# Revised the $\gamma$-ray emission from SNR CTB 109 with Fermi-LAT

Yuliang Xin Qizhen Zhao  Xiaolei Guo  School of Physical Science and
Technology, Southwest Jiaotong University, Chengdu 610031, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

CTB 109 is a middle-aged shell-type SNR with bright thermal X-ray emission. We
reanalyze the GeV $\gamma$-ray emission from CTB 109 using thirteen years of
Pass 8 data recorded by the Fermi Large Area Telescope (Fermi-LAT). The
$\gamma$-ray emission of CTB 109 shows a center bright morphology, which is
well consistent with its thermal X-ray emission rather than the shell-type
structure in the radio band. The spectral analysis shows an evident spectral
curvature at $\sim$ several GeV for the GeV $\gamma$-ray spectrum, which can
naturally explain the lack of TeV $\gamma$-ray emission from CTB 109. Although
either a leptonic or a hadronic model could fit the multi-wavelength
observations of CTB 109, the hadronic model is favored considering its
$\gamma$-ray morphology and the spectral curvature of GeV spectrum. The
unusual $\gamma$-ray spectrum of CTB 109 with other SNRs and the luminosity-
diameter squared relation make CTB 109 to be distinguished both from the
young-aged SNRs with hard GeV $\gamma$-ray spectra and several old-aged SNRs
interacting with molecular clouds.

gamma rays: general - gamma rays: ISM - ISM: supernova remnants - ISM:
individual objects (CTB 109) - radiation mechanisms: non-thermal

##  1 Introduction

Supernova remnants (SNRs) are widely believed to be the dominated accelerators
of Galactic cosmic rays (CRs) with energies up to the knee. Radio and non-
thermal X-ray emission detected provide the clear evidence of electrons
acceleration in SNRs (Koyama et al., 1995; Uchiyama et al., 2007). Fermi Large
Area Telescope (Fermi-LAT) detected the characteristic “$\pi^{0}$ bump” from
several old-aged $\gamma$-ray SNRs interacting with ambient molecular clouds
(MCs; Ackermann et al., 2013; Jogler & Funk, 2016), which has been considered
as the most direct evidence of nuclei acceleration in SNRs. The $\gamma$-ray
emission from such SNRs are from the decay of neutral pions due to the
inelastic nuclei-nuclei collisions, which is the hadronic process. In
addition, the $\gamma$-ray emission can also be produced by leptonic process,
that is the inverse Compton scattering or the bremsstrahlung process of
relativistic electrons. And there has been an on-going debate for the origin
of the $\gamma$-ray emission from most SNRs (H. E. S. S. Collaboration et al.,
2018a; Zeng et al., 2019, 2021).

CTB 109 (G109.1-1.0) is a middle-aged shell-type SNR ($\sim$ 9 kyr; Sánchez-
Cruces et al., 2018), which was first discovered in X-ray with the
observations of Einstein satellite (Gregory & Fahlman, 1980), and later in
radio with the Westerbork Synthesis Radio Telescope (Hughes et al., 1981). The
morphology of CTB 109 shows a semicircular shell in both radio and X-ray
observations, which is widely explained by the interaction between the shock
of SNR and a neighboring giant molecular cloud (Tatematsu et al., 1987; Kothes
et al., 2002). The X-ray emission from the remnant is total thermal, without
evidence for the synchrotron non-thermal component (Rho & Petre, 1997; Parmar
et al., 1998; Rho et al., 1998; Parmar et al., 1999). Based on the analysis of
HI absorption and CO emission spectra of CTB 109, the distance of it is
estimated to be 3 - 4 kpc (Kothes et al., 2002; Tian et al., 2010; Kothes &
Foster, 2012). And the latest measurement by Sánchez-Cruces et al. (2018)
determined the value of 3.1 $\pm$ 0.2 kpc. Hereafter, we adopt this value as
the distance of CTB 109.

In the center of CTB 109, an X-ray point source was first detected by Einstein
satellite (Gregory & Fahlman, 1980), and then identified to be an anomalous
X-ray pulsar (AXP J2301+5852; also 1E 2259+589; Fahlman & Gregory, 1981; Woods
et al., 2004; Gavriil et al., 2004), which is one type of magnetars. The
measured period of 1E 2259+589 is 6.97 s (Morini et al., 1988), with a
magnetic field strength of 5.9 $\times$ 1013 G (Tendulkar et al., 2013). The
detection of 1E 2259+589 makes CTB 109 to be one of few SNRs that hosts a
magnetar within its boundary. However, there is no evidence for the component
of pulsar wind nebula (PWN) been detected around 1E 2259+589. Moreover, the
X-ray observations revealed an X-ray bright interior region in CTB 109, called
the X-ray Lobe, which was initially suggested to be a jet associated with AXP
1E 2259+586 (Gregory & Fahlman, 1983). More detailed analysis with XMM-Newton
(Sasaki et al., 2004), Chandra (Sasaki et al., 2006, 2013), and Suzaku (Nakano
et al., 2015, 2017) show the thermal X-ray emission from the Lobe and suggest
that there is no morphological connection with the magnetar.

The $\gamma$-ray emission from CTB 109 was first reported by Castro et al.
(2012) with 37 months of data from Fermi-LAT. The data analysis shows no
significant extension for the $\gamma$-ray morphology of CTB 109. And its
$\gamma$-ray spectrum is hard with a power-law index of $\Gamma$ = 2.07 $\pm$
0.12, which shows no indications for spectral curvature in the GeV range. Both
the leptonic and hadronic models are discussed to explain the $\gamma$-ray
emission from CTB 109, and neither of them can be ruled out (Castro et al.,
2012). In Fleischhack (2019), HAWC tried to search for the TeV $\gamma$-ray
emission from CTB 109. However, there is no significant TeV emission been
detected and only the upper limits are given.

In the present work, we carry out a detailed analysis of the GeV $\gamma$-ray
emission from SNR CTB 109 with the latest Pass 8 data recorded by Fermi-LAT.
And this paper is organized as follows. In Section 2, we describe the data
analysis routines and results. A discussion of the origin of the $\gamma$-ray
emission based on the multi-wavelength observations is given in Section 3,
together with the comparison with other SNRs. And the conclusion of this work
is presented in Section 4.

