# The Highest Energy HAWC Sources are Likely Leptonic and Powered by Pulsars

Takahiro Sudoh,11footnotetext: Corresponding author. Tim Linden  Dan Hooper

###### Abstract

The HAWC Collaboration has observed gamma rays at energies above 56 TeV from a
collection of nine sources. It has been suggested that this emission could be
hadronic in nature, requiring that these systems accelerate cosmic-ray protons
or nuclei up to PeV-scale energies. In this paper, we instead show that the
spectra of these objects favor a leptonic (inverse Compton) origin for their
emission. More specifically, the gamma-ray emission from these objects can be
straightforwardly accommodated within a model in which $\sim\mathcal{O}(10\%)$
of the host pulsar’s spindown power is transferred into the acceleration of
electrons and positrons with a power-law spectrum that extends to several
hundred TeV or higher. The spectral break that is observed among these sources
is naturally explained within the context of this simple model, and occurs at
the energy where the timescale for energy losses matches the age of the
pulsar. In contrast, this spectral feature cannot be straightforwardly
accommodated in hadronic scenarios. Furthermore, hadronic models predict that
these sources should produce more emission at GeV-scale energies than is
observed. In light of these considerations, we conclude that HAWC’s highest
energy sources should be interpreted as TeV halos or pulsar wind nebulae,
which produce their emission through inverse Compton scattering, and are
powered by the rotational kinetic energy of their host pulsar.

##  1 Introduction

The cosmic-ray spectrum is thought to be dominated by Galactic sources up to
energies of $\sim$ 1 PeV, corresponding to the spectral feature known as the
“knee”. The nature of the Milky Way’s so-called “PeVatrons” remains an open
and widely debated question. Among the proposed candidates, supernova remnants
have long been the most popular, and gamma-ray measurements support the
conclusion that these objects produce high-energy protons [1, 2]. That being
said, it has also been argued that such sources may be unable to accelerate
protons beyond a few hundred TeV [3, 4]. Other PeVatron candidates include the
Milky Way’s supermassive black hole [5, 6, 7], and clusters of young and
massive stars [8]. The sources of the highest energy Galactic protons are
expected to generate gamma rays through the production and decay of neutral
pions, resulting in a power-law gamma-ray spectrum that extends to $\sim$100
TeV.

Pulsars can also accelerate electrons and positrons up to energies of at least
$\sim$100 TeV. Due to the Klein-Nishina suppression associated with inverse-
Compton scattering, electrons and positrons in this energy range lose much of
their energy to synchrotron emission, suppressing the leptonic production of
$\sim$100 TeV-scale gamma rays. Through this distinction, very high-energy
gamma-ray telescopes provide us with one of the most powerful ways to
discriminate between accelerators of hadronic and leptonic cosmic rays.

The High Altitude Water Cherenkov (HAWC) observatory has recently produced a
catalog of nine gamma-ray sources detected at energies above 56 TeV. Three of
these sources have been observed above 100 TeV, making this the highest energy
gamma-ray catalog reported to date [9].222The Tibet air shower array has also
reported the detection of emission above 100 TeV from the Crab Nebula [10].
Given that all nine of these sources are located within $0.5^{\circ}$ of a
known pulsar, it appears likely that they are associated with this class of
objects. Furthermore, eight of these nine pulsars are quite young
($t_{c}\equiv P/2\dot{P}\sim 1-50\,{\rm kyr}$), and have exceptionally high
spindown power ($\dot{E}>10^{36}\,{\rm erg/s}$). This information suggests two
possible interpretations. On the one hand, the gamma-ray emission from these
sources could be leptonic in nature, powered by the host pulsars’ rotational
kinetic energy. Alternatively, the observed emission could be hadronic,
revealing these systems’ supernova remnants to be among the Milky Way’s long-
sought-after PeVatrons.

In this paper, we examine the luminosity, spectrum, and morphology of the very
high-energy sources observed by HAWC in order to evaluate whether they are
more likely to be leptonic sources powered by the rotational kinetic energy of
the young pulsar, or hadronic PeVatrons powered by the supernova remnant. We
find that the former interpretation is favored by three factors. First, the
spectra of these sources can be easily accommodated by simple models in which
very high-energy electrons and positrons are accelerated with a power-law
spectrum. In contrast, hadronic models cannot straightforwardly account for
the spectra observed from several of HAWC’s highest energy sources. Second,
the gamma-ray luminosities observed from these sources are well-matched to the
past integrated spindown power of their host pulsars. And third, the spectral
break observed among these systems at $E_{\gamma}\sim\mathcal{O}(10\,{\rm
TeV})$ is naturally explained by the guaranteed suppression of the inverse
Compton scattering cross section by Klein-Nishina effects, and the energy
dependence of the electron/positron energy-loss time-scale, which is smaller
than the pulsar age for the highest-energy leptons.

In light of these considerations, we conclude that HAWC’s highest energy
sources are likely to be TeV halos and/or pulsar wind nebulae, with gamma-ray
emission that is 1) powered by the rotational kinetic energy of the host
pulsar, and 2) produced through inverse Compton scattering.

##  2 TeV Halos and Pulsar Wind Nebulae

Observations by HAWC and Milagro have detected diffuse multi-TeV emission from
the regions surrounding the nearby Geminga and Monogem pulsars [11, 12, 13,
14]. The spectrum and intensity of this emission indicate that these sources
convert a significant fraction ($\sim$ $10\%$) of their total spindown power
into very high-energy electron-positron pairs. Furthermore, each of these TeV
halos exhibits an angular extension of $\sim$ $2^{\circ}$ (corresponding to
$\sim$ $25\,{\rm pc}$), indicating that cosmic-ray propagation in the vicinity
of these pulsars is much less efficient than is typically experienced
elsewhere in the interstellar medium [15, 16, 17, 18, 19, 20, 21, 22].

Looking beyond the specific examples of Geminga and Monogem, observations by
HAWC (and HESS [23, 24]) have led to the identification of a new class of
spatially extended, multi-TeV gamma-ray sources, powered by the rotational
kinetic energy of pulsars, and which produce their observed emission through
the inverse Compton scattering of very high-energy electrons and positrons on
the surrounding radiation field [25, 26]. A large fraction of the sources
detected by HAWC [11, 12, 27] have been shown to be spatially coincident with
a pulsar, and all indications suggest that TeV halos are a generic feature of
middle-aged pulsars (whether or not TeV halos also accompany millisecond
pulsars is an open question [28]). These observations suggest that nearby TeV-
halos are likely responsible for the observed cosmic-ray positron excess [15,
29, 30, 29, 31, 32] (for earlier work, see Refs. [33, 34, 35]), as well as the
diffuse TeV excess observed by Milagro [36], and could plausibly dominate the
TeV-scale emission observed from the Galactic Center by HESS [37] (as opposed
to the hypothesis that this emission is produced by a Galactic Center PeVatron
[7]). Extrapolating to the Milky Way’s larger pulsar population, we expect
HAWC and the Cherenkov Telescope Array (CTA) [38] to ultimately detect $\sim$
$50-240$ TeV halos [26], including many whose pulsed radio and gamma-ray
emission is not beamed in the direction of Earth [25].

When referring to TeV halos, we adopt a definition for this source class which
requires that the high-energy electrons and positrons responsible for the
observed gamma-ray emission propagate via diffusion, rather than convection or
advection. This distinguishes TeV halos from pulsar wind nebulae, for which
advection plays an important and often dominant role (for a review, see Ref.
[39]). TeV halos are also more spatially extended than typical pulsar wind
nebulae. Pulsar wind nebulae are created when the energetic outflow from a
pulsar collides with the ambient medium (supernova ejecta or interstellar
medium), resulting in a shockwave surrounding a diffuse plasma of electrons
and positrons. Like TeV halos, the emission from a pulsar wind nebula is
powered by its pulsar’s rotational kinetic energy, and is leptonic in nature.
We consider it to be plausible that HAWC’s highest energy sources could be a
combination of TeV halos, pulsar wind nebulae, and objects that are currently
in a transitional state between these two classifications.333An alternative
definition has been put forth by Giacinti et al. [40] which classifies a
region as a TeV halo if it contains an overdensity of relativistic electrons
and positrons around a pulsar, and if the pulsar and associated supernova
remnant does not dominate the dynamics or composition of the interstellar
medium in that region. Compared to our definition, this choice leads Giacinti
et al. to classify many objects that we would call TeV halos as pulsar wind
nebulae, despite the fact that the dynamics assumed by both groups are
similar.

