# Hydrodynamic Evolution of Sgr A East: The Imprint of A Supernova Remnant in
the Galactic Center

Mengfei Zhang,1,2 Zhiyuan Li,1,2 Ziqian Hua1,2 Mark R. Morris3
1School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China
2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210023, China
3Department of Physics and Astronomy, University of California, Los Angeles,
CA 90095, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform three-dimensional numerical simulations to study the hydrodynamic
evolution of Sgr A East, the only known supernova remnant (SNR) in the center
of our Galaxy, to infer its debated progenitor SN type and its potential
impact on the Galactic center environment. Three sets of simulations are
performed, each of which represents a represent a certain type of SN explosion
(SN Iax, SN Ia or core-collapse SN) expanding against a nuclear outflow of hot
gas driven by massive stars, whose thermodynamical properties have been well
established by previous work and fixed in the simulations. All three
simulations can simultaneously roughly reproduce the extent of Sgr A East and
the position and morphology of an arc-shaped thermal X-ray feature, known as
the “ridge”. Confirming previous work, our simulations show that the ridge is
the manifestation of a strong collision between the expanding SN ejecta and
the nuclear outflow. The simulation of the core-collapse SN, with an assumed
explosion energy of $5\times 10^{50}$ erg and an ejecta mass of 10 M⊙, can
well match the X-ray flux of the ridge, whereas the simulations of the SN Iax
and SN Ia explosions underpredict its X-ray emission, due to a smaller ejecta
mass. All three simulations constrain the age of Sgr A East to be
$\lesssim$1500 yr and predict that the ridge should fade out over the next few
hundred years. We address the implications of these results for our
understanding of the Galactic center environment.

###### keywords:

ISM: supernova remnants – X-rays: ISM – methods: numerical – stars: winds,
outflows – Galaxy: centre

††pagerange: Hydrodynamic Evolution of Sgr A East: The Imprint of A Supernova
Remnant in the Galactic Center–LABEL:lastpage

##  1 Introduction

Thanks to its proximity, the Galactic center offers a unique opportunity for
disentangling the interplay between a supermassive black hole (SMBH) and its
environment in unparalleled detail. The Galactic center SMBH, commonly known
as Sgr A*, is currently extremely underluminous for its well-determined
gravitational mass of $4\times 10^{6}\rm~{}M_{\odot}$ (Do et al., 2019;
Gravity Collaboration et al., 2020). Surrounding and physically coupled to Sgr
A* is the Nuclear Star Cluster (NSC), which mainly consists of $\sim 10^{7}$
low-mass, old stars occupying the central parsecs (Feldmeier-Krause et al.,
2017). At the center of the NSC, lying within $\sim$0.5 pc of Sgr A*, is the
young nuclear cluster (YNC) containing more than a hundred massive stars. The
YNC is the result of the latest episode ($\sim$ 4–6 Myr ago) of star formation
around Sgr A* (Genzel et al., 2010; Lu et al., 2013). Most prominent among the
YNC are $\sim$30 Wolf-Rayet (WR) stars, each producing strong winds at a mass
loss rate of $\sim 10^{-5}\rm~{}M_{\odot}~{}yr^{-1}$ and a terminal velocity
of $\sim 1000\rm~{}km~{}s^{-1}$ (Martins et al., 2007). The mutual collisions
of these stellar winds create strong shocks, leading to rapid thermalization
of the wind kinetic energy (e.g., as observed in the case of IRS 13E; Zhu et
al., 2020; Wang et al., 2020). This ultimately results in a complex network of
hot gas with temperatures of $\sim 10^{7}$ K pervading the colliding wind
zone, as demonstrated by dedicated hydrodynamic simulations (Rockefeller et
al., 2004; Cuadra et al., 2006; Cuadra et al., 2008; Russell et al., 2017;
Ressler et al., 2018). It is generally accepted that Sgr A* is currently
mainly fed by the shocked winds, i.e., a Bondi-like accretion, at an extremely
sub-Eddington yet significantly variable level (Coker & Melia, 1997; Cuadra et
al., 2005; Calderón et al., 2020; Ressler et al., 2020a; Ressler et al.,
2020b).

Outside the effective Bondi radius of $\lesssim 0.1$ pc, the hot gas is not
gravitationally bound, but ultimately forms an outflow (Quataert, 2004; Cuadra
et al., 2006). This outflow, hereafter dubbed the nuclear outflow, carries
away the bulk mass, momentum and energy released by the massive stars, thereby
providing an effective shielding for Sgr A* from being heavily fed by the
circumnuclear medium. Therefore, a quasi-steady inflow-outflow system is
established in the central parsecs around Sgr A* (Shcherbakov & Baganoff,
2010), which, in the absence of violent outbursts from Sgr A* itself, should
be only mildly modulated by the orbital motion of the WR stars on a timescale
of $\lesssim 1000$ yr.

One expects that this quasi-steady state will be eventually broken when one or
more of the WR stars evolves to a core-collapse supernova (CCSN), the kinetic
energy and momentum of which can greatly reshape the circumnuclear gas.
Theoretical studies show that, depending on the strength of the SN and its
distance from the SMBH, the SN shock wave may sweep over the central volume
otherwise shielded by the stellar winds (Yalinewich et al., 2017) and may pump
gas into the close vicinity of the SMBH, potentially boosting its accretion
level (Palouš et al., 2020; Barna et al., 2021).

In fact, strong evidence exists for a SN shock currently encroaching upon the
vicinity of Sgr A*. High-resolution Chandra observations have revealed an arc-
shaped feature of diffuse X-ray emission to the east (in the sense of Galactic
coordinates) of Sgr A* (Maeda et al., 2002). This feature, highlighted in
Figure 1 and hereafter referred to as the “X-ray ridge” (Rockefeller et al.,
2005), spans a radial range of 0.36–0.6 pc from Sgr A* and an azimuth of
$\sim$ $170^{\circ}$. Rockefeller et al. (2005) proposed that the X-ray ridge
originates from an ongoing collision between the cumulative winds of massive
stars around Sgr A* and the forward shock of Sgr A East, the prominent shell-
like, non-thermal radio source that is widely believed to be a supernova
remnant (SNR; Ekers et al., 1983; Maeda et al., 2002). Rockefeller et al.
(2005) and Fryer et al. (2006) verified this scenario using smoothed particle
hydrodynamic (SPH) simulations, finding a reasonable agreement with Chandra
observations on the morphology and X-ray luminosity of the ridge and inferring
a dynamical age $\lesssim 2000$ yr for Sgr A East.

Sgr A East certainly plays an active and unique role in transforming the
immediate environment of Sgr A*. In this work, we revisit the evolution of Sgr
A East and its interaction with the nuclear outflow using hydrodynamic
simulations. We are motivated by several recent developments. The first
motivation is related to the type of SN explosion that created Sgr A East. For
decades since the identification of Sgr A East as a SNR, a core-collapse
progenitor has been favored. This is chiefly owing to (i) the fact that
numerous massive stars exist in the Galactic center, (ii) a high metal
abundance inferred from the X-ray spectrum of Sgr A East, presumably dominated
by the reverse shock-heated SN ejecta (Maeda et al., 2002; Sakano et al.,
2004), and (iii) a candidate neutron star, known as the “Cannonball” (Park et
al., 2005), whose current position and measured proper motion suggest a
physical association with Sgr A East (Zhao et al., 2013). However, this
consensus was recently challenged by Zhou et al. (2021), who, based on an
updated X-ray spectroscopic measurement of metal abundances and abundance
ratios of Sgr A East and a detailed comparison with modern SN nucleosynthesis
models, favored a Type Iax supernova (SN Iax). Compared to CCSNe, a SN Iax is
characterized by a relatively low explosion energy and a small ejecta mass,
which could strongly affect the subsequent hydrodynamic evolution. Hence it is
timely to investigate the hydrodynamic evolution of a SN Iax in the vicinity
of Sgr A* and to contrast it with the case of a CCSN, as was assumed by
Rockefeller et al. (2005).

A second motivation arises from significant recent progress that has led to a
good understanding of the thermodynamic properties of the nuclear outflow and
the gravitational potential around Sgr A* (see details in the following
sections), thanks to dedicated multi-wavelength observations and comprehensive
numerical simulations conducted over the past two decades. This facilitates
the use of numerical simulations to follow the hydrodynamic evolution of
different types of SNe in the vicinity of Sgr A*, which in turn allows us to
place an independent constraint on the explosive progenitor by confronting the
key observed properties of Sgr A East. Last but not least, a deliberated case
study of the currently only known SNR in the vicinity of Sgr A* is highly
complementary to the theoretical investigations of Yalinewich et al. (2017),
Palouš et al. (2020) and Barna et al. (2021), which survey the general
behavior of circumnuclear SNRs.

The remainder of this paper is organized as follows. We outline the key
observational constraints in Section 2 and describe the setup of the
hydrodynamic (HD) simulations in Section 3. The simulation results are
presented in Section 4. Implications of the results as well as caveats of our
simulations are discussed in Section 5. We assume a canonical distance of 8
kpc to Sgr A* ($1\arcsec$ corresponds to 0.039 pc) in this work.

##  2 Observational Constraints

We consider three key observed properties of Sgr A East as robust constraints
on the modelling of its evolution. The first is the current physical size of
Sgr A East, which is essentially defined by the extent of its radio shell. In
the left panel of Figure 1, we plot radio continuum intensity contours of the
Sgr A complex, obtained from the VLA 5.5 GHz map of Zhao et al. (2013),
against the X-ray emission from the same region, as seen by a deep Chandra
image originally presented in Zhu et al. (2020) and Zhou et al. (2021). We
approximate the radio shell with a cyan ellipse of a size of $\rm
8.2~{}pc\times 5.8~{}pc$, which will be used to guide the SN forward shock in
the HD simulations. We note that the radio shell has a clear elliptical
morphology on its eastern side, whereas a precise determination of its western
extent is complicated by the presence of Sgr A West (Herrnstein & Ho, 2005)
and the presumed interaction between the forward shock and the nuclear
outflow. Despite this uncertainty, the geometric center of the radio shell can
be readily inferred, which is assumed to be the SN explosion center in this
work, though Lau et al. (2015) questioned this assumption (see further
discussion in Section 5.1). However, this assumption is reinforced by the
second observational aspect, i.e., the X-ray-bright core of Sgr A East, which
is approximately delineated by a magenta ellipse in Figure 1. The X-ray
spectra of this region reveal strong metal lines, especially from highly-
ionized Fe, which are generally thought to be due to the SN ejecta (Maeda et
al., 2002; Sakano et al., 2004; Park et al., 2005; Koyama et al., 2007; Ono et
al., 2019; Zhou et al., 2021). The third and the most peculiar aspect is the
presence of the X-ray ridge, as highlighted in the right panel of Figure 1.
Its current position, morphology and surface brightness should provide a
strong constraint on the properties of the incoming SN shock, given the
thermodynamics properties of the nuclear outflow (see Section 3.2). A
satisfactory HD simulation is expected to reproduce the above three
observational aspects.

