Iterative variable selection for high-dimensional data with binary outcomes
Nilotpal Sanyal 1
1 Indian Statistical Institute
<EMAIL_ADDRESS>

## Introduction

High-dimensional data with binary outcomes are common in biological sciences
and related fields such as omics, epidemiology, and healthcare, environmental
sciences, physical and engineering sciences, and business and policy making.
Often, such data are sparse so that only a small number of all available
variables truly affect the binary outcome. Several frequentist and Bayesian
methods exist in the literature that provide a high-dimensional variable
selection for binary outcomes (see [1]). Recently, [2] proposed a high-
dimensional variable selection scheme for continuous outcomes in the context
of genome-wide association studies. They introduce the concept of a
‘structured screen-and-select’ strategy and examined the use of non-local
priors within the same. Here, we build upon their work and propose an
iterative variable selection scheme for high-dimensional data with binary
outcomes.

## Methods

We consider the problem of variable selection with binary outcomes for $n$
subjects and $p$ independent variables where $p$ can be much larger than $n$
($p\gg n$). Let ${\bf y}^{n\times 1}=(y_{1},y_{2},\ldots,y_{n})$ denote the
vector of binary outcomes and ${\bf X}^{n\times p}$ denote the design matrix
whose columns correspond to the independent variables.

We propose an iterative scheme for variable selection that uses non-local
priors. In each iteration, a smaller number of variables are screened from all
candidate variables using a measure of association, and non-local prior-based
Bayesian model selection is performed with the screened variables. We call
these two steps respectively the _screening step_ and the _selection step_ of
an iteration. First, we describe the data distribution and the priors used for
the selection steps and afterward, describe the full iterative scheme.

### Data distribution and priors

The selection steps perform Bayesian model selection using non-local priors.
Here, a model consists of a collection of variables. Let us denote by $p_{s}$
the number of screened variables undergoing the $s$th selection step. The
number of all possible models at the $s$th selection step is $2^{p_{s}}$.
Suppose for model ${\bf k}$, ${\bf k}=1,\ldots,2^{p_{s}}$, ${\bf X}_{{\bf
k}}^{s}$ denotes the design matrix whose $i$th row, ${\bf x}_{i{\bf k}}^{s}$,
corresponds to the $i$th subject and $\mbox{\boldmath$\beta$}_{{\bf
k}}^{s}=(\beta_{1,{\bf k}}^{2},\ldots,\beta_{|{\bf k}|,{\bf k}}^{2})$ denotes
the vector of regression coefficients where $|{\bf k}|$ is the number of
variables in model ${\bf k}$. For any ${\bf k}$, given ${\bf x}_{i{\bf
k}}^{s}$, we assume a logistic regression model for the binary phenotype
$y_{i}$, i.e.,

$Pr(y_{i}=1|{\bf x}_{i{\bf k}}^{s})=\frac{\exp({{\bf x}_{i{\bf
k}}^{s}}^{T}\mbox{\boldmath$\beta$}_{{\bf k}}^{s})}{1+\exp({{\bf x}_{i{\bf
k}}^{s}}^{T}\mbox{\boldmath$\beta$}_{{\bf k}}^{s})}.$ $None$

Model selection involves two types of uncertainties–the uncertainty regarding
selecting a model out of all possible models forming the model space, and the
uncertainty regarding the estimation of the parameters of a selected model
from the parameter space. In the Bayesian context, one needs to specify prior
distributions over both the model space and the parameter space.

We consider two different choices for non-local priors over the parameter
space—the product moment prior (pMOM prior) and the product inverse moment
prior (piMOM prior) [3]. The pMOM prior is the product of individual moment
(MOM) priors for the regression parameters $\mbox{\boldmath$\beta$}_{{\bf
k}}^{s}$ and is given by

$\pi(\mbox{\boldmath$\beta$}_{\bf k}^{s}\;|\;r,\tau_{1})\;=\;M_{|{\bf
k}|}^{-1}\;(\phi\tau_{1})^{-|{\bf k}|/2-r|{\bf k}|}\;\prod_{j=1}^{|{\bf
k}|}{\beta_{j,{\bf
k}}^{s}}^{2r}\;\exp\left[-\frac{1}{2\phi\tau_{1}}\sum_{j=1}^{|{\bf
k}|}{\beta_{j,{\bf k}}^{s}}^{2}\right],$ $None$

where $\tau_{1}$ is the common scale parameter and $r$ is the common order of
the MOM priors, $\phi$ is a dispersion parameter and $M_{|{\bf k}|}$ is a
marginalizing constant given by $M_{|{\bf k}|}=(2\pi)^{-|{\bf
k}|/2}\left\\{\prod_{l=1}^{r}(2l-1)\right\\}^{|{\bf k}|}$.

The piMOM prior is the product of individual inverse moment (iMOM) priors for
the regression parameters $\mbox{\boldmath$\beta$}_{{\bf k}}^{s}$ and is given
by

$\pi(\mbox{\boldmath$\beta$}_{\bf
k}^{s}\;|\;\nu,\tau_{2})\;=\;\frac{(\phi\tau_{2})^{\nu|{\bf
k}|/2}}{(\Gamma({\nu\over 2}))^{|{\bf k}|}}\;\prod_{j=1}^{|{\bf
k}|}|\beta_{j,{\bf
k}}^{s}|^{-(\nu+1)}\;\exp\left(-\phi\tau_{2}\sum_{j=1}^{|{\bf
k}|}{1\over{\beta_{j,{\bf k}}^{s}}^{2}}\right),$ $None$

where $\tau_{2}$ is the common dispersion parameter and $\nu$ is the common
shape parameter of the iMOM priors.

For the model space, we assume a beta-binomial prior [3], given by $\pi({\bf
k}|\gamma)=\gamma^{|{\bf k}|}\;(1-\gamma)^{p_{s}-|{\bf k}|}\text{ where
}\gamma\sim beta(1,1)$ distribution. In the Bayesian paradigm, the posterior
probability of each model is computed and the model with the highest posterior
probability is generally considered as the selected model.

### The proposed iterative scheme

The proposed iterative variable selection scheme uses the structured screen-
and-select strategy [2] whereby in each iteration, smaller subsets of
candidate variables are screened using measures of association between the
outcome and the independent variables, and among the independent variables.
Screened variables are evaluated for being included in the final selection
based on non-local prior-based Bayesian model selection. The scheme is
described below.

  * •

In the first iteration ($s=1$), the association of all candidate variables
with the binary outcome is ascertained in terms of maximum marginal likelihood
estimates (MMLE) [4]. For the $j$th candidate variable, the MMLE is given by

$\widehat{\beta_{j}^{1}}_{mmle}=\text{the last element of
}\operatorname*{arg\,min}_{\beta
0,\beta_{j}^{1}}\sum_{i=1}^{n}L(y_{i},\beta_{0}+x_{ij}^{1}\beta_{j}^{1}),$

where $L$ is the usual logarithmic loss function for the logistic regression
of $y_{i}$ on independent variable $x_{j}$ and an intercept. The $k_{0}$
variables that have the largest absolute values of the association are called
the _leading variables_. For each leading variable, we collect all candidate
variables that have an absolute Pearson correlation coefficient value $\geq r$
with that leading variable where $r$ is a given threshold; this collection is
called the _leading set_. For each leading set, non-local prior-based model
selection is performed according to the method in [3], using the data
distribution and non-local priors described in Equations 1-3. The variables
contained in the higher posterior probability model of each leading set are
included in the final selection. The remaining variables of each leading set
are excluded from subsequent analysis.

  * •

Subsequent iterations, $s=2,3,\ldots$, proceed as before except for the
evaluation of the association between candidate variables and the outcome
which is done as follows. Suppose ${\bf X}_{sel}^{(s-1)}$ is the set of
variables selected in iterations 1 through $(s-1)$, and ${\bf
x}_{i,sel}^{(s-1)}$ is its $i$th row. In iteration $s$, the association of the
remaining candidate variables with the binary outcome is ascertained in terms
of their MMLEs in presence of the variables ${\bf X}_{sel}^{(s-1)}$ in the
model. For the $j$th candidate variable of iteration $s$, the MMLE is given by

$\widehat{\beta_{j}^{s}}_{mmle}=\text{the last element of
}\operatorname*{arg\,min}_{\beta_{0},\mbox{\boldmath$\beta$}_{sel}^{(s-1)},\beta_{j}^{s}}\sum_{i=1}^{n}L(y_{i},\beta_{0}+{\bf
x}_{i,sel}^{(s-1)}\mbox{\boldmath$\beta$}_{sel}^{(s-1)}+x_{ij}^{s}\beta_{j}^{s}),$

where $\mbox{\boldmath$\beta$}_{sel}^{(s-1)}$ are the regression coefficients
for variables ${\bf x}_{i,sel}^{(s-1)}$.

  * •

Variables are selected through this structured screen-and-select strategy
until we reach a desired number, $m$, of selected variables, or until the
number of iterations that select no variables reaches a maximum allowed value,
$maxno$.

The proposed scheme has four tuning parameters $k_{0}$, $r$, $m$ and $maxno$
that determine the number of selected variables. In the following application,
we set them using the heuristic guidelines provided in [2].