##  3 Associating Very High-Energy HAWC Sources With Known Pulsars

We begin by describing the known pulsars that could potentially be responsible
for powering the nine sources found in the eHWC ($>$ $56\,{\rm TeV}$) catalog
[9]. In Table 1, we list some of the selected characteristics of these
pulsars, as reported in the Australia Telescope National Facility (ATNF)
pulsar catalog [41]. This list of pulsars was identified in Ref. [9] based on
their locations (within $0.5^{\circ}$ of the corresponding HAWC sources), and
their high spindown power. In some cases, other nearby pulsars are not listed,
primarily when observations indicate that they have substantially lower values
of $\dot{E}$.

Comparing the values of the spindown luminosity of these pulsars,
$\dot{E}/4\pi d^{2}$, to their integrated gamma-ray flux, $F_{\gamma}$, it is
clear that their rotational kinetic energy is more than sufficient to produce
the very high-energy gamma-ray emission reported by HAWC.444Note that the
values of $F_{\gamma}$ given in Table 1 are based on an extrapolation to
energies lower than those measured by HAWC, and thus may somewhat overestimate
the total gamma-ray flux above 0.1 TeV. More quantitatively, this comparison
suggests that between 0.5% and 20% of these pulsars’ spindown power goes into
the production of gamma rays above 0.1 TeV (consistent with the range of
values required to explain the TeV halos of Geminga and Monogem [15, 25, 26]).
The only exception to this is eHWC J0534+220, which would be far brighter if
the spindown power of its pulsar was transferred into gamma rays with similar
efficiency. Given that this source is associated with the Crab Nebula, we do
not find this result particularly surprising. In particular, the magnetic
field of the Crab pulsar wind nebula is significantly stronger than that found
among typical pulsar wind nebulae (or TeV halos), causing a large fraction of
its spindown power to be transferred into the production of synchrotron
emission [42, 43, 44, 45, 46].

HAWC Source | Pulsar Candidate | Distance | $\dot{E}$ | $\dot{E}/4\pi d^{2}$ | $F_{\gamma}$ |  $F_{\gamma}$ / ($\dot{E}/4\pi d^{2}$) | $P$ | $\dot{P}$ | $t_{c}\equiv P/2\dot{P}$ | Radio
---|---|---|---|---|---|---|---|---|---|---
|  | $({\rm kpc})$ | $({\rm erg/s})$ | $({\rm TeV/cm^{2}/s})$ | $({\rm TeV/cm^{2}/s})$ |  | $({\rm s})$ | $\times 10^{14}$ | $({\rm kyr})$ |
eHWC J0534+220 | PSR J0534+2200 | 2.00 | $4.5\times 10^{38}$ | $5.9\times 10^{-7}$ | $1.8\times 10^{-10}$ | 0.0003 | 0.033 | 42.1 | 1.26 | ${\rm Yes}$
eHWC J1809-193 | PSR J1809-1917 | 3.27 | $1.8\times 10^{36}$ | $8.8\times 10^{-10}$ | $8.5\times 10^{-11}$ | 0.1 | 0.083 | 2.55 | 51.4 | ${\rm Yes}$
– | PSR J1811-1925 | 5.00 | $6.4\times 10^{36}$ | $1.3\times 10^{-9}$ | – | 0.07 | 0.065 | 4.40 | 23.3 | ${\rm No}$
eHWC J1825-134 | PSR J1826-1334 | 3.61 | $2.8\times 10^{36}$ | $1.1\times 10^{-9}$ | $2.3\times 10^{-10}$ | 0.2 | 0.101 | 7.53 | 21.4 | ${\rm Yes}$
– | PSR J1826-1256 | 1.55 | $3.6\times 10^{36}$ | $7.8\times 10^{-9}$ | – | 0.03 | $0.110$ | 12.1 | 14.4 | ${\rm No}$
eHWC J1839-057 | PSR J1838-0537 | 2.0 | $6.0\times 10^{36}$ | $7.8\times 10^{-9}$ | $4.1\times 10^{-10}$ | 0.05 | 0.146 | 47.2 | 4.89 | ${\rm No}$
eHWC J1842-035 | PSR J1844-0346 | 2.4 | $4.2\times 10^{36}$ | $3.8\times 10^{-9}$ | $7.6\times 10^{-11}$ | 0.02 | 0.113 | 15.5 | 11.6 | ${\rm No}$
eHWC J1850+001 | PSR J1849-0001 | 7.00 | $9.8\times 10^{36}$ | $1.0\times 10^{-9}$ | $4.5\times 10^{-11}$ | 0.05 | 0.039 | 1.42 | 43.1 | ${\rm No}$
eHWC J1907+063 | PSR J1907+0602 | 2.37 | $2.8\times 10^{36}$ | $2.6\times 10^{-9}$ | $4.6\times 10^{-11}$ | 0.02 | 0.107 | 8.68 | 19.5 | ${\rm Yes}$
eHWC J2019+368 | PSR J2021+3651 | 1.80 | $3.4\times 10^{36}$ | $5.5\times 10^{-9}$ | $2.7\times 10^{-11}$ | 0.005 | 0.104 | 9.57 | 17.2 | ${\rm Yes}$
eHWC J2030+412 | PSR J2032+4127 | 1.33 | $1.5\times 10^{35}$ | $4.4\times 10^{-10}$ | $5.1\times 10^{-11}$ | 0.1 | 0.143 | 1.13 | 201 | ${\rm Yes}$

Table 1: Properties of the pulsars potentially associated with the highest
energy HAWC sources [9], as reported in the ATNF Catalog [41]. Note that eHWC
J1809-193 and eHWC J1825-134 each have two possible pulsar associations. When
possible, we show the distance determinations as provided in the ATNF catalog,
and for the cases of PSR J1838-0537, PSR J1844-0346, and PSR J1849-0001, we
show those from Refs. [47], [48], and [49], respectively. The quantity
$F_{\gamma}$ is the integrated gamma-ray flux between 0.1 and 100 TeV,
adopting the best-fit power law parameters as reported in Ref. [12]. In the
rightmost column, we report “Yes” if the ATNF catalog reports a detection of
emission at any of 0.4, 1.2, or 2 GHz. Figure 1: The locations and spatial
extent of the nine very high-energy HAWC sources described in Ref. [9]. The
black ‘$e$’ and the surrounding black circle in each frame denotes the best-
fit location and 68% containment of the source, as reported in the eHWC
catalog (no circle is shown in the case of eHWC J0534+220, as its morphology
is consistent with that of a point source). The symbol ‘2’ represents the
best-fit center of the source as reported in the previous 2HWC catalog. The
symbol ‘P’ (and in cases with multiple possible associations, the symbol ‘S’)
represents the location of the associated pulsars (see Table 1). Also shown
are the location and spatial extent of any nearby TeV gamma-ray sources (red),
as reported by HESS, VERITAS, and/or MAGIC, as well as the GeV counterparts as
detected by Fermi (blue) [50, 51]. The dotted blue circles represent the best-
fit spatial extent of the GeV emission, as reported in Ref. [52]. Figure 2:
The gamma-ray spectra of the nine very-high energy HAWC sources reported in
the eHWC catalog [9] (black stars), as well as the best-fit power law from the
earlier 2HWC catalog [12] (gray dashed). Also shown are the spectra of the
potential counterparts measured by HESS, VERITAS, and Fermi [53, 54, 50, 52,
51]. We note that the gamma-ray fluxes reported by different telescopes (such
as HAWC and HESS) can in some cases be different, likely resulting from the
larger (typically $\sim 2^{\circ}$) angular extension adopted in the analysis
of the HAWC Collaboration. Note that for eHWC sources which only have an
integrated flux above 56 TeV reported, we have adopted a spectral index of 2.5
in this figure.

