands from the shock oscillation model, we recalculated 2D
radiation MHD simulations during $t$= 1.9237 – 2.0 $\times 10^{7}$ s, using
further smaller time step of 20$R_{\rm g}/c$ ($\sim$ 13 min). Figs. 8 and 9
show two cases of monochromatic luminosity curves of $L_{\rm 22}$ at 22 GHz
(blue square), $L_{\rm 43}$ at 43 GHz (orange Delta), $L_{\rm 350}$ at 350 GHz
(black diamond) and $R_{\rm s}$ (green circle) during $t$= 1.925– 1.960
$\times 10^{7}$ s and 1.966– 1.986 $\times 10^{7}$ s, respectively, over a
shock oscillation cycle. The “arrow” shows local peaks or dips on the
luminosity curves where A1– 12, B1 – B12 and C1– C12 are the corresponding
event names in $L_{\rm 22}$, $L_{\rm 43}$ and $L_{\rm 350}$ curves,
respectively. The number within the bracket of B and C shows the delay point
number relative to the 22 GHz flare. Here, one point time interval is
$20R_{\rm g}/c$ ($\sim$ 13 min). The scale of $L_{\rm 350}$ is shown inside on
the right vertical axis. $L_{\rm 22}$ delays $L_{\rm 43}$ by $\sim$ 1 – 2
points time interval (13 – 26 min) on the shock ascending branch and
conversely $L_{\rm 43}$ delays $L_{\rm 22}$ by 1 – 4 points time interval
except for event A1 and A8 on the shock descending branch. $L_{\rm 350}$
delays $L_{\rm 22}$ variously by $\sim$ 1 – 11 points time interval (13 min –
2.5 h) positively on the shock descending branch and negatively on the shock
ascending branch, respectively.

---
Figure 8: Monochromatic luminosity curves of $L_{\rm 22}$ (erg ${\rm
s}^{-1}{\rm Hz}^{-1}$) at 22 GHz (blue square), $L_{\rm 43}$ (erg ${\rm
s}^{-1}{\rm Hz}^{-1}$) at 43 GHz (orange Delta) , $L_{\rm 350}$ (erg ${\rm
s}^{-1}{\rm Hz}^{-1}$) at 350 GHz (black diamond) and oscillating shock
location $R_{\rm s}$ (green circle) on the equator during $t$= 1.925– 1.960
$\times 10^{7}$ s in model Rad1. The “arrow” shows local peaks or dips on the
luminosity curves where A1– A6, B1 – B6 and C1– C6 are the corresponding event
names in $L_{\rm 22}$, $L_{\rm 43}$, and $L_{\rm 350}$ curves, respectively,
the number within the bracket of B and C shows the delay point number relative
to $L_{\rm 22}$ and negative number means time delay to 22 GHz flare. The
scale of $L_{\rm 350}$ is shown inside on the right vertical axis. Here, one
point time interval corresponds to $20R_{\rm g}/c$ ($\sim$ 13 min).
---
Figure 9: Same as Fig. 8 but for events A7 – A12, B7 – B12 and C7– C12 during
$t$= 1.966 – 1.986 $\times 10^{7}$ s.
---
Figure 10: Monochromatic radiation flux (erg ${\rm s}^{-1}{\rm cm}^{-2}{\rm
Hz}^{-1}$) (left-hand panel) and optical thickness (right-hand panel) at the
disc surface in the frequency of 22 GHz (solid line), 43 GHz (dashed line) and
350 GHz (dash-dot-dot line) at $t=1.936\times 10^{7}$ s for model Rad1.
---
Figure 11: Same as Fig. 10 but for $t=1.980\times 10^{7}$ s.

These time delays are understood from consideration of the frequency-dependent
effective radii of $R_{\rm 22}$, $R_{\rm 43}$ and $R_{\rm 350}$, at whicht the
optical thickness $\tau_{\rm R,\nu}$ at frequency $\nu$ becomes $\sim$ unity.
Figs. 10 and 11 show the radial dependence of vertical radiation flux $F_{\rm
R,\nu}$ and optical thickness $\tau_{\rm R,\nu}$ at the accretion disc surface
for $t=1.936\times 10^{7}$ and 1.980 $\times 10^{7}$ s, which correspond to
event A2 in Fig. 8 and event R6 in Fig. 6, respectively. We find here that the
radiation fluxes are negligibly small beyond the frequency-dependent effective
radii $R_{\rm 22}$, $R_{\rm 43}$ and $R_{\rm 350}$. That is, most of the
monochromatic luminosities are produced within the frequency-dependent
effective radii. $R_{\rm 22}$, $R_{\rm 43}$ and $R_{\rm 350}$ are $\sim$ 22.0,
20.0 and 4.0 at $t=1.936\times 10^{7}$ s and 31.0, 30.0 and 7.0 at $t$ = 1.980
$\times 10^{7}$ s, respectively, from Figs. 10 and 11. These effective radii
are dependent only on $\kappa_{\rm\nu}$, that is, the density and the
temperature, but not the magnetic field. Then, the delay times between 22 and
43 GHz flares and between 22 and 350 GHz flares are estimated as the transit
time of the sound velocity, ($R_{\rm 22}$ \- $R_{\rm 43})/V_{\rm s}$ and
($R_{\rm 22}$ \- $R_{\rm 350})/V_{\rm s}$, respectively, which are roughly 20
and 180$R_{\rm g}/c$ ($\sim$ 13 min and 2 h) with $V_{\rm s}=0.1c$ at
$t=1.936\times 10^{7}$, and 10 and 240$R_{\rm g}/c$ ($\sim$ 8 min and 2.5 h)
with $V_{\rm s}\sim 0.1c$ at $t=1.980\times 10^{7}$ from the velocity profiles
in Fig. 7. Figs. 10 and 11 also show the existence of the optically thick
region in the radio frequencies. These optically thick regions appear at 22 –
350 GHz emissions considerd here only in the inner region of the accretion
flow.