## Application

We used the proposed method to select single nucleotide polymorphisms (SNPs)
for binary phenotypes. The genotype data were simulated using methods
described in [2] such that the linkage disequilibrium structure of the
genotype matrix resembled that of real genotyped SNPs in chromosome 1 of the
human genome. The data contain genotypes for $p=20000$ SNPs for $n=2000$
subjects. 20 SNPs were randomly chosen as causal SNPs and their standardized
effect sizes were randomly generated from the N(0,1) distribution. The
phenotype data were generated according to the model in Equation 1.

We analyze the data using our proposed method as well as LASSO, elastic net
($\alpha=0.75,0.5,0.25$), and ISIS with SCAD penalty (see references within
[1] and [2]). For our method, we set $\phi=1$, $\tau_{1}=\tau_{2}=0.2$,
$k_{0}=1$, $r=0.3$ and $maxno=3$. For the other methods, default cross-
validation tuning was used. The performance of all the methods was compared in
terms of true positive rate (TPR) and false discovery rate (FDR), and shown in
the following figure. Our proposed method has evidently shown a very
competitive profile compared to the best of the alternatives.

Figure 1: Comparison of the proposed method (pMOM and piMOM) with LASSO,
Elastic net, and ISIS in terms of true positive rate and false discovery rate
based on genomic SNP data with $p=20000$ SNPs and $n=2000$ subjects and 20
truly causal SNPs. The labels in the x-axis show the names of the various
methods and the number of selected variables within parentheses.

## Conclusion

We propose a competitive variable selection method for high-dimensional data
with binary outcomes. The proposed method has been implemented within the
existing R package GWASinlps. In this work, we have considered setting the
tuning parameters based on heuristics. In the future, we wish to develop
objective criteria for optimally estimating these tuning parameters from the
data.

REFERENCE

## References

  * [1] Nikooienejad, A., Wang, W., & Johnson, V. E., Bayesian variable selection for binary outcomes in high-dimensional genomic studies using non-local priors. Bioinformatics, 2016, N. 32(9), p. 1338–1345.
  * [2] Sanyal, N., Lo, M. T., Kauppi, K., Djurovic, S., Andreassen, O. A., Johnson, V. E., & Chen, C. H., GWASinlps: non-local prior based iterative SNP selection tool for genome-wide association studies. Bioinformatics, 2019, N. 35(1), p. 1–11.
  * [3] Johnson, V. E., & Rossell, D., Bayesian model selection in high-dimensional settings. Journal of the American Statistical Association, 2012, N. 107(498), p. 649–660.
  * [4] Fan, J., Samworth, R., & Wu, Y., Ultrahigh dimensional feature selection: beyond the linear model. The Journal of Machine Learning Research, 2009, N. 10, p. 2013–2038.

Annotation:

Nilotpal Sanyal (Kolkata, India)

High-dimensional variable selection for binary data

Abstract. We propose an iterative variable selection scheme for high-
dimensional data with binary outcomes. The scheme adopts a structured screen-
and-select framework and uses non-local prior-based Bayesian model selection
within the same. The structured screening is based on the association of the
independent variables with the outcome which is measured in terms of the
maximum marginal likelihood estimator. Performance comparison with several
well-known methods in terms of true positive rate and false discovery rate
shows that our proposed method stands to be a competitive alternative for
sparse high-dimensional variable selection with binary outcomes. The method
has been implemented within the R package GWASinlps.

Key words: _High-dimensional, variable selection, nonlocal prior_


# Typhon: a polar stream from the outer halo raining through the Solar
neighborhood

Wassim Tenachi Université de Strasbourg, CNRS, Observatoire astronomique de
Strasbourg, UMR 7550, F-67000 Strasbourg, France  Pierre-Antoine Oria
Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR
7550, F-67000 Strasbourg, France  Rodrigo Ibata Université de Strasbourg,
CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg,
France  Benoit Famaey Université de Strasbourg, CNRS, Observatoire
astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France  Zhen Yuan
Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR
7550, F-67000 Strasbourg, France  Anke Arentsen Université de Strasbourg,
CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg,
France  Nicolas Martin Université de Strasbourg, CNRS, Observatoire
astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France  Akshara
Viswanathan Kapteyn Astronomical Institute, University of Groningen, Postbus
800, 9700 AV, Groningen, the Netherlands

(Received June 21, 2022; Accepted August 02, 2022)

###### Abstract

We report on the discovery in the Gaia DR3 astrometric and spectroscopic
catalog of a new polar stream that is found as an over-density in action
space. This structure is unique as it has an extremely large apocenter
distance, reaching beyond 100 kpc, and yet is detected as a coherent moving
structure in the Solar neighborhood with a width of $\sim 4{\rm\,kpc}$. A sub-
sample of these stars that was fortuitously observed by LAMOST has a mean
spectroscopic metallicity of $\langle{\rm[Fe/H]}\rangle=-1.60^{+0.15}_{-0.16}$
dex and possesses a resolved metallicity dispersion of
$\sigma({\rm[Fe/H]})=0.32^{+0.17}_{-0.06}$ dex. The physical width of the
stream, the metallicity dispersion and the vertical action spread indicate
that the progenitor was a dwarf galaxy. The existence of such a coherent and
highly radial structure at their pericenters in the vicinity of the Sun
suggests that many other dwarf galaxy fragments may be lurking in the outer
halo.

Galaxy: halo – Galaxy: kinematics and dynamics – stellar streams

††journal: ApJL

##  1 Introduction

One of the principal goals of the Gaia space mission (Gaia Collaboration et
al., 2016) is to survey the Milky Way, so as to allow us to understand how our
home galaxy was built up over cosmic time. Although we only observe the end
state of this majestic structure, fortunately the processes of formation and
growth have left copious amounts of evidence in the form of debris that is now
scattered throughout our Galaxy (Belokurov et al., 2006; Shipp et al., 2018;
Ibata et al., 2021b; Malhan et al., 2022). Some of these residues are due to
the accretion of small galaxies and globular clusters, which disrupted under
the action of tidal forces, leaving long stellar streams. In some cases they
can still remain as elongated structures, many billion years after the
dissolution of their progenitors (Helmi, 2008). Studying these structures is
of great importance since their trajectories probe the galactic acceleration
field and the underlying dark matter distribution (e.g., Koposov et al., 2010;
Sanders & Binney, 2013; Malhan & Ibata, 2019; Ibata et al., 2021a).

A particularly powerful means to uncover such fossil remnants is by searching
for groups of stars with common integrals of motion. Action coordinates are
perhaps the best choice for this, as they are adiabatic invariants that will
have been preserved along orbits if the Milky Way’s potential evolved only
slowly through time (Binney & Tremaine, 2011). However, to transform our
stellar measurements into actions (and their conjugate angles), we require the
full six-dimensional positions and velocities. With present instrumentation
this is only achievable close to the Solar position in the Galaxy.

Figure 1: Actions and total energy of Typhon members (indicated by star-
symbols) and of the 573 pre-selected stars having $\varpi/\delta\varpi>10$,
$r_{apo}>75{\rm\,kpc}$ and $d_{\odot}<4{\rm\,kpc}$ (denoted by circles). Stars
are colored by their apocenter values in the upper panel and and by their
vertical action values in the lower row of panels. Upper panel:
$(J_{\phi},J_{z})$ plane used for the selection where the overdensity was
discovered. The most significant detection obtained using a Hough transform
technique (Illingworth & Kittler, 1988) on stars with
$J_{z}>1000{\rm\,kpc}{\rm\,km\,s^{-1}}$ (i.e. with large departures from the
Galactic mid-plane) is shown with a red line. This line runs through the
Typhon structure. The parallelogram selection of the structure is depicted in
a solid line encompassing 16 stars, and is defined by:
$J_{z}\in[2000,3100]{\rm\,kpc}{\rm\,km\,s^{-1}}$ and
$3.3J_{\phi}+3500{\rm\,kpc}{\rm\,km\,s^{-1}}<J_{z}<3.3J_{\phi}+5000{\rm\,kpc}{\rm\,km\,s^{-1}}$.
The symmetric (retrograde) selection with respect to the $J_{\phi}=0$ line is
shown with a dashed line. Bottom-left and bottom-right panels respectively
show the $(J_{\phi},J_{r})$ and $(J_{\phi},E_{tot})$ planes.

The Gaia mission has recently made accessible its third data release (DR3)
(Gaia Collaboration, 2022) of its all-sky survey. It contains approximately 33
million stars with mean radial velocities down to $G\sim 15$, which,
complemented with the excellent proper motions and parallaxes published in the
earlier EDR3 release (Gaia Collaboration et al., 2021), provide the required
phase-space constraints. Because the DR3 radial velocity limit is quite
shallow, it almost exclusively probes the very nearby regions of the Galaxy
(the median distance of the sample with $10\sigma$ parallaxes is only
$1.26{\rm\,kpc}$). In the vicinity of the Sun, dynamical times are short and
tidal debris are expected to phase-mix rapidly (Helmi et al., 1999, 2003),
erasing any initial stream-like coherence that might have been present.

Figure 2:  Positions and velocity vectors in galactic Cartesian coordinates of
Typhon sample members. Velocity vectors scale $1:3\times 10^{3}$. The sample
shows very clear streaming motion. For reference, the position and velocity
vector of the Sun is also shown in red.

In this contribution we show that, surprisingly and contrary to those
expectations, the Solar neighborhood contains a very wide yet kinematically
coherent metal poor stellar stream, which we name Typhon111The serpent Typhon
is the child of Gaia and Tartarus (the deep abyss) in Greek myth., whose
apocenter reaches out to $>100{\rm\,kpc}$ – the edge of the Galactic halo.