In Fig. 1, we show the locations of the nine very-high-energy HAWC sources,
along with the positions of any pulsars and gamma-ray sources that are
potentially associated with them. In each frame, we show the location of the
HAWC source as reported in Ref. [9], as well as in the earlier 2HWC catalog
[12]. Following Ref. [12], we also show any nearby TeV gamma-ray sources, as
reported by HESS, VERITAS, and/or MAGIC. In addition, we show any GeV
counterparts555We disregard Fermi-LAT sources that are identified as GeV
pulsars, as the pulsed spectrum from these sources falls off rapidly above 10
GeV, and does not appreciably contribute to the TeV emission. that are
associated with a TeV source according to Fermi’s 4FGL catalog [50] (unless
otherwise noted below), as well as those very recently reported in Ref. [52].
When there is no counterpart listed in the 4FGL catalog, we include any other
4FGL gamma-ray sources that lie within 0.5∘ of a given eHWC source. Note that
the circles shown in this figure do not represent the uncertainties pertaining
to a given source’s location, but rather the best-fit angular extent as
reported by each collaboration. Also note that in the case of J0534+220 (the
Crab Nebula), the gamma-ray emission is observed to be point-like and
coincident with the radio pulsar PSR J0534+2200. While it is very likely that
most of these pulsars are authentically associated with the eHWC source in
question, it would not be surprising if one or two were the result of chance
alignment, and not physically related to the corresponding gamma-ray source.

We will now comment on these associations on a source-by-source basis:

  * •

eHWC J0534+220 is associated with the Crab Nebula [12]. This source has been
detected over a wide range of wavelengths [42, 43, 44, 45], and is a point-
like and well-localized gamma-ray source, coincident with the radio pulsar PSR
J0534+2200.

  * •

eHWC J1809-193 is associated with the unidentified TeV gamma-ray source HESS
J1809-193 [12], and with the extended GeV source 4FGL J1810.3-1925e [50].

  * •

eHWC J1825-134 is associated with HESS J1825-137 [12], which is known to be a
spatially extended pulsar wind nebula. This source is also close to another
source (HESS J1826-130). However, this second HESS source is dimmer than the
2HWC source by nearly an order of magnitude (see Fig. 2), suggesting that the
association is unlikely. This HAWC source is also associated with the extended
Fermi source 4FGL J1824.5-1351e [50]. We additionally show the extent of this
source as reported in Ref. [52], using their “IEM-4FGL” interstellar emission
model and the 30-100 GeV energy bin.

  * •

eHWC J1839-057 is associated with HESS J1841-055 [12], which is a complex
region containing two supernova remnants, three bright pulsars, and one X-ray
binary. While HESS J1837-069 is also considered as a potential counterpart to
the 2HWC source [12], this HESS source is located relatively far ($\sim$
$1^{\circ}$) from the best-fit position of eHWC source. Since this separation
is larger than the HAWC angular resolution and extension of the eHWC source
($\sim$ $0.3^{\circ}$), this association seems unlikely. HESS J1841-055 is
also associated with 4FGL J1840.9-0532e [50].

  * •

eHWC J1842-035 is associated with the unidentified source HESS J1843-033 [12].
Although the 2HWC paper also considers HESS J1844-030 as a potential
association [12], the flux of this HESS source is dimmer than HAWC’s
measurement by nearly an order of magnitude (see Fig. 2), making this
association unlikely. eHWC J1842-035 is located within $0.5^{\circ}$ of three
Fermi sources: 4FGL J1842.5-0359c, 4FGL J1842.2-0328, and 4FGL J1843.7-0326.

  * •

eHWC J1850+001 is associated with the pulsar wind nebula HESS J1849-000 [12]
and is located within $0.5^{\circ}$ of the Fermi source 4FGL J1851.8-0007c.

  * •

eHWC J1907+063 is associated with the pulsar wind nebula MGRO J1908+06 [12],
which is also known as HESS J1908+063, and is also associated with 4FGL
J1906.2+0631 [50]. We additionally show the extent of this source as reported
in Ref. [52], using their “IEM-4FGL” interstellar emission model. The flux
reported in Ref. [52] for this source is very different from that listed in
the 4FGL catalog, due to this source’s significant spatial extension. In
presenting our results, we will use the spectrum for this source as reported
in Ref. [52].

  * •

eHWC J2019+368 is associated with the source VER J2019+368 [12], an extended
source that covers two pulsars and one star-forming region. We present the
spectra of this source as reported in Ref. [54]. There are no sources in the
4FGL catalog located near eHWC 2019+368, nor is there any extended emission
reported in Ref. [52].

  * •

eHWC J2030+412 is associated with the pulsar wind nebula TeV J2031+4130 [12].
We present the spectra reported by the MAGIC Collaboration in Ref. [53]. The
flux measured by HAWC comes from a larger angular region and is much brighter
than that measured by VERITAS and MAGIC, suggesting a contribution from
additional components. Although there are no sources near eHWC 2030+412 in the
4FGL catalog, a potential GeV counterpart (identified using Fermi data) has
been reported in Ref. [51].

In Fig. 2, we show the spectra of these nine very-high-energy HAWC sources as
reported in the eHWC catalog [9], as well as the best-fit power law from the
earlier 2HWC catalog [12]. Also shown in these frames are the spectra of the
potential counterparts measured by HESS, VERITAS, and Fermi. In most cases,
these measurements lead to a consistent picture across a wide range of
energies. We note that measurements by different telescopes (such as HAWC and
HESS) can in some cases be different, due to the treatment of these sources’
spatial extension.

##  4 Pulsars and Inverse Compton Emission

In this section, we describe our calculation of the gamma-ray spectrum
produced through the inverse Compton scattering of a population of very high-
energy electrons and positrons, injected with a given spectrum and over a
given time profile.

Very high-energy electrons and positrons lose energy through a combination of
inverse Compton scattering and synchrotron processes, leading to the following
energy loss rate [55]:

$\displaystyle-\frac{dE_{e}}{dt}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{4}{3}\sigma_{T}u_{i}S_{i}(E_{e})\bigg{(}\frac{E_{e}}{m_{e}}\bigg{)}^{2}+\frac{4}{3}\sigma_{T}u_{\rm
mag}\bigg{(}\frac{E_{e}}{m_{e}}\bigg{)}^{2}$ (4.1) $\displaystyle\equiv$
$\displaystyle b(E_{e})\,\bigg{(}\frac{E_{e}}{{\rm TeV}}\bigg{)}^{2},$

where $\sigma_{T}$ is the Thomson cross section and

$\displaystyle b$ $\displaystyle\approx$ $\displaystyle 1.02\times
10^{-13}\,{\rm TeV}/{\rm s}\,$ (4.2) $\displaystyle\times$
$\displaystyle\bigg{[}\sum_{i}\frac{u_{i}}{{\rm eV}/{\rm
cm}^{3}}\,S_{i}(E_{e})+\frac{u_{\rm mag}}{{\rm eV}/{\rm cm}^{3}}\bigg{]}.$

The sum in this expression is carried out over the various components of the
radiation backgrounds, consisting of the cosmic microwave background (CMB),
infrared emission (IR), and starlight (star). We take each of these radiation
components to have a blackbody spectrum and adopt the following values for
their energy densities and temperatures: $u_{\rm CMB}=0.260$ eV/cm3, $u_{\rm
IR}=0.30$ eV/cm3, $u_{\rm star}=0.3$ eV/cm3, $T_{\rm CMB}=2.7$ K, $T_{\rm
IR}=20$ K, and $T_{\rm star}=5000$ K [56]. For the energy density of the
magnetic field, we adopt as our default value $u_{\rm mag}=0.224$ eV/cm3,
corresponding to $B\simeq 3\,\mu$G. At relatively low electron energies, these
parameters correspond to a value of $b\simeq 1.2\times 10^{-13}$ TeV/s
($S_{i}\approx 1$). At very high energies ($E_{e}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}m^{2}_{e}/2T$),
however, the inverse Compton scattering will be well outside of the Thomson
regime, and Klein-Nishina suppression will play an important role. For our
calculations, we utilize the full Klein-Nishina cross-section formula, as
calculated in Ref. [55] and as implemented in the publicly available code
naima [57] (see also Refs. [58, 59]). For illustrative purposes, however, the
key features of Klein-Nishina suppression can be seen more clearly using the
following approximate expression [60]:

$S_{i}(E_{e})\approx\frac{45\,m^{2}_{e}/64\pi^{2}T^{2}_{i}}{(45\,m^{2}_{e}/64\pi^{2}T^{2}_{i})+(E^{2}_{e}/m^{2}_{e})}.$
(4.3)

For electrons of a given energy, the effects of Klein-Nishina suppression are
most pronounced for the highest-energy target photons. For the very-high-
energy ($E_{e}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}\,{\rm TeV}$) electrons/positrons of most interest to
this study, energy losses from inverse Compton scattering are dominated by
scattering with the IR background, as well as the CMB. At energies greater
than $\sim$ $50\,{\rm TeV}$, the CMB alone dominates this process.

