# GRB Progenitor Classification from Gamma-Ray Burst Prompt and Afterglow
Observations

P. Nuessle Department of Physics, The George Washington University, 725 21st
St. NW, Washington, DC 20052  Astrophysics Science Division, NASA Goddard
Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771  J. L. Racusin
Astrophysics Science Division, NASA Goddard Space Flight Center, 8800
Greenbelt Rd, Greenbelt, MD 20771  N.E. White Department of Physics, The
George Washington University, 725 21st St. NW, Washington, DC 20052 P. Nuessle
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using an established classification technique, we leverage standard
observations and analyses to predict the progenitors of gamma-ray bursts
(GRBs). This technique, utilizing support vector machine (SVM) statistics,
provides a more nuanced prediction than the previous two-component Gaussian
mixture in duration of the prompt gamma-ray emission. Based on further
covariance testing from Fermi-GBM, Swift-BAT, and Swift-XRT data, we find that
our classification based only on prompt emission properties gives perspective
on the recent evidence that mergers and collapsars exist in both “long” and
“short” GRB populations.

Gamma-ray bursts (629), Support vector machine (1936), Astrostatistics
techniques (1886), Classification (1907)

##  1 Introduction

Until recently, GRB data and theory generally supported a Gaussian mixture of
GRB properties corresponding to two classes due to massive stellar collapse of
hydrogen- and helium-stripped stars, either triggered by supernova or
collision (collapsars, Crowther, 2008; Dessart et al., 2011; Woosley, 2011;
Woosley & Heger, 2012) and compact mergers including at least one neutron star
(mergers, Jespersen et al., 2020; Salmon et al., 2022; Tarnopolski, 2022;
Zhang et al., 2009). The prompt gamma-ray emission periods from mergers are
generally shorter than some instrument-dependent duration boundary, (generally
estimated $\sim$2 seconds) while those from collapsars are longer (Woosley,
1993). The collapsar mechanism was first confirmed by the association of SN
1998bw, a Type Ic supernova, with GRB 980425, a long burst (Galama et al.,
1998; Wang & Wheeler, 1998). Since then, dozens of supernova-associated long
GRBs have been discovered (Cano et al., 2014; Dainotti et al., 2022; Hjorth et
al., 2003; Pian et al., 2006). The merger hypothesis was originally supported
by the discovery of short bursts like GRB 050509B, which lay on the outskirts
of old elliptical galaxies (Gehrels et al., 2005; Lee et al., 2005). This
progenitor scenario was confirmed by the coincident detections of GW170817/GRB
170817A/AT 2017gfo, a binary neutron star merger and the association of an
r-process kilonova (Abbott et al., 2017a, b). GRB 170817A spurred a search for
more archival kilonova signatures following bursts, and some were found (Bulla
et al., 2023; Jin et al., 2020; Tanvir et al., 2013; Troja et al., 2022, 2019,
2018; Wang et al., 2017; Yang et al., 2015). However, there are now directly
observed classifications where kilonova signatures followed long-duration
bursts (long mergers, Bulla et al., 2023; Levan et al., 2023; Rastinejad et
al., 2022; Troja et al., 2018, 2022; Yang et al., 2022) and where one
supernova signature followed a short-duration bursts (short collapsars,
Ahumada et al., 2021a; Zhang et al., 2021).

We used these unexpected associations combined with the more familiar long
GRB-supernovae and short GRB-kilonovae to train our new classifier so that it
would not be wholly based only on the duration or hardness of the burst, but
rather trained on known associations via this list. The formation and
transformation of the progenitor system should cause profound observational
effects on the resulting GRB, making these ”muddled” bursts separable based on
other factors. A compact binary system is more likely to have migrated out of
its star-forming region during the long period between when the stars collapse
to form compact objects and when they merge, leaving the binary with little to
no surrounding gas. This causes the afterglow to be suppressed, as it is the
interactions with the environment that create broadband afterglow emission
(Gehrels et al., 2008; Metzger, 2017). This just one of many features we could
use to identify the progenitor system. It is unlikely that the two burst types
have different emission mechanisms given that they appear to have similar
relationships between their observed flux; luminosity; and peak, isotropic,
and collimation-corrected energies, but it is clear that higher peak energies
are generally associated with merger events (Ghirlanda et al., 2011; Zhang et
al., 2018).

The motivation to build one or more burst classifiers came directly from these
exceptions to the paradigm which assumed that collapsars were long and mergers
were short. Some recent methods have focused on the short kilonovae alone or
on searching for third (or fourth) classes of GRBs to explain the property
overlap (Dimple et al., 2023; Horváth et al., 2010; Jespersen et al., 2020).
We instead examine $E_{p,prompt}S_{\gamma,\text{prompt}}^{-1}$ vs.
$\text{T}_{\text{90}}$ as in Goldstein et al. (2010) for discriminatory power
(Dainotti & Amati, 2018). We find evidence for a continuum between the two
progenitor classes of GRBs.

In Sec. 2, we define our known-progenitor training sample and the rest of the
sample selection. In Sec. 3, we outline the support vector machine (SVM)
machine learning tools used in this paper. In Sec. 4, we determine the best
discriminator and used those factors to create a probabilistic SVM model and
attempt to quantify its covariates and accuracy. In Sec. 5, we discuss our
results and suggest where further work may be necessary. In Sec. 6, we
conclude.

The Hubble constant is assumed to be $H_{0}$=70 km s-1 Mpc-1. The significance
threshold is $\alpha=0.01$, which is approximately $2.32\sigma$
111https://www.scribbr.com/statistics/statistical-significance/.