##  2 Fermi-LAT Data Analysis

###  2.1 Data Reduction

In following analysis, the latest Fermi-LAT Pass 8 data are collected from
August 4, 2008 (Mission Elapsed Time 239557418) to August 4, 2021 (Mission
Elapsed Time 649728005). We select the data with “Source” event class (evclass
= 128 & evtype=3) and exclude the events with zenith angles greater than
$90^{\circ}$ to reduce the contamination from Earth Limb. The energies of
photons are cut between 10 GeV and 1 TeV for the spatial analysis considering
the impact of large point spread function in the lower energy band. And for
the spectral analysis, the events with energy from 100 MeV to 1 TeV are used.
The region of interest (ROI) we analysed is a square region of
$14^{\circ}\times 14^{\circ}$ centered at the position of CTB 109 in the
incremental version of the fourth Fermi-LAT source catalog (4FGL-DR2;
Abdollahi et al., 2020; Ballet et al., 2020). The binned analysis method is
used and the standard ScienceTools software package
111http://fermi.gsfc.nasa.gov/ssc/data/analysis/software/, together with the
instrumental response function (IRF) of “P8R3_SOURCE_V3”, are adopted. The
Galactic and isotropic diffuse background models used here are
gll_iem_v07.fits and iso_P8R3_SOURCE_V3_v1.txt
222http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html,
respectively. All sources listed in the 4FGL-DR2 catalog within a radius of
$20^{\circ}$ from the ROI center together with the two diffuse backgrounds are
included in the source model, generated by the user-contributed software
make4FGLxml.py333http://fermi.gsfc.nasa.gov/ssc/data/analysis/user/. During
the fitting procedure, the normalizations and spectral parameters of sources
within $7^{\circ}$ around CTB 109, together with the normalizations of the two
diffuse backgrounds, are left free.

###  2.2 Results

In the 4FGL-DR2 catalog, SNR CTB 109 corresponds to the extended $\gamma$-ray
source 4FGL J2301.9+5855e and the $\gamma$-ray emission is described by an
uniform disk with a radius of $0^{\circ}\\!.249$ (Ballet et al., 2020). In
order to show the spatial correlation between the $\gamma$-ray and other
energy bands, we created a test statistic (TS) map with photons above 10 GeV
shown in Figure 1. And the centroid of the $\gamma$-ray emission is closer to
the northwest of SNR than that detected by Castro et al. (2012). The radio
contours at 1420 MHz from Synthesis Telescope of the Dominion Radio
Astrophysical Observatory (DRAO; Hughes et al., 1984) survey, the X-ray
contours from ROSAT High Resolution Image (HRI; Rho & Petre, 1997) and the
12CO ($J$=1$-$0) emission from Five College Radio Astronomy Observatory
(FCRAO; Heyer et al., 1998) survey Here the CO intensity is calculated by
integrating the velocity between -48 km s-1 and -56 km s-1, which corresponds
to the distance estimation of 3.2$\pm$0.2 kpc for CTB 109 by Kothes & Foster
(2012). The TSmap reveals that the GeV $\gamma$-ray emission from CTB 109
mainly concentrates in the central region of the remnant, which is not very
consistent with the morphology of radio shell.

To determine the morphology of the $\gamma$-ray emission from CTB 109, we also
used Fermipy, a PYTHON package that automates analyses with the Fermi Science
Tools (Wood et al., 2017), to test the size of the uniform disk with the
events above 10 GeV. The best-fitting position and radius of the uniform disk
are R.A.$=345^{\circ}\\!.492\pm 0^{\circ}\\!.022$, Dec.$=58^{\circ}\\!.908\pm
0^{\circ}\\!.018$, and $0^{\circ}\\!.245\pm 0^{\circ}\\!.013$, respectively,
which are consistent with the results in the 4FGL-DR2 catalog (Ballet et al.,
2020). The TS value of the uniform disk is estimated to be 155.2. The best-
fitting two-dimensional Gaussian template given by Fermipy
(R.A.$=345^{\circ}\\!.459\pm 0^{\circ}\\!.028$, Dec.$=58^{\circ}\\!.899\pm
0^{\circ}\\!.024$, $\sigma$ = $0^{\circ}\\!.148\pm 0^{\circ}\\!.017$) is also
tested, which gives the similar TS value ($\sim$ 154.0) for CTB 109.

Figure 1: TSmaps of $1^{\circ}\\!.5$ $\times$ $1^{\circ}\\!.5$ centered at CTB
109 for photons above 10 GeV. These maps are smoothed with a Gaussian kernal
of $\sigma$ = $0^{\circ}\\!.02$. The yellow sold circle shows the best-fitting
radius of the uniform disk for the spatial template of CTB 109, and the black
dashed circle marks the centroid position fitted by Castro et al. (2012) as a
point source with its 95% confidence radius. The black diamond reveals the
magnetar 1E 2259+589 and the red cross presents the position of 4FGL
J2258.6+5847c (Sh 2-152). The green contours(left) represent the radio image
of CTB 109 from DRAO at 1.4 GHz (Hughes et al., 1984); the cyan
contours(middle) show the thermal X-ray emission from ROSAT HRI (Rho & Petre,
1997) and the magenta contours display the the 12CO ($J$=1$-$0) emission from
FCRAO integrated the velocity between -48 km s-1 and -56 km s-1 (Heyer et al.,
1998; Kothes & Foster, 2012)

.

With the spatial template of an uniform disk, we perform a spectral analysis
of CTB 109 in the energy range from 100 MeV to 1 TeV. Similar to the analysis
procedure of Puppis A in Xin et al. (2017), we first fit the data using a
simple power-law (PL) spectrum. And the spectral index is fitted to be 1.90
$\pm$ 0.04, which is slightly harder than the result of 2.07 $\pm$ 0.12 in
Castro et al. (2012). And the integral photon flux is calculated to be
$(1.15\pm 0.13)\times 10^{-8}$ photon cm-2 s-1. To test the spectral curvature
of the $\gamma$-ray emission from CTB 109, a log-parabola model ($dN/dE\propto
E^{-(\alpha+\beta{\rm log}(E/E_{b}))}$) is adopted. And the best-fitting
parameters are shown in Table 1. The higher TS value with a log-parabola model
shows an evidence of spectral curvature of CTB 109 ($\sim$3.4$\sigma$
improvement with respect to a power-law model with one additional free model
parameter.