Over a period of time in which an electron or positron of energy $E_{e}$ loses
a small quantity of energy, $\Delta E_{e}$, that particle will generate the
following spectrum of inverse Compton emission:

$\displaystyle\frac{dN_{\gamma}}{dE_{\gamma}}(E_{\gamma},E_{e})$
$\displaystyle=$ $\displaystyle A(E_{e},\Delta E_{e})\,f_{\rm
ICS}(E_{e})\,l_{e}$ $\displaystyle\times$
$\displaystyle\int\frac{dn}{d\epsilon}(\epsilon)\,\frac{d\sigma_{ICS}}{dE_{\gamma}}(\epsilon,E_{\gamma},E_{e})\,d\epsilon,$

where $dn/d\epsilon$ is the spectrum of target radiation, which we take to the
be the sum of the blackbody distributions described above. The quantity $A$ is
set by requirement that $\Delta E_{e}=\int
dE_{\gamma}\,E_{\gamma}\,dN_{\gamma}/dE_{\gamma}$, and $f_{\rm ICS}(E_{e})$ is
the fraction of the electron or positron’s energy losses that are from inverse
Compton scattering (as opposed to synchrotron). The differential cross section
for inverse Compton scattering is given by [61]:

$\displaystyle\frac{d\sigma_{ICS}}{dE_{\gamma}}(\epsilon,E_{\gamma},E_{e})$
$\displaystyle=$ $\displaystyle\frac{3\sigma_{T}m_{e}^{2}}{4\epsilon
E_{e}^{2}}\,\bigg{[}1+\bigg{(}\frac{z^{2}}{2(1-z)}\bigg{)}$ (4.5)
$\displaystyle+\bigg{(}\frac{z}{\beta(1-z)}\bigg{)}$ $\displaystyle-$
$\displaystyle\bigg{(}\frac{2z^{2}}{\beta^{2}(1-z)}\bigg{)}-\bigg{(}\frac{z^{3}}{2\beta(1-z)^{2}}\bigg{)}$
$\displaystyle-$
$\displaystyle\bigg{(}\frac{2z}{\beta(1-z)}\bigg{)}\ln\bigg{(}\frac{\beta(1-z)}{z}\bigg{)}\bigg{]},$

where $z\equiv E_{\gamma}/E_{e}$ and $\beta\equiv 4\epsilon E_{e}/m_{e}^{2}$.
At energies within the range measured by HAWC, inverse Compton scattering
generally yields photons with energies not very far below that of the incident
electrons and positrons, $E_{\gamma}\sim E_{e}$.

As time passes, pulsars slow down and lose rotational kinetic energy,
transferring much of this energy into the acceleration of particles which
produce the radio, gamma-ray, and other non-thermal emission that is observed
from these objects. From the measured quantities $P$ and $\dot{P}$, we can
define the pulsar’s characteristic age, $t_{c}$:

$t_{c}\equiv\frac{P}{2\dot{P}}=\frac{n-1}{2}(t_{\rm age}+\tau),$ (4.6)

where $n$ is the braking index, $t_{\rm age}$ is the age of the pulsar, and
$\tau$ is its spindown timescale. From the spindown equations, we can write
the spindown timescale as

$\tau=\frac{2t_{c}}{n-1}\left(\frac{P_{0}}{P}\right)^{n-1},$ (4.7)

where $P_{0}$ is the initial period of the pulsar. For a given set of $P_{0}$
and $n$, these equations determine $\tau$ and $t_{\rm age}$. The spindown
power of a pulsar evolves as follows:

$\dot{E}(t)=4\pi^{2}I\frac{\dot{P}}{P^{3}}=\dot{E}_{0}\left(1+\frac{t}{\tau}\right)^{-\frac{n+1}{n-1}},$
(4.8)

where $\dot{E}_{0}$ is the initial spindown power, given by

$\dot{E}_{0}=4\pi^{2}I\frac{\dot{P}}{P^{3}}\left(1+\frac{t_{\rm
age}}{\tau}\right)^{\frac{n+1}{n-1}}.$ (4.9)

These equations leave us with $P_{0}$, $n$, and $I$ as free parameters. Unless
otherwise stated, we will adopt $I=10^{45}$ g cm2 and $n=3$ throughout this
study.

##  5 Results

Figure 3: The spectrum of inverse Compton emission predicted from the pulsars
PSR J1826-1256, PSR J1826-1334, PSR J1907+0602, and PSR J2021+3651, compared
to the measured gamma-ray spectra from the associated HAWC sources (see Fig.
2). We have parameterized the injected electron/positron spectrum as
$dN_{e}/dE_{e}\propto E_{e}^{-\gamma}\exp(-E_{e}/E_{\rm cut})$ and show
results for three values of $E_{\rm cut}$. We adopt a spectral index of
$\gamma=2.1$ in this figure, except for in the bottom left frame, where we
have used $\gamma=2.0$. For each pulsar, we have selected values for the
electron/positron efficiency ($\eta_{e}$) and initial period ($P_{0}$) which
lead to reasonable agreement with the observed spectrum and intensity of each
source. We also show the value of each pulsar’s age, as calculated from the
value of $P_{0}$. We emphasize that the cutoff observed in these spectra above
$\sim$ $1-10\,{\rm TeV}$ is an unavoidable consequence of the model, and
occurs when the age of a pulsar exceeds the timescale for electron/positron
energy losses.

In Fig. 3, we show the spectra of inverse Compton emission predicted from the
pulsars PSR J1826-1256, PSR J1826-1334, PSR J1907+0602, and PSR J2021+3651,
comparing our results with the gamma-ray observations of each associated HAWC
source. In each case, we have parameterized the injected electron/positron
spectrum as a power-law with an exponential cutoff, $dN_{e}/dE_{e}\propto
E_{e}^{-\gamma}\,\exp(-E_{e}/E_{\rm cut})$. Along with $\gamma$ and $E_{\rm
cut}$, we treat as free parameters each pulsar’s initial period, and the
fraction of its spindown power that goes into the production of electrons and
positrons integrated above 10 GeV, $\eta_{e}$. For each pulsar’s distance,
period, and rate of change of its period, we adopt the values reported in the
Australia Telescope National Facility (ATNF) pulsar catalog [41] (as shown in
Table 1). We adopt $\gamma=2.0$ in the bottom left frame of Fig. 3, and 2.1 in
the other three frames. In each frame, we show results for three choices of
$E_{\rm cut}$, and have selected values of $\eta_{e}$ and $P_{0}$ (obtaining
the corresponding value of $t_{\rm age}$) which, when possible, lead to
reasonable agreement with the observed spectral shape and intensity of each
source.

Figure 4: As in Fig. 3, but for selected parameter variations, and focusing on
the case of eHWC J1825-134/PSR J1826-1334. In the upper left frame, we
consider three different combinations for the values of the initial period and
pulsar age, while in the lower left frame we show results for three choices of
the energy density of the magnetic field. In the upper right frame, we
consider scenarios in which the electrons and positrons are able to escape the
emitting region on a timescale given by $t_{\rm esc}=t_{\rm
diff,K}\times(E_{e}/{\rm TeV})^{-1/3}$, corresponding to the energy dependence
predicted for Kolmogorov diffusion. In the lower right frame, we show results
for four choices of the pulsar’s braking index, $n$. In each panel, unless
stated otherwise, we have adopted $P_{0}=30\,{\rm ms}$ ($t_{\rm age}=19\,{\rm
kyr}$), and $\eta_{e}=0.7$.

As seen in Fig. 3, the gamma-ray spectrum that is produced through inverse
Compton scattering is automatically suppressed at energies above $\sim$
$10\,{\rm TeV}$, for which the age of these pulsars exceeds the timescale for
electron/positron energy losses, $t_{\rm age}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}(bE_{e})^{-1}$.
Around this energy, the spectrum of the ambient electrons and positrons
transitions from the injected index ($\gamma$) to a significantly softened
index ($\gamma-1$). Note that this suppression occurs even if the injected
spectrum does not have a cutoff in the relevant energy range ($E_{\rm cut}\gg
100\,{\rm TeV}$). Klein-Nishina effects also influence the exact shape of the
high-energy spectrum. In these results, there is no indication of a cutoff in
the injected spectrum of electrons and positrons, suggesting that these
sources accelerate such particles to at least several hundred TeV. At lower
energies, $E_{e}\ll(b\,t_{\rm age})^{-1}\sim 20-50\,{\rm TeV}$, the electrons
and positrons that have been injected from the pulsar over its history have
not lost much of their initial energy. In this limit, the normalization of the
gamma-ray spectrum is set by the total integrated energy that has been
injected from the pulsar in the form of electrons and positrons, which is
proportional to $\eta_{e}\,t_{\rm age}/P^{2}_{0}$.