##  2 Sample Selection

Prompt emission data were utilized from the Fermi-Gamma Ray Burst Monitor
(GBM) GRB catalog from 2017 August through 2023 May (von Kienlin et al.,
2020). This instrument was chosen due to its ability to characterise the
prompt emission energy spectrum over a wide bandpass and its large uniform
sample data set. We chose to use the X-ray afterglow (where available) to
study potential selection effects. This afterglow sample runs through the same
period of time and came from the Neil Gehrels Swift Observatory X-Ray
Telescope (XRT) (Evans et al., 2009). We included this data as a check on our
support vector machine for the two progenitors, as characterization of the
early afterglow, including the important plateau period, may be related to
continued energy injection into the circumburst environment (Gehrels et al.,
2008; Racusin et al., 2011). These samples are cross-associated through the
trigger times as matched through the Swift-Burst Alert Telescope (BAT) GRB
catalogs. When both published GRB catalogs and preliminary characterization
are included, all three data sets are complete through 2023 May
222https://swift.gsfc.nasa.gov/results/batgrbcat/#Contact,
https://swift.gsfc.nasa.gov/archive/grb_table/. We refer to this as the GBM-
BAT-XRT sample. If more XRT data were needed or we wanted to perform our own
data analysis directly on the Swift burst data, it was retrieved using the
swifttools python package as a burst analyzer
333https://www.swift.ac.uk/API/ukssdc/data/GRB.md. If more detailed GBM data
was required, it was downloaded from the GBM archive and processed using the
gbm-data-tools python package (Goldstein et al., 2022).

If only the prompt emission was needed, for instance when tentatively
predicting the progenitor of all bursts in the GBM GRB catalog, only the GBM
sample was used. However, when testing for distance limitations on the
classifier, a redshift-associated GBM sample combined with a subsample of the
overlapped GBM-BAT-XRT sample that has a measured redshift was used. We also
used an overlapped GBM-BAT-XRT sample with afterglow associations as a stand-
in test for the GRB environments (and therefore at least some burst
progenitors). These redshifts were found using the BAT redshift table and J.
Greiner’s GRB table 444https://swift.gsfc.nasa.gov/results/batgrbcat/#Contact,
https://www.mpe.mpg.de/~jcg/grbgen.html.

We refer duration to mean $\text{T}_{\text{90}}$ the time between which 5 and
95% of a burst’s fluence over 50 to 300 keV ($S_{\gamma,\text{prompt}}$) is
observed. All bursts referred to as long in this work are taken to have a
prompt emission duration longer than 4.2 seconds in von Kienlin et al. (2020),
and duration shorter than 4.2 seconds as ”short”. Many authors (e.g. Levan et
al. (2023)) now recognize that the duration separation between a ”long” and a
”short” burst may not always be two seconds as determined for the Compton
Gamma-ray Observatory (CGRO) Burst and Transient Spectrometer Experiment
(BATSE) sample (Fishman et al., 1985; Kouveliotou et al., 1993; Schaefer et
al., 1994). Instead, it may depend on the instrument, its trigger criteria,
energy range, and sensitivity as hypothesized by Bromberg et al. (2013). It is
also known to depend on the burst’s redshift, as time dilation affects the
duration, fluence, and counts detected above background. These authors found
that the $\text{T}_{\text{90}}$ at which a GRB is equally likely to be a
collapsar or merger is 0.8, 1.7, and 3.1 seconds in the BAT, the GBM, and
BATSE data, respectively (Barthelmy et al., 2005; von Kienlin et al., 2020;
Lien et al., 2016; Meegan et al., 2009). Taking this under advisement, we used
4.2 seconds, which the authors of the Fourth Fermi-GBM GRB Catalog fitted
using a lognormal fit to the $\text{T}_{\text{90}}$ distribution (von Kienlin
et al., 2020).

###  2.1 Detected Progenitors

In this section, we describe how we selected our detected progenitors. Each
class is treated separately.

####  2.1.1 Collapsar Selection

The presence of a localization-consistent supernova or late afterglow optical
light curve bump typical of a supernova is taken to represent a collapsar
origin for the gamma-ray burst. We used these characteristics to compile a
list of 42 typical ”long collapsars”, 1 ”short collapsar” (GRB 200826A), and
one notable ”exotic burst” which seemed to have features of both progenitor
types (GRB 210704A) by searching the literature and the GCN Circulars
555https://gcn.nasa.gov/circulars. The large number of supernovae can be
attributed to their relative brightness and extensive literature of detection.
The vast majority of collapsar bursts were taken from in Table 7 of Dainotti
et al. (2022). The four exceptions are GRB 221009A (Blanchard et al., 2023);
GRB 211023A, (Belkin et al., 2021); GRB 150210A(Jiang et al., 2023a); and GRB
200826A, which is the only short collapsar (Ahumada et al., 2021a). In total,
there were 43 bursts with associated supernovae used to train our classifier
our sample (Table A).

####  2.1.2 Merger Selection

The presence of an optical or infrared kilonova signature is taken to
represent a neutron star merger origin, which can be observed with either
distinct spectral features or a weak bump in the infrared light curve. We also
select short GRBs in the outskirts of their host galaxies as indicators of a
possible merger. There are fewer known merger events and their associations
can be more preliminary. Each of our selected merger associations has a unique
associated paper or GCN Circular (Table LABEL:kilonova_table). Nine of those
have direct kilonovae detections, meaning that a kilonova model normalized by
AT2017gfo could be fit to their infrared or optical data. These are the
progenitors of which we are most certain. The three bursts fit with a weak
kilonova may only have a bump in the IR associated with r-process
nucleosynthesis. Along with the burst with a short spectral lag (another
common feature of short GRBs, see: Joens, 2023; Ukwatta et al., 2010), these
are the bursts we have some confidence of being mergers. Finally, the eight
position inferred mergers are those of which we are least certain.