##  2 Selection

From the Gaia DR3 catalogue, we select the 25,355,580 stars with well-
constrained distances ($\varpi/\delta\varpi>10$), radial velocities measured
by Gaia’s Radial Velocity Spectrometer (RVS) instrument (Recio-Blanco et al.,
2022), having at least a 5-parameter astrometric solution, and with magnitudes
in the range $0\leq G\leq 22$, $0\leq G_{BP}\leq 30$, $0\leq G_{RP}\leq 30$.
To convert the apparent motions to motions in a frame222Throughout the paper,
we use a right-handed Galactic Cartesian coordinate system. at rest with
respect to the Galaxy, we assume that the Sun is located at
$(x,y,z)_{\odot}=(-8.2240,0,0.0028){\rm\,kpc}$ (Solar radius from Bovy 2020
and $z$-position of the Sun from Widmark et al. 2021), and that it moves with
a peculiar velocity
$(v_{x},v_{y},v_{z})_{\odot}=(11.10,7.20,7.25){\rm\,km\,s^{-1}}$ (Schönrich et
al. 2010, with the $\phi$-direction velocity from Bovy 2020), and we take the
circular velocity at the Solar radius to be $243{\rm\,km\,s^{-1}}$ (Bovy,
2020). We use the resulting phase space measurements to derive the orbital
parameters of the stars, including the pericenter and apocenter distances, as
well as action-angle coordinates calculated using the AGAMA package (Vasiliev,
2019) in a realistic potential model (McMillan, 2017) for the Milky Way. Since
we are particularly interested in finding debris from the outer halo that
could be associated to ancient merger events, we impose an apocenter cut at
$r_{apo}>75{\rm\,kpc}$, which yields a sub-sample of 870 stars.

Further analysis is performed in the space of actions
$(J_{r},J_{\phi},J_{z})$, which encode, respectively, the amplitude of orbital
motion in the radial, azimuthal, and vertical directions. In particular, we
plot the $(J_{\phi},J_{z})$ projection colored by $r_{apo}$ in Figure 1.
There, a polar structure can be spotted as a tight, almost vertical, linear
grouping between $(J_{\phi}\sim-650{\rm\,kpc}{\rm\,km\,s^{-1}},J_{z}\sim
2100{\rm\,kpc}{\rm\,km\,s^{-1}})$ and
$(J_{\phi}\sim-400{\rm\,kpc}{\rm\,km\,s^{-1}},J_{z}\sim
3000{\rm\,kpc}{\rm\,km\,s^{-1}})$. We find that this feature is most striking
when the sample is limited to stars with heliocentric distances
$d_{\odot}<4{\rm\,kpc}$, approximately at the limit of useful 6-D phase-space
data in the DR3 catalog. In particular, performing the Hough transformation
(Illingworth & Kittler, 1988) line detection technique on the stars in the
$(J_{\phi},J_{z})$ plane (binning the action data into pixels of size
$30{\rm\,kpc}{\rm\,km\,s^{-1}}$ on a side and adopting a $1^{\circ}$
discretization for the angle of the fitted lines), we find that the most
significant linear grouping of stars with
$J_{z}>1000{\rm\,kpc}{\rm\,km\,s^{-1}}$ (i.e. that experience large excursions
from the Galactic mid-plane) corresponds to this quasi-linear overdensity (red
line in Figure 1). These 16 stars possess similar apocenter distances
($r_{apo}\approx 100{\rm\,kpc}$), and are also highly correlated in the angle
coordinates $(\theta_{r},\theta_{\phi},\theta_{z})$ conjugate to the actions.

We then separate this structure from the bulk of the data by applying a simple
parallelogram selection in the $(J_{\phi},J_{z})$ plane, as follows:
$J_{z}\in[2000,3100]$ and $3.3J_{\phi}+3500<J_{z}<3.3J_{\phi}+5000$, which
results in a final sample of 16 stars. This selection box is displayed as a
solid black parallelogram in Figure 1.

Furthermore, it should be noted that the symmetric control selection around
$J_{\phi}=0$, shown as a dashed parallelogram, encompasses only two stars and
they do not possess homogeneous dynamical properties. Assuming that the halo
is symmetric in angular momentum, there is no a priori reason for the prograde
selection to contain significantly more stars than the symmetric retrograde
selection as is the case here, other than the selection containing a coherent
dynamical group. Taking the symmetric selection as a control sample we
estimate the significance of the detection to be $\approx 3.5\sigma$. We note
in passing that the Gaia Universe Model Snapshot (GUMS, Robin et al. 2012,
updated for Gaia DR3) contains no artificial stars with the selection criteria
used to detect Typhon, suggesting that Typhon is a coherent structure that can
only be explained by an external body not included in that simulation.

##  3 Characteristics

The positions and velocities of the sample members of the Typhon stream are
shown in Figure 2. We find that member stars of this polar stream are spread
out all around us, passing through the Solar neighborhood with a high vertical
velocity, and exiting the disk at an angle of $\sim 50\degree$ with respect to
it.

In Figure 3 we show the result of integrating Typhon members backwards in time
for $5{\rm\,Gyr}$ in the (McMillan, 2017) Milky Way potential model. Although
the stars were selected from a small region in the $(J_{\phi},J_{z})$ plane
(but with no constraint on $J_{r}$), and so should therefore possess similar
orbits, there was no a-priori reason for the sample to be in phase, as is
clearly the case from an inspection of Figure 3. The sample is dynamically
coherent, with very similar orbital parameters: $r_{peri}=6.0\pm
0.5{\rm\,kpc}$, $r_{apo}=99\pm 15{\rm\,kpc}$, $J_{r}=6400\pm
1000{\rm\,kpc}{\rm\,km\,s^{-1}}$, $J_{\phi}=-560\pm
110{\rm\,kpc}{\rm\,km\,s^{-1}}$, $J_{z}=2500\pm
300{\rm\,kpc}{\rm\,km\,s^{-1}}$ and eccentricity $e=0.88\pm 0.02$.

We estimate the 3-dimensional velocity dispersion of the stream to be
$\sigma_{v,3D}\approx 13{\rm\,km\,s^{-1}}$ by considering the velocity
differences of the stars to the computed orbit of the star with Gaia ID
3939346894405032576 (whose orbit through the Solar neighborhood appears
closest to the middle of the sample). Assuming isotropy, the one-dimensional
velocity dispersion is then $\sigma_{v}\approx 7.5{\rm\,km\,s^{-1}}$.

Figure 3: Trajectories of the sample members of Typhon during a $5{\rm\,Gyr}$
backward integration in the (McMillan, 2017) potential in galactic Cartesian
coordinates. Trajectories of the 7 stars whose metallicity is available
through LAMOST DR8 (Wang et al., 2022) are colored in yellow.

We cross-matched our sample with the LAMOST DR8 (Wang et al., 2022) catalog,
in particular the “FEH_PASTEL” column which covers a wide range of
metallicities especially on the very metal-poor regime, enabling us to obtain
high quality spectroscopic metallicities for 7 stars of the Typhon stream (the
stellar parameters of which lie within the reliable range of the PASTEL
catalog). These measurements span between ${\rm[Fe/H]}=-2.23\pm 0.06$ dex and
${\rm[Fe/H]}=-1.25\pm 0.09$ dex.

As shown in Figure 3, where we color orbits of stars of known metallicity in
yellow, these stars are dynamically representative of the full sample. In
Figure 4 (left panel), we show the likelihood distribution (black contour
lines) for the mean metallicity and for the intrinsic dispersion of the
metallicity distribution (correcting for the LAMOST uncertainty estimates,
assuming that they are reliable). We find
$\langle{\rm[Fe/H]}\rangle=-1.60^{+0.15}_{-0.16}$ dex, and
$\sigma({\rm[Fe/H]})=0.32^{+0.17}_{-0.06}$ dex, which indicates that the
system has a resolved dispersion in metallicity. We note however, that this
result depends on the inclusion of the most metal-poor star in the sample; if
it is removed (although we have no a-priori reason to do so) these values
become $\langle{\rm[Fe/H]}\rangle=-1.41^{+0.05}_{-0.09}$ dex, and
$\sigma({\rm[Fe/H]})=0.06^{+0.17}_{-0.06}$ dex, consistent with no dispersion
at the $1\sigma$ level.

These metallicities are consistent with the color magnitude diagram shown in
Figure 4, where we use the 3D extinction estimates by Anders et al. 2022 to
deredden the stars. In addition, based on the PARSEC stellar population models
(Bressan et al., 2012), and using the canonical two-part-power law initial
mass function corrected for unresolved binaries (Kroupa, 2001), and Gaia’s
detection limit, we compute the order of magnitude of the density of the
Typhon stream to be of $\sim 25{\rm\,M_{\odot}}/{\rm\,kpc}^{-3}$ in the
$d_{\odot}<1.5{\rm\,kpc}$ solar vicinity fragment. However, without further
information we refrain from extrapolating this value out to compute the mass
of the full stream structure.