From the lower frames of Fig. 3, we see that the pulsars PSR J1907+0602 and
PSR J2021+3651 can produce the emission observed by HAWC and Fermi, in each
case requiring an efficiency similar to that of Geminga or Monogem,
$\eta_{e}\sim 0.1$. In the upper frames, we see that either PSR J1826-1256 or
PSR J1826-1334 (or some combination thereof) could be responsible for the
gamma-ray emission attributed to eHWC J1825-134, although the latter would
require a high value of $\eta_{e}\sim 0.7$, and neither of these pulsars
provides a particularly good fit in the $\sim 1-10$ TeV range.

In Fig. 4, we consider some variations regarding our parameter choices,
focusing on the case of eHWC J1825-134 and its corresponding PSR J1826-1334.
In the upper left frame, we consider three different combinations for the
values of the initial period and pulsar age. As described above, this does not
impact the spectrum at high energies, where only the current power of the
injected electrons/positrons determines the normalization. At lower energies,
however, the normalization scales as $\eta_{e}t_{\rm age}/P^{2}_{0}$,
corresponding to the total energy injected into high-energy electrons and
positrons over the life of the pulsar. In the lower-left frame, we consider
variations to the energy density of the magnetic field, showing results for
$u_{\rm mag}=0.224$ eV/cm3 (our default value), 0.5 eV/cm3, 5.0 eV/cm3 and 20
eV/cm3, corresponding to $B=3.0$ $\mu$G, 4.5 $\mu$G, 14.2 $\mu$G and 28.3
$\mu$G, respectively. By increasing the energy density of the magnetic field,
a larger fraction of the energy in electrons and positrons is lost to
synchrotron, suppressing the gamma-ray emission that is produced through
inverse Compton scattering.

Thus far in our calculations, we have assumed that the electrons and positrons
remain within the TeV halo or pulsar wind nebula, and do not escape via
diffusion. This corresponds to one or both of the following conditions being
satisfied: $t_{\rm esc}\gg t_{\rm age}$ or $t_{\rm esc}\gg(bE_{e})^{-1}$,
where $t_{\rm diff}$ is the timescale for particles to escape the TeV halo via
diffusion. In the upper right frame of Fig. 4, we consider a class of
scenarios in which the electrons/positrons instead escape on a timescale given
by $t_{\rm esc}=t_{\rm diff,K}\times(E_{e}/{\rm TeV})^{-1/3}$, corresponding
to the energy dependence predicted for Kolmogorov diffusion. More
quantitatively, we reduce the number of electrons and positrons within the
emission region by a factor of $e^{-\delta t/t_{\rm esc}}$ in each timestep of
length $\delta t$. The impact of diffusion is significant only when $t_{\rm
esc}$ is smaller than both the age of the pulsar (which, in this case, is 19
kyr), and the timescale for energy losses (which is $\sim$ $10^{3}\,{\rm yr}$
at the highest energies shown, and $\sim$ $10^{5}\,{\rm yr}$ at TeV-scale
energies). This could, in principle, significantly suppress the predicted
gamma-ray emission, but only in scenarios with very rapid diffusion (much
faster than favored by the spectra of Geminga and Monogem [15]). We do not
expect diffusion to play an important role in most of the sources under
consideration in this study.

Lastly, in the lower right frame of Fig. 4, we show results for four choices
of the pulsar’s braking index, $n$. The spectrum of this particular source is
somewhat better fit for lower values of the braking index.

Figure 5: The fraction of each pulsar’s spindown power that must go into the
production of electrons and positrons, $\eta_{e}$, in order to explain the
intensity of the gamma-ray emission observed by HAWC for each of the pulsars
potentially associated with a source in the eHWC catalog. These results are
shown for two choices of the injected spectral index, and as a function of
either the energy density of the magnetic field, or the timescale for
diffusion. In each case, we have adopted $P_{0}=30$ ms, and $E_{\rm cut}=1$
PeV. For reference, we show as vertical lines the values of these quantities
as measured in the local interstellar medium, assuming a 10 pc emitting region
for our calculation of the diffusion timescale. From these results, it is
clear that the intensity of the observed gamma-ray emission can be produced
for reasonable efficiencies, $\eta_{e}\sim\mathcal{O}(0.1)$, so long as 1) the
magnetic field is not much stronger than in the local interstellar medium
($u_{\rm mag}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}1$ eV/cm3), 2) diffusion is highly suppressed in the
region of inverse Compton scattering ($t_{\rm diff}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}10$ kyr), and 3)
the injected spectral index is somewhat hard ($\gamma\sim 2$). These
characteristics are consistent with those observed from the Geminga and
Monogem TeV halos.

In Fig. 5, we show the values of $\eta_{e}$ that are required to explain the
intensity of the gamma-ray emission observed by HAWC from each of the sources
in the eHWC catalog, for each of the potentially associated pulsars listed in
Table 1 (with the exception of the Crab Pulsar, which requires a significantly
smaller value of $\eta_{e}$ for a given value of $u_{\rm mag}$). We show
results for two choices of the injected spectral index ($\gamma=2.0$, 2.3),
and present these results as a function of either the energy density of the
magnetic field, or the timescale for diffusion. In each case, we have adopted
$P_{0}=30$ ms, and $E_{\rm cut}=1$ PeV. For reference, we show as vertical
lines the values of these quantities as measured in the local interstellar
medium.

From Fig. 5, it is clear that in the case of $\gamma=2$, the intensity of the
observed gamma-ray emission can be produced for reasonable efficiencies,
$\eta_{e}\sim\mathcal{O}(0.1)$, so long as 1) the magnetic field is not much
stronger than in the local interstellar medium ($u_{\rm mag}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1$ eV/cm3), and 2)
diffusion is highly suppressed in the region of inverse Compton scattering
($t_{\rm diff}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10$ kyr), as is known to be the case for both the
Geminga and Monogem TeV halos. Comparing this to the results found in the
$\gamma=2.3$ case, it is clear that somewhat hard spectral indices are also
required to produce the observed emission, again consistent with that observed
from Geminga and Monogem. Note that in calculating the values of $\eta_{e}$,
we have adopted a power-law injected spectrum of electrons and positrons,
integrated to a minimum energy of 10 GeV. Multiwavelength studies of pulsar
wind nebulae often require the electrons/positrons to be injected with a
broken power-law spectrum, with $E_{\rm br}\sim$ 0.1 TeV (see, for example,
Ref. [62]). Adopting such a function can reduce the required efficiency by a
factor of approximately $\sim(E_{\rm br}/10~{}\rm GeV)^{\gamma-2}$.

##  6 Comparison of Hadronic and Leptonic Models

Figure 6: A comparison of the gamma-ray spectra observed from four of the
sources in the eHWC catalog to that predicted from both leptonic and hadronic
models. For three of these four sources, the hadronic emission that is
predicted at GeV-scale energies significantly exceeds that observed by Fermi.
We have adopted a spectrum of protons that is described by a power-law with an
exponential cutoff, $dN/dE\propto E^{-p}\exp(-E/E_{\rm cut})$, with $E_{\rm
cut}=1$ PeV and extending to a minimum energy of 10 GeV. In each frame, we
have adopted a value of $p$ which best accommodates the spectra reported by
HAWC.

The spectral feature that is observed from these sources around $\sim$1-10 TeV
is a natural consequence of our leptonic model, and occurs at the energy where
the timescale for energy losses matches the age of the pulsar. In hadronic
models, no such feature is expected. Furthermore, hadronic models that can
explain the spectrum observed from these sources at very high energies
generally predict far more emission than is observed in the GeV range [50, 52,
51]. This disfavors models in which these sources produce their gamma-ray
emission primarily through hadronic processes.

