Chilingarian et al. Dynamical masses of UDGs

17 2023 10.1017/xxxxx

Proceedings of IAU Symposium 379

# Dark matter content and dynamical masses of ultra-diffuse galaxies in the
Coma cluster

Igor V. Chilingarian 1,2 Kirill A. Grishin3,2 Anton V. Afanasiev4,2 Anton
Mironov5 Daniel Fabricant1 Sean Moran1 Nelson Caldwell1 Ivan Yu. Katkov6,2
Irina Ershova5 1Center for Astrophysics – Harvard and Smithsonian, 60 Garden
St. Cambridge, MA, 02138 USA
2Sternberg Astronomical Institute, Moscow State University, 13 Universitetsky
pr., Moscow, Russia
3Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France
4LESIA, Observatoire de Paris, 5 place Jules Janssen, 92195, Meudon, France
5Faculty of Space Research, Moscow State University, 1 Leninskie Gory, bld.
52, Moscow, Russia
6New York University Abu Dhabi, Saadiyat Island, PO Box 129188, Abu Dhabi, UAE

###### Abstract

Ultra-diffuse galaxies (UDGs) are spatially extended, low surface brightness
stellar systems with regular elliptical-like morphology found in large numbers
in galaxy clusters and groups. Studies of the internal dynamics and dark
matter content of UDGs have been hampered by their low surface brightnesses.
We identified a sample of low-mass early-type post-starburst galaxies, ‘future
UDGs’ in the Coma cluster still populated with young stars, which will
passively evolve into UDGs in the next 5–10 Gyr. We collected deep
observations for a sample of low-mass early-type galaxies in the Coma cluster
using MMT Binospec, which includes present-day and future UDGs. We derived
their dark matter content within a half-light radius (70–95 %) and total
dynamical masses ($M_{200}=5.5\cdot 10^{9}-1.4\cdot 10^{11}M_{\odot}$)
assuming the Burkert density profile and assess how different proposed
evolutionary channels affect dark and visible matter in UDGs. We also discuss
observational methodology of present and future UDG studies.

###### keywords:

ultradiffuse galaxies, dwarf galaxies, dynamics of galaxies, stellar
populations

††editors: P. Bonifacio, M.-R. Cioni, F. Hammer, M. Pawlowski, and S. Taibi,
eds.

##  1 Introduction

Ultra-diffuse galaxies (UDGs) are low-surface brightness
($\langle\mu_{r}\rangle>24$ mag arcsec-2) extended ($R_{e}$ up-to 10 kpc)
stellar systems. Initially discovered in the Virgo cluster in the 80s (Sandage
& Binggeli, 1984), they were nearly forgotten for almost two decades until
Caldwell (2006) resolved red giant branch stars in two of them. Another decade
later, UDGs attracted a lot of attention, when hudreds of them were discovered
in the Coma cluster (van Dokkum et al., 2015; Koda et al., 2015) and progress
in astronomical instrumentation enabled their spectroscopic studies in
integrated light. The main unanswered question is whether UDGs represent an
extension of dwarf elliptical (dE) galaxy population to larger sizes and lower
surface brightnesses (Conselice, 2018) and share common evolutionary paths, or
whether they form a separate galaxy class with different formation scenarios
(see e.g. Yozin & Bekki, 2015).

The proposed UDG formation scenarios can be confirmed or ruled out by detailed
assessment of their internal dynamics, dark matter content, and star formation
histories. Here we review our recent results on dynamical modelling of UDGs
and put them into the context of their formation and evolution.

##  2 UDG Observations and Data Reduction: Methodological Considerations

Dynamical modelling of collisionless stellar systems requires knowledge of the
stellar density distribution and internal kinematics (stellar velocities,
velocity dispersions, possibly higher order moments of the line-of-sight
velocity distribution (LOSVD), and proper motions of individual stars for
nearby galaxies and star clusters). Therefore, an input observational dataset
usually contains images and spectra, or, in rare cases, a fully calibrated
spectral data cube, which can be used for stellar density measurements, even
though direct images are typically more reliable in the areas of low surface
brightness.

###  2.1 Imaging and Light Profile Decomposition

Figure 1: Improvement in sky background subtraction for HyperSuprimeCam data
for the central region of the Coma cluster. We show contrast-enhanced images
obtained from the HyperSuprimeCam Legacy Archive (left) and using our own data
reduction (right). The background over-subtraction in the left panel reults in
dark rings around the two dominant cluster galaxies.

When observed from the ground, UDG central surface brightnesses fall below
that of the dark moonless sky, and one typically needs to trace light profiles
to $\mu_{r}>26$ mag arcsec-2 to confidently measure an effective radius and
Sérsic concentration. Therefore, the quality of sky subtraction is the key
quality factor for light profiles. Unfortunately, standard mosaic imaging data
processing pipelines use local sky background modelling, leading to background
oversubtraction near extended targets and to artefacts in galaxies close to
the edges of individual detectors in the array. For this project, we modified
the Subaru HyperSuprimeCam data reduction pipeline (Bosch et al., 2018) to
construct a global sky background model (see Fig. 1), which allowed us to
mitigate these problems and confidently reach the surface brightnesses
$\mu_{g}=28.5\dots 29.0$ mag arcsec-2.

Relatively nearby galaxies (to a distance of $\sim$20 Mpc) can be resolved
into stars using Hubble and the James Webb Space telescopes. Star counts allow
a more precise determination of the outer light profiles. Because resolved
images don’t reach the faintest stars we need to renormalize the star count
density profile using ground-based data (Afanasiev et al., 2023). Even though
crowding might become an issue in the inner parts of a galaxy we are able to
define a region where both unresolved and resolved profiles overlap for
renomalization. At the Virgo cluster distance ($\sim$16 Mpc) at a surface
brightness of $\mu_{r}=23.5\dots 26.0$ mag arcsec-2 confusion does not affect
the population of red giant stars.

###  2.2 Optical Spectroscopy

Obtaining integrated-light spectra of UDGs and using them to extract stellar
kinematics, ages, and metallcities remains challenging. However, even
relatively low signal-to-noise spectra (S/N$\sim$3 pix-1) can yield good
measurements of internal dynamics with an appropriate observing strategy
(Chilingarian et al., 2019).

The key factors are spectral resolution, choice of wavelength range, and slit
width for long-slit or multi-slit observations. Three main rules apply. (1)
The slit width should not exceed the seeing FWHM; otherwise the galaxy light
profile will define the spectral line shape making internal kinematics
measurements impossible. (2) The spectral resolution
($\sigma_{\mathrm{inst}}$) should match the expected velocity dispersion of
the galaxy ($R\approx 5000$ for UDGs). If the resolution is too low measuring
velocity dispersions and radial velocities will be difficult or impossible. If
the resolution is too high, the LOSVD will be oversampled, leading to no
improvement in $v_{*}$ and $\sigma_{*}$ uncertainties. (3) The chosen
wavelength range must contain many strong absorption lines. This selection
affects the quality of kinematics more than any other factor (Chilingarian &
Grishin, 2020). Additionally, to make stellar population measurements
possible, one should include age- (e.g. Balmer lines) and metallicity-
sensitive features in the spectral range, even though many more spectral lines
can be used to determine stellar ages (Chilingarian, 2009) than typically
understood. The best (optical) wavelength range for studies of internal
kinematics and stellar populations of faint quiescent galaxies is between
3900–5400 Å.

##  3 Observations, Data Reduction and Analysis

We observed UDGs and diffuse post-starburst galaxies (PSGs) or ‘future UDGs’
in the Coma$=$Abell 1656 ($z=0.023$) and Abell 2147 ($z=0.035$) galaxy
clusters using the high-throughput multi-object optical spectrograph Binospec
(Fabricant et al., 2019) at the f/5 focus of the 6.5 m MMT. The wavelength
range $3700<\lambda<5300$ Å at a spectral resolving power $R=4800$ was chosen
to target low-surface brightness absorption-line galaxies. We used deep
archival imaging of the Coma cluster from HyperSuprimeCam at the 8-m Subaru
telescope.

We reduced spectroscopic data using the standard Binospec data reduction
pipeline (Kansky et al., 2019). We reduced HSC images using a customized
version of the HSC data reduction pipeline were we added a block to construct
a global sky model.

We use the NBursts full spectum fitting code (Chilingarian et al., 2007) with
$R=10000$ simple stellar population models computed with the pegase.hr code
(Le Borgne et al., 2004) and a dedicated grid of synthetic spectra to model a
ram-pressure-induced starburst (Grishin et al., 2019) for diffuse PSGs (Fig.
2). We analyze images with galfit (Peng et al., 2010) using up to three light
profile components. We perform Jeans axisymmetric modelling including a
spherical dark matter halo in addition to the stellar component (Afanasiev et
al., 2023).

Figure 2: A sample of diffuse PSGs (8 ‘future UDGs’ and 2 ‘future dEs’ in the
Coma and Abell 2147 galaxy clusters, for which we performed full dynamical
modelling using internal kinematics out to $1.5-3R_{e}$. Color images are
assembled from archival Canada-France-Hawaiian Telescope images. For GMP 2640
we show a Subaru Suprimecam $R$-band image.

##  4 Dynamical Masses of UDGs

Full details of the Binospec data analysis for the ‘bona fide UDGs’ and for
diffuse PSGs are presented in Chilingarian et al. (2019) and Grishin et al.
(2021) respectively. Here we discuss the derived dark matter content within
$R_{e}$, how it is related to full (extrapolated) dynamical masses, and how
assumptions about the dark matter profiles affect this extrapolation.

The derived dark matter content within 1 $R_{e}$ (70–95 %) derived for UDGs
using sparse measurements with $\sigma_{*}$ profiles including between 1 and 5
data points and extending barely to 1 $R_{e}$, agree remarkably well with the
measurements of ‘future UDGs’ from well sampled kinematics extending to 1.5-3
$R_{e}$. There seems to be a trend of increased DM for smaller stellar surface
densities but this trend should be verified with a larger sample. The
agreement between the two approaches validates the low-S/N measurements and
proves that diffuse PSGs are indeed ‘future UDGs’ despite their current high
surface brightness arising from abundant young stars.

At the same time, the total virial masses $M_{200}$ derived under assumptions
of NFW (Navarro et al., 1997) and Burkert (1995) DM profiles derived from
stellar kinematics within the central 1–3 $R_{e}$ are different by up-to an
order of magnitude (Burkert profiles being lighter). This phenomenon is best
illustrated by the case of KDG 64, a small UDG or a large dwarf spheroidal
galaxy in the nearby M 81 group where despite very low surface brightness, the
profiles of internal kinematics are well sampled within 1 $R_{e}$ (Afanasiev
et al., 2023).

Figure 3: Left: Total stellar-to-halo mass ratio vs total extrapolated halo
mass for a sample of diffuse PSGs in Coma and Abell 2147. Right: Stellar-to-
dark matter mass ratio within $R_{e}$ vs total extrapolated halo mass. Red and
blue dots mark galaxies, which will passively evolve into UDGs and dEs
respectively.