Figure 1: Left: A Chandra X-ray image of the inner few parsec region of the
Galactic center (from Zhou et al., 2021), overlaid by VLA 5.5 GHz continuum
intensity contours (from Zhao et al., 2013). The cyan ellipse delineates the
radio shell of Sgr A East, while the magenta ellipse highlights the X-ray-
bright interior. Right: A zoom-in view of the central 1.6 pc$\times$1.6 pc
region outlined by the green rectangle in the left panel. The positions of
known WR stars, which are responsible for the nuclear outflow, are marked by
the ‘x’ symbols, while Sgr A* is marked by the ‘+’ symbol. The X-ray ridge is
delineated by the green wedge. Galactic coordinates are adopted in both
panels.

##  3 Simulation

We use the publicly available, modular HD code
PLUTO111http://plutocode.ph.unito.it/ (Mignone et al., 2007, 2012) to simulate
the early evolution of Sgr A East. This grid-based HD code, with a second-
order Runge–Kutta time integrator and a Harten-Lax-van Leer Riemann solver for
middle contact discontinuities, is well suited for simulating the interaction
between the SN shock and the nuclear outflow induced by the WR stars
surrounding Sgr A*.

###  3.1 Basic configuration

The HD simulations for the SNR evolution are performed on a three-dimensional
(3D) Cartesian frame with an Adaptive Mesh Refinement (AMR) grid. The AMR is
desired due to the substantial dynamic range involved: Sgr A East currently
has a physical size of 8.2 pc $\times$ 5.8 pc, the nuclear outflow is launched
from a region spanning a few tenths of a parsec around Sgr A*, and the
interaction between the SN shock and the nuclear outflow is resolved by
Chandra on a scale of $\lesssim$0.04 pc. Moreover, the interaction zone is
highly dynamic, which precludes the use of a static mesh refinement. In the
AMR setup, we use a base grid (level 0) of $128^{3}$ and three refinement
levels (levels 1, 2 and 3) having an equivalent grid of $256^{3}$, $512^{3}$
and $1024^{3}$, respectively. Grid refinement takes place following a
substantial change in the total energy density. The physical volume of the
simulation box is set as $\rm 10^{3}~{}pc^{3}$ to cover the entire radio shell
of Sgr A East. Hence the linear resolution of levels 0, 1, 2 and 3 is $\sim$
0.08, 0.04. 0.02 and 0.01 pc pixel-1, respectively. In addition, we perform
seed simulations for the nuclear outflow (Section 3.2) on a 3D Cartesian frame
with a fixed grid of $512^{3}$, which are then expanded to $1024^{3}$ by using
linear interpolation.

We set the $z$-axis to be perpendicular to the Galactic disk (north as
positive), the $y$-axis to run along decreasing Galactic longitude, and the
$x$-axis to be parallel to the line-of-sight (the observer at the positive
side). The three-dimensional offset of the SN explosion center with respect to
Sgr A* is not known a priori and cannot be fully determined by observations.
Nevertheless, guided by the roughly elliptical shape of the radio shell
(Figure 1) and the commonly accepted fact that the bulk of Sgr A East is
located behind Sgr A* along the line-of-sight (e.g., Herrnstein & Ho, 2005),
we estimate the position of the explosion center and set it as the origin of
the simulation box. This places Sgr A* at $(x,y,z)_{\rm BH}$ = (1.0, 1.8, 1.5)
pc. The uncertainty of the explosion center could be a few tenths of a parsec
in the $y-z$ plane and somewhat larger in the $x$-direction. However, we note
that the dynamical age of Sgr A East, which is primarily constrained by the
projected size of the radio shell and the recent formation of the X-ray ridge
(see Section 4), is insensitive to the line-of-sight position of the explosion
center. In Appendix A, we present several test simulations that explore the
effect of a different position of the explosion center.

The simulations neglect magnetic fields, which are prevalent in the Galactic
center (Guenduez et al., 2020) and can in principle affect the dynamics of SN-
driven flows (Zhang et al., 2021). However, both the strength and orientation
of the magnetic field in the innermost parsecs of the GC are currently not
well constrained. Hence in the present study we assume that the dynamic
evolution of Sgr A East is not significantly affected by the magnetic field.
In this case, the fluid evolution can be described by the ideal HD equations,

$\displaystyle\frac{\partial\rho}{\partial
t}+\nabla\cdot(\rho\bf{v})=\dot{\rho}_{\rm wind}+\dot{\rho}_{\rm SN},$ (1)
$\displaystyle\frac{\partial(\rho\bf{v})}{\partial
t}+\nabla\cdot\left[\rho\bf{vv}+\bf{1}p\right]^{T}=-\rho\nabla\Phi+\dot{\Pi}_{\rm
wind}+\dot{\Pi}_{\rm SN},$ (2) $\displaystyle\frac{\partial E_{t}}{\partial
t}+\nabla\cdot\left[\left(E_{t}+p+\rho\Phi\right)\bf{v}\right]=-\frac{\partial\left(\rho\Phi\right)}{\partial
t}+\dot{E}_{\rm wind}+\dot{E}_{\rm SN},$ (3)

where $\rho$ is the mass density, $p$ the thermal pressure, $\bf{v}$ the
velocity, $\bf{1}$ the dyadic tensor, $\Phi$ the gravitational potential, and
$E_{t}$ the total energy density defined as

$\displaystyle E_{t}=\rho\epsilon+\frac{\rho\bf{v}^{2}}{2},$ (4)

where $\epsilon$ is the internal energy. We adopt an ideal equation of state,
i.e., $\rho\epsilon=p/(\Gamma-1)$, in which the ratio of specific heats
$\Gamma$ = 5/3. The source terms, ($\dot{\rho}_{\rm wind},\dot{\Pi}_{\rm
wind},\dot{E}_{\rm wind})$ and ($\dot{\rho}_{\rm SN},\dot{\Pi}_{\rm
SN},\dot{E}_{\rm SN})$, consist of the continuous input from the WR star winds
and the instantaneous input from the SN explosion (see Section 3.2 for details
of implementation).

The gravitational potential in the simulation volume is assumed to be static
and completely determined by the SMBH and the NSC. A point mass of $4\times
10^{6}\rm~{}M_{\odot}$ is taken to represent the SMBH. For the NSC, we adopt a
spherically symmetric distribution following Chatzopoulos et al. (2015,
Equation 5 therein), which has an enclosed mass of $\rm 6.5\times
10^{6}~{}M_{\odot}$ within a radius of 5 pc. Although a spheroidal model of
the NSC can better explain the velocity dispersion outside the central 5 pc
(Chatzopoulos et al., 2015), a spherical model is sufficiently accurate for
the stellar mass distribution within our simulation box of 103 pc3.

The simulations neglect viscosity but include thermal conduction and radiative
cooling. For the latter, we adopt the TABULATED cooling function implemented
in PLUTO, which is generated by Cloudy (Ferland et al., 2017) for an optically
thin plasma of a given elemental abundance. We impose a minimum cooling
temperature of 104 K, the exact choice of which does not significantly affect
the HD evolution and the resultant X-ray emissivity. We also adopt a floor for
the gas number density ($n_{\rm floor}=10^{-6}\rm~{}cm^{-3}$) to prevent
numerical artifacts that may lead to negative values in this quantity.

###  3.2 Nuclear outflow and supernova explosion

The mutual interaction of the strong winds from $\sim$30 WR stars in the
central half-parsec has been extensively studied by analytic models and
numerical simulations (Quataert, 2004; Rockefeller et al., 2004; Cuadra et
al., 2006; Cuadra et al., 2008; Ressler et al., 2018). These studies
demonstrate that wind-wind collisions lead to rapid thermalization of the wind
kinetic energy, creating a complex network of hot gas that ultimately escapes
in the form of a bulk outflow. Since we are mainly interested in the
interaction of this outflow with Sgr A East, our simulations neglect the
details of wind-wind collision and the chaotic gas motions within the wind
colliding zone. Instead, we implement a smooth and isotropic radial outflow by
employing an internal boundary in PLUTO, which approximates the gas density,
temperature and velocity profiles in a spherical shell between $r=0.08-0.2$ pc
following the results of Quataert (2004) and Ressler et al. (2018): $n\propto
r^{-1}\rm~{}cm^{-3}$, $T=1.5r^{-0.56}\rm~{}keV$, $v/v_{\rm out}=0.64{\rm
log}(r)+0.62$, where $r$ is the radial coordinate from Sgr A* in units of pc,
and the terminal velocity $v_{\rm out}=1000\rm~{}km~{}s^{-1}$. The
normalization of the density is such that the asymptotic mass loss rate
$\dot{M}_{\rm out}=1\times 10^{-3}\rm~{}M_{\odot}~{}yr^{-1}$, a value adopted
by Ressler et al. (2018) and compatible with the cumulative mass loss from the
$\sim 30$ WR stars. In this shell, the resolution is 0.01 pc pixel-1, or 12
grid points within the internal boundary. The velocity setting ensures that
the gas can flow both inward and outward. Gas within $r\lesssim 0.04$ pc is
bound by the gravity of Sgr A*. To mimic the SMBH accretion and avoid
unphysical gas pileup, we reset the gas density of the innermost pixels to a
constant value ($10^{-3}\rm~{}cm^{-3}$) and the gas temperature to $10^{4}$ K
after every time step. Such a treatment does not affect the outward
propagation of the nuclear outflow and its interaction with the SN shock. To
convert the gas number density into a mass density, we have assumed an element
composition of ($X_{\odot}$=0, $Y_{\odot}$=0.96, $Z_{\odot}$=0.04), i.e.,
hydrogen-free and a three-times solar abundance for elements heavier than
helium. This is consistent with the simulations of Ressler et al. (2018) and
motivated by the expectation that the WR star winds are hydrogen-depleted but
carbon- and nitrogen-enriched.