The predicted delay time between 22 and 43 GHz flares agree well the numerical
ones. However, the delay times between very different frequencies such as 22
and 350 GHz seem to differ from the actual numerical delay time. This may be
partly related to the broader optically thick region between $R_{\rm 22}$ and
$R_{\rm 350}$ because the transit time of the light passing through the
optically thick region is not specified definitly. While, in the transit
region between $R_{\rm 22}$ and $R_{\rm 43}$, the optically thick gas is
confined in very narrow region and the predicted delay time may agree with the
numerical ones.

####  3.2.3 Time delay between 8 and 10 GHz flares

Remarking on the recently detected simultaneous observation at 8 and 10 GHz
with VLA ( Michail, Yusef-Zadeh & Wardle 2021), we examined the time delay
between 8 and 10 GHz flares. Fig. 12 shows the monochromatic luminosity curves
$L_{\rm 8}$ at 8 GHz (blue square), $L_{\rm 10}$ at 10 GHz (orange circle) and
the oscillating shock location $R_{\rm s}$ (green circle) on the equator
during $t$= 1.940 – 1.950 $\times 10^{7}$ s in the shock oscillation model.
The “arrow” shows local peaks on the luminosity curves where A1– A5 and B1 –
B5 are the corresponding event names in $L_{\rm 8}$ and $L_{\rm 10}$,
respectively, the number in the bracket of 10 GHz event B shows the delay
point number relative to $L_{\rm 8}$. Although the flares show marginal one
point delay time except for 5 points delay time in event B3, we consider them
to be significant from comparison and analogy with the time delay in 23 – 43
GHz flare and conclude the delay time between 8 and 10 GHz flare as 1 point
time interval ( $\sim$ 13 min). The 8 GHz flare delays 10 GHz flare on the
shock ascending branch but precedes 10 GHz flare on the shock descending
branch as well as the time-delay relation between 22 and 43 GHz flares. For
instance, taking the event A2 at $t=1.9412\times 10^{7}$, the effective radii
$R_{\rm 8}$ at 8 GHz and $R_{\rm 10}$ at 10 GHz are found to be 35.5 and 33.0,
respectively. Then, the predicted delay time between 8 and 10 GHz flare is 25
$R_{\rm g}/c\sim$ 16 min with $V_{\rm s}\sim 0.1c$ which agrees well with the
numerical delay time of $\sim$ 13 min.

---
Figure 12: Monochromatic luminosity curves of $L_{\rm 8}$ (erg ${\rm
s}^{-1}{\rm Hz}^{-1}$) at 8 GHz (blue square), $L_{\rm 10}$ (erg ${\rm
s}^{-1}{\rm Hz}^{-1}$) at 10 GHz (orange circle) and oscillating shock
location $R_{\rm s}$ (green circle) on the equator during $t$= 1.940– 1.950
$\times 10^{7}$ s in model Rad1. The “arrow” shows local peaks on the
luminosity curves where A1– A5 and B1 – B5 are the corresponding event names
in $L_{\rm 8}$ and $L_{\rm 10}$ curves, respectively. The number in the
bracket of 10 GHz flare B shows the delay point number relative to $L_{\rm
8}$. The delay time between 8 and 10 GHz flare is 1 point time interval (
$\sim$ 13 min) except 5 points interval in event B3. The 8 GHz flare delays 10
GHz flare on the shock ascending branch but precedes 10 GHz flare on the shock
descending branch as well as the flare relation between 22 and 43 GHz. Here
one point time interval corresponds to $20R_{\rm g}/c$ ($\sim$ 13 min).
Figure 13: Schematic light curves of X-ray, 350 GHz, 43 GHz and 23 GHz bands
on 2006 July 17 for Sgr A* taken from Fig. 1 in the paper by Yusef-Zadeh et
al. (2008). a2, b2, b3, c1, c2, d1 and d2 show flares at each energy bands.
Only horizontal axis of time is roughly shown for time delay discussion. Arrow
shows rough flare peak time. We suggest that a series of flares a2, b2. c2 and
d2 may be associated each other.  Figure 14: Schematic light curves of 850
$\mu m$ (350 GHz) and 7 mm at the second epoch on 2004 September for Sgr A*
taken from Fig. 8 in the paper by Yusef-Zadeh et al. (2006a), where A1 – A6
and B3 – B6 show flares at 850 $\mu m$ and 7 mm in radio bands, respectively.
Only horizontal axis of time is roughly shown for time delay discussion and
the flares appear roughly with a period of 1d.

####  3.2.4 Comparison with the Plasmon model

The expanding hot plasmon model (van der Laan 1966) has been proposed to
explain the time delay of 20 – 40 min between the flare peaks observed at 22
and 43 GHz in several different epochs using the VLA (Yusef-Zadeh et al.
2006b, 2008). Following van der Laan (1966), the synchrotron flux from a
homogeneous blob can be written as

$S_{\nu}(R)=S_{0}\left(\frac{\nu}{\nu_{0}}\right)^{2.5}\left(\frac{R}{R_{0}}\right)^{3}\,\frac{1-\exp(-\tau_{\rm\nu})}{1-\exp(-\tau_{0})},$
(25)

where $R(t)$ is the radius of the expanding hot blob and $R_{\rm 0}$ is the
initial blob radius, and the optical depth

$\tau_{\rm\nu}=\tau_{0}\left(\frac{\nu}{\nu_{0}}\right)^{-(K+4)/2}\left(\frac{R}{R_{0}}\right)^{-(2K+3)}.$
(26)

Here $K$ is the index of the relativistic particle energy spectrum as
$n(E)\propto E^{-K}$, $\tau_{0}$ is the critical optical depth at the maximum
of the light curve at any frequency, which satisfies

$e^{\tau_{0}}-(2K/3+1)\tau_{0}-1=0$ (27)

and $\nu_{0}$ is the frequency at which this occurs when $R=R_{0}$. This
equation is derived from the condition $\partial_{t}S(\nu,t)=0$ and $\tau_{0}$
ranges 0.55 to 1.9 as K ranges from 0.5 to 3.