Using the measurements presented in Grishin et al. (2021) we can compare total
stellar mass $M_{*}$ to a total dark matter halo mass $M_{200}$ computed for
the Burkert profile, and perform the same comparison for the quantities within
$R_{e}$. Interestingly, there seems to be a clear trend of the total
$M_{*}/M_{200}$ ratio as a function of $M_{200}$ while no correlation is seen
for the $M_{*}/M_{DM}$ within $R_{e}$ (see Fig. 3). The total mass ratio trend
goes in the opposite direction from expectation for the low-mass end of the
galaxy mass function in the ‘star formation efficiency’ diagram from Behroozi
et al. (2019). For our sample of ‘future UDGs’ the stellar-to-halo mass ratio
increases when the halo mass decreases. We also notice that our values of
$M_{*}/M_{\mathrm{halo}}$ are over an order of magnitude higher than those
from Behroozi et al. (2019). This discrepancy may be reduced if we use a NFW
halo density profile.

##  5 Discussion

Figure 4: An artist’s impression of an evolutionary path of a normal low-mass
disk galaxy into a UDG/dE in a cluster. (a) An unperturbed galaxy falls into
the cluster following an orbit with a substantial tangential component (shown
by a dashed line). (b) Interaction with the intergalactic medium induces a
burst of star formation and causes gas stripping, creating a ‘jellyfish’
galaxy. (c) Star formation and stripping remove all of the gas, quenching
further star formation and causing the disk expansion. (d) The galaxy evolves
into a UDG or dE inside the cluster for billions of years.

Which phenomena are responsible for the trend of $M_{*}/M_{200}$ vs $M_{200}$?
Does this result from assumptions made during dynamical modelling? Grishin et
al. (2021) demonstrated statistically that almost half the brightest UDGs in
the Coma cluster were likely formed by ram pressure stripping of gas by the
hot intracluster medium (Gunn & Gott, 1972) from low-mass disky progenitors.
Moreover, prior to stripping, a burst of star formation is induced by
compression of the gas inside an infalling galaxy, and up-to 50% of the total
stellar mass is formed during this event. Later on, gas from a galaxy gets
stripped entirely, and star formation quenches. In a gas-rich galaxy, a
substantial fraction of the mass can be stripped in the form of gas, so that
the gravitational potential in the galactic disk changes, and quasi-circular
orbits of stars become elliptical leading to the disk expansion. Later, the
disk slowly expands even further as the galaxy passively evolves, losing
stellar mass over time as the galaxy interacts with other members of the
cluster. This process is illustrated in Fig. 4.

The dark matter halo helps to stabilize the process of ram pressure stripping
preventing a galaxy from a complete dissipation as a result of substantial
mass loss. The distributions of both dark matter and baryons can change as a
result of mass loss, potentially affecting the dynamical modelling results
from our spectroscopic observations ($<2\dots 3R_{e}$), where we have assumed
a fixed halo density profile.

Another possibility is halo truncation from tidal interactions with other
galaxies in the cluster. In this case the extrapolation of the dynamical
properties derived from the ‘dense’ baryonic part of the galaxy becomes
invalid at $R_{200}$. However, we do not expect this process to be important
for galaxies infalling into the cluster for the first time, such as the
diffuse PSGs we observe.

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#  Shadow and deflection angle of asymptotic, magnetically-charged, non-
singular black hole

Yashmitha Kumaran<EMAIL_ADDRESS>Physics Department, Eastern
Mediterranean University, Famagusta, 99628 North Cyprus via Mersin 10, Turkey.
Ali Övgün<EMAIL_ADDRESS>https://aovgun.weebly.com Physics Department,
Eastern Mediterranean University, Famagusta, 99628 North Cyprus via Mersin 10,
Turkey.

(August 31, 2024)

###### Abstract

In this paper, we presents a detailed analysis of an asymptotic, magnetically-
charged, non-singular black hole. By utilizing the Gauss-bonnet theorem and
null geodesic methods, we aim to unravel the intricate astrophysics associated
with this unique black hole. The study explored various aspects including the
black hole’s gravitational field, intrinsic properties, light bending, and the
shadow of the black hole. Through rigorous calculations and simulations, we
derive the mass function, metric function, and deflection angle of the AMCNS
black hole. We study the geodesics and present raytracing around the black
hole. Additionally, we investigate the impact of the plasma medium on the
deflection angle and examined the distinctive features of the black hole’s
shadow. Our findings not only deepen our understanding of gravitational
lensing but also pave the way for future improvements in black hole theories
by minimizing restrictive assumptions and incorporating a more realistic
representation of these cosmic phenomena.

Deflection of light; Gravitational lensing; Massive photon; black hole; Gauss-
Bonnet theorem;

###### pacs:

04.40.-b, 95.30.Sf, 98.62.Sb

##  I INTRODUCTION

The evolution of black hole physics can be traced all the way back to the year
1784 when the English philosopher John Michell michell. He theorized that all
light emitted by an invisible central body having a radius more than five
hundred times that of the sun but with a density equal to that of the sun
would return to itself due to its own gravity. Essentially, when an object
falls from infinity into the central body, it acquires a velocity greater than
that of light at the surface of the central body owing to the latter’s force
of attraction.

Laplace followed this by proposing an invisible star that could potentially be
the largest and highly gravitating luminous object in the universe in the year
1796 de1799exposition. His speculations were connected to the observations on
an astronomical body with a density equal to that of the Earth but with a
diameter more than 250 times that of the sun, not allowing its light to reach
us, thanks to its strong attraction.

Einstein’s ground-breaking theory formulated in 1915 suggests that the
apparent gravitational force originates from the curvature of the fabric of
spacetime Einstein:1916vd. Schwarzschild gave the first solution for the
simplest black holes in the year 1916 depicting the gravitational field from a
spherical mass with no angular momentum, charge, or universal cosmological
constant Schwarzschild:1916uq. Between then and the year 1918, the combined
works of four European Physicists discovered the Reissner-Nordström metric
solved using the Einstein-Maxwell equations representing an electrically
charged black holes 1918KNAB...20.1238N.

Centuries of research have enabled humanity to understand the concept of a
star collapsing due to its own intense gravity into a black hole to an extent
that nothing can escape its gravitational pull beyond its horizon. Being a
consequence of General Relativity, black holes are enigmatic objects of the
universe that started off as a bunch of mathematical equations which were
written down only to take form in the sky hundred years later, only to be
confirmed with recent observations by Event Horizon Telescopes (EHT)
EventHorizonTelescope:2019dse; EventHorizonTelescope:2022xnr;
EventHorizonTelescope:2021dqv; Vagnozzi:2022moj.

Now that their existence is proven, the notion of singularity inside a black
hole becomes the next question of interest Penrose:1964wq. Quantum Field
Theory is sought for answers in this regard with the novelty idea of Hawking
radiation for a stationary black hole suggesting that the black holes shrink
until the point of singularity is reached at its center. This stems from a
semi-classical approximation that radiation has a negative flux of energy
contradicting the singularity assumptions Hawking:1975vcx.

When there is no existing singularity, a ”regular” black hole is a good
prospect to proceed with, as pioneered by Bardeen. These non-singular black
holes are deemed to possess regular centers with a core of collapsed charged
matter instead of singularity bardeen1968proceedings; Carleo:2022ukm. The
ensuing Einstein tensor is not only reasonable for a physical domain but also
satisfies the necessary conditions that eventually give rise to a static,
spherically symmetric, and asymptotically flat metric. A non-singular
spacetime is defined locally for a black hole forming initially from a vacuum
region and evaporating subsequently to a vacuum region, with its quiescence
described as a static region Hayward:2005gi.

The invisibility of a black hole has not stopped its distant observers from
seeing it, thanks to the existence of a shadow Luminet:1979nyg. The accretion
disk that is formed due to the immense gravity of a black hole attracting
everything in its path such as dust, photons, radiation, etc., constitutes the
shadow shape and size, hence, indirectly making it visible and observable
Falcke:1999pj; Bronzwaer:2021lzo. The shadow is imagined to be a circular ring
of light around its host; however, in reality, this is not the case. It is a
region that is geometrically thick and optically thin, especially for regular
black holes, that is not so circular. In relativistic models, the size and
shape of a shadow is found to be dependant on the geometry of the spacetime,
rather than the characteristics of the accretion Narayan:2019imo.

The extreme gravitational pull of the black hole compels the light rays to be
deflected towards the singularity, causing those that skim the photon sphere
to start looping around it. When a photon ends up precisely on the photon
sphere, it will continue to encircle the black hole forever. This phenomenon
occurring for the light rays passing in the neighborhood of the unstable
photon region abets the intensity of the original source through the extended
path length of the light rays, thus, increasing its brightness around the
shadow’s edge; consecutively, the cloud’s brightness just outside the shadow
appears to be enhanced as well. The captured light rays spiraling into a black
hole and the scattered light rays veering away from it are separated by a
shadow, giving it the facade of a bright surrounding illuminating a dark disk.
Therefore, it is also known as the critical curve that manifests as an
isotropic and homogenous emission ring. For a given black hole, its intrinsic
parameters play a major role in determining the size of the shadow, whereas,
the light rays and the instabilities of their orbits in the photosphere affect
its contour. Numerous researchers have explored the distinct imprints that
alternative gravitational theories leave behind by studying the phenomenon of
shadows.Symmergent-bh3; Symmergent-bh; Symmergent-bh2; Ghosh:2020spb;
Allahyari:2019jqz; Bambi:2019tjh; Vagnozzi:2022moj; Prashant2021;
Ovgun:2018tua; Ovgun:2020gjz; Ovgun:2019jdo; Kuang:2022xjp; Kumaran:2022soh;
Mustafa:2022xod; Okyay:2021nnh; Atamurotov:2022knb; Abdikamalov:2019ztb;
Abdujabbarov:2016efm; Atamurotov:2015nra; Papnoi:2014aaa; Abdujabbarov:2012bn;
Atamurotov:2013sca; Cunha:2018acu; Gralla:2019xty; Belhaj:2020okh;
Belhaj:2020rdb; Konoplya2019; Wei2019; Ling:2021vgk; Kumar:2020hgm;
Kumar2017EPJC; Cunha:2016wzk; Cunha:2016bpi; Cunha:2016bjh; Zakharov:2014lqa;
Tsukamoto:2017fxq; Chakhchi:2022fls; Li2020; EventHorizonTelescope:2021dqv;
Vagnozzi:2022moj; Pantig:2022ely; Pantig:2022gih; Lobos:2022jsz;
Uniyal:2022vdu; Ovgun:2023ego; Uniyal:2023inx; Panotopoulos:2021tkk;
Panotopoulos:2022bky; Khodadi:2022pqh; Khodadi:2021gbc; Meng:2023unt;
Pantig:2022whj; Pantig:2022sjb; Pantig:2023yer; Wang:2019skw; Roy:2020dyy;
Konoplya:2021ube; Anjum:2023axh; Hou:2018bar; Lambiase:2023fbd.

