∎

BerkeleyPresent address: University of California, Berkeley, Department of
Physics, CA 94720, Berkeley, USA LNGSGPresent address: INFN Laboratori
Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy MilanoaPresent address:
Dipartimento di Fisica, Universitá degli Studi e INFN Milano-Bicocca, 20126
Milano, Italy PadovaPresent address: Dipartimento di Fisica e Astronomia dell’
Universitá di Padova and INFN Sezione di Padova, Padova, Italy RomaaPresent
address: Istituto Superiore per la Protezione e la Ricerca Ambientale, 00144
Roma, Italy

11institutetext: The Borexino Collaboration11institutetext: Physik-Department,
Technische Universität München, 85748 Garching, Germany 22institutetext:
Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany
33institutetext: Dipartimento di Fisica, Università degli Studi e INFN, 20133
Milano, Italy 44institutetext: Chemical Engineering Department, Princeton
University, Princeton, NJ 08544, USA 55institutetext: INFN Laboratori
Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy 66institutetext: Physics
Department, Princeton University, Princeton, NJ 08544, USA 77institutetext:
Dipartimento di Fisica, Università degli Studi e INFN, 16146 Genova, Italy
88institutetext: Lomonosov Moscow State University Skobeltsyn Institute of
Nuclear Physics, 119234 Moscow, Russia 99institutetext: St. Petersburg Nuclear
Physics Institute NRC Kurchatov Institute, 188350 Gatchina, Russia
1010institutetext: National Research Centre Kurchatov Institute, 123182
Moscow, Russia 1111institutetext: APC, Université de Paris, CNRS,
Astroparticule et Cosmologie, Paris F-75013, France 1212institutetext: Gran
Sasso Science Institute, 67100 L’Aquila, Italy 1313institutetext: Joint
Institute for Nuclear Research, 141980 Dubna, Russia 1414institutetext:
Institute of Physics and Excellence Cluster PRISMA+, Johannes Gutenberg-
Universität Mainz, 55099 Mainz, Germany 1515institutetext: M. Smoluchowski
Institute of Physics, Jagiellonian University, 30348 Krakow, Poland
1616institutetext: Institute for Nuclear Research of NAS Ukraine, 03028 Kyiv,
Ukraine 1717institutetext: RWTH Aachen University, 52062 Aachen, Germany
1818institutetext: National Research Nuclear University MEPhI (Moscow
Engineering Physics Institute), 115409 Moscow, Russia 1919institutetext:
Dipartimento di Chimica, Biologia e Biotecnologie, Università degli Studi e
INFN, 06123 Perugia, Italy 2020institutetext: Amherst Center for Fundamental
Interactions and Physics Department, UMass, Amherst, MA 01003, USA
2121institutetext: Department of Physics, Technische Universität Dresden,
01062 Dresden, Germany 2222institutetext: Physics Department, Virginia
Polytechnic Institute and State University, Blacksburg, VA 24061, USA
2323institutetext: Department of Physics, Royal Holloway, University of
London, Department of Physics, School of Engineering, Physical and
Mathematical Sciences, Egham, Surrey, TW20 OEX, UK 2424institutetext:
Institute of Nuclear Research (Atomki), Debrecen, Hungary

# Search for Low-Energy Signals from Fast Radio Bursts with the Borexino
Detector

S. AppelMunchen  Z. BagdasarianBerkeley,Juelich  D. BasilicoMilano  G.
BelliniMilano  J. BenzigerPrincetonChemEng  R. BiondiLNGS  B. CaccianigaMilano
F. CalapricePrinceton  A. CaminataGenova  A. ChepurnovLomonosov  D.
D’AngeloMilano  A. DerbinPeters,Kurchatov  A. Di GiacintoLNGS  V. Di
MarcelloLNGS  X.F. DingPrinceton  A. Di LudovicoPrinceton  L. Di NotoGenova
I. DrachnevPeters  D. FrancoAPC  C. GalbiatiPrinceton,GSSI  C. GhianoLNGS  M.
GiammarchiMilano  A. GorettiLNGSG,Princeton  A.S. GöttelJuelich,RWTH  M.
GromovLomonosov,Dubna  D. GuffantiMilanoa,Mainz  Aldo IanniLNGS  Andrea
IanniPrinceton  A. JanyKrakow  V. KobychevKiev  G. KorgaLondon,Atomki  S.
KumaranJuelich,RWTH  M. LaubensteinLNGS  E. LitvinovichKurchatov,Kurchatovb
P. LombardiMilano  I. LomskayaPeters  L. LudhovaJuelich,RWTH  G.
LukyanchenkoKurchatov  I. MachulinKurchatov,Kurchatovb  J. MartynMainz  E.
MeroniMilano  L. MiramontiMilano  M. MisiaszekKrakow  V. MuratovaPeters  R.
NugmanovKurchatov,Kurchatovb  L. OberauerMunchen  V. OrekhovMainz  F.
OrticaPerugia  M. PallaviciniGenova  L. PelicciJuelich,RWTH  Ö. PenekJuelich
L. PietrofacciaPrinceton  N. PilipenkoPeters  A. PocarUMass  G.
RaikovKurchatov  M.T. RanalliLNGS  G. RanucciMilano  A. RazetoLNGS  A.
ReMilano  M. RedchukPadova,Juelich,RWTH  N. RossiLNGS  S. SchönertMunchen  D.
SemenovPeters  G. SettantaRomaa,Juelich  M. SkorokhvatovKurchatov,Kurchatovb
A. SinghalJuelich,RWTH  O. SmirnovDubna  A. SotnikovDubna  R. TartagliaLNGS
G. TesteraGenova  E. UnzhakovPeters  A. VishnevaDubna  R.B. VogelaarVirginia
F. von FeilitzschMunchen  M. WojcikKrakow  M. WurmMainz  S. ZavatarelliGenova
I. ZhutikovKurchatov,Kurchatovb  K. ZuberDresda  G. ZuzelKrakow

(Received: date / Accepted: date)

###### Abstract

The search for neutrino events in correlation with several of the most intense
fast radio bursts (FRBs) has been performed using the Borexino data. We have
searched for signals with visible energies above $250$ keV within a time
window of $\pm$1000 s corresponding to the detection time of a particular FRB.
We also applied an alternative approach based on searching for specific shapes
of neutrino-electron scattering spectra in the full exposure spectrum of the
Borexino detector. In particular, two incoming neutrino spectra were
considered: the monoenergetic line and the spectrum expected from supernovae.
The same spectra were considered for electron antineutrinos detected through
the inverse beta-decay reaction. No statistically significant excess over the
background was observed. As a result, the strongest upper limits on FRB-
associated neutrino fluences of all flavors have been obtained in the $0.5-50$
MeV neutrino energy range.

###### Keywords:

Fast radio bursts neutrino Borexino

##  1 Introduction

A fast radio burst (FRB) is a millisecond radio transient observed at the
extragalactic or cosmological distance. Although FRBs were discovered almost
15 years ago Lor2007 , the nature of their source remains unclear. Numerous
models with a wide variety of physical processes have been proposed to explain
the origin of FRBs (see review articles Pop2018 ; Pet2019 ; Cor2019 ; Pla2020
). The most popular class of models postulates a production mechanism
associated with arising activity of magnetars Pop2013 ; Lyu2014 . These models
have received support from the repeating behavior of some FRBs, especially
after detection of FRB200428 in temporal and spatial coincidence with the
X-ray burst from magnetar SGR 1935+2154 in the Milky Way galaxy Boc2020 ;
And2020 .

Single-burst FRB models involve processes of supernova evolution, mergers, and
collapses of neutron stars Rav2014 ; Fal2014 ; Tot2013 ; Wan2016  with
emission of neutrinos (and possibly axions) which could be potentially
detected by large-volume Cherenkov or scintillation detectors. The IceCube
Neutrino Observatory has searched for the spatial and temporal correlation
between events with energies above $50$ GeV, as well as the temporal
correlation between MeV events and 28 FRBs. It has set upper limits on the
neutrino fluences associated with them Khe2017 ; Aar2018 ; Aar2020 . The
ANTARES Neutrino Telescope looked for TeV-PeV high-energy neutrinos spatially
and temporally coincident with the FRBs detected during 2013-2017, but no
coincident neutrino candidate was observed Alb2019 .

Borexino, a real-time liquid scintillator detector designed for solar neutrino
spectroscopy, is located at the Gran Sasso National Laboratory, in Italy
Ali2002 ; Ali2009 ; Bel2014 ; Ago2019 . Due to its extremely low background
level, large target mass, and low energy threshold, the Borexino detector has
been successfully used for studying low-energy neutrino fluxes from such
transients as $\gamma$-ray bursts (GRBs), gravitational wave (GW) events and
solar flares Bel2011 ; Ago2017a ; Ago2017b ; Ago2021 .

Since modern radio telescopes have a small field of view, they are able to
register only a fraction of the actual FRB amount. The total expected all-sky
event rate of FRBs with a fluence above $\sim 2$ Jy$\cdot$ms111Jy (Jansky) is
a non-SI unit of spectral flux density, $1\ \rm{Jy}=10^{-26}$
W$\cdot$m${}^{-2}\cdot$Hz-1 is roughly $2\times 10^{3}$ per day Pet2019 ;
Bha2018 . Scintillation detectors lack the directional sensitivity to incoming
neutrinos and therefore can not be used for the temporal analysis in a wide
time window due to the high rate of FRBs (in contrast to GRBs or GW events)
and a possible delay of the low-energy neutrino signal arriving from
extragalactic distances.

