# BROADBAND SPECTRO-TEMPORAL STUDY ON BLAZAR TXS 1700+685

Anuvab Banerjee1, Prantik Nandi1,5, Raj Prince2, Rukaiya Khatoon3,4, Debanjan
Bose1
1S. N. Bose National Centre for Basic Sciences, Block JD, Salt Lake, Kolkata
700106, India
2Center for Theoretical Physics, Polish Academy of Sciences, Lotników 32/46,
Warsaw, Poland
3Inter-University Center for Astronomy and Astrophysics, Post Bag 4,
Ganeshkhind, Pune-411007, India
4Tezpur University, Napaam-784028, Assam, India
5Physical Research Laboratory, Navrangpura, Ahmedabad, 380009, India E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We attempt to present a multiwavelength variability and correlation study as
well as detailed multi-waveband spectral characteristics of the May 2021
$\gamma$-ray flare of the blazar source TXS 1700+685. The multi-wavelength
observation from Fermi-LAT, Swift-XRT/UVOT as well as radio archival data are
used for our spectro-temporal investigation. We estimate the variability time-
scale of the source from the flux doubling time in different flaring region
detected in Fermi-LAT observation and the shortest variability time is used to
put a constraint on the minimum Doppler factor and on the size of the emission
region. We have detected a statistically significant quasi-periodic
oscillation feature (QPO) at $\sim$ 17 days. The broad-band emission is
satisfactorily represented during its flaring state with a leptonic
synchrotron and inverse Compton component. From the broadband spectral
modelling, we observe the external Comptonization of the seed photons
originating in the broad line region to be dominant compared to the dusty
torus. This is further supported by the fact that the emission region is also
found to be residing within the BLR. The equipartition value implies the
energy density of the magnetic field in the jet comoving frame is weak, and
that is also reflected in the magnetic field and low power corresponding to
the magnetic field component of the jet. In order to produce the high energy
hump, we need the injection of a large population of high energy electrons
and/or the presence of strong magnetic field; and we observe the later
component to be sub-dominant in our case. The flat rising and steep falling
profile in the $\gamma$-ray SED as well as the break or spectral curvature at
$\sim$ 1 GeV are in commensuration with the flat-spectrum radio quasar (FSRQ)
nature of the source.

###### keywords:

galaxies: active – galaxies: jets – quasars: individual: TXS 1700+685 – gamma-
rays: galaxies – ultraviolet: galaxies – X-rays: galaxies

††pubyear: 2022††pagerange: BROADBAND SPECTRO-TEMPORAL STUDY ON BLAZAR TXS
1700+685–LABEL:lastpage

##  1 Introduction

Blazar sources are a subclass of active galactic nuclei (AGNs) that emit
copiously in all frequencies extending from the radio to very high-energy
(more than a few tens of GeV) $\gamma$ ray band, and their relativistic jets
are oriented almost along the line of sight (angles less than $\sim
14^{\circ}$) to the observer (Urry & Padovani, 1995). Owing to the complex
characteristics of AGNs as well as the lack of high-resolution instruments,
the exact origin of such high energy emission as well as the complete physical
mechanism driving such energetic outflowing matter has remained unresolved for
several decades. The association between very high energy emission and jet
activity became evident after the Energetic Gamma-ray Experiment Telescope
(EGRET) observations onboard the Compton Gamma-Ray Observatory (CGRO; Hartman
et al. (1999)). This has been more comprehensively established by Fermi/LAT
(Fermi Gamma-Ray Space Telescope/Large Area Telescope) extensive all-sky
survey.

Another defining characteristic of blazar sources is their large amplitude
flux variability across different wavebands which has been used previously to
approximate the location and size of the emission regions (see Hovatta &
Lindfors (2019) for a review). In many of the cases, the optical flares are
found to be correlated with $\gamma$-ray flares (Chatterjee et al., 2012;
Carnerero et al., 2015). However, this may not always be the case and
$\gamma$-ray flares may not even be associated with an optical counterpart
(Vercellone et al., 2011; MacDonald et al., 2015; Cohen et al., 2014; Rajput
et al., 2019). Further, the short time scale of high energy $\gamma$-ray flux
variation is indicative of the fact that such emission is originated from very
compact regions in the jet.

A double hump broad-band spectral energy distribution (SED) is apparent in
blazar, where the low energy hump peaks at IR - X-ray band and the higher-
energy component peaks in the MeV-GeV band (Fossati et al., 1998; Mao et al.,
2016). The low energy hump is now believed to be originated from the
synchrotron emission of the relativistic electrons in the jet. However, the
origin of higher energy hump is still debated, and there could be a range of
physical mechanisms that contribute to the genesis of this hump. In the
leptonic scenario, the high-energy hump is attributed to the inverse Compton
(IC) scattering of low-energy photons within the jet. The seed photons for IC
scattering can be supplied by the jet itself (synchrotron self-Compton or SSC;
Konigl (1981); Marscher & Gear (1985); Ghisellini & Tavecchio (2009)), or the
origin of the seed photons can be external to the jet (external Compton or
EC). In EC process, the seed photons can be supplied by the disc (Dermer &
Schlickeiser, 1993; Böttcher et al., 1996, 1997), the dusty torus (Ghisellini
& Tavecchio, 2008), or the broad-line emission (BLR) region (Sikora et al.,
1994).