The simulations begin with an initial uniform ambient density of $\rm
0.1~{}cm^{-3}$ and a uniform temperature of 106 K. After $\sim$ 1 Myr, the
outflow propagates throughout the simulation box, resulting in distributions
of density, temperature and radial velocity consistent with the overall
profiles of Quataert (2004) and Ressler et al. (2018). This quasi-steady
environment for the subsequent explosion and evolution of the SN is completely
determined by the assumed properties of the nuclear outflow. We note that this
effectively neglects cold gas components, in particular, the circumnuclear
disk (CND) of dense molecular gas, a ring-like structure roughly aligned with
the Galactic plane and encompassing Sgr A West (Ferrière, 2012; Hsieh et al.,
2017). The potential effect of the CND is discussed in Section 5.3.

Once the quasi-steady pre-SN conditions are established, the SN is implemented
at the aforementioned explosion center in the form of injected energy ($E_{\rm
SN}$) and ejecta mass ($M_{\rm SN}$). Motion of the progenitor star relative
to Sgr A* is neglected, which should be small (order 100 km s-1) compared to
the SN shock velocity. We consider three combinations of $E_{\rm SN}$ and
$M_{\rm SN}$, which represent three possible types of supernova explosion. The
first type is SN Iax, as suggested by Zhou et al. (2021) for Sgr A East, which
has $E_{\rm SN}=5\times 10^{50}\rm~{}erg$ and $M_{\rm SN}=1.3\rm~{}M_{\odot}$.
The second type is SN Ia, for which we adopt $E_{\rm SN}=1\times
10^{51}\rm~{}erg$ and $M_{\rm SN}=1.3\rm~{}M_{\odot}$. The last type is CCSN,
for which we adopt $E_{\rm SN}=5\times 10^{50}\rm~{}erg$ and $M_{\rm
SN}=10\rm~{}M_{\odot}$. We note that SNe Iax can have a $E_{\rm SN}$ as low as
$10^{50}\rm~{}erg$, while CCSNe can have $E_{\rm SN}$ up to $10^{51}\rm~{}erg$
and a wide range of $M_{\rm SN}$. The values chosen here are such that the SN
Iax and CCSN simulations differ from the SN Ia simulation by just one of the
two key parameters, which helps to discern the main effect of either explosion
energy or ejecta mass. In Appendix B, we discuss a test simulation with
$E_{\rm SN}=5\times 10^{51}\rm~{}erg$ and $M_{\rm SN}=10\rm~{}M_{\odot}$. A
more thorough exploration of the parameter space in $E_{\rm SN}$ and $M_{\rm
SN}$ is deferred to a future work.

$E_{\rm SN}$ and $M_{\rm SN}$ are injected at time $t=0$ within an effective
radius of 0.2 pc (20 grid points) around the explosion center, following the
analytic density, velocity and temperature profiles of Blondin & Ellison
(2001). The corresponding age of the SN at this radius is $\sim$ 10 yr (Leahy
& Williams, 2017), which is small compared to the evolution time up to the
formation of the X-ray ridge (see Section 4). Specifically, for the SN Ia and
Iax, we adopt a uniform core, a power-law ejecta density profile with a power-
law index $n=7$ and a constant ambient density profile with a power-law index
$s=0$, while for the CCSN, we adopt $n=7$ and $s=2$. Our test simulations
indicate that the results are insensitive to the exact values of $n$ and $s$
within reasonable ranges. For simplicity, we assume a chemical composition for
the SN ejecta same as for the nuclear outflow. Although in reality the ejecta
could be significantly more metal-enriched, this choice has little effect on
the HD evolution, nor does it significantly affect the synthetic X-ray map,
since the simulations show that the ejecta contribute little to the emission
from the X-ray ridge (see Section 4). This is further confirmed by our test
simulation with a strongly metal-enriched ejecta ($Z_{\odot}$=0.08), which is
detailed in Appendix C.

Once injected, the ejecta is in contact with the nuclear outflow and the two
fluids eventually partially mix with each other. We introduce a tracer
parameter, $Q$, which is evaluated at each pixel in the simulation and obeys a
simple conservation law:

$\displaystyle\frac{\partial(\rho Q)}{\partial t}+\nabla\cdot(\rho
Q\bf{v})=0.$ (5)

$Q$ has a value of 1 for pure SN ejecta and 0 for the unpolluted nuclear
outflow, and values in between to indicate a mixed gas. The physical
conditions at $t=0$ are illustrated in Figure 2, which plots one-dimensional
profiles of the gas density, temperature, velocity and ejecta tracer parameter
along the line passing through the SN explosion center and Sgr A*.

Figure 2: One-dimensional profiles of density (blue), temperature (orange),
velocity with respect to Sgr A* (purple; positive as away from Sgr A*) and
ejecta tracer (red) along the line connecting the explosion center (zero-
point) and Sgr A* (marked by a vertical dotted line), immediately after the SN
explosion. Here the tracer parameter assumes a lower bound of 0.001 for ease
of visualization.

###  3.3 Simulation runs and synthetic X-ray emission

With the above settings, we run a series of simulations. Our fiducial
simulation is denoted by run E5M1.3, which represents an SN Iax explosion, as
defined in Section 3.2, and serves as an illustration of the basic behavior of
the SNR evolution (Section 4.1). The two other runs, E10M1.3 and E5M10,
represent the case of SN Ia and CCSN, respectively, and differ from the
fiducial run in either the explosion energy or the ejecta mass. The main
parameters of the different runs are summarized in Table 1. For all three
runs, the total elapsed time since the injection of the supernova is set to be
2 kyr, which is roughly the time when the simulation box is almost filled by
the ejecta in the fiducial run. The time step is adaptive and ranges between
$0.1-10$ yr.

Table 1: Main parameters of the simulation runs

Run | $E_{\rm SN}$ | $M_{\rm SN}$ | $t$ | $S_{2-8}$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
E5M1.3 | 5 | 1.3 | 700 | 2.8
E10M1.3 | 10 | 1.3 | 500 | 4.9
E5M10 | 5 | 10 | 1400 | 6.0

  * •

(1) Simulation run. (2) Explosion energy, in units of $10^{50}$ erg. (3)
Ejecta mass, in units of $\rm M_{\odot}$. (4) The epoch (in units of yr) when
the morphology of the synthetic X-ray ridge best matches the observed one
(Section 4). (5) The predicted 2–8 keV surface brightness of the ridge at this
epoch, in units of $10^{-15}\rm~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$.

To facilitate comparison with the observations, we generate synthetic X-ray
intensity maps for selected snapshots of each simulation run. The X-ray
emissivity of an optically-thin thermal plasma in collisional ionization
equilibrium (CIE; Smith et al., 2001) is extracted from
ATOMDB222http://www.atomdb.org, version 3.0.9, for which we adopt the
elemental composition specified in Section 3.2. The X-ray intensity maps are
derived by integrating along the $x$-axis. In calculating the synthetic X-ray
emission, the electron temperature is assumed to be equal to the ion
temperature. This is justified by the estimated equipartition time of
$\sim$100 yr between electrons and ions, due to Coulomb interaction (Masai,
1994), for density (10 cm-3), ion temperature (108 K) and electron temperature
(106 K) conditions characteristic of the X-ray ridge, which is comparable to
the dynamical timescale. On the other hand, the timescale for CIE is $\sim
300(\rm cm^{-3}/n)$ yr for He and longer for heavier elements (Smith & Hughes,
2010). Hence the assumption of CIE is not strictly valid, but this should have
little effect on the resultant X-ray flux (Schure et al., 2009; Zhang &
Chevalier, 2019). An a posteriori comparison also finds a reasonable agreement
between the emissivity-weighted ion temperature predicted by our simulations
and the mean electron temperature inferred from the observed X-ray spectrum of
the ridge (Hua et al., 2023).

We note, in reality, the evolution of Sgr A East and its interaction with the
nuclear outflow are determined by many factors, including the SN explosion
energy, ejecta mass, the site and environment of the explosion, the nuclear
outflow properties, etc. With a primary focus on the formation of the X-ray
ridge, the exploration of the parameter space in this work is necessarily
limited. In particular, conditions of the medium at and near the SN explosion,
such as the presence of dense cold cloudlets, may significantly affect the
subsequent evolution. However, the exact properties of such cloudlets are
rather difficult, if at all possible, to accurately constrain. Thus, we do not
consider them in this work (see further discussions in Section 5.3), but note
that Rockefeller et al. (2005) and Ehlerová et al. (2022) have performed
simulations to consider some specific conditions of the surrounding medium,
which are complementary to our study here. A more thorough exploration of the
parameter space would be an interesting future work.

##  4 Results

In this section, we present the simulation results. We first describe in
detail the evolution of the SNR and the formation of the X-ray ridge in the
fiducial run (Section 4.1). We then examine these processes in the two
additional runs, one with a higher explosion energy (Section 4.2) and the
other with a higher ejecta mass (Section 4.3), to illustrate how the change in
one of two key parameters would affect the evolution of the SNR and the
formation of the X-ray ridge.

###  4.1 The run in the fiducial set

The evolution of Sgr A East and its subsequent interaction with the nuclear
outflow in run E5M1.3 are illustrated by the sequential snapshots presented in
Figure 3 and 4.