Here, we have performed the temporal correlation analysis between Borexino
events with visible energies above $0.25$ MeV and some of the most intensive
FRBs assuming the direct connection between radio and neutrino fluences of an
FRB. Another approach was based on the search for the characteristic shape of
($\nu,e$)-scattering in the high-statistic Borexino spectrum. Finally, we have
taken into account that no events were observed with an energy greater than a
certain value in the Borexino spectrum. Two different spectra of incoming
neutrinos ($\nu_{e,\mu,\tau}$ and $\bar{{\nu}}_{e,\mu,\tau}$) were used for
the analysis: the monoenergetic line and the spectrum expected from
supernovae. The same $\bar{{\nu}}_{e}$-neutrino spectra were considered for
detection with the inverse beta-decay reaction (IBD).

##  2 Borexino Detector

Borexino is a real-time liquid scintillator detector for solar neutrino
spectroscopy. It is located underground at the Gran Sasso Laboratory (Italy)
at a depth of $3400$ m.w.e. Its main goal is to measure low-energy solar
neutrinos via ($\nu,e$) scattering in an ultrapure liquid scintillator. The
extremely high radiopurity of the detector, its large mass, and low energy
threshold also allow addressing other fundamental problems of astroparticle
physics.

The inner vessel of the detector (IV) comprises $278$ tons of purified organic
liquid scintillator confined in a transparent nylon sphere of $4.25$ m in
radius. The scintillator was produced from petrochemical organics extracted
from underground to fulfill high radiopurity requirements. The scintillator
compound is based on pseudocumene (PC, $\rm{C_{9}H_{12}}$) doped with $1.5$
g/L of PPO ($\rm{C_{15}H_{11}NO}$). It is surrounded by two concentric PC
buffers ($323$ and $567$ tons, respectively) doped with a small amount of
light quencher (dimethylphthalate, DMP) intended for light yield reduction.
The buffer partitioning is performed in order to reduce the diffusion of radon
into the scintillator bulk. The IV and the buffers are contained inside a
stainless steel sphere (SSS) with a diameter of $6.75$ m fixed in position by
a stainless steel support structure. The SSS is enclosed in a cylinder with a
hemispheric top with a diameter of $18$ m and height of $16.9$ m.

The water tank (WT) is constructed of stainless steel with high radiopurity
and contains $2100$ tons of ultrapure water as additional shielding against
external $\gamma$-rays and neutrons. The WT is equipped with $208$ 8-inch PMTs
and serves as the Cherenkov muon veto (outer detector, OD) for identification
of residual muons crossing the detector. The scintillation light is detected
by nominally $2212$ 8-inch PMTs of the inner detector (ID) uniformly
distributed on the inner surface of the SSS.

Borexino detects charged particles via scintillation light produced in the
liquid scintillator. Each event occurring in the detector is characterized by
a number of fired PMTs whose pulse amplitudes and arrival times are recorded.
The number of active PMTs has slowly declined over the course of the Borexino
data taking. For each FRB event considered in this analysis, the true number
of active PMTs was included in the modeling of the detector response and the
signal normalized appropriately. These data are used to reconstruct the energy
and spatial coordinates of the event and to identify the type of the particle
$(e,\alpha,\mu)$. Both energy and spatial resolutions of the detector were
studied with radioactive sources placed at different positions inside the
inner vessel Bac2012 . The energy and position resolutions are
$\sigma_{E}\approx 50$ keV and $\sigma_{X}\approx 10$ cm at $1$ MeV with 2000
PMTs, respectively; both scaling with the energy of the event as $\sim
1/\sqrt{E}$ at low energies. The Borexino detector is unable to provide
sufficient directional information about a single event due to the nearly
isotropic emission of scintillation light (see, however, Ago2021e ).

A more detailed description of the Borexino detector can be found in the
following papers Ali2002 ; Ali2009 ; Bel2014 ; Ago2019 .

Neutrinos ($\nu_{x},x=e,\mu,\tau$) and antineutrinos ($\bar{\nu_{x}}$) are
interacted by means of their elastic scattering on electrons:

$\nu_{x}+e^{-}\rightarrow\nu_{x}+e^{-}\,,\quad\bar{\nu}_{x}+e^{-}\rightarrow\bar{\nu}_{x}+e^{-}\,.$
(1)

For a given neutrino energy, the maximum electron recoil energy is given by
the Compton formula:

$E_{e_{max}}=2E_{\nu}^{2}/(2E_{\nu}+m_{e}),$ (2)

where $E_{\nu}$ is the (anti)neutrino energy and $m_{e}$ is the electron mass.
The interaction between the scattered electron and scintillator molecules
produces photons which are registered by PMTs.

Electron antineutrinos $\bar{\nu}_{e}$ can also be detected via the inverse
beta-decay (IBD) reaction with an energy threshold of $1.8$ MeV:

$\bar{\nu}_{e}+p\rightarrow n+e^{+}.$ (3)

The visible energy of the positron and two annihilation photons is related to
the antineutrino energy as $E_{vis}=E_{\bar{\nu}_{e}}-~{}0.784~{}$MeV. The
neutron capture on protons produces a $2.22$ MeV $\gamma$-ray providing a
delayed signal with the mean capture time of $\sim 260$ $\mu$s Bel2011d . In
contrast to the total cross section of $(\nu_{e},e)$-scattering which is
proportional to $E_{\nu}$ when $E_{\nu}\gg m_{e}$, the cross section of the
IBD is proportional to $\sim E_{\bar{\nu}_{e}}^{2}$.

Borexino was the first experiment which to detect and then precisely measure
the ${}^{7}\rm{Be}$ solar neutrino flux Arp2008 ; Bel2011a  as well as the
$\rm{{}^{8}B}$-neutrino rate with $3$ MeV threshold Bel2010 ; Ago2020 . It
also observed $pep$-neutrinos for the first time Bel2012  and made the first
spectral measurement of $pp$-neutrinos Bel2014a ; Ago2018 ; Ago2019 , and
provided the first experimental evidence of solar neutrinos produced in the
CNO cycle Ago2020a ; Ago2020e . The Borexino detector also registered
antineutrinos $\bar{\nu}_{e}$ emitted in the decay of radionuclides naturally
occurring in the Earth Ago2015 ; Ago2020b ; Bel2010g ; Bel2013g .

Due to its excellent radiopurity, large target mass and low energy threshold,
Borexino is perfectly suited for the study of other fundamental problems, as
well as searching for rare and exotic processes in particle physics and
astrophysics.

The Borexino experiment ruled out any significant day–night asymmetry of the
7Be neutrino interaction rate Bel2012a  and set new limits on the effective
magnetic moment of solar neutrinos Arp2008 ; Ago2017 , possible violation of
the Pauli exclusion principle Bel2010a , on the flux of high-energy solar
axions Bel2012b , the flux of $\bar{\nu_{e}}$ from the Sun and different
unknown sources Bel2011c ; Ago2021 , heavy sterile neutrino mixing in the
$\rm{{}^{8}B}~{}\beta^{+}$-decay Bel2013 , the stability of the decay of an
electron into a neutrino and a photon Ago2015a , non-standard neutrino
interactions Aga2020 , neutrino and antineutrino fluences in temporal
correlation with $\gamma$-ray bursts Ago2017a , and gravitational wave events
Ago2017b , and some other rare processes.

##  3 Data Selection

Since a FRB is a very frequent event that is detected in a small fraction of
cases, the analysis was conducted via two different approaches: a transient
event search in a fixed time window with respect to the FRB detection moment
and a generic search for extra neutrino-induced electron recoils or IBD
components in the detector energy spectrum.

###  3.1 Transient Approach Data Selection

The first approach does not require a well-described spectral shape due to the
limited number of events in the time window and does not need any profound
background reduction techniques, although the background is to be kept
substantially low. Electron neutrino scattering events which are of interest
in the current analysis lack any characteristic interaction signature. Thus,
the background reduction has to be performed in a generic manner so as
decrease in the detector count rate per unit of exposure. The background of
the Borexino detector includes the following main components:

  * •

Short-lived cosmogenic backgrounds ($\tau\sim 0.31$ s) produced inside the
detector fiducial volume, such as 12B, 8He, 9C, 9Li etc.

  * •

Other cosmogenic backgrounds produced inside the detector fiducial volume,
such as longer-lived isotopes 11Be, 10C, 11C etc.

  * •

External gamma-background associated with natural radioactivity in detector
construction materials and PMTs.

  * •

Backgrounds of the inner nylon vessel associated with radioactivity of the
210Pb and uranium/thorium decay chains.

  * •

Natural backgrounds contained in the bulk of the detector fluid, such as 14C,
85Kr, 210Bi, and 210Pb.

  * •

Solar neutrino recoil electrons of the $pp$-chain and the CNO cycle.

These backgrounds can be suppressed by using information from the processed
detector data, such as ID/OD coincidences as well as the energy and position
reconstruction. Cosmogenic backgrounds can be reduced by applying the detector
time vetoes after each muon event that can be discriminated through
coincidence with the outer veto as well as via pulse-shape discrimination
Bel2011d ; Ago2021a . A veto length of $0.3$ s following a muon event is
applied to suppress 12B to a statistically insignificant level and reduce 8He,
9C, and 9Li by a factor of 3 with a live time loss of $1$ %.

Background components contained in the bulk can be reduced by applting a cut
on the visible energy. This is important specifically due to the presence of
14C in the scintillator. The carbon isotope 14C produces a $\beta$-spectrum
with an endpoint of $156$ keV and has an activity of roughly $110$ Bq in the
whole inner vessel. The presence of this spectral component sets the lower
threshold of the analysis at $250$ keV of the visible energy222The visible
energy spectrum of 14C is broadened up to this value due to the detector
energy resolution and its pile-up.