Apart from accretion, the other factor that enables the visibility of the
shadow is a phenomenon that occurs due to the deflection of light rays by the
gravity fields of a massive object in their path to a distant observer. Called
gravitational lensing, the light rays from the source in the background are
distorted owing to the massive object acting as a lens Virbhadra:1999nm;
Virbhadra:2002ju; Adler:2022qtb; Bozza:2001xd; Bozza:2002zj; Perlick:2003vg;
He:2020eah. When the lensing is weak, the magnifications are too minute to be
detected, yet, enable the observer to study the structural aspects in its
periphery. The effects of lensing were seen to influence the shadow
dramatically through the radiation from the accretion Falcke:1999pj;
Bronzwaer:2021lzo. In the realm of astrophysics, the determination of object
distances plays a pivotal role in unraveling their inherent characteristics.
Remarkably, Virbhadra demonstrated that solely by observing relativistic
images, devoid of any knowledge regarding masses and distances, it is feasible
to accurately establish an upper limit on the compactness of massive dark
entities Virbhadra:2022ybp. Furthermore, Virbhadra unveiled a distortion
parameter, leading to the cancellation of the algebraic sum of all singular
gravitational lensing images. This groundbreaking discovery has undergone
rigorous testing using Schwarzschild lensing within both weak and strong
gravitational fields Virbhadra:2022iiy. Weak gravitational lensing exhibits
the property of differential deflection and can be utilized to distinguish
between mass distributions. The extent of the light deflection and lensing is
denoted by the deflection angle or the bending angle. In order to calculate
the weak deflection angle, the Gauss-Bonnet theorem is used to derive the
optical geometry of the corresponding spacetime by Gibbons and Werner in 2008
Gibbons:2008rj. Subsequently, this methodology has been extensively employed
to investigate a wide array of phenomena Ovgun:2022opq; Kumaran:2021rgj;
Javed:2020pyz; Kumaran:2019qqp; Ovgun:2018fnk; Ovgun:2019wej; Ovgun:2018oxk;
Javed:2019ynm; Werner2012; Ishihara:2016vdc; Ono:2017pie; Li:2020dln;
Li:2020wvn; Belhaj:2022vte; Pantig:2022toh; Javed:2023iih; Javed:2023IJGMMP;
Javed:2022fsn; Javed:2022gtz. The specialty of this formula is the
relationship it establishes between the surface curvature and the underlying
topology for a given differential geometry as Gibbons:2008rj:

$\iint_{D_{R}}\mathcal{K}\mathrm{\leavevmode\nobreak\ d}S+\oint_{\partial
D_{R}}\hat{k}\mathrm{\leavevmode\nobreak\
d}\sigma+\sum_{j}\tilde{\theta}_{j}=2\pi\leavevmode\nobreak\
\mathcal{X}(D_{R}),$ (1)

where $\mathcal{K}$ is the Gaussian optical curvature, $\hat{k}$ is the
geodesic curvature, $\tilde{\theta}_{j}$ is an exterior angle at the vertex
$j$, and $\sigma$ is the line element along the boundary $D_{R}$. For a two-
dimensional manifold, a regular domain $D_{R}$ aligned by the two-dimensional
surface $S$ with the Riemannian metric $\hat{g}_{ij}$ is considered, along
with its piece-wise smooth boundary $\partial D_{R}=\gamma_{g}\cup C_{R}$.
Also, for a regular domain, it is known that the Euler characteristic is
unity, $\mathcal{X}_{D_{R}}=1$.

### The Gibbons and Werner method

If $\gamma$ is taken to be a smooth curve in the said domain, then
$\dot{\gamma}$ becomes the unit-speed vector. while $\hat{k}$ is termed as a
geodesic curvature and is given by Gibbons:2008rj; Kumaran:2021rgj:

$\hat{k}=g^{opt}(\nabla_{\dot{\gamma}}\dot{\gamma},\ddot{\gamma}),$ (2)

having the unit-speed condition $g^{opt}(\dot{\gamma},\dot{\gamma})=1$. Here,
$\ddot{\gamma}$ is the unit-acceleration vector that is perpendicular to
$\dot{\gamma}$. When $R\rightarrow\infty$, the relevant jump angles are
considered to be $\pi/2$; that is to say, the angles respective to the source
and the observer sum up as
$\tilde{\theta}_{S}+\tilde{\theta}_{O}\rightarrow\pi$. Putting all this
together, Eq. (1) now becomes:

$\iint\limits_{\mathcal{D}_{R}}\mathcal{K}\,\mathrm{\leavevmode\nobreak\
d}S+\oint\limits_{C_{R}}\kappa\,\mathrm{\leavevmode\nobreak\
d}t\overset{{R\rightarrow\infty}}{=}\iint\limits_{\mathcal{D}_{\infty}}\mathcal{K}\,\mathrm{\leavevmode\nobreak\
d}S+\int\limits_{0}^{\pi+\hat{\alpha}}\mathrm{\leavevmode\nobreak\
d}\phi=\pi.$ (3)

where, $\phi$ is the angular coordinate centered at the lens and
$\hat{\alpha}$ is the small, positive, non-trivial, asymptotic angle of
deflection. Knowing that $\gamma_{\tilde{g}}$ is a geodesic and the geodesic
curvature offers no contribution i.e., $\hat{k}(\gamma_{\tilde{g}})=0$, the
curve $C_{R}$ is pursued which contributes in such a manner that:

$\kappa(C_{R})=\mid\nabla_{\dot{C}_{R}}\dot{C}_{R}\mid.$ (4)

Considering $C_{R}:=r(\phi)=R=\text{constant}$, $R$ is the distance from the
origin of the selected coordinate system. The radial component of the geodesic
curvature is stated as:

$(\nabla_{\dot{C}_{R}}\dot{C}_{R})^{r}=\dot{C}_{R}^{\phi}(\partial_{\phi}\dot{C}_{R}^{\phi})+\Gamma^{\,r}_{\phi\phi}(\dot{C}_{R}^{\phi})^{2}.$
(5)

Evidently, the first term vanishes from the presumptions and the second term
can be obtained using the unit-speed condition. Then:

$\lim_{R\to\infty}\kappa(C_{R})=\lim_{R\to\infty}\left|\nabla_{\dot{C}_{R}}\dot{C}_{R}\right|\to\frac{1}{R}\quad\text{and
at higher limits of radial
distance,}\quad\lim_{R\to\infty}\mathrm{\leavevmode\nobreak\ d}t\to
R\mathrm{\leavevmode\nobreak\ d}\phi$

When gravitational lensing is approached geometrically, this proves effective
in finding the deflection caused by any curved surface. In the weak lensing
regime, it becomes possible to relate the geometry and the topology with the
Gauss-Bonnet theorem to determine the bending angle through the optical metric
because:

  * •

the metric evolves from the surface curvature relating it to its geometry

  * •

the geodesics of the metric are spatial light rays regarding the focused light
rays as a topological effect.

Gibbons and Werner obtained the bending angle as a consequence of weak lensing
using the Gauss-Bonnet theorem with the Gaussian curvature of the optical
metric directed outwards from the light ray Gibbons:2008rj. For a simply
connected and an asymptotically flat domain $D_{R}$ in which the lens is not
an element of the regime but the source and the observer are, with the
differential domain consisting of a bounding geodesic at the source, the weak
deflection angle is evaluated as Gibbons:2008rj; Kumaran:2021rgj:

$\hat{\alpha}=-\iint_{D_{R}}\mathcal{K}\mathrm{\leavevmode\nobreak\ d}S$ (6)

These are the foundations of this paper with the target of finding the weak
deflection angle for an asymptotic magnetically charged non-singular (AMCNS)
black hole. The paper is organized as follows: § IIthe solution of the AMCNS
black hole is analyzed, followed by its application in § III using the Gauss-
Bonnet theorem and the Gibbons and Werner method. In § IV, Geodesics are
discussed using the raytracing method. § V deals with the influence of plasma
on the bending angle. The greybody factor is investigated in § VI. We then
carry on to explore the shadow of the AMCNS black hole in § VII before
concluding in § VIII.

##  II ASYMPTOTIC, MAGNETICALLY-CHARGED, NON-SINGULAR BLACK HOLE SOLUTION

Extensive research has been conducted on the solutions that depict black holes
within the framework of nonlinear electrodynamics in general relativity
Breton:2003tk; Hendi:2013dwa; Kruglov:2015yua; Kruglov:2016ymq;
Kruglov:2016ezw; Kruglov:2017fck; Ovgun:2021ttv. To describe the solution for
a magnetically charged black hole, consider the exponential nonlinear
electrodynamics Kruglov:2017fck where the Lagrangian density is written as:

$\pounds=-P\exp(-\beta P)$ (7)

where,
$P=\nicefrac{{1}}{{4}}\left(F_{\mu\nu}\,F^{\mu\nu}\right)=\nicefrac{{1}}{{2}}\left(\mathbf{B}^{2}-\mathbf{E}^{2}\right),\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.$ $F^{\mu\nu}$ is the
electromagnetic field tensor, $\mathbf{B}$ is the magnetic field, $\mathbf{E}$
is the electric field, $A^{\mu}$ is the four-potential, and $\beta$ is a
parameter with the dimensions of [Length]4 having an upper bound of
$\left(\beta\leq 1\times 10^{-23}\text{T}^{-2}\right)$ The Euler-Lagrange
equation is formulated as:

$\partial_{\mu}\left(\frac{\partial\pounds}{\partial\left(\partial_{\mu}A_{\nu}\right)}\right)-\frac{\partial\pounds}{\partial
A_{\nu}}=0$

where $\mu,\leavevmode\nobreak\ \nu\in 0,1,2,3$. Thus, the field equations
come to be:

$\partial_{\mu}\left[\left(\beta P-1\right)\exp\left(-\beta
P\right)F^{\mu\nu}\right]=0$ (8)

The energy-momentum tensor takes the form of:

$\tau^{\mu\upsilon}=H^{\mu\lambda}F_{\lambda}^{\upsilon}-g^{\mu\upsilon}\pounds$
(9)

where, $g^{\mu\upsilon}={1}/{g_{\mu\upsilon}}$ and:

$H^{\mu\lambda}\equiv\frac{\partial\pounds}{\partial
F_{\mu\lambda}}=-\left(1-\beta P\right)\exp\left(-\beta
P\right)F^{\mu\lambda}$ (10)

Thus, the energy-momentum tensor derived from the Lagrangian density is:

$\tau^{\mu\upsilon}=\exp\left(-\beta P\right)\left[\left(\beta
P-1\right)F^{\mu\lambda}F_{\lambda}^{\upsilon}+g^{\mu\upsilon}P\right]$ (11)

whose trace is:

$\tau=4\beta P^{2}\exp\left(-\beta P\right)$ (12)

The instance of $\beta\rightarrow 0$ not only signifies weak limits, but also
reverts to classical electrodynamics as $\pounds\rightarrow-P$ and $\tau=0$ in
Eq. (12).

