# Extreme & High Synchrotron Peaked Blazars at the limit of Fermi-LAT
detectability: the $\gamma$-ray spectrum of 1BIGB sources

B. Arsioli1,2,5, U. Barres de Almeida3,5, E. Prandini4, B. Fraga3,5, L.
Foffano4,
1Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas
(UNICAMP), Rua Sérgio Buarque de Holanda 777, 13083-859 Campinas, Brazil
2Science Data Center della Agencia Spaziale Italiana, SSDC - ASI, Rome, Italy
3Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150,
22290-180 URCA, Rio de Janeiro, Brasil
4University of Padova, Department of Physics and Astronomy, and INFN sez.
Padova, Italy
5ICRANet-Rio, CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 URCA, Rio de Janeiro,
Brazil E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received: April 23, 2018)

###### Abstract

We present the 1-100 GeV spectral energy distribution for a population of 148
high-synchrotron-peaked blazars (HSPs) recently detected with Fermi-LAT as
part of the First Brazil-ICRANet Gamma-ray Blazar catalogue (1BIGB). Most of
the 1BIGB sources do not appear in previous Fermi-LAT catalogues and their
$\gamma$-ray spectral properties are presented here for the first time,
representing a significant new extension of the $\gamma$-ray blazar
population. Since our sample was originally selected from an excess signal in
the 0.3 - 500 GeV band, the sources stand out as promising TeV blazar
candidates, potentially in reach of the forthcoming very-high-energy (VHE)
$\gamma$-ray observatory, CTA. The flux estimates presented here are derived
considering PASS8 data, integrating over more than 9 years of Fermi-LAT
observations. We also review the full broadband fit between 0.3-500 GeV
presented in the original 1BIGB paper for all sources, updating the power-law
parameters with currently available Fermi-LAT dataset. The importance of these
sources in the context of VHE population studies with both current instruments
and the future CTA is evaluated. To do so, we select a subsample of 1BIGB
sources and extrapolate their $\gamma$-ray SEDs to the highest energies,
properly accounting for absorption due to the extragalactic background light.
We compare those extrapolations to the published CTA sensitivity curves and
estimate their detectability by CTA. Two notable sources from our sample,
namely 1BIGB J224910.6-130002 and 1BIGB J194356.2+211821, are discussed in
greater detail. All $\gamma$-ray SEDs, which are shown here for the first
time, are made publicly available via the Brazilian Science Data Center (BSDC)
service, maintained at CBPF, in Rio de Janeiro.

###### keywords:

active galactic nuclei – blazars – gamma rays – very high-energy

††pubyear: 2018††pagerange: Extreme & High Synchrotron Peaked Blazars at the
limit of Fermi-LAT detectability: the $\gamma$-ray spectrum of 1BIGB sources–A

##  1 Introduction

The Fermi-LAT sources and specially the high-energy catalogues, 2FHL (50 GeV,
Ackermann et al., 2016) and 3FHL (10 GeV, Ajello et al., 2017), provide the
best unbiased proxy to the very-high-energy (VHE) extragalactic sky.
Meaningful extrapolations can be derived therefrom, as to what are the
expectations for future studies in this extreme observational window.

Complementary to those works is the 1BIGB catalogue (Arsioli & Chang, 2017),
which contains 150 extragalactic $\gamma$-ray sources associated to excess
signals $\rm>3\sigma$ in the 0.3-500 GeV band as observed with Fermi-LAT.
Bearing nearly no overlap with the previous Fermi-LAT catalogues, the 1BIGB
presents sources which were found through targeted analysis of a group of
candidate VHE blazars obtained from the 1WHSP and 2WHSP catalogues (Arsioli et
al. (2015) and Chang et al. (2017)), selected on the basis of their IR and
X-ray spectral energy distribution (SED) synchrotron properties. Collectively,
the objects in the 1BIGB catalogue represent a unique sample from a population
of extragalactic objects at the limit of Fermi-LAT detectability, and which
are at the extreme of blazar phenomenology.

Blazars are active galactic nuclei believed to produce $\gamma$ rays inside
relativistic jets which beam the emission towards the observer with bulk
Lorentz factors of $\sim 10$ or larger (Urry & Padovani, 1995). The
$\gamma$-ray production mechanism in blazars is not completely understood,
evidence suggesting that most of the emission has a leptonic origin, although
a significant but yet unquantified contribution from hadronic processes cannot
be excluded (Cerruti et al., 2017). Good sampling of the $\gamma$-ray SED of
these objects is therefore fundamental for blazar studies (Böttcher et al.,
2013). A related example is the recent effort to unveil new $\gamma$-ray Low-
Synchrotron-Peaked (LSP) blazars (Arsioli & Polenta, 2018), previously
expected to be $\gamma$-ray quiet.

The SEDs from blazars can extend along many decades in energy, from a few GHz
in radio, up to TeV $\gamma$ rays, and are usually characterized by the
presence of a synchrotron and an inverse-Compton (IC) bumps in the Log
(${\nu})$) vs. Log ($\rm\nu f_{\nu}$) plane (Padovani et al., 2017). The peak-
power associated to the synchrotron bump tells us at which frequency most of
the AGN electromagnetic power is being released, the parameter Log
($\rm\nu_{peak}^{syn}$) being extensively used to classify blazars. Following
discussions from Padovani & Giommi (1995), Abdo et al. (2010b) and Ghisellini
(1999), objects with Log($\rm\nu_{peak}^{syn}$) < 14.5, in between 14.5 to
15.0, and $>$ 15.0 [Hz] have been called, respectively, low-, intermediate-,
and high-synchrotron-peak blazars (LSP - ISP - HSP).

Recent observational data in hard X-rays (Costamante et al., 2017) support the
existence of a population of extreme high-energy synchrotron peaked sources
(EHSPs). The class had been first proposed by Costamante & Ghisellini (2002),
being composed of sources with a synchrotron peak frequency above $10^{17}$
Hz. Clear indications or evidence for the synchrotron peak actually reaching
the MeV range is nevertheless still under debate, as in the following works:
Chang et al. (2017); Tanaka et al. (2014a); Kaufmann et al. (2011); Tavecchio
et al. (2011).

Extreme HSPs are characterized by low luminosity and limited variability, in
contrast to typical HSP blazars; the weak variability, however, could be
biased by the limited sensitivity of current instruments, rather than to
intrinsic characteristics of the sources. The high synchrotron peak (and
consequently high IC peak) and low luminosity of EHSPs seem to support the
blazar sequence paradigm (Fossati et al., 1998; Ghisellini et al., 2017).
However, the expected shift of both SED peaks to higher energies has not been
firmly established yet. We should also call attention to the fact that nearly
half of the 1BIGB sources (63 objects) have no measured redshift and another
21 only have an estimated lower-limit redshift. Therefore, as typical of HSP
blazars, many 1BIGB sources have no derived luminosity, which could introduce
strong bias to the blazar sequence paradigm, as extensively discussed in
Arsioli et al. (2015) with focus to HSP blazars.

Since EHSPs are possibly characterized by an IC peak at VHE $\gamma$ rays,
they are not easy to observe, being hardly detected in current $\gamma$
surveys as well. For example, one of the brightest such sources in X-rays, and
the best-studied extreme blazar to date, 1ES 0229+200, was detected by Fermi-
LAT only in the 3FGL, after 4 years of exposure. The HE and VHE $\gamma$-ray
spectra for this object were shown to connect smoothly and to be very hard,
with no apparent break up to TeV energies (Aharonian et al., 2007).

EHSPs draw increasing attention given the possibility that blazars might be
associated to astrophysical neutrinos (Padovani et al., 2016; Padovani et al.,
2018), and ultra-high-energy cosmic rays (Resconi et al., 2017). Extreme
blazars have also large impact as probes of the extragalactic background light
(EBL) and cosmic magnetic fields (Bonnoli et al., 2015). Given the broad
context in which blazars play an important role in future astro-particle
physics, describing new $\gamma$-ray spectral properties of HSP and EHSP-
candidates becomes of major relevance.

An unbiased population study of blazars at VHE $\gamma$ rays is made difficult
by the fact that these objects are characterised by intense flaring activity.
In fact, many of the known VHE $\gamma$-ray blazars have been detected via
triggers based on high states from other bands, rather than through surveys.
To produce good estimates on the size and nature of the blazar population
potentially detectable at VHEs is therefore an important task. Furthermore,
completing the information available on the faint and extreme sector of blazar
phenomenology is specially useful, with implications for the determination of
their luminosity function (the $\rm\log{N}-\log{S}$ distribution), and the
estimates of the total $\gamma$-ray background (Inoue & Ioka, 2012; Ackermann
et al., 2015).

Observations with the current class of VHE instruments, as well as Fermi-LAT
(Tanaka et al., 2014a), seem to corroborate with the existence of a population
of EHSPs with IC peak above 100 GeV in the $\gamma$-ray band. Fermi-LAT, for
which sensitivity is already greatly decreasing above 100 GeV, might
nevertheless be probing only the low-energy side of the IC component, which
could easily reach beyond the TeV range. As a result, these putative EHSPs are
clearly undersampled in current Fermi-LAT catalogues, with just three well-
studied sources in the local universe up to date – Mkn 421 and Mkn 501 (at
$z\sim 0.03$) and 1ES 0229+200 (at $z$ = 0.14).

By virtue of its selection methodology, the blazar population in the 1BIGB
catalogue is composed of faint HSP blazars and extreme blazars. Until now, a
source-by-source discrimination between the two classes has not yet been
possible though, due to the limited information available on the synchrotron
SED and on the 0.3-500 GeV $\gamma$-ray excess. The fact that most of the
objects presented is this work do not show up in previous Fermi-LAT
catalogues, points to a significant new extension of the blazar population,
with important implications to the extragalactic science case of currently
active VHE $\gamma$-ray observatories, as well as for the future Cherenkov
Telescope Array, CTA (CTA Consortium et al., 2017; Sol et al., 2013).

In this work we analyze the complete sample of 1BIGB sources, using the latest
PASS8 Fermi-LAT data release (Atwood et al., 2013), and present, for the first
time, their 1-100 GeV $\gamma$-ray spectra. The final list results in the
richest high-energy spectral sample of faint HSPs and candidate extreme
blazars to date, providing useful new information for the classification of
these objects, given that only low-energy SED information was previously
available. As we discuss in following Section 5, the catalogue constitutes
also a potential list of targets for current and future VHE observational
programmes.

The paper is organised as follows. In Section 2, we describe in detail the
1BIGB sample and calculate updated 0.3-100 GeV broadband fitting for all
sources, integrating over 9.3 yrs of currently available data. In Section 3 we
present for the first time, the 1-100 GeV $\gamma$-ray SEDs of its 148
sources. In Section 4 we stack the $\gamma$-ray SEDs, providing a common view
of the source population, discuss about the Fermi-LAT sensitivity limit at 10
GeV, and present two notable sources from our sample. Finally, in Section 5 we
briefly compare our results to the published sensitivity curves of MAGIC,
H.E.S.S., and CTA, in order to estimate the detectability and the impact of
our source population to current and future VHE $\gamma$-ray studies.

##  2 The 1BIGB sample description

The 1BIGB sample was built based on the selection of 400 $\gamma$-ray
candidates from the 1WHSP and 2WHSP samples. Those candidates comprise HSPs
with the brightest synchrotron peaks Log ($\rm\nu f_{\nu}$) > -12.1 erg/cm2/s,
but having no counterpart in previous Fermi-LAT catalogues – 1FGL, 2FGL and
3FGL (Abdo et al., 2010a; Nolan et al., 2012; Acero et al., 2015). A
likelihood analysis in the 0.3-500 GeV band, integrating over 7.2 years of
Fermi-LAT observations, and using the PASS8 data release, unveiled a total of
150 $\gamma$-ray signatures, of which 85 had high-significance, with Test
Statistics (TS, Mattox et al., 1996) ranging from 25 up to 130 (all >$\rm
5\sigma$ detections) and the remaining 65 had lower-significance, with TS in
between 10 to 25 ($\rm 3\ to\ 5\sigma$ detections). Many low and high-
significance detections were studied with the help of $\gamma$-ray TS maps,
showing a typical point-like source emerging from a smooth background. In
addition, the sample showed $\gamma$-ray spectral properties as expected for
HSP blazars, with photon spectral index $\rm\langle\Gamma\rangle\sim 1.94$.
Given that each of those 150 signatures is associated to a powerful HSP
counterpart, they represent robust detections at the edge of $\it Fermi$-LAT
sensitivity (as well described by Figure 8 from Arsioli & Chang, 2017),
grouped under the acronym 1BIGB.