Background components contained in the nylon of the IV and other construction
materials cannot be removed by any kind of purification and therefore are
$10^{2}-10^{3}$ times higher than within the bulk of the scintillator. The
most dangerous components are the 214Bi and 208Tl decays. These nuclides
undergo $\beta$ and $\beta+\gamma$ decay processes with continuous spectra
overlapping with the energy region used in this analysis. The only way to
overcome this kind of background is to perform a geometrical cut on events,
selecting those within the fiducial volume. In our case the fiducial volume is
defined such that all events within $75$ cm to the IV are discarded, which
corresponds to 3 standard deviations of position reconstruction uncertainty at
the lowest energy threshold. The corresponding fiducial volume has a mass of
$145$ t. The Borexino dataset from May 15, 2007, (corresponding to the
detector operation start) to June 21, 2021, was used for the temporal
correlation analysis between the Borexino signals and the most intensive FRBs.

###  3.2 Spectral Approach Data Selection

The spectral approach to data selection is more complicated since background
components should not only be sufficiently suppressed, but also well described
in terms of spectral shapes. Minimization of the number of these spectral
components also benefits the final result due to suppression of their spectral
correlations fit procedure, even if it comes at the expense of exposure loss.
Thus, we modify the data selection procedure of the transient analysis in the
following way:

  * •

We use the most radiopure dataset acquired from January 01, 2013,
(corresponding to the detector stabilization after the water extraction
procedure finished at the end of 2011) to November 31, 2020.

  * •

We apply an advanced system of the cosmogenic veto based on time and position
reconstruction of muons and neutrons in the post-muon gate that strongly
reduces cosmogenic backgrounds.

The advanced cosmogenic veto system dedicated to the discrimination of short-
lived cosmogenic nuclides is based on the information from the muon trigger
and post-muon trigger of the Borexino detector. It is organized in the
following way:

  * •

$120$ s full detector veto after each muon that crossed the ID and has more
than 20 neutron-like daughters within the following $1.6$ ms trigger gate.

  * •

$20$ s veto on the cylinder with a radius of $0.8$ m aligned with the muon
track in case the track is reconstructed with OD signals.

  * •

$120$ s spherical veto with a radius of $1.3$ m on each reconstructed neutron
position in the $1.6$ ms post-muon trigger gate.

  * •

full detector veto of $4$ s after each muon crossing the ID.

This veto system comes at the cost of $15.8$% exposure loss that is checked
with toy Monte Carlo events produced uniformly within the ID with the constant
rate of $100$ Hz.

The obtained spectrum is followed by the statistical subtraction of external
background based on the radial distribution fit of each energy bin with the
function described as:

$N(R)=N_{0}(R)\times(A+B*exp(\lambda R))\,,$ (4)

where $N_{0}(R)$ is the radial distribution of toy Monte Carlo events within
the fiducial volume, $\lambda$ is a free parameter that describes external
gamma-background attenuation, and A and B correspond to the internal and
external signal, respectively. Validity of this procedure was tested using the
full detector Monte Carlo data Ago2018a  and provides reliable separation of
external backgrounds which comes at the cost of increased uncertainty in each
bin. The spectrum obtained corresponds to $298.4$
$\mathrm{kt}\cdot\mathrm{day}$ of exposure and has a significantly reduced
background composition that can be described by known background components.

##  4 Analysis and Results

As mentioned above, two different approaches have been used for studying the
neutrino fluence associated with FRBs. First, the excess of the number of
Borexino events was searched for in temporal correlation with the most intense
FRBs selected from the existing databases. Then, the Borexino energy spectrum
with high statistics was analyzed in order to determine the possible
additional unaccounted contribution from $(\nu,e)$-scattering and inverse
beta-decay reactions.

###  4.1 Selection of FRBs from Databases

We used the `chime-frb.ca` database accumulated by the CHIME Radio Telescope
Ami2018 ; Ami2021  and the `frbcat.org` database that collected and summarized
the data from several other telescopes, such as Parkes Yan2021 , Arecibo
Arecibo , Green Bank GRBNK , UTMOST utmost1 ; utmost2 , ASKAP Ban2017 , FAST
fst , Apertif apertif ; apertif1 , VLA vla , DSA-19 dsa , and Pushchino
pushchino . These databases contain information about the FRB time, duration,
energy spectrum, intensity, and redshift value (available only for some FRBs)
.

Within the period of interest (December 2007 to June 2021), we selected $42$
FRBs with radio fluence values $\Phi_{\mathrm{FRBi}}$ above $40$ $\rm{(Jy\cdot
ms)}$.

The temporal correlation method is based on the fact that radio fluence (which
is supposed to be related to the neutrino fluence) from the time window of the
most intense FRBs must be greater than the radio fluence from the window of
background determination. The expected total radio fluence from the given time
window is proportional to its time span $\Delta t$ and the average radio flux
from all FRBs $F_{all}$. In the windows of the most intense FRBs, the radio
fluence will be increased by the average fluence $\Phi_{40}$.

In order to calculate the $F_{all}$ and $\Phi_{40}$ values, we used a power
law index $\alpha=-1.4$ for cumulative fluence distribution and the all-sky
FRB rate of $N_{5}=\rm{818~{}sky^{-1}day^{-1}}$ above the fluence of 5
$\rm{(Jy\cdot ms)}$ obtained in Ami2021 . A small additional contribution to
$F_{all}$ from FRBs with the fluence $\leq 5\rm{(Jy\cdot ms)}$ was made by
adding $\simeq 1200$ uniformly distributed fluences in the $(0-5)\rm{(Jy\cdot
ms)}$ range. As a result, the average flux from all FRBs turned out to be
$F_{all}$ = $\rm{0.16~{}(Jy\cdot ms)s^{-1}}$ (or $7.0$ $\rm{(Jy\cdot ms)}$ per
FRB) while the average fluence of FRBs with the fluence $\geq
40~{}\rm{(Jy\cdot ms)}$ amounted to $\Phi_{40}$=74 $\rm{(Jy\cdot ms)}$.

The similar results were obtained from the analysis of specific FRBs.
According to the CHIME data Ami2021 , the average radio fluence of all $536$
FRBs is $\Phi_{all}=7.0$ $\rm{(Jy\cdot ms)}$ per FRB, while for $12$ FRBs with
the fluence above $40$ $\rm{(Jy\cdot ms)}$, the average equals to
$\Phi_{40}=61.3$ $\rm{(Jy\cdot ms)}$. Consequently, the average flux should be
expressed as $F_{all}=\Phi_{all}N_{all}/T$ s-1, where $N_{all}$ is the number
of all-sky FRB events per day ($T=24\rm{h}=86400~{}\rm{s}$).

Thus, the excess of the expected neutrino events corresponding to the most
intense FRB time intervals is defined by the factor $r=\Phi_{40}/(\Delta
tF_{all})$. Since in our study we chose the length of the time window $\Delta
t=2000$ s, this ratio turns out to be $r=0.2$ of the average neutrino flux.

Independently, we analysed the Borexino signals in coincidence with FRB 200428
from magnetar SGR 1935+2154 . This event occurred on April 28, 2020 at the
intragalactic distance of $9.5$ kpc, thus yielding a very high fluence of
$1.5\times 10^{6}$ $\rm{(Jy\cdot ms)}$.

The biggest redshift $z=0.66$ was observed for FRB 190523. We have considered
the coincidence time window $\Delta t=2000$ s centered at the FRB observation
time with a width of $\pm 1000~{}$s covering a possible delay of sub-MeV
neutrinos propagating at the sublight speed. For a distance corresponding to
the $z=0.66$ redshift, the delay should reach $1000$ s in case of $0.6$ MeV
neutrinos with a rest mass of $70$ meV, which is the upper limit on the
heaviest neutrino mass state from the Planck 2015 data and oscillation mass
squared differences Ago2017b .

The FRB arrival time could have its own delay associated with propagation
through intergalactic plasma. The delay depends on the registered dispersion
measure and the frequency $\omega$ at which the signal was recorded as
$\omega^{-2}$ Lor2007 ; Pop2018 ; Lor2004 .

Only the Pushchino telescope pushchino  operating at a sufficiently low
frequency of $109-113$ MHz was able to register FRB 160920 with a
significantly large delay of $620$ s. Among $11$ FRBs detected by the
Pushchino telescope, only $6$ FRBs have a signal delay exceeding $100$ s.
Higher working frequencies (up to $1$ GHz) of all other telescopes result in
delays below $100$ s. However, these delays were calculated and taken into
account in our analysis.

All selected FRBs had the data taking time above $95$% of the corresponding
time interval $\Delta t$.

Figure 1: Borexino events with an energy above $0.25$ MeV occurring within
$\pm 5000$ s of FRB 200428 detection time. Figure 2: Borexino energy spectrum
of singles in correlation with FRBs in the $\pm 1000$ s time window (Line 1
with dots). Line 2 shows the normalized background spectrum measured in
$\left[{-5000}\dots{-1000}\right]$ s and $\left[{1000}\ldots{5000}\right]$ s
intervals. The inset shows the difference between these spectra in terms of
standard deviations (SD). Figure 3: 90% C.L. upper limits on mono-energetic
neutrino fluences obtained through the temporal correlation analysis for the
most intensive FRBs with the fluence $\Phi_{\rm{FRB}}\geq 40~{}\rm{Jy~{}ms}$:
1 – $\nu_{e}$, 2 – $\nu_{\mu,\tau}$, 3 – $\bar{\nu}_{e}$, 4 –
$\bar{\nu}_{\mu,\tau}$.

###  4.2 Neutrino Spectra

As noted above, the origin of FRBs is largely unknown. It is not even clear
whether all of the observed radio bursts belong to the same type of physical
processes. Among many different FRB models, there are several, such as neutron
star mergers or supramassive neutron star collapses, which predict collateral
neutrino and axion radiation Pop2018 ; Pla2020 . In case of high-energy
$\mathrm{GeV}-\mathrm{PeV}$ neutrinos produced in hadronic processes, one
would expect the power law spectrum $\sim E_{\nu}^{-2}$. There is no reliable
theory of the low-energy part of the FRB neutrino emission spectrum.