Fermi Gamma-ray Space Telescope onboard the Large Area Telescope (LAT)
detected enhanced $\gamma$-ray activity on March 22, 2009 from the blazar TXS
1700+685 (Gasparrini & Hill, 2009). This source is also known as 4FGL
J1700.0+6830 (Abdollahi et al., 2020) and is located at redshift z = 0.301
(Henstock et al., 1997) with R.A. = 255.038 deg. and Decl. = +68.5019 (Petrov
et al., 2008). Recently this source has shown renewed $\gamma$-ray activity on
March 15, 2021 when the daily averaged gamma-ray flux ($E>100$ MeV) rising to
$(1.1\pm 0.2)\times 10^{-6}\text{ erg cm}^{-2}\text{s}^{-1}$, with the
detection of a 29 GeV photon (Cheung & Angioni, 2021). This daily average flux
is a factor $\sim$ 20 more compared to the 10-year average flux reported in
the fourth Fermi LAT source catalog Data Release 2 (4FGL-DR2; Ballet et al.
(2020)). Subsequently on May 13, more prominent flaring activity in
$\gamma$-ray was registered by Fermi-LAT, when daily average $\gamma$-ray flux
rose to $(1.8\pm 0.2)\times 10^{-6}\text{ erg cm}^{-2}\text{s}^{-1}$; a flux
increase by a factor $\sim$ 30 relative to the daily average flux. On May 14
and 15, it showed a daily averaged gamma-ray flux ($E>100$ MeV) $(1.0\pm
0.2)\times 10^{-6}\text{ erg cm}^{-2}\text{s}^{-1}$ (Ciprini, 2021).

In this paper, we have attempted to investigate the possible correlation
between the different wavebands, as well as the accretion and jet behavior
from the multi-wavelength spectral analysis during May 2021 flare,
corresponding to which simultaneous Swift XRT/UVOT observations are present.
The organization of the paper is as follows: In Section 2, we describe the
data reduction and analysis techniques; in Section 3, we describe our result
on the multi-waveband spectro-temporal analysis, and in Section 4, we draw our
conclusions.

##  2  Multi-waveband observations and data analysis

###  Fermi-LAT observations

Fermi-LAT is a pair conversion $\gamma$-ray detector in orbit with a field of
view (FOV) of about 2.4 sr, which is sensitive to photon energies between 20
MeV - 500 GeV (Atwood et al., 2009). The FOV covers $\sim$ 20% of the sky at
any given instant, and it scans the whole sky in every three hours. At 100
MeV, LAT’s single-photon resolution is $<3.5^{\circ}$, but at higher energies
($>1$ GeV), this resolution improves to $<0.6^{\circ}$. In order to identify
the potential high energy $\gamma$-ray sources within a given region of
interest (ROI), a likelihood analysis is needed to be undertaken using an
input model using the gtlike task (Cash, 1979; Mattox et al., 1996). In order
to perform our analysis, we have primarily used
fermitools111https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/
package, maintained by the Fermi-LAT collaboration. Usage of
fermipy222https://fermipy.readthedocs.io/en/latest/quickstart.html was also
required for some specific aspects during spectral analysis (Wood et al.,
2017).

We collected 0.1–300 GeV Fermi-LAT data within the time-span between MJD 58970
(May 1, 2020) to 59487 (September 30, 2021) for the purpose of our analysis. A
circular region of $10^{\circ}$ around the source position was chosen to be
the ROI, within which the presence of high energy $\gamma$ ray was searched.
As recommended by the Fermi-LAT team in fermitools documentation, the user-
specified cut ‘evclass=128’ and ‘evtype=3’ were applied for the purpose of the
selection of rows from the input event data and the filter
(DATA_QUAL>0)&&(LAT_CONFIG==1) was applied to update the good time interval
(GTI) information based on spacecraft specifications. In order to filter any
contamination due to the Earth’s limp, a zenith angle cut of $90^{\circ}$ was
applied. The latest instrument response function P8R3_SOURCE_V6 was selected
for the analysis, and in order to take care of the isotropic and the diffuse
background, we used iso_P8R3_SOURCE_V3_v1.txt and gll_psc_v21.xml
respectively, available from Fermi Science Support Center (FSSC). The spectral
profiles, as well as parameters, were initialized as per the values published
in the 4FGL catalog, and beyond 3 degrees of ROI center, all sources were kept
frozen. The $\gamma$-ray spectral profile was compared with three different
empirical models:

  1. 1.