###  2.1 Updating the 1BIGB broadband fit between 0.3-500 GeV

The 1BIGB detections considered a broadband likelihood analysis between
0.3-500 GeV, integrating data from August 04 2008 up to November 04 2015
($\sim$7.2 years). Since that time, Fermi-LAT has accumulated more than 2
additional years of observation time (at the time of writing, we have
available data up to December 05 2017). Taking advantage of a larger exposure
time with Fermi-LAT, we thus refine the power-law fitting for each 1BIGB
source by performing a likelihood analysis similar to Arsioli & Chang (2017),
applying the same setup and data quality cuts.

Table 1: Table showing the model description for 13 of the 148 1BIGB $\gamma$-ray signatures, all detected with TS$>$100\. Note that a complete table with all the 148 source is available on-line. The first three columns show respectively the 1BIGB source names, right ascension R.A. and declination Dec. in degrees (J2000). The fourth column shows the reported redshifts from literature (Shaw et al., 2013b; Pita et al., 2014; Furniss et al., 2013; Danforth et al., 2010; Shaw et al., 2013a; Masetti et al., 2013; Sbarufatti et al., 2005; Massaro et al., 2015). The flag ? is used for values reported as uncertain; lower limits are marked with “$>$" (all lower-limits shown here were derived in Arsioli et al., 2015; Chang et al., 2017), and sources with currently absent redshift were given 0. value. The $\gamma$-ray model parameters from the Fermi Science Tools assume a power-law to describe the spectrum within the studied energy range 0.3-500 GeV. The parameter $\rm N_{0}$ (see Equation 1) is given in units of ph/cm2/s/MeV, and $\Gamma$ is the spectral photon index, which are direct outputs from the likelihood analysis over 9.3 years of Fermi-LAT data in the 0.3-500 GeV band; those results consider the pivot energy fixed as E0 = 1 GeV. The column Flux gives the photon counts in units of ph/cm2/s calculated by integrating Equation 1 along the energy range 1-100 GeV, column E-Flux corresponds to the energy flux in units of MeV/cm2/s. For the columns Flux and E-Flux, upper and lower case values represent positive and negative errors, respectively. 1BIGB Source name | R.A. (deg) | Dec. (deg) | z | $\Gamma$ | N0 (10-15) | TS | Flux${}_{1-100\,GeV}^{(\times 10^{-10})}$ | E-Flux${}_{1-100\,GeV}^{(\times 10^{-13})}$
---|---|---|---|---|---|---|---|---
1BIGBJ021631.9+231449 | 34.13333 | 23.24722 | 0.288 | 1.88$\pm$0.10 | 2.49$\pm$0.52 | 116.4 | 2.76${}^{0.98}_{-0.79}$ | 15.0${}^{+4.7}_{-3.3}$
1BIGBJ044240.6+614039 | 70.66917 | 61.6775 | 0. | 2.01$\pm$0.09 | 4.00$\pm$0.71 | 145.0 | 3.92${}^{+1.12}_{-0.95}$ | 17.9${}^{+4.3}_{-3.2}$
1BIGBJ050335.3-111506 | 75.8975 | -11.25167 |  $>$0.57 | 1.85$\pm$0.11 | 2.30$\pm$0.53 | 102.6 | 2.63${}^{+1.03}_{-0.82}$ | 15.0${}^{+5.1}_{-3.5}$
1BIGBJ050727.1-334635 | 76.86333 | -33.77639 | 0. | 1.79$\pm$0.11 | 1.84$\pm$0.43 | 125.0 | 2.25${}^{+0.93}_{-0.72}$ | 14.1${}^{+5.0}_{-3.5}$
1BIGBJ075936.1+132116 | 119.90042 | 13.35472 | 0. | 1.75$\pm$0.09 | 2.12$\pm$0.44 | 152.6 | 2.71${}^{+0.98}_{-0.79}$ | 17.9${}^{+5.5}_{-4.0}$
1BIGBJ082904.7+175415 | 127.27 | 17.90417 | 0.089 | 2.25$\pm$0.10 | 4.47$\pm$0.55 | 136.8 | 3.55${}^{+0.79}_{-0.67}$ | 12.1${}^{+2.7}_{-2.0}$
1BIGBJ113755.6-171041 | 174.48167 | -17.17833 | 0.6 | 1.71$\pm$0.10 | 1.90$\pm$0.43 | 126.9 | 2.55${}^{+1.00}_{-0.79}$ | 18.0${}^{+6.0}_{-4.2}$
1BIGBJ121510.9+073203 | 183.79542 | 7.53444 | 0.137 | 1.64$\pm$0.11 | 1.26$\pm$0.37 | 105.5 | 1.87${}^{+0.97}_{-0.72}$ | 14.9${}^{+6.2}_{-4.1}$
1BIGBJ144236.4-462300 | 220.65167 | -46.38361 | 0.103 | 1.92$\pm$0.10 | 3.14$\pm$0.64 | 107.7 | 3.33${}^{+1.14}_{-0.93}$ | 17.1${}^{+5.0}_{-3.6}$
1BIGBJ154202.9-291509 | 235.5125 | -29.2525 | 0. | 1.78$\pm$0.08 | 2.62$\pm$0.50 | 143.2 | 3.27${}^{+1.04}_{-0.86}$ | 20.8${}^{+5.4}_{-4.1}$
1BIGBJ194356.2+211821 | 295.98417 | 21.30611 | 0. | 1.44$\pm$0.08 | 2.01$\pm$0.57 | 194.7 | 3.93${}^{+1.84}_{-1.44}$ | 42.9${}^{+13.2}_{-9.8}$
1BIGBJ200204.0-573644 | 300.51708 | -57.6125 | 0. | 2.08$\pm$0.10 | 2.93$\pm$0.49 | 105.5 | 2.69${}^{+0.78}_{-0.64}$ | 11.2${}^{+3.0}_{-2.2}$
1BIGBJ224910.6-130002 | 342.29458 | -13.00056 |  $>$0.5 | 2.33$\pm$0.01 | 87.01$\pm$1.43 | 11240.5 | 64.8${}^{+1.93}_{-1.88}$ | 202.8${}^{+6.2}_{-6.0}$

We update the 1BIGB power-law parameters and re-evaluate the TS values
considering the current 9.3 years of observation, aiming to deliver the most
significant description up to date. For this broadband analysis we integrate
over the entire 0.3-500 GeV energy band, using PASS8 data release, and
assuming that the $\gamma$-ray spectrum of each $\gamma$-ray source can be
described by a power-law:

$\rm\hskip
85.0pt\frac{dN}{dE}\,\text{=}\,N_{0}\;\left(\frac{E}{E_{0}}\right)^{-\Gamma}\;,$
(1)

where $\rm E_{0}$ is a scale parameter (also known as pivot-energy), $\rm
N_{0}$ is the prefactor (normalization) corresponding to the flux density in
units of ph/cm2/s/MeV at the pivot energy $\rm E_{0}$, and $\rm\Gamma$ is the
photon spectral index for the energy range considered. Both $\rm\Gamma$ and
$\rm N_{0}$ are set as free parameters and further adjusted by the likelihood
fitting routine. Source positions and E0 = 1000 MeV are set as fixed
parameters, kept constants for the analysis. In the source-input xml file, all
sources within 10∘ from the candidate had both $\rm\Gamma$ and $\rm N_{0}$
parameters flagged as free111In this regard we are following recommendations
from Fermi Science Tools user guide
https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/binned_likelihood_tutorial.html
which advise users to set free parameters at least within 7∘ from the source
of interest. This is a consequence of the large point spread function (PSF)
specially at the low-energy threshold, which can overlap with nearby sources.
Therefore, in order to get a confident description of a particular source, we
need to properly fit and adjust the whole environment around it., and
therefore their 3FGL models that are based on 4 years of observations will be
adjusted. This particular choice increases the computational burden of the
analysis, but is crucial for adapting the model maps to the extra 5.3 years of
exposure being considered. A sub-sample of the results, listing only 13 cases
detected with TS$>$100 is shown in Table 1; a list with the total 148 cases
which had good convergence will be available as on-line material.

Figure 1: Comparison between the photon spectral index ($\Gamma$) integrating
over for 7.2 years of exposure, with the updated values calculated with 9.3
years of observations with Fermi-LAT. The $\Gamma$ factor considers a power-
law fit to the entire 0.3-500 GeV energy band, and its corresponding error
bars are represented in gray. The red-dashed line is a curve with slope equal
one.

For most cases the $\Gamma$ parameter estimates obtained by integrating over
7.2 ($\Gamma_{7.2}$) and 9.3 ($\Gamma_{9.3}$) years are relatively stable,
since the intrinsic variability is well contained within the error bars
(compared to the red-dashed line in Figure 1, for $\rm\Gamma_{7.2}$ =
$\rm\Gamma_{9.3}$). There were only two cases which did not show good
likelihood-convergence when integrating along 9.3 yrs, namely 1BIGB
J080135.8+463824 and 1BIGB J145508.2+192014, which were removed from the 1BIGB
SED list. We would like to note that those sources had TS close to 10 in the
previous 1BIGB sample (7.2 years). Few non-detections are expected when
integrating over different time-windows, specially owing to the uncertainty
induced by source variability.

##  3 The $\gamma$-ray spectral energy distribution

The 1BIGB sample is composed of $\gamma$-ray sources which are, on average,
faint and close to the detectability limit of Fermi-LAT. In the process of
building the $\gamma$-ray SEDs, we had to deal with the problem of poor photon
counts when integrating over a short energy bin. In most cases, simply
dividing the broadband energy (between 1 GeV and 500 GeV) in equally spaced
logarithmic bins result in low TS detections per bin, which most of the times
only become upper limits for the SED. This is the usual way SEDs are built,
and inevitably a considerable amount of relevant information available via
broadband analysis is not being incorporated.

In order to capture the shape of the $\gamma$-ray spectrum and estimate
reliable SED data points, we integrate each energy bin over a large energy
band (larger than ‘equally spaced logarithmic bins’) and evaluate the flux at
a specific energy for each bin. In other words, we take into account
information from a broadband analysis by accepting superposed energy channels.

Table 2 describes the energy bands over which we integrate in order to
estimate fluxes at specific pivot-energies ($\rm E_{0}$), later used to build
the $\gamma$-ray SEDs. We choose those E0 values to be close to equally spaced
bins in Log(E) scale, with increments of the order of $\sim$100.25. For
example, starting from the 1 GeV SED point we integrate over 1.0 to 5.0 GeV,
adjusting a power-law (see Equation 1) to this interval and setting E0 = 1
GeV, so that N0 represents the flux at 1 GeV when taking into account the
spectral trend in the selected energy-bin. For the later SED point at 1.7 GeV,
we increment the energy band by a factor $\sim$100.25, therefore integrating
over 1.0 to 10 GeV, so to capture the power-law trend when estimating an SED
point at higher energy. We proceed in this way up to the 5 GeV SED point, and
for the next ones we start incrementing both the low and high-energy band by a
factor $\sim$100.25, till we reach the 100 GeV SED data point, with a high-
energy threshold of 500 GeV.

Table 2: Definition of energy bins used for the broadband analysis, to estimate fluxes N0 at each pivot energy $\rm E_{0}$. The estimate is done via likelihood analysis, adjusting a power-law to each energy bin. E0 [GeV] | Integrate over [GeV]
---|---
1.0 | 1.0 - 5.0
1.7 | 1.0 - 10.0
3.0 | 1.0 - 17.0
5.0 | 1.0 - 30.0
10.0 | 1.7 - 50.0
17.0 | 3.0 - 100.0
30.0 | 5.0 - 170.0
50.0 | 10.0 - 300.0
100.0 | 30.0 - 500.0

Considering the total of 1350 likelihood analysis (integrating over 9.3 yrs)
needed to be computed for building the entire SED catalogue, we limit the low
energy band to 1 GeV to improve the computation time. Given that the photon
counts at 1 GeV and higher energy levels are relatively reduced with respect
to lower energy bands, working with a cut at 1 GeV allows the likelihood to
converge in a reasonable time (so that a large scale analysis could be
performed). The cut in energy also adjusts well to the hard $\gamma$-ray
spectral slope expected for HSP sources; taking into account that the detector
PSF improves with increasing energy, the energy cut also helps to avoid
contamination from nearby sources.