From the above relations, the initially optically thick ($\tau_{\rm\nu}\gg 1$)
synchrotron flux $S_{\nu}(R)$ at a fixed frequncy $\nu$ increases with
increasing radius as $\propto R^{3}$ but turns to decrease with further
increasing radius as $\propto R^{-2K}$ when the optical thickness becomes
small as $\tau_{\rm\nu}\ll 1$. If we consider the synchrotron flares at
frequencies $\nu_{1}$ and $\nu_{2}$ ($\nu_{1}<\nu_{2}$) and the frequency-
dependent radii of $R_{\rm\nu_{1}}$ and $R_{\rm\nu_{2}}$ at which the optical
thickness is $\tau_{0}$, $R_{\rm\nu_{1}}$ must be larger than $R_{\rm\nu_{2}}$
from the relation of equation (20). Accordingly, the low frequency flare (the
maximum flux at the low frequency) appears after the high frequency flare (the
maximum flux at the high frequency) because the hot blob is expanding. In this
respet, our analytical method for the time delay prediction on the shock
ascending branch is identical to the idea of the plasmon model because the
radiation flux has same functional form of the optical depth in both cases. We
compare the frequency-dependent radii of $R_{\rm\nu_{22}}$, $R_{\rm\nu_{43}}$
and $R_{\rm\nu_{350}}$ at 22, 43 and 350 GHz by the plasmon model with the
numerical ones in the shock oscillation model. The frequency-dependent radius
$R_{\rm\nu}$ is given by, from equation (20)

$R_{\rm\nu}=\left(\nu\over\nu_{0}\right)^{-(K+4)\over{2(2K+3)}}\;R_{0}.$ (28)

As the result, $R_{\rm\nu_{22}}$, $R_{\rm\nu_{43}}$ and $R_{\rm\nu_{350}}$ are
8.0, 6.2 and 2.7 with K=3, $\nu_{0}$=130 GHz and $R_{0}$ = 4, and 19.0, 13.0
and 4.0 with K=0.5, $\nu_{0}$ = 350 GHz and $R_{0}$=4, respectively (see Fig.
3 in Yuan-Zadeh et al. 2006b and Fig. 6b in Yusef-Zadeh et al. 2008). These
values of $R_{\rm\nu_{22}}$, $R_{\rm\nu_{43}}$ and $R_{\rm\nu_{350}}$ are
somewhat small compared with the frequency-dependent effective radii of 22.0,
20.0 and 4.0 at $t=1.936\times 10^{7}$ and 31.0, 30.0 and 7.0 at $t=1.98\times
10^{7}$ s in our simulations. This may be partly due to that the frequency-
dependent radius in our model is the radius at which the optical depth is
unity but not $\tau_{0}$. However, the remarkable difference between the
plasmon model and the shock oscillation model is that the former consider only
the expanding hot blob, while in the latter the time delay of flares between
the radio and X-ray and between the narrow radio bands appear on the shock
ascending branch as well as the plasmon model but occurs inversely also on the
shock descending branch.

##  4 Comparison with observations of Sgr A*

We compare the time lag between the synchrotron and bremsstrahlung emissions
in the shock oscillation model with that between radio and X-ray emission
observed in Sgr A*. Through the simultaneous radio, near-infrared (NIR), and
X-ray observation campaign with HST NICMOS, XMM-Newton and VLA, the
observations have shown that radio flares delays X-ray flares by $\sim$ 2 h on
2006 July (Yusef-Zadeh et al. 2008), up to $\sim$ 5 h on 2007 April (Yusef-
Zadeh et al. 2009), $\leq$ 80 min on 2013 July, $\sim 2$ h on 2013 September,
$\leq$ 7.5 h or $\leq$ 3.9 h on 2013 October and $\sim$ 30 min on 2014 May
(Capellupo et al. 2017), $\sim$ 30 min on 2019 July (Michail et al. 2021), and
20 – 30 min on 2019 July (Boyce et al. 2022). Most of the delay time is 0.5 –
5 h and agrees roughly with the time delay of 2 – 3 h obtained from the
numerical simulations. However, our simulations predict that radio flare
delays X-ray flare on the shock descending branch with inflow but X-ray delays
radio flare on the ascending branch with outflow.

Ponti et al. (2015, 2017) found a very bright flare from Sgr A*, which is by
more than two orders of magnitude higher than the bremsstrahlung emission in
usual flares, starts in NIR and then an X-ray flare follows after $\sim
10^{3}$ s. They confirm the origin of the very bright X-ray flare as the
synchrotron nature. If the NIR flare is simultaneous with radio flare by the
original synchrotron emission, the very bright flare may appear as Type C as
is found in event R6 in Fig. 6. The time delay is comparable with the time
delay of $\leq$ 0.55 h predicted from the shock oscillation model.

Yusef-Zadeh et al. (2008) report that 22 GHz flare delays 43 GHz by 20 – 30
min. The delay time is comparable to 13 – 26 min in our simulations. Michail,
Yusef-Zadeh & Wardle (2021) report the detection and analysis of radio flares
across narrow 16 frequency bands between 8 and 10 GHz on 2014 April and find
that the flare delays successively with decreasing frequencies and 8 GHz light
curve delays 10 GHz light curve by 18 min, which agrees well with $\sim$13 min
obtained from the present simulations. The positive or negative time delay
relation is dependent on the shock location branch as well as the case of 22
and 43 GHz flare.