##  4 Discussion and conclusions

Although the search for new stellar streams is currently a very active field,
to the best of our knowledge the structure discussed here (Typhon) that we
isolated thanks to the new and excellent Gaia DR3 data was never identified
before. It should be noted that although Typhon is very close to the DTG-11
stream identified in (Yuan et al., 2020) in the $(J_{\phi},J_{z})$ plane, we
verified that Typhon is a distinct structure. In particular, we see that
Typhon members have much higher apocenters ($\approx 100{\rm\,kpc}$ vs.
$\approx 15{\rm\,kpc}$ for DTG-11) which becomes obvious when comparing their
very different $J_{r}$ values. In addition, we compared our sample to the
thorough Malhan et al. (2022) atlas of stellar streams and found no previously
mapped equivalent structure. We note that the discovery of the Typhon
structure was confirmed by Dodd et al. (2022) shortly after the first
submission of our paper using a formal clustering metric.

Figure 4: Left: Likelihood contours of the mean metallicity and metallicity
dispersion of the spectroscopic sample, shown for the full 7 star sample
(black lines), and removing the most metal poor star (grey lines). Right:
Color magnitude diagram of the sample members of Typhon. For reference, the
grey line shows a PARSEC isochrone model (Bressan et al., 2012) of age
$12.5{\rm\,Gyr}$ and of metallicity ${\rm[Fe/H]}=-1.60$ dex. The reasonable
correspondence of this model shows that the population is predominantly very
old.

In addition, a follow up study focusing on the chemical abundances of Typhon
was published by Ji et al. (2022). That contribution presents high resolution
spectra for 7 Typhon members chosen solely based on observability, including 3
members whose metallicities are not available in LAMOST DR8, which
nevertheless show consistent metallicities with the LAMOST sub-sample, thereby
supporting our conclusions regarding the metallicity distribution of the
structure.

The characteristics of Typhon members given in Section 3 lead us to believe
that Typhon is likely the tidal remnant of a dwarf galaxy. In particular the
metallicity spread, vertical action spread and structure width appear
completely incompatible with a globular cluster progenitor. With metallicities
reaching ${\rm[Fe/H]}\sim-1.3$ dex, and with a mean of ${\rm[Fe/H]}\sim-1.6$
dex, the mass-metallicity relation of dwarf galaxies (Kirby et al., 2013)
suggests that the progenitor likely possessed a luminosity of $10^{6}$ –
$10^{7}{\rm\,L_{\odot}}$, perhaps similar to the Sculptor dSph. Ji et al.
(2022) concur with us on this point. The estimated velocity dispersion value
of $\sigma_{v}\approx 7.5{\rm\,km\,s^{-1}}$ lies between that of the Orphan
Stream ($\sigma_{v}\approx 5{\rm\,km\,s^{-1}}$, Koposov et al. 2019) and the
stream of the Sagittarius dwarf galaxy ($\sigma_{v}\approx
13{\rm\,km\,s^{-1}}$, Gibbons et al. 2017), suggesting that the mass of the
Typhon progenitor likely exceeded $10^{8}{\rm\,M_{\odot}}$ (an estimate for
the mass of Orphan Stream progenitor, Fardal et al. 2019), but was not as
massive as the Sagittarius dwarf.

We noticed that although in the heavy McMillan (2017) gravitational potential
$(M_{\rm vir}=1.3\times 10^{12}{\rm\,M_{\odot}})$ all stars in the sample are
bound, in the lighter MWPotential2014 (Bovy 2015, $M_{\rm vir}=8\times
10^{11}{\rm\,M_{\odot}}$), half of the Typhon stream members are
unbound333Note that none of the Typhon stars were flagged as hyper-velocity
stars in the Marchetti et al. (2019) census.. This underlines how having
constraints on the trajectories of streams such as Typhon is of great value as
the trajectories of these streams are very dependant on the acceleration field
of the Milky Way and its underlying dark matter distribution.

We also checked whether the Typhon members could have close encounters with
the Large Magellanic Cloud (LMC) or the Sagittarius dwarf galaxy. Taking the
trajectories of the two satellites from Vasiliev et al. (2021), we find that
the LMC remains always very distant ($\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}40{\rm\,kpc}$). However, the Typhon
stars probably did experience a relatively close flyby of Sagittarius ($\sim
20{\rm\,kpc}$, $0.10{\rm\,Gyr}$ ago). We note that Typhon and Sagittarius
share very similar orbital planes, although they possess opposite angular
momentum vectors (i.e. the direction of motion in the plane is opposite). The
interaction between Typhon and Sagittarius will be interesting to analyse with
$N$-body simulations, but we defer that investigation to a future
contribution.

The identification of this high apocenter polar stream passing so close to the
Sun raises many questions. Assuming that the Solar vicinity is not special and
is representative of an average location in the disk, the present detection
could be used to place constraints on the number of highly radial accretions
that took place during the formation of the Milky Way. The picture suggested
by Typhon is that there may be a large population of outer halo dwarf galaxies
or dwarf galaxy fragments residing near their apocenters, akin to the “Oort
Cloud” around the Sun. A more thorough survey of local phase space for other
Typhon-like structures and also deeper next-generation sky surveys (with LSST,
for instance) that might detect them in place in the outer halo will help
quantify this possibility.

This discovery also underlines the relevance of stream research in the Solar
vicinity where great quantities of high quality data are available in addition
to spatially wider searches. This poses several challenges and may require the
development of new algorithmic approaches suited to exploit Gaia era data for
nearby structures with incomplete astrometry (e.g. missing line of sight
velocities) as sections of streams passing near us are not easily identifiable
as streams when projected onto a sky map.

In future work, it will be very useful to attempt to extend the detections
along the stream so as to chart it out further in its orbit through the
Galaxy. As we alluded to above, such stars may provide very useful dynamical
probes for the Milky Way’s dark halo, and they will be invaluable to inform
follow-up simulation studies attempting to model the N-body evolution of the
system. Similarly, having full metallicity information for the member stars
would be of great value in order to confirm the present hypothesis regarding
the nature of the progenitor.

## Data Availability

The final sample is provided in Table 1 with a more complete table available
at DOI 10.5281/zenodo.6979887.

Table 1: The Typhon sample. Orbital parameters derived using a (McMillan, 2017) potential. Metallicities from LAMOST DR8 (PASTEL column) (Wang et al., 2022) are listed when available.  Gaia source ID | RA | DEC | $J_{r}$ | $J_{\phi}$ | $J_{z}$ | $r_{\rm peri}$ | $r_{\rm apo}$ | ${\rm FeH}$
---|---|---|---|---|---|---|---|---
| [deg] | [deg] | $[{\rm\,kpc}{\rm\,km\,s^{-1}}]$ | $[{\rm\,kpc}{\rm\,km\,s^{-1}}]$ | $[{\rm\,kpc}{\rm\,km\,s^{-1}}]$ | $[{\rm\,kpc}]$ | $[{\rm\,kpc}]$ | [dex]
291927350856672768 | 23.65 | 24.43 | 5467.78 | 2109.67 | -678.85 | 5.64 | 85.13 | -
1255095276181144320 | 218.71 | 25.17 | 5500.85 | 2153.71 | -641.67 | 5.65 | 85.59 | $-1.42\pm 0.11$
1264504793612855808 | 226.50 | 24.27 | 5880.03 | 2536.48 | -561.03 | 6.11 | 92.86 | $-1.50\pm 0.41$
1303595799235740288 | 243.67 | 25.97 | 5957.51 | 2210.55 | -625.38 | 5.63 | 91.92 | -
1470463353223683840 | 205.82 | 33.33 | 8919.22 | 2929.18 | -555.25 | 6.65 | 138.32 | $-1.94\pm 0.22$
1485076859188949760 | 212.45 | 37.57 | 5789.44 | 2253.55 | -641.30 | 5.81 | 90.15 | -
1765600930139450752 | 327.45 | 10.81 | 7675.31 | 2069.85 | -643.15 | 5.17 | 114.35 | $-2.24\pm 0.06$
3013164238138435712 | 82.50 | -10.56 | 6320.46 | 2733.22 | -343.02 | 6.15 | 98.75 | -
3029220853122930432 | 115.02 | -14.82 | 6971.03 | 2793.03 | -391.92 | 6.34 | 108.47 | -
3573787693673899520 | 181.02 | -13.37 | 6106.32 | 2288.46 | -607.18 | 5.75 | 94.38 | -
3736372993468775424 | 197.96 | 11.29 | 4584.30 | 2469.84 | -651.10 | 6.44 | 76.04 | $-1.50\pm 0.09$
3793377208170393984 | 173.49 | -2.47 | 6185.63 | 2884.76 | -441.23 | 6.65 | 98.78 | $-1.25\pm 0.09$
3891712266823336192 | 181.86 | 1.68 | 5794.96 | 3007.81 | -400.13 | 6.87 | 94.16 | -
3913243629368310912 | 178.32 | 10.63 | 7589.72 | 2826.15 | -502.54 | 6.47 | 118.11 | -
3939346894405032576 | 200.01 | 19.69 | 6128.64 | 2412.30 | -602.82 | 5.99 | 95.58 | $-1.35\pm 0.07$
4537774136693362944 | 280.74 | 26.22 | 7212.08 | 2163.32 | -670.81 | 5.49 | 108.86 | -

The authors thank the anonymous referee for their very helpful comments and
acknowledge funding from the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme (grant
agreement No. 834148) and from the Agence Nationale de la Recherche (ANR
projects ANR-18-CE31-0006, ANR-18-CE31-0017 and ANR-19-CE31-0017). This work
has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## References