In the source-input xml file, all sources within 10∘ from the candidate are
set free to vary their spectral fitting parameters, just as used in previous
broadband analysis (Section 2.1). Both $\Gamma$ and $\rm N_{0}$ are set as
free parameters and further adjusted by the gtlike fitting routine. The source
position and the scaling $\rm E_{0}$ are set as fixed parameters, with $\rm
E_{0}$ changing only according to the SED point being calculated. At this
point, we call attention to the choice of the pivot-energy $\rm E_{0}$, which
corresponds to the energy where the differential flux $\rm N_{0}$
[ph/MeV/cm2/s] is going to be estimated. Later, we compute the flux
[erg/cm2/s] at a given $\rm E_{0}$ simply converting222The factor
1.602$\times$10-6 is used to convert MeV to erg.: $\text{flux = }\rm
N_{0}\cdot E_{0}^{2}\cdot 1.602\times 10^{-6}$, and cases showing TS $<$ 6.0
per bin are given upper limits. This is how we combine a broadband analysis
with the flux calculation for each energy bin, to deal specifically with faint
sources.

For the highest energy channels 50 GeV and 100 GeV, a broadband binned
analysis suffers from rather low photon counts, to which an unbinned analysis
is much better adapted. Therefore, for those two channels we have applied
unbinned analyses, assuring a good SED agreement between the few 1BIGB sources
which are also part of the 2FHL and 3FHL catalogues. As an example333Please,
consult Arsioli & Chang (2017) for a complete list of 1BIGB sources having
2FHL counterparts we cite the cases of 1BIGB J044240.6+614039, 1BIGB
J030433.9-005403 and 1BIGB J113046.0-313807.

##  4 Average $\gamma$-ray behavior and blazar populations

The 1BIGB catalogue contains average spectral information about the sources
derived from a 0.3-500 GeV broadband analysis. With the purpose of
investigating in more detail the spectral properties of these faint objects,
here we present the 1-100 GeV spectral points resulting from the analysis of
9.3 years of Fermi-LAT data as reported in the previous Section. As can be
seen in Figure 2, the spectral points populate the full energy window under
consideration, except for the highest energy bin, around 100 GeV, where only
13% of the blazars present a significant signature.

When considering an individual power-law fit to the $\gamma$-ray spectrum, the
derived photon spectral index covers a wide range from 1.26 to 2.81, with an
average error on the slope of the order of 10%. The large spread in spectral
slopes suggests that the objects composing this catalogue do not present any
common spectral feature at the HE $\gamma$-ray band. No characteristic
spectral turnover, nor spectral break, correlating with other properties such
as synchrotron-peak frequency or flux level was found.

At first glance such results may look at odds with expectations from a
description following the blazar sequence (Costamante et al., 2001); we
remind, nevertheless, the fact that an effective population study is hardly
possible here, given the limited sample size and the lack of homogeneity in
the sample selection, which is composed of objects distinguished only by being
at the very limit of Fermi-LAT detectability. The 1BIGB source population is
therefore likely composed of a mixture of faint HSPs and extreme-HSP blazars.

Within the population, two sources stand out showing a $\gamma$-ray spectrum
which clearly exceed most objects in flux. All the remaining sources are
clustered in a region of $\nu F_{\nu}$ between 10-13 and 10-12 erg/cm2/s, with
no clear common spectral trend correlated to synchrotron peak information. The
two peculiar sources will be further discussed in Section 4.3.

We fitted the clustered points for the remaining sources with a power-law (red
line in Figure 2) and found an average photon spectrum slope of
$\Gamma\sim$1.86. This suggests that for the 1BIGB population as a whole, the
peak of the second SED bump is located at energies larger than those covered
by our analysis, falling within the VHE $\gamma$-ray range. This result is
consistent with expectations for EHSPs, which should have a hard spectrum at
high-energies, such as those measured for 1ES 0229+200 and 1ES 1011+496
(Aharonian et al., 2007; Ahnen et al., 2016), the two hardest spectrum sources
ever detected in the VHEs. The large spread in spectral points and fit slopes,
however, make these considerations on the average trend not representative of
all the sample, which appears non-uniform except for the faintness in flux.

Figure 2: Stacking of the 148 1BIGB SEDs in the 1-100 GeV energy band. Red dashed lines enclose the main flux density interval covered by the measured fluxes and its uncertainties, showing the main region in Log ($\rm\nu$) [GeV] vs. Log ($\rm\nu f_{\nu}$) [eg/cm2/s] covered by the new $\gamma$-ray spectrum. Red line is a average power-law fit considering all the 148 SEDs. Table 3: Most extreme 1BIGB sources (highest synchrotron peak frequency). The index is calculated by fitting a power-law to the SED points in the range 1-100 GeV. Name | $\Gamma_{1-100\,GeV}$ | Log $\nu_{peak}^{syn}$
---|---|---
1BIGB J194356.2+211821 | $1.68\pm 0.23$ | 18.1
1BIGB J151618.7-152344 | $2.08\pm 0.07$ | 18.0
1BIGB J225147.5-320611 | $2.04\pm 0.07$ |  $>$18.0
1BIGB J020412.9-333339 | $1.60\pm 0.08$ | 17.9
1BIGB J032056.2+042447 | $2.65\pm 0.18$ | 17.9
1BIGB J050419.5-095631 | $2.21\pm 0.08$ | 17.9
1BIGB J055716.7-061706 | $1.81\pm 0.05$ | 17.9
1BIGB J125341.2-393159 | $1.72\pm 0.09$ | 17.9
1BIGB J132541.8-022809 | $1.83\pm 0.15$ | 17.9
1BIGB J160519.0+542058 | $2.18\pm 0.14$ | 17.9

###  4.1 Population considerations

Based on synchrotron peak values reported in the 2WHSP catalogue, the 1BIGB
sample holds 52 EHSP candidates, of which 11 objects do not have a firm
determination of the peak. We classify the remaining 96 objects of the 1BIGB
as (faint) HSP blazars for the purposes of this study.

In order to investigate the properties of EHSP and HSP populations within the
catalogue, we have compared the SEDs of the two groups of sources. We find
that both the average slope of the EHSP candidates and their spanned fluxes
are comparable to those of the HSPs. Therefore, the two populations have no
clear signature in the SEDs analyzed which would allow distinguishing them on
the basis of their Fermi-LAT $\gamma$-ray properties alone.

This result suggests that there are EHSPs whose IC peak behaves like standard
HSPs, and therefore a shift of the synchrotron peak location to extreme
energies does not always reflect in a shift of the second SED peak.
Variability most certainly plays an important role on the detectability of
these faint sources and a bias towards the presence of high states can
contaminate the sample mean, affecting also its spectral properties.

For cases where the second peak is shifted, this could be detected in the VHE
regime with the current IACTs or the forthcoming CTA. In any case, the sub-
class of EHSPs displaying an IC peak location in the VHE $\gamma$-ray range is
certainly one of the primary targets of VHE $\gamma$-ray observations, and
might represent the only class of blazar sources partially undetected by
Fermi-LAT, but well within the reach of IACTs.

###  4.2 Sensitivity limit

The 1BIGB sources give us the opportunity to study the sensitivity limit for
the $\gamma$-ray detection of HSP and EHSP blazars with Fermi-LAT. Figure 3
shows the flux density at 10 GeV as a function of the synchrotron peak
position taken from the 2WHSP catalogue. Except for two peculiar cases, all
sources are contained within a region of flux density at 10 GeV comprised
between 10-12.1 erg/cm2/s and 10-13.2 erg/cm2/s. The sources presented in the
1BIGB catalogue explore a new range of sensitivity, previously not accessible
to Fermi-LAT catalogs due to the lower exposure time (4 years, in case of the
3FGL).

Figure 3: Logarithm of flux density (erg/cm$2$/s) at 10 GeV vs. logarithm of
synchrotron peak frequency (Hz) for the 148 1BIGB sources considered in the
study. Dashed lines show the previous (gray) and improved (blue) sensitivity
limits.

The cut at the bottom left corner seems to represent the increasing difficulty
for detecting sources with the lowest synchrotron Log ($\nu^{Syn}_{peak}$)
values. A possible explanation is that for low synchrotron-peak values, the IC
peak may move to lower energies, hindering the detections; but also this could
be connected to incompleteness of the 2WHSP sample close to the synchrotron
peak cut, at $\nu\sim$1015 Hz. As reported in Chang et al. (2017) the
incompleteness might be induced by the multi-frequency selection criteria
which was used to build the 2WHSP sample.

(a)

(b)

Figure 4: Left plot: SED of 1BIGBJ224910.6-130002, showing the mean
$\gamma$-ray SED obtained by integrating over 9.3 years in blue, and during
the 25 days flare peak (MJD 57662.65-57687.65) in black. Right plot: Light
curve of 1BIGBJ224910.6-130002 considering one month time bins, integrating
over the broadband of 900 MeV up to 500 GeV. Flux calculated only for >
3$\sigma$ bins (red points), error bars and upper limits (black cross) are
calculated using Integral method considering 95% of confidence level.

###  4.3 Notable sources

The stacked $\gamma$-ray SEDs displayed in Figure 2 present two distinct
outliers, namely the sources 1BIGB J224910.6-130002 and 1BIGB
J194356.2+211821.

The source 1BIGB J224910.6-130002, also known as RBS 1899, is a blazar of
unknown redshift classified as an EHSP due to its synchrotron peak position
estimated at 1017.5 Hz. Interestingly the Fermi-LAT spectrum obtained with
over 9.3 years of data shows a steep decaying behavior in Log ($\nu$) versus
Log ($\nu F_{\nu}$) space, with photon spectral index $\Gamma\sim 2.33$ and
relatively bright in $\gamma$ rays, as seen in Figure 4a. However this unusual
behavior is probably due to the averaging of the quiescent state with a series
of $\gamma$-ray flares which happened along 2016 to 2017, as can be seen in
the light curve reported in Figure 4b. During these flaring episodes, the flux
increased of $\sim$40 times with respect to the quiescent phase. This may also
explain why the source was undetected in previous FGL and FHL catalogues. This
is an important example of a $\gamma$-ray blazar which could represent a major
target for the future CTA during periods of extreme activity (as detected
during MJD 57662 - 57687, Figure 4a in black). It also illustrates the strong
impact of flaring activity in attempting a population study of extreme or
faint blazars, since the most extreme $\gamma$-ray activity is smoothed over
time when integrating over years of Fermi-LAT observations.

The second peculiar source, 1BIGB J194356.2+211821 also known as HESS
J1943+213, is the most extreme blazar of the 1BIGB catalogue, with Log
($\rm\nu_{p}$) = 18.1. It was serendipitously discovered by the H.E.S.S.
Collaboration (H.E.S.S. Collaboration et al., 2011) at VHE $\gamma$ rays
during a galactic plane survey and later confirmed with VERITAS observations
(Shahinyan & VERITAS Collaboration, 2015). Due to the source’s position on the
galactic plane and its multi-wavelength properties, the classification was
difficult and three competing hypotheses were proposed: a pulsar wind nebula
(PWN), a $\gamma$-ray binary, and a blazar, with the latter being favored.
Previous studies tried to analyse Fermi-LAT data in order to determine the
nature of the source: Tanaka et al. (2014b) analysed Pass 7 data on the
position of the HESS source, finding a weak detection (TS=22.3) in the 10-300
GeV range, with a rather soft spectral index of $\Gamma\approx 2.4$; analysis
of the Pass 7 Reprocessed data (Peter et al., 2014) shows a detection near the
position determined by HESS with TS=36.0 in the range 1-300 GeV, and a
spectral index of $\Gamma$ = 1.59 (in agreement with the value reported here
considering the errors). Both these studies favor the blazar interpretation
based on multiwavelength data and further observations seem to corroborate
this (see e.g. Straal et al. (2016) and Akiyama et al. (2016)), making this
the first blazar serendipitously detected by VHE $\gamma$-ray ground-based
instruments. Interestingly, the optical and UV properties alone would not be
enough to classify this source as a blazar, showing the importance of the
multi-wavelength approach. The SED in Figure 5 shows that the VHE $\gamma$-ray
data is complemented nicely by our Fermi-LAT points, making the IC peak
location clearly above few hundred GeV.