A composite light curve of flaring activity in the X-ray, 350 GHz, 43 GHz and
22 GHz on 2006 July reported by Yusef-Zadeh et al. (2008) is very interesting
because we are able to interpret the observational result with our time delay
analysis from the simulations. Fig. 13 shows schematic light curves of X-ray,
350 GHz, 43 GHz and 23 GHz bands for Sgr A* taken from Fig. 1 in the paper by
Yusef-Zadeh et al. (2008). Here, a2, b2, b3, c1, c2, d1 and d2 show the flare
events at each energy bands. Only horizontal axis of time is roughly shown for
the time delay interpretation. A strong X-ray flare (a2) and weak and strong
350 GHz flares (b2 and b3) are detected at $\sim$ 6:10, 5:45 and 7:30 hr UT on
2006 July. Two flares c1 and c2 at 43 GHz occur at $\sim$ 4:25 and 6:30 – 7:00
hr UT, respectively and 22 GHz flares (d1, d2) delay the 43 GHz flare (c1, c2)
by $\sim$ 20 – 30 min. From the plasmon model and our flare analysis on the
shock ascending branch, 22, 43 and 350 GHz flares should successively delay in
this order. Then we may recognize the flares b2, c2 and d2 as a relevant
group. The strong flare 350 GHz (b3) perhaps correspond to next relevant group
at 22 and 43 GHz flare although it may be ambiguous at the final phase of the
light curve. Then, the 43 GHz flare (c2) delays the 350 GHz flare (b2) by
$\sim$ 45 – 75 min. Such delay time between 43 and 350 GHz is observed as
$\sim$ 100 min at events A4 and A11 in Figs. 8 and 9 of the simulations. If
X-ray flare (a2) at the top panel in Fig. 13 is associated with the radio
flare of 350 GHz flare (b2), the time delay of $\sim$ 25 min between X-ray and
radio flare is reasonable compared with $\sim$ 30 min in events R4 and R7 in
Figs. 5 and 6 on the ascending branch in the shock oscillation model. Finally,
we suggest that a series of flares a2, b2. c2 and d2 may be associated each
other, which appears as Type C flare.

The plasmon model predicts that the low frequency flare always delays the high
frequency flare, However, there exist an important observation of flares at
radio bands, where the high frequency flare delays the low frequency flare.
Yusef-Zadeh et al. (2006a) showed simultaneous X-ray, 850 $\mu$m (0.85 mm),
450 $\mu$m, 3mm and 7 mm flare light curves at two epochs on 2004 March –
September of Sgr A*. Fig. 14 shows schematic light curves of 850 $\mu m$ (350
GHz) and 7 mm at the second epoch taken from Fig. 8 in the paper by Yusef-
Zadeh et al. (2006a), where A1 – A6 and B3 – B6 show flare points at 850 $\mu
m$ and 7 mm, respectively. Here, consecutive four pairs of flare (A3, B3) –
(A6, B6) with a period of $\sim$ 1d are regarded to correlate to each other
simultaneously and 850 $\mu m$ flares delay 7 mm flares roughly by $\sim$ 4 h.
Although another pairs of (A3, B4) and (A4, B5) may be considered to associate
together, the delay time of $\sim$ 20 h is too long to explain the time delay
from the analysis of the shock oscillation model. Similarly, from Fig. 7 in
Yusef-Zadeh et al. (2006a), we suggest that the successive three 850 $\mu m$
flares delay 7 mm flars about by $\sim$ 1 h. These positive delays of the high
frequency to the low frequency flare contradict the plasmon model but are
naturally expected in the shock oscillation model.

##  5 Summary and Discussion

We examined the time delay relations of flares between radio and X-ray and
between narrow radio frequency bands in Sgr A*, from analyses of the
synchrotron and bremsstrahlung luminosity and monochromatic luminosity
$L_{\rm\nu}$ curves obtained from the time-dependent 2D relativistic radiation
MHD simulations based on the shock oscillation model. Here, ion and electron
temperatures are obtained by solving the radiation energy equilibrium equation
that Coulomb collisions transfer energy from ions to electrons equals to the
sum of synchrotron and bremsstrahlung cooling rates in electrons. The results
can be summarized as follows:

(1) We find three types of time delay between the synchrotron and
bremsstrahlung emissions which correspond to radio and X-ray emissions,
respectively; Type A with a time delay of 2 - 3 h, Type B with no time delay
and Type C with an inverse negative time delay of $\leq$ 0.5 – 1 h. Type A
flare occurs on the shock descending branch, Type B appears when the shock is
far away from the center and Type C occurs only when the oscillating shock
contracts up to the minimum location, an expanding hot blob triggered by the
sporadic magnetic field is incorporated into the oscillating shock, and as a
result a strong flare is formed.

(2) The synchrotron emission is always dominant in a core region of 5$R_{\rm
g}$ size, while most of bremsstrahlung emission mainly originates in a region
of 10 – 20$R_{\rm g}$ at the minimum shock location but comes from a distant
region of $\leq 50R_{\rm g}$ at the maximum shock location. The time delay of
Type A and Type C is interpreted as the transit time of the Alfvén and
acoustic waves between the synchrotron and bremsstrahlung dominant regions,
respectively. The delay time of Type A is $\sim 250R_{\rm g}/c\sim$ 2.5 h with
the Alfvén velocity $\sim 0.02c$ but in Type B there is actually no
interaction between the synchrotron and bremsstrahlung dominant regions
because the transit time is too long. On the other hand, the delay time of
Type C is small as $\sim$ 0.5 h because the sound velocity far exceeds the
Alfvén velocity.

(3) 22 GHz flare delays 43 GHz flare by $\sim$ 13 – 26 min on the shock
ascending branch and conversely precedes 43 GHz flare by the same delay time
on the shock descending branch. The delay time between 22 and 350 GHz ranges
extensively up to 4 h. The delay time between 8 and 10 GHz flare is $\sim$ 13
min as well as the time delay relation between 22 and 43 GHz. These time
delays between 22 and 43 GHz and between 8 and 10 GHz agree well with 20 – 30
min and 18 min, respectively, observed in Sgr A*. The time delay of flares
between the narrow radio frequency bands is interpreted as the transit time of
the acoustic wave between the frequency-dependent effective radii
$R_{\rm\tau_{\rm\nu}=1}$ at which the optical depth $\tau_{\rm\nu}$ is unity.