We calculated fluence limits on two different kinds of possible neutrino
spectra: the monoenergetic line and the SN low-energy continuous spectrum. The
latter was assumed as a quasi-thermal spectrum with the mean energy $\langle
E\rangle$ and deviation from thermal distribution characterized by the
pinching parameter $\alpha=3$ for all neutrino flavors
($\nu_{x},\bar{\nu}_{x}$) Tam2012 ; Luj2014 ; Mir2016 . The emitted neutrino
spectrum $S(E_{\nu})$ depends on the neutrino energy $E_{\nu}$ as:

$S(E_{\nu})\sim(E_{\nu}/T)^{\alpha}e^{(-E_{\nu}/T)},$ (5)

where $T=\langle E\rangle/(\alpha+1)$ is the effective temperature, which was
considered to be the same for all neutrino flavors.

###  4.3 Temporal Correlations for the Most Intensive FRBs

The goal of this analysis was to search for an excess of the selected events
above the measured background, in coincidence with FRBs in a time window of
$\Delta t=2000~{}$s centered at the FRB arrival time. We calculated the
overall number of candidate events above $250$ keV in the $\Delta t$ interval,
which met the requirements for selection cuts of the described data.

For reference, the Borexino events with an energy above 0.25 MeV produced
within $\pm 5000~{}$s of the most intensive Galactic FRB 200428 detection time
are shown in Figure 1. The closest events, with energies of $1.13$ MeV and
$1.49$ MeV, occurred $105$ s before and $539$ s after the FRB arrival,
respectively. There were only three events in the $\pm 1000$ s interval with
the energy in the $0.25-15$ MeV range, while $4.4\pm 0.1$ solar neutrino and
background events were obtained within the same time/energy window from the
weekly run containing FRB 200428. All detected events were in agreement with
the expected solar neutrino and background count rate.

Figure 2 shows the energy spectrum measured for the integrated time exposure
$N_{\mathrm{FRB}}\times\Delta t$ in the
$250~{}\mathrm{keV}-3.5~{}\mathrm{MeV}$ energy range. There is only a single
$6.8$ MeV event outside this energy interval. For comparison, the same Figure
2 contains the background spectrum measured in two adjacent time intervals,
$\left[{-5000}\ldots{-1000}\right]$ s and $\left[{1000}\ldots{5000}\right]$ s.
No statistically significant excess of the difference between these spectra
for any energy interval was observed.

We calculated the upper limits on fluences $\Phi_{\nu_{x},\bar{\nu_{x}}}$ for
monoenergetic (anti)neutrinos with the energy $E_{\nu}$ as:

$\Phi_{\nu_{x},\bar{\nu_{x}}}=\frac{N_{90}(E_{\nu},n_{\mathrm{obs}},n_{\mathrm{bkg}})}{rN_{e}\sigma(E_{\mathrm{th}},E_{e_{\mathrm{max}}})},$
(6)

where $N_{90}(E_{\nu},n_{\mathrm{obs}},n_{\mathrm{bkg}})$ is the 90% C.L.
upper limit for the number of FRB-correlated events in the
$(E_{\mathrm{th}},E_{e_{\mathrm{max}}})$ interval per single FRB, $N_{e}$ is
the number of electrons in $145$ t of the Borexino scintillator. The factor
$\sigma(E_{\mathrm{th}},E_{e_{\mathrm{max}}})$ represents the cross section
for detected neutrinos $(\nu_{x},\bar{\nu_{x}})$ with the energy $E_{\nu}$
without oscillations while recoil electrons are detected in the interval
($E_{\mathrm{th}},E_{e_{\mathrm{max}}}$) Ago2017a . The recoil electron
detection efficiency was taken as 1, with the accuracy corresponding to the
precision of the fiducial volume definition ($\simeq 1\%$).

The numerator $N_{90}(E_{\nu},n_{\mathrm{obs}},n_{\mathrm{bkg}})$ was
calculated taking into account the statistical compatibility between the
correlated and uncorrelated FRB spectra and assuming Poisson statistics for
the number of detected events in the energy interval
($E_{\mathrm{th}},E_{e_{\mathrm{max}}}$). Here, $n_{\mathrm{obs}}$ and
$n_{\mathrm{bkg}}$ denote overall numbers of events in the energy interval
($E_{\mathrm{th}},E_{e_{\mathrm{max}}}$) detected in the time periods
$N_{\mathrm{FRB}}\times\Delta t_{\mathrm{obs}}$ and
$N_{\mathrm{FRB}}\times\Delta t_{\mathrm{bkg}}$ normalized by their respective
times. The intervals $\Delta t_{\mathrm{obs}}$ and $\Delta t_{\mathrm{bkg}}$
were taken as $\pm 1000$ s and summa of $\left[-10000\ldots-1000\right]$ s and
$\left[1000\ldots 10000\right]$ s, correspondingly. The longer interval for
background detection was chosen in order to reduce the error of
$n_{\mathrm{bkg}}$ that plays an important role in the Feldman-Cousins method.
The value $n_{\mathrm{bkg}}$ was normalized by the overall time ratio taking
into account the actual live time of the detector within these time windows.

In order to have the best ratio of the expected effect with respect to the
background, the lower integration limit $E_{\mathrm{th}}$ was optimized for
different neutrino energies taking into account the decrease in the energy
spectrum shape with the energy (Figure 2). In the analysis, in addition to the
0.25 MeV threshold, the 3.0 MeV threshold was used for higher neutrino
energies. The upper limits on neutrino and antineutrino fluences of different
flavors normalized per single FRB are shown in Figure 3. These are the first
constraints on the MeV neutrino fluxes obtained from the neutrino-electron
scattering reaction. The average radio fluence of the most intensive FRBs and
the limits on the neutrino fluence shown in Figure 3 allow us to obtain the
limit on the ratio $(\Phi_{\nu}/\Phi_{\mathrm{FRB}})$ in
$(\nu\cdot\rm{cm^{-2}/Jy\cdot ms})$ units.

Since there is no reliable model for the low-energy neutrino spectrum for
FRBs, we perform calculations of neutrino emission from a supernova collapse
Tam2012 ; Luj2014 ; Mir2016 . Assuming quasi-thermal distributions (5) with a
mean energy $\langle E\rangle=15.6$ MeV and the parameter $\alpha=3$ and
integrating over the analyzed electron recoil energy interval $0.25-15.0$ MeV,
we get the limits on the total electron neutrino fluence per single FRB:
$\Phi(\nu_{e})\leq 3.69\times 10^{10}\rm{cm^{-2}}$ (90% C.L.) that is about
three times weaker than the limit obtained for monoenergetic neutrinos with
the same energy. The values of the limits on other neutrino flavors obtained
from the $(\nu,e)$-scattering channel, as well as from the IBD reaction for
$\bar{\nu}_{e}$, are given in Table 1.

We also calculated the upper limit on the electron antineutrinos
($\bar{\nu}_{e}$) fluence using the IBD reaction from the relation (6) but
replacing $N_{e}$ with the number of protons $N_{p}$, considering cross-
section of IBD reaction and using the data acquired between December 2007 and
October 2017 Ago2021 . The data period considered includes only 19 FRB events
above the threshold $40$ $\rm{(Jy\cdot ms)}$. No IBD events were observed in
the $\pm 1000$ s interval around the selected FRBs and the expected background
was almost zero Ago2021 ; Ago2020b  that allowed us to use the conservative
value of $N_{90}(E_{\nu},n_{\mathrm{obs}},n_{\mathrm{bkg}})$ = 2.44 in the
analysis.

Table 1: Upper limits on fluences per single FRB for all neutrino flavors obtained from the temporal correlation analysis in $10^{9}~{}\rm{cm^{-2}}$ units (90% C.L.) calculated for monoenergetic neutrinos and the SN spectrum with $\langle E\rangle=15.6$ MeV. $E_{\nu}$ | ${\Phi}_{\nu_{e}}$ | ${\Phi}_{\bar{\nu}_{e}}$ | ${\Phi}_{\nu_{\mu,\tau}}$ | ${\Phi}_{\bar{\nu}_{\mu,\tau}}$ | IBD
---|---|---|---|---|---
2 | $4620$ | $12750$ | $24250$ | $29000$ | $2475$
6 | $157$ | $890$ | $890$ | $1280$ | $40.7$
10 | $125$ | $475$ | $770$ | $970$ | $12.2$
14 | $77.5$ | $255$ | $474$ | $590$ | $5.88$
$<15.6>$ | $157$ | $367$ | $900$ | $1070$ | $11.6$
Figure 4: Spectral fit of the selected Borexino data after statistical
subtraction of external backgrounds. The spectral components considered are: 1
– 210Po $\alpha$-peak, 2 – recoiled electrons from the solar 7Be, 3– recoiled
electrons from the solar CNO and $pep$-neutrinos, 4 – 210Bi $\beta$-spectrum,
5 – solar 8B neutrino electron recoils, 6 – 11C $\beta^{+}$-decay, 7 – 10C
$\beta^{+}$-decay. The inset shows the data in the energy range [0.5 – 4] MeV.
The fit shows a good statistical agreement with this spectral composition
model and with the fluxes that are in agreement with Bel2014 ; Ago2018 .

###  4.4 Limits on the $\nu_{e,\mu,\tau}$ and $\bar{{\nu}}_{e,\mu,\tau}$
Fluences from the Spectral Fit

Figure 4 shows the Borexino spectrum after applying data selection cuts from
Section 4.2 for $298.39$ $\mathrm{kt\cdot day}$ statistics or $2058$ days of
live time. The spectrum is dominated by 14C in the region below $250$ keV,
electron recoil from solar 7Be neutrinos in the $0.25-1$ MeV interval, by
cosmogenic 11C in the $1-2$ MeV region, and by 8B solar neutrinos above $2$
MeV. The background used in the fit procedure was described with the actual
spectral components, such as 210Po, 85Kr, solar neutrino recoil electrons,
11C, and 10C. External gammas were statistically subtracted and thus were not
included in the fit that was performed with the standard $\chi^{2}$ likelihood
function. The detector energy response was described according to Smi2008
with additional empiric consideration of light quenching and Cherenkov
radiation emission. The energy scale calibration and light quenching for
$\alpha$\- and $\beta^{\pm}$-particles were left as free parameters of the
likelihood function Bel2014 .