Table 4: Five sources selected for the VHE $\gamma$-ray extrapolation study. 1BIGB J Source Name | $z$ | log $\nu_{p}$ | $\Gamma$ | Cut-off
---|---|---|---|---
090802.2$-$095936 | 0.053 | 17.6 | $1.65\pm 0.13$ | 600 GeV
151041.0+333503 | 0.114 |  $>$17.5 | $1.92\pm 0.15$ | 300 GeV
225147.5$-$320611 | 0.139 |  $>$18 | $2.04\pm 0.07$ | 1 TeV
220155.8$-$170700 | 0.169 | 17.7 | $0.90\pm 0.15$ | 150 GeV
223301.0+133601 | 0.214 |  $>$17 | $1.76\pm 0.15$ | 1 TeV
Figure 5: SED of 1BIGB J194356.2+211821. In the $\gamma$-ray band we have data
points corresponding to our new Fermi-LAT detection (in blue), 3FHL (in
magenta), and H.E.S.S. (in light green). Notice at 100 GeV the good agreement
between 1BIGB (highest energy point) and 3FHL (lowest energy point).

##  5 Potential for VHE $\gamma$-ray observations

The individual SEDs of the 1BIGB sources can be exploited to estimate the
detectability of these blazars at VHE $\gamma$ rays. To illustrate this
possibility, we have selected five promising objects. The choice of sources
was done considering only those objects with known redshift, in order to
correctly estimate the absorption of VHE $\gamma$-rays due to the interaction
with the extragalactic background light, EBL (Stecker et al., 1992). Among
those, we considered only blazars with a hard Fermi-LAT photon spectral index
($\Gamma$ < 2.0). The selected sources are listed in Table 4, and ordered by
redshift. The table also reports the synchrotron peak location, the Fermi-LAT
slope obtained from our fit to the 1-100 GeV points, and the cut-off energy of
the intrinsic spectrum assumed in our extrapolations.

We would like to note that all selected sources are EHSPs, and mostly this
bias result from the synchrotron peak shifting towards high energies, so that
the host galaxy thermal component emerge at optical frequencies, allowing a
precise redshift measurement.

In Figure 6 we show the $\gamma$-ray SEDs of the selected sources up to the
highest available energies. We extrapolate the _Fermi_ -LAT data points
considering a power-law function with exponential cut-off. The cut-off energy
for each source was estimated by taking into account all data points and upper
limits within the 1-100 GeV energy band. We considered absorption due to
interaction with EBL based on $\tau_{(E,z)}$ values as predicted by Domínguez
et al. (2011).

Depending on the location of each source, we have considered detectability
with CTA-North and MAGIC, or CTA-South and H.E.S.S. . Each $\gamma$-ray SED is
represented before (dashed line) and after (dotted-dashed line) EBL
absorption. It is evident that the redshift is a very important parameter that
can affect sensibly the detectability of blazar candidates. This is
particularly true for EHSPs with an IC peak exceeding TeV energies, since in
this case, already at relatively small redshifts such as those considered
here, the absorption strongly affects the spectrum. The effect is evident in
the two sources with the highest assumed cut-off energy, namely 1BIGB
J225147.5$-$320611 and 1BIGB J223301.0+133601 , located at redshift 0.139 and
0.214. Thanks to the relatively high intrinsic emission, despite the EBL
absorption severely affecting the observed flux, the extrapolations are still
confidently within CTA detectability in both cases.

For 1BIGB J220155.8$-$170700 the upper limit of $4.19\cdot 10^{-13}$ erg/cm2/s
at 100 GeV derived from Fermi-LAT data severely constrains the location of the
IC peak despite the negligible EBL absorption effect, and make this a
difficult target even for the expected sensitivity of CTA-North. The last two
sources, 1BIGB J151041.0+333503 and 1BIGB J090802.2-095936, present a spectrum
compatible with an intrinsic IC peak location at hundreds GeV, and seems on
the reach of CTA-North at least in the energy interval close to the SED peak.

All the extrapolated fluxes shown in Figure 6 are well below the sensitivity
of current generation of IACTs, indicating that a VHE $\gamma$-ray emission
from these sources could be detected only in case of extremely bright states.

##  6 Conclusions

We have presented for the first time the 1-100 GeV spectrum of a population of
$\gamma$-ray faint EHSP sources, recently detected with Fermi-LAT in the 1BIGB
catalogue. The results presented here are an update of the previous 1BIGB
work, in the sense that we now include new PASS8 data integrated over 9.3
instead of 7.2 years of observations. This is the largest catalogue of extreme
and weak blazars observed in $\gamma$ rays. All the data presented here is
made available in VO tables via the Brazilian Science Data Center (BSDC)
service (www.bsdc.cbpf.br), maintained at CBPF, and the BSDC Virtual
Observatory (vo.bsdc.icranet.org). It is also accessible via the Open Universe
Initiative Portal (www.openuniverse.asi.it) and through the SSDC SED Builder
tool (tools.asdc.asi.it).

The 1BIGB sample discussed in this work consists of both HSP and EHSP blazars,
whose spectral signatures in the Fermi-LAT band were shown to be similar among
the two classes, and likely dependent on the source state. No spectral
template for the classes was derivable, nevertheless we call attention to fact
that the sample is limited in size and inhomogeneous.

The average observed spectral index in the 1-100 GeV band ranges between 1.3
and 2.8 indicating a predominantly hard spectrum. The average flux of the
extreme HSP sample is relatively low, being composed of weak sources at the
limit of the Fermi-LAT sensitivity. For these reasons we argue that EHSPs may
escape Fermi-LAT detection (or be detected with low significance only), but
still be prime candidates for observations with Cherenkov telescopes, due to
the expected high VHE $\gamma$-ray emission. As we showed, it is expected for
at least part of the objects, that the IC peak will fall well within the VHE
band, above 100 GeV. Methods of selecting EHSPs not based exclusively on
Fermi-LAT detections or extrapolations are therefore extremely interesting for
completing the extragalactic population to be targeted by CTA.

In fact, we performed a detailed study of the potential for VHE $\gamma$-ray
observations of the strongest 1BIGB sources with known redshift, indicating
that they are well within the reach of future CTA capabilities. These analyses
show that future VHE observations are in a good position to probe this poorly
known, extreme class of HSP sources, expanding our understanding of the blazar
phenomenon and bringing new elements to the framework of the blazar sequence
scenario.

## Acknowledgements

During this work, BA was supported by São Paulo Research Foundation (FAPESP)
with grant n. 2017/00517-4. UBdA acknowledges support from a FAPERJ Young
Scientist Fellowship n. E10/2016-226465 and a CNPq Level 2 Fellowship n.
310827/2016-7. BF is supported by FAPERJ grants n. 202.687/2016 and
202.688/2016. We would like to thanks Prof. Marcelo M. Guzzo and Prof. Orlando
L. G. Peres for the full support endorsing the author’s partnership with
FAPESP, and granting access to the Feynman Cluster from CCJDR Data Center at
IFGW Unicamp, Campinas-Brazil. We also thank IcraNet, Prof. Remo Ruffini and
Prof. Carlo Bianco for the cooperation granting access to Joshua Computer
Cluster (Rome-Italy). The availability of computational resources was key to
the development of our work. We make use of archival data and bibliographic
information obtained from the NASA-IPAC Extragalactic Database (NED), data,
and software facilities from the Space Science Data Center (SSDC) from the
Italian Space Agency. The VO publication of our data (vo.bsdc.icranet.org) is
made by the Brazilian Science Data Center (BSDC) service maintained at CBPF,
Rio de Janeiro, and accessible through the United Nations Open Universe
Initiative at http://www.openuniverse.asi.it. We thank Mr. Carlos Brandt for
his support on this task. This research has made use of the CTA instrument
response functions provided by the CTA Consortium and Observatory, see
http://www.cta-observatory.org/science/cta-performance/ (version prod3b-v1)
for more details.

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: Extrapolation of the $\gamma$-ray spectrum of five selected sources
of the 1BIGB catalogue. The intrinsic spectrum assumed is a power-law with an
exponential cut-off (dashed line). The resulting spectra once corrected for
EBL absorption are displayed with dashed-dotted lines. CTA, MAGIC (Aleksić et
al., 2016), and H.E.S.S. (Aharonian et al., 2006) sensitivities for 50 h of
observations are also reported in the plots.