In Fig. 6, we compare the gamma-ray spectra observed from four of the sources
in the eHWC catalog to that predicted in both leptonic and hadronic models. As
we did to calculate the emission from inverse Compton scattering, we made use
of the publicly available code naima to determine the spectra of hadronic
gamma-ray emission [57] (see also Refs. [63]). For three of these four
sources, the hadronic emission that is predicted at GeV-scale energies
significantly exceeds that observed by Fermi. In this figure, we have adopted
a spectrum of protons that is described by a power-law with an exponential
cutoff, $dN/dE\propto E^{-p}\exp(-E/E_{\rm cut})$, extending from a minimum
energy of 10 GeV, and with $E_{\rm cut}=1$ PeV. In each frame, we have adopted
values of $p$ which best accommodate the spectrum reported by HAWC ($p=2.65$
in the upper left, 2.15 in the upper right, 2.2 in the bottom left, and 2.0 in
the bottom right). In the case of the Crab Nebula (upper left), we adopt a
braking index of $n=2.5$ in the leptonic model, and adopt a large value for
the strength of the magnetic field, $B=90~{}\mu$G (in order to be compatible
with the emission observed in the $\sim\mathcal{O}(0.1\,{\rm GeV})$ range,
which is attributed to synchrotron). In the case of the Crab Nebula, we have
also included synchrotron photons as targets of inverse Compton scattering,
within a region taken to be 2 parsecs in radius. For PSR J1825-1334 we have
adopted $n=1.5$, while we have retained our default choice of $n=3$ for the
three remaining pulsars. For each curve, we adopt a normalization to match the
HAWC data.

Since each eHWC source is located in a region where young pulsars, and hence
recent star formation and supernova explosion exist, there may also be
contributions from hadronic processes to the gamma-ray flux. Comparing our
model curves with GeV data indicates that, hadronic component could also
produce a significant fraction of the observed flux for eHWC J2019+368, while
likely at most $\sim$10$\%$ level for the other sources. We note that our
hadronic models assume that protons are injected with a single power law. If
we assume a broken power law to reduce the energy injected into GeV-scale
protons, more contributions from hadronic processes could be allowed without
violating the Fermi data. Such a hard spectrum could be realized in a scenario
where very-high-energy protons that escape early from the SNR travel into
massive gas clouds, producing gamma rays there, while lower-energy protons
remain confined in the accelerator (e.g., [64]). However, the eHWC sources
shown in Figure 6, except for eHWC 1825-134, have not been reported to have a
clear spatial correlation with gases, which challenges this scenario.
Regarding eHWC 1825-134, mixed hadronic/leptonic contributions is a plausible
scenario. ([65], see also Sec. 7)

##  7 Discussion and Summary

The nature of the highest energy HAWC sources is a subject of considerable
interest, which has recently been discussed by a number of authors and
collaborations. In particular, the HAWC Collaboration has used multiwavelength
data to argue that the gamma-ray emission from eHWC J2019+368 is leptonic in
origin [66], in agreement with our assessment of this source. More recently,
HAWC has performed a stacking analysis of ten pulsars that are not associated
with any eHWC sources, identifying evidence of gamma-ray emission at energies
above 56 TeV [67]. More broadly speaking, they conclude from this information
that high-spindown power pulsars universally produce extremely high energy
photons. In Ref. [65], members of the HAWC Collaboration argued that eHWC
J1825-134 can be separated into four components: diffuse Galactic emission,
HAWC J1826-128 (the counterpart to HESS J1826-130), HAWC J1825-138 (the
counterpart to HESS J1825-137), and the newly discovered source HAWC
J1825-134. The spectrum of the emission associated with HAWC J1825-134, and
its spatial correlation with dense gas, favors a hadronic interpretation for
this emission. In contrast, the other two HAWC sources that contribute to eHWC
J1825-134 are likely leptonic in origin.

Beyond the HAWC Collaboration, Di Mauro et al. [52] have shown that the
spectra of three eHWC sources (eHWC J1825-134, J1907+063, and J2019+368) can
be well fit by leptonic models, in concordance with our conclusions (see also,
Ref [68]). On similar grounds, Fang et al. [69] have argued that eHWC
J2019+368 is likely leptonic in nature. In contrast, the authors of Ref. [70]
have claimed that HESS J1809-193 (associated with eHWC J1809-193) is likely to
be a hadronic source.

In this paper, we have studied each of the nine gamma-ray sources contained in
the eHWC catalog, expanding on the previous work described above, and
identifying significant evidence that their emission is likely leptonic in
origin. In particular, the gamma-ray emission from these sources can be
straightforwardly accommodated within a model in which $\sim\mathcal{O}(10\%)$
of the host pulsar’s spindown power is transferred into the acceleration of
electrons and positrons with a simple power-law spectrum. The spectral break
that is observed among these sources is an unavoidable consequence of this
model.

In contrast, the spectral feature that is observed from these sources is not
expected in hadronic scenarios, which also predict far more emission at GeV-
scale energies than is observed. For the three eHWC sources with detailed
spectral information, we can rule out scenarios in which a significant
fraction of their observed emission is hadronic in origin. While it remains
possible that one or more of the other six eHWC sources could produce hadronic
emission (see, for example, Ref. [70]), we stress that nothing in our analysis
differentiates any of these sources from those that are clearly leptonic in
nature. This disfavors an interpretation of these sources as the long-sought-
after Galactic PeVatrons.

Furthermore, all nine sources in the eHWC catalog can be powered by the
rotational kinetic energy of their host pulsar, requiring efficiencies that
are similar to those of the Geminga and Monogem TeV halos. Also like Geminga
and Monogem, diffusion appears to be suppressed within the emission regions of
these sources, and electrons and positrons are injected into these regions
with a relatively hard spectral index, $\gamma\sim 2$.

In light of the considerations described in the paragraphs above, we conclude
that HAWC’s highest energy sources are likely to be TeV halos or pulsar wind
nebulae, which produce their gamma-ray emission through inverse Compton
scattering, and which are powered by the rotational kinetic energy of their
host pulsar. We find no evidence that this class of sources produces
significant gamma-ray emission through hadronic processes, or accelerates
protons to PeV-scale energies.

Acknowledgments. We would like to thank Mattia Di Mauro for providing us with
the data from Ref. [52]. TS is supported by a Research Fellowship of Japan
Society for the Promotion of Science (JSPS) and by JSPS KAKENHI Grant No. JP
18J20943. TL is partially supported by the Swedish Research Council under
contract 2019-05135, the Swedish National Space Agency under contract 117/19
and the European Research Council under grant 742104. DH is supported by the
Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the
U.S. Department of Energy, Office of High Energy Physics. In this work, we
have made use of naima [57], gammapy666https://www.gammapy.org[71, 72],
astropy777http://www.astropy.org [73, 74], matplotlib [75], numpy [76], and
scipy [77].

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# Towards Sustainable Development: A Novel Integrated Machine Learning Model
for Holistic Environmental Health Monitoring

Anirudh Mazumder Texas Academy of Mathematics & Science Denton, United States
of America<EMAIL_ADDRESS>Sarthak R. Engala Texas Academy of
Mathematics & Science Denton, United States of America<EMAIL_ADDRESS>Aditya Nallapuraju Texas Academy of Mathematics & Science Denton, United
States of America<EMAIL_ADDRESS>
###### Abstract

Urbanization enables economic growth but also harms the environment through
degradation. Traditional methods of detecting environmental issues have proven
inefficient. Machine learning has emerged as a promising tool for tracking
environmental deterioration by identifying key predictive features. Recent
research focused on developing a predictive model using pollutant levels and
particulate matter as indicators of environmental state in order to outline
challenges. Machine learning was employed to identify patterns linking areas
with worse conditions. This research aims to assist governments in identifying
intervention points, improving planning and conservation efforts, and
ultimately contributing to sustainable development.

###### Index Terms:

Machine learning, environmental sustainability, air quality, water quality

##  I Introduction

Urbanization has become a global phenomenon in the 21st century, marking a new
era of global modernity [1]. The migration of people to urban centers has
brought about significant economic opportunities, increased social mobility,
and improved infrastructure in the form of transportation, sanitation, and
technology networks [2]. However, rapid and uncontrolled urbanization has also
resulted in intensive environmental exploitation and degradation, posing a
complex sustainability challenge for municipalities [3]. Traditional methods
of detecting environmental problems rely heavily on direct observation, manual
data collection, and periodic reporting, which are often inefficient, costly,
and provide only fragmentary insights. Consequently, separate indices have
been created for different environmental quality metrics, such as air and
water quality [4], to address these issues; however, there have yet to be
concerted efforts to integrate these indices to provide a comprehensive,
holistic view of overall environmental health.