  * Anders et al. (2022) Anders, F., Khalatyan, A., Queiroz, A. B. A., et al. 2022, A&A, 658, A91, doi: 10.1051/0004-6361/202142369
  * Belokurov et al. (2006) Belokurov, V., Zucker, D. B., Evans, N. W., et al. 2006, ApJ, 642, L137, doi: 10.1086/504797
  * Binney & Tremaine (2011) Binney, J., & Tremaine, S. 2011, Galactic dynamics, Vol. 13 (Princeton university press)
  * Bovy (2020) Bovy. 2020, arXiv e-prints, arXiv:2012.02169.  https://arxiv.org/abs/2012.02169
  * Bovy (2015) Bovy, J. 2015, ApJS, 216, 29, doi: 10.1088/0067-0049/216/2/29
  * Bressan et al. (2012) Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127, doi: 10.1111/j.1365-2966.2012.21948.x
  * Dodd et al. (2022) Dodd, E., Callingham, T. M., Helmi, A., et al. 2022, arXiv e-prints, arXiv:2206.11248.  https://arxiv.org/abs/2206.11248
  * Fardal et al. (2019) Fardal, M. A., van der Marel, R. P., Sohn, S. T., & del Pino Molina, A. 2019, MNRAS, 486, 936, doi: 10.1093/mnras/stz749
  * Gaia Collaboration (2022) Gaia Collaboration. 2022, Astronomy & Astrophysics, doi: 10.1051/0004-6361/202243940
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1, doi: 10.1051/0004-6361/202039657
  * Gibbons et al. (2017) Gibbons, S. L. J., Belokurov, V., & Evans, N. W. 2017, MNRAS, 464, 794, doi: 10.1093/mnras/stw2328
  * Helmi (2008) Helmi, A. 2008, A&A Rev., 15, 145, doi: 10.1007/s00159-008-0009-6
  * Helmi et al. (1999) Helmi, A., White, S. D. M., de Zeeuw, P. T., & Zhao, H. 1999, Nature, 402, 53, doi: 10.1038/46980
  * Helmi et al. (2003) Helmi, A., White, S. D. M., & Springel, V. 2003, MNRAS, 339, 834, doi: 10.1046/j.1365-8711.2003.06227.x
  * Ibata et al. (2021a) Ibata, R., Diakogiannis, F. I., Famaey, B., & Monari, G. 2021a, ApJ, 915, 5, doi: 10.3847/1538-4357/abfda9
  * Ibata et al. (2021b) Ibata, R., Malhan, K., Martin, N., et al. 2021b, ApJ, 914, 123, doi: 10.3847/1538-4357/abfcc2
  * Illingworth & Kittler (1988) Illingworth, J., & Kittler, J. 1988, Computer vision, graphics, and image processing, 44, 87
  * Ji et al. (2022) Ji, A. P., Naidu, R. P., Brauer, K., Ting, Y.-S., & Simon, J. D. 2022, arXiv e-prints, arXiv:2207.04016.  https://arxiv.org/abs/2207.04016
  * Kirby et al. (2013) Kirby, E. N., Cohen, J. G., Guhathakurta, P., et al. 2013, ApJ, 779, 102, doi: 10.1088/0004-637X/779/2/102
  * Koposov et al. (2010) Koposov, S. E., Rix, H.-W., & Hogg, D. W. 2010, ApJ, 712, 260, doi: 10.1088/0004-637X/712/1/260
  * Koposov et al. (2019) Koposov, S. E., Belokurov, V., Li, T. S., et al. 2019, MNRAS, 485, 4726, doi: 10.1093/mnras/stz457
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Malhan & Ibata (2019) Malhan, K., & Ibata, R. A. 2019, MNRAS, 486, 2995, doi: 10.1093/mnras/stz1035
  * Malhan et al. (2022) Malhan, K., Ibata, R. A., Sharma, S., et al. 2022, ApJ, 926, 107, doi: 10.3847/1538-4357/ac4d2a
  * Marchetti et al. (2019) Marchetti, T., Rossi, E. M., & Brown, A. G. A. 2019, MNRAS, 490, 157, doi: 10.1093/mnras/sty2592
  * McMillan (2017) McMillan, P. J. 2017, MNRAS, 465, 76, doi: 10.1093/mnras/stw2759
  * Recio-Blanco et al. (2022) Recio-Blanco, A., de Laverny, P., Palicio, P. A., et al. 2022, arXiv e-prints, arXiv:2206.05541.  https://arxiv.org/abs/2206.05541
  * Robin et al. (2012) Robin, A. C., Luri, X., Reylé, C., et al. 2012, A&A, 543, A100, doi: 10.1051/0004-6361/201118646
  * Sanders & Binney (2013) Sanders, J. L., & Binney, J. 2013, MNRAS, 433, 1826, doi: 10.1093/mnras/stt816
  * Schönrich et al. (2010) Schönrich, R., Binney, J., & Dehnen, W. 2010, MNRAS, 403, 1829, doi: 10.1111/j.1365-2966.2010.16253.x
  * Shipp et al. (2018) Shipp, N., Drlica-Wagner, A., Balbinot, E., et al. 2018, ApJ, 862, 114, doi: 10.3847/1538-4357/aacdab
  * Vasiliev (2019) Vasiliev, E. 2019, MNRAS, 482, 1525, doi: 10.1093/mnras/sty2672
  * Vasiliev et al. (2021) Vasiliev, E., Belokurov, V., & Erkal, D. 2021, MNRAS, 501, 2279, doi: 10.1093/mnras/staa3673
  * Wang et al. (2022) Wang, C., Huang, Y., Yuan, H., et al. 2022, ApJS, 259, 51, doi: 10.3847/1538-4365/ac4df7
  * Widmark et al. (2021) Widmark, A., de Salas, P. F., & Monari, G. 2021, A&A, 646, A67, doi: 10.1051/0004-6361/202039852
  * Yuan et al. (2020) Yuan, Z., Myeong, G. C., Beers, T. C., et al. 2020, ApJ, 891, 39, doi: 10.3847/1538-4357/ab6ef7


Misra Ghosh

1]Aryabhatta Research Institute of Observational Sciences (ARIES), Manora
Peak, Nainital-263002, India. 2]Department of Physics, Deen Dayal Upadhyaya
Gorakhpur University, Gorakhpur-273009, India. 3]School of Studies in Physics
and Astrophysics, Pandit Ravishankar Shukla University, Raipur 492010,
Chattisgarh, India

<EMAIL_ADDRESS>
# Insights into the properties of GRBs with TeV emission

Kuntal  Dimple  Ankur  [  [  [

(13th October 2020)

###### Abstract

This study investigates the environments and characteristics of Gamma-Ray
Bursts (GRBs) exhibiting very high energy (VHE) emission. Recent detections of
VHE emission, up to TeV energies, challenge synchrotron-only emission models
and particle acceleration concepts in GRBs. Until now, only a handful of GRBs
have been detected in the VHE range. We compare the number densities of the
circumburst medium of VHE-detected GRBs to check if the environment impacts
the VHE emission. This shows that these GRBs have environments similar to the
larger population of GRBs. We employ machine learning algorithms to create
two-dimensional embeddings of GRB prompt emission light curves from the Swift-
BAT catalog. VHE-detected GRBs are located across the map, indicating that VHE
emission does not favour any particular cluster. These findings indicate that
VHE-detected GRBs do not show any peculiar characteristics other than the
observational detection of VHE photons. Future detections will increase the
sample size required for a rigorous understanding of the origin of VHE
emission in GRBs.

###### keywords:

GRBs, VHE emission, emission mechanisms, environments, Machine Learning

##  1 Introduction

Gamma-Ray Bursts (GRBs) are brief, luminous flashes of gamma-rays arising from
extreme cataclysmic events in the universe. Though GRBs have been studied
extensively for decades, the emission mechanisms powering them remain
uncertain. The relativistic fireball shock model is widely accepted, where a
relativistic outflow dissipates its kinetic energy through internal or
external shocks, producing GRB emission (Piran, 2005; Kumar and Zhang, 2015).
But details of the microphysical dissipation and radiation processes are still
debated. Suggested emission mechanisms invoke dissipation through internal or
external shocks. Broadband spectra and variability of GRB prompt emission
indicate that non-thermal radiation mechanisms involving relativistic
particles play a crucial role.

Recently, TeV emission has been detected from GRB afterglows using the High
Energy Stereoscopic System (H.E.S.S.) and Major Atmospheric Gamma Imaging
Cherenkov (MAGIC) telescopes. Observations reveal the complex nature of the
afterglow, which does not follow expected relations. In the case of GRB
190114C, the model required the shock microphysical parameters to evolve with
time to explain the afterglow evolution (Misra et al., 2021). Detection of VHE
emission from some GRBs, extending into the TeV range, presents novel
challenges for GRB models and particle acceleration concepts. Existing
afterglow models cannot explain the production of TeV photons. Proposed
scenarios for the VHE emission include synchrotron self-Compton, proton
synchrotron radiation, or the decay of secondary particles from photohadronic
interactions (Zhang and Mészáros, 2001; Beniamini et al., 2015; Vietri, 1997;
Böttcher and Dermer, 1998). It remains unclear if these mechanisms can fully
explain VHE-detected GRBs and if they possess exceptionally distinct
properties. Comparing the environments and cluster structure of GRBs with and
without VHE emission could determine what makes these VHE bursts unique.