##  Appendix A The 1BIGB complete table with broadband fitting information

Table 5: Table showing the 148 1BIGB $\gamma$-ray signatures. The first three columns show respectively the 1BIGB source names, right ascension R.A. and declination Dec. in degrees (J2000). The fourth column shows the reported redshifts from literature (Shaw et al., 2013b; Pita et al., 2014; Furniss et al., 2013; Danforth et al., 2010; Shaw et al., 2013a; Masetti et al., 2013; Sbarufatti et al., 2005; Massaro et al., 2015), flag ? is used for values reported as uncertain; lower limit are marked with “$>$" (all lower-limits shown here were derived in Arsioli et al., 2015; Chang et al., 2017), and sources with currently absent redshift were given 0. value. The $\gamma$-ray model parameters from Fermi Science Tools assume a power law to describe the spectrum within the studied energy range 0.3-500 GeV. The parameter $N_{0}$ (see eq. 1) is given in units of [ph/cm2/s/MeV], and $\Gamma$ is the spectral photon index, which are direct outputs from the likelihood analysis over 7.2 years of Fermi-LAT data in the 0.3-500 GeV band; those results consider the pivot energy fixed as E0 = 1 GeV. The column Flux gives the photon counts in units of ph/cm2/s calculated by integrating eq. 1 along the energy range 1-100 GeV, column E-FLux corresponds to the energy flux in units of MeV/cm2/s. For the columns Flux and E-flux, upper and lower case values represent positive and negative errors, respectively. 1BIGB Source name | R.A.(deg) | Dec.(deg) | z | $\Gamma$ |  $N_{0}$ (10-15) | TS | Flux${}_{1-100GeV}^{(\times 10^{-10})}$ | E-Flux${}_{1-100GeV}^{(\times 10^{-13})}$
---|---|---|---|---|---|---|---|---
1BIGBJ000949.6-431650 | 2.45708 | -43.28056 |  $>$0.56 | 2.19$\pm$0.13 | 2.29$\pm$0.40 | 86.2 | 1.92${}^{+0.63}_{-0.50}$ | 7.0${}^{+2.3}_{-1.5}$
1BIGBJ001328.8+094929 | 3.37 | 9.825 | 0. | 2.07$\pm$0.19 | 1.56$\pm$0.57 | 24.6 | 1.44${}^{+0.95}_{-0.67}$ | 6.0${}^{+3.5}_{-1.9}$
1BIGBJ001527.8+353638 | 3.86625 | 35.61083 |  $>$0.57 | 1.90$\pm$0.23 | 0.94$\pm$0.48 | 35.3 | 1.02${}^{+0.99}_{-0.62}$ | 5.4${}^{+4.7}_{-2.2}$
1BIGBJ002928.6+205332 | 7.36917 | 20.8925 | 0. | 1.66$\pm$0.19 | 0.59$\pm$0.31 | 29.5 | 0.85${}^{+0.87}_{-0.53}$ | 6.5${}^{+5.3}_{-2.7}$
1BIGBJ004146.9-470136 | 10.44583 | -47.02667 | 0. | 1.79$\pm$0.27 | 0.33$\pm$0.21 | 10.8 | 0.40${}^{+0.55}_{-0.30}$ | 2.5${}^{+3.0}_{-1.2}$
1BIGBJ005816.6+172312 | 14.56958 | 17.38694 | 0. | 1.79$\pm$0.24 | 0.73$\pm$0.44 | 24.0 | 0.89${}^{+1.07}_{-0.62}$ | 5.6${}^{+5.7}_{-2.5}$
1BIGBJ010250.8-200158 | 15.71208 | -20.03278 |  $>$0.38 | 1.49$\pm$0.23 | 0.25$\pm$0.17 | 20.4 | 0.45${}^{+0.68}_{-0.35}$ | 4.6${}^{+5.2}_{-2.2}$
1BIGBJ011501.6-340027 | 18.75708 | -34.0075 | 0.48 | 1.65$\pm$0.16 | 0.82$\pm$0.36 | 70.8 | 1.19${}^{+0.98}_{-0.64}$ | 9.2${}^{+6.1}_{-3.4}$
1BIGBJ012657.1+330730 | 21.73833 | 33.125 | 0. | 2.19$\pm$0.29 | 1.38$\pm$0.57 | 22.2 | 1.15${}^{+0.99}_{-0.61}$ | 4.2${}^{+3.9}_{-1.6}$
1BIGBJ014040.8-075849 | 25.17 | -7.98028 |  $>$0.49 | 1.77$\pm$0.14 | 1.12$\pm$0.37 | 45.0 | 1.41${}^{+0.85}_{-0.61}$ | 9.1${}^{+4.7}_{-2.9}$
1BIGBJ020106.1+003400 | 30.27542 | 0.56667 | 0.298 | 1.70$\pm$0.24 | 0.47$\pm$0.30 | 19.9 | 0.65${}^{+0.84}_{-0.47}$ | 4.7${}^{+5.0}_{-2.2}$
1BIGBJ020412.9-333339 | 31.05375 | -33.56111 | 0.617 | 1.85$\pm$0.19 | 0.87$\pm$0.35 | 31.7 | 1.00${}^{+0.78}_{-0.51}$ | 5.7${}^{+4.1}_{-2.1}$
1BIGBJ021205.6-255757 | 33.02375 | -25.96611 | 0. | 1.99$\pm$0.14 | 1.67$\pm$0.41 | 58.9 | 1.67${}^{+0.74}_{-0.56}$ | 7.8${}^{+3.3}_{-2.1}$
1BIGBJ021216.8-022155 | 33.07 | -2.36528 | 0. | 1.92$\pm$0.19 | 1.16$\pm$0.44 | 31.8 | 1.23${}^{+0.89}_{-0.60}$ | 6.3${}^{+4.3}_{-2.3}$
1BIGBJ021631.9+231449 | 34.13333 | 23.24722 | 0.288 | 1.88$\pm$0.10 | 2.49$\pm$0.52 | 116.4 | 2.76${}^{+0.98}_{-0.79}$ | 15.0${}^{+4.7}_{-3.3}$
1BIGBJ022048.4-084250 | 35.20167 | -8.71389 |  $>$0.43 | 1.88$\pm$0.20 | 1.00$\pm$0.43 | 34.1 | 1.11${}^{+0.90}_{-0.59}$ | 6.0${}^{+4.3}_{-2.2}$
1BIGBJ023340.9+065611 | 38.42042 | 6.93639 | 0. | 2.03$\pm$0.14 | 3.23$\pm$0.90 | 98.0 | 3.08${}^{+1.45}_{-1.12}$ | 13.6${}^{+5.4}_{-3.5}$
1BIGBJ023430.5+804336 | 38.6275 | 80.72694 | 0. | 1.55$\pm$0.14 | 0.37$\pm$0.16 | 32.4 | 0.63${}^{+0.49}_{-0.34}$ | 5.8${}^{+3.2}_{-1.9}$
1BIGBJ030103.7+344100 | 45.26542 | 34.68361 | 0.24 | 2.33$\pm$0.16 | 3.26$\pm$0.63 | 56.3 | 2.43${}^{+0.87}_{-0.68}$ | 7.6${}^{+2.7}_{-1.7}$
1BIGBJ030330.1+055429 | 45.87542 | 5.90833 | 0.196 | 1.50$\pm$0.23 | 0.33$\pm$0.26 | 24.5 | 0.60${}^{+0.99}_{-0.51}$ | 6.0${}^{+7.0}_{-3.0}$
1BIGBJ030433.9-005403 | 46.14125 | -0.90111 | 0.511 | 1.69$\pm$0.16 | 0.79$\pm$0.35 | 38.1 | 1.09${}^{+0.87}_{-0.59}$ | 7.9${}^{+5.0}_{-2.8}$
1BIGBJ030544.1+403509 | 46.43375 | 40.58611 | 0. | 1.88$\pm$0.25 | 0.77$\pm$0.46 | 19.0 | 0.85${}^{+0.98}_{-0.58}$ | 4.6${}^{+4.5}_{-2.0}$
1BIGBJ031103.1-440227 | 47.76333 | -44.04111 | 0. | 1.96$\pm$0.35 | 0.72$\pm$0.43 | 16.4 | 0.73${}^{+1.03}_{-0.52}$ | 3.5${}^{+5.5}_{-1.8}$
1BIGBJ031423.8+061955 | 48.59958 | 6.33222 | 0.62? | 1.78$\pm$0.11 | 2.25$\pm$0.58 | 92.4 | 2.78${}^{+1.22}_{-0.95}$ | 17.6${}^{+6.4}_{-4.4}$
1BIGBJ032009.1-704533 | 50.03833 | -70.75917 | 0. | 1.77$\pm$0.18 | 0.63$\pm$0.26 | 32.1 | 0.79${}^{+0.63}_{-0.41}$ | 5.1${}^{+3.5}_{-1.9}$
1BIGBJ032037.9+112451 | 50.15833 | 11.41444 | 0. | 2.36$\pm$0.24 | 3.12$\pm$0.99 | 24.8 | 2.27${}^{+1.35}_{-0.95}$ | 6.9${}^{+3.9}_{-2.1}$
1BIGBJ032056.2+042447 | 50.23458 | 4.41333 | 0. | 2.70$\pm$0.22 | 3.03$\pm$0.61 | 30.1 | 1.78${}^{+0.69}_{-0.52}$ | 4.1${}^{+1.5}_{-0.9}$
1BIGBJ032647.2-340446 | 51.69708 | -34.07972 | 0. | 2.04$\pm$0.12 | 1.96$\pm$0.38 | 76.7 | 1.86${}^{+0.64}_{-0.51}$ | 8.1${}^{+2.7}_{-1.8}$
1BIGBJ032852.6-571605 | 52.21917 | -57.26806 | 0. | 1.55$\pm$0.18 | 0.36$\pm$0.19 | 36.3 | 0.61${}^{+0.65}_{-0.39}$ | 5.6${}^{+4.6}_{-2.4}$
1BIGBJ033623.7-034738 | 54.09875 | -3.79389 | 0.162 | 1.44$\pm$0.19 | 0.36$\pm$0.23 | 41.6 | 0.71${}^{+0.93}_{-0.52}$ | 7.8${}^{+7.1}_{-3.5}$
1BIGBJ033831.9-570447 | 54.63333 | -57.08 | 0. | 1.89$\pm$0.27 | 0.43$\pm$0.25 | 10.6 | 0.48${}^{+0.56}_{-0.32}$ | 2.5${}^{+2.8}_{-1.1}$
1BIGBJ035856.1-305447 | 59.73375 | -30.91306 | 0.65? | 1.92$\pm$0.14 | 1.43$\pm$0.40 | 59.6 | 1.53${}^{+0.76}_{-0.57}$ | 7.9${}^{+3.6}_{-2.2}$
1BIGBJ041112.2-394143 | 62.80125 | -39.69528 |  $>$0.7 | 1.74$\pm$0.29 | 0.23$\pm$0.19 | 6.7 | 0.31${}^{+0.53}_{-0.26}$ | 2.1${}^{+3.0}_{-1.1}$
1BIGBJ041238.3-392629 | 63.16 | -39.44139 | 0. | 1.91$\pm$0.30 | 0.34$\pm$0.25 | 6.7 | 0.37${}^{+0.54}_{-0.29}$ | 1.9${}^{+2.5}_{-9.5}$
1BIGBJ042900.1-323641 | 67.25042 | -32.61139 |  $>$0.51 | 2.04$\pm$0.17 | 1.20$\pm$0.36 | 27.9 | 1.13${}^{+0.63}_{-0.45}$ | 4.9${}^{+2.6}_{-1.5}$
1BIGBJ043517.7-262121 | 68.82375 | -26.35611 | 0. | 2.60$\pm$0.28 | 1.64$\pm$0.43 | 20.2 | 1.02${}^{+0.54}_{-0.38}$ | 2.5${}^{+1.4}_{-0.7}$
1BIGBJ044127.4+150454 | 70.36417 | 15.08194 | 0.109 | 2.10$\pm$0.18 | 3.03$\pm$1.18 | 43.0 | 2.74${}^{+1.81}_{-1.30}$ | 11.1${}^{+6.0}_{-3.4}$
1BIGBJ044240.6+614039 | 70.66917 | 61.6775 | 0. | 2.01$\pm$0.09 | 4.00$\pm$0.71 | 145.0 | 3.92${}^{+1.12}_{-0.95}$ | 17.9${}^{+4.3}_{-3.2}$
1BIGBJ044328.3-415156 | 70.86833 | -41.86556 |  $>$0.39 | 1.96$\pm$0.15 | 1.58$\pm$0.44 | 66.8 | 1.63${}^{+0.83}_{-0.61}$ | 8.0${}^{+3.8}_{-2.3}$
1BIGBJ050335.3-111506 | 75.8975 | -11.25167 |  $>$0.57 | 1.85$\pm$0.11 | 2.30$\pm$0.53 | 102.6 | 2.63${}^{+1.03}_{-0.82}$ | 15.0${}^{+5.1}_{-3.5}$
1BIGBJ050419.5-095631 | 76.08125 | -9.94222 |  $>$0.46 | 2.14$\pm$0.23 | 1.29$\pm$0.54 | 16.6 | 1.12${}^{+0.86}_{-0.58}$ | 4.3${}^{+3.0}_{-1.5}$
1BIGBJ050601.6-382054 | 76.50667 | -38.34861 | 0.182 | 1.95$\pm$0.11 | 2.06$\pm$0.40 | 96.5 | 2.13${}^{+0.73}_{-0.59}$ | 10.5${}^{+3.4}_{-2.4}$
1BIGBJ050727.1-334635 | 76.86333 | -33.77639 | 0. | 1.79$\pm$0.11 | 1.84$\pm$0.43 | 125.0 | 2.25${}^{+0.93}_{-0.72}$ | 14.1${}^{+5.0}_{-3.5}$