In the top and middel panels of Figure 3, we show the density and temperature
distributions in the $x=1$ pc plane (i.e., the plane across Sgr A*) at $t=$
200, 500, 800 and 1100 yr after the SN explosion. The artifacts shown in the
panels can be attributed to the AMR, which arise primarily at the site of
level transition. The expanding ejecta drive a strong forward shock at an
initial velocity of $\rm\sim 10^{4}~{}km~{}s^{-1}$, compressing and heating up
the ambient gas to temperatures $\gtrsim 10^{8}$ K. In the meantime, a reverse
shock develops back into the ejecta. This results in the familiar morphology
seen at $t=200$ yr, where a nearly spherical shell of shock-heated hot gas
encompasses the cold (due to adiabatic expansion), unshocked ejecta
(delineated by the colored contours of the trace parameter in the density
maps). As it approaches Sgr A*, the forward shock is decelerated by the ram
pressure of the nuclear outflow. This results in a significant deviation of
the shell from a round shape on the side facing the nuclear outflow by the
time of $t=500$ yr. In fact, the ram pressure of the nuclear outflow is so
strong that the forward shock can never penetrate into the region of
$r\lesssim 0.3$ pc. This is evident in the snapshot of $t=800$ yr, where the
forward shock bends around the central outflow zone. In contrast, the forward
shock has expanded almost freely on the opposite (i.e., southeast) side,
reaching a radius comparable to the radio extent of Sgr A East (indicated by
the pink ellipse) at $t=800$ yr. At later times, the forward shock continues
to bend on the northwest side and progressively engulfs the central outflow
zone, while on the southeast side it has moved out of the simulation box by
the time of $t=1100$ yr. The bottom panels of Figure 3 show the synthetic 2–8
keV X-ray surface brightness ($S_{2-8}$) distribution in the different epochs.
Generally, the synthetic X-ray morphology traces the expanding shell. High
surface brightness ($S_{2-8}\gtrsim
10^{-15}\rm~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$) is found in the central
outflow zone and the surrounding regions where the outflow interacts with the
forward shock. A low surface brightness cavity is coincident with the
unshocked (hence cold) ejecta. Notably, this is in sharp contrast with the
observed X-ray-bright interior of Sgr A East (delineated by the red ellipse).
This discrepancy will be further addressed in Section 5.3.

Figure 3: Simulation run E5M1.3. The top and middle panels show snapshots of
density and temperature distributions in the $x=1$ plane at 200, 500, 800 and
1100 yr (from left to right) after the SN explosion at the origin, while the
bottom panels show the corresponding 2–8 keV X-ray surface brightness
distribution. Sgr A* is located at $(x,y,z)=(1.0,1.8,1.5)$ pc and marked by a
black ‘+’. The pink ellipse, red ellipse and red wedge outline the observed
position of the radio shell, X-ray-bright interior and X-ray ridge,
respectively, as in Figure 1. The extent of the SN ejecta is outlined by the
white, red and black contours in the top panels, which correspond to an ejecta
tracer parameter of 0.9, 0.5 and 0.1, respectively. The white arrows in the
middle panels indicate the projected velocity vector. Some peculiar filaments
seen in the central region of the temperature maps are artifacts due to the
adaptive grid refinement.

.

Figure 4 provides a closeup view of the interaction between the forward shock
and the nuclear outflow, in which the upper (middle) two rows depict the
density (temperature) distribution in the central 1.6 pc $\times$ 1.6 pc
region around Sgr A*, for eight consecutive snapshots from $t=400$ yr to 1100
yr. At $t=400$ yr, the forward shock is approaching from the southeast,
followed by the SN ejecta (indicated by the tracer contours in the density
maps), but its interaction with the nuclear outflow is still weak at this
time. At $t=500$ yr, the shock front comes closer to Sgr A*, and clearly its
shape becomes flattened due to the ram pressure of the nuclear outflow. The
intrusion of the ejecta lags the forward shock. Notably, the ejecta tracer
contours exhibit finger-shaped patterns, which are understood as due to the
Rayleigh-Taylor instability. At $t=600$ yr, the shock front reaches a distance
of $\sim$0.5 pc from Sgr A* and starts to bend around an apex where the ram
pressure of the nuclear outflow approximately equals that of the shock. This
effectively builds up a dynamic interaction zone between the nuclear outflow
and the SN ejecta, which is sandwiched by two standing shocks.

The stratified structure of this interaction zone is highlighted in Figure 5,
in which the one-dimensional profiles of gas density, temperature, velocity
and ejecta tracer parameter between the explosion center and Sgr A* are
plotted for the epochs of $t=600$, 700 and 800 yr. At $t=600$ yr, the two
standing shocks are clearly captured by coincident temperature, density and
velocity jumps, one separating the unshocked and shocked ejecta at an offset
of $\sim$1.5 pc from the explosion center, and the other separating the
unshocked and shocked outflow at an offset of $\sim$2.1 pc. Both the shocked
ejecta and the shocked outflow have their temperatures strongly raised to
$\sim 10^{7}-10^{8}$ K. The main contact discontinuity between the shocked
ejecta and shocked outflow is located at an offset of $\sim$1.6 pc, but along
this particular line the ejecta actually intrudes deeper due to the Rayleigh-
Taylor instability, which helps to mix the shocked ejecta with the shocked
outflow. Nevertheless, the ejecta generally do not reach within $\sim$0.6 pc
from Sgr A*. At later times ($t=$ 700 and 800 yr), the inner standing shock
gradually weakens and is pushed back toward the unshocked ejecta by the over-
pressured shocked ejecta, whereas the outer standing shock roughly maintains
its position.

Figure 4: Similar to Figure 3, but zooming into a 1.6 pc $\times$ 1.6 pc
region centered on Sgr A* and for snapshots from $t=$ 400 yr to 1100 yr in
steps of 100 yr, in the simulation run E5M1.3. The central dip is due to an
artificial sink of gas to mimic accretion onto the SMBH. The white plus and
red arc indicate Sgr A* and the X-ray ridge, respectively. The white arrow
points in the direction of the SN explosion center. Color scales in the X-ray
surface brightness maps (the bottom two rows) are adjusted to suppress the
central outflow zone at radii of 0.2 pc from Sgr A* for an optimal
illustration of the ridge.

.

Figure 5: One-dimensional profiles of density (blue), temperature (orange),
velocity with respect to Sgr A* (purple; positive as away from Sgr A*) and
ejecta tracer (red) along the line connecting the explosion center (zero-
point) and Sgr A* (marked by a vertical dotted line) at $t=$ 600, 700 and 800
yr in the simulation run E5M1.3. Here the tracer parameter assumes a lower
bound of 0.001 for ease of visualization.

The bottom two panels of Figure 4 again show the synthetic X-ray surface
brightness distribution at the different epochs. Compared to Figure 3, the
color scale here is adjusted such that the central outflow zone is suppressed
to enhance visualization of the outer regions. A high surface brightness ridge
(in yellow and orange color, with ${\rm log}S_{2-8}\gtrsim-14.7$) is formed as
early as $t=400$ yr, which arises from the freshly shocked outflow. This ridge
gradually moves toward Sgr A* until $t\approx 700$ yr, as the nuclear outflow
is being pushed back by the SN forward shock. Between $t=600-800$ yr, both the
location and morphology of this ridge roughly match the observed X-ray ridge
(delineated by the red wedge), especially its lower (southeastern, between
position angles $85^{\circ}-175^{\circ}$) and brighter half (Figure 1).
However, the upper (northeastern, between position angles
$5^{\circ}-85^{\circ}$) half of the observed X-ray ridge, which is about half
as bright as the lower half, is not reproduced in the synthetic X-ray map.
Moreover, the simulation-predicted mean surface brightness of the ridge at
$t=700$ yr is only $\rm 2.8\times
10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$, which is substantially lower
than the observed value of $\rm 6.1\times
10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$ for the part of the ridge
between position angles $85^{\circ}-175^{\circ}$. We address possible causes
for this discrepancy in Section 5.1. From $t=900$ yr and beyond, the X-ray
surface brightness of the ridge rapidly fades away, as the result of the
weakening interaction between the ejecta and the nuclear outflow.

Figure 6: Synthetic 2–8 keV X-ray surface brightness distribution of the 1.6
pc$\times$1.6 pc region centered on Sgr A*, smoothed with a
0$\aas@@fstack{\prime\prime}$5 Gaussian kernel and for snapshots from $t=$ 600
yr to 900 yr at a step of 100 yr, in the simulation run E5M1.3. The top panels
show the total surface brightness, while the middle and bottom panels show the
contribution from the outflow-dominated and ejecta-dominated components, which
are defined as having $Q<0.001$ and $Q>0.5$, respectively.

Figure 6 further displays the synthetic X-ray surface brightness maps between
$t=$ 600 yr to 900 yr (top panel), this time smoothed with a Guassian kernel
of FWHM = 0$\aas@@fstack{\prime\prime}$5 to mimic the Chandra point-spread
function. Meanwhile, we decompose the X-ray surface brightness into two
components according to the tracer parameter $Q$: the outflow-dominated
component with $Q<0.001$ (middle panel) and the ejecta-dominated component
with $Q>0.5$ (bottom panel). Obviously, it is the shocked outflow that
dominates the bulk of the high-surface-brightness regions, in particular the
ridge. The shocked ejecta have a minor contribution to the surface brightness,
mainly as clumps and filaments seen in the southeastern portion (most obvious
at $t=600$ yr), never reaching a surface brightness above $5\times
10^{-16}\rm~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$. These features, likely
due to the Rayleigh-Taylor instability, gradually disappear after $t=$ 700 yr,
consistent with the above notion that the ejecta is pushed back by the
outflow. That the shocked ejecta have a minor X-ray contribution in the ridge
holds true even if a substantially higher (thus more realistic) metallicity
were adopted to calculate the ejecta’s X-ray emission (Appendix C).

###  4.2 The run with higher explosion energy

Figure 7: Similar to Figure 3, but for simulation run E10M1.3 at $t=$ 200,
500, 800 and 1100 yr after the SN explosion.