To increase the sample of known mergers, we included events from Fong et al.
(2022), who performed a detailed study of the host galaxies associated with
short GRBs. This has been proven in previous analyses such as Li et al. (2016)
and Li et al. (2020) to meaningfully distinguish between two populations of
gamma-ray bursts in the Greiner catalog. These authors similarly found that no
combination of factors may cleanly predict a burst’s progenitor, and the
primary relationship they find is redshift-dependent (p. 28 Li et al., 2016).
Their choices of effective amplitude (scaling down the signal to noise ratio
of the peak flux to the background to make the duration less than two seconds)
and brightness fraction (another measure of how active the star formation in
the GRB’s area of the galaxy is) rather than fluence and peak energy are also
themselves redshift-dependent. The calculations to transform the first through
different frames of reference are a corollary of the calculations in Sec. 4.
However, when they use a Naive Bayes classifier trained on the Greiner catalog
definition of Type I and II GRBs, they remove this dependence in favor of a
high-dimensional model in the prompt emission and galactic information (p. 6,
Li et al., 2020). We chose to instead simplify our model to three factors so
we could train it on the few known progenitors. From Ch. 14.9 of Tauris & van
den Heuvel (2023), we also assume that the majority of merger GRBs occur at
the outskirts of their host galaxies due to the large kick velocities from one
or both of the supernovae the formed the neutron stars. We therefore assumed
that short GRBs found more than a certain distance away from the center of
their host were likely mergers. For each detected host galaxy, the authors
performed positional and spectral analyses between the GRB and a catalog of
galaxies to determine a probability of chance coincidence between the GRB
afterglow and the host galaxy itself. The percent chance of coincident
association between the GRB and its host galaxy were low for our sample but
ranged up to 5% instead of 1%. We assume that if a GRB was more than 1.96
radii away from the center of its host galaxy, it was more likely to be in a
region of inactive star formation (Kruijssen et al., 2019). This value was
chosen because if the radial stellar light profile of a galaxy were Gaussian,
then 95% of the stellar mass density would be within this radius. While the
mass density of a galaxy is best represented by a Navarro–Frenk–White profile,
Portinari & Salucci (2010) found that a constant stellar mass over luminosity
ratio for an exponential radial luminosity profile would be a good zeroth-
order approximation to the real kinematics. This also means the luminosity
profile would not be too skewed towards the edges, meaning that a Gaussian
would be a good approximation. As a result, we could not test many GRBs, as
the effective radius of their host galaxy is unmeasured. As for the rest, it
was unclear if they simply had not moved from their area of formation or if
they were true short collapsars. We chose not to take them as mergers them
without more information on their origin. This netted us 8 further short-
merger type GRBs.

##  3 Methods

The goal of this classification is to differentiate between the two accepted
progenitor classes of GRBs. As there are a large number of possible input
variables but only a small amount of training objects (and two output
classes), simpler methods and statistical models are mathematically preferred.
To that end, we explored multiple machine learning methods before settling on
a support vector machine (SVM, Cortes & Vapnik, 1995; Vapnik, 1995). For
completeness, we include some of them here as possibilities for future work.
The primary classification method used in this paper is SVM as it has many
useful features in our case, as well as being a tested method in astrophysics
(Canty, 2009; Graff et al., 2016; Hartley et al., 2017; Mobina Hosseini et
al., 2023). An SVM can be understood as a classification based upon the
creation of a ($n-1$)-dimensional hyperplane in the $n$-dimensional geometry
created by the $n$ features of the objects to be classified (Vapnik, 1995).
This plane is defined by a kernel function that determines its shape and
properties–we used a radial function as the kernel to approximate the creation
of a neural network without requiring a large training set. SVM is ideal for a
large data set with a small training set, as it minimizes the training set as
part of its algorithm (Vapnik, 1995). Per Siemers & Bajorath (2023), it acts
consistently over many sizes of training sets. The GRBs with known
classification are then chosen to be the training data. Those members of the
training set that lie closest to the hyperplane become its ”support vectors,”
which dictate the hyperplane’s angle, curvature, and placement. SVM is not a
precise method, and it has no inherent classification confidence, though there
are affiliated methods such as Platt scaling and the relevance vector machine
which make this possible, at least in the binary case (Bishop & Tipping, 2013;
Platt, 2000). We chose this method with the application of Platt scaling
because in the binary case, it requires the fewest assumptions while
approximating more complex algorithms.

Figure 1: The prompt energy ratio
($E_{p,prompt}S_{\gamma,\text{prompt}}^{-1}$) is strongly correlated with
$T_{90}$, is a good separator of events with know progenitors, making it the
basis of our classification. The 4.2 second long/short duration demarcator
derived by the Fourth GBM GRB catalog as a dotted green line (von Kienlin et
al., 2020) demonstrates how duration alone is an insufficienct classifier.
Top: The unknown long and short bursts are unsorted. Middle: Our SVM binary
classification is applied to the sample. Bottom: We apply Platt scaling to the
classifier to make it probabilistic rather than binary, and highlight other
notable bursts with known progenitors. As the error bars make it difficult to
distinguish individual events, a version of this plot with error bars is in
App. B.

Other classification methods were also explored including random forest, which
shows promise as a future method when the training sample is significantly
larger. In this algorithm, thousands of decision trees are created and
evaluated based on how many groups they split the data into (Ukwatta et al.,
2016). The ”deeper” a tree is, (i.e. the more decisions it includes) the more
subtrees it includes, and the less each individual deep tree is weighted in
the result (Yang, 2016). This means that the algorithm can suffer from
overfitting if the trees are allowed to become too deep or there are too many.
However, boosting the classifier by making it a weighted linear combination of
multiple algorithms makes it more robust to this error (Canty, 2009). Random
forest is also computationally intensive when calculating a large number of
trees, and it is strongly suggested that the training sample be a significant
fraction of the test sample, as the training sample is used to bootstrap
smaller ”sub-training” samples for each tree to ensure that they are more
random than the overall model (Yang, 2016). We chose to discuss it even though
we did not have enough training data to properly utilize its power in testing.
As more known progenitors are found, it may be possible to use random forest
or gradient boosting trees to create more complex models than the one
presented in this paper. Some authors such as Luo et al. (2023), are already
using it. They use the Greiner table with some interpretation to distinguish
between Type I and Type II GRBs, rather than directly training on known
progenitors. This allows them enough data to tune the model. Our relative lack
of training data also limited our ability to use neural networks.

##  4 Analysis

Amati et al. (2002) observed a correlation between $E_{p,prompt}$ and
$E_{\gamma,iso}$ using rest-frame prompt spectral properties of GRBs. Updates
to the model later proposed that long and short GRBs might follow different
linear relationships (Amati, 2006). There is debate in the literature if the
so-called Amati relation is due to intrinsic or observational selection
effects (Collazzi et al., 2012; Kocevski, 2012), though some review articles
take it to be at least partially intrinsic to the system (Parsotan & Ito,
2022). However, when taken to the observer frame, Goldstein et al. (2010)
found that the Amati relation in BATSE data had a suggestive relationship to
features also known to partially differentiate between progenitors such as
burst duration and hardness ratio. We take this ”prompt energy ratio” feature
to be $E_{p,prompt}S_{\gamma,\text{prompt}}^{-1}$. After observing the
progenitor classes’ visual separation in Fig. 1, we created an SVM based on
the prompt emission variables $E_{p,prompt}S_{\gamma,\text{prompt}}^{-1}$ and
duration for easier visualization in two dimensions. In three dimensions, it
can be more easily manipulated as $E_{p,prompt}$, $S_{\gamma,\text{prompt}}$,
and duration.