1BIGBJ053626.8-254748 | 84.11167 | -25.79667 | 0. | 2.06$\pm$0.16 | 1.70$\pm$0.44 | 47.9 | 1.58${}^{+0.76}_{-0.56}$ | 6.7${}^{+3.1}_{-1.9}$
1BIGBJ053645.2-255841 | 84.18875 | -25.97806 | 0. | 1.92$\pm$0.18 | 1.20$\pm$0.44 | 33.8 | 1.28${}^{+0.88}_{-0.60}$ | 6.6${}^{+4.2}_{-2.3}$
1BIGBJ055716.7-061706 | 89.32 | -6.285 | 0. | 1.95$\pm$0.15 | 2.51$\pm$0.86 | 52.2 | 2.59${}^{+1.52}_{-1.12}$ | 12.7${}^{+6.3}_{-3.8}$
1BIGBJ060714.2-251859 | 91.80958 | -25.31639 | 0.275 | 1.94$\pm$0.14 | 1.54$\pm$0.40 | 51.9 | 1.61${}^{+0.75}_{-0.57}$ | 8.1${}^{+3.4}_{-2.2}$
1BIGBJ062149.6-341148 | 95.45667 | -34.19694 | 0.529 | 2.53$\pm$0.12 | 4.77$\pm$0.59 | 93.3 | 3.09${}^{+0.68}_{-0.58}$ | 8.1${}^{+1.6}_{-1.2}$
1BIGBJ062626.2-171045 | 96.60917 | -17.17944 |  $>$0.7 | 2.00$\pm$0.18 | 1.80$\pm$0.66 | 34.6 | 1.78${}^{+1.17}_{-0.82}$ | 8.2${}^{+4.9}_{-2.7}$
1BIGBJ063014.9-201236 | 97.5625 | -20.21 | 0. | 1.56$\pm$0.33 | 0.37$\pm$0.47 | 16.2 | 0.62${}^{+1.79}_{-0.72}$ | 5.6${}^{+11.0}_{-3.3}$
1BIGBJ065932.8-674350 | 104.88708 | -67.73056 | 0. | 1.63$\pm$0.17 | 0.55$\pm$0.29 | 39.4 | 0.82${}^{+0.78}_{-0.50}$ | 6.5${}^{+4.6}_{-2.5}$
Table 6: Continued. 1BIGB Source name | R.A.(deg) | Dec.(deg) | z | $\Gamma$ |  $N_{0}$ (10-15) | TS | Flux${}_{1-100GeV}^{(\times 10^{-10})}$ | E-Flux${}_{1-100GeV}^{(\times 10^{-13})}$
---|---|---|---|---|---|---|---|---
1BIGBJ071745.0-552021 | 109.4375 | -55.33944 | 0. | 2.12$\pm$0.14 | 2.32$\pm$0.53 | 60.9 | 2.05${}^{+0.83}_{-0.65}$ | 8.1${}^{+3.0}_{-2.0}$
1BIGBJ073152.6+280432 | 112.96958 | 28.07583 | 0.248 | 1.90$\pm$0.18 | 1.06$\pm$0.39 | 29.9 | 1.15${}^{+0.79}_{-0.54}$ | 6.1${}^{+3.8}_{-2.1}$
1BIGBJ073329.5+351542 | 113.37292 | 35.26167 | 0.177 | 2.56$\pm$0.23 | 1.71$\pm$0.42 | 21.0 | 1.09${}^{+0.51}_{-0.37}$ | 2.8${}^{+1.2}_{-0.7}$
1BIGBJ075936.1+132116 | 119.90042 | 13.35472 | 0. | 1.75$\pm$0.09 | 2.12$\pm$0.44 | 152.6 | 2.71${}^{+0.98}_{-0.79}$ | 17.9${}^{+5.5}_{-4.0}$
1BIGBJ080015.4+561107 | 120.06458 | 56.18528 | 0. | 1.92$\pm$0.12 | 1.54$\pm$0.34 | 78.9 | 1.65${}^{+0.65}_{-0.51}$ | 8.5${}^{+3.1}_{-2.1}$
1BIGBJ082904.7+175415 | 127.27 | 17.90417 | 0.089 | 2.25$\pm$0.10 | 4.47$\pm$0.55 | 136.8 | 3.55${}^{+0.79}_{-0.67}$ | 12.1${}^{+2.7}_{-2.0}$
1BIGBJ083724.5+145819 | 129.3525 | 14.97222 | 0.278 | 1.75$\pm$0.15 | 1.07$\pm$0.37 | 50.6 | 1.37${}^{+0.87}_{-0.61}$ | 9.1${}^{+4.9}_{-2.9}$
1BIGBJ085749.8+013530 | 134.4575 | 1.59167 | 0.281 | 2.57$\pm$0.23 | 1.81$\pm$0.48 | 18.6 | 1.15${}^{+0.56}_{-0.42}$ | 2.9${}^{+1.3}_{-0.7}$
1BIGBJ090802.2-095936 | 137.00917 | -9.99361 | 0.053 | 1.79$\pm$0.26 | 0.68$\pm$0.49 | 22.5 | 0.83${}^{+1.17}_{-0.65}$ | 5.2${}^{+5.8}_{-2.4}$
1BIGBJ090953.2+310602 | 137.47167 | 31.10083 | 0.272 | 1.81$\pm$0.21 | 0.62$\pm$0.32 | 22.3 | 0.75${}^{+0.73}_{-0.45}$ | 4.5${}^{+3.7}_{-1.8}$
1BIGBJ091322.3+813305 | 138.34292 | 81.55139 | 0.639? | 1.40$\pm$0.15 | 0.20$\pm$0.10 | 41.5 | 0.42${}^{+0.42}_{-0.26}$ | 4.9${}^{+3.4}_{-1.9}$
1BIGBJ091651.8+523827 | 139.21625 | 52.64111 | 0.19 | 1.97$\pm$0.17 | 1.17$\pm$0.35 | 60.6 | 1.18${}^{+0.67}_{-0.48}$ | 5.7${}^{+3.1}_{-1.8}$
1BIGBJ093239.2+104234 | 143.16375 | 10.70972 | 0.361 | 2.00$\pm$0.14 | 1.95$\pm$0.49 | 58.5 | 1.92${}^{+0.87}_{-0.66}$ | 8.9${}^{+3.7}_{-2.4}$
1BIGBJ093430.1-172120 | 143.62542 | -17.35583 | 0. | 1.75$\pm$0.22 | 0.79$\pm$0.50 | 20.9 | 1.02${}^{+1.22}_{-0.72}$ | 6.7${}^{+6.4}_{-2.9}$
1BIGBJ095224.1+750212 | 148.10042 | 75.03694 | 0.181 | 1.43$\pm$0.15 | 0.29$\pm$0.14 | 62.4 | 0.57${}^{+0.54}_{-0.34}$ | 6.3${}^{+4.2}_{-2.4}$
1BIGBJ095507.9+355100 | 148.78292 | 35.85 | 0.834 | 2.00$\pm$0.25 | 0.82$\pm$0.38 | 24.1 | 0.80${}^{+0.74}_{-0.46}$ | 3.7${}^{+3.3}_{-1.5}$
1BIGBJ095628.2-095719 | 149.1175 | -9.95528 | 0. | 1.80$\pm$0.26 | 0.60$\pm$0.40 | 16.1 | 0.73${}^{+0.98}_{-0.54}$ | 4.5${}^{+5.2}_{-2.1}$
1BIGBJ095849.8+703959 | 149.7075 | 70.66639 | 0. | 1.98$\pm$0.23 | 0.99$\pm$0.43 | 37.1 | 0.99${}^{+0.84}_{-0.54}$ | 4.7${}^{+3.7}_{-1.8}$
1BIGBJ102100.3+162554 | 155.25125 | 16.43167 | 0.556 | 2.16$\pm$0.23 | 1.48$\pm$0.52 | 27.9 | 1.27${}^{+0.86}_{-0.58}$ | 4.8${}^{+3.2}_{-1.6}$
1BIGBJ104303.7+005420 | 160.76583 | 0.90556 | 0. | 1.75$\pm$0.14 | 1.30$\pm$0.43 | 56.5 | 1.67${}^{+1.00}_{-0.72}$ | 11.0${}^{+5.5}_{-3.4}$
1BIGBJ104857.6+500945 | 162.24 | 50.1625 | 0.402 | 2.37$\pm$0.15 | 2.01$\pm$0.35 | 53.1 | 1.46${}^{+0.47}_{-0.38}$ | 4.4${}^{+1.4}_{-0.9}$
1BIGBJ105534.3-012616 | 163.89292 | -1.43778 | 0. | 1.66$\pm$0.11 | 1.53$\pm$0.43 | 99.0 | 2.20${}^{+1.08}_{-0.82}$ | 16.9${}^{+6.6}_{-4.5}$
1BIGBJ111717.5+000633 | 169.32292 | 0.10917 | 0.451 | 1.92$\pm$0.15 | 1.87$\pm$0.53 | 57.4 | 2.00${}^{+1.04}_{-0.76}$ | 10.3${}^{+4.9}_{-3.0}$
1BIGBJ112317.9-323217 | 170.825 | -32.53833 | 0. | 2.06$\pm$0.22 | 1.25$\pm$0.48 | 22.8 | 1.16${}^{+0.85}_{-0.57}$ | 4.9${}^{+3.5}_{-1.7}$
1BIGBJ112611.8-203723 | 171.54958 | -20.62333 | 0. | 2.12$\pm$0.27 | 1.15$\pm$0.59 | 11.6 | 1.02${}^{+0.99}_{-0.62}$ | 4.0${}^{+3.5}_{-1.6}$
1BIGBJ113046.0-313807 | 172.69208 | -31.63528 | 0.151 | 1.29$\pm$0.20 | 0.13$\pm$0.10 | 24.8 | 0.33${}^{+0.56}_{-0.28}$ | 4.7${}^{+5.0}_{-2.3}$
1BIGBJ113105.2-094405 | 172.77167 | -9.735 | 0. | 1.66$\pm$0.14 | 0.99$\pm$0.35 | 64.1 | 1.43${}^{+0.92}_{-0.65}$ | 10.9${}^{+5.7}_{-3.5}$
1BIGBJ113444.6-172900 | 173.68625 | -17.48361 | 0.571 | 1.68$\pm$0.17 | 0.79$\pm$0.35 | 36.9 | 1.10${}^{+0.91}_{-0.60}$ | 8.2${}^{+5.4}_{-3.0}$
1BIGBJ113755.6-171041 | 174.48167 | -17.17833 | 0.6 | 1.71$\pm$0.10 | 1.90$\pm$0.43 | 126.9 | 2.55${}^{+1.00}_{-0.79}$ | 18.0${}^{+6.0}_{-4.2}$
1BIGBJ121158.6+224233 | 182.99417 | 22.70917 | 0.45 | 1.83$\pm$0.17 | 0.89$\pm$0.34 | 30.2 | 1.04${}^{+0.74}_{-0.50}$ | 6.0${}^{+3.8}_{-2.1}$
1BIGBJ121510.9+073203 | 183.79542 | 7.53444 | 0.137 | 1.64$\pm$0.11 | 1.26$\pm$0.37 | 105.5 | 1.87${}^{+0.97}_{-0.72}$ | 14.9${}^{+6.2}_{-4.1}$
1BIGBJ121603.1-024304 | 184.01333 | -2.71778 | 0.169 | 2.22$\pm$0.16 | 2.81$\pm$0.67 | 56.3 | 2.28${}^{+0.97}_{-0.74}$ | 8.0${}^{+3.2}_{-2.0}$
1BIGBJ124141.4+344029 | 190.4225 | 34.675 |  $>$0.7 | 1.88$\pm$0.15 | 1.19$\pm$0.37 | 46.8 | 1.33${}^{+0.76}_{-0.55}$ | 7.3${}^{+3.8}_{-2.3}$
1BIGBJ125015.4+315559 | 192.56458 | 31.93306 | 0. | 1.82$\pm$0.31 | 0.55$\pm$0.39 | 16.8 | 0.65${}^{+1.02}_{-0.51}$ | 3.9${}^{+5.7}_{-2.0}$
1BIGBJ125341.2-393159 | 193.42167 | -39.53306 | 0.179 | 1.89$\pm$0.17 | 1.29$\pm$0.51 | 39.8 | 1.42${}^{+1.00}_{-0.69}$ | 7.6${}^{+4.6}_{-2.6}$
1BIGBJ125847.9-044744 | 194.7 | -4.79583 | 0.586? | 1.93$\pm$0.11 | 2.50$\pm$0.62 | 61.2 | 2.64${}^{+1.10}_{-0.87}$ | 13.5${}^{+4.7}_{-3.2}$
1BIGBJ130145.6+405623 | 195.44 | 40.94 | 0.652 | 2.18$\pm$0.15 | 2.04$\pm$0.40 | 71.5 | 1.71${}^{+0.63}_{-0.49}$ | 6.3${}^{+2.3}_{-1.5}$
1BIGBJ130713.3-034430 | 196.80542 | -3.74194 | 0. | 2.05$\pm$0.26 | 1.14$\pm$0.57 | 19.0 | 1.08${}^{+1.03}_{-0.64}$ | 4.7${}^{+4.2}_{-1.9}$
1BIGBJ132541.8-022809 | 201.42417 | -2.46944 | 0.8? | 1.93$\pm$0.18 | 1.13$\pm$0.42 | 27.3 | 1.19${}^{+0.80}_{-0.56}$ | 6.0${}^{+3.6}_{-2.0}$
1BIGBJ132617.7+122957 | 201.57375 | 12.49944 | 0.204 | 2.08$\pm$0.20 | 1.57$\pm$0.50 | 32.2 | 1.44${}^{+0.87}_{-0.61}$ | 6.0${}^{+3.6}_{-1.9}$
1BIGBJ132833.4+114520 | 202.13958 | 11.75556 | 0.811 | 1.92$\pm$0.17 | 1.31$\pm$0.46 | 37.5 | 1.40${}^{+0.90}_{-0.63}$ | 7.1${}^{+4.1}_{-2.3}$
1BIGBJ133612.1+231958 | 204.05042 | 23.33278 | 0.267 | 1.97$\pm$0.15 | 1.66$\pm$0.42 | 62.5 | 1.68${}^{+0.79}_{-0.59}$ | 8.0${}^{+3.6}_{-2.3}$
1BIGBJ135328.0+560056 | 208.36667 | 56.01556 | 0.404 | 2.25$\pm$0.15 | 2.13$\pm$0.42 | 71.0 | 1.69${}^{+0.61}_{-0.48}$ | 5.7${}^{+2.0}_{-1.3}$
1BIGBJ140629.9-393508 | 211.625 | -39.58583 | 0.37 | 1.54$\pm$0.16 | 0.56$\pm$0.27 | 42.7 | 0.96${}^{+0.87}_{-0.56}$ | 9.0${}^{+5.9}_{-3.3}$