(4) Many delay type between radio and X-ray flares observed in Sgr A* belongs
to Type A with time lags of $\sim$ 0.5 – 5 h and Type C is rarely observed
with time lag of $\leq$ 20 min, as is found on 2006 July 17 (Yusef-Zadeh et
al. 2008) and 2014 August 30 (Ponti et al. 2015, 2017). Similarly, it is
likely that the flare at the higher radio frequency delays that at the lower
radio frequency by $\sim$ 10 – 30 min and also the inverse delay occurs, as is
confirmed in the observed flares on 2004 September (Yusef-Zadeh et al. 2006a)
and on 2014 April 17 (Michail, Yusef-Zadeh & Wardle 2021), respectively.

The shock oscillation model for the long-term flare of Sgr A* explains well
the observations of the flaring rate and also the delay time of flares between
radio and X-ray and between the narrow frequency radio bands. Of course, the
energy spectra of Sgr A* do not consist of only the simple synchrotron and
bremsstrahlung emissions and are reproduced by various radiation mechanisms
such as the synchrotron, the inverse Compton and the synchrotron self-Compton
emissions including a hybrid electron population consisting of both thermal
and nonthermal particles (e.g. Yuan, Quataert & Narayan 2003, 2004). However,
the present estimate of time delay between the radio and X-ray emissions and
between the narrow radio bands may not be altered largely.

The expanding hot plasmon model explains not only the time delay between the
peak emission of 43 and 22 GHz but also other available measurements of the
flux values in near-IR, submillmeter, milimeter, and radio wavelengths, using
the adquate parameters fitted to the observed light curves. To explain the
submillimeter light curves correlated with the NIR and possible delays of
$\sim$ 30 min between 22 and 43 GHz , Witzel et al. (2021) proposed a simple
source model with three-step-process of electron injection with a nonthermal
energy distribution into a spherical central region, compression of magnetic
field lines and increasing magnetic flux, and adiabatic expansion with no
injection of electrons, and modeled the three-step process as a cyclic process
in a single zone. The model is similar to the plasmon model but differs from
that it considers the cyclic expansion and contraction processes of the blob.

In the shock oscillation model, the monochromatic radiation energy flux
$F_{\rm\nu}(R,\tau_{\rm\nu})$ on the shock descending branch begins to
brighten abruptly when the optical depth $\tau_{\rm\nu}$ becomes unity. Since
$\tau_{\rm\nu}\propto\kappa_{\rm\nu}\propto 1/\nu^{2}$ for radio waves, a
larger optical depth $\tau_{\rm\nu}$ at a smaller frequency $\nu$ leads to a
larger effective radius $R_{\rm\tau_{\rm\nu}=1}$ because the optical depth
decreases with increasing radius. This means that a brightening of radiation
flux $F_{\rm\nu}(R,\tau_{\rm\nu})$ begins progressively from a flare with the
smallest frequency at inflow phase of the accretion flow. Contrarily, when the
intermittently increasing magnetic field near the event horizon yields to an
outflow and expanding hot blob as Type C flare, the extinction of the
brightened optically thick flux starts in a flare with the largest frequency
and ends with the smallest frequency flare which delays most. Therefore, the
oscillating shock model predicts that, when radio flare delays X-ray flare, 43
GHz flare necessarily delays 22 GHz flare, and vice versa. In section 4, we
already confirmed an existence of observation for Sgr A* such that the high
frequency flare delays the low frequency flare, which contradicts the plasmon
model but is natural in the shock oscillation model. Another observational
constraint on the shock oscillation model is that the positive delay of radio
to X-ray flare have to appear together with the positive delay of high
frequency to low frequency flare, if these flares occur simultaneously. At
present, such confirmatory observation on the oscillating shock model has not
been met yet, including denial observation. We hope future simultaneous
observations in radio to X-ray and in narrow radio bands for Sgr A* confirm
them.

## Acknowledgments

CBS is supported by the National Natural Science Foundation of China under
grant no. 12073021. The numerical computations were conducted on the Yunnan
University Astronomy Supercomputer.

## Data Availability

The numerically created data that support the findings of this study are
available from the corresponding authors upon reasonable request.

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# A Simple Baseline for Knowledge-Based Visual Question Answering

Alexandros Xenos1 Themos Stafylakis2,3 Ioannis Patras1 Georgios Tzimiropoulos1
1Queen Mary University of London 2Athens University of Economics and Business
3Omilia - Conversational Intelligence, Athens, Greece
<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author.

###### Abstract

This paper is on the problem of Knowledge-Based Visual Question Answering (KB-
VQA). Recent works have emphasized the significance of incorporating both
explicit (through external databases) and implicit (through LLMs) knowledge to
answer questions requiring external knowledge effectively. A common limitation
of such approaches is that they consist of relatively complicated pipelines
and often heavily rely on accessing GPT-3 API. Our main contribution in this
paper is to propose a much simpler and readily reproducible pipeline which, in
a nutshell, is based on efficient in-context learning by prompting LLaMA (1
and 2) using question-informative captions as contextual information. Contrary
to recent approaches, our method is training-free, does not require access to
external databases or APIs, and yet achieves state-of-the-art accuracy on the
OK-VQA and A-OK-VQA datasets. Finally, we perform several ablation studies to
understand important aspects of our method. Our code is publicly available at
https://github.com/alexandrosXe/A-Simple-Baseline-For-Knowledge-Based-VQA

##  1 Introduction

Knowledge-based VQA (KB-VQA) is a recently introduced VQA task Wang et al.
(2017, 2018); Marino et al. (2019); Shah et al. (2019) where the image alone
is not sufficient to answer the given question, but effective utilization of
external knowledge resources is additionally required. To solve such a task, a
model would need not only strong visual perception but also reasoning
capabilities while also being able to effectively incorporate world knowledge
from external KBs (e.g. Wikipedia, etc) and LLMs. Systems capable of answering
general and diverse questions about the visual world find a wide range of
applications: from personal assistants to aids for the visually impaired and
robotics 111https://www.adelaide.edu.au/aiml/our-research/machine-
learning/vqa-vision-and-language.