Typically, when $\beta$ is not zero and the energy-momentum tensor has a non-
zero trace, it entails that the scale invariance is violated. That’s why, if
any forms of nonlinear electrodynamics incorporate a dimensional parameter,
they would also break the scale invariance, prompting the dilation current to
be non-trivial: $\partial_{\upsilon}\,x^{\mu}\,\tau_{\mu}^{\upsilon}=\tau$

This can be overruled by the general principles of causality together with
unitarity according to which the group velocity of the background
perturbations are less than $c$. The sustaining of the causality principle
necessitates that $\pounds_{P}=\partial\pounds/\partial P\leq
0\Rightarrow\beta P\leq 1$. For a purely magnetic field,
$B\leq\sqrt{\nicefrac{{2}}{{\beta}}}$ ought to be fulfilled. Besides, the
constraint $\pounds_{P}+2P\pounds_{PP}\leq 0$ allows the unitarity principle
to hold provided $\pounds_{PP}\geq 0$. While Eq. (7) construes $\beta P\leq
0.219$ agreeing with , it is clear that the limitations for both the causality
and unitarity principles incur for $\beta P\leq 0.219$ giving:

$\mathbf{B}\leq\sqrt{\frac{5-\sqrt{17}}{2\beta}}\simeq\frac{0.66}{\sqrt{\beta}}$

Deriving the metric of a static AMCNS black hole in a spherically symmetric
spacetime is our next goal. For a pure magnetic field, the governing equations
are as follows:
The invariant $P$ is deliberated in terms of an electric charge $q$ to be:

$P=\frac{q^{2}}{2r^{4}}$ (13)

The generic line element of a spherically symmetric black hole with a time
coordinate $t$, radial coordinate $r$, co-latitude $\theta$ and longitude
$\phi$ both outlined for a point on the two-sphere, and mass $M$ is given by:

$\mathrm{\leavevmode\nobreak\ d}s^{2}=-f(r)\mathrm{\leavevmode\nobreak\
d}t^{2}+f(r)^{-1}\mathrm{\leavevmode\nobreak\
d}r^{2}+r^{2}(\mathrm{\leavevmode\nobreak\
d}\theta^{2}+\sin^{2}\theta\mathrm{\leavevmode\nobreak\ d}\phi^{2})$ (14)

Computing the function $f(r)$ specific to the AMCNS black hole is necessary to
go on. To pursue this, putting:

$f=1-\frac{2mr^{2}}{r^{3}+2ml^{2}}$ (15)

engenders the Hayward metric Hayward:2005gi of a non-singular black hole that
is static, has no charge, and emerges as the solution of a certain modified
gravity theory. The quantity $l$ is an approximate parameter in the length-
scale under which the corollaries of the cosmological constant prevail.
If $M$ is assumed to vary with $r$, then:

$M(r)=\int_{0}^{r}\rho(r)r^{2}\mathrm{\leavevmode\nobreak\
d}r=m-\int_{r}^{\infty}\rho(r)r^{2}\mathrm{\leavevmode\nobreak\ d}r$ (16)

where, $m=\int_{0}^{\infty}\rho(r)r^{2}\mathrm{\leavevmode\nobreak\ d}r$
describes the magnetic mass of the black hole that is responsible for
screening the magnetic interactions so that the perturbative divergences of
magnetostatics can be eliminated.
The energy density, in the absence of an electric field, ensues from Eq. (11)
to be:

$\rho=\frac{q^{2}}{2r^{4}}\exp\left(\frac{-\beta q^{2}}{2r^{4}}\right)$ (17)

This transforms the mass function to:

$M(r)=\frac{q^{2}}{2}\int_{0}^{r}\exp\left(\frac{-\beta
q^{2}}{2r^{4}}\right)\frac{\mathrm{d}r}{r^{2}}=\frac{{q^{3/2}}\,\Gamma\left(\frac{1}{4},\frac{\beta
q^{2}}{2r^{4}}\right)}{2^{11/4}\,\beta^{1/4}}$ (18)

where, the incomplete gamma function is characterized by:

$\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\mathrm{\leavevmode\nobreak\ d}t$
(19)

Therefore, rewriting the magnetic mass as:

$m=M(\infty)=\frac{q^{\frac{3}{2}}\Gamma\left(\frac{1}{4}\right)}{2^{\frac{11}{4}}\beta^{\frac{1}{4}}}\simeq\frac{0.54q^{\frac{3}{2}}}{\beta^{\frac{1}{4}}}$
(20)

Consolidating all of the above, the metric function is discovered to be
Ali:2019bcn:

$f\left(r\right)=1-\frac{r^{2}q^{\frac{3}{2}}\Gamma\left(\frac{1}{4},\frac{\beta
q^{2}}{2r^{4}}\right)2^{\frac{-7}{4}}\beta^{\frac{-1}{4}}}{r^{3}+l^{2}q^{\frac{3}{2}}2^{\frac{-7}{4}}\beta^{\frac{-1}{4}}\Gamma\left(\frac{1}{4},\frac{\beta
q^{2}}{2r^{4}}\right)}$ (21)

The metric function in the circumstance of $l=0$ reduces to Reissner-Nordström
solution Kruglov:2017fck. In the vicinity of radial infinity, its asymptotic
value is attained using the expansion of the series:

$\Gamma(s,z)=\Gamma(s)-z^{s}\left[\frac{1}{s}-\frac{z}{s+1}+\frac{z^{2}}{2\left(s+2\right)}+O\left(z^{3}\right)\right],\text{
\ }z\rightarrow 0$

Ultimately, the metric function $f(r)$ at $r\rightarrow\infty$ is structured
as Ali:2019bcn:

$f\left(r\right)=1-\frac{r^{2}\left[2m-\frac{q^{2}}{r}+\frac{\beta
q^{4}}{20r^{5}}+O(r^{-9})\right]}{r^{3}+l^{2}\left[2m-\frac{q^{2}}{r}+\frac{\beta
q^{4}}{20r^{5}}+O(r^{-9})\right]}$ (22)

This will be used for the subsequent computations of the AMCNS black hole. The
event horizon radius $r_{h}$ of a black hole is the larger root of the above
equation - where the outer horizon is located - with $f(r)=0$. As shown in
Fig. (1), the number of horizons are dependant on the parameters of $l$.

Figure 1: Left figure shows the lapse function $f(r)$ as a function of $r$ for
$m=1$, $q=0.5$, $\beta=0.5$ and for the different values of $l$. Right figure
shows the lapse function $f(r)$ as a function of $r$ for $m=1$, $q=0.5$,
$l=0.5$ and for the different values of $\beta$.

To calculate the Hawking temperature of the black hole, we obtain the
4-velocity as

$u=u^{t}\partial_{t},$ (23)

by the satisfying normalization condition

$1=u^{\mu}u_{\mu},$ (24)

where $u^{t}={1}/\sqrt{g_{tt}}$. Then the particle acceleration $a_{p}^{\mu}$
is given by Wald:1984rg

$a^{\mu}=-g^{\mu v}\partial_{v}\ln u^{t},$ (25)

as the metric components exist as functions of $r$ and $\theta$. The surface
gravity $(\kappa)$ is defined as

$\kappa=\lim_{r\rightarrow r_{h}}\frac{\sqrt{a^{\mu}a_{\mu}}}{u^{t}},$ (26)

which gives the Hawking temperature of the black hole

$\kappa_{sg}=\left.\frac{1}{2}\partial_{r}f^{2}\right|_{r_{h}},\quad
T=\frac{\kappa}{2\pi},$ (27)

as shown in Fig.2 for different values of $\beta$. The horizon area is plainly
calculated as

$A=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\sqrt{-g}d\theta=4\pi r_{h}^{2},$ (28)

giving the black hole entropy to be

$S_{BH}=\frac{A}{4}=\pi r_{h}^{2}.$ (29)

Figure 2: Hawking temperature $T$ versus $r$ for $m=1$, $q=0.5$, $\beta=05$
and for the various values of $\l$.

##  III WEAK DEFLECTION ANGLE USING THE GAUSS-BONNET THEOREM

The deflection angle of a photon approaching the said black hole with the
distance of closest approach $r_{0}$ corresponds to Virbhadra:1998dy;
Weinberg:1972kfs; Lu:2019ush:

$\hat{\alpha}(r_{0})=-\pi+2\int_{r_{0}}^{\infty}\mathrm{\leavevmode\nobreak\
d}r\frac{\sqrt{f(r)^{-1}}}{r\sqrt{\frac{r^{2}}{r_{0}^{2}}\frac{f(r_{0})}{f(r)}-1}}$
(30)

The condition $M/r_{0}\ll 1$ is satisfied for weak lensing since the
deflection angle will be too small keeton. As $r_{0}$ verges on the photon
sphere, $\hat{\alpha}$ increases until it diverges, hence, giving rise to
strong lensing. Considering the equatorial plane in which $\theta=\pi/2$
equation Eq. (14) reduces to the optical metric for null geodesics:

$\mathrm{\leavevmode\nobreak\ d}t^{2}=\frac{\mathrm{\leavevmode\nobreak\
d}r^{2}}{f(r)^{2}}+\frac{r^{2}}{f(r)}\mathrm{\leavevmode\nobreak\ d}\phi^{2}$
(31)

The Gaussian curvature is computed from the non-zero Christoffel symbols due
to its proportionality to the Ricci scalar as $\mathcal{K}={R}/{2}$ and is
found to be:

$\begin{split}\mathcal{K}=-\frac{1}{4}&\left\\{\left(1-\frac{r^{2}\zeta}{r^{3}+l^{2}\zeta}\right)\left[\frac{4\zeta}{r^{3}+l^{2}\zeta}+\frac{8\,r\left({\frac{q^{2}}{r^{2}}}-{\frac{\beta\,{q}^{4}}{4r^{6}}}\right)}{r^{3}+l^{2}\zeta}-\frac{8\,r\zeta\left\\{3\,r^{2}+l^{2}\left(\frac{q^{2}}{{r}^{2}}-\frac{\beta\,q^{4}}{4r^{6}}\right)\right\\}}{\left({r}^{3}+{l}^{2}\zeta\right)^{2}}+\frac{2\,r^{2}\left(\frac{-2q^{2}}{r^{3}}+\frac{3\beta\,q^{4}}{2r^{7}}\right)}{r^{3}+l^{2}\zeta}\right.\right.\\\
&-\frac{\left(4q^{2}-\frac{\beta\,q^{4}}{r^{4}}\right)\left\\{3\,{r}^{2}+{l}^{2}\left(\frac{q^{2}}{r^{2}}-\frac{\beta\,q^{4}}{4r^{6}}\right)\right\\}}{\left(r^{3}+l^{2}\zeta\right)^{2}}+\frac{4r^{2}\zeta\left\\{3\,r^{2}+l^{2}\left(\frac{q^{2}}{r^{2}}-\frac{\beta\,q^{4}}{4r^{6}}\right)\right\\}^{2}}{\left({r}^{3}+{l}^{2}\zeta\right)^{3}}-\frac{2\,r^{2}\zeta\left\\{6\,r+{l}^{2}\left(\frac{-2q^{2}}{r^{3}}+\frac{3\beta\,q^{4}}{2r^{7}}\right)\right\\}}{\left({r}^{3}+{l}^{2}\zeta\right)^{2}}\\\
&\left.\left.+\frac{1}{r^{3}+l^{2}\zeta-r^{2}\zeta}\left(-\frac{2\,r\zeta}{r^{3}+l^{2}\zeta}-\frac{q^{2}-\frac{\beta\,q^{4}}{4\,r^{4}}}{r^{3}+l^{2}\zeta}+\frac{r^{2}\zeta\left\\{3\,{r}^{2}+{l}^{2}\left(\frac{{q}^{2}}{{r}^{2}}-\frac{\beta\,{q}^{4}}{4\,{r}^{6}}\right)\right\\}}{\left({r}^{3}+{l}^{2}\zeta\right)^{3}}\right)\right]\right\\}-\frac{2m}{r^{3}}\end{split}$
(32)