Machine learning techniques hold promise for addressing complex classification
problems involving multidimensional, evolving data streams [6]. Using
traditional statistical methods, machine learning models can identify
intricate patterns that are not easily detectable. They also can adapt to new
data over time continuously. These characteristics make machine learning well-
suited for developing scalable, flexible classification systems. Given
environmental health data’s complex, multifaceted nature, machine learning
presents a promising avenue for developing an integrated environmental
classification model to promote effective sustainability planning.

Therefore, it was hypothesized that a machine learning model could be
developed to classify environments into different labels that holistically
define overall environmental health, integrating across the distinct indices
for water, air, soil, biodiversity, and other facets, which would help address
the key limitations of existing environmental monitoring methods, relying on
fragmented indices and incomplete data sources. Current methods also often
require extensive manual data collection and analysis, which is inefficient
and costly. A machine learning model could be trained on large,
multidimensional environmental datasets to identify intricate patterns and
relationships between different aspects of environmental health. Such a model
could provide a more holistic, automated assessment of current conditions and
emerging threats. Additionally, machine learning techniques can continuously
adapt to new data over time, enabling the model to detect novel environmental
issues that static statistical methods may miss. Overall, the strengths of
machine learning in handling complex, evolving classification problems make it
a promising approach to pursue developing an integrated, efficient, and
proactive environmental health monitoring system.

##  II Methodology

###  II-A Materials

Python was the primary programming language utilized for developing and
testing the machine learning algorithm throughout the research. Indian Air
Quality and Indian Water Quality datasets were the primary data sources.

###  II-B Pipeline

####  II-B1 Algorithm

The machine lear˜ning pipeline was split up into multiple different steps, to
train a machine learning model which is able to generate high accuracy and
precise outputs. The layout of the different steps can be seen in Figure 1

Figure 1: Flow chart showing the novel pipeline that was built

####  II-B2 Data Preprocessing

The algorithm’s initial stage involved preprocessing the data in the datasets.
For the Indian Air Quality dataset, the primary metrics were chemical
concentrations and particles present in the air. These metrics were utilized
to calculate the air quality index for each data point in the dataset. The
same process was repeated for the Indian Water Quality dataset to enable the
classification of air and water quality.

####  II-B3 Concatenation

To ensure the alignment of the data, a similarity between the datasets was
identified, which was found in the location of the metrics within the
datasets. Since the alignment was based on the state, one of the datasets
needed to be averaged for each metric by state. In this case, the water
quality dataset was chosen for averaging as it contained more data for each
state, resulting in a more representative and generalizable value for the
dataset compared to the air quality dataset. After averaging the water quality
dataset by state, it was aligned with the air quality dataset. This alignment
process was crucial to facilitate the concatenation of the two datasets,
ensuring that random state datasets were not simply combined. Once the data
was aligned, the datasets were concatenated to create a unified dataset
suitable for our specific task.

####  II-B4 Labeling

In order to create a machine learning model capable of accurately assessing
the environmental challenges faced by cities, relevant data needed to be
labeled. Without labels, the model would have had to rely solely on
unsupervised learning techniques, which allow models to find patterns in data
without guidance. However, supervised learning models are more accurate for
specialized classification tasks [JMLR:v24:22-0019]. To enable supervised
learning, data about cities worldwide was manually reviewed, and each city was
assigned a label describing its environmental conditions. Six labels were
defined, ranging from very negative environmental conditions (severe) to very
positive ones (good). The labeling process involved examining factors related
to water quality, including contaminant levels, pollution sources, and
treatment methods, as well as air quality data covering pollution levels,
emissions sources, and respiratory illness rates. By comparing the labeled
dataset, the model could learn which environmental characteristics matched
each descriptive label, allowing it to classify new, unlabeled city data based
on patterns from the labeled examples. The labels enabled supervised learning
that produced a significantly more accurate classification model than
unsupervised methods could have.

####  II-B5 Machine Learning

To effectively learn from the data, the dataset was split into two sets: 80%
for training and 20% for testing. A stacking ensemble model was employed,
consisting of three machine learning models: Random Forest Classifier (RFC),
Support Vector Classifier (SVC), and Logistic Regression. The RFC was chosen
for its ability to utilize multiple decision trees, specified to be ten
predictors, to predict the best classifier based on a list of metrics about
the environment. The SVC was selected for its ability to separate all data
points into their respective classes based on the feature space, allowing the
algorithm to identify outliers and prevent them from influencing the decision
made by the algorithm. Lastly, the logistic regression model was chosen for
its ability to understand how many independent variables can influence a
specific dependent variable, making it an accurate predictor of the label to
which the data is classified.

The unique abilities of these different models were combined to maximize the
predictor’s accuracy. Ensemble models, which combine multiple models to create
a more robust prediction model, were employed. Since the base models were all
different machine learning algorithms, a stacking-based ensemble was
performed. This approach finds the optimal combination of multiple other
machine learning models and can utilize multiple machine learning algorithms.
A stacking algorithm was used instead of other ensembling methods because it
can utilize all the different machine learning algorithms while other models
cannot. Additionally, it is based on a hierarchical structure rather than a
flat equal-level model, which is essential because it allows the algorithm to
use the strengths of all the base models. Most importantly, stacking
algorithms can improve the accuracy of the overall machine-learning algorithm.

####  II-B6 Feature Selector

To perform feature selection, the labels were converted into numerical
representations. Specifically, the values 1 were assigned for ”good,” 2 for
”satisfactory,” 3 for ”fair,” 4 for ”poor,” 5 for ”bad,” and 6 for ”severe.”
This numerical representation allowed for the computation of the correlation
between the highest values and the features contributing to this correlation
[7].

###  II-C Labeling Algorithm

The dataset used in the study had two unique features: the water quality index
and the air quality index. A labeling classifier was built to develop a
specific environmental quality classifier based on these features. The air
quality index was classified into six categories: good, satisfactory,
moderate, poor, very poor, and severe. In contrast, the water quality index
was classified into five categories: excellent, good, satisfactory, poor, and
severe.

To create a holistic label that describes the environment, an approximate
average between the feature values from the two labels was calculated. The
labeler was classified using a specific structure, which involved considering
the following scenarios:

  * •

If air and water quality are the same (e.g., both ’Good’), the overall
classification is set to the same value (i.e., ’Good’ in this case).

  * •

If air quality is ’Good’ and water quality is ’Excellent,’ the overall
classification is set to ’Excellent.’

  * •

If air quality is ’Good’ and water quality is ’Good,’ the overall
classification is set to ’Good.’

  * •

If air quality is ’Good’ and water quality is ’Satisfactory,’ the overall
classification is set to ’Good.’

  * •

If air quality is ’Good’ and water quality is ’Poor,’ the overall
classification is set to ’Fair.’

  * •

If air quality is ’Good’ and water quality is ’Severe,’ the overall
classification is set to ’Bad.’

The model incorporated both air quality and water quality labels, but with a
higher weight placed on air quality. This weighting scheme was used because
the air quality labels provided more granular scoring compared to the broader
water quality labels. The air quality data enabled finer distinctions between
different pollution levels. In contrast, the water labels categorized water
quality in broader terms. Since the air quality data offered greater
complexity and ability to discern subtle differences, it received a higher
weightage in the model. This helped account for its higher granularity while
still allowing the model to take advantage of its informative scoring. At the
same time, the water quality labels provided useful complementary signals,
just weighted lower than air quality. By giving air quality higher influence,
the model could benefit from its granular detail while seamlessly integrating
the water quality data as well. The weighted, multi-faceted labeling enabled
robust predictions that took advantage of the strengths of both data sources.

##  III Results

###  III-A Accuracy

To rigorously evaluate the performance of the proposed machine learning model,
a standard train-test split technique was utilized. The original dataset was
divided such that 80% of the data was allocated to a training set for model
optimization, while the remaining 20% was held out as a test set. This testing
data was not used in model training and provided an unbiased estimate of
generalizability and accuracy. The trained model was applied to the test set,
and predictions were generated for all test points. These predictions were
compared to the known ground truth labels, allowing the calculation of the
test accuracy. The model achieved an overall accuracy of 0.99% on the unseen
test data. This high accuracy indicates excellent generalization ability and
suggests that the patterns learned during training generalize very well to
new, unseen data. To gain further insight, a confusion matrix was generated to
break down the prediction performance for each class. As seen in the total
confusion matrix in Figure 2, the exceptionally high diagonal and low off-
diagonal values confirm the high accuracy across all classes. Out-of-sample
testing helps prevent overfitting, and the decisive test performance
demonstrates that the model has learned robust representations that go beyond
just fitting the training data. This rigorous train-test evaluation provides
confidence that the model will maintain high accuracy when deployed in
practice. It also highlights the importance of proper evaluation techniques to
ensure that the model’s performance is not overestimated.