In this work, we study whether GRBs that exhibit VHE emission have
distinguishing characteristics by probing their environments and examining
similarities between their light curves using machine learning techniques.

##  2 Are VHE-detected GRBs similar?

Thus far, VHE emission has been reported in six GRBs (Fraija et al., 2019;
MAGIC Collaboration et al., 2019; H. E. S. S. Collaboration et al., 2021;
Blanch et al., 2020a, b; Dzhappuev et al., 2022). The redshift z, the $T_{90}$
duration in sec, isotropic equivalent energy ($E_{iso}$) in erg, the maximum
energy of the photon detected, and the facility which made the detection are
listed in Table 1. Understanding the physical mechanisms that generate VHE
emission in these sources can provide valuable insights into the phenomenon of
this emission from GRBs in general. A key question is whether GRBs that
exhibit emission at such extreme energies share characteristics that
distinguish them from the overall GRB population.

To investigate this, we studied and compared the properties and environments
of the VHE-detected GRBs summarised in Table 1. Further, machine learning
algorithms can examine the fine structures in the GRB light curves and cluster
them together based on similarities and dissimilarities between them. We
analysed the locations of these GRBs in two-dimensional embeddings created
using machine learning algorithms applied to Swift-BAT catalog
data111https://swift.gsfc.nasa.gov/results/batgrbcat/ (Lien et al., 2016). The
locations of the GRBs in these embeddings allow us to determine if GRBs with
VHE emission tend to lie within the same cluster. Any preferential clustering
would suggest common underlying factors that can enable their VHE emission
(Dimple et al., 2023).

Table 1: Properties of the TeV detected GRBs

GRB | redshift | $T_{90}$ | $E_{\rm iso}$ | Maximum photon | TeV detection | References
---|---|---|---|---|---|---
|  |  |  | energy | facility |
| (z) | (s) | (erg) | (TeV) |  |
GRB 180720B | 0.654 | $51.1\pm 3.0$ | $6.82_{-0.22}^{+0.24}\times 10^{53}$ | 0.44 | H.E.S.S. | Vreeswijk et al. (2018); Fraija et al. (2019)
GRB 190114C | 0.425 |  $\sim$ 116 | $3.5\pm 0.1\times 10^{53}$ | 1 | MAGIC | MAGIC Collaboration et al. (2019)
GRB 190829A | 0.078 | $57\pm 3$ | $\sim 2.0\times 10^{50}$ | 3.3 | H.E.S.S. | H. E. S. S. Collaboration et al. (2021)
GRB 201015A | 0.426 | $9.78\pm 3.47$ | $\sim 3.86\times 10^{51}$ | - | MAGIC | Blanch et al. (2020a)
GRB 201216C | 1.1 | $29.95\pm 0.57$ | $\sim 6.32\times 10^{53}$ | - | MAGIC | Blanch et al. (2020b)
GRB 221009A† | 0.151 | $1068.40\pm 13.38$ | $\sim 1.2\times 10^{55}$ | 251 | Carpet-2 | Dzhappuev et al. (2022)

† Also detected by MAGIC

###  2.1 Environments

The densities of the circumburst medium in GRB environments can vary greatly
in the VHE regime. GRBs 180720B, 190114C and 221009A have circumburst medium
densities around or below 0.1 cm-3 (Guarini et al., 2023). Tentative
interpretation of radio, optical, and X-ray data suggest even lower
circumburst medium densities of approximately 0.1 cm-3. In contrast, GRB
201015A has a mild-relativistic jet surrounded by a very dense medium
(n=1202.3 cm-3, Zhang et al. 2023), while GRB 201216C has an ultra-
relativistic jet surrounded by a moderately dense medium (n=5 cm-3, Zhang et
al. 2023). GRB 190829A also has a moderate environmental density (n=15 cm-3,
Zhang et al. 2021). Figure 1 shows the relation between the number density and
kinetic energy of GRBs with the VHE detected GRBs highlighted with different
colors. We notice that these GRBs follow the general trend of the GRB
population, do not exhibit any peculiar behavior, and show no preference for a
specific environment. This suggests that VHE GRBs behave similarly to other
GRBs regarding their number density and kinetic energy relationship. These
findings emphasise the need for further observations and analysis to
comprehend the properties of their environment and the implications for
understanding these captivating astrophysical phenomena.

Figure 1: The relation between the number density and kinetic energy of GRBs
is shown in this plot, where the VHE-detected GRBs are indicated with
different colors. These GRBs are seen to follow the general trend along with
the larger GRB population.

###  2.2 Locations in two-dimensional embeddings

Studying populations of GRBs detected by an instrument and analysing their
corresponding prompt emission light curves may help identify any structures
within GRB populations. This could provide clues on different GRB progenitor
classes. Analyzing large numbers of GRB light curves makes the analysis
difficult due to high dimensionality. Machine learning algorithms like
t-distributed Stochastic Neighbor Embedding (tSNE; van der Maaten and Hinton,
2008; van der Maaten, 2014) and Uniform Manifold Approximation and Projection
(UMAP; McInnes et al., 2018) with Principal Component Analysis (PCA;
Hotelling, 1933) initialisation can be used to reduce the dimensionality of
the data as well as to visualise the local and global structures within the
data. tSNE is a nonlinear dimensionality reduction method that visualises data
by embedding high-dimensional neighborhoods stochastically. It minimises the
differences between probability distributions in high and lower-dimensional
spaces using Kullback-Leibler divergence. On the other hand, UMAP utilises
concepts from topology to construct a high-dimensional representation of the
data using a similarity matrix. It then leverages these topological
relationships to find a low-dimensional projection that preserves distances
between points, effectively capturing both local and global structures in the
data.

Figure 2: The location of VHE-detected GRBs in two-dimensional embedding
obtained using PCA-UMAP over the Swift-BAT sample. The different colors
indicate the clusters identified by AutoGMM.

We employ both tSNE and UMAP with PCA initialisation (PCA-tSNE/PCA-UMAP) to
cluster GRB prompt emission light curves. For this, we use the light curves
from the Swift-BAT catalog 2022 available in four energy bins (15-25, 25-50,
50-100, and 100-350 keV) with a temporal resolution of 64 ms. This catalog
includes light curves of 1525 GRBs, detected between December 17, 2004, and
July 15, 2022. The catalog for 2022 consists of light curves of 1525 GRBs
detected between December 17, 2004, and July 15, 2022. The data show
variations in burst duration, fluence, and start times that can impact machine
learning algorithms. Therefore, we standardised the data, which involves
normalising the light curves with fluence, shifting them to the same start
time, padding them with zeros to match length, and performing the Discrete-
time Fourier Transform (DTFT) to retain time delay information between light
curves. We could standardise the data only for 1450 GRBs since fluence
information was sometimes missing. Next, we performed PCA on the standardised
dataset and estimated the number of PCs, preserving around 99% of the
variance, and utilised those for further dimensionality reduction using tSNE
and UMAP algorithms. These algorithms were optimised by tuning their key
hyperparameters, such as perplexity in tSNE and n_neighbors and min_dist in
UMAP. We used a perplexity=25, n_neighbors=25, and min_dist=0.01, obtained
after several iterations, to generate two-dimensional embeddings. The
embeddings were further subjected to the Auto Gaussian Mixture Model (AutoGMM,
Athey et al. 2019) algorithm to identify different clusters present. The
detailed description can be found in Dimple et al. (2023).

Figure 2 shows the two-dimensional embeddings obtained using PCA-UMAP with the
locations of VHE-detected GRBs. Similar results are found with PCA-tSNE. The
GRBs are located all over the map, giving indications that these GRBs do not
cluster in a particular subgroup and do not bear similarities with each other.
However, their spread in location all over the map hints that these could be
distributed amongst the GRB population, with the only difference being their
detection in very high energy regimes.

##  3 Summary

In this work, we investigated the properties and environments of GRBs that
exhibit VHE emission, a phenomenon that challenges current GRB models and
particle acceleration concepts. We have compared the number density and
kinetic energy of these GRBs with the overall GRB population and found no
significant difference or any peculiar behavior. We have applied machine
learning techniques tSNE and UMAP with PCA initialisation (PCA-tSNE/PCA-UMAP)
to cluster GRB prompt emission light curves. We created two-dimensional
embeddings of GRB prompt emission light curves from the Swift-BAT catalog and
located the VHE-detected GRBs on these maps. We have found that these GRBs are
distributed across the map and do not cluster in any particular subgroup,
indicating that they do not share similarities among themselves or with any
specific progenitor class. Our findings suggest that VHE-detected GRBs behave
similarly to other GRBs in terms of their number density, kinetic energy, and
light curve morphology and that any distinctive environmental or physical
factors do not influence their VHE emission. A larger sample of VHE-detected
GRBs is needed to confirm these results and to explore further the origin and
mechanism of their high energy emission extending out to TeV bands. Future
detections of VHE in GRBs with sensitive detectors will provide more data to
advance our understanding of these GRBs.

###### Acknowledgements.

The authors thank the referee for providing constructive comments on the
manuscript. The authors thank Prof. K. G. Arun and Prof. L. Resmi for their
useful discussions.

0000-0003-1637-267XKuntalMisra 0000-0001-9868-9042Dimple
0000-0003-2265-0381AnkurGhosh

All authors in this work have made significant contributions.

The authors declare no conflict of interest.