We now turn to simulation run E10M1.3, which has two times higher explosion
energy and the same ejecta mass compared to that assumed in the fiducial run,
E5M1.3. Similar to Figure 3, Figure 7 displays snapshots of density,
temperature and X-ray surface brightness at $t=$ 200, 500, 800 and 1100 yr in
run E10M1.3. In this case, the overall evolution of the SNR is similar to the
fiducial case. Due to the higher explosion energy, the forward shock acquires
a higher velocity, thus reaching the same physical extent at an earlier time
than in the fiducial run. In particular, except in the direction toward Sgr
A*, the forward shock reaches an extent comparable to the observed size of the
radio shell as early as $t=500$ yr. In the meantime, on its side approaching
Sgr A*, the forward shock is interacting with the nuclear outflow, forming
again a ridge of dense, high-temperature gas with high X-ray surface
brightness, which is illustrated by the closeup view in Figure 8, for epochs
from $t=400$ yr to 700 yr. However, the location of this ridge comes
significantly closer to Sgr A* than in run E5M1.3 (compare Figure 4), which
can be understood as due to the stronger momentum of the ejecta in run
E10M1.3.

At $t=$ 500 yr, when the overall morphology of the synthetic ridge agrees well
with the observed one, we estimate a mean surface brightness of $\rm 4.9\times
10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$. This is higher than E5M1.3,
which is mainly due to the fact that the nuclear outflow has a higher density
at a smaller distance from Sgr A* and that the X-ray emissivity is
proportional to density squared. However, this is still lower than the
observed value and rapidly drops after $t=$ 600 yr.

The middle and bottom panels of Figure 8 indicate that the overall X-ray
emission in this region is also dominated by the shocked outflow. The shocked
ejecta contribute little to the X-ray emission from the ridge, although they
reach a somewhat higher surface brightness in the southeastern portion of the
map, when compared to the case of E5M1.3 (Figure 6), likely due to the higher
ejecta velocity that leads to a higher post-shock temperature.

Figure 8: Similar to the X-ray maps in Figure 6, but for simulation run
E10M1.3 at snapshots from $t=$ 400 yr to 700 yr at a step of 100 yr. The maps
have been smoothed with a 0$\aas@@fstack{\prime\prime}$5 Gaussian kernel. The
top panels show the total surface brightness, while the middle and bottom
panels show the contribution from the outflow-dominated and ejecta-dominated
components, which are defined as having $Q<0.001$ and $Q>0.5$, respectively.

###  4.3 The run with higher ejecta mass

Figure 9: Similar to Figure 3, but for simulation run E5M10 at $t=$ 800, 1100,
1400 and 1700 yr after the SN explosion.

The simulation run E5M10 has the same explosion energy but a $\sim$8 times
higher ejecta mass compared to that assumed in the fiducial run. Figure 9
displays snapshots of density, temperature and X-ray surface brightness in run
E5M10 at epochs of 800, 1100, 1400 and 1700 yr, which show that the overall
evolution of the SNR is also similar to the fiducial case. However, the higher
ejecta mass, without an increase in the total energy, results in a
substantially lower velocity of the forward shock and the ejecta.
Consequently, the SNR expands much slower, and not until $t=$ 1400 yr that the
forward shock expands to a size roughly matching that of the radio shell. In
the meantime, the unshocked ejecta form a denser core compared to the fiducial
case, again manifesting themselves as a cavity of little X-ray emission.
Similar to the two previous runs, a ridge of high X-ray surface brightness is
formed due to interaction between the forward shock and the nuclear outflow,
which is illustrated by the closeup view in Figure 10 for epochs between $t=$
1300 yr and 1600 yr. The location of the ridge is closer to Sgr A* than in the
fiducial run (Figure 6), which can also be understood as due to a stronger
momentum of the ejecta. Notably, the northeastern half of the observed ridge
is again not reproduced.

At $t=$ 1400 yr, when the overall morphology of the synthetic ridge well
matches the observed one, we estimate a mean X-ray surface brightness of the
ridge as $\rm 6.0\times 10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$,
which is quite comparable to the observed value. That the mean X-ray surface
brightness is substantially higher than in the fiducial case can be explained
by a denser and slower ejecta (because of a higher $M_{\rm SN}$), which
results in a higher thermal pressure in the shocked ejecta holding up a denser
shocked outflow.

The middle and bottom panels of Figure 10 again compare the relative
contributions of the shocked outflow and the shocked ejecta. Notably, the
shocked ejecta has a significantly higher fractional contribution at the
apparent location of the ridge compared to the cases of E5M1.3 and E10M1.3.
More quantitatively, the ejecta contribute $\sim$ 14% of the X-ray emission
from the ridge region. However, by inspecting the 3D distribution of the
tracer parameter, we find that the ejecta emission mostly comes from behind
the ridge. In the meantime, the shocked ejecta also produces significant
emission at the southeastern corner, i.e., immediately outside the ridge. This
appears coincident with the observed excess emission at a similar position, as
evident in the right panel of Figure 1.

Figure 10: Similar to the X-ray maps in Figure 6, but for simulation run E5M10
at snapshots from $t=$ 1300 yr to 1600 yr at a step of 100 yr. The maps have
been smoothed with a 0$\aas@@fstack{\prime\prime}$5 Gaussian kernel. The top
panels show the total surface brightness, while the middle and bottom panels
show the contribution from the outflow-dominated and ejecta-dominated
components, which are defined as having $Q<0.001$ and $Q>0.5$, respectively.

##  5 Discussion and Conclusion

In the previous sections, we have presented 3D simulations of the early-stage
hydrodynamic evolution of a SNR in the central parsecs of the Galactic center,
incorporating necessarily simplified yet sufficiently realistic physical
conditions of both the SN explosion and the environment in which it evolves.
The three main simulation runs, representing the evolution of a SN Iax
(E5M1.3), a SN Ia (E10M1.3) or a CCSN (E5M10), have a varied degree of success
and failure in reproducing the main observed properties of Sgr A East (Section
2). In this section, we discuss the results and implications for our
understanding of the enigmatic Galactic center ecosystem.

###  5.1 Formation of the X-ray ridge and the progenitor supernova of Sgr A
East

All three simulations can reproduce a ridge-like structure, which is
unambiguously identified as the site of dynamic interaction between the
nuclear outflow and the expanding SNR, thus confirming the proposal of
Rockefeller et al. (2005). We further demonstrate that the X-ray emission from
the ridge is dominated by the shocked outflow and that the SN ejecta can
barely penetrate into the ridge. Among the three simulations, E5M10 produces a
ridge a bit too close to Sgr A*, whereas in E5M1.3 the ridge finds itself at a
location somewhat further away from Sgr A* compared with the observation
(Figures 6 and 10). This is more concisely illustrated in Figure 11, which
compares the X-ray radial surface brightness profiles of the three
simulations, constructed from consecutive wedges centered on Sgr A* and
covering position angles between $85^{\circ}-175^{\circ}$, for the epoch when
the ridge has the best observation-matching morphology. These synthetic
profiles are to be contrasted with the observed one, which has been extracted
from the deep Chandra image (Figure 1) in the same manner, with additional
care in background subtraction and masking of interloping point sources (Z.
Hua et al. 2023, submitted). Comparing the three profiles, a remarkable
agreement between the observed and synthetic X-ray surface brightness of the
ridge is clearly found in E5M10, except that the peak of its profile is found
at $\sim$0.4 pc from Sgr A*, which is $\sim 0.08$ pc closer than actually
observed. In contrast, the profiles of E5M1.3 and E10M1.3 are substantially
lower and flatter than observed.

Figure 11: Observed (black data points with error bars) and synthetic (colored
curves) 2–8 keV surface brightness profiles, constructed from consecutive
wedges centered on Sgr A* and covering position angles between
$85^{\circ}-175^{\circ}$, at a certain epoch indicated by the insert.

The above comparison between the synthetic and observed properties of the
X-ray ridge thus favors a CCSN as the progenitor supernova of Sgr A East. This
is in line with the suggestion or assumption of various previous work (e.g.,
Maeda et al., 2002; Sakano et al., 2004; Park et al., 2005; Zhao et al., 2013;
Lau et al., 2015), but in tension with the recent work of Zhou et al. (2021),
which, based on X-ray spectroscopic evidence of a low ratio of intermediate-
mass elements to Fe and large Mn/Fe and Ni/Fe ratios in the X-ray-emitting
ejecta of Sgr A East, disfavored a CCSN but favored a SN Iax origin.

From the simulation point of view, there are potential ways to alter the
synthetic X-ray emission from the ridge for a given type of SN explosion. The
X-ray surface brightness of the ridge is strongly affected by both $E_{\rm
SN}$ and $M_{\rm SN}$. Current models for SN Iax (Foley et al., 2013) leave
little room for changing $M_{\rm SN}$ (i.e., not significantly below the
Chandrasekhar limit). Meanwhile our assumption of $E_{\rm SN}=5~{}\times
10^{50}\rm~{}erg$ is probably already close to the upper end for these under-
luminous explosions. Properties of the nuclear outflow are equally important
in determining the X-ray surface brightness, as it is demonstrated in Section
4 that X-ray emission from the ridge is dominated by the shocked outflow. Our
three simulations share a default nuclear outflow, for which we have adopted
basic parameters consistent with the dedicated numerical simulations that
establish the hydrodynamic behavior of the nuclear outflow (Ressler et al.,
2018; Calderón et al., 2020). In particular, a total mass loss rate of
$\dot{M}_{\rm out}=1\times 10^{-3}\rm~{}M_{\odot}~{}yr^{-1}$ is assumed, which
is inferred from near-infrared spectroscopic observations of the $\sim$30
individual WR stars (Martins et al., 2007). A higher $\dot{M}_{\rm out}$, for
instance $3\times 10^{-3}\rm~{}M_{\odot}~{}yr^{-1}$ as suggested by older
observations (Gatley et al., 1986) and adopted by Rockefeller et al. (2005),
is expected to produce a substantially higher X-ray flux from the ridge.
However, a higher $\dot{M}_{\rm out}$ also means a stronger ram pressure of
the nuclear outflow, and consequently a much larger offset of the ridge from
Sgr A* (Yalinewich et al., 2017). We have performed a test simulation with
$\dot{M}_{\rm out}=3\times 10^{-3}\rm~{}M_{\odot}~{}yr^{-1}$, with other
settings identical to E5M1.3, to confirm this expectation. Other default
parameters, such as the chemical composition of the outflow and the ejecta, or
the assumed CIE emissivity, are not expected to significantly alter the
synthetic X-ray emission. Therefore, it appears difficult to reconcile the
$\sim$2 times lower X-ray surface brightness in E5M1.3, without sacrificing
the agreement found in the ridge location and the consensus on the WR star
wind properties by previous observational and theoretical work. The same above
arguments can in fact be applied to the E5M10 case, suggesting that the
putative CCSN has only a moderate explosion energy of $5\times
10^{50}{\rm~{}erg}$. Raising this to a more canonical energy of
$10^{51}{\rm~{}erg}$, for instance, would come at the price of a higher
nuclear outflow rate, in order to reconcile with the ridge location (see
Appendix B).