For the sake of simplicity, the parameter $\zeta\equiv
2m-\nicefrac{{q^{2}}}{{r}}+\nicefrac{{\beta q^{4}}}{{20r^{5}}}$ has been
defined. Employing the straight-line approximation as $r=b/\sin\phi$, where
$b$ is the impact parameter, the Gauss-Bonnet theorem suggests that
Gibbons:2008rj; Kumaran:2021rgj:

$\hat{\alpha}=-\int_{0}^{\pi}\int_{{b}/{\sin\phi}}^{\infty}\mathcal{K}\mathrm{\leavevmode\nobreak\
d}S$ (33)

where $\mathrm{\leavevmode\nobreak\ d}S=r\mathrm{\leavevmode\nobreak\
d}r\mathrm{\leavevmode\nobreak\ d}\phi$. Owing to the complexity of this
calculation, the Gaussian curvature was reduced to
$\mathcal{O}\left(m^{2}\right)$ and the integral was simplified by ignoring
the higher-order terms $\mathcal{O}\left(q^{5}\right)$. Thus, the deflection
angle due to weak lensing for an asymptotic, magnetically charged, non-
singular black hole is estimated to be:

$\begin{split}\hat{\alpha}=\frac{{4m}}{{b}}+\frac{{3\pi
q^{2}}}{{4b^{2}}}-\frac{{3\pi
m^{2}}}{{4b^{2}}}-\frac{{8mq^{2}}}{{3b^{3}}}+\frac{{8m^{3}}}{{3b^{3}}}+\frac{{15\pi
q^{4}}}{{64b^{4}}}-\frac{{75\pi m^{2}}}{{8b^{4}}}-\frac{{45\pi
m^{2}q^{2}}}{{32b^{4}}}+\frac{{32mq^{2}}}{{b^{5}}}+\frac{{12mq^{4}}}{{5b^{5}}}\\\
-\frac{{64m^{3}}}{{b^{5}}}+\frac{{16m^{3}q^{2}}}{{5b^{5}}}-\frac{{64l^{2}m^{3}}}{{25b^{5}}}-\frac{{175\pi
q^{4}}}{{64b^{6}}}+\frac{{7\pi\beta q^{4}}}{{128b^{6}}}+\frac{{125\pi
m^{2}q^{2}}}{{8b^{6}}}+\frac{{525\pi m^{2}q^{4}}}{{256b^{6}}}\end{split}$ (34)

for the weak-field limit. The range of values for $q$ is chosen according to
the acquirable potentialities of a charged particle in the black hole locale.
From this result, the deflection angle is expected to change significantly
because of the parameters that govern the black hole, reducing it to the case
of a Schwarzschild black hole when $\beta=0$, $q=0$, and $l=0$. This relation
is depicted in Fig. 3. The curve is seen to fall from positive infinity to
take a dip toward negative infinity before climbing back up to a peak and
declining like in the Schwarzschild case. More the value of $q$, the lesser
the dip (with the trough stepping up into the positive vertical axis when
$q^{2}$ is equal to its root retrieved from the quadratic equation solved for
$\hat{\alpha}=0$), holding a local minimum for $1.5<q<2$. The deviating curve
shows that the deflection angle is less than the Schwarzschild case when there
is an electric charge, and it keeps decreasing as $q$ increases. For $q<1$,
there is a contribution from $l$ that is seemingly increasing the deflection
angle. Additionally, the deflection angle increases briskly with $m$ and the
effect of $\beta$ is seen to be obsolete in the lower orders of magnitude of
the impact parameter and, therefore, too trivial. Since $q$ exists as a
square, these are true for $-2<q<0$ as well.

Figure 3:  Weak deflection angle plotted against the impact parameter. The
solid black line denotes the Schwarzschild case and the gradient of grey shows
the case of the asymptotic, magnetically-charged, non-singular black hole,
with $m=1$, $\beta=0.99$, $l=0.5$, and $0<q<2$.

The presence of the charge $q$ appears to be counteracted by the black hole’s
gravity. This suggests that the distortions will be weaker, less evident, and
could contain more information about the gravitational field of the black hole
than its structure, the properties of the surrounding matter, or the light
source.

##  IV GEODESICS

Looking back at some rudimentary physics, Newton stated that a moving object
has no reason whatsoever to change its direction in the absence of an external
force. When a particle follows a natural trajectory, it tends to move in a
straight line without any acceleration. This makes it easy for an observer to
predict the path that the particle would take at a particular time in the
future by extrapolating its velocity vector along the trajectory. Known as a
geodesic, this type of trajectory obtained by transposing velocity is followed
by all the objects in our universe Gralla:2019xty.

It gets interesting when the surface that a particle is moving along is not
flat: the base vector indicating the direction of the velocity is now
consistent with the geometry involved, and hence, not necessarily straight.
(It is important to note that this property of changing coordinates and their
behavior with respect to the surface grid illustrates Christoffel symbols.) A
long-haul flight traveling from one city to another in a trajectory that is a
straight line along the curvature of the Earth forms a geodesic that is a
curve on a flat frame of reference. Owing to the plane’s orientation to the
latitudes and longitudes of our planet, the corresponding coordinates are
curved with their axes being straight lines on a sphere - arcs that are not
actually straight.

###  IV.1 Raytracing

An observer perceives the objects of the physical world due to the scattering
of light as the photons from a source bounce off of a surface to form an
image. Reversing this notion, shooting out light rays at various angles from
the point of perspective of the observer such that each of the light rays
corresponds to a pixel in the image of the object is called raytracing
Gralla:2019xty; Qin:2020xzu. The resulting image is traced out by straight
lines in non-relativistic settings. On the other hand, the image obtained
through raytracing requires one to calculate the full geodesics of the ray in
GR.

Raytracing for a black hole is especially practical as it can demonstrate the
properties of the black hole and how light behaves around it by reason of
intense gravity, curvature, and thus, geodesics. In order to achieve this for
the asymptotic, magnetically-charged, non-singular black hole, take the Euler-
Lagrange equation:

$\frac{\mathrm{d}\mathcal{L}}{\mathrm{d}x^{\mu}}=\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{\mathrm{\leavevmode\nobreak\
d}\mathcal{L}}{\mathrm{\leavevmode\nobreak\ d}\dot{x}^{\mu}}\right)$

where, the Lagrangian $\mathcal{L}=g_{\mu\nu}x^{\mu}\dot{x}^{\nu}$ and
$\nicefrac{{\mathrm{d}}}{{\mathrm{d}s}}$ and $\dot{x}$ signify derivatives in
terms of some parameter $s$, with the metric independent of $\phi$ or $t$.
This approach obviates the need to determine all the Christoffel symbols
individually. Two conserved quantities, equivalent to energy $E$ and angular
momentum $L$, are derived. After substituting the Euler-Lagrange equation with
these quantities, the expression turns out to be:

$\left(\frac{\mathrm{d}r}{\mathrm{d}\phi}\right)^{2}=\frac{E^{2}r^{2}}{L^{2}}-f(r)\left(\frac{r^{4}}{L^{2}}+r^{2}\right)$

Transforming this as $u=1/r$ and applying the massless limit, followed by
differentiating it with respect to $\phi$ (to acquire a second-order
differential equation while eliminating the square derivative), it is found
that:

$\ddot{u}=-u\left[f(r)+\frac{u}{2}f^{\prime}(u)\right]$

This is computable: the solution is plotted against the Schwarzschild case in
Fig. 4 to highlight the abrupt change in the trajectory of this black hole
caused by the magnetic charges.

Figure 4:  Raytracing of the Schwarzschild black hole in comparison to the
asymptotic, magnetically-charged, non-singular black hole. The light rays are
seen to originate from a point on the Schwarzschild radius. The central mass,
$m=1$; $q=0.5$, $\beta=0.99$, and $l=0.5$.

The subsidizing parameters of the AMCNS black hole are visibly seen to modify
the photon trajectory of the in-falling ray to exhibit more curvature than the
Schwarzschild black hole instigated by the auxiliary electromagnetic
component.

Consequently, this is plotted for more light rays in the case of our black
hole in Fig. 5. The number of dispersing light rays seems to be more than the
in-falling rays. Fig. 6 shows the raytracing when the rays are shot out from
twice the Schwarzschild radius. The light rays from infinity heading towards
the black hole are traced in Fig. 7. The overlay of the light could be a
possible reason for the intensity decay of the photon sphere for a distant
observer pertaining to the classic case of destructive interference.

Figure 5:  Raytracing of the asymptotic, magnetically-charged, non-singular
black hole when the light rays originate from a point on the Schwarzschild
radius for $m=1$, $q=0.5$, $\beta=0.99$, and $l=0.5$. Figure 6:  Raytracing of
the asymptotic, magnetically-charged, non-singular black hole when the light
rays originate from a point at a distance that is twice the Schwarzschild
radius for $m=1$, $q=0.5$, $\beta=0.99$, and $l=0.5$. Figure 7:  Raytracing of
the asymptotic, magnetically-charged, non-singular black hole when the light
rays originate from infinity for $m=1$, $q=0.5$, $\beta=0.99$, and $l=0.5$.

##  V WEAK DEFLECTION ANGLE IN PLASMA MEDIUM USING THE GAUSS-BONNET THEOREM

The dispersive properties of light rays are predominant in the presence of a
medium compared to that of a vacuum. Plasma (which is essentially a hot and
ionized medium of gas), in particular, besieges deflection divulging non-
trivial components for the bending angle which arises due to more deflection
fostered by refraction. This change of medium given by the index of refraction
accounts for the non-trivial terms in the corrected expression of the
deflection angle when light travels from vacuum to plasma. Plasma is a
dispersive medium in itself and this characteristic feature infers the
dependency of the refractive index on frequency. Modeling the propagation of
light as rays is an approach in which all the plasmatic effects with respect
to absorption, wave optics and related manifestations can be disregarded, as a
consequence of sensing its existence through its influence on the space-time
trajectories of the rays alone.