1BIGBJ143342.7-730437 | 218.42792 | -73.07722 | 0. | 1.33$\pm$0.20 | 0.15$\pm$0.11 | 24.6 | 0.36${}^{+0.55}_{-0.29}$ | 4.7${}^{+4.6}_{-2.2}$
1BIGBJ143825.4+120418 | 219.60625 | 12.07167 | 0. | 2.11$\pm$0.21 | 1.35$\pm$0.49 | 25.2 | 1.20${}^{+0.82}_{-0.56}$ | 4.8${}^{+3.1}_{-1.6}$
1BIGBJ144236.4-462300 | 220.65167 | -46.38361 | 0.103 | 1.92$\pm$0.10 | 3.14$\pm$0.64 | 107.7 | 3.33${}^{+1.14}_{-0.93}$ | 17.1${}^{+5.0}_{-3.6}$
1BIGBJ145543.6-760051 | 223.93167 | -76.01444 | 0. | 1.61$\pm$0.12 | 0.63$\pm$0.22 | 43.7 | 0.96${}^{+0.59}_{-0.43}$ | 7.9${}^{+3.6}_{-2.3}$
1BIGBJ145603.5+504825 | 224.015 | 50.80722 |  $>$0.49 | 2.09$\pm$0.24 | 1.23$\pm$0.55 | 27.2 | 1.11${}^{+0.92}_{-0.60}$ | 4.5${}^{+3.4}_{-1.6}$
1BIGBJ150637.0-054004 | 226.65458 | -5.66778 | 0.518 | 1.72$\pm$0.17 | 0.94$\pm$0.41 | 35.2 | 1.25${}^{+1.00}_{-0.67}$ | 8.7${}^{+5.5}_{-3.1}$
1BIGBJ151041.0+333503 | 227.67125 | 33.58444 | 0.114 | 1.59$\pm$0.30 | 0.27$\pm$0.24 | 15.2 | 0.43${}^{+0.91}_{-0.40}$ | 3.7${}^{+6.4}_{-2.1}$
1BIGBJ151136.8-165326 | 227.90375 | -16.89056 |  $>$0.56 | 2.15$\pm$0.35 | 1.22$\pm$0.74 | 13.1 | 1.06${}^{+1.36}_{-0.74}$ | 4.0${}^{+5.3}_{-1.8}$
1BIGBJ151618.7-152344 | 229.07792 | -15.39556 |  $>$0.54 | 1.98$\pm$0.18 | 1.50$\pm$0.58 | 27.2 | 1.51${}^{+1.02}_{-0.72}$ | 7.1${}^{+4.1}_{-2.3}$
1BIGBJ151826.5+075222 | 229.61083 | 7.87278 | 0.41 | 1.74$\pm$0.20 | 0.66$\pm$0.33 | 24.6 | 0.85${}^{+0.81}_{-0.51}$ | 5.7${}^{+4.5}_{-2.3}$
1BIGBJ151845.7+061355 | 229.69042 | 6.23222 | 0.102 | 1.96$\pm$0.21 | 1.36$\pm$0.56 | 31.8 | 1.39${}^{+1.09}_{-0.72}$ | 6.8${}^{+5.0}_{-2.5}$
1BIGBJ152646.6-153025 | 231.69417 | -15.50722 |  $>$0.43 | 2.04$\pm$0.13 | 2.61$\pm$0.64 | 58.4 | 2.47${}^{+1.06}_{-0.82}$ | 10.8${}^{+4.1}_{-2.7}$
1BIGBJ152913.5+381216 | 232.30625 | 38.20472 |  $>$0.59 | 2.04$\pm$0.19 | 1.22$\pm$0.39 | 34.6 | 1.16${}^{+0.69}_{-0.49}$ | 5.1${}^{+2.9}_{-1.6}$
1BIGBJ154202.9-291509 | 235.5125 | -29.2525 | 0. | 1.78$\pm$0.08 | 2.62$\pm$0.50 | 143.2 | 3.27${}^{+1.04}_{-0.86}$ | 20.8${}^{+5.4}_{-4.1}$
1BIGBJ154625.0-285723 | 236.60417 | -28.95639 |  $>$0.6 | 1.73$\pm$0.18 | 0.67$\pm$0.36 | 18.7 | 0.88${}^{+0.84}_{-0.54}$ | 6.0${}^{+4.3}_{-2.3}$
1BIGBJ155053.2-082245 | 237.72167 | -8.37944 | 0. | 1.87$\pm$0.14 | 1.96$\pm$0.63 | 64.9 | 2.20${}^{+1.24}_{-0.91}$ | 12.2${}^{+5.8}_{-3.6}$
Table 7: Continued. 1BIGB Source name | R.A.(deg) | Dec.(deg) | z | $\Gamma$ |  $N_{0}$ (10-15) | TS | Flux${}_{1-100GeV}^{(\times 10^{-10})}$ | E-Flux${}_{1-100GeV}^{(\times 10^{-13})}$
---|---|---|---|---|---|---|---|---
1BIGBJ155432.5-121324 | 238.63542 | -12.22361 | 0. | 1.69$\pm$0.13 | 1.04$\pm$0.39 | 34.3 | 1.45${}^{+0.94}_{-0.68}$ | 10.6${}^{+5.3}_{-3.3}$
1BIGBJ160218.0+305108 | 240.575 | 30.8525 |  $>$0.47 | 2.44$\pm$0.27 | 1.83$\pm$0.47 | 33.8 | 1.27${}^{+0.70}_{-0.48}$ | 3.6${}^{+2.3}_{-1.1}$
1BIGBJ160519.0+542058 | 241.32917 | 54.34972 | 0.212 | 1.88$\pm$0.13 | 1.13$\pm$0.29 | 64.0 | 1.25${}^{+0.59}_{-0.45}$ | 6.8${}^{+3.0}_{-1.9}$
1BIGBJ160618.4+134532 | 241.57667 | 13.75889 | 0.29 | 2.05$\pm$0.21 | 1.76$\pm$0.64 | 33.8 | 1.65${}^{+1.14}_{-0.77}$ | 7.1${}^{+4.7}_{-2.5}$
1BIGBJ161327.1-190835 | 243.36292 | -19.14333 | 0. | 2.14$\pm$0.15 | 3.15$\pm$0.82 | 34.1 | 2.73${}^{+1.25}_{-0.95}$ | 10.5${}^{+4.3}_{-2.7}$
1BIGBJ162115.1-003140 | 245.31333 | -0.52778 |  $>$0.52 | 1.80$\pm$0.21 | 0.77$\pm$0.41 | 14.4 | 0.93${}^{+0.93}_{-0.58}$ | 5.6${}^{+4.7}_{-2.3}$
1BIGBJ162330.4+085724 | 245.87708 | 8.95667 | 0.533 | 1.91$\pm$0.20 | 1.23$\pm$0.52 | 35.5 | 1.33${}^{+1.05}_{-0.69}$ | 7.0${}^{+4.9}_{-2.5}$
1BIGBJ162646.0+630047 | 246.69167 | 63.01333 | 0. | 1.91$\pm$0.17 | 1.00$\pm$0.33 | 51.6 | 1.07${}^{+0.65}_{-0.46}$ | 5.5${}^{+3.1}_{-1.8}$
1BIGBJ164220.2+221143 | 250.58458 | 22.19528 | 0.592 | 2.19$\pm$0.19 | 1.76$\pm$0.55 | 32.1 | 1.46${}^{+0.83}_{-0.60}$ | 5.3${}^{+2.8}_{-1.6}$
1BIGBJ164419.9+454644 | 251.08333 | 45.77889 | 0.225 | 1.65$\pm$0.16 | 0.48$\pm$0.20 | 39.1 | 0.70${}^{+0.56}_{-0.37}$ | 5.4${}^{+3.5}_{-2.0}$
1BIGBJ165517.8-224045 | 253.82458 | -22.67917 | 0. | 1.99$\pm$0.21 | 1.64$\pm$0.75 | 17.4 | 1.64${}^{+1.38}_{-0.91}$ | 7.7${}^{+5.6}_{-2.8}$
1BIGBJ171108.5+024403 | 257.78583 | 2.73444 | 0. | 1.83$\pm$0.18 | 0.93$\pm$0.42 | 18.3 | 1.09${}^{+0.89}_{-0.60}$ | 6.4${}^{+4.2}_{-2.3}$
1BIGBJ174419.7+185218 | 266.0825 | 18.87167 | 0. | 1.58$\pm$0.21 | 0.41$\pm$0.26 | 15.5 | 0.65${}^{+0.85}_{-0.48}$ | 5.7${}^{+5.6}_{-2.6}$
1BIGBJ174702.5+493800 | 266.76042 | 49.63361 | 0.46? | 2.27$\pm$0.22 | 1.74$\pm$0.49 | 37.3 | 1.37${}^{+0.75}_{-0.53}$ | 4.5${}^{+2.6}_{-1.4}$
1BIGBJ184822.4+653656 | 282.09375 | 65.61583 | 0.364 | 1.59$\pm$0.15 | 0.50$\pm$0.21 | 52.8 | 0.79${}^{+0.63}_{-0.42}$ | 6.8${}^{+4.2}_{-2.4}$
1BIGBJ185023.9+263153 | 282.6 | 26.53139 | 0. | 1.63$\pm$0.13 | 1.20$\pm$0.45 | 88.6 | 1.80${}^{+1.18}_{-0.85}$ | 14.5${}^{+7.1}_{-4.5}$
1BIGBJ185813.3+432451 | 284.55583 | 43.41417 | 0. | 2.12$\pm$0.16 | 1.89$\pm$0.48 | 43.3 | 1.67${}^{+0.77}_{-0.58}$ | 6.6${}^{+2.9}_{-1.8}$
1BIGBJ193412.7-241919 | 293.55292 | -24.32222 | 0. | 1.60$\pm$0.12 | 0.88$\pm$0.30 | 53.5 | 1.36${}^{+0.81}_{-0.59}$ | 11.5${}^{+5.2}_{-3.4}$
1BIGBJ194356.2+211821 | 295.98417 | 21.30611 | 0. | 1.44$\pm$0.08 | 2.01$\pm$0.57 | 194.7 | 3.93${}^{+1.84}_{-1.44}$ | 42.9${}^{+13.2}_{-9.8}$
1BIGBJ200204.0-573644 | 300.51708 | -57.6125 | 0. | 2.08$\pm$0.10 | 2.93$\pm$0.49 | 105.5 | 2.69${}^{+0.78}_{-0.64}$ | 11.2${}^{+3.0}_{-2.2}$
1BIGBJ201200.9-771219 | 303.00375 | -77.20528 | 0. | 2.10$\pm$0.30 | 0.78$\pm$0.52 | 9.8 | 0.71${}^{+0.89}_{-0.52}$ | 2.8${}^{+3.0}_{-1.2}$
1BIGBJ205242.4+081040 | 313.17708 | 8.17778 | 0. | 1.51$\pm$0.23 | 0.25$\pm$0.20 | 15.3 | 0.45${}^{+0.73}_{-0.37}$ | 4.3${}^{+5.1}_{-2.2}$
1BIGBJ214533.3-043438 | 326.38875 | -4.5775 | 0.069 | 2.64$\pm$0.31 | 1.34$\pm$0.47 | 9.7 | 0.82${}^{+0.55}_{-0.37}$ | 1.9${}^{+1.2}_{-0.6}$
1BIGBJ215214.0-120540 | 328.05875 | -12.09472 | 0.121 | 2.26$\pm$0.14 | 3.77$\pm$0.76 | 81.3 | 2.98${}^{+1.06}_{-0.84}$ | 10.0${}^{+3.3}_{-2.2}$
1BIGBJ220107.3-590639 | 330.28042 | -59.11111 | 0. | 1.93$\pm$0.18 | 0.78$\pm$0.28 | 21.3 | 0.82${}^{+0.55}_{-0.38}$ | 4.1${}^{+2.4}_{-1.4}$
1BIGBJ220155.8-170700 | 330.4825 | -17.11667 | 0.169 | 1.32$\pm$0.37 | 0.16$\pm$0.22 | 33.5 | 0.38${}^{+1.63}_{-0.47}$ | 5.1${}^{+14.1}_{-3.5}$
1BIGBJ221029.5+362159 | 332.62333 | 36.36639 | 0. | 2.17$\pm$0.32 | 1.19$\pm$0.75 | 13.3 | 1.01${}^{+1.23}_{-0.72}$ | 3.7${}^{+4.1}_{-1.6}$
1BIGBJ221108.2-000302 | 332.78458 | -0.05056 | 0.326 | 1.84$\pm$0.14 | 1.43$\pm$0.47 | 48.7 | 1.67${}^{+0.98}_{-0.71}$ | 9.7${}^{+4.8}_{-2.9}$
1BIGBJ223301.0+133601 | 338.25458 | 13.60028 | 0.214 | 1.51$\pm$0.24 | 0.24$\pm$0.19 | 15.4 | 0.43${}^{+0.73}_{-0.37}$ | 4.2${}^{+5.1}_{-2.1}$
1BIGBJ223626.2+370713 | 339.10958 | 37.12028 | 0. | 1.85$\pm$0.20 | 0.98$\pm$0.48 | 29.1 | 1.12${}^{+1.01}_{-0.66}$ | 6.3${}^{+4.8}_{-2.4}$
1BIGBJ224910.6-130002 | 342.29458 | -13.00056 |  $>$0.5 | 2.33$\pm$0.01 | 87.01$\pm$1.43 | 11240.5 | 64.8${}^{+1.93}_{-1.88}$ | 202.8${}^{+6.2}_{-6.0}$
1BIGBJ225147.5-320611 | 342.94792 | -32.10333 | 0.246 | 2.07$\pm$0.19 | 1.60$\pm$0.51 | 48.6 | 1.48${}^{+0.90}_{-0.63}$ | 6.2${}^{+3.7}_{-2.0}$
1BIGBJ225613.3-330338 | 344.05542 | -33.06056 | 0.243 | 2.56$\pm$0.21 | 2.23$\pm$0.42 | 39.7 | 1.42${}^{+0.54}_{-0.41}$ | 3.6${}^{+1.4}_{-0.8}$
1BIGBJ230634.9-110347 | 346.64583 | -11.06333 | 0. | 1.69$\pm$0.21 | 0.62$\pm$0.34 | 28.5 | 0.86${}^{+0.94}_{-0.56}$ | 6.3${}^{+5.7}_{-2.7}$
1BIGBJ232039.7-630918 | 350.16583 | -63.155 | 0.2 | 1.84$\pm$0.16 | 0.84$\pm$0.30 | 43.6 | 0.98${}^{+0.64}_{-0.45}$ | 5.7${}^{+3.2}_{-1.8}$
1BIGBJ233112.8-030129 | 352.80375 | -3.025 | 0. | 2.05$\pm$0.15 | 1.95$\pm$0.54 | 40.8 | 1.84${}^{+0.90}_{-0.67}$ | 7.9${}^{+3.5}_{-2.2}$
1BIGBJ235320.9-145856 | 358.3375 | -14.9825 | 0. | 1.83$\pm$0.28 | 0.55$\pm$0.37 | 13.3 | 0.65${}^{+0.92}_{-0.49}$ | 3.8${}^{+4.9}_{-1.9}$