Nevertheless, it is worth emphasizing that the spectroscopic evidence for a SN
Iax origin as suggested by Zhou et al. (2021) is totally independent of the
hydrodynamical arguments presented here. Further, the remarkable match between
the E5M10 simulation and the observations of the morphology and X-ray flux of
the ridge should be considered at least a partial coincidence, in view of the
uncertainty in the assumed explosion energy, ejecta mass and outflow rate, as
well as the uncertainty in the line-of-sight geometry. Therefore, a final word
on the progenitor SN must await further studies. For instance, the ejecta of a
SN Iax may have a higher degree of spatial uniformity than that of a CCSN,
which can be examined by a spatially-resolved X-ray spectral analysis (Lopez
et al., 2011).

The future evolution of the ridge might provide an alternative diagnostic. As
shown in Section 4 and concisely illustrated in Figure 12, the mean surface
brightness of the ridge is predicted to decline shortly after the maximum
value is reached in all three simulations. From the synthetic light curves in
Figure 12, we estimate a fractional decay of 20% and 4% over the next 100 yr,
followed by rapid decay afterwarsds, for the case of E5M1.3 and E10M1.3,
respectively. In the case of E5M10 the mean surface brightness is predicted to
remain nearly constant for 100 yr, followed by a much slower decay afterwards.
Observationally, it is found that the X-ray surface brightness of the ridge
exhibited no significant variability over the past 20 years as seen by the
Chandra observations (Z. Hua et al. 2023, submitted), which further supports
the case of E5M10. Future monitoring observations of the ridge into the next
decades will provide a more definite answer. In a few hundred years, the
nuclear outflow is expected to completely resume its dynamical dominance in
the central parsec, whereas the shocked ejecta will be pushed away and, on a
longer timescale, cool and fragment until fully mixed with the circumnuclear
medium.

We shall remark on the rather puzzling aspect that all three simulations fail
to reproduce the northeastern half (position angles between
$5^{\circ}$–$85^{\circ}$) of the X-ray ridge (Section 4.1). We note that this
northeastern half is unlikely an interloping foreground/background feature, in
view of its morphology that smoothly joins with the northwestern half of the
ridge. The most obvious way to generate an azimuthally-wider ridge is to have
the SN ejecta coming from smaller position angles, but our test simulations
show that this requires that the explosion center be placed 1 pc north of the
currently assumed position (see more discussions in Appendix A). Consequently,
the position of the forward shock would look very different from that of the
radio shell (Figure 13). Moreover, due to the symmetry of the problem, a
shifted explosion center cannot automatically lead to a factor of $\sim$2
difference in the observed X-ray surface brightness between the northeastern
and southeastern halves of the ridge (Figure 1). We conclude that other
factors are responsible for the wider ridge. One such factor may be an
anisotropic nuclear outflow. For ease of implementation we have assumed an
isotropic, radial outflow, having azimuthally-averaged thermodynamic
properties derived from the dedicated 3D simulations of wind-wind interactions
within the central half-parsec region (Ressler et al., 2018). However, in
reality the WR stars have a non-uniform and anisotropic distribution (Figure
1), and most of them orbit about Sgr A* counterclockwise (Genzel et al., 2010)
in a period of $\sim 10^{4}$ years. These might have temporarily resulted in a
weaker outflow at position angles between $5^{\circ}$–$85^{\circ}$, allowing
the SN ejecta to reach a closer distance from Sgr A* than in the isotropic
case. A collimated outflow driven by Sgr A* may cause additional anisotropy
(e.g., Li et al., 2013). A similar argument might also be true for the SN
ejecta, such that it temporarily carries a stronger momentum and intrudes to a
closer distance along the desired orientation. Alternatively and/or
simultaneously, a significant angular momentum carried by the nuclear outflow,
which is inherited from the counterclockwise orbital motion of the WR stars,
might facilitate a laminar motion along the ridge, effectively increasing its
azimuthal extent as observed. Confirming these possibilities would require
dedicated simulations incorporating the 3D distribution and motion of the WR
stars, which we defer to future work.

We note in passing that G359.95-0.04, an X-ray-bright, comet-like, non-thermal
feature (Figure 1), which has long been considered a pulsar wind nebula (Wang
et al., 2006), may find its physical connection to the ridge. Recently, Zhao
et al. (2022) detected the long-sought compact radio counterpart of
G359.95-0.04, which exhibits a proper motion consistent with the latter’s
head-tail morphology. Interestingly, the relative position of G359.95-0.04 and
the ridge suggests that the putative pulsar is probably running into the
ridge. In this case, the ridge as a dense structure may provide the necessary
ram pressure to form the observed head-tail morphology of G359.95-0.04, which
is otherwise hard to explain, as the direction of the proper motion is such
that the putative pulsar travels along, rather than against, the nuclear
outflow. It would be interesting to study how the forward shock of a SN
propagating in the central parsec might “light up” hidden pulsars, which are
expected to be numerous in the Galactic center (Wharton et al., 2012).

Figure 12: The mean 2–8 keV surface brightness of the ridge as a function of
time. The light curves of E10M1.3 and E5M10 have been artificially shifted for
better alignment. A shadow between 700-800 yr is used to highlight the peak
and a subsequent mild decay.

###  5.2 The age of Sgr A East

All three runs can match the observed average size of Sgr A East, but at a
substantial different epoch after the SN explosion, due to the different
explosion energy and ejecta mass assumed. Remarkably, this is achieved
approximately at the same time as the formation of the X-ray ridge, in
particular in simulations E5M1.3 ($t=$ 700 yr) and E5M10 ($t=$ 1400 yr).
Notably from Figures 3, 7 and 9, the eastern half of the shell, which is
unaffected by the nuclear outflow, appears rounder than actually observed.
This can be understood as due to the intrinsic symmetry assumed in both the SN
injection and the ambient gas. In reality, SN ejecta are often found to
exhibit a certain degree of anisotropy. In addition, it is known that the
forward shock of Sgr A East is running into and interacting with the so-called
50 km s-1 Cloud to the southeast (Lee et al., 2008; Ferrière, 2012; Figure 1),
which is at least partially responsible for the flattened morphology. We note
also that we have not attempted to calculate a synthetic radio synchrotron
intensity for the shell, which requires knowledge about the local magnetic
field and energy density of relativistic electrons accelerated by the forward
shock. We defer to future work a more dedicated simulation to self-
consistently and quantitatively reproduce the radio shell. Nevertheless, we
consider that all three simulations are able to reproduce the radio shell of
Sgr A East. This alone constrains the age of Sgr A East between $\tau_{\rm
SN}\sim$ 500 yr to $\sim$ 1500 yr, depending on whether the progenitor SN was
a SN Ia or a CCSN. We note that the latter case is in agreement with
Rockefeller et al. (2005), who inferred an age of $\sim 1700$ yr, based on SPH
simulations of a CCSN interacting with the nuclear outflow, although their
very different initial conditions compared to ours necessarily led to very
different evolutionary paces. More recently, Ehlerová et al. (2022) also
inferred an age of 1–2 kyr for Sgr A East, in their simulations with a low
density medium, although the age can reach 10 kyr in their simulations with a
dense medium near the explosion center. However, they and we do not reproduce
the ”X-ray ridge” in the case with an age of 10 kyr, so the age derived from
our simulations is favoured on this point.

That $\tau_{\rm SN}\lesssim$ 1500 yr calls into serious question an often
assumed physical association between Sgr A East and the Cannonball (Park et
al., 2005; Zhao et al., 2013). Given the current position of the Cannonball
(Figure 1) and its measured transverse velocity of $500\pm
100\rm~{}km~{}s^{-1}$ (Zhao et al., 2013), a physical association with Sgr A
East requires a dynamical age of at least 6400 yr, which is highly unlikely
given the inferred $\tau_{\rm SN}$. We note that a similar conclusion was
first drawn by Yalinewich et al. (2017), who used hydrodynamic simulations to
study SNR evolution in the Galactic center under more general conditions. This
leaves the relic of the progenitor SN an open question, although the existence
of a relic neutron star is neither expected (in the case of SN Ia or Iax) nor
required (in the case of a CCSN giving birth to a black hole). Even assuming
that a relic neutron star acquired a transverse velocity similar to that of
the Cannonball, which is already high among observed pulsar kicks (Faucher-
Giguère & Kaspi, 2006), it would have travelled a projected distance of only
0.5($\tau_{\rm SN}$/1000 yr) pc, i.e., it should still be found deep inside
the SNR. This invites a target search, perhaps most beneficial among known
compact X-ray sources (Zhu et al., 2018) and radio sources in the region (Zhao
et al., 2020). On the other hand, our simulations predict that the forward
shock has a current expansion velocity of $\sim 2000\rm~{}km~{}s^{-1}$. Thus
in the near future the SNR shell is expected to catch up with and encompasses
the Cannonball, certainly in projection and plausibly in real space.
Interestingly enough in the latter case, the Cannonball would become the
Cuckoo Egg.

###  5.3 The unexplained X-ray-bright interior

There is yet another important observational aspect that all three simulations
fail to reproduce, namely, the X-ray-emitting interior of Sgr A East, based
primarily on which elemental abundances of the SN ejecta were measured and the
type of progenitor SN were debated (Maeda et al., 2002; Sakano et al., 2004;
Zhou et al., 2021).