Any matter that emits light escaping the photon sphere travels at a momentous
fraction of $c$, irrespective of its destiny towards the event horizon, the
eternal circumnavigation, or plasma ejection of the black hole. So, the
velocity $v$ of the photons through non-magnetized plasma begets the
refractive index as Crisnejo:2018uyn; Javed:2022fsn:

$n(r)\equiv\frac{c}{v}=\frac{1}{\mathrm{\leavevmode\nobreak\
d}r/\mathrm{\leavevmode\nobreak\ d}t}$ (35)

Here, $c$ is naturally the speed of light. For an asymptotic, magnetically
charged, non-singular black hole, the refractive index can be written as:

$n(r)=\sqrt{1-\frac{\omega_{e}^{2}}{{\omega_{\infty}^{2}}}\left[1-\frac{r^{2}\left(2m-\frac{q^{2}}{r}+\frac{\beta
q^{4}}{20r^{5}}\right)}{r^{3}+l^{2}\left(2m-\frac{q^{2}}{r}+\frac{\beta
q^{4}}{20r^{5}}\right)}\right]}$ (36)

where $\omega_{e}$ is the electron plasma frequency and $\omega_{\infty}$ is
the photon frequency measured by an observer at infinity.The line element in
equation (14) can be re-written with the optical metric as Crisnejo:2018uyn;
Ovgun:2019wej; Javed:2019ynm:

$\mathrm{\leavevmode\nobreak\
d}\sigma^{2}=g_{ij}^{\mathrm{opt}}\mathrm{\leavevmode\nobreak\
d}x^{i}\mathrm{\leavevmode\nobreak\
d}x^{j}=\frac{n^{2}(r)}{f(r)}\left[\frac{\mathrm{\leavevmode\nobreak\
d}r^{2}}{f(r)}+r^{2}\mathrm{\leavevmode\nobreak\ d}\phi^{2}\right].$ (37)

Hence, the weak deflection angle of an asymptotic, magnetically charged, non-
singular black hole surrounded by plasma restricted to the second order of $m$
is:

$\begin{split}\hat{\alpha}=\frac{{4m}}{{b}}+\frac{{3\pi
q^{2}}}{{4b^{2}}}-\frac{{3\pi
m^{2}}}{{4b^{2}}}-\frac{{8mq^{2}}}{{3b^{3}}}+\frac{{15\pi
q^{4}}}{{64b^{4}}}-\frac{75\pi m^{2}}{8b^{4}}-\frac{{45\pi
m^{2}q^{2}}}{{32b^{4}}}+\frac{{32mq^{2}}}{{b^{5}}}+\frac{{12mq^{4}}}{{5b^{5}}}\\\
+\frac{{7\pi\beta q^{4}}}{{128b^{6}}}-\frac{{175\pi
q^{4}}}{{64b^{6}}}+\frac{{125\pi m^{2}q^{2}}}{{8b^{6}}}+\frac{{525\pi
m^{2}q^{4}}}{{256b^{6}}}\\\
+\frac{{\omega_{e}^{2}}}{{\omega_{\infty}^{2}}}\left[\frac{{2m}}{{b}}+\frac{{\pi
q^{2}}}{{2b^{2}}}-\frac{{7\pi
m^{2}}}{{4b^{2}}}-\frac{{2mq^{2}}}{{b^{3}}}+\frac{{3\pi
q^{4}}}{{16b^{4}}}-\frac{25\pi m^{2}}{16b^{4}}-\frac{{315\pi
m^{2}q^{2}}}{{64b^{4}}}+\frac{80mq^{2}}{3b^{5}}\right.\\\
\left.+\frac{{2mq^{4}}}{{3b^{5}}}+\frac{{3\pi\beta
q^{4}}}{{64b^{6}}}-\frac{{75\pi q^{4}}}{{32b^{6}}}+\frac{{1125\pi
m^{2}q^{2}}}{{32b^{6}}}+\frac{{1085\pi
m^{2}q^{4}}}{{256b^{6}}}\right]\end{split}$ (38)

The terms are collected together in an attempt to emphasize their dainty
nature as they evolve distinctly in the presence of plasma. Studying this
expression graphically, Fig. 8 shows the drastic effects of refraction and its
obvious aggravating consequence on the deflection angle as it decreases, just
like the vacuum case. The contribution from $\beta$ is yet again infinitesimal
and, as $\mathcal{O}\left(m^{3}\right)$ is neglected, the consequence of $l$
is omitted altogether. The choice of varying the parameter
$z\equiv\nicefrac{{\omega_{e}^{2}}}{{{\omega_{\infty}^{2}}}}$ is to emphasize
the subtlety of plasma on $f(r)$ which in turn impacts the black hole
inevitably. Although it has a momentous offsetting from the Schwarzschild case
in vacuum, its nature is identical to Fig. 3 for the Schwarzschild case in
plasma.

Figure 8:  Weak deflection angle plotted against the impact parameter in the
presence of plasma. The solid black line denotes the Schwarzschild case, the
dashed black line represents the Schwarzschild black hole in the presence of
plasma with $z=0.9$, the blue line represents the AMCNS black hole in vacuum,
and the gradient of orange shows the case of the asymptotic, magnetically-
charged, non-singular black hole in the presence of plasma with varying $z$,
$m=1$, $q=0.5$, and $\beta=0.99$.

It is obligatory for the frequency of the plasma to be small compared to that
of light, primarily based on the size of the black hole shadow, which is
probed further in the next section. By definition, the influences of plasma is
tangible only in the radio regime. The dominating central gravity will coerce
each of the realistic emission regions to either rotate or dissipate smoothing
out the encircling inhomogeneities of the plasma by virtue of its potential
over some dynamical timescales.

##  VI GREYBODY FACTOR

The equivalence of black holes to black bodies inspires curiosity about their
relationship. This section is devoted to inspecting that correlation. A black
body is an idealized object that absorbs all radiation that strikes it and
emits radiation without incurring any loss at all wavelengths and at the
maximum frequency; for a given spectrum, although the wavelength is dependent
on temperature, its efficiency is 100%. Whereas, black holes absorb all
radiation and matter inside the event horizon and emit Hawking radiation that
is dependent on the mass of the black hole preserving the quantum information
of the engulfed particles Hawking:1975vcx; Akhmedov:2006pg. While the gravity
field and the matter encompassing a black hole deteriorates its energy
eventually impairing the Hawking radiation emitted by the black hole from
being fully efficient, its resemblance to a pure black body at its horizon,
when isolated (resulting from quantum effects), is impeccable.

The deteriorating aspects of a radiating black hole are given by the greybody
factor: the nomenclature underlines the departure from the black body
behavior. It is a function of frequency and angular momentum that typifies the
degree of deviation in the emission spectrum of a black hole from a perfect
black body spectrum. The main culprit for this digression is the scattering of
the radiation by the geometry of the black hole itself relating to the
captured/radiated particles that fall back into the extreme force. This causes
the spectrum emitted by the black hole to deviate and can be observed as the
greybody factor Maldacena:1996ix; Harmark:2007jy; Rincon:2018ktz;
Rincon:2020cos; Panotopoulos:2018pvu; Panotopoulos:2017yoe; Klebanov:1997cx;
Liu:2022ygf; Fernando:2004ay; Konoplya:2019hlu; Javed:2022rrs; Javed:2022kzf;
Yang:2022ifo; Javed:2021ymu; Mangut:2023oou; Kanzi:2023itu; Al-Badawi:2021wdm;
Al-Badawi:2022uwh; Al-Badawi:2022aby.

By the scattering of a wave packet off a black hole, the greybody factor can
be determined with the help of the classical Klein-Gordon equation – the
Schrödinger-like equation that governs the modes for a wave function
$\varphi(r)$ relating the position of an electron to its wave amplitude is
written as:

$\left(\frac{\mathrm{d}^{2}}{\mathrm{d}r_{*}^{2}}+\omega^{2}-{V}(r)\right)\varphi(r)=0$

where, $r_{*}$ is commonly known as the tortoise coordinate. Its purpose is to
evolve to infinity in a manner that counters the singularity of the metric
under scrutiny. The time of an object looming towards the event horizon in the
constructed system of coordinates grows to infinity… so does $r_{*}$ but at a
rate that is appropriate. Mathematically:

$\mathrm{d}r_{*}=\frac{1}{f(r)}\mathrm{\leavevmode\nobreak\ d}r$

and the potential ${V}(r)$ for the azimuthal quantum number $\ell$ specifying
the mode can accordingly be formulated as:

${V}(r)=\frac{f(r)f^{\prime}(r)}{r}+\ell(\ell+1)\frac{f(r)}{r^{2}}$

This embodies the potential energy of the particle when there is an external
force/field. The information about the wavefunction, the bound and scattering
states, etc. can be begotten from this quantity.

Assuming that the energy of the particles emitted by the black hole falls in
the range $\hbar\omega$ to $\hbar(\omega+\mathrm{d}\omega)$ for a given wave
frequency $\omega$, let $\Upsilon$ be a positive function that obeys
$\Upsilon(-\infty)=\sqrt{\omega^{2}-V_{-\infty}}$ and
$\Upsilon(\infty)=\sqrt{\omega^{2}-V_{\infty}}$ when
$V_{\pm\infty}=V\pm\infty$. To find the lower bound, the condition
$V_{\pm\infty}=0$ must be satisfied, implying that $\Upsilon=\omega$ and the
transmission probability is deduced as follows Visser:1998ke; Boonserm:2008zg;
Boonserm:2017qcq:

$\displaystyle T$
$\displaystyle\geq\mathrm{sech}^{2}\left(\int_{-\infty}^{\infty}{V}(r)\mathrm{\leavevmode\nobreak\
d}r_{*}\right)$
$\displaystyle=\mathrm{sech}^{2}\left[\frac{1}{2\omega}\int_{r_{H}}^{\infty}\left(\frac{f^{\prime}(r)}{r}+\frac{\ell(\ell+1)}{r^{2}}\right)\mathrm{\leavevmode\nobreak\
d}r\right]$

This expression for $T$ offers insight into the probability of a particle with
a specific energy and angular momentum escaping the black hole and being
detected at infinity. Here, $r_{H}$ refers to the black hole horizon radius,
the distance measured from the center of the black hole to its event horizon.
It affects the details of the emitted radiation spectrum and can be evaluated
by finding the roots of $f(r)=0$.

Because of the intricacy of solving for the polynomial, we procure $r_{H}$
numerically. Taking $m=1$, $q=0.5$, $l=0.5$, and $\beta=0.99$, the lower bound
was simplified to:

$T\gtrsim\mathrm{sech}^{2}\left[\frac{1}{2\omega}\left(-0.264+\frac{\ell(\ell+1)}{1.69}\right)\right]$

The term $\ell(\ell+1)$ depicts the discrete robustness of quantum mechanics
in general relativity by providing a measure of the angular momentum carried
by the particles that the black hole emits. Also called the angular momentum
quantum number, the value that $\ell$ takes says quite a lot about the emitted
particles and spectrum:

  * •

$\leavevmode\nobreak\ \ell=0$ insinuates that the emitted particles have zero
angular momentum and are moving in a direction away from the black hole. The
probability of a particle escaping the black hole in this case is maximum at
low energies and it decreases as the energy of the particle increases. The
spectrum of emitted radiation for the particles in this state is typically
dominated by low-energy photons.

  * •

$\leavevmode\nobreak\ \ell=1$ insinuates that the emitted particles have some
angular momentum and are moving in a circular orbit around the black hole. The
probability of a particle escaping the black hole in this case is maximum at
intermediate energies and it decreases at both low and high energies. The
spectrum of emitted radiation for the particles in this state typically peaks
at intermediate energy.