#  $\sigma$-Stable Matrices

Michael A.S. Thorne  British Antarctic Survey, Cambridge<EMAIL_ADDRESS>
###### Abstract.

$\sigma$-Stable matrices are introduced and it is shown that the real roots of
the polynomials comprising the coefficients of the characteristic polynomial
indicate the coefficient sign changes. A proof of Obrechkoff is then used to
show that the largest real root from the coefficients is the point of
stability when the maximal eigenvalue of the $\sigma$-stable matrix is in
$\mathbb{R}$. Some implications of the coefficient behaviour for a scaling
relation are then discussed.

###### 1991 Mathematics Subject Classification:

93D15

Let $\mathbf{M}$ be an $n\times n$ square matrix with real-valued entries,
$m_{i,j}\in\mathbb{R}$ $\forall$ $i,j$. Then, multiply the diagonal entries
$\mathbf{D}$ of $\mathbf{M}$ by a variable $\sigma\in\mathbb{C}$,

$\mathbf{M}_{\sigma}=\mathbf{M}-(1-\sigma)\cdot\mathbf{D}=\begin{pmatrix}\sigma
m_{1,1}&m_{1,2}&\cdots&m_{1,n}\\\ m_{2,1}&\sigma m_{2,2}&\cdots&m_{2,n}\\\
m_{3,1}&m_{3,2}&\cdots&m_{3,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{n,1}&m_{n,2}&\cdots&\sigma m_{n,n}\end{pmatrix}.$

If a $\sigma>0$ can be found for $\mathbf{M}_{\sigma}$ such that the maximal
(largest real-part) eigenvalue is $0$,
$\operatorname{Re}(\lambda_{max(\mathbf{M}_{\sigma})})=0$, and any larger
real-part of $\sigma$ results in
$\operatorname{Re}(\lambda_{max(\mathbf{M}_{\sigma})})<0$, then
$\mathbf{M}_{\sigma}$ is $\sigma$-stable.

The characteristic polynomial, $P$, of $\mathbf{M}_{\sigma}$ is a monic
polynomial in $x$, each of whose coefficients is a polynomial in $\sigma$:

$P_{\mathbf{M}_{\sigma}}=\sum_{i=0}^{n}p_{i}x^{i}=\sum_{i=0}^{n}(\sum_{j=0}^{n-i}q_{j}\sigma^{j})x^{i}$

with $p_{n}=1$ and

$p_{i}=\sum_{j=0}^{n-i}q_{j}\sigma^{j}$

where the $q_{j}$ consist of terms of length $n-i$ from elements of
$\mathbf{M}$. In each $p_{i}$, the $(n-i-1)^{st}$ term in the polynomial is
$0$, with other terms also potentially absent (but never the $(n-i)^{th}$
term). If each $p_{i}$ has a leading coefficient that is positive, then
$\mathbf{M}_{\sigma}$ can be made $\sigma$-stable through diagonal dominance
[1]. This can occur for the highest order of $\sigma$ in each $p_{i}$ if each
of the sums of the $\binom{n}{i}$ sets of the negative of the diagonal
elements $\\{\sum_{i=1}^{n}(-m_{i,i}),\sum_{i\neq
l}(-m_{i,i})(-m_{l,l}),...\\}$ are positive.

The polynomials $p_{i}$ provide an indication of the behaviour of each
coefficient in the characteristic polynomial of $\mathbf{M}_{\sigma}$
depending on the value of $\sigma$ in relation to the individual real-valued
roots of each $p_{i}$ ($\sigma_{i,1}...\sigma_{i,k}$ for some $k$ specific to
each $p_{i}$).

######  Theorem 1.

The real-valued roots $({\sigma_{i,1},...,\sigma_{i,k}}\in\mathbb{R})$ of each
$p_{i}$ indicate the point of respective coefficient sign change in the
characteristic polynomial given the value of $\sigma$ of
$\mathbf{M}_{\sigma}$. Furthermore, the largest real-valued root for each
$p_{i}$ indicates when the coefficient becomes positive and it remains
positive for any larger value of $\sigma$ (restricted to $\mathbb{R}$) applied
to $\mathbf{M}_{\sigma}$.

###### Proof.

A necessary condition for stability is that all the coefficients of a
characteristic polynomial are positive (a point that can be deduced for
$\sigma$-stable matrices through the sign of the trace and Descartes‘ rule of
signs [2]). Further, any $\sigma$-stable matrix can be made stable through
diagonal dominance. This implies that eventually, given the appropriate
$\sigma$, all the coefficients will become positive. Each coefficient of the
characteristic polynomial in $x$, which is in the field of $\mathbb{R}$, is
itself a polynomial, $p_{i}$, in $\sigma$ and, again by Descartes‘ rule of
signs, the real roots reflect the changing sign of the $p_{i}$ and therefore
of the coefficients of the characteristic polynomial in $x$. Diagonal
dominance ensures that any $\sigma$ larger than the largest real root of each
$p_{i}$ will produce a positive coefficient. Any and all smaller real roots of
$p_{i}$ indicate the points at which, for the appropriate $\sigma$, the sign
of the coefficient alternates. ∎

While a necessary condition for stability is that all the coefficients of a
characteristic polynomial are positive, it is not sufficient. This point,
along with Theorem 1, leads to the following theorem and corollary. For these,
we define the set
$\Omega=\\{\\{\sigma_{0,1},...,\sigma_{0,k_{0}}\\},...,\\{\sigma_{i,1},...,\sigma_{i,k_{i}}\\},...,\\\
\\{\sigma_{n-1,1},...,\sigma_{n-1,k_{n-1}}\\}\\}$ consisting of all the real-
valued roots ($\Omega\in\mathbb{R}$) of all the $p_{i}$ polynomials from the
characteristic polynomial of a given $\mathbf{M}_{\sigma}$.

######  Theorem 2.

If, for a given $\sigma$, the leading eigenvalue,
$\lambda_{max(\mathbf{M}_{\sigma})}$, of $\mathbf{M}_{\sigma}$ is $0$, and
$\lambda_{max(\mathbf{M}_{\sigma})}\in\mathbb{R}$
($\operatorname{Re}(\lambda_{max(\mathbf{M}_{\sigma})})=0$,
$\operatorname{Im}(\lambda_{max(\mathbf{M}_{\sigma})})=0$), then
$max(\Omega)=\sigma$.

###### Proof.

By the necessary condition of stability, all coefficients must be positive,
which means that $\sigma\geq max(\Omega)$. By the proof of Obrechkoff [3], if
a polynomial has positive coefficients and has a root in the right-half plane,
then the root cannot lie on the real axis. This implies that if
$\sigma\in\mathbb{R}$ then $\sigma\leq max(\Omega)$. Therefore,
$\sigma=max(\Omega)$. ∎

######  Corollary 1.

If, for a given $\sigma$, the real-part of the leading eigenvalue,
$\lambda_{max(\mathbf{M}_{\sigma})}$, of $\mathbf{M}_{\sigma}$ is $0$ but with
a non-zero imaginary part
($\operatorname{Re}(\lambda_{max(\mathbf{M}_{\sigma})})=0$,
$\operatorname{Im}(\lambda_{max(\mathbf{M}_{\sigma})})\neq 0$), then
$max(\Omega)\leq\sigma$.

###### Proof.

By the necessary condition of stability, all coefficients of the
characteristic polynomial must be positive. Therefore $max(\Omega)\leq\sigma$.
∎

A scaling operation, first introduced in [4], related the $\sigma$ of
$\mathbf{M}_{\sigma}$ with the maximal eigenvalue of a matrix,
$\bar{\mathbf{M}}_{0}$, defined by

$\bar{\mathbf{M}}_{0}=-\mathbf{D}^{-1}\mathbf{M}+\mathbf{I}.$

The characteristic polynomial of $\bar{\mathbf{M}}_{0}$ in $x$ is the same
polynomial as that for the last coefficient $p_{0}$ in $\sigma$ of the
characteristic polynomial of $\mathbf{M}_{\sigma}$
($P_{\bar{\mathbf{M}}_{0}}=p_{0}$, $p_{0}=(-1)^{n}$
det$(\mathbf{M}_{\sigma})$). Therefore when
$\sigma=\lambda_{max(\bar{\mathbf{M}}_{0})}$, $p_{0}$ vanishes, forcing
$\mathbf{M}_{\sigma}$ to have an eigenvalue of $0$.

References

[1] S. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix. Izv.
Akad. Nauk. USSR Otd. Fiz.-Mat, 6: 749-754, 1931.

[2] R. Descartes, La Géométrie. Paris, 1637.

[3] N. Obrechkoff, Sur un problème de Laguerre. C.R. Acd. Sci. Paris, 177:
223-235, 1923.

[4] A-M. Neutel, M. Thorne, Interaction strengths in balanced carbon cycles
and the absence of a relation between ecosystem complexity and stability.
Ecology Letters, 17 (6): 651-661, 2014.