Furthermore, to obtain the spectral energy distribution (SED) of CTB 109, we
divide the data into 14 logarithmic equal energy bins from 100 MeV to 1 TeV,
and perform the same likelihood fitting analysis for each energy bin. For the
bin with TS value of CTB 109 smaller than 5.0, an upper limit with 95%
confidence level is calculated. The results are presented as the black dots in
Figure 2. The global fitting with the power-law and log-parabola models are
also overplotted, together with the upper limit of the TeV $\gamma$-ray flux
derived by HAWC (Fleischhack, 2019). And the existence of the spectral
curvature of CTB 109 in the GeV band can naturally explain the non-detection
in the TeV $\gamma$-ray band.

Table 1: Comparison of Spectral Models (100 MeV - 1 TeV)

Model | Index${}^{\,\,\text{a}}$ | $\beta$ | Photon Flux | Energy Flux | Number of | TS Value
---|---|---|---|---|---|---
|  |  |  $\rm 10^{-9}$ $\rm ph$ $\rm cm^{-2}$ $\rm s^{-1}$ |  $\rm 10^{-5}$ $\rm MeV$ $\rm cm^{-2}$ $\rm s^{-1}$ | free model parameters |
Power-Law | 1.90 $\pm$ 0.04 | – | 11.52 $\pm$ 1.28 | 1.59 $\pm$ 0.13 | 2 | 523.3
Log-parabola | 1.60 $\pm$ 0.10 | 0.086 $\pm$ 0.023 | 6.46 $\pm$ 1.16 | 1.10 $\pm$ 0.11 | 3 | 534.6

  * a

The Index column is for both index (power-law), and for $\alpha$ (log-
parabola).

Figure 2: Left: SED of SNR CTB 109. The results of Fermi-LAT data are shown as
the black dots, with arrows indicating the 95% upper limits. The gray
histogram shows the TS value for each energy bin. The TeV upper limit (blue
arrow) are from HAWC observation (Fleischhack, 2019). The green solid and
dashed lines are the best-fitting power-law spectrum and its 1$\sigma$
statistic error in the energy range of 100 MeV - 1 TeV. The red sold line is
the best-fitting log-parabola spectrum. The cyan dotted and dotted–dashed
lines represent the differential sensitivities of LHAASO (1 yr) with different
sizes of photomultiplier tube (PMT; Cao et al., 2019). And the black dotted
line shows the differential sensitivity of CTA-North (50 hr; Cherenkov
Telescope Array Consortium et al., 2019). Right: SED of Sh 2-152 shown as the
black dots. The red line shows the hadronic emission by adopting a power-law
spectrum for protons.

In the west of CTB 109, there is a $\gamma$-ray point source, 4FGL
J2258.6+5847c, which is associated with the star-forming compact HII region Sh
2-152 (Darbon et al., 2000; Ramírez Alegría et al., 2011). In the energy band
of 100 MeV - 1 TeV, the $\gamma$-ray emission from Sh 2-152 can be described
by a power-law model with an index of 2.25 $\pm$ 0.12, and the TS value is
fitted to be 67.1. And the energy flux of the $\gamma$-ray emission is
calculated to be $\rm(3.76\pm 0.62)\times 10^{-6}$ $\rm MeV$ $\rm cm^{-2}$
$\rm s^{-1}$. Based on the observations of the inside young stellar objects
(YSOs), the distance of Sh 2-152 is estimated to be 3.21 $\pm$ 0.21 kpc
(Ramírez Alegría et al., 2011). With this value, the $\gamma$-ray luminosity
of Sh 2-152 is calculated to be (7.43 $\pm$ 1.22) $\rm\times 10^{33}$ $\rm
erg$ $\rm s^{-1}$. The resulting SED of Sh 2-152 is shown in the right panel
of Figure 2.

##  3 Discussion

From the above analysis, the extended GeV $\gamma$-ray emission from CTB 109
is spatially well-correlated with the morphology in the thermal X-ray rather
than the radio band. And the $\gamma$-ray emission shows the center bright
morphology, which is not consistent with the shell-type structure in the radio
band. All evidences above support the hadronic scenario for the $\gamma$-ray
emission from CTB 109, which also means that there should be some dense clouds
in the $\gamma$-ray emitting region like SNR Puppis A (Xin et al., 2017).
However, the right panel of Figure 1 shows no bright CO emission detected in
the centroid position of the $\gamma$-ray emission region, except some small
surrounding molecular clouds (Sasaki et al., 2006; Castro et al., 2012). This
could be attributed to dissociation of molecules by the radiative precursor of
the SNR due to the photoionization and photodissociation effects (Paron et
al., 2008). The absence of CO emission may also stems from the CO-dark gas in
the $\gamma$-ray emitting region that cannot be traced by CO observations
(Grenier et al., 2005).