  * •

$\ell=2$ insinuates that the emitted particles have substantial angular
momentum and are moving in a rather elliptical orbit around the black hole.
The probability of a particle escaping the black hole in this case is maximum
at high energies and it decreases as the energy of the particle decreases. The
spectrum of emitted radiation for the particles in this state typically peaks
at high energy.

Therefore, the value of $\ell$ particularizes the detailed shape of the
spectrum of the radiation that a black hole emits, granting information about
the features of the black hole itself.

The graphical representation of the results acquired for $V(r)$ and $T$ is
presented in Fig. 9 for a Schwarzschild black hole and an AMCNS black hole.
Plotted to show the variation of the potential $V(r)$ in terms of the radial
coordinate $r$ for $\ell=0$, $\ell=1$, and $\ell=2$, Fig. 9(b) is perceived to
show negative potential for low values of $r$ and the peak appears to be
smaller than the peak in Fig. 9(a) when $\ell=0$. However, for $\ell=1$ and
$\ell=2$, Fig. 9(b) has a peak that is five times more than that of Fig. 9(a),
which is attributable to the parameters characterizing the AMCNS black hole.

(a) Schwarzschild black hole

(b) AMCNS black hole

Figure 9: Potential plotted against radial distance for $m=1$, $q=0.5$,
$l=0.5$, and $\beta=0.99$.

Fig. 10 illustrates the variation of the Transmission probability $T$ in terms
of the frequency $\omega$ for a Schwarzschild black hole and an AMCNS black
hole. Whilst the graph traits of Figs. 10(a) and 10(b) are analogous to each
other with pretty much the same probability for $\ell=0$, the initial offset
for $\ell=1$ and $\ell=2$ is conspicuous; it can be inferred that any and
every kind of matter in the proximity of a black hole gives rise to innate
frequency.

(a) Schwarzschild black hole

(b) AMCNS black hole

Figure 10: Transmission probability plotted against frequency for $m=1$,
$q=0.5$, $l=0.5$, and $\beta=0.99$.

Nevertheless, the overall nature of the AMCNS black hole plots is adequately
comparable to that of the Schwarzschild case and to Figs. 2 and 5 in in which
the author handles a regular Hayward black hole.

##  VII BLACK HOLE SHADOW

In this section, the shadow that is formed by a black hole that has a thin-
accretion disk is examined. An accretion disk refers to the structure that
arises from the gravitational pull of a central gravitating object on the
adjacent matter like gas or dust Bambi:2012tg; Bambi:2013nla;
Jaroszynski:1997bw; Gralla:2019xty. As this matter moves toward the central
object, it gains kinetic energy and generates a rapidly rotating disk around
it. The temperature and density of the disk can cause it to emit radiation in
different forms, such as X-rays, visible light, or radio waves. Accretion
disks serve as an essential component in the development and progression of
gravitating objects since they facilitate the transfer of mass and angular
momentum.

When the accretion disk concurs with the lensing effect, produces the
appearance of a shadow of a black hole. In essence, the phenomenon of
gravitational lensing is the key factor that distinguishes the black hole
shadow in the Newtonian case in contrast to that of the general-relativistic
case, thus, exaggerating the shadow with the bending of light. Illustrated as
the critical curve that separates the capture orbits that spiral into the
black hole from the scattering orbits that swerve away from it, the shadow is
formed by a geometrically thick, optically thin region filled with emitters
and is associated with a distant, homogeneous, isotropic emission ring. The
former property is important while talking about an emission region since the
intensity takes a dip that is coincident with the shadow, making it observable
through this strong visual signature. The size of the shadow is mainly
determined by the inherent parameters of the black hole, and its contour is
shape-shifting owing to the orbital instability of the light rays from the
photon sphere. For a distant observer, the shadow appears as a dark, two-
dimensional disk, which is illuminated by its bright, uniform surroundings.

For the line element given by Eq. (14), introducing the function $h(r)$ such
that:

$h^{2}(r)=\frac{r^{2}}{A(r)}$

which, in Schwarzschild gravity, is equivalent to the effective potential for
photon motion. Tracing back to the Euler-Lagrange equation in § LABEL:geodm,
Eq. (39) can be rearranged and rewritten as:

$\left(\frac{\mathrm{d}r}{\mathrm{d}\phi}\right)^{2}=\frac{r^{4}}{B(r)}\left(\frac{1}{b^{2}A(r)}-\frac{1}{r^{2}}\right)=\frac{r^{2}}{B(r)}\left(\frac{r^{2}}{b^{2}A(r)}-1\right)$
(39)

where, the impact parameter $b\equiv L/E$ yet again. This equation is
analogous to the traditional energy-conservation law described for one-
dimensional motion in classical mechanics:

$\left(\frac{\mathrm{d}r}{\mathrm{d}\phi}\right)^{2}+V(r)=0$

with $\phi$ taking the place of the temporal variable and the effective
potential $V(r)$ displaying its dependence on $r$, and by extension, $b$. This
is also known as the orbit equation.

To determine the circular trajectories, the equations $V=0$ and
$\nicefrac{{\mathrm{d}V}}{{\mathrm{d}r}}=0$ can be solved. For a light ray
approaching the center, if there exists a point at which it turns back around
to exit the orbit after the ray reaches the minimum radius $R$, then the
condition $\nicefrac{{\mathrm{d}r}}{{\mathrm{d}\phi}}|_{R}=0$ needs to be
complied with, yielding the orbit equation to be:

$\frac{1}{b^{2}}=\frac{A(R)}{R^{2}}$

This relation between the constant of motion and $R$ instigates the following
relationship:

$b=h(R)$ (40)

which in turn yields:

$\left(\frac{\mathrm{d}r}{\mathrm{d}\phi}\right)^{2}=\frac{r^{2}}{B(r)}\left(\frac{h^{2}(r)}{h^{2}(R)}-1\right)$
(41)

Say, a static observer located at a radial coordinate $r_{o}$ shoots one light
ray to the past. The dimensionless quantity $\psi$ is identified in terms of
the cotangent to be:

$\cot\psi=\left.\sqrt{\frac{g_{rr}}{g_{\phi\phi}}}\,\frac{\mathrm{d}r}{\mathrm{d}\phi}\right|_{r=r_{O}}=\left.\frac{\sqrt{B(r)}}{r}\,\frac{\mathrm{d}r}{\mathrm{d}\phi}\right|_{r=r_{O}}$

as represented by Fig. 11 to be the measurable angle between the light ray and
the radial coordinate which is directed towards the point of closest approach.
It is termed as the angular radius of the shadow.

Figure 11:  Description of the turning point with the minimum radius $R$ and
the shadow’s angular radius $\psi$ in the Schwarzschild spacetime when a light
ray is sent to the past by an observer located at $r_{o}$ in the present.

Squaring this equation and substituting Eqs. (40) and (41) in it gives:

$\cot^{2}\psi=\frac{h^{2}r_{o})}{h^{2}(R)}-1\quad\quad\text{i.e.,}\quad\quad\sin^{2}\psi=\frac{h^{2}(R)}{h^{2}\left(r_{o}\right)}$

Here, the light ray approaching the circular (with radius $r_{ph}$), unstable
photon orbits asymptotically is oriented to the past, and so is the shadow’s
boundary curve. In the limit where where $R\rightarrow r_{ph}$, the above
expression can be rewritten as:

$\sin^{2}\psi_{sh}=\frac{h^{2}\left(r_{ph}\right)}{h^{2}\left(r_{o}\right)}\quad\quad\text{and}\quad\quad
b_{cr}=h\left(r_{ph}\right)$ (42)

This implies:

$\sin^{2}\psi_{sh}=\frac{b_{cr}^{2}}{h^{2}(r_{o})}=\frac{b_{\mathrm{cr}}^{2}}{r_{o}^{2}/A(r_{o})}$

In the pursuit of calculating $r_{ph}$ for the metric in Eq. (14) next, it is
essential to realize that both $\mathrm{d}r/\mathrm{d}\phi$ and
$\mathrm{d}^{2}r/\mathrm{d}\phi^{2}$ should simultaneously be zero.
Differentiating Eq. (39), all terms vanish following the conditions except the
third term acquired from the parentheses which are equated to zero resulting
in:

$0=\frac{\mathrm{d}}{\mathrm{d}r}h^{2}(r)$

The range of values that $r$ can take is extensive inferring that there could
be multiple photon orbits – stable and unstable, existing together, with the
light rays oscillating and spiraling respectively – impinging on the
construction of the black hole shadow. For $r=r_{ph}$, the shadow can be
determined for any distance, small or large, in a static, spherically-
symmetric, asymptotically-flat spacetime with $h(r)$ from Eq. (42) by:

$b_{cr}=\frac{r_{ph}}{\sqrt{A\left(r_{ph}\right)}}\quad\quad\text{and}\quad\quad\psi_{sh}\approx\frac{b_{cr}}{r_{o}}\sqrt{A(r_{o})}.$

In the numerical plot depicted in Fig. 12, the radius of the black hole’s
shadow ($R_{sh}$) is calculated for different values of $l$. It is intriguing
to observe the exponential and inverse relationship exhibited by $l$,
irrespective of the range within which the parameters vary. In Fig. 12, the
upper limits of $l$ are presented based on EHT observations. According to the
$68\%$ confidence level (C.L.) Vagnozzi:2022moj, the upper limit for $l$ is
$0.89$.

Figure 12: Constraints from EHT horizon-scale image of SgrA* at
$1\sigma$Vagnozzi:2022moj, after averaging the Keck and VLTI mass to distance
ratio priors for SgrA* with $m=1$, $q=0.5$, $\beta=0.5$ and varying $l$.