Recently, several works on KB-VQA Gui et al. (2022); Lin et al. (2022) have
emphasized the significance of incorporating both explicit and implicit
knowledge. However, such approaches usually require complicated pipelines.
Firstly, a KB (e.g. wikidata) covering world knowledge needs to be maintained
and used for knowledge retrieval which is time-consuming and very sensitive to
noise. Secondly, powerful LLMs such as GPT-3 Brown et al. (2020) or OPT-175B
Zhang et al. (2022) are leveraged due to the huge amount of implicit knowledge
stored in their parameters and their powerful reasoning capabilities through
few-shot in-context learning. However, the computational or even actual
monetary cost (e.g. cost for API access) associated with accessing such models
renders them unaffordable for many researchers. Thirdly, it is crucial to
train a fusion mechanism that can effectively reason by combining the
retrieved explicit and implicit knowledge.

Main contributions: We present a simple yet powerful pipeline for KB-VQA which
by-passes the need for using most of the components of the above-mentioned
systems. Specifically, the proposed system is simply based on few-shot
prompting of LLaMA-13B Touvron et al. (2023a, b). The key component of our
method is the implementation of effective in-context learning using question-
informative captions as contextual information which, as we show, results in
large accuracy boosts.

The proposed system features several advantages: (1) it is entirely training-
free, requiring only a few examples for in-context learning; (2) it is based
on the open-source LLaMA-13B Touvron et al. (2023a, b) (considerably smaller
than the widely-used GPT-3); (3) it is straightforward to reproduce; and (4)
achieves state-of-the-art (SOTA) accuracy on the widely-used OK-VQA Marino et
al. (2019) and A-OK-VQA datasets Schwenk et al. (2022).

##  2 Related Work on KB-VQA

Methods Without LLMs: Several methods have been proposed including KRISP
Marino et al. (2021) which uses a multi-modal pretrained BERT Devlin et al.
(2019), MAVEx Wu et al. (2022) which proposes to validate promising answer
candidates based on answer-specific knowledge retrieval, and DPR which uses
pseudo-relevance labels integrated with answer generation for end-to-end
training. Typically, these systems are not as competitive as the ones based on
LLMs.

Methods based on LLMs: PICa Yang et al. (2022) is the first method to adopt
GPT-3 for solving the KB-VQA task in a few-shot manner by just providing a few
in-context VQA examples. Gui et al. (2022) proposed to use both implicit (i.e.
GPT-3) and explicit (i.e. KBs) knowledge based on CLIP retrieval Radford et
al. (2021) which are combined by a novel fusion module called KAT (based on T5
or Bart). Lin et al. (2022) proposed to integrate local visual features and
positional information (bounding box coordinates), retrieved external and
implicit knowledge (using a GPT-3) into a transformer-based question-answering
model. Hu et al. (2023) proposed PromptCap, a novel task-aware captioning
model that uses a natural language prompt to control the generation of the
visual content that can be used in conjunction with GPT-3 in-context learning.
Img2Prompt Guo et al. (2023) is a zero-shot VQA method that generates image-
relevant exemplar prompts for the LLM. Their key insight is that synthetic
question-answer pairs can be generated using image captioning and question-
generation techniques as in-context exemplars from the provided image. Prophet
Shao et al. (2023) proposes to prompt GPT-3 with answer heuristics (answer
candidates and answer-aware examples) that are encoded into the prompts to
enable GPT-3 to better comprehend the task, thus enhancing its capacity.

##  3 Methodology

While explicit knowledge retrieval focuses on semantic matching between an
image and knowledge entries, it lacks implicit commonsense knowledge (e.g.
Lemons are sour) which can be found in LLMs Gui et al. (2022). LLMs are
critical in extracting implicit knowledge due to the vast amount of implicit
information embedded in their parameters, and their powerful reasoning
capacity through few-shot in-context learning. Different from previous work
Yang et al. (2022); Gui et al. (2022); Lin et al. (2022) we leverage the open-
source LLM LLaMA-13B Touvron et al. (2023a, b) instead of GPT-3 as an implicit
language knowledge base and treat VQA as an open-ended text generation task.

Our method builds upon the pipeline of PICa, which is the pioneering work that
utilizes GPT-3 for few-shot in-context learning in order to address the KB-VQA
task. GPT-3 is a decoder-only autoregressive LLM of 175B parameters, trained
on a diverse range of data sources, including Common Crawl, webtexts, books,
and Wikipedia Brown et al. (2020). During inference, in-context few-shot
learning involves formulating a novel downstream task as a text sequence
generation task using the frozen GPT-3 model. When provided with a testing
input $x$, the target $y$ is predicted based on a formatted prompt
$p(h,C,E,c,x)$. In this prompt, $h$ represents a prompt head or instruction
that describes the task, while $E=\\{e_{1},e_{2},...,e_{n}\\}$ represents a
set of $n$ in-context examples (shots), where $e_{i}=(x_{i},y_{i})$ represents
an input-target pair of the task, where $x_{i}$ and $y_{i}$ are the input and
target, respectively. These pairs are constructed manually or sampled from the
training set. $C=\\{c_{1},c_{2},...,c_{n}\\}$ represents a set of generic
image captions describing each $x_{i}$ since images cannot be inputted to
GPT-3. The caption for the test input is labeled as $c$. The target $y$ is
denoted as a text sequence consisting of $L$ tokens, expressed as
$y=(y^{1},y^{2},...,y^{L})$. At each decoding step $t$, the following
conditions apply:

$\hat{y}^{t}=\operatorname*{argmax}_{y^{t}}p_{LLM}(y^{t}|p,\hat{y}^{<t})$ (1)

In order to utilize any LLM for the knowledge-based VQA task, the crucial step
is to design suitable prompts. When given a question $q_{i}$ and an image
$v_{i}$ as inputs, the VQA task’s objective is to predict the corresponding
answer $a_{i}$. However, since LLMs do not inherently comprehend images, it
becomes necessary to convert the image into a caption $c_{i}$ using a pre-
existing captioning model. While SOTA pretrained captioning models have
demonstrated impressive performance, they are primarily optimized to generate
generic image captions. Unfortunately, these captions often fail to capture
all the specific details required to accurately answer a given question about
the image. In this work, instead of generic captions, we generate question-
guided informative image captions using the Plug-and-Play VQA (PNPVQA)
framework Tiong et al. (2022) which identifies the most related image patches
to the question with a saliency map-based interpretability technique and
generates captions from these patches only.