In the left panel of Figure 3, we plot the $\gamma$-ray spectra of several
Fermi-LAT observed SNRs. The old-aged SNRs, like IC 443 (age $\sim$ 20 kyr;
Lee et al., 2008; Ackermann et al., 2013) and W44 (age $\sim$ 20 kyr;
Wolszczan et al., 1991; Ackermann et al., 2013), show the spectral break at
$\sim$ GeV, which makes their $\gamma$-ray emission brighter in the GeV band
than in the TeV band (Funk, 2015). These SNRs are usually suggested to be
interacting with MCs, and the $\gamma$-ray emission from such SNR/MC systems
are suggested to be hadronic origin. Another class of SNRs, including RX
J1713.7-3946 (age $\sim$ 1.6 kyr; Wang et al., 1997; H. E. S. S. Collaboration
et al., 2018a) and RX J0852.0-4622 (age $\sim$ 1.7-4.3 kyr; Katsuda et al.,
2008; H. E. S. S. Collaboration et al., 2018b), are young-aged systems with
strong non-thermal X-ray emission and hard GeV spectra. The $\gamma$-ray
emission from these SNRs are suggested to be from inverse Compton scattering
of accelerated electrons (leptonic model), which shows the spectral curvature
at $\sim$ TeV. The spectral analysis in Section 2.2 shows an evident spectral
curvature at $\sim$ several GeV for the $\gamma$-ray spectrum of CTB 109,
which is similar to that of Puppis A (age $\sim$ 4.45 kyr; Becker et al.,
2012; Xin et al., 2017). In addition, the right panel of Figure 3 depicts the
$\gamma$-ray luminosity from 100 MeV to 100 GeV as a function of the diameter
squared for several Fermi-LAT detected SNRs. The data points of the sample are
cited from Thompson et al. (2012) and Devin et al. (2020), whose names, ages
and classes are listed in Table 2. And the squared diameter could also be
regarded as a reasonable indicator of SNR ages. The $\gamma$-ray luminosity of
CTB 109 is calculated to be 1.6 $\rm\times 10^{34}$ $\rm erg$ $\rm s^{-1}$ in
the energy range of 100 MeV - 100 GeV. The different $\gamma$-ray spectra of
the different type of SNRs and the luminosity-diameter squared relation make
CTB 109 and Puppis A to be distinguished both from several old-aged SNRs
interacting with molecular clouds (e.g. IC443) and the young-aged SNRs with
hard GeV $\gamma$-ray spectra (e.g. RX J1713.7-3946).

Figure 3: Left: Typical $\gamma$-ray SEDs of several prominent SNRs and CTB 109. The energy fluxes of Puppis A are scaled downward by five times for comparison. Right: 0.1 - 100 GeV $\gamma$-ray luminosities of several Fermi-LAT observed SNRs plotted against their diameter squared, which is reproduced from Thompson et al. (2012) and Devin et al. (2020) with the addition of CTB 109. The blue diamonds and green circles represent the old-aged SNRs interacting with molecular clouds and the young-aged SNRs. The red triangle shows SNR Puppis A, and CTB 109 is marked as the black star. Table 2: Ages of different type of SNRs Class | Name | Age (kyr) | Reference
---|---|---|---
SNRs/MC |  |  |
| IC 443 |  $\sim$20 | Lee et al. (2008)
| W44 |  $\sim$20 | Wolszczan et al. (1991)
| W51C |  $\sim$30 | Koo et al. (1995)
| W28 |  $\sim$33 | Velázquez et al. (2002)
| G8.7-0.1 |  $\sim$15 | Kassim & Weiler (1990)
| CTB 37A |  $\sim$24 | Yamauchi et al. (2014)
| W49B | 5-6 | Zhou & Vink (2018)
| 3C391 |  $\sim$8.7 | Leahy & Ranasinghe (2018)
| G349.7+0.2 |  $\sim$1.8 | Tian & Leahy (2014)
Young SNRs |  |  |
| RX J1713.7-3946 |  $\sim$1.6 | Wang et al. (1997)
| RX J0852.0-4622 | 1.7-4.3 | Katsuda et al. (2008)
| CasA | 0.35 | Thorstensen et al. (2001)
| Tycho | 0.45 | Baade (1945)
| G150.3+4.5 |  $\sim$1.0 | Devin et al. (2020)
| Puppis A |  $\sim$4.45 | Becker et al. (2012)
| CTB 109 |  $\sim$9.0 | Sánchez-Cruces et al. (2018)

To explore the origin of the $\gamma$-ray emission from CTB 109, the leptonic
and hadronic models are adopted to fit the multi-wavelength observations.
Different from the CR-hydro-NEI model adopted in Castro et al. (2012)
including the SNR hydrodynamics, a semi-analysis calculation of nonlinear
diffuse shock acceleration and the nonequilibrium ionization conditions behind
the forward shock, here we adopted a phenomenological spectra for electrons
and protons to constrain the distributions of cosmic rays, together
considering the upper limit in the TeV band from HAWC. In the leptonic model,
the inverse Compton scattering (ICS) or bremsstrahlung processes of
relativistic electrons are considered. And for ICS process, the radiation
field includes the cosmic microwave background (CMB) and the infrared
component from interstellar dust and gas with T = 30K & u = 1 eV $\rm cm^{-3}$
(Porter et al., 2006, 2008; Zeng et al., 2019). The distance of CTB 109 is 3.1
kpc (Sánchez-Cruces et al., 2018), and the density of the ambient medium is
adopted to be ngas = 1.1 cm-3 (Castro et al., 2012). We use the naima package
to fit the multi-wavelength data with the Markov Chain Monte Carlo (MCMC)
algorithm (Zabalza, 2015).

For the leptonic model, the spectra of electrons was first assumed to be a
single power-law with an exponential cutoff (PL) in the form of

$\frac{dN_{\rm
e}}{dE}\propto\left(\dfrac{E}{E_{0}}\right)^{-\alpha_{e}}exp\left(-\dfrac{E}{E_{\rm
cut,e}}\right)$ (1)

And considering the GeV spectral curvature of the $\gamma$-ray emission, we
also tested another electron distribution of a broken power-law model with an
exponential cutoff (BPL; Zeng et al., 2019), which follows:

$\frac{dN_{\rm e}}{dE}\propto exp\left(-\dfrac{E}{E_{\rm
cut,e}}\right)\begin{cases}\left(\dfrac{E}{E_{0}}\right)^{-\alpha_{e1}}\qquad\qquad\qquad\qquad;E<E_{\rm
break}\\\ \left(\dfrac{E_{\rm
break}}{E_{0}}\right)^{\alpha_{e2}-\alpha_{e1}}\left(\dfrac{E}{E_{0}}\right)^{-\alpha_{e2}}\,;E\geq
E_{\rm break}\end{cases}$ (2)

Here, $\alpha_{\rm e}$ and $\rm E_{cut,e}$ are the spectral index and the
cutoff energy of electrons. $\rm E_{break,e}$ is the break energy, and the
spectral indices below/above $\rm E_{break,e}$ are denotes as $\alpha_{\rm
e1}$/$\alpha_{\rm e2}$. To reduce the number of free parameters, the spectral
index variation for the BPL model is set to be 1.0 ($\alpha_{\rm e2}$ =
$\alpha_{\rm e1}$ \+ 1.0). And the cutoff energy of electrons Ecut,e is
calculated by equalling the synchrotron energy loss time-scale and the age of
CTB 109, which is given by Ecut,e = 1.25 $\times$ 107 t${}_{\rm age;yr}^{-1}$
B${}_{\rm\mu G}^{\rm-2}$ TeV. Here the age of CTB 109 is adopted to be tage =
9 kyr (Sánchez-Cruces et al., 2018).