Figure 2: Confusion matrix showing the predicted vs the actual values

###  III-B Feature Selector

In this research, a feature selection was performed using a correlation-based
approach to determine which water and air quality parameters were most
predictive of an unfavorable environment. The Pearson correlation coefficient
of each feature with the target variable of bad water quality was computed,
and features were ranked based on the absolute value of the correlation which
can be seen in Figure 3. The feature selection results showed that fecal
coliform levels had the highest positive correlation with bad water quality,
indicating that higher fecal coliform counts are associated with a
deteriorating environment. Total suspended solids (spm) and biochemical oxygen
demand (BOD) were the most correlated parameters, followed by nitrate, total
coliform, pH, and conductivity. Notably, dissolved oxygen (DO) exhibited a
negative correlation, implying that lower DO relates to worsening conditions.

Feature selection is a critical modeling step that can significantly improve
the performance and interpretability of environmental quality assessment. By
systematically ranking the predictive capacity of each feature, one can focus
on just those variables that have the strongest relationship with
environmental health. This data-driven approach helps weed out all the non-
informative ”noise” features that offer little signal. Identifying the
critical predictive indicators provides vital insights that can guide targeted
remediation efforts to improve environmental conditions. For example, knowing
fecal coliform is strongly associated with degraded water environments
suggests wastewater treatment interventions could have a high impact. Reducing
organic pollution and nitrate levels also appears promising based on the
strength of their correlations. Isolating the prime factors driving
environmental deterioration allows policymakers and sustainability
professionals to allocate resources more effectively to initiatives and
technologies capable of addressing the most influential issues. The power of
feature selection is not just improving predictive modeling but revealing
actionable opportunities for meaningful interventions to restore the ecosystem
and community wellbeing on a broader scale.

Figure 3: Feature selection matrix showing correlations

##  IV Discussion

###  IV-A Conclusion

As the world continues to urbanize at an unprecedented rate, the environment
is deteriorating under the immense strains that cities place on natural
resources, with no way to quantify or identify which areas are most at risk
accurately. Expanding populations and industrialization results in polluting
air and waterways. Ultimately, correctly identifying emerging environmental
degradation patterns is imperative to maintaining ecological balance and
protecting public health in urban communities. By training models on available
datasets of regional water and air quality indicators, a Pearson correlation
coefficient calculator provides a data-driven approach to not only accurately
identify the areas with environmental deterioration but also pinpoint the
specific features - suspended particulate matter (SPM) and fecal coliform -
that most directly lead to pollution and other problems.

In conclusion, the hypothesis was proven true as a comprehensive machine
learning pipeline was developed that could assess overall environmental
quality by combining air and water quality data, while achieving high
accuracy. The pipeline involved key steps of preprocessing the raw data,
aligning and concatenating the datasets, labeling each data point with a
categorical quality rating, training an ensemble model, and performing feature
selection. Through this process, a robust classifier was created that
demonstrated strong performance in distinguishing between favorable and
challenged environments. The feature correlations uncovered provide actionable
insights into the primary factors tied to environmental degradation. By fusing
disparate datasets and leveraging advanced machine learning, this work enables
data-driven monitoring and diagnosis of ecological health. The pipeline and
models constructed lay the groundwork for developing predictive tools to guide
policies and interventions to safeguard communities. Incorporating additional
data sources and applying techniques like deep learning could further enhance
predictive accuracy. Overall, this research highlights the potential of AI to
drive evidence-based decisions for environmental sustainability.

###  IV-B Future Works

The current predictive modeling pipeline for environmental quality provides a
robust starting point. However, to enhance model accuracy and
comprehensiveness, incorporating diverse complementary metrics capturing
additional ecological perspectives is necessary. Specifically, adding
agricultural data such as soil health, crop patterns, and vegetation would
provide insights into a critical human impact area. Climate and weather data
could help control for periodic and geographic variations. Biodiversity
statistics lend a biological view of ecosystem stability. Furthermore,
integrating social and economic factors like demographics and development
would offer a human lens regarding the societal drivers and implications of
environmental challenges. Blending these multidimensional datasets within the
flexible modeling framework constructed here would allow for a more nuanced,
holistic understanding of intricate ecological issues, leading to more
targeted, effective sustainability policies and interventions. This future
work highlights the power of an interdisciplinary data science approach that
unifies statistical and domain expertise to address society’s most pressing
environmental concerns [8].

Another area for future improvement is enhancing the machine learning
techniques used. While the current pipeline relies on conventional methods
like random forests and support vector machines, adopting modern deep-learning
approaches could improve performance. Graph neural networks could precisely
better model the complex interdependencies in ecological systems by
representing relationships between entities like species, nutrients, and
pollutants. Recurrent networks could identify temporal dynamics by analyzing
time series data. Multi-modal models fusing numeric and image data could
incorporate satellite imagery to assess vegetation and land use changes.
Additionally, active learning techniques could be leveraged to achieve more
accurate labeling and reduce the labeled data needed. By selectively sampling
uncertain and informative points to be manually labeled, far fewer labeled
examples would be required. Furthermore, transfer learning from related tasks
could provide proper initialization and representations to enable training
with limited environment-quality data. In summary, applying cutting-edge
profound learning advances in architecture, temporal modeling, multi-modality,
active learning, and transfer learning could significantly boost model
capabilities.

Another critical area for future work is model interpretation and result
visualization. While predictive accuracy is essential, generating trusted and
actionable insights from AI is equally crucial. Techniques like LIME could
highlight the features most influential to predictions, enabling an
understanding of model behavior. Interactive visualizations could be developed
to allow filtering and drilling down on regions and factors of concern.
Automated report generation summarizing project conclusions using natural
language techniques would make results more accessible to diverse
stakeholders. Critically evaluating results for potential biases and
uncertainties is essential to ensure fair, transparent conclusions. Overall,
substantial opportunities exist to enhance model interpretability, interactive
analysis, and responsible, transparent reporting to convert predictions into
meaningful, trustworthy insights for sustainability decision-making.

One exciting area for future development is applying the insights from this
modeling work to design effective filtration systems that can actively remove
key contaminants degrading environmental quality. Since influential factors
like nitrate, phosphate, and particulate matter have been identified through
feature selection, we can strategically target filtration mechanisms for these
harmful substances. For example, the model results suggest that deploying
nitrate and phosphate adsorbent materials at agricultural runoff sites could
have an outsized impact on water quality. Additionally, high-efficiency
particulate air (HEPA) filtration systems in industrial facilities could help
lower ambient particulate matter levels in surrounding communities. By
quantitatively linking contaminants to environmental quality deterioration, we
can develop technical remediation solutions tailored to address the most
impactful pollution sources. New filtration materials and technologies like
nanofibers and reactive membranes show promise for removing specific threats
identified by the model. An interdisciplinary approach combining cutting-edge
data science with expertise in environmental engineering can help design the
next generation of intelligent, sustainable filtration infrastructure to
improve ecological health actively.

Another important next step is to integrate this predictive modeling pipeline
with real-time sensing and monitoring infrastructure. The current model
provides powerful batch analytics on static historical data. However, dynamic
real-time environment quality assessment could be enabled by connecting these
models to live sensor feeds. Sensors for parameters identified as highly
predictive by the model, like nitrate levels or particulate matter, could be
deployed in bodies of water or air quality monitoring stations. The incoming
sensor data could then be fed into the optimized AI models to generate
frequent environment quality updates, alerts for deteriorating conditions, and
anticipate future states.

Furthermore, connecting the pipeline outputs to visualization dashboards would
allow stakeholders to monitor overall environmental health continually and
dive into granular details as needed. Investing in this end-to-end
infrastructure would transform broad batch analytics into precisely-targeted,
responsive decision support. Policymakers could rapidly respond to emerging
threats before they escalate. Scientists could discern the environmental
impacts of perturbations in near real-time. Ultimately, transitioning the
advanced analytics developed here into an operationalized system would
maximize its value for continuous sustainability progress.

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