Figure 1: Inference-time of our method for n-shot VQA. The input prompt to
LLaMA consists of a prompt head $h$ (blue box), $n$ in-context examples
($\\{c_{i},x_{i},y_{i}\\}^{n}_{i=1}$) (red boxes), and the VQA input
$\\{c,x\\}$ (green box). The answer $y$ is produced in an open-ended text
generation manner. In this example we use two question-informative captions
per example (separated by commas).

For each image-question pair, we first generate 50 question-guided informative
image captions from the image $v_{i}$ using PNPVQA. We then employ BLIP’s Li
et al. (2022) text encoder to encode all the image captions and BLIP’s image
encoder to encode the image $v_{i}$. We rank the image captions per image
$v_{i}$ according to their cosine similarity with the image $v_{i}$ and keep
the top-$m$ most similar captions $c_{i}$ per example. After extracting the
top-$m$ most similar captions per image $v_{i}$ we construct a carefully
designed text prompt consisting of a general instruction sentence, the
captions $C$, the question, the test input’s captions $c$, and a set of
context-question-answer triplets (shots) taken from the training dataset that
are semantically most similar to the current image-question pair (see Fig. 1).
Then this text prompt is passed to a frozen LLaMA-13B model and in-context
few-shot learning is performed in order to obtain its output as a promising
answer candidate to the current image-question pair.

###  3.1 Selecting Informing Examples For Few-Shot In-Context Learning

As Yang et al. (2022) notes, feeding more in-context examples to GPT-3 yields
better few-shot performance. However, the maximum input length of the model
constrains the maximum number of examples $n$ in the prompt. To better use
these available examples we: (i) improve the example quality by careful in-
context example selection Liu et al. (2022); Gui et al. (2022); Shao et al.
(2023), and (ii) use more examples via multi-query ensemble.

In-context Example Selection tries to search for the best examples for each
inference-time input $x$ among all available examples Yang et al. (2022). We
consider in-context examples that have similar question features as $x$. More
specifically, given an inference-time question, we use BLIP’s text encoder to
obtain its textual feature and compute its cosine similarity with the
questions in all available in-context examples. We then average the question
text similarity with the image visual similarity to guide the example
selection similarly to Yang et al. (2022). We select the top-$n$ questions
with the highest similarity and use the corresponding examples as the in-
context examples.

Multi-query ensemble: Given an inference-time example $x$, we use $k\times n$
in-context examples to generate $k$ prompts. This way, we prompt LLaMA-13B for
$k$ times and obtain $k$ answer predictions instead of 1 similar to Yang et
al. (2022), where $k$ is the number of queries to ensemble. Finally, among the
$k$ answer predictions, we select the one with the most occurrences (majority
vote).

##  4 Experimental Results

Method | Knowledge Resources | Acc (%)
---|---|---
KRISP | Wikipedia+ConceptNet | 38.35
MAVEx | Wikipedia+ConceptNet+Google Images | 39.4
Unified-IO (2.8B) | Multimodal Pretraining | 54
Flamingo (80B) | Multimodal Pretraining | 57.8
PICa-Full | Frozen GPT-3 (175B) | 48.0
KAT_base (single) | Wikidata+Frozen GPT-3 (175B) | 50.58
KAT_large (single) | Wikidata+Frozen GPT-3 (175B) | 53.09
KAT_large (ensemble) | Wikidata+Frozen GPT-3 (175B) | 54.41
REVIVE_large (single) | Wikidata+Frozen GPT-3 (175B) | 56.6
REVIVE_large (ensemble) | Wikidata+Frozen GPT-3 (175B) | 58.0
Prophet | Frozen GPT-3 (175B) | 61.1
Ours | Frozen LLaMA (13B) | 58.69
Ours + MCAN | Frozen LLaMA (13B) | 60.02
Ours | Frozen LLaMA 2 (13B) | 59.07
Ours + MCAN | Frozen LLaMA 2 (13B) | 61.2

Table 1: Comparison with other methods on the OK-VQA dataset: Our method with
9 question-informative captions achieves state-of-the-art performance.

Comparative results on OK-VQA: Table 1 summarizes the results of various
methods on OK-VQA including our best method (last row) which uses 9 question-
informative captions and 5 query ensembles. When using LLaMA our approach
outperforms all methods and achieves comparable results with Prophet
especially when using the same shot selection strategy based on MCAN Yu et al.
(2019). Moreover, it performs better than Unified-IO and the 80B Flamingo
which have been pre-trained with multimodal objectives. When compared to
methods that rely on GPT-3 for implicit knowledge extraction, our approach
outperforms PICa-Full which only uses generic image captions by 12.02% while
outperforming the SOTA supervised methods KAT and REVIVE by 5.61% and 2.02%
respectively. Finally, when using LLaMA 2 and MCAN-based shot selection
strategy, our method achieves state-of-the-art accuracy of 61.2%.

Method | DA |  | MC |
---|---|---|---|---
| Val | Test | Val | Test
ClipCap | 30.9 | 25.9 | 56.9 | 51.4
ViLBERT | 30.6 | 25.9 | 49.1 | 41.5
LXMERT | 30.7 | 25.9 | 51.4 | 41.6
KRISP | 33.7 | 27.1 | 51.9 | 42.2
GPV-2 | 48.6 | 40.7 | 60.3 | 53.7
Unified-IO | - | 45.2 | - | -
Prophet | 58.2 | 55.7 | 59.3 | 57.3
Ours (LLaMA) | 54.4 | 53.8 | - | -
Ours + MCAN (LLaMA) | 57.4 | 55.0 | - | -
Ours (LLaMA 2) | 57.1 | 55.4 | - | -
Ours + MCAN (LLaMA 2) | 58.6 | 57.5 | - | -

Table 2: Comparison with other methods on the A-OK-VQA dataset: Our method
with 9 question-informative captions achieves state-of-the-art performance at
the direct answer (DA) setting. Note that our method does not support
multiple-choice (MC).