Figure 2: Each of the greyscale curves is a different simulated burst with the
same spectral model, but a different $\text{T}_{\text{90}}$ in the rest frame.
We simulated their observed properties from $0.01\leq\text{z}\leq 8$. This
simulation demonstrates that distance alone does not explain all observed
progenitor relationships in Fig. 1 nor most of the uncertainty. It does so as
the majority of the simulated data is found to run perpendicular to the
relationship, rather than along it. The spectral values chosen ($E_{p}$=150
keV, $E_{\gamma,iso}=3\times 10^{52}$ ergs, and modeled using a Band spectrum
with $\alpha=-0.8$ and $\beta=-2.76$) were the median GBM catalog values for
short bursts fitted with a Band or SBPL function.

The advantage of the prompt energy ratio-$\text{T}_{\text{90}}$ correlation is
that it includes many bursts from both progenitor classes because the
selection criteria are very broad and only come from the observed-frame prompt
emission. The model is difficult to interpret physically but is strongly
related to the Amati and Ghirlanda relations (Goldstein et al., 2010).
However, both relations may be instrument-dependent (Collazzi et al., 2012;
Kocevski, 2012; Dainotti & Amati, 2018), making it possible that the proposed
classifier should be retrained each time it is applied to an instrument with a
different bandpass or spectral fit. As derived by Band & Preece (2005),
$E_{p,prompt}^{2}/S$ should be weakly related to redshift, z. Goldstein et al.
(2010) then simplifies this ratio to
$E_{p,prompt}S_{\gamma,\text{prompt}}^{-1}$, finding that it creates a
Gaussian mixture in BATSE bursts. When we then plot this ratio for GBM bursts
against the $\text{T}_{\text{90}}$, we see that there is a two-dimensional
Gaussian mixture in the duration and spectral features of the burst. A
successful classifier should be able to identify if an object has had a merger
or collapsar progenitor, regardless of the burst’s duration, on both the
training and test data sets. Therefore, it appears suggestive that this
distribution is mixed even within the same duration.

In searching all GBM bursts (Section 2), we find that 13 of the 2310 bursts
with a measured peak energy also have a known progenitor (Fig. 1) (9 known
collapsar systems and 4 known merger systems). We used these 13 known
progenitors to create an SVM classifier based on the prompt energy ratio and
the $\text{T}_{\text{90}}$ of these bursts. The full SVM model is available
for use and transformation as needed 66610.5281/zenodo.11107782. The
hyperplane is a line segment (a 1-D hyperplane) that initially appears mostly
straight, though strongly tilted according to a burst’s peak energy, based on
6 support vectors of the 13 known progenitors. We then applied this
classification to all bursts in the test data to extract interesting features.
(Due to the error bars’ relative size, a version of the plots including them
can be found in App. B.) We find that the border between merger and collapsar
bursts is tilted in the peak energy-time axis and slightly above the 4.2s
divider given in von Kienlin et al. (2020).

We examined several potential observational selection effects, including the
real distribution of distance, simulated scaling of distance effects on other
parameters, and selection effects on prompt and afterglow fluence, that could
be covariates for our classifier–that is, secondary factors that could
influence or better predict the progenitor of a GRB. After noticing that many
papers that had shown a strong redshift-dependence in their correlations used
redshift-dependent variables, (including ours, Li et al., 2016) we chose to
simulate if a single long or short burst with or without error was simulated
at different redshifts could reproduce our correlation. Instead, it predicted
a different burst property distribution than the one found in the GBM catalog.
These bursts were simulated using $E_{p}=350$ keV, isotropic energy ($3\times
10^{52}$ ergs), and median values of the Band spectral indices of bursts in
the GBM catalog shorter than 4.2 seconds best fit by either a Band or SBPL
function ($-0.800,-2.76$). From that and duration at a redshift of zero of
0.05, 0.3, 3, 9, and 27 seconds, we were able to simulate the peak energy,
fluence, and observed duration of each burst at a list of randomized distances
between z=0.01 and 8. These values were simulated as: for $z_{sim}\in[0.01,8]$

$E_{p}=\frac{E_{p,0}}{1+z_{sim}}$ (1)
$\begin{split}\text{T}_{\text{90}}&=\text{T}_{\text{90},0}\times\int_{t_{a}}^{t_{b}}N(0,\text{T}_{\text{90},0}(1+z_{sim}))-\\\
&0.4\frac{e^{-erfinv(\times\text{T}_{\text{90}}^{2})\text{T}_{\text{90}}^{4}}}{\text{T}_{\text{90}}}dt\end{split}$
(2)

where

$N(0,\text{T}_{\text{90},0}(1+z_{sim}))|_{t_{a}}=-100$ (3)
$N(0,\text{T}_{\text{90},0}(1+z_{sim}))|_{t_{b}}=100$ (4)
$\begin{split}S_{\gamma,\text{prompt}}&=\frac{E_{\gamma,iso,0}(1+z_{sim})}{(4\pi
kd_{lum}^{2})}\times\int_{t_{a}}^{t_{b}}N(0,\text{T}_{\text{90},0}(1+z_{sim}))-\\\
&0.4\frac{e^{-erfinv(0.9\text{T}_{\text{90}}^{2})\text{T}_{\text{90}}^{4}}}{\text{T}_{\text{90}}}dt\end{split}$
(5)

Where $d_{lum}$ is the luminosity distance in centimeters of an object at
$z_{sim}$, 0.4 is the normalization of the normal distribution, and k is the
k-correction of a Band function–that is, the change in fluence due to spectral
changes with redshift and instrument sensitivity. We also extrapolate the
fluence from our received range of 10 to 100 keV to 1 to 10000 keV:

$k=\frac{\int_{10\ keV}^{1000\
keV}f(E,E_{p}=350keV,\alpha=-0.8,\beta=-2.76)dE}{\int_{\frac{1\
keV}{1+z_{sim}}}^{\frac{10000\ keV}{1+z_{sim}}}f(E,350,-0.8,-2.76)dE}$ (6)

where $f(E,E_{p},\alpha,\beta)$ is the Band spectral function as laid out in
Band et al. (1993). The peak energy, ($E_{p}$) low energy power law index,
($\alpha$) and high energy power law index ($\beta$) values were selected as
the averages of Fermi-GBM bursts shorter than five seconds and best fit by
Band functions:

$f(E)=\left\\{\begin{array}[]{lr}\left(\frac{E}{100}\right)^{\beta}e^{\beta-\alpha}\left(\frac{(\alpha-\beta)E_{p}}{100\alpha}\right)^{\alpha-\beta},\text{
if }E\leq\frac{(\alpha-\beta)E_{p}}{\alpha}\\\
\frac{E}{100}^{\alpha}e^{-\frac{\alpha*E}{break_{E}}},\text{ if
}E>\frac{(\alpha-\beta)E_{p}}{\alpha}\end{array}\right\\}$ (7)

Rather than resembling the real burst distribution, these simulations created
a distinct distribution roughly perpendicular to the observed data, as can be
seen in Fig. 2. These simulations appear to predict that distance explains
some of the scatter in bursts with similar properties rather than the full
distribution.

As another test to see if redshift dominates the correlation, we split our
sample in two by high and low redshift, here defined to be 1.5, roughly both
the median and mean of the redshift sample, and found that they showed no
significant difference in the way they were correlated at a significance level
of 0.01. This and all further tests on the difference between two populations
were performed using a Student-T test with a significance threshold of 0.01.
All tests on the similarity of two populations were done using a power
analysis with effect size 0.5 (meaning we expect a moderate different in the
populations) and power 0.99 (meaning the analysis will detect that moderate
effect with 99% accuracy if there is one). We examined the observed redshift
and the simulated distance separately as redshift would indicate if our
classification showed limitations dependent on distance, whereas distance
simulations would indicate if distance was a significant predictor of the
classifier. The sample with redshift could not be rejected as representative
of the entire GRB sample at a significance level of 0.01, though it also could
not be accepted as representative of the entire sample at that level. We
postulate that this classification is not well explained by observed redshift
or simulated distance alone, but rather by multiple physical factors.

Next, we calculated an afterglow fluence, as it is considered a secondary test
of the progenitor system. We did this by assuming that the integrated X-ray
afterglow flux (i.e. fluence) over 0.3-10 keV in Swift-XRT during the
”plateau” phase of the early afterglow would be related to the environment of
a GRB, and that the environments of merger and collapsar-type bursts would be
relatively distinct. We used the following to calculate this fluence:

$S_{\text{X},AG}=Q*\int_{t_{start}}^{t_{stop}}g(t,E_{i},\alpha_{i})dt$ (8)

Where $Q$ is a normalization factor found by setting the function equal to the
afterglow flux at 11 hours, $g(t,E_{i},\alpha_{i})$ is a power law in time
dependent on the number of breaks present in the afterglow lightcurve, and the
start and end times are measured from the XRT lightcurve. We found that when
we split our full afterglow sample in two by high and low afterglow fluence at
the our calculated median value of $6.6\times 10^{-7}\text{ergs cm}^{-2}$, the
data show only a slight difference in the way they were correlated at a
significance level of 0.01. The duration and prompt energy ratio of this
subsample was found to differ from the main sample at a significance level of
0.01, so we must reject the null hypothesis that these bursts represent the
underlying distribution. Instead, we hypothesize that their afterglows are
brighter than average, meaning there are selection effects on which bursts
have detectable afterglows. This would not negate the fact that this test
predicts two slightly different afterglow (and therefore environment)
populations that are classified slightly differently. It rather implies that
it only does so on a small fraction of the data. We therefore assert that the
classifier appears to differentiate based on a burst’s environment as a result
of progenitor classification on at least some data.

As a test of the limits of our classifier, we split the sample into high and
low prompt fluence subsamples across the GBM catalog median value of $3\times
10^{-6}\text{ erg cm}^{-2}$. This fluence was measured at 10-1000 keV over the
burst duration, which we take to be time between the start and end time of the
fluence spectrum. The $\text{T}_{\text{90}}$ is not used as it is related to
the fluence over 50-300 keV. We find cuts across this boundary lead to
significantly different classification fits and distributions. Furthermore,
one of these distributions is significantly different from the main
distribution; both correlations are significantly different from each other.
We take this to indicate that the classifier strongly depends on a burst’s
prompt fluence. Furthermore, despite having roughly equal numbers of ”high”
and ”low” fluence bursts, it appears that the classification distribution as a
whole is more similar to the low fluence bursts. Meanwhile, the classification
is very different across all three groups. This could mean that the classifier
works better on high-fluence bursts, as it changes so much when they are
subtracted from the distribution.

Figure 3: When performing time-resolved spectral fits classifying only the
short hard spike, and repeating their parametrizations, the burst observations
moved within the classification parameter space. This is a source of
uncertainty for many of bursts both near and distant from the classification
border, and it is worth deeper study.

Due to the small number of known progenitors, we also choose to test selection
effects within those. To do so, we randomize the number of collapsars that
”exist” in the sample between 1 (to make sure two classes would be present)
and 18 (twice the number that are present in the original sample). We do not
randomize the mergers as 3 of the 4 of them are used as support vectors in the
classifier. We then select that many collapsars randomly with replacement from
the original collapsar sample. Each ”new collapsar” is then assigned a random
the prompt energy-fluence ratio and the burst duration using a Gaussian with
mean 0 and standard deviation of the uncertainty of the measurement (0.15 and
0.12, respectively). These two steps would approximate the residual bootstrap
of a linear/nonlinear model, where the data sample and residuals are resampled
with replacement, then assigned randomly. This simulation indicated that our
previous analysis using assumptions for an t-distributed variable was limited
at best, as the distributions of the test variables were not found to be
normal. However, we did find that the percentage of collapsars, and the
classification percentages of GRBs 170817A and 230307A were near the peak of
their distributions. This indicates that the classifier is possibly limited by
its statistical method and physical model, rather than the sample size. This
was predicted by our choice of classifier (which would minimize the number of
used progenitors) as well as work performed by Siemers & Bajorath (2023). When
more progenitors become available, it should be possible to revisit this
analysis using a more self-correcting method like gradient boosting trees, as
we mentioned in Sec. 3.