The obtained count rates of all spectral components are in a good statistical
agreement with previous publications Bel2014 ; Ago2018 ; Ago2020a ; Ago2020e ,
but have larger uncertainties caused by the statistical subtraction procedure.

The additional component responsible for the potential FRB signal was added to
the background and had a spectral shape of the monoenergetic line with the
energy $E_{\nu}$ or the supernova spectrum given by (5) with different
$\langle E\rangle$ values. The limit on the number of events from the
additional component was derived using the $\chi^{2}$ profile as the value
that corresponds to increase by $(1.64)^{2}$ with respect to the minimal value
or the value at zero count rate in cases when the best-fit count rate value
turned out to be negative. This limit $N_{90}(E_{\nu})$ corresponds to a
confidence level of 90% and could be converted into limit on the FRB fluence
as:

$\Phi=\frac{N_{90}(E_{\nu})}{N_{e}\sigma(E_{\mathrm{th}},E_{\nu})},$ (7)

where $N_{e}$ is the number of electrons in the FV scintillator and
$\sigma(E_{\mathrm{th}},E_{\nu})$ is the $(\nu,e)$-scattering cross section.

The limits on the total fluence (time-integrated neutrino flux) during $2058$
days of measurements obtained with this procedure for monoenergetic neutrinos
of all flavors are shown in Figure 5. Assuming an expected all-sky FRBs rate
of $N_{all}=2\times 10^{3}$ per day Pet2019 ; Bha2018 , the limits per single
FRB are $4\times 10^{6}$ times stronger than those shown in Figure 5.
Comparing the limits obtained from the temporal correlation analysis (Figure
3) and from the spectral fit (Figure 5), one can see that the latter are about
$(1-3)\times 10^{4}$ times more stringent although they depend on the assumed
all-sky FRB count rate $N_{all}$.

The same analysis was applied to the neutrino spectrum from a supernova given
by eq. (5). Figure 6 shows the 90% C.L. upper limits on fluences for supernova
neutrino spectra with different mean energies $\langle E\rangle$. For a value
of $\langle E\rangle=15.6$ MeV, the upper limit per total fluence of $\nu_{e}$
is $\Phi_{\nu_{e}}\leq 1.66\times 10^{13}$ $\mathrm{cm}^{-2}$. Taking into
account the number of FRBs expected in $2058$ days, the obtained limit turns
out to be $9\times 10^{3}$ times more stringent than the one obtained from the
temporal analysis (Table 1).

Figure 5: Upper limits on the fluences of monoenergetic $\nu_{e,\mu,\tau}$ and
$\bar{\nu}_{e,\mu,\tau}$ obtained from the spectral fit (90% C.L.): 1 –
$\nu_{e}$, 2 – $\nu_{\mu,\tau}$, 3 – $\bar{\nu}_{e}$, 4 –
$\bar{\nu}_{\mu,\tau}$. Figure 6: Upper limits on the fluences of
$\nu_{e,\mu,\tau}$ and $\bar{\nu}_{e,\mu,\tau}$ with supernova neutrino
spectra obtained from the spectral fit (90% C.L.): 1 – $\nu_{e}$, 2 –
$\nu_{\mu,\tau}$, 3 – $\bar{\nu}_{e}$, 4 – $\bar{\nu}_{\mu,\tau}$.

###  4.5 Limits Obtained from the Absence of the $(\nu,e)$-Events above 13.6
MeV

As for the neutrinos with higher energies, the limits on $\nu_{e,\mu,\tau}$
and $\bar{\nu}_{e,\mu,\tau}$ were alternatively obtained using the fact that
no events were observed above $13.6$ MeV. The right boundary of the interval
for the analysis was set to $16.8$ MeV in accordance with the verified energy
calibration of the data acquisition system. The different values of the
monoenergetic neutrino energy $E_{\nu}$ and the supernova neutrino mean energy
$\langle E\rangle$, the expected spectra of recoil electrons and the number of
events $N_{x}$ in the interval $13.6-16.8$ MeV were used in the calculations.

According to the Feldman-Cousins approach to the case of no observed events
with the conservative zero background, the $90$% C.L. upper limit in the
$13.6-16.8$ MeV interval is $N_{\mathrm{lim}}=2.44$. The relation $N_{x}\leq
N_{\mathrm{lim}}$ was converted into the obtained fluence limits for all
neutrino flavors given in Figure 7 for monoenergetic neutrinos and in Figure 8
for supernova neutrinos. The figures 7 and 8 also show the expected spectra of
recoil electrons for the $(\nu,e)$-elastic scattering reactions for
$E_{\nu_{x}}=30$ MeV and $\langle E\rangle=15.6$ MeV.

The region of the neutrino energy up to $50$ MeV was chosen in accordance with
the characteristic energies of neutrinos appearing in the pion decay at rest.
A sharp decrease in sensitivity to the neutrino fluence at the energies below
$17$ MeV occurs when the neutrino energy enters the $13.6-16.8$ MeV interval.
At higher energies, the limit becomes almost constant since the
$(\nu,e)$-scattering cross section is proportional to $E_{\nu}$ and the
spectrum of recoil electrons weakly depends on the electron energy (Figure 7).

Figure 7: The upper limits on the fluences of monoenergetic $\nu_{e,\mu,\tau}$ and $\bar{{\nu}}_{e,\mu,\tau}$ with 1 MeV step (90% C.L.). The inset shows the Borexino data and the expected $(\nu_{x},e)$-scattering spectra for $30$ MeV neutrinos. Figure 8: The same as in Figure 7, but for the supernova spectra. The inset shows the expected electron recoil spectra for the case $\langle E\rangle=15.6$ MeV. Table 2: Borexino $90$% C.L. upper limits for the total FRB fluences of all neutrino flavours, obtained through the study of $(\nu,e)$ elastic scattering of monoenergetic neutrinos ($2058$ d) and IBD reaction ($2485$ d). $E_{\nu}$ is given in MeV units, ${\Phi}_{\nu_{x},\bar{\nu}_{x}}$ – in $10^{12}~{}\rm{cm^{-2}}$ units.  $E_{\nu}$ | ${\Phi}_{\nu_{e}}$ | ${\Phi}_{\bar{\nu}_{e}}$ | ${\Phi}_{\nu_{\mu,\tau}}$ | ${\Phi}_{\bar{\nu}_{\mu,\tau}}$ | IBD
---|---|---|---|---|---
2 | $2040$ | $11700$ | $11400$ | $14200$ | $30.1$
6 | $74.8$ | $584$ | $662$ | $678$ | $0.173$
10 | $61.6$ | $310$ | $444$ | $574$ | $0.027$
14 | $10.2$ | $116$ | $112$ | $126$ | $0.013$
18 | $1.74$ | $14.5$ | $12.3$ | $16.8$ | $0.008$
30 | $1.70$ | $5.08$ | $10.8$ | $13.0$ | $-$
50 | $1.66$ | $2.97$ | $9.41$ | $10.5$ | $-$

The results of the spectral fit and the case of no observed events are also
shown in Table 2 for monoenergetic neutrinos. One can see the validity of
using different analysis methods. The spectral fit gives a better sensitivity
to lower neutrino energies in the $0.5-14.0$ MeV range while the second method
allows expanding the neutrino energy range up to $50$ MeV.

In case of the supernova neutrino spectrum, the fluence constraints based on
the absence of events above $13.6$ MeV becomes stronger than the monoenergetic
neutrino limit starting from the mean energies above $\langle E\rangle\geq 10$
MeV. The resulting most stringent limits are given in Table 3.

Table 3: Borexino 90% C.L. upper limits for the total FRB fluences obtained through the study of $(\nu,e)$ scattering ($2058$ d) and the IBD reaction ($2485$ d) with supernova neutrino spectra. The mean energy $\langle E\rangle$ is given in MeV units, ${\Phi}_{\nu_{x},\bar{\nu}_{x}}$ – in $10^{12}~{}\rm{cm^{-2}}$ units.  $\langle E\rangle$ | ${\Phi}_{\nu_{e}}$ | ${\Phi}_{\bar{\nu}_{e}}$ | ${\Phi}_{\nu_{\mu,\tau}}$ | ${\Phi}_{\bar{\nu}_{\mu,\tau}}$ | IBD
---|---|---|---|---|---
2 | $1220$ | $4060$ | $5840$ | $6640$ | $528$
6 | $119$ | $259$ | $779$ | $943$ | $0.117$
10 | $12.4$ | $85.6$ | $86.6$ | $113$ | $0.036$
14 | $4.8$ | $26.1$ | $32.8$ | $41.8$ | $0.030$
18 | $3.11$ | $13.8$ | $20.8$ | $26.0$ | $0.034$
30 | $2.00$ | $5.72$ | $12.6$ | $14.9$ | $0.073$
50 | $1.73$ | $3.49$ | $10.1$ | $11.4$ | $0.245$

###  4.6 Limits on the $\widetilde{\nu_{e}}$-Fluence from the IBD Reaction

As already mentioned, electron antineutrinos can be also detected in Borexino
via the neutron inverse $\beta$-decay (IBD) reaction on protons with an energy
threshold of $1.8$ MeV. The cross section of this process is much higher than
the one for $(\bar{\nu}_{e},e)$ elastic scattering. Additionally, the IBD
offers a unique signature given by temporal and spatial coincidence of two
correlated events associated with the detection of the positron and the
neutron. The prompt positron event with a visible energy of $E_{\nu}-0.784$
MeV accompanied by $\gamma$-rays from neutron capture mostly on protons or
carbon nuclei with a small probability. As a result, the rate of the events
selected as IBD candidates is much lower with respect to the rate of single
electron-like events.