As illustrated in Figures 3, 7 and 9, the central region of the SNR remains
cold (with temperatures $\lesssim 10^{4}$ K), due to adiabatic expansion, and
produces essentially no X-ray emission. In general, the expanding SN ejecta
can be heated up to X-ray-emitting temperatures ($\gtrsim 10^{6}$ K) by the
reverse shock. In previous work (Maeda et al., 2002; Sakano et al., 2004),
which took Sgr A East as an evolved SNR with an age of $\sim$5000–10000 yr,
the reverse shock is supposed to have propagated back to the explosion center,
heating the entire ejecta to X-ray-emitting temperatures. This situation was
also implicitly assumed by Zhou et al. (2021).

However, in our simulations, the age of Sgr A East is constrained to be
$\lesssim$1500 yr. In this case, the bulk of the ejecta are still unaffected
by the reverse shock and remain cold, as illustrated in Figures 3, 7 and 9. A
simple estimate following the model of Leahy & Williams (2017) suggests that
an ambient density of at least $\sim 1\rm~{}cm^{-3}$ around the explosion
center is required to have the reverse shock propagating back to the center in
2000 yr after the explosion. However, in our simulations, the initial gas
density in the vicinity of the explosion center is only $\rm\sim
0.1~{}cm^{-3}$, which is essentially determined by the uninterrupted nuclear
outflow (Figure 2). The initial ambient gas density remains lower than $\rm
1~{}cm^{-3}$ even if the explosion center moves closer to Sgr A* until about 1
pc away, which is unlikely given the current shell position of Sgr A East. In
fact, in Figure 11 we already observe a deficiency of our predicted surface
brightness outside the ridge (i.e., at a projected distance $\gtrsim$0.6 pc),
which is understood to be dominated by the shocked ejecta. This means that
even at the site where the back reaction to the ejecta is strongest, the
observed X-ray emission arising from the ejecta is largely unaccounted for.

A much higher ambient gas density than offered by the nuclear outflow can in
principle help to reconcile with the observed X-ray-bright, metal-enriched
ejecta of Sgr A East. One possibility is that the SN exploded in a dense
circumstellar wind-blown bubble, for which case a massive progenitor star
(hence a CCSN) is most likely responsible. Alternatively, the SN might have
exploded inside the dense cloudlets of molecular gas. Molecular gas is not
rarely seen in the Galactic center. However, the hot and fast nuclear outflow
is expected to blow up or evaporate small cloudlets within the central
parsecs. Thus the chance of a SN exploding inside some cloudlets should be
rather slim. Nevertheless, we show a test simulation in Appendix D to
illustrate the potential effect of dense cloudlets surrounding the SN
explosion center, but caution that it is practically difficult to constrain
the extra properties (e.g., density, size and number) of such cloudlets.
Moreover, the lopsided morphology of the X-ray interior, in the sense that the
part facing the direction of Sgr A* is substantially brighter (Figure 1),
suggests some casual relation with the nuclear outflow or Sgr A* itself,
whereas an initially high ambient density, whether due to the progenitor star
or a dense molecular cloudlet, is not expected to introduce such a
lopsidedness. In addition, the CND may partly impede and/or reflect the
forward shock (Zhao et al., 2016). A CND-like component was included in the
simulation of Rockefeller et al. (2005). We have similarly performed a test
simulation with the CND included, which is mimicked by many clumps of dense
cold gas (with $n\sim 10^{4}\rm~{}cm^{-3}$ and $T\sim 100$ K) distributed
along a circular orbit with a diameter of 3 pc and aligned with the Galactic
disk (Ferrière, 2012). We find that, consistent with Rockefeller et al.
(2005), the CND can only slightly affect the shock propagation, but has no
appreciable effect in either the formation of the ridge or the reverse shock-
heating of the ejecta.

The current location of the reverse shock is also closely related to the
detection of infrared-emitting dust within Sgr A East (Lau et al., 2015). It
was suggested that the dust formed in the ejecta of a CCSN and survived the
subsequent destruction caused by the passage of the reverse shock (Lau et al.,
2015). However, if Sgr A East were as young as our models imply, then the dust
found deep inside the SNR – if it indeed formed from the ejecta – would not
have encountered a reverse shock, according to our simulations. Therefore, one
cannot argue using Sgr A East that dust can survive a reverse shock, which
would call the conclusion of Lau et al. (2015) into question.

We speculate that the X-ray-bright ejecta are mainly attributed to
photoionization rather than the conventional collisional ionization by the
reverse shock. Signatures of a photoionized and recombining plasma in the
X-ray spectrum of Sgr A East were recently suggested by Ono et al. (2019).
Such an intriguing situation can occur in the unique Galactic center
environment, in the presence of a SMBH. While being currently extremely under-
luminous, Sgr A* is known to exhibit a bolometric luminosity orders of
magnitude higher just several hundred years ago (Ponti et al., 2013; Clavel et
al., 2013). This activity coincides with the early evolutionary stage of Sgr A
East, and to some extent would inevitably irradiate the expanding SN ejecta
with copious hard X-ray photons. If true, such a scenario can provide a
natural understanding to the lopsided morphology of the X-ray-bright ejecta,
despite the absent role of the reverse shock.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(grants 12225302, 11873028) and the National Key Research and Development
Program of China (grant 2017YFA0402703). We thank Dr. Ping Zhou and Fangzheng
Shi for helpful discussions. We acknowledge the cosmology simulation database
(CSD) in the National Basic Science Data Center (NBSDC) and its funds the
NBSDC-DB-10.

## Data Availability

The simulation data underlying this article may be shared upon reasonable
request to the corresponding author.

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##  Appendix A The effect of changing the explosion center

As discussed in Section 2 and Section 3.1, the exact position of the explosion
center of Sgr A East is subject to some uncertainty, both on the sky plane and
along the line-of-sight. To explore the effect of a different position of the
explosion center (with respect to Sgr A*), we have performed several test
observations. Specifically, we adopt the following coordinates for Sgr A*:
$(x,y,z)_{\rm BH}$ = (1.0, 1.8, 0.5), (1.0, 2.5, 1) and (0.5, 1.8, 1.5) pc.
These are offset from the fiducial site, $(x,y,z)_{\rm BH}$ = (1.0, 1.8, 1.5).
The remaining parameters of each run are identical to run E5M1.3.

As expected, the resultant morphology of the SNR and the X-ray ridge varies
significantly with the assumed explosion center. In particular, in the
simulation with $(x,y,z)_{\rm BH}=(1.0,1.8,0.5)$ pc, i.e., the explosion
center moves northward relative to Sgr A*, the predicted X-ray ridge can
better reproduce the observed one, in particular the northeastern half, which
is a notable discrepancy in the three main simulations (Section 5.1). This is
illustrated in Figure 13. However, this improvement in reproducing the X-ray
ridge comes at the price of a substantially offset SNR, in accordance with the
new explosion center, which cannot reproduce the observed location of the
radio shell (nor for the location of the X-ray-bright core), as again
illustrated in Figure 13. Therefore, it is unlikely that the true explosion
site of Sgr A East is close to this northern position.

In the cases where the explosion center moves along both the $y$\- and
$z$-axis, i.e., $(x,y,z)_{BH}$ = (1.0, 2.5, 1) pc, the resultant morphology is
clearly inconsistent with the observed, as illustrated in Figure 14. In
particular, the ridge appears much further away from Sgr A* in this case, due
to a larger offset of the explosion center.

Lastly, when the explosion center is placed further behind Sgr A*, i.e.,
$(x,y,z)_{BH}$ = (0.5, 1.8, 1.5) pc, the resultant ridge is still found close
to the observed position, but the ridge appears thinner and brighter than
observed (Figure 15), because in this case the interaction zone between the
nuclear outflow and the SN shock projects into a narrower region.

Figure 13: The 2–8 keV X-ray surface brightness distribution of a test
similuation similar to E5M1.3, but with Sgr A* placed at
$(x,y,z)_{BH}=(1.0,1.8,0.5)$ pc. The top panels show the snapshots in the
whole simulation box at 400 – 700 yr (from left to right) after the SN
explosion, while the bottom panels show the corresponding zoom-in images
surrounding Sgr A*, which is marked by a black ‘+’. The pink ellipse, red
ellipse and red wedge outline the observed position of the radio shell, X-ray-
bright interior and X-ray ridge, respectively, as in Figure 1.

.

Figure 14: Similar to Figure 13, but with Sgr A* placed at (x,y,z)BH = (1.0,
2.5, 1.0) pc.

.

Figure 15: Similar to Figure 13, but with Sgr A* placed at $(x,y,z)_{\rm
BH}=(0.5,1.8,1.5)$ pc and for epochs of 600 – 900 yr.

.

##  Appendix B The case of a core-collapse supernova with a higher explosion
energy

We have also performed the simulation E10M10, which assumes an explosion
energy of $10^{51}$ erg and an ejecta mass of 10 $\rm M_{\odot}$, with other
settings identical to run ${\it E5M10}$. However, in the present case the
X-ray ridge forms earlier, due to the higher explosion energy (thus also a
higher momentum). As a result, the age of the system upon forming the ridge is
significantly younger than in the case of E5M10. Figure 16 shows the X-ray
view of the ridge at several epochs in E10M10. In this case, the X-ray
emission from the ridge is still dominated by the nuclear outflow, with only a
small ($\sim$15%) contribution from the ejecta at $t=900$ yr (at this time the
ridge looks most similar to the observed one). A mean X-ray surface brightness
of $\rm 7.3\times 10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$ is found
for the ridge at $t=$ 900 yr, which is somewhat higher than the observed value
($\rm 6.1\times 10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$).

Figure 16: Similar to the X-ray maps in Figure 6, but for simulation run
E10M10 at snapshots from $t=$ 700 yr to 1000 yr at a step of 100 yr. The maps
have been smoothed with a 0$\aas@@fstack{\prime\prime}$5 Gaussian kernel.The
top panels show the total surface brightness, while the middle and bottom
panels show the contribution from the outflow-dominated and ejecta-dominated
components, which are defined as having $Q<0.001$ and $Q>0.5$, respectively.