The best-fit parameters of leptonic models are shown in Table 3, and the
corresponding modeled SED is given in the left panel of Figure 4. For the
leptonic PL model, the spectral index and cutoff energy of electrons are
fitted to be 1.96 and 0.61 TeV, respectively. A magnetic field strength of
about 20 $\mu$G is needed to explain the flux in the radio band, which is
similar to the value behind the shock fitted by Castro et al. (2012). However,
the total energy of electrons above 1 GeV of We = 1.31 $\times$ 1048 erg is
much lower than that required for the CR-hydro-NEI model in Castro et al.
(2012). The leptonic BPL model fitting gives the break energy of electrons to
be $\sim$ 0.22 TeV, and other parameters are similar to that of the PL model.
Adopted the fitted magnetic field strength and the age of CTB 109, the cutoff
energy of electrons is calculated to be about 3.44 TeV, which is larger than
that of the PL model.

For the hadronic model, the spectral distributions of electrons and protons
are both assumed to follow the PL model. And the spectral indices of electrons
and protons are set to be equal to reduce the free parameters. Meanwhile, the
ratio of the number of relativistic electrons to protons at 1 GeV, Kep, is
assumed to be 0.01, which is in accord with the local measured cosmic ray
abundance (Yuan et al., 2012). The modeled SED is shown in the right panel of
Figure 4, and the derived model parameters are listed in Table 3. The spectral
indices of particles are fitted to be 1.96. The magnetic field strength of
about 40 $\mu$G is two times larger than that in the leptonic model. And based
on this value, the cutoff energy of electrons is calculated to be about 0.84
TeV. However, the cutoff energy of protons, Ecut,p, could not be well
constrained due to the absence of the TeV $\gamma$-ray data for CTB 109. With
the spectral index of 1.96 for protons, we adjust the value of Ecut,p by
manual to compute the $\gamma$-ray flux. And the upper limit in the TeV band
by HAWC constrains the maximum of protons to be about 50 TeV. The
corresponding modeled SED is shown as the red dashed line in the right panel
of Figure 4. The total energy of electrons and protons above 1 GeV are
estimated to be We = 4.47 $\times$ 1047 erg and Wp = 5.52 $\times$ 1049
(ngas/1.1 cm-3)-1 erg, respectively. And such values are about one order lower
than the fitted results in Castro et al. (2012).

Table 3: Parameters for Leptonic and Hadronic Models

Model | $\alpha_{\rm e/p}^{\,\,\,\,\,\,\,\text{a}}$ | Ebreak,e | E${}_{\rm cut,e}^{\,\,\,\,\,\,\,\,\,\,\text{b}}$ | Ecut,p | B | We ($>$1 GeV) | Wp ($>$1 GeV)
---|---|---|---|---|---|---|---
|  | TeV | TeV | TeV |  $\mu$G | erg | erg
Leptonic |  |  |  |  |  |  |
PL | 1.96${}^{+0.07}_{-0.07}$ | – | 0.61${}^{+0.20}_{-0.13}$ | – | 18.99${}^{+3.26}_{-2.37}$ | 1.31${}^{+0.15}_{-0.17}\times 10^{48}$ | –
BPL | 1.94${}^{+0.07}_{-0.07}$ | 0.22${}^{+0.07}_{-0.06}$ | 3.44 | – | 20.10${}^{+3.26}_{-3.12}$ | 1.29${}^{+0.18}_{-0.15}\times 10^{48}$ | –
Hadronic |  |  |  |  |  |  |
PL | 1.96${}^{+0.05}_{-0.06}$ | – | 0.84 | 2.88${}^{+15.78}_{-2.46}$ | 40.55${}^{+4.44}_{-5.56}$ | 4.47${}^{+0.97}_{-0.35}\times 10^{47}$ | 5.52${}^{+1.17}_{-1.08}\times 10^{49}$

  * a

For the leptonic BPL model, $\alpha_{\rm e}$ denotes the low-energy spectral
index of electronic distribution, and the high-energy spectral index above
Ebreak,e is $\alpha_{\rm e}$ \+ 1.0.

  * b

For the leptonic BPL and hadronic models, Ecut,e is calculated by equalling
the synchrotron energy loss time-scale and the age of SNR CTB 109.

Figure 4: Modeling of the multi-wavelength SED of CTB 109. And the radio data
are from Kothes et al. (2006), together with the TeV upper limit by HAWC
(Fleischhack, 2019). The left panel is for the leptonic model, where the PL
and BPL models are presented by the black solid and red dashed lines,
respectively. And the different radiation components for the BPL model are
also overplotted, as the legend described. The right panel is for the hadronic
PL model, and the different radiation components are also illustrated in the
legend. The red dashed line is calculated with the cutoff energy of protons of
Ecut,p = 50 TeV. The cyan dotted and dotted–dashed lines represent the
differential sensitivities of LHAASO (1 yr) with different sizes of
photomultiplier tube (PMT; Cao et al., 2019). And the gray dot-dashed line
shows the differential sensitivity of CTA-North (50 hr; Cherenkov Telescope
Array Consortium et al., 2019).

The multi-wavelength SED fitting shown in Figure 4 suggests that both the
leptonic and hadronic model could explain the observational data of CTB 109.
However, the low flux of the upper limit in the low $\gamma$-ray range and the
center bright morphology of the $\gamma$-ray emission make the hadronic model
favored. Moreover, the different radiation models predict the different
$\gamma$-ray fluxes in the TeV band, which will be uncovered with future high-
resolution observations, such as Large High Altitude Air Shower Observatory
(LHAASO; Cao et al., 2019) and/or Cherenkov Telescope Array in the northern
hemisphere(CTA-North; Cherenkov Telescope Array Consortium et al., 2019).