The present study of the electron antineutrino flux associated with FRBs is
based on the data acquired between December 2007 and October 2017. The
procedure of IBD events selection and the energy spectrum of prompt positron
events are described in detail in Ago2021 . After the application of all
selection cuts the total live time decreases to $2485$ days and the final
efficiency of IBD reaction detection turns out to be $0.85$ Ago2021 . All
electron antineutrino candidates were identified with the main DAQ system as
well as the FADC system that provides a linear dynamic range up to $\sim 50$
MeV. In this analysis, we used the same $16.8$ MeV upper boundary of the
antineutrino energy range as in the case of the $(\nu,e)$-scattering analysis.

In order to estimate a limit on the $\bar{\nu}_{e}$ fluence, we also exploited
the fact that no events were observed with a prompt visible energy exceeding
$7.8$ MeV. Assuming two different neutrino energy spectra of monoenergetic and
supernova neutrinos and no observed events within the $7.8-16.8$ MeV interval,
we obtain the upper limits on the $\bar{\nu}_{e}$ fluences presented in Figure
9, Figure 10, Table 2 and Table 3 (sixth column in each Table).

The limits on the fluences of monoenergetic $\bar{\nu}_{e}$ with the energies
below $7.8$ MeV are based on the data from Ago2021  in which the upper limits
on the $\bar{\nu}_{e}$ flux in the $1.8-7.8$ MeV range were established for
$1$ MeV bins. The resulting conservative limits obtained by the Feldman-
Cousins procedure with the expected background $n_{\rm{bkg}}$ (excluding the
cosmogenic component Ago2021 ) are shown in Figure 9 and Table 2.

The limits in Figures 9 and 10 are the limits on the total antineutrino
fluence over a period of $2485$ days. Since the expected number of FRBs during
this time is $\sim 5\times 10^{6}$, the reduced limits per single FRB will be
$5\times 10^{6}$ times stronger.

The Figure 10 shows the IceCube upper limit on the $\bar{\nu}_{e}$ fluence of
the supernova spectrum with the mean neutrino energy $\langle E\rangle=15.6$
MeV and pinching the parameter $\alpha$=3 (shown with a circle) based on a
collective increase in the rate of hits in the detector in coincidence with 28
FRBs Aar2020 . The Borexino detector is capable of detecting single
$\bar{\nu}_{e}$ practically from the threshold of the IBD reaction that leads
to significantly higher sensitivity to low energy electron antineutrino
fluxes.

Figure 9: Upper limits on the fluences of monoenergetic $\widetilde{\nu_{e}}$
(90% C.L.). The inset shows the Borexino spectrum of
$\widetilde{\nu_{e}}$-like events Ago2021 . Figure 10: 90% C.L. upper limits
on the $\widetilde{\nu_{e}}$ fluence obtained with the SN neutrino spectrum
with respect to the mean neutrino energy $\langle E\rangle$. The bold point
shows the IceCube result Aar2020 . The inset shows the neutrino spectrum for
$\langle E\rangle=15.6$ MeV (1) and expected Borexino spectrum (2). Arrows
mark the analysis interval.

##  5 Conclusion

We looked for an excess in the number of events detected by Borexino produced
by neutrino–electron elastic scattering and the inverse beta-decay on protons
correlated to the most intense FRBs between 2007 and 2021. We found no
statistically significant increase in the number of events, with the visible
energy above $0.25$ MeV within time windows of $\pm 1000$ s centered at the
time of FRB arrivals. As a result, new limits on the fluence of monochromatic
neutrinos of all flavors were set for neutrino energies in the range of
$0.5-15$ MeV.

Another approach was based on the search for specific shapes of neutrino-
electron scattering in the high statistic Borexino spectrum. The strongest
limits on the fluences of monoenergetic neutrinos with energies in the range
of $0.5-50$ MeV and of the supernova neutrino spectrum given by the modified
Fermi-Dirac distribution were obtained for different effective neutrino
temperatures. Additionally, the inverse beta-decay reaction was considered to
set a new limit on the fluence of electron antineutrinos related to FRBs.

###### Acknowledgements.

The Borexino program is possible by funding from INFN (Italy), NSF (USA), DFG
and HGF (Germany), RFBR (Grant 19-02-00097 A), RSF (Grant 21-12-00063)
(Russia), and NCN (Grant No. UMO 2013/10/E/ST2/00180) (Poland). We acknowledge
the generous hospitality and support of the Laboratori Nazionale del Gran
Sasso (Italy).

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# Revisiting Taylor and the Trinity Test

Elizabeth Mone  College of Sciences, Georgia Institute of Technology, Atlanta,
Georgia  Pranay Seshadri  College of Engineering, Georgia Institute of
Technology, Atlanta, Georgia

## Abstract

The atomic bomb uses fission of heavy elements to produce a large amount of
energy. It was designed and deployed during World War II by the United States
military. The first test of an atomic bomb occurred in July of 1945 in New
Mexico and was given the name Trinity; this test was not declassified until
1949. In that year Geoffrey Ingram Taylor released two papers, detailing his
process in calculating the energy yield of the atomic bomb from pictures of
the Trinity explosion alone. Many scientists made similar calculations
concurrently, though Taylor is often accredited with them. Since then many
scientists have also attempted a to calculate a yield through various methods.
This paper walks through these methods with a focus on Taylor’s method—based
on first principles—as well as redoing the calculations that he performed with
modern tools. In this paper we make use of the state-of-the-art computer
vision tools to find a more precise measurement of the blast radius, as well
as using curve fitting and numerical integration methods. With more precise
measurements we are able to follow in Taylor’s footstep towards a more
accurate approximation.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Notation
    2. 2.2 Dimensional analysis approach
    3. 2.3 Taylor’s first principles approach
  3. 3 Revisiting Taylor’s Analysis
  4. 4 Discussion
  5. 5 Conclusions

##  1 Introduction

In most introductory engineering fluid mechanics and thermodynamic courses,
the work of Sir Geoffrey Ingram Taylor [14, 16] is presented as an important
application of dimensional analysis: using the Buckingham-Pi theorem [17] to
estimate the power of an atomic bomb by studying a few images of the blast.
Over the years there have been numerous surveys of Taylor’s work [5, 3, 4],
all emphasizing that Taylor _did not_ invoke the dimension reduction analysis
he is often incorrectly credited with. Arguably his work was far more
profound; starting with underlying equations and introducing a series of
simplifications in concert with observations to work out a chosen quantity of
interest. The approach Taylor takes is particularly salient and relevant for
the broad (beyond fluids and thermodynamics) Data-Centric Engineering (DCE)
audience, as it offers insights into studying data-driven physical processes.
In this brief survey article, the work of Taylor is revisited and communicated
using a series of illustrations, state-of-the-art computer vision software,
and commentary—particularly on uncertainty in equations and data. The aim is
to make Taylor’s original papers more accessible, and in doing so highlight
how much information regarding a problem can be obtained without resorting to
computational simulations. The exposition here is distinct from the
aforementioned surveys and is intended for the DCE readership that cuts across
engineering, statistics, and machine learning.

##  2 Background

Figure 1: Picture of the Trinity Explosion [1]

The Trinity explosion, shown in Figure 1, was the name for the first test of
the atomic bomb that occurred on July $16^{th}$, 1945 in New Mexico, just a
month before the bombs were dropped into Hiroshima and Nagasaki. The test bomb
released a huge amount of energy causing radioactive fallout in the area and a
blast that could be seen from space, reaching 40,000 feet in only 7 minutes
[13]. The atomic bomb would go on to end the war and have devastating
consequences for the civilians of Japan, and raise profound questions on the
very nature of our existence and the use of weapons of such cataclysmic
destruction. Those involved in the Manhattan project111We note that our
submission coincides with the release of Christopher Nolan’s _Oppenheimer_ ,
that details the life of the chief scientist of the Manhattan project, Robert
J. Oppenheimer., i.e., the name of the team that developed the atomic bomb, as
well as others such as Taylor, spent much time actualizing the idea of a
fission bomb and understanding the effects of such a device.

There are multiple approaches one can take when calculating the energy of a
blast, such as one from an atomic bomb. During the time of the Trinity test
and World War II there were three main minds working on the problem. Taylor,
the focus of this paper, is often cited with the calculation as he was the
only one who gave proof of the validity of a point source model, as well as
calculating the energy within a range of error. Two others, John von Neumann
and Russian Leonid Sedov also performed similar calculations at the time, and
actually achieved better accuracy than Taylor. Taylor’s approach was to use
partial differential forms of the equations for motion, continuity, and state
as well as integral forms of energy to arrive at an approximate formula for
the energy of the blast wave. Sedov and von Neumann followed similar
approaches that began with dimensional analysis. The two then determined
integral forms for energy and computed them analytically, though neither
directly calculated energy. A more in-depth exploration of their calculations
are provided in their own papers [10, 9] and in the work of Deakin [3]. All
three did not find exact values as the radiative energy was not taken into
account [3].

Table 1: Summary of Trinity test calculations. Author | Energy (kilotons TNT)
---|---
Taylor [16] | 16.8
Sedov, von Neumann [3] | 16.9
Deakin [3] | 17.5
Hanson [7] | 22.1
Truman [2] | 20
Groves [6] | 15-20

Table 1 shows the energy estimates of various authors including Taylor’s
estimate. Taylor’s, Deakin’s, Sedov’s, and von Neumann’s calculations are all
under the assumption that the specific heat ratio $\gamma$ is constant and
equal to $1.4$. Hanson’s calculations—done more recently—use physical samples
from the explosion to calculate the amount of plutonium and other elemental
constituents of the bomb, from which the yield is determined [7]. President
Truman and General Groves, who was involved in the Manhattan project, released
the values presented in the table once the information was declassified [3,
4].