###  VII.1 Spherically infalling accretion

In this section, we study the spherically free-falling accretion model on the
BH from infinity using the method defined in Jaroszynski:1997bw; Bambi:2012tg.
This method gives us realistic visualization of the shadow cast with the
accretion disk. In reality, the actual image of the BH can not be seen as an
apparent boundary in the universe. Moreover, it is not realistic to use a
static accretion disk model because there is a moving accretion disk around a
black hole, and also it provides a synchrotron emission from the accretion. To
do so, we first study the specific intensity observed at the observed photon
frequency $\nu_{\text{obs}}$ solving this integral along the light ray:

$I(\nu_{\text{obs}},b_{\gamma})=\int_{\gamma}g^{3}j(\nu_{e})dl_{\text{prop}}.$
(43)

It is noted that $b_{\gamma}$ is the impact parameter, $j(\nu_{e})$ is the
emissivity/volume, $dl_{\text{prop}}$ is the infinitesimal proper length and
$\nu_{e}$ stands for the photon frequency of the emitter. Here we define the
redshift factor for the infalling accretion as follows:

$g=\frac{k_{\mu}u^{\mu}_{o}}{k_{\mu}u^{\mu}_{e}},$ (44)

where the 4-velocity of the photon is $k^{\mu}=\dot{x}_{\mu}$ and 4-velocity
of the distant observer is $u^{\mu}_{o}=(1,0,0,0)$. Moreover, the
$u^{\mu}_{e}$ stands for the 4-velocity of the infalling accretion

$u_{\mathrm{e}}^{t}=\frac{1}{A(r)},\quad
u_{\mathrm{e}}^{r}=-\sqrt{\frac{1-A(r)}{A(r)B(r)}},\quad
u_{\mathrm{e}}^{\theta}=u_{\mathrm{e}}^{\phi}=0.$ (45)

Using the relation of $k_{\alpha}k^{\alpha}=0$, one can derive $k_{r}$ and
$k_{t}$ which is a constant of motion for the photons:

$k_{r}=\pm k_{t}\sqrt{B(r)\left(\frac{1}{A(r)}-\frac{b^{2}}{r^{2}}\right)}.$
(46)

Note that sign of $\pm$ shows that the photon gets close to (away from) the
black hole. Then the redshift factor $g$ and proper distance $dl_{\gamma}$ can
be written as follows

$g=\Big{(}u_{e}^{t}+\frac{k_{r}}{k_{t}}u_{e}^{r}\Big{)}^{-1},$ (47)

and

$dl_{\gamma}=k_{\mu}u^{\mu}_{e}d\lambda=\frac{k^{t}}{g|k_{r}|}dr.$ (48)

We then consider only the monochromatic emission for the specific emissivity
with rest-frame frequency $\nu_{*}$ as follows:

$j(\nu_{e})\propto\frac{\delta(\nu_{e}-\nu_{*})}{r^{2}}.$ (49)

Afterwards, the intensity equation given in (43) becomes

$F(b_{\gamma})\propto\int_{\gamma}\frac{g^{3}}{r^{2}}\frac{k_{e}^{t}}{k_{e}^{r}}dr.$
(50)

We investigate the shadow cast with the thin-accretion disk of the black hole.
First of all, we solve the above equation numerically using Einsteinpy
shreyas_bapat_2021_4739508, however one can alternatively do it using the
Mathematica notebook package Okyay:2021nnh, (also used in Chakhchi:2022fls;
Kuang:2022xjp; Uniyal:2022vdu; Pantig:2022ely) and this integration of the
flux show the effects of the parameters of the NED $\beta$ on the specific
intensity seen by a distant observer for an infalling accretion in Figures.

The figures in Fig. 13 show the stereographic projections of the black hole
described specifically by Eqs. (14) and (22) following the method of
Bambi:2013nla; Jaroszynski:1997bw; Gralla:2019xty. The shadow of the same
black hole is investigated more thoroughly with intensity plots for varied
fields of view. As we zoom out, each layer around the black hole shadow gives
us more and more information about our black hole:

  * •

Fig. 13(a): This depicts a bronze border as the accretion disc fades swiftly
implying that the latter is not as dense as it would be in the case of a
Schwarzschild black hole resulting from the significant influences of $q$,
$\beta$, and $l$. The emissivity climbs up smoothly from the circumference as
one moves toward the event horizon.

  * •

Fig. 13(b): The wider the bronze-colored region gets, the thinner the photon
orbits around the shadow appear to be. This is on account of the photons
either being pulled abruptly into the black hole or being sharply deflected
away from the black hole, with very few continuing to loop around the black
hole. Nevertheless, its intensity plot brings out the actual luminance of the
photosphere irrespective of how thin it is.

  * •

Fig. 13(c): The rings around the shadow that steadily change shades give us a
good idea about how profound even a small fluctuation can be in the appearance
of a black hole shadow by crisping out every orbit. The intensity plot peaks
with an intriguingly smooth climb that is attributable to the steady color
gradient.

(a) Shadow 1: 10 km

(b) Shadow 2: 20 km

(c) Shadow 3: 30 km

Figure 13: Stereographic projection of the asymptotic, magnetically-charged,
non-singular black hole (top) and their respective Intensity plots with the
intensity plotted against the impact parameter (bottom) for three fields of
view. The specific intensity $I_{\text{obs}}$ seen by a distant observer for
an in-falling accretion at fixed $m=1$, $q=0.5$, $\beta=0.99$, and $l=0.5$.

Moreover, one look at this black hole relative to the Schwarzschild black hole
from the point of maximum field of view (practically speaking) as in Fig.
14(a) instantly advocates the sharp differences in the circular light bands
around the shadow. The reddish hue that is prominent around the black hole
shadow in Fig. 14(b) indicates the effect of the parameters $q$, $\beta$, and
$l$ in contrast to a Schwarzschild black hole. This is coherent with the
interpretation from raytracing in § IV.1. As denoted by Fig. 4, the prompt in-
falling of the light ray for the AMCNS black hole against the Schwarzschild
case explains the presence of the additional bands of photon orbits in Fig.
14(a), appearing to be brighter. Its intensity plot shows the not-so-smooth
increase in the journey directed to the event horizon.

The difference in the appearance of the shadow size between Figs. 14(a) and
14(b) is hard to miss, given that both black holes have the same mass. The
crucial factor that brings about this striking distinction is singularity. The
Schwarzschild black hole is singular at its center meaning that it is a point
of infinite density and zero volume. Among all the properties of a black hole
that are disturbed by this (such as curvature, geometry, density profile,
etc.), the one that is sizably impacted is the event horizon which is a
mathematical construct that assumes the black hole is a point singularity. For
a non-singular black hole, this assumption does not hold; in that case, the
event horizon becomes an apparent boundary beyond which light cannot escape to
infinity, but can still escape to some finite distance. This leads the non-
singular black hole to have a smaller event horizon than the Schwarzschild
black hole, which in turn explains the former’s smaller shadow size.

Irrespective of any and every perfect presumption of vacuum, the emission
region circling a black hole rendering the shadow to be visible at all depends
on plasma astrophysics. When there is plasma enveloping the black hole, more
deflection occurs as a result of refraction. The effect of plasma on black
hole shadow is particularly interesting since it is a medium that can be
treated analytically for several spacetimes, while most media need to be
analyzed numerically, especially when it is electron-ion plasma that is non-
magnetized and pressure-less. Plasma can scatter and absorb some of the light
that would otherwise form the shadow of the black hole, and the scattered
light can also interfere with the light that forms the shadow, leading to
additional distortion, changes in the intensity and polarization of the light
that forms the shadow. Hence, the size of the black hole shadow in the
presence of plasma is larger than that in a vacuum. Fig. 14 plainly reveals
this. It is remarkable to notice the comparability of the shadow sizes of the
non-singular black hole to that of the Schwarzschild black hole. Though the
photon sphere is denser in Fig. 14(a), it looks rather blurred unlike Fig.
14(c) which is well-defined.

(a) Schwarzschild black hole

(b) AMCNS black hole

(c) AMCNS black hole in plasma medium

Figure 14: Stereographic projection of the black hole shadow (top) and their
respective Intensity plots with the intensity plotted against the impact
parameter (bottom) for three different black holes at a field of view of 50
km. The specific intensity $I_{\text{obs}}$ seen by a distant observer for an
in-falling accretion at fixed $m=1$, $q=0.5$, $\beta=0.99$, $l=0.5$, and
$n=0.9$.

In summary, any entity that skirts around a black hole rendering a time-
dependent appearance of the shadow will inexorably give rise to distortions,
big or small, contingent on its characteristics. Also, the singularity at the
center of a black hole can cause its space-time geometry to curve more
bringing about a wider event horizon and a bigger shadow size. On the other
hand, a non-singular black hole would have a smoother geometry and a smaller
horizon that results in a smaller shadow size. Furthermore, plasma intensifies
the extent of distortions because of dispersion.

##  VIII CONCLUSION

In this article, we have attempted to analyze the astrophysics of an
asymptotic, magnetically-charged, non-singular black hole. o find its
solution, we commenced with the Euler-Lagrange equation for nonlinear
electrodynamics and applied it to the Hayward metric through the trace of the
energy-momentum tensor. Establishing its connection to the energy density
eased us in uncovering the mass function and, with additional computations,
the metric function of the AMCNS black hole.

This illation was then employed in the Gauss-Bonnet theorem and modified with
the Gibbons and Werner method to obtain the weak deflection angle of the AMCNS
black hole. This is done with the aim of examining the black hole’s
gravitational field, its intrinsic attributes such as mass, spin, etc., the
nature of the light bending, and, ultimately, the validity of Einstein’s
theory deeper. These help us understand the dynamics of black holes better and
prepares Observational Astrophysicists on what to or not to expect. The
deflection angle that was obtained was observed to be vastly variant for the
AMCNS black hole triggered by its governing parameters.

Raytracing was achieved through this aiding us in comprehending the stark
contrast in behaviors between the Schwarzschild black hole and the AMCNS black
hole. This was extrapolated further for more rays shot from three different
points so as to comprehend more about the light deflection. We then used the
Geodesic method as an alternative approach to find the deflection angle and we
saw that the result was in agreement with the previous one in the linear
order.

We then proceeded to calculate the effects of the plasma medium on the
deflection angle. Numerous theoretical and experimental endeavors are adopted
to bridge the gap between physics and astrophysics; incorporating the
consequences of plasma majorly subsidizes that. Plasma plays a key role in the
intensified regime of black holes accreting and jets launching. The plasma
medium was perceived to increase the deflection more, and for the AMCNS balc
hole, the slightest drop in the refractive index generated an immense upshot.

The greybody factor was then investigated using the Schrödinger-like equation.
The potential and the transmission probability were derived for the AMCNS
black hole and were plotted against a Schwarzschild black hole. One can
discern the sizable differences in each of the curves for every case of the
azimuthal quantum number.

Finally, the black hole shadow was probed for the AMCNS case. Although many
parameters affect the shadow size, the factors assumed here for this
particular black hole display highly distinguished traits, making the black
hole fairly distinctive from a Schwarzschild black hole. With the assistance
of the Euler-Lagrange equation once again, the critical curve and the shadow’s
angular radius were determined. They were scrutinized thoroughly with
stereographic projections; the singularity at the center of the Schwarzschild
black hole curves and distorts its spacetime geometry more than that of a non-
singular black hole, therefore, having a larger horizon and shadow size. The
smoother spacetime geometry of the AMCNS black hole gives it a relatively
smaller horizon and shadow size.

Meticulous methodologies of maximal parametric inclusiveness facilitate us in
refining our understanding of gravitational lensing. The distortions that
every black hole produces offer so much information about the source of the
light rays along with the black hole, which in turn motivates us to come up
with new hypotheses that could polish and perfect our perception: our future
work is directed toward the same. we intend to gather as many contributing
factors that affect our grasp of black hole theories as possible and enhance
them, no matter how complex it is, so that both theoretical and observational
astrophysics can use them as a resource for improvement. We hope to attain
ways that could minimize our assumptions like zero spin, staticity, spherical
symmetry, and so many other constraints, and deal with the black hole
realistically instead to the greatest degree feasible.

###### Acknowledgements.

A. Ö. would like to acknowledge the contribution of the COST Action CA18108 -
Quantum gravity phenomenology in the multi-messenger approach (QG-MM). A. Ö.
is funded by the Scientific and Technological Research Council of Turkey
(TUBITAK).