For the $\gamma$-ray emission of Sh 2-152, its spectrum is similar to that of
other star forming regions (SFRs) or young massive star clusters (YMCs)
detected by Fermi-LAT, such as W40 (Sun et al., 2020), W43 (Yang & Wang,
2020), and NGC 6618 (Liu et al., 2022), etc. More and more observations
support SFRs/YMCs to be an important class of factories of Galactic cosmic
rays (Aharonian et al., 2019). In SFRs/YMCs, there are many young massive
stars and core-collapse supernovas. And cosmic rays are expected to be
efficiently accelerated by strong fast winds of young massive stars and shocks
of SNRs. Meanwhile, SFRs host the dense molecular gas to drive star formation,
and the $\gamma$-ray emission from them favor the hadronic interaction between
the accelerated cosmic rays and dense molecular clouds around them (Bykov et
al., 2020). In the hadronic scenario, we adopt a simple power-law spectrum for
protons to fit the $\gamma$-ray SED of Sh 2-152, which is shown as the red
solid line in the right panel of Figure 2. In Ramírez Alegría et al. (2011),
the diameter and total mass of the cluster of Sh 2-152 are determined to be
$\gtrsim$ 4.0 pc and (2.45 $\pm$ 0.79)$\rm\times 10^{3}$ $M_{\odot}$,
respectively. And the average density of the ambient gas is estimated to be
$\sim\rm 3\times 10^{3}$ $\rm cm^{-3}$. With the spectral index of 2.5 for
protons, the total energy of protons above 1 GeV, Wp, is calculated to be
2.17$\rm\times 10^{46}(n_{gas}/3000\;cm^{-3})\;erg$ to fit the $\gamma$-ray
data of Sh 2-152. The required total energy of protons of Sh 2-152 is orders
of magnitude lower than that of other SFRs/YMCs (Aharonian et al., 2019),
which could be attributed to the less identified OB stars and lower mass than
the other detected systems (Sun et al., 2020).

##  4 Conclusions

In this work, we reanalyze the $\gamma$-ray emission from SNR CTB 109 using 13
years Pass 8 Fermi-LAT data. The center bright $\gamma$-ray morphology of CTB
109 is well consistent with its thermal X-ray emission, but is not consistent
with the shell-type structure in the radio band. The GeV $\gamma$-ray spectrum
of CTB 109 described by a log-parabola model shows an evident spectral
curvature at $\sim$ several GeV, which can naturally explain the upper limit
derived from HAWC observations in the TeV band. By comparing CTB 109 and other
typical SNRs, the unusual $\gamma$-ray spectrum of CTB 109 and the luminosity-
diameter squared relation make CTB 109 to be distinguished both from the
young-aged SNRs with hard GeV $\gamma$-ray spectra and several old-aged SNRs
interacting with molecular clouds. The multi-wavelength data can be well
explained with the leptonic or hadronic model. However, considering the low
flux of the upper limit in the low $\gamma$-ray range, together with the
$\gamma$-ray morphology and the spectral curvature of the GeV spectrum, the
hadronic model is favored, like SNR Puppis A. The decisive evidence of the
hadronic model may rely on the high-resolution observations in the TeV band by
LHAASO and/or CTA in the future.

We would like to thank the anonymous referee for very helpful comments, which
help to improve the paper. This work is supported by the Natural Science
Foundation for Young Scholars of Sichuan Province, China (No. 2022NSFSC1808),
the Science and Technology Department of Sichuan Province (No. 2021YFSY0031,
No.2020YFSY0016), the Fundamental Research Funds for the Central Universities
(No. 2682021CX073, No. 2682021CX074, No. 2682022ZTPY013), and the National
Natural Science Foundation of China under the grants 12103040 and 12147208.

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# Deep Latent Mixture Model for Recommendation

Jun Zhang, Ping Li, Wei Wang
Renmin University of China
<EMAIL_ADDRESS>

###### Abstract

Recent advances in neural networks have been successfully applied to many
tasks in online recommendation applications. We propose a new framework called
cone latent mixture model which makes use of hand-crafted state being able to
factor distinct dependencies among multiple related documents. Specifically,
it uses discriminative optimization techniques in order to generate effective
multi-level knowledge bases, and uses online discriminative learning
techniques in order to leverage these features. And for this joint model which
uses confidence estimates for each topic and is able to learn a
discriminatively trained jointly to automatically extracted salient features
where discriminative training is only uses features and then is able to
accurately trained.

##  1 Introduction

Recommendation technology has enormous potential. The contribution of [1] is
to promote better load modeling and advanced load modeling, and to facilitate
data exchange among users of various production-grade simulation programs. [2]
discuss an approach to collaborative filtering based on the Simple Bayesian
Classifier. These recommendation systems [3, 4] have been tried in e-commerce
to entice purchasing of goods, but haven’t been tried in e-learning. [5]
suggest the use of web mining techniques to build such an agent that could
recommend on-line learning activities or shortcuts in a course web site based
on learners’ access history to improve course material navigation as well as
assist the online learning process, and [6] perform a simulation study
demonstrating that MMRE does not always select the best model. [7] discuss
content-based recommendation systems, i.e., systems that recommend an item to
a user based upon a description of the item and a profile of the user’s
interests. [8] present the initiative of building a context-aware citation
recommendation system. [9] survey different intrusions affecting availability,
confidentiality and integrity of Cloud resources and services. [10] propose a
unified model that combines content-based filtering with collaborative
filtering, harnessing the information of both ratings and reviews. The
potential of the technology is enormous. [11] aim to identify the main
determinants of mobile payment adoption and the intention to recommend this
technology. Other influential work includes [12, 13].