As changing the known progenitors of the classifier would either have no
effect (if we did not change a support vector) or a profound effect, we chose
to test the meaning of each classification more indirectly. We instead
examined five GRBs in different areas of interest on the plot (Fig. 3). Two
(GRBs 081109 and 111010B) were high indicator variable outliers. When we split
them by features in their light curves and performed time-resolved spectral
analyses and remeasured their durations, we found that the listed peak
energies in the catalog were poorly constrained because they were originally
fit by a simple power law. When the classifier selected a secondary fit to use
its peak energy, the derived value was not properly constrained, existing at
an upper limit. We therefore conclude that bursts best fit by a power law are
sometimes poor candidates for this method. However, when all bursts best fit
by a power law are removed from the model, all but one merger are removed as
well. From here forward, we used the second most likely fit.

Two GRBs (110820C and 100907A) lie near the center of the distribution, one
with a slight classification preference and one without (36% and 45% predicted
similarity to a merger, respectively). When only what appeared to be their
main emission was selected, refit, and reclassified, their progenitor
predictions become less certain (now 53% and 56% predicted similarity,
respectively). Finally, GRB 120308B was chosen as it was a typical long
collapsar. However, when its spectrum was refit on a shortened duration around
the first pulse of the burst, it became slightly more similar to a merger than
a collapsar.

We also included the known progenitor bursts 221009A and 230307A as checks on
the method (see Fig. 1). They appear on the far bottom right of the
distribution with most estimations of the peak energy, as spectral fits
published by teams other than the GBM team for GRB 230307A are fit with an
evolving 2SPBL fit and that of GRB 221009A with an evolving Band plus power
law component (Levan et al., 2023; Lesage et al., 2023). Both are classified
as long collapsars, where one is a known long collapsar and the other a long
merger, which we suspect to be due to their extraordinary fluence. However, we
also cannot rule out that there is separation line evolution with fluence due
to its relationship with $\text{T}_{\text{90}}$ as seen in our prompt fluence
separation test. It is also probable that any classification based on prompt
emission alone would be limited, as the central engine and environment are
needed to fully determine information about a burst’s progenitor. We include
these progenitors as a indicator that further refinement on this classifier is
a topic for future work, including our maintained GitHub. As a field, GRB
science is frequently surprised by energetic and duration outliers such as
these. They may indicate a gap in understanding that future observations or
better analysis and machine learning techniques may help to resolve. These
instruments may help us to detect more of these objects, increasing our
knowledge of which ones are outliers and which represent new physics, as well
as to coordinate observations of their multiwavelength components with other
observatories. This parameterization of the prompt emission still shows
promise as a classifier between two burst classes as seen in Fig. 1. We do not
find any evidence of accessory classes in line with the results of previous
machine learning studies (Bhave et al., 2022; Salmon et al., 2022;
Tarnopolski, 2022).

##  5 Discussion

Considering the results of our analysis of our prompt energy ratio vs.
$\text{T}_{\text{90}}$ model, we find that the SVM classifier successfully
predicts GRB progenitors with some high-fluence and spectral model
limitations. We find no probabilistic reason to split our burst distribution
into more than two types based on the features. Our main motivator for this
assertion is the lack of a third class of progenitor system, though the return
of a Gaussian mixture in three dimensions also contributes. There is also the
fact that the classifier correctly classifies all its training data. The only
exceptions to its correct classification of known progenitors was 230307A,
which we believe to have exceeded some as yet unknown brightness or axial
limitation. In this situation, a simpler model should be favored, here meaning
one with fewer classes.

We find that the $\text{T}_{\text{90}}$ cutoff between collapsar and merger
bursts is not only slightly above 4.2s but also dependent on the prompt energy
ratio of the burst. This is better centralized in our tests than having it
depend on the hardness ratio of the burst. It indicates that there are some
MeV-peak bursts still characterized as merger-type, and some middle-fluence
bursts which are undoubtedly collapsars. We are unsure of the cause of these,
but we did try to model them using our known progenitors list (Table
LABEL:progenitor_table). Many of those progenitors never made it into this or
any other sample as they were not seen by GBM or BAT or XRT. We can only
speculate that these shortened collapsars may be either burst precursors or
have weak jets (Ahumada et al., 2021b; Wang et al., 2022), the long mergers
may have gone through a rotationally-supported hypermassive neutron star phase
(Rowlinson et al., 2014), or the exotic burst could be anything from a white-
dwarf-neutron star merger to a particularly unusual supernova (Becerra et al.,
2023). In fact, papers numerically modelling short and intermediate-length
engines for collapsars such as Hamidani et al. (2017) have suggested that
their jet lightcurves may be strongly biased towards the precursor jet phase
and highly variable. With the data as taken, it is difficult to determine
which interpretations show promise. However, we did create this classifier
with the ability to return a probabilistic classification, as seen in Table
LABEL:fig:_unnecessarily_long_table. Given that most bursts have a fairly
certain classification, we interpret that most GRBs fall squarely into our
known progenitor systems of either a high-mass stripped star or a colliding
neutron star binary per the predictions of Woosley (1993). As more ”short”
collapsars are added to this model, it may become easier to determine if it is
a physically meaningful classifier given that the line of determination is
neither straight nor denoted in the unlabelled graph. We could also try adding
features that characterize the environment, variability timescale, and
spectral lag which might refine our models. The two paths to accomplish this
will be adding more known progenitors and creating physical models that
include more variables. While SVM minimizes the number of training data points
it uses to build the model, it is very likely that the new data will include
one or many points that are more significant than the ones currently present.
It is also highly likely that when we use models that predict a progenitor’s
behavior in more than two (or three) dimensions, the SVM itself will separate
the bursts differently as the higher dimension plane will have a different
minimal path. We believe these two paths serve as a path forward, and prove
this distribution to be a first step rather than a finalize model.