##  Appendix C The effect of a higher metallicity in the SN ejecta

In the three main simulations we have assumed that the metallicity of the SN
ejecta is the same as that of the nuclear outflow. In reality, the SN ejecta
may have a substantially higher abundance for certain metals. We have
performed a test simulation in which a metallicity of six times solar is
assumed for the SN ejecta, a value based on the iron abundance obtained by
Zhou et al. (2021). The other settings are identical to run E5M1.3.

Comparing with E5M1.3 (Figure 4), it is found that in this simulation (Figure
17) the overall evolution of the SNR and the formation of the ridge are rather
insensitive to the metallicity. The mean X-ray surface brightness of the ridge
is $\rm 2.9\times 10^{-15}~{}erg~{}s^{-1}~{}cm^{-2}~{}arcsec^{-2}$ at $t=700$
yr, essentially identical to the case of E5M1.3. This is understandable,
because the ejecta has little contribution to the X-ray emission from the
ridge even with the currently adopted high metallicity.

Figure 17: Similar to Figure 4, but with a higher metallicity (6 Z⊙) for the
ejecta of Sgr A East.

.

##  Appendix D The effect of a clumpy environment around the supernova
explosion center

If there were only few small and low-density cloudlets around the explosion
center, it would not significantly influence the evolution of Sgr A East and
the formation of the X-ray ridge. However, if many large and dense cloudlets
produced an envelope surrounding the explosion center, the results would be
totally different. To illustrate this, we randomly add 10 cloudlets within the
spherical region with a radius of 1.5 pc centered on the explosion center, and
each cloudlet has a radius of 0.2 pc and a density of 10 cm-3. The result is
shown in Figure 18. It can be seen that the resultant morphology is
dramatically different from our standard simulations and also disagrees with
the observation. Moreover, it cannot reproduce the “X-ray” ridge. In this
regard, a clumpy environment around the explosion center is strongly
disfavored. We note that the recent simulations of Ehlerová et al. (2022) also
included the influence of clouds, but they only considered the large M50 cloud
and limited physical conditions.

Figure 18: Similar to Figure 3, but including some dense cloudlets surrounding
the explosion center.

.


# Music Generation Using Three-layered LSTM

1st Vaishali Ingale  Army Institute of Technology
Pune, India
<EMAIL_ADDRESS>2nd Anush Mohan  Army Institute of Technology
Pune, India
<EMAIL_ADDRESS>3rd Divit Adlakha  4th Krishan Kumar  Army
Institute of Technology
Pune, India
<EMAIL_ADDRESS>Army Institute of Technology
Pune, India
<EMAIL_ADDRESS>5th Mohit Gupta  Army Institute of
Technology
Pune, India
<EMAIL_ADDRESS>
###### Abstract

This paper explores the idea of utilising Long Short-Term Memory neural
networks (LSTMNN) for the generation of musical sequences in ABC notation. The
proposed approach takes ABC notations from the Nottingham dataset and encodes
it to be fed as input for the neural networks. The primary objective is to
input the neural networks with an arbitrary note, let the network process and
augment a sequence based on the note until a good piece of music is produced.
Multiple calibrations have been done to amend the parameters of the network
for optimal generation. The output is assessed on the basis of rhythm,
harmony, and grammar accuracy.

###### Index Terms:

Music, RNN, LSTM, ABC, Adam, Tensorflow, Keras

##  I Introduction

It is a myth that you need to be a musician to generate music. Even an music
enthusiast can produce a musical piece through the means of technology. Until
recently, all music generation was done manually by means of analogue signals.
However, now, music production is done through technology, assisted by humans.
The task that has been accomplished in the paper is the construction of neural
network architectures that can efficiently portray the complex details of
harmony and melody without the need for human intervention. A brief summary of
the precise details of music and its mechanisms has been provided in the paper
with appropriate citations where required. The primary objective of the papers
was to devise an algorithm that can be used to create musical notes utilizing
Long Short-Term Memory (LSTM) [9] networks in Recurrent Neural Networks (RNNs)
[2]. The output data obtained is in ABC notation.

To train the model we have chosen to work on ABC notations. ABC notation is
one of the ways to represent music. It consists of two parts. First part
represents the meta data which comprises various characteristics of tune such
as index, time signature, default note length and type of tune. And second
part represents the actual tune which is nothing but a sequence of characters.
The devised algorithm learns the sequences of monophonic musical notes over
three single layered LSTM network.

The following sections cover other work that has been conducted in the field,
the details of the approach taken and the subsequent results thus obtained.

##  II Literature Review

Music generation has been at the epicenter of attention of members of the
research community and has thus been studied upon a lot. Many who did try to
generate music have done so using different approaches. Hence there exists
numerous ways in which music can be generated and an amalgamation of such
approaches can be used to create and design a new yet competent model. These
approaches have been divided into two main categories – Traditional[4] and
Autonomous [3]. Traditional approach uses algorithms working on already
defined functions to make music, whereas Autonomous Model learn from the prior
iterations of the notations and then generates new ones. Algebra founded upon
the usage of the tree structure to enforce grammar constitutes one of the
earlier attempts [4]. Markov chains[5] can be used to design such a model [2].
Many models and approaches have been documented in the field of artificial
intelligence soon after the field experienced a massive boom. Such models
include probabilistic models which use variants of RNN, namely Char RNNs,
Anticipation RNNs [7] [2]. One method that is actively being used to generate
musical notes is the Generative adversarial networks (GANs) which contains two
neural networks – discriminator network and the generator network [8]. The
generator network and the discriminator network work in tandem to evaluate
authenticity of the generated data against the original dataset. Research
studies reveal that LSTM outclasses the GAN in terms of fixating on certain
sequences, that is, LSTM is superior when it comes to uncovering specific
patterns and then recycling them throughout the course of the output sequence.
The models based upon LSTM were able to get out of certain note loops and
shift into other notes[9]. When it came to GAN-based models, they were only
able to pickup on basic concepts, albeit better, and they exhibited shorter
training times [8].

##  III The Methodology

###  III-A Objectives and Technical Challenges

The initial challenge faced when dealing with music is that of its
representation. Signals, MIDI, notations, etc. are all possible
representations. Due to the higher efficacy of notations for the task at hand,
the model used in this paper is fed input and generates output in ABC
notations. The ABC notation uses 7 letters (A to G) with other symbols
representing features such as – flat, sharp, note length, key, etc. to
represent the given notes.

However, the subsequent issue is that if the output is in the form of
notations, it is not readily eligible by the majority of humans. To make the
output understandable by an amateur, the notations can be easily converted
into the preferred audio format. In our testings, we converted the notation-
based sequence into MIDI and MP3 formats.

###  III-B Design

RNNs fall victim to the vanishing/exploding gradient problem due to their use
of back-propagation[2], to remedy this we have employed the use of LSTMs. At
each timestep of the RNN, the individual LSTM[9] cell is fed a value, the cell
then calculates the hidden vector and outputs is to the next timesteps. The
current input, and the previous hidden and memory states are given as input to
the cell. Similarly, the current hidden state and the current memory state are
the outputs. Taking the current timestep as t, the current hidden vector ht is
found using the current input at and the previous timestep’s hidden vector ht
– 1. This is how RNNs process data sequentially.

Figure 1: The figure depicts the flow of data through the layers present in
the model.

###  III-C Data Processing

The model is designed to interpret the musical notes in the form of 87 unique
characters present in a dictionary, using integer encoding. The dataset is
hot-encoded as well, to convert the labels into binary vectors. This is then
fed into the LSTM units in batches. The specification of the batches used here
are as follows: Batch Size = 16, Sequence Length = 64.

###  III-D Architecture

The model is built around 3 LSTM [2] layers, acting as the core. Due to the
multi-class classification nature of the problem statement, the SoftMax
activation function is deployed as well. Dropout layer is also present to aid
in the avoidance of overfitting. The Adam optimizer is used since the model
deals with RNNs [2]. Also, to process the outputs at each timestep, Time
Distributed Dense Layer is utilized here.

Figure 2: The figure shows the architecture, that is the different layers,
their input and output sizes etc.

##  IV Results & Discussion

The model is implemented using 90 epochs which yields in less training loss
per epoch with increasing number of epochs which in turn result in training
accuracy of 94%. Values of some epoch’s loss and accuracy are shown in the
table [Fig3].

Tensorboard outputs (visualizations) represents these values in the form of
graphs. First graph [Fig4] describes the training accuracy values of the model
with corresponding epochs. The initial training accuracy level is observed to
be 59%, which then linearly increases 5th epoch onwards, that is from 75% to
94% linear increase in span of 85 epochs.

The second graph [Fig5] have variables epoch and its corresponding training
loss. Here, the gradual reduction in training loss is clearly visible with
higher number of epochs. With 1.4359 training loss in the 1st epoch to the
final 0.1737 loss at the last epoch, the reduction can clearly be seen in Fig.
3.

Figure 3: The table is the output of running the model for 90 epochs, yielding
upto 94% accuracy. Figure 4: The graph represents the Tensorboard output of
the training accuracy over time. X-Axis: Time Steps. Y-Axis: Accuracy Figure
5: The graph represents the Tensorboard output of the training loss over time.
X-Axis: Time Steps. Y-Axis: Loss

##  V Conclusions

LSTM[9] RNNs propose a promising approach for automated sequence generation.
These networks excel at predicting the next member of the sequence by making
decisions based on context. The saving of weights during back propagation
during each epoch has led to greater accuracy and lower losses. The
performance of this model can be more impressive if the dataset is altered to
include more tunes, in variety, and that of multiple instruments. The
training, if done more rigorously, can yield better results as well.

The monophonic music generated here is pleasant to the ear and has high
accuracy. Training the model with polyphonic data can help make the output
sequence more appreciable to the normal person.

The model is trained on monophony, that is, the dataset is comprised of a
single instrument. Further progressions can be made by delving into multiple
instruments, that is, polyphony.

The dataset can also be further expanded to comprise of more tunes, in
variety, so that the model can have a more robust training.

## References

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