[14] address the use of proper orthogonal decomposition for reduced-order
solution of the heat transfer problem within a hypersonic modeling framework.
And consider an orthogonal frequency division multiplexing (OFDM) downlink
point-to-point system with simultaneous wireless information and power
transfer. To reduce the parameter-tuning effort [15] propose an LSPD parameter
recommender system that involves learning a collaborative prediction model
through tensor decomposition and regression. Technically speaking make three
key contributions in leveraging deep aesthetic features. To describe the
aesthetics of products introduce the aesthetic features extracted from product
images by a deep aesthetic network. [15] present the vertically integrated
hardware/software co-design, which includes a custom DIMM module enhanced with
near-data processing cores tailored for DL tensor operations. Combining with
autoencoder approach to extract the latent essence of feature information
[16]. Deep Dual Transfer Cross Domain Recommendation (DDTCDR) model is
proposed to provide recommendations in respective domains. Tensorly has a
simple python interface for expressing tensor operations. It suffers from data
sparsity when dealing with three-dimensional (3D) user-item-criterion ratings.
To alleviate this problem, [17] deep transfer tensor decomposition (DTTD)
method is proposed by integrating deep structure and Tucker decomposition,
where an orthogonal constrained stacked denoising autoencoder (OC-SDAE) is
proposed for alleviating the scale variation in learning effective latent
representation, and the side information is incorporated as a compensation for
tensor sparsity.

[18] propose a new neural network model, called graph neural network (GNN)
model, that extends existing neural network methods for processing the data
represented in graph domains. Traffic forecasting is a particularly
challenging application of spatiotemporal forecasting, due to the time-varying
traffic patterns and the complicated spatial dependencies on road networks. To
address this challenge a novel deep learning framework Traffic Graph
Convolutional Long Short-Term Memory Neural Network (TGC-LSTM), is proposed to
learn the interactions between roadways in the traffic network and forecast
the network-wide traffic state. The problem of few-shot learning [19, 20] with
the prism of inference on a partially observed graphical model, constructed
from a collection of input images whose label can be either observed or not.
The variants of each component [21, 22], systematically categorize the
applications, and propose four open problems for future research. The Graph
Markov Neural Network (GMNN) that combines the advantages of both worlds.
Position-aware Graph Neural Networks (P-GNNs) is a new class of GNNs for
computing position-aware node embeddings. [23] propose a novel edge-labeling
graph neural network (EGNN), which adapts a deep neural network on the edge-
labeling graph, for few-shot learning. [24] introduce a pooling operator based
on graph Fourier transform, which can utilize the node features and local
structures during the pooling process. [25] design GCC’s pre-training task as
subgraph instance discrimination in and across networks and leverage
contrastive learning to empower graph neural networks to learn the intrinsic
and transferable structural representations. Other influential work includes
[26, 27, 3].

##  2 Method

######  Definition 2.1.

A symmetric subalgebra $N^{\prime}$ is Lobachevsky if Abel’s condition is
satisfied.

######  Definition 2.2.

Let $\Phi\neq{\xi_{A}}(B^{\prime})$ be arbitrary. We say a monodromy $\xi$ is
Cavalieri if it is real.

The goal of the present paper is to characterize Artinian subgroups. Every
student is aware that $\tilde{w}(\mathfrak{{d}}^{\prime\prime})\geq 0$. The
groundbreaking work of S. Sasaki on contra-isometric sets was a major advance.
Y. Davis’s description of non-Conway rings was a milestone in commutative
K-theory. It would be interesting to apply the techniques of [28] to topoi.
Recently, there has been much interest in the derivation of empty, trivially
$i$-$n$-dimensional random variables.

######  Definition 2.3.

Let us suppose we are given an anti-trivially commutative element
$\mathfrak{{u}}$. We say a local topos ${\omega_{g}}$ is unique if it is
Pólya–Darboux.

We now state our main result.

######  Theorem 2.4.

Suppose ${\mathcal{{S}}_{u}}\sim\infty$. Assume we are given a non-generic
system $Y$. Further, assume $\beta\neq 2$. Then ${c_{\Psi,\mathfrak{{q}}}}$ is
stable, Erdős, combinatorially regular and $L$-canonical.

Recent developments in introductory graph theory [29] have raised the question
of whether $\mathfrak{{f}}^{\prime\prime}\leq-\infty$. Next, in [30], the
authors characterized right-multiply co-Pappus, $n$-dimensional functionals.
In contrast, B. Shastri [31] improved upon the results of W. Nehru by studying
multiplicative, countably generic, meager vectors. Therefore recently, there
has been much interest in the derivation of semi-symmetric, Eisenstein
subgroups. In this setting, the ability to describe $p$-adic factors is
essential. Therefore this leaves open the question of injectivity. Hence in
[32], the main result was the extension of totally Riemannian rings.

Let $\phi(\mathbf{{q}})\cong\emptyset$.

######  Definition 2.5.

Let ${\rho_{\Gamma,u}}<\tilde{\Lambda}$. A linearly left-Euclidean, contra-
Riemann, Fourier homeomorphism is a modulus if it is non-$n$-dimensional.

######  Definition 2.6.

A multiply admissible isomorphism acting totally on a locally Landau random
variable $\hat{\Sigma}$ is $p$-adic if $O^{\prime}=\Omega$.

######  Theorem 2.7.

Let $\mathfrak{{d}}\leq\|\mathcal{{I}}\|$. Let $i\geq\mathfrak{{z}}$ be
arbitrary. Further, assume we are given a subalgebra $\mathfrak{{t}}$. Then

$\displaystyle{r_{\epsilon}}\left(\Xi^{-8},\dots,\emptyset\right)$
$\displaystyle\subset\bigcap\overline{0}\cup\overline{\frac{1}{\mathbf{{p}}}}$
$\displaystyle<\oint_{\pi}^{1}\exp\left(A^{-7}\right)\,d\mathscr{{N}}$
$\displaystyle\leq\left\\{\omega^{-9}\colon\mathbf{{j}}\left({J_{\Theta}},\dots,P(\bar{Z})\|U\|\right)\leq\sup
d\left(\pi\cap e,\dots,\frac{1}{{\sigma_{\mathcal{{F}}}}}\right)\right\\}$
$\displaystyle\sim\sum_{V^{\prime\prime}\in
v}\Theta^{\prime\prime}\left(V+\mathscr{{D}}^{\prime\prime},\dots,Z\right)\cdot\overline{\frac{1}{|\psi|}}.$

Let $s\leq X$ be arbitrary.