In the interest of collecting more data about merger environments, we
recommend that broadband observations of long mergers similar to GRB 211211A
be undertaken. Papers that performed similar work would include Ren & Dai
(2022); Ren et al. (2023); Duncan et al. (2023). This burst is close enough
that the rapidly fading afterglow X-ray flux was constrained at late time
(Minaev et al., 2021). We suspect in addition that the burst’s environment may
be unusually dense, making it a good candidate for radio follow-up. If a radio
afterglow signal is detectable, it is possible that the GRB environment was
relatively dense, changing our understanding of where long mergers evolve.
However, if the afterglow was difficult to observe in the radio, it may be
possible that long mergers are instead a result of time-dilation of the
duration. We do not believe this to have a strong effect on the classifier,
but rather recommend it to be investigated as a possible cause of GRB duration
lengthening.

Our classifier appears to classify some classical short GRBs with extended
emission as mergers. However, it is unclear why others like GRB 120308B would
be confidently classified as mergers considering that their lightcurves appear
more similar to typical long bursts. This demonstrates that our model could be
a new avenue for identifying possible candidates for future progenitor
confusion studies.

We also reiterate our recommendation that supernovae be searched for after
”short” bursts as well as ”long”, as we suspect from our classifier that there
exist some collapsar GRBs whose $\text{T}_{\text{90}}$ is well below 4.2
seconds. This includes possibly using a similar classifier to predict which
progenitor we may detect, rather than sorting bursts as long or short as many
instruments currently do. Whether these short collapsars have jets weaker than
typical Type Ic supernovae (Ahumada et al., 2021b) or are only the precursor
of their events (Wang et al., 2022), we know that at least one exists (GRB
200826A Ahumada et al., 2021a). If that remains the only one found after a
long period of searching, it becomes more likely that only short-lived central
engines form ”short collapsars.” If ”short collapsars” become more common as
they are searched for, it becomes more likely that they are a ”common rarity”
that requires models to explain the link between the progenitor and the
central engine’s shorter lifespan. However, it will be difficult to judge
without a second event with which to compare GRB 200826A.

##  6 Conclusions

In this paper, we outline the potential for a GRB progenitor classifier based
on standard prompt emission properties. We found that the ratio of the peak
energy over the prompt fluence could be preliminarily used as a progenitor
classifier when combined with the $\text{T}_{\text{90}}$. It appeared to split
bursts at a generally longer duration than the accepted value of 4.2s for the
GBM catalog, which we suspect to be due to the presence of ”long mergers” (von
Kienlin et al., 2020). We suggest further studies be performed when more
progenitors are correlated with data from instruments like GBM that can
perform highly specific spectral studies of burst prompt emission. These
studies may significantly affect the classifier, including potentially
necessitating the use of more complex functions to explore the boundary or
inclusion of more variables as progenitor prediction models become more
complex. We also recommend that future studies examine how this relationship
changes for the very highest fluence bursts when possible. If the predictions
of Burns et al. (2023) are correct, there may be up to 3 of these bursts per
decade (assuming that the high-fluence cutoff is $\sim 10^{-3}$ erg cm-2.

##

This work made use of data supplied by the UK Swift Science Data Centre at the
University of Leicester. We gratefully acknowledge Phil Evans and Israel
Martinez Castellanos for helpful discussions. We also thank the referees for
their helpful commentary.

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##  Appendix A Processed Data

###  A.1 Known Progenitors

Table 1: The following two tables demonstrate the very unbalanced nature of our known progenitor sample. While there are dozens of GRB-associated supernova detections, even within our time constraints, there are very few GRB-associated mergers. Furthermore, our exotic and ”short collapsar” burst progenitors are unique. This limits our ability to interpret extensions of our model to unusual bursts. Even only requiring GBM to view a known progenitor eliminates a large number of known kilonovae and supernovae from Swift-BAT and INTEGRAL.  Name | Alt_Name | Available Data | Source
---|---|---|---
Long Collapsars
GRB 091127 | GRB 091127976 | GBM | Dainotti et al. (2022)
GRB 101219B | GRB 101219686 | BAT, XRT, GBM, redshift | Dainotti et al. (2022)
GRB 140606B | GRB 140606133 | GBM | Dainotti et al. (2022)
GRB 150210A | GRB 150210935 | GBM | Jiang et al. (2023a)
GRB 180720B | GRB 180720598 | BAT, XRT, GBM, redshift | Dainotti et al. (2022)
GRB 180728A | GRB 180728728 | BAT, XRT, GBM, redshift | Dainotti et al. (2022)
GRB 190829A | GRB 190829830 | BAT, XRT, GBM, redshift | Dainotti et al. (2022)
GRB 200826A | GRB 200826187 | GBM | Dainotti et al. (2022)
Short Collapsars
GRB 200826A | GRB 200826187 | GBM | Ahumada et al. (2021a)
Table 2: Each row gives the reference to a paper that described the observational characteristic that we used to identify the object as a merger, if it did not perform it directly. Some mergers-associations in this table are performed by directly observing the infrared spectrum of r-process nucleosynthesis. Those which are only detected as a bump in the infrared where r-process nucleosynthesis might be creating new elements are labelled as weak kilonova fit. Those short mergers which occurred in the outer reaches of their galaxies are labelled as position inferred. Finally, one short GRB also had a very small spectral lag, and is thus suspected to be a merger.  Name | Source | Merger Asso. Method
---|---|---
GRB 050509B | Fong et al. (2022) | position inferred
GRB 050709A | Fong et al. (2022) | position inferred
GRB 051210A | Fong et al. (2022) | position inferred
GRB 060614 | Yang et al. (2015) | kilonova detected
GRB 070714B | Fong et al. (2022) | position inferred
GRB 070809 | Jin et al. (2020) | kilonova detected
GRB 071227A | Fong et al. (2022) | position inferred
GRB 080503 | Zhou et al. (2023) | weak kilonova fit
GRB 080905A | Fong et al. (2022) | position inferred
GRB 090515A | Fong et al. (2022) | position inferred
GRB 111005A | Wang et al. (2017) | kilonova detected
GRB 120304B | Jiang et al. (2023b) | spectral lag indicated
GRB 130603B | Tanvir et al. (2013) | kilonova detected
GRB 150101B | Troja et al. (2018) | kilonova detected
GRB 160624A | O’Connor et al. (2021) | weak kilonova fit
GRB 160303A | Fong et al. (2022) | position inferred
GRB 160821B | Troja et al. (2019) | kilonova detected
GRB 170817A | Troja et al. (2017) | kilonova detected
GRB 200522A | O’Connor et al. (2021) | weak kilonova fit
GRB 211211A | Troja et al. (2022) | kilonova detected
GRB 230307A | Bulla et al. (2023) | kilonova detected