Comparative results on A-OK-VQA: Table 2 summarizes the results of various
methods on A-OK-VQA including our best method (last row) which uses 9
question-informative captions and 5 query ensembles. We compare our method to
the strong baselines in Schwenk et al. (2022) and the current state-of-the-art
method Prophet Shao et al. (2023). When employing LLaMA, our approach
surpasses all other methods on the DA setting and achieves comparable results
to Prophet, particularly when employing the same shot selection strategy based
on MCAN. Finally, with LLaMA 2 and MCAN our method attains state-of-the-art
performance on both the validation and test sets, achieving 58.6% and 57.5%
accuracy respectively, demonstrating the effectiveness and robust
generalization of our proposed method.

##  5 Ablation Studies

We conduct several ablations on OK-VQA to better understand the key components
of our method.

Captions |  |  | $n$ | $k$ | Acc (%)
---|---|---|---|---|---
Generic |  |  | 14 | 5 | 43.35
Question-informative |  |  | 14 | 5 | 57.56

Table 3: Generic vs. question-informative captions.

Effect of question-informative captions: Table 3 shows the performance of our
method when using generic captions vs question-informative captions for in-
context learning which is the key component of our system. Following Yang et
al. (2022); Shao et al. (2023) we leverage the OSCAR+ Zhang et al. (2021) as
the captioning model. The results suggest using question-informative captions
results in huge accuracy boosts (43.35% vs 57.56%).

Shot Selection Strategy | Captions | $m$ | $n$ | $k$ | Acc (%)
---|---|---|---|---|---
Random | Question-informative | 1 | 14 | 5 | 53.19
Avg. Question and Image Sim. | Question-informative | 1 | 14 | 5 | 56.50
MCAN latent space | Question-informative | 1 | 14 | 5 | 57.56

Table 4: Accuracy when using different shot selection strategies. Avg.
question and image sim. strategy retrieves shots based on the average cosine
similarity between the test sample’s question and image, and the training
examples’ question and image. MCAN latent space strategy retrieves shots that
are closer to the test sample in the trained MCAN’s latent space.

Effect of shot selection strategy: Table 4 shows that selecting random shots
during in-context learning hurts the accuracy, confirming the findings of Yang
et al. (2022). Retrieving shots based on the similarity between the test
sample and the training examples yields a significant accuracy boost.
Prophet’s shot selection strategy based on MCAN also seems to be effective but
we note that it is based on pre-training a vanilla VQA model on a different
dataset (VQA-v2).

Effect of number of question-informative captions: Fig. 2 (a) shows the
accuracy when we increase the number of captions per sample in the prompt
during in-context learning. Here, we are using $k=5$, and $n=10$ when using
1-10 captions, and $n=5$ when using more than 10 captions due to max. sequence
length constraints. More captions provide more information for each example
helping the model to make a more accurate prediction based on context. As
shown in the figure, the validation accuracy keeps increasing up to 60.02%.
When using more than 10 captions, the accuracy decreases but this also can be
attributed to the fact that we are also decreasing the number of shots to 5.

Effect of multi-query ensemble: Fig. 2 (b) shows the accuracy as the number of
prompts, $k$, increases. As anticipated, employing multiple prompts of LLaMA
instead of just one yields improved accuracy. However, beyond $k=6$, the
accuracy begins to fluctuate. It is important to note that this fluctuation
could be attributed to the retrieval of noisy (irrelevant to the question)
context examples as the value of $k$ increases.

|
---|---
(a) | (b)
Figure 2: (a) Accuracy vs number of question informative captions used per
shot during few shot in-context learning. (b) Accuracy vs number of prompts
$k$ used during in-context learning.

Effect of explicit knowledge: We also tried to use KAT’s Gui et al. (2022) KB
and trained a T5 Raffel et al. (2020) in order to integrate explicit knowledge
into our model. For each image, we used BLIP to extract explicit knowledge via
image-to-text retrieval. We used $40$ retrieved passages and LLaMA predictions
as explicit and implicit knowledge, respectively. We achieved an accuracy of
58.70% which shows that our model does not benefit from such an approach.

Effect of size of LLM: We also used a LLaMA-7B model using 9 question-
informative captions, $n=10$ and $k=5$. Reducing the size of the LLM leads to
decreased accuracy but the drop is not large, still obtaining 57.99% accuracy.

##  6 Conclusions

We proposed a simple yet effective baseline for KB-VQA. Our training-free
method is based on in-context few-shot learning of the open-source LLaMA using
question-informative captions. We show that this is sufficient to achieve SOTA
results on the widely used OK-VQA and A-OK-VQA datasets.

## Limitations

It is important to acknowledge that we have not explored the utilization of
any other medium-sized LLMs apart from LLaMA, which presents a limitation of
our study. Lastly, due to limitations in resources, we were unable to conduct
experiments with larger sizes beyond 13B. However, it would indeed be
intriguing to observe the performance when employing LLaMA models of sizes
such as 30B or 65B.

## Ethics Statement

The authors of this paper recognize the importance of responsible AI in both
research and development endeavors. We are committed to ensuring that the
model we develop is not only accurate but also fair and unbiased. We
understand the potentially significant impact of VQA technology on society
and, therefore, pledge to maintain transparency by sharing our findings and
progress with relevant researchers and stakeholders.

## Acknowledgements

Alexandros Xenos is funded by Queen Mary Principal’s PhD Studentships.

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##  Appendix A Example Appendix

###  A.1 Implementation Details

We used the Huggingface Transformers
library222https://huggingface.co/transformers/ in order to run LLaMA models.
We used beam search with beam size = 2 during generation and max new tokens =
5 while using the default values for all the other parameters in the generate
method. We run our model on a 40-GB VRAM A-100 GPU card.