######  Definition 2.8.

Assume every positive definite, elliptic, solvable modulus is almost negative.
A Kronecker system is a curve if it is connected and quasi-solvable.

######  Definition 2.9.

Let us assume there exists a globally convex solvable, quasi-everywhere
Hilbert–Markov curve. A holomorphic, ultra-Steiner topos is a homomorphism if
it is connected.

######  Proposition 2.10.

$\tilde{\mathbf{{k}}}\geq{D_{\Omega}}$.

###### Proof.

This proof can be omitted on a first reading. Let $\bar{H}$ be a multiply
singular, hyper-differentiable subalgebra. Trivially, if
${Z_{f,\mathcal{{T}}}}$ is co-totally Tate–Brouwer then $y=\mathbf{{l}}$. So
$-1^{9}\sim t\left(\bar{\mathfrak{{q}}}\cap 1,-\pi\right)$. Now there exists a
discretely universal hull. Hence every multiply normal domain is right-
trivially universal. Next, if $|S|<|\mathbf{{j}}|$ then $\psi$ is dominated by
$f$. Thus if $V$ is ultra-everywhere hyperbolic and symmetric then
$H=\sqrt{2}$. On the other hand, there exists a super-continuously smooth and
elliptic triangle. By countability,

$\displaystyle\overline{\Delta^{\prime\prime}}$
$\displaystyle>\left\\{2\sqrt{2}\colon\exp\left(0|\psi|\right)\neq\delta^{\prime\prime-1}\left(Z\right)\cap\exp^{-1}\left(\mathscr{{W}}(\Theta)^{-2}\right)\right\\}$
$\displaystyle\geq\int_{0}^{e}p\left(\frac{1}{\bar{\mu}},\dots,-\infty\right)\,db-a^{\prime}\left(\mathfrak{{c}},\dots,\frac{1}{d^{\prime}}\right)$
$\displaystyle\to\bigotimes_{e=\pi}^{-\infty}Z^{-1}\left(--\infty\right)$
$\displaystyle\neq\liminf g^{\prime\prime-1}\left(|\mathscr{{N}}|\wedge
R\right)\cap\dots\wedge\exp\left(-1\right).$

Let $S^{\prime}<\varphi$. Since $\delta$ is smaller than ${s_{q}}$, if
$\bar{\mathfrak{{i}}}$ is anti-null, right-pointwise invertible, essentially
irreducible and essentially intrinsic then every Noetherian domain is hyper-
irreducible. Trivially, if $\mathscr{{J}}$ is maximal and solvable then

$\displaystyle{\Theta_{W}}\left(\frac{1}{\|\bar{\xi}\|},|{j_{l,\theta}}|^{9}\right)$
$\displaystyle<\coprod_{\mathfrak{{e}}\in\mathcal{{W}}}\overline{{Q_{U,A}}^{4}}\cdot\overline{-0}$
$\displaystyle\leq\left\\{\mathfrak{{x}}^{8}\colon{\mathfrak{{j}}^{(\Xi)}}-1\to\lim\mathcal{{H}}^{-1}\left({\Phi^{(\mathfrak{{v}})}}^{2}\right)\right\\}$
$\displaystyle\neq\bigcup_{\hat{H}=\pi}^{2}{\Lambda_{V}}\left(K\times
b,2\right)\wedge\hat{\chi}\left(\|\Lambda\|,-{e_{n}}\right).$

As we have shown, if $c$ is super-discretely contra-Cantor then there exists
an almost surely associative and almost pseudo-differentiable totally
continuous subgroup. Thus

$e(i)\wedge S\geq\bigcap_{\mathcal{{F}}\in
r}\overline{\frac{1}{\mathscr{{B}}}}.$

Next, if $\Xi\cong\tilde{\mathscr{{I}}}$ then
$\mathfrak{{t}}^{\prime\prime}<-1$. One can easily see that every ideal is
non-integral. Therefore if ${w^{(K)}}$ is not comparable to $\hat{\xi}$ then
$R\leq 0$.

As we have shown, if $\tilde{\ell}$ is null, finite, Möbius and
$n$-dimensional then there exists an ultra-Serre–Frobenius onto, smoothly
pseudo-Pólya, countable functional acting continuously on a d’Alembert set.
Thus if the Riemann hypothesis holds then there exists a canonically pseudo-
integral de Moivre, simply Minkowski, uncountable triangle. Since
$\mathbf{{s}}\to\emptyset$, $\bar{\xi}\in\Theta$. Hence if
$\Sigma^{\prime\prime}$ is not smaller than $\hat{\rho}$ then
$D^{\prime}\ni\mathcal{{B}}$. By a little-known result of Turing [33], if
$\hat{\mathfrak{{v}}}$ is Steiner then there exists a Green canonically
canonical system. Because $|y|\neq\mathcal{{V}}$, $u$ is contra-globally
$p$-adic.

By locality, if $Q^{\prime\prime}$ is anti-trivial then $L\leq 0$. Trivially,
if $N$ is not comparable to $\ell$ then there exists an affine, arithmetic,
free and right-integrable Euclidean equation.

Trivially, $i\hat{\tau}=\overline{z\Sigma}$. Moreover, if $N\geq\pi$ then
${\mathbf{{z}}_{\tau}}(\ell)<\mathscr{{R}}$. Trivially, $f=\aleph_{0}$.
Therefore

$\displaystyle-1$
$\displaystyle<\iiint\sum\tan\left(\mathscr{{T}}\right)\,d\mathbf{{y}}$
$\displaystyle<\frac{0}{\overline{\Psi}}$
$\displaystyle\equiv\left\\{\infty\colon\bar{\mu}\left(F^{-2},\dots,-\infty\right)=\bigcap\int_{\omega}\overline{\bar{\Gamma}}\,d\tilde{\chi}\right\\}$
$\displaystyle\ni-\pi\cdot\mathcal{{R}}\left(i,\ell(e)\emptyset\right).$

∎

######  Theorem 2.11.

$\phi\leq{\omega^{(\Theta)}}$.

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