## References

  * Athey et al. (2019) Athey, T. L., Liu, T., Pedigo, B. D. and Vogelstein, J. T. (2019) AutoGMM: Automatic and Hierarchical Gaussian Mixture Modeling in Python.  arXiv e-prints, arXiv:1909.02688.
  * Beniamini et al. (2015) Beniamini, P., Nava, L., Duran, R. B. and Piran, T. (2015) Energies of GRB blast waves and prompt efficiencies as implied by modelling of X-ray and GeV afterglows.  MNRAS, 454(1), 1073–1085.  https://doi.org/10.1093/mnras/stv2033.
  * Blanch et al. (2020a) Blanch, O., Gaug, M., Noda, K., Berti, A., Moretti, E., Miceli, D., Gliwny, P., Ubach, S., Schleicher, B., Cerruti, M., Stamerra, A. and MAGIC Collaboration (2020a) MAGIC observations of GRB 201015A: hint of very high energy gamma-ray signal.  GCN, 28659, 1.
  * Blanch et al. (2020b) Blanch, O., Longo, F., Berti, A., Fukami, S., Suda, Y., Loporchio, S., Micanovic, S., Green, J. G., Pinter, V., Takahashi, M. and MAGIC Collaboration (2020b) GRB 201216C: MAGIC detection in very high energy gamma rays.  GCN, 29075, 1.
  * Böttcher and Dermer (1998) Böttcher, M. and Dermer, C. (1998) High Energy Gamma Rays from Ultrahigh Energy Cosmic Ray Protons in Gamma Ray Bursts.  In APS April Meeting Abstracts, APS Meeting Abstracts, p. R8.06.
  * Dimple et al. (2023) Dimple, Misra, K. and Arun, K. G. (2023) Evidence for Two Distinct Populations of Kilonova-associated Gamma-Ray Bursts.  ApL, 949(2), L22.  https://doi.org/10.3847/2041-8213/acd4c4.
  * Dzhappuev et al. (2022) Dzhappuev, D. D., Afashokov, Y. Z., Dzaparova, I. M., Dzhatdoev, T. A., Gorbacheva, E. A., Karpikov, I. S., Khadzhiev, M. M., Klimenko, N. F., Kudzhaev, A. U., Kurenya, A. N., Lidvansky, A. S., Mikhailova, O. I., Petkov, V. B., Podlesnyi, E. I., Pozdnukhov, N. A., Romanenko, V. S., Rubtsov, G. I., Troitsky, S. V., Unatlokov, I. B., Vaiman, I. A., Yanin, A. F. and Zhuravleva, K. V. (2022) Swift J1913.1+1946/GRB 221009A: detection of a 250-TeV photon-like air shower by Carpet-2.  ATel, 15669, 1.
  * Fraija et al. (2019) Fraija, N., Dichiara, S., Pedreira, A. C. C. d. E. S., Galvan-Gamez, A., Becerra, R. L., Montalvo, A., Montero, J., Betancourt Kamenetskaia, B. and Zhang, B. B. (2019) Modeling the Observations of GRB 180720B: from Radio to Sub-TeV Gamma-Rays.  ApJ, 885(1), 29.  https://doi.org/10.3847/1538-4357/ab3e4b.
  * Guarini et al. (2023) Guarini, E., Tamborra, I., Bégué, D. and Rudolph, A. (2023) Probing gamma-ray bursts observed at very high energies through their afterglow.  MNRAS, 523(1), 149–162.  https://doi.org/10.1093/mnras/stad1421.
  * H. E. S. S. Collaboration et al. (2021) H. E. S. S. Collaboration, Abdalla, H., Aharonian, F., Ait Benkhali, F., Angüner, E. O., Arcaro, C., Armand, C., Armstrong, T., Ashkar, H., Backes, M., Baghmanyan, V., Barbosa Martins, V., Barnacka, A., Barnard, M., Becherini, Y., Berge, D., Bernlöhr, K., Bi, B., Bissaldi, E., Böttcher, M., Boisson, C., Bolmont, J., de Bony de Lavergne, M., Breuhaus, M., Brun, F., Brun, P., Bryan, M., Büchele, M., Bulik, T., Bylund, T., Caroff, S., Carosi, A., Casanova, S., Chand, T., Chandra, S., Chen, A., Cotter, G., Curyło, M., Damascene Mbarubucyeye, J., Davids, I. D., Davies, J., Deil, C., Devin, J., Dirson, L., Djannati-Ataï, A., Dmytriiev, A., Donath, A., Doroshenko, V., Dreyer, L., Duffy, C., Dyks, J., Egberts, K., Eichhorn, F., Einecke, S., Emery, G., Ernenwein, J. P., Feijen, K., Fegan, S., Fiasson, A., Fichet de Clairfontaine, G., Fontaine, G., Funk, S., Füßling, M., Gabici, S., Gallant, Y. A., Giavitto, G., Giunti, L., Glawion, D., Glicenstein, J. F., Grondin, M. H., Hahn, J., Haupt, M., Hermann, G., Hinton, J. A., Hofmann, W., Hoischen, C., Holch, T. L., Holler, M., Hörbe, M., Horns, D., Huber, D., Jamrozy, M., Jankowsky, D., Jankowsky, F., Jardin-Blicq, A., Joshi, V., Jung-Richardt, I., Kasai, E., Kastendieck, M. A., Katarzyński, K., Katz, U., Khangulyan, D., Khélifi, B., Klepser, S., Kluźniak, W., Komin, N., Konno, R., Kosack, K., Kostunin, D., Kreter, M., Lamanna, G., Lemière, A., Lemoine-Goumard, M., Lenain, J. P., Leuschner, F., Levy, C., Lohse, T., Lypova, I., Mackey, J., Majumdar, J., Malyshev, D., Malyshev, D., Marandon, V., Marchegiani, P., Marcowith, A., Mares, A., Martí-Devesa, G., Marx, R., Maurin, G., Meintjes, P. J., Meyer, M., Mitchell, A., Moderski, R., Mohrmann, L., Montanari, A., Moore, C., Morris, P., Moulin, E., Muller, J., Murach, T., Nakashima, K., Nayerhoda, A., de Naurois, M., Ndiyavala, H., Niemiec, J., Oakes, L., O’Brien, P., Odaka, H., Ohm, S., Olivera-Nieto, L., de Ona Wilhelmi, E., Ostrowski, M., Panny, S., Panter, M., Parsons, R. D., Peron, G., Peyaud, B., Piel, Q., Pita, S., Poireau, V., Priyana Noel, A., Prokhorov, D. A., Prokoph, H., Pühlhofer, G., Punch, M., Quirrenbach, A., Raab, S., Rauth, R., Reichherzer, P., Reimer, A., Reimer, O., Remy, Q., Renaud, M., Rieger, F., Rinchiuso, L., Romoli, C., Rowell, G., Rudak, B., Ruiz-Velasco, E., Sahakian, V., Sailer, S., Salzmann, H., Sanchez, D. A., Santangelo, A., Sasaki, M., Scalici, M., Schäfer, J., Schüssler, F., Schutte, H. M., Schwanke, U., Seglar-Arroyo, M., Senniappan, M., Seyffert, A. S., Shafi, N., Shapopi, J. N. S., Shiningayamwe, K., Simoni, R., Sinha, A., Sol, H., Specovius, A., Spencer, S., Spir-Jacob, M., Stawarz, Ł., Sun, L., Steenkamp, R., Stegmann, C., Steinmassl, S., Steppa, C., Takahashi, T., Tam, T., Tavernier, T., Taylor, A. M., Terrier, R., Thiersen, J. H. E., Tiziani, D., Tluczykont, M., Tomankova, L., Tsirou, M., Tuffs, R., Uchiyama, Y., van der Walt, D. J., van Eldik, C., van Rensburg, C., van Soelen, B., Vasileiadis, G., Veh, J., Venter, C., Vincent, P., Vink, J., Völk, H. J., Wadiasingh, Z., Wagner, S. J., Watson, J., Werner, F., White, R., Wierzcholska, A., Wong, Y. W., Yusafzai, A., Zacharias, M., Zanin, R., Zargaryan, D., Zdziarski, A. A., Zech, A., Zhu, S. J., Zorn, J., Zouari, S., Żywucka, N., Evans, P. and Page, K. (2021) Revealing x-ray and gamma ray temporal and spectral similarities in the GRB 190829A afterglow.  Sci, 372(6546), 1081–1085.  https://doi.org/10.1126/science.abe8560.
  * Hotelling (1933) Hotelling, H. (1933) Analysis of a complex of statistical variables into principal components.  JEP, 24(6), 417–441.  https://doi.org/10.1037/h0071325.
  * Kumar and Zhang (2015) Kumar, P. and Zhang, B. (2015) The physics of gamma-ray bursts & relativistic jets.  PhR, 561, 1–109.  https://doi.org/10.1016/j.physrep.2014.09.008.
  * Lien et al. (2016) Lien, A., Sakamoto, T., Barthelmy, S. D., Baumgartner, W. H., Cannizzo, J. K., Chen, K., Collins, N. R., Cummings, J. R., Gehrels, N., Krimm, H. A., Markwardt, C. B., Palmer, D. M., Stamatikos, M., Troja, E. and Ukwatta, T. N. (2016) The Third Swift Burst Alert Telescope Gamma-Ray Burst Catalog.  ApJ, 829(1), 7.  https://doi.org/10.3847/0004-637X/829/1/7.