###  2.1 Notation

Table 2 contains the variables used both in this paper and by Taylor in [14,
16], and will be used throughout this brief paper.

Table 2: Nomenclature. Symbol | Quantity | Symbol | Quantity
---|---|---|---
$p$ | pressure | $p_{0}$ | stagnation pressure
$\rho$ | density | $\rho_{0}$ | stagnation density
$u$ | radial velocity | $t$ | time
$E$ | energy | $\gamma$ | heat capacity ratio
$R$ | shock radius | $r$ | radial position

###  2.2 Dimensional analysis approach

Prior to detailing Taylor’s approach, it will be useful to touch upon
dimensional analysis with the Buckingham Pi theorem. This theorem states that
with $N$ variables and $K$ fundamental units there are $P=\left(N-K\right)$
dimensionless quantities [17]. By using this theorem we can identify the unit-
less quantities that will enable us to calculate the energy of the explosion.
To complete such an analysis the three scientists, Taylor, Sedov, and von
Neumann, reduced the explosion to starting from a point source and expanding
outwards in a spherical shock wave, rapidly releasing large amounts of energy.
The use of a shock wave suggests that the state of the air can be expressed in
two categories: internal shock wave properties and external conditions of the
undisturbed atmosphere. Therefore, we can devise seven variables: $E$ the
energy, $R$ the radius, $t$ the time, $p$ the static pressure of the wave,
$p_{0}$ the pressure of the undisturbed atmosphere or stagnation pressure,
$\rho$ the static density of the shock wave, and $\rho_{0}$ the undisturbed
density of the atmosphere or stagnation density. All variable units, when
broken down into their base units, contain three fundamental units: mass,
length, and time. In terms of the Buckingham Pi theorem there are $N=7$
variables and $K=3$ fundamental quantities, therefore this problem contains
$P=4$ dimensionless quantities [3].

We wish to focus on a quantity that involves the target variable of energy, as
well as easily measurable quantities which in this case are radius, time, and
the density of stagnant air. To find the exact form of the dimensionless
quantity we can use dimensional analysis to solve for energy. In SI units
energy is recorded in Joules $(J)$ which is equivalent to a Newton-meter
$(N\cdot m)$ or kilogram-meter-squared-per-second-squared $\left(kg\cdot
m^{2}/s^{2}\right)$. Therefore, an equation for energy must contain units of
mass $(kg)$, length $(m)$, and time $(s)$. Mass is not directly used within
the context of the problem, instead density (units of $kg/m^{3}$) takes its
place. In Taylor’s paper he is interested in the time since the blast, the
density of the undisturbed atmosphere (stagnation), and the radius of the
shock wave. Thus, the equation for energy has the form

$E=K\rho_{0}^{a}R^{b}t^{c}$ (2.1)

where $a,b,c,K$ are constant [5].

There are a few important points to note regarding the above. Firstly, energy
is per second squared and since neither density nor radius contain units of
time, the equation for energy must contain time, leading to the negative
second power, or $c=-2$. Secondly, density is the only variable that contains
units of mass, and since both energy and density contain mass only to the
first power, then $a$ must equal 1. However, the inclusion of density adds an
additional $m^{-3}$ to the equation that must be corrected in order to get
$m^{2}$ in the final form. Therefore, to fix this we need a factor of $m^{5}$
which can be provided by the radius, meaning $b=5$ and we have that

$E=K\rho_{0}R^{5}t^{-2},$ (2.2)

where $K$ is a constant and proven to be a function of $\gamma$ by Taylor
[14]. Dimensional analysis is taken from [5].

###  2.3 Taylor’s first principles approach

While many believe that Taylor used the above approach [3], he instead used
first principles and similarity assumptions to arrive at an equation for
energy. He eventually arrives at an expression of the same form as (2.2) but
is able to define the constant $K$. Taylor’s process and assumptions are
captured in the flow chart in Figure 2.

(1a) Pressure assumption:
$\frac{p}{p_{0}}=y=R^{-3}f_{1}$$f_{1}=f_{1}(\eta)$(1b) Density Assumption:
$\frac{\rho}{\rho_{0}}=\psi$, $\psi=\psi(\eta)$(1c) Velocity Assumption:
$u=R^{-3/2}\phi_{1}$$\phi_{1}=\phi_{1}(\eta)$$\eta=\frac{r}{R}$(2) Start with
Navier-Stokes: $\rho\frac{Du}{Dt}=-\nabla p+\rho g+\mu\nabla^{2}u$Assume
pressure, density, and velocity are only radially changingAssume no
gravitational or viscous effects and use (1a)$\rho\frac{Du}{Dt}=-\nabla
p$$\rho\left(\frac{\partial u}{\partial t}+(u\cdot\nabla)u\right)=-p_{0}\nabla
y$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial
r}=-\frac{p_{0}}{\rho}\frac{\partial y}{\partial r}$Use assumptions (1a-c)(3)
Assume a solution of the form $\frac{dR}{dt}=AR^{-3/2}$(3a) Use (3) to solve
for A: $At=\frac{2}{5}R^{-5/2}\rightarrow
A^{2}=\frac{4}{25}R^{5}t^{-2}$$-\left(\frac{3}{2}\phi_{1}+\eta\phi_{1}^{\prime}\right)R^{-5/2}\frac{dR}{dt}+R^{-4}\left(\phi_{1}\phi_{1}^{\prime}+\frac{p_{0}}{\rho_{0}}\frac{f_{1}^{\prime}}{\psi}\right)=0$$-A\left(\frac{3}{2}\phi_{1}+\eta\phi_{1}^{\prime}\right)+\phi_{1}\phi_{1}^{\prime}+\frac{p_{0}}{\rho_{0}}\frac{f_{1}^{\prime}}{\psi}=0$(4)
Assume continuity of mass: $\frac{\partial\rho}{\partial
t}+u\frac{\partial\rho}{\partial r}+\rho\left(\frac{\partial u}{\partial
r}+\frac{2u}{r}\right)=0$(5) Assume perfect gas:
$p\rho^{-\gamma}=constant\rightarrow\left(\frac{\partial}{\partial
t}+u\frac{\partial}{\partial r}\right)\left(p\rho^{-\gamma}\right)=0$(6)
Nondimensionalize $f_{1}$ and $\phi_{1}$ by using (3a), the speed of sound
$a$, and dimensional analysis$f=f_{1}\frac{a^{2}}{A^{2}}$,
$\phi=\frac{\phi_{1}}{A}$(7) Rankine-Hugoniot relations can be assumed so that
$y_{1}$, the value of $\frac{p}{p_{0}}$ at $\eta=1$, is large in order to
satisfy
(1a-c)$\frac{\rho_{1}}{\rho_{0}}=\frac{\gamma-1+(\gamma+1)y_{1}}{\gamma+1+(\gamma-1)y_{1}}\approx\frac{\gamma+1}{\gamma-1}$$\frac{U^{2}}{a^{2}}=\frac{1}{2\gamma}(\gamma-1+(\gamma+1)y_{1})\approx\frac{\gamma+1}{2\gamma}y_{1}$222Note:
There is a typographical error in [14], where Taylor flips this ratio; the end
result is the same and this does not lead to any
errors.$\frac{2(y_{1}-1)}{\gamma-1+(\gamma+1)y_{1}}\approx\frac{2}{\gamma+1}$$\psi\approx\frac{\gamma+1}{\gamma-1}$$f\approx\frac{2\gamma}{\gamma+1}$$\phi\approx\frac{2}{\gamma+1}$Use
(6)(8) Using (2, 5-7) we can find approximate forms of $\phi$, $f$, and
$\psi$$\phi=\frac{\eta}{\gamma}+\alpha\eta^{n}$$\textrm{ln}\left(f\right)=\textrm{ln}\left(\frac{2\gamma}{\gamma+1}\right)-\frac{2\gamma^{2}+7\gamma-3}{7-\gamma}\textrm{ln}\left(\frac{\gamma+1-\eta^{n-1}}{\gamma}\right)$$\textrm{ln}\left(\psi\right)=\textrm{ln}\left(\frac{\gamma+1}{\gamma-1}\right)+\frac{3}{\gamma-1}\textrm{ln}\left(\eta\right)-2\frac{\gamma+5}{7-\gamma}\left(\frac{\gamma+1-\eta^{n-1}}{\gamma}\right)$
Figure 2: Introduction of non-dimensional pressure, density and velocity along
with approximate ratios for large blast radii.