###  A.2 Progenitors as Sorted by Classifier

Table 3: This table contains a small smaple of the probabilities that each of our GBM bursts is associated with either a merger or collapsar event, as derived by our SVM algorithm. This algorithm was fitted using a radial kernel on $E_{p,prompt}S_{\gamma,prompt}^{-1}$ versus the $\text{T}_{\text{90}}$ of the known progenitors of Fermi GRBs 211211549, 200826187, 190829830, 180728728, 180720598, 171010792, 160624477, 150101641, 150210A, 140606133, 130427324, 130215063, 120304248, 101219686, 091127976, 090618353, and 80905499. As one can see, most bursts strongly prefer one progenitor over the other, leading us to assert that the two-progenitor system is most likely correct for most bursts. As we only fitted a small number of bursts and applied that fit to the much larger sample, this model must be interpreted cautiously. We suggest that future groups investigate adding more progenitors, especially in the ”short” collapsar category. For more information, please see ”Goldstein_Classification_sorted.csv.”  Designation | $\text{T}_{\text{90}}$ | $E_{p,prompt}S_{\gamma,\text{prompt}}^{-1}$ | $P_{collapsar}$ | $P_{merger}$
---|---|---|---|---
GRB 101219686 | 51.009 | 2.07E+07 | 3.56E-01 | 6.44E-01
GRB 091127976 | 8.701 | 1.07E+06 | 8.43E-01 | 1.57E-01
GRB 171010792 | 107.266 | 2.18E+05 | 5.97E-01 | 4.03E-01
GRB 180720598 | 48.897 | 2.13E+06 | 7.06E-01 | 2.94E-01
GRB 130215063 | 143.746 | 4.06E+06 | 5.30E-01 | 4.70E-01
GRB 130427324 | 138.242 | 3.35E+05 | 4.83E-01 | 5.17E-01
GRB 140606133 | 22.784 | 7.61E+07 | 1.89E-01 | 8.11E-01
GRB 090618353 | 112.386 | 5.55E+05 | 5.42E-01 | 4.58E-01
GRB 120304248 | 5.376 | 1.31E+08 | 4.30E-01 | 5.70E-01
GRB 150101641 | 0.08 | 5.25E+08 | 8.96E-01 | 1.04E-01
GRB 160624477 | 0.384 | 2.94E+09 | 6.49E-01 | 3.51E-01
GRB 120403857 | 4.288 | 1.29E+11 | 2.12E-01 | 7.88E-01
GRB 120227725 | 17.408 | 5.87E+06 | 6.91E-01 | 3.09E-01
GRB 141205018 | 13.056 | 1.40E+08 | 3.86E-01 | 6.14E-01
GRB 180630467 | 12.032 | 2.57E+07 | 8.41E-01 | 1.59E-01
GRB 170116238 | 9.216 | 1.62E+08 | 9.14E-01 | 8.64E-02
GRB 091026550 | 8.96 | 3.48E+08 | 7.44E-01 | 2.56E-01
GRB 150312403 | 0.32 | 2.15E+09 | 9.52E-01 | 4.80E-02
GRB 110520302 | 12.288 | 2.81E+08 | 6.20E-01 | 3.80E-01
GRB 121216419 | 9.216 | 1.52E+09 | 8.91E-01 | 1.09E-01
GRB 101227195 | 95.488 | 6.63E+07 | 5.67E-01 | 4.33E-01
GRB 160326062 | 19.456 | 1.69E+08 | 5.55E-01 | 4.45E-01
GRB 101116481 | 0.576 | 1.44E+09 | 2.22E-01 | 7.78E-01
GRB 170130697 | 29.184 | 2.34E+08 | 8.26E-01 | 1.74E-01
GRB 110205027 | 5.376 | 8.44E+08 | 7.22E-01 | 2.78E-01

##  Appendix B Error Propagation

The model-dependent uncertainty in the peak energy is given in the GBM
catalog. We assumed that the spectroscopic model was correct, though this is
itself known to have a level of uncertainty. The prompt fluence uncertainty
was also taken from the catalog, as above, making the error bars in the peak
energy over the fluence:

$\sigma_{\frac{E_{P}}{S}}=\sqrt{\left(\frac{\sigma_{E_{P}}}{S}\right)^{2}+\left(\frac{E_{P}\sigma_{S}}{S^{2}}\right)^{2}}=\left(\frac{E_{P}}{S}\right)\sqrt{\left(\frac{\sigma_{E_{P}}}{E_{p}}\right)^{2}+\left(\frac{\sigma_{S}}{S}\right)^{2}}$
(B1)

where each error is taken to be the average of its left and right values. The
error in $\text{T}_{\text{90}}$ could be taken directly from the GBM catalog.

Figure 4: In this version of the sorted data, it is all but impossible to view
the progenitor classes concerning the geometric division of the data. It also
becomes much more obvious that the uncertainty in this model is very high.
This model only worked as well as it did because GBM (and BATSE before it) can
specify the peak energy of the prompt much better than BAT. We caution that
this model should not be applied using instruments with narrow prompt
bandpasses that cannot analyze the prompt spectrum in as much detail. In our
testing, it did not work well, if at all, with even a relatively large sample
from BAT of bursts that were detected as having a cutoff power law.

For the figure that records the prompt fluence versus the afterglow flux at 11
hours, we again took the prompt fluence to have the GBM cataloged uncertainty.
We estimated the error in the X-ray flux at 11 hours as the change in the
average flux. This should have been a clear overestimation, but even with
three points recorded as having fluxes well below what XRT can nominally
measure, their uncertainties were not noticeably different from their
neighbors.