  * MAGIC Collaboration et al. (2019) MAGIC Collaboration, Acciari, V. A., Ansoldi, S., Antonelli, L. A., Engels, A. A., Baack, D., Babić, A., Banerjee, B., Barres de Almeida, U., Barrio, J. A., Becerra González, J., Bednarek, W., Bellizzi, L., Bernardini, E., Berti, A., Besenrieder, J., Bhattacharyya, W., Bigongiari, C., Biland, A., Blanch, O., Bonnoli, G., Bošnjak, Ž., Busetto, G., Carosi, R., Ceribella, G., Chai, Y., Chilingaryan, A., Cikota, S., Colak, S. M., Colin, U., Colombo, E., Contreras, J. L., Cortina, J., Covino, S., D’Elia, V., da Vela, P., Dazzi, F., de Angelis, A., de Lotto, B., Delfino, M., Delgado, J., Depaoli, D., di Pierro, F., di Venere, L., Do Souto Espiñeira, E., Dominis Prester, D., Donini, A., Dorner, D., Doro, M., Elsaesser, D., Fallah Ramazani, V., Fattorini, A., Ferrara, G., Fidalgo, D., Foffano, L., Fonseca, M. V., Font, L., Fruck, C., Fukami, S., García López, R. J., Garczarczyk, M., Gasparyan, S., Gaug, M., Giglietto, N., Giordano, F., Godinović, N., Green, D., Guberman, D., Hadasch, D., Hahn, A., Herrera, J., Hoang, J., Hrupec, D., Hütten, M., Inada, T., Inoue, S., Ishio, K., Iwamura, Y., Jouvin, L., Kerszberg, D., Kubo, H., Kushida, J., Lamastra, A., Lelas, D., Leone, F., Lindfors, E., Lombardi, S., Longo, F., López, M., López-Coto, R., López-Oramas, A., Loporchio, S., Machado de Oliveira Fraga, B., Maggio, C., Majumdar, P., Makariev, M., Mallamaci, M., Maneva, G., Manganaro, M., Mannheim, K., Maraschi, L., Mariotti, M., Martínez, M., Mazin, D., Mićanović, S., Miceli, D., Minev, M., Miranda, J. M., Mirzoyan, R., Molina, E., Moralejo, A., Morcuende, D., Moreno, V., Moretti, E., Munar-Adrover, P., Neustroev, V., Nigro, C., Nilsson, K., Ninci, D., Nishijima, K., Noda, K., Nogués, L., Nozaki, S., Paiano, S., Palatiello, M., Paneque, D., Paoletti, R., Paredes, J. M., Peñil, P., Peresano, M., Persic, M., Moroni, P. G. P., Prandini, E., Puljak, I., Rhode, W., Ribó, M., Rico, J., Righi, C., Rugliancich, A., Saha, L., Sahakyan, N., Saito, T., Sakurai, S., Satalecka, K., Schmidt, K., Schweizer, T., Sitarek, J., Šnidarić, I., Sobczynska, D., Somero, A., Stamerra, A., Strom, D., Strzys, M., Suda, Y., Surić, T., Takahashi, M., Tavecchio, F., Temnikov, P., Terzić, T., Teshima, M., Torres-Albà, N., Tosti, L., Vagelli, V., van Scherpenberg, J., Vanzo, G., Vazquez Acosta, M., Vigorito, C. F., Vitale, V., Vovk, I., Will, M., Zarić, D., Nava, L., Veres, P., Bhat, P. N., Briggs, M. S., Cleveland, W. H., Hamburg, R., Hui, C. M., Mailyan, B., Preece, R. D., Roberts, O. J., von Kienlin, A., Wilson-Hodge, C. A., Kocevski, D., Arimoto, M., Tak, D., Asano, K., Axelsson, M., Barbiellini, G., Bissaldi, E., Dirirsa, F. F., Gill, R., Granot, J., McEnery, J., Omodei, N., Razzaque, S., Piron, F., Racusin, J. L., Thompson, D. J., Campana, S., Bernardini, M. G., Kuin, N. P. M., Siegel, M. H., Cenko, S. B., O’Brien, P., Capalbi, M., Daı, A., de Pasquale, M., Gropp, J., Klingler, N., Osborne, J. P., Perri, M., Starling, R. L. C., Tagliaferri, G., Tohuvavohu, A., Ursi, A., Tavani, M., Cardillo, M., Casentini, C., Piano, G., Evangelista, Y., Verrecchia, F., Pittori, C., Lucarelli, F., Bulgarelli, A., Parmiggiani, N., Anderson, G. E., Anderson, J. P., Bernardi, G., Bolmer, J., Caballero-García, M. D., Carrasco, I. M., Castellón, A., Castro Segura, N., Castro-Tirado, A. J., Cherukuri, S. V., Cockeram, A. M., D’Avanzo, P., di Dato, A., Diretse, R., Fender, R. P., Fernández-García, E., Fynbo, J. P. U., Fruchter, A. S., Greiner, J., Gromadzki, M., Heintz, K. E., Heywood, I., van der Horst, A. J., Hu, Y. D., Inserra, C., Izzo, L., Jaiswal, V., Jakobsson, P., Japelj, J., Kankare, E., Kann, D. A., Kouveliotou, C., Klose, S., Levan, A. J., Li, X. Y., Lotti, S., Maguire, K., Malesani, D. B., Manulis, I., Marongiu, M., Martin, S., Melandri, A., Michałowski, M. J., Miller-Jones, J. C. A., Misra, K., Moin, A., Mooley, K. P., Nasri, S., Nicholl, M., Noschese, A., Novara, G., Pandey, S. B., Peretti, E., Pérez Del Pulgar, C. J., Pérez-Torres, M. A., Perley, D. A., Piro, L., Ragosta, F., Resmi, L., Ricci, R., Rossi, A., Sánchez-Ramírez, R., Selsing, J., Schulze, S., Smartt, S. J., Smith, I. A., Sokolov, V. V., Stevens, J., Tanvir, N. R., Thöne, C. C., Tiengo, A., Tremou, E., Troja, E., de Ugarte Postigo, A., Valeev, A. F., Vergani, S. D., Wieringa, M., Woudt, P. A., Xu, D., Yaron, O. and Young, D. R. (2019) Observation of inverse Compton emission from a long $\gamma$-ray burst.  Natur, 575(7783), 459–463.  https://doi.org/10.1038/s41586-019-1754-6.
  * McInnes et al. (2018) McInnes, L., Healy, J. and Melville, J. (2018) UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction.  arXiv e-prints, arXiv:1802.03426.
  * Misra et al. (2021) Misra, K., Resmi, L., Kann, D. A., Marongiu, M., Moin, A., Klose, S., Bernardi, G., de Ugarte Postigo, A., Jaiswal, V. K., Schulze, S., Perley, D. A., Ghosh, A., Dimple, Kumar, H., Gupta, R., Michałowski, M. J., Martín, S., Cockeram, A., Cherukuri, S. V., Bhalerao, V., Anderson, G. E., Pandey, S. B., Anupama, G. C., Thöne, C. C., Barway, S., Wieringa, M. H., Fynbo, J. P. U. and Habeeb, N. (2021) Low frequency view of GRB 190114C reveals time varying shock micro-physics.  MNRAS, 504(4), 5685–5701.  https://doi.org/10.1093/mnras/stab1050.
  * Piran (2005) Piran, T. (2005) The physics of gamma-ray bursts.  RvMP, 76, 1143–1210.  https://doi.org/10.1103/RevModPhys.76.1143.
  * van der Maaten (2014) van der Maaten, L. (2014) Accelerating t-sne using tree-based algorithms.  JMLR, 15(93), 3221–3245.
  * van der Maaten and Hinton (2008) van der Maaten, L. and Hinton, G. (2008) Visualizing data using t-sne.  JMLR, 9(86), 2579–2605.
  * Vietri (1997) Vietri, M. (1997) Gev photons from ultrahigh energy cosmic rays accelerated in gamma ray bursts.  PhRvL, 78, 4328–4331.  https://doi.org/10.1103/PhysRevLett.78.4328.
  * Vreeswijk et al. (2018) Vreeswijk, P. M., Kann, D. A., Heintz, K. E., de Ugarte Postigo, A., Milvang-Jensen, B., Malesani, D. B., Covino, S., Levan, A. J. and Pugliese, G. (2018) GRB 180720B: VLT/X-shooter redshift.  GCN, 22996, 1.
  * Zhang and Mészáros (2001) Zhang, B. and Mészáros, P. (2001) High-Energy Spectral Components in Gamma-Ray Burst Afterglows.  ApJ, 559(1), 110–122.  https://doi.org/10.1086/322400.
  * Zhang et al. (2021) Zhang, L.-L., Ren, J., Huang, X.-L., Liang, Y.-F., Lin, D.-B. and Liang, E.-W. (2021) Nearby SN-associated GRB 190829A: Environment, Jet Structure, and VHE Gamma-Ray Afterglows.  ApJ, 917(2), 95.  https://doi.org/10.3847/1538-4357/ac0c7f.
  * Zhang et al. (2023) Zhang, L.-L., Ren, J., Wang, Y. and Liang, E.-W. (2023) Very-High-Energy Gamma-Ray Afterglows of GRB 201015A and GRB 201216C.  arXiv e-prints, arXiv:2305.00847.  https://doi.org/10.48550/arXiv.2305.00847.