With this process Taylor used an equation of motion derived from the Navier-
Stokes under the assumption of incompressible flow, and no gravitational or
viscous effects, leading to the incompressible Euler equations. This process
lead him to three equations for the derivatives of the non-dimensionless
ratios $f$, $\psi$, and $\phi$ of pressure, density, and velocity
respectively. These are expressed as

$f^{\prime}\left(\left(\eta-\phi\right)^{2}-\frac{f}{\gamma}\right)=f\left(-3\eta+\phi\left(3+\frac{1}{2}\gamma\right)-2\frac{\gamma\phi^{2}}{\eta}\right)$
(2.3)

and

$\phi^{\prime}\left(\eta-\phi\right)=\frac{1}{\gamma}\frac{f^{\prime}}{\psi}-\frac{3}{2}\phi.$
(2.4)

$\frac{\psi^{\prime}}{\psi}=\frac{\phi^{\prime}+\frac{2\phi}{\eta}}{\eta-\phi}$
(2.5)

From (2.3), (2.4), and (2.5) Taylor performs a step-by-step calculation for
the values of $f$, $\phi$, and $\psi$ for a non-dimensional radius
$\eta\in[0,1]$, setting the initial conditions to be at $\eta=1$ and found by
the Rankine-Hugonoit solutions, which are calculated in box (7) in Figure 2.
However, these equations cannot be directly solved unless $f^{\prime}$ is
assumed small. Using that assumption he substitutes equation 2.5 into 2.3 and
solves for an approximate form of $\phi$. He notes this form, shown in blue in
Figure 3, does not agree well with the step calculations above $\eta\approx
0.8$, so he suggests the approximate form would not be linear. Instead, the
equation would contain a non-linear term of order $n$ which he calculates
using the initial conditions set forth by the Rankine-Huginoit solutions at
$\eta=1$. From the corrected approximate form of $\phi$ he used equations
(2.3), (2.4), and (2.5) to arrive at approximate forms for $\psi$ and $f$ as
well [14], shown in step (8) of Figure 2. These step-by-step values as well as
the approximated values are shown in Figure 3. One can observe that the step-
by-step calculations (black) have relatively good agreement with the
approximate forms (red), with almost complete agreement for $\psi$.

Figure 3: $f$, $\phi$, and $\psi$ vs $\eta$

Once he found these approximate forms he was able to calculate the constant
$K$ in his energy equation. In Figure 4 we show his process for formulating
the equation for energy. This marks the end of what he accomplished in his
first paper in regards to calculating the energy of the blast [14], as values
for radius and time are still needed. Once the Trinity test was declassified
he obtained those values and in his second paper [16] he linearly interpolates
data on the log scale plot of $R^{5/2}$ vs. $t$ and uses the intercept value
for $R^{5}t^{-2}$ in his energy calculations.

(9) Assume total energy consists of heat and kinetic
energy$K.E.=4\pi\int_{0}^{R}\frac{1}{2}\rho
u^{2}r^{2}dr$$H.E.=4\pi\int_{0}^{R}\frac{pr^{2}}{\gamma-1}dr$Use (1a-c) and
(3)$E=K.E.+H.E.=\rho_{0}A^{2}\left(2\pi\int_{0}^{1}\psi\phi^{2}\eta^{2}d\eta+\frac{4\pi}{\gamma(\gamma+1)}\int_{0}^{1}f\eta^{2}d\eta\right)$$E=\rho_{0}R^{5}t^{-2}K$
where
$K=\frac{4}{25}\left(2\pi\int_{0}^{1}\psi\phi^{2}\eta^{2}d\eta+\frac{4\pi}{\gamma(\gamma-1)}\int_{0}^{1}f\eta^{2}d\eta\right)$Implies
K is a function of $\gamma$ only and therefore
$E=S(\gamma)\rho_{0}R^{5}t^{-2}$, which follows from dimensional analysis
Figure 4: Energy calculations

##  3 Revisiting Taylor’s Analysis

In this paper we take the same approach as Taylor but make use of modern
techniques to improve upon his calculation. We started by reviewing his work
as documented in the previous section, to get a full understanding of his
approach and assumptions.

Figure 5: Estimating the blast radius from images taken at different times and
plotting the data on a logarithmic scale

To emulate his radius measurements we made use of the open source Computer
Visualization Annotation Tool (CVAT) [11] to annotate the sets of photos by
Mack [8] and the others presented by Taylor in his second paper [16], though
we could not find an image for $t=3.26$ ms. With this tool we made an outline
of the blast for earlier times as well as annotating two circles that
represent the smallest enclosing circle and the largest enclosed circle for
all times. For later time stamps, an arrow is provided by Mack [8] that
designates the shock front, and this is used as the maximum circle for these
times. The minimum enclosing circle was checked using the Welzl method [19]
and the maximum enclosed circle by testing circles centered at Voronoi points
[18] of the polygon created by the blast outline. An example of this method
along with the radii that Taylor provided for $t=0.36$ and $t=3.53$ ms is
shown in figure 5 (a) and (b). The larger of the two circles is closer to the
approximation that Taylor made, and is the more likely radius as the shock
wave is the radius we are interested in, which occurs in front of the fire
blast.

The assumption that Taylor makes based on his first principles analysis, shown
in figure 2 step (3a), is that the radius and time are related by
$At=\frac{2}{5}R^{-5/2}$ where A is a constant. Therefore, the radius and time
of the blast should correspond to:

$\frac{5}{2}\textrm{log}(R)=\textrm{log}(t)+\textrm{log}(n)$ (3.1)

Where $n$ is constant and $n^{2}$ corresponds to the value of $R^{5}t^{-2}$.
Using the radii found in the previous paragraph, we plot the logarithmic-scale
graph of $R^{5/2}$ vs $t$ as Taylor did in his second paper [16]; this is
shown in Figure 5. This figure contains three lines with a slope of one whose
y-intercepts correspond to values of $R^{5}t^{-2}$ for the minimum, maximum,
and average radii based on equation 3.1. These values are calculated using
using standard linear least-squares interpolation to fit a linear
approximation. Figure 5 also displays a zoomed-in view of the calculations
from $t\in[3.53,4.61]$ ms as the difference between our values and Taylor’s
lead to large discrepancy in the comparison of energy values shown in the next
section.

To calculate $K$ in the energy equation shown in Figure 4 we used the
equations for $\phi$, $\psi$, and $f$ put forth by Taylor in his first paper
[14] and shown in Figure 2 step (8). These integrals do not have an analytical
solution, so the open-source code equadratures [12] was used to
computationally estimate these integrals, i.e.,

$K\approx\frac{4\omega_{j}}{25}\left(2\pi\sum_{j=1}^{J=20}\psi\left(\eta_{j}\right)\phi^{2}\left(\eta_{j}\right)\eta_{j}^{2}+\frac{4\pi}{\gamma(\gamma-1)}\sum_{j=1}^{J=20}f\left(\eta_{j}\right)\eta_{j}^{2}\right)$
(3.2)

where $\left\\{\eta_{j},\omega_{j}\right\\}_{j=1}^{J=21}$ are Gauss-Legendre
quadrature points and weights with order 20 across the support
$\eta\in\left[0,1\right]$. This order was found to be sufficiently high for
numerical precision. We calculated $K$ using (3.2) for 10 values of $\gamma$
in the range $\left[1.2,1.667\right]$. The values calculated with this method
show minor differences with those Taylor calculated; see Table 3. It is
difficult to ascertain what approach Taylor took in calculating the integrals
as he simply states in [14] that he evaluates the integrals and uses the step-
by-step calculations.

$\gamma$ | This Paper | Taylor
---|---|---
1.2 | 1.720 | 1.727
1.3 | 1.145 | 1.167
1.4 | 0.852 | 0.856
1.667 | 0.495 | 0.487
Table 3: $K$ Values computed using (3.2) contrasted with those reported by
Taylor

Using the values for $R^{5}t^{-2}$ and K calculated in (3.1) and (3.2)
respectively, the total energy was estimated using the same equation as
Taylor, shown in figure 4 step (9). This was done for both the minimum and
maximum radii. These values are contrasted to those that Taylor presents in
his paper [16].

##  4 Discussion

Figure 6: Energy graphs: (left) energy estimates from this paper (blue error
bars) across different values of $\gamma$; (right) energy estimates from this
paper (box plot) for $\gamma=1.4$ at different times

Figure 6 shows the results laid out in the previous section. Figure 6(a) shows
ten energy values at $\gamma\in[1.2,1.7]$ where the $R^{5}t^{-2}$ value is
taken from the average of the two circles representing the blast, with the
blue error bars encompassing the standard deviation. The red shaded rectangle
in both (a) and (b) shows the energy range that was given by General Groves in
Table 1; his values were used as the range due to the fact that he was
directly involved in the Manhattan Project, and therefore the Trinity test.
Figure 6(b) shows the energy value using the $R$, $t$ pairs and the box plot
of the energy for those pairs. The scale of the x-axis is not linear and is
set to have equal spacing between time measurements to have a better
perspective of the box plots. The agreement with the actual energy estimate is
best between 1 to 3 ms. This is as expected as the point source solution is
invalid at earlier times such as the first time stamp of 0.10 ms.
Additionally, the spherical model is no longer valid at later times,
approximations at these times can be seen in figure 7.

From figure 6 we can see that our calculations, specifically for the larger of
the two radii, are in very good agreement with those of Taylor. As well as
being within the acceptable range of Energy values for
$\gamma\approx[1.2,1.4]$, which Taylor stated was a likely range for the value
of $\gamma$. In (b) however between 3.53-4.61 ms the energy values we
calculated are different from those calculated using Taylor’s radius and time
stamps. This is due to differences in the radii measured as these images are
unclear and the window is not wide enough to view the whole blast so a certain
radius has been assumed, which must have been different for us and Taylor.

Figure 7: CVAT Annotations Compared With Taylor’s Values at Later Times

Additionally, small differences in radii can lead to large differences in
energy as the radii is raised to the 5th power. Thus, where the radii
difference between Taylor and this paper for $t=3.53$ ms differ by only
$~{}2.3\ m$, the energy difference is around $3000$ tons.

##  5 Conclusions

Taylor’s approximation was, for the time and even today, a very good
approximation of the energy. He did all this using basic principles and a
series of assumptions in order to make all the calculations simpler to
compute. The github with all the code/annotation files is published here:
https://github.com/03emone/Trinity-Taylor-Files.

More broadly, the fusion of governing equations with data—inferred from a
series of images—for estimating a physical quantity of interest is a task that
likely underscores many DCE efforts. Our hope in writing this relatively short
paper is that the methods detailed serve as a useful example of how one can:
(i) leverage governing equation without necessarily resorting to costly
numerical simulations, and (ii) utilize dimensional analysis to gain valueable
quantitative insight into a problem.

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