Power-law profile, defined as

$dN(E)/dE=N_{0}(E/E_{0})^{-\alpha},$ (1)

where $E_{0}$ is is the energy scaling factor and $\alpha$ is the spectral
index.

  2. 2.

A log-parabola (LP), defined as

$dN(E)/dE=N_{0}(E/E_{0})^{-\alpha-\beta\ln{(E/E_{0})}},$ (2)

where $E_{0}$ and $\alpha$ stand as before, and $\beta$ is the curvature
index.

  3. 3.

A broken power-law (BPL), defined as

$dN(E)/dE=N_{0}(E/E_{\text{break}})^{-\alpha_{i}},$ (3)

with $i=1$ for $E<E_{\text{break}}$ and $i=2$ for $E>E_{\text{break}}$.

### X-ray observations of Swift-XRT/UVOT

TXS 1700+685 was observed seven times between 16th May 2021 and 7th June 2021
by Swift-XRT/UVOT (details of the observations are provided in Table 1) with
nonzero exposure. We used FTOOLS task xrtpipeline version 0.13.5 to produce
cleaned events. The HEASoft version 6.28 and XSPEC version 12.11.1 have been
used for the purpose of our analysis. Cleaned event files corresponding to the
Photon Counting mode were considered for analysis and circular regions of
radius 20 arcsec centered at the source and slightly away from the source were
chosen for the source and the background regions respectively. The source and
background spectra were produced using xselect task. The redistribution matrix
file (RMF) was obtained from the latest HEASARC calibration database (CALDB)
and the ancillary response files (ARF) were produced using the task  xrtmkarf.
Using grppha, the source event file, background event file, response file, and
ancillary response files were tied together and binned in such a way that each
bin contains a minimum 10 counts. The spectra were fitted using simple power-
law model, with the galactic absorption column density $n_{H}$ kept fixed at
$4.0\times 10^{20}~{}\text{cm}^{-2}$.

The UV-optical data were obtained via The Swift Ultraviolet/Optical Telescope
(Swift-UVOT, Roming et al. (2005)) which gathers data in six filters: U, V, B,
W1, M2, and W2. The source image was extracted from a region of 5 arcsec
radius centered at the source and the background image was extracted from a
region of 10 arcsec centered away from the source.

### Radio observations

We did not find any publicly available contemporaneous radio observation
corresponding to the 2021 flare. However, for reference, we add non-
simultaneous radio data released by the Planck mission as a part of their
second release of the Planck Catalogue of Compact Sources (PCCS). Even though
the primary objective of Planck mission is to measure the cosmic microwave
background (CMB) anisotropy, as a by-product of their operation detailed
catalogue of compact sources as well as galactic diffusion maps are released,
and we use the data between 2009 August 12 and 2010 November 27 for the
purpose of our analysis (Planck Collaboration et al., 2014, 2016). The data is
processed by the low-frequency instrument (LFI; 30–70 GHz) and high-frequency
instrument (HFI; 100–857 GHz) Data Processing Centres (DPCs) and is downloaded
via the publicly available SED Builder tool333https://tools.ssdc.asi.it/
developed at the Space Science Data Center (SSDC).

Table 1: Log of Swift XRT/UVOT observations used in our analysis Instrument | Observation ID | Start Time | XRT Exposure | UVOT Exposure
---|---|---|---|---
|  | (MJD) | (ks) | (ks)
SWIFT-XRT/UVOT | 00031385003 | 59350.15 | 3.271 | 3.211
SWIFT-XRT/UVOT | 00031385005 | 59352.48 | 1.221 | 1.152
SWIFT-XRT/UVOT | 00031385006 | 59357.31 | 2.602 | 2.560
SWIFT-XRT/UVOT | 00031385007 | 59360.35 | 2.225 | 2.134
SWIFT-XRT/UVOT | 00031385008 | 59363.41 | 1.590 | 1.544
SWIFT-XRT/UVOT | 00031385009 | 59365.21 | 1.334 | 1.293
SWIFT-XRT/UVOT | 00031385011 | 59372.29 | 0.731 | 0.711

##  3 Results

In this section, we describe our results obtained from the spectro-temporal
analysis and we discuss the significance of the results in multi-waveband SED
modelling.

### Light curve analysis-Gamma ray

In Figure-1, we provide the $\gamma$-ray lightcurve for the source in 0.1-300
GeV corresponding to the time between May 1, 2020 - September 30, 2021. In
panels (b) and (c), we also show the same light curve for the lower energy
band (LEB, 0.1-1 GeV) and higher energy band (HEB, 1-300 GeV). In Section 3.1,
we describe the variability properties and the rise as well as decay times of
the prominent flares that we calculated. We provide the correlation features
between the LEB and HEB in Section 3.2, which hints at possible delays of
onset of these flares in different wavebands and in turn can indicate the
possible emission mechanisms.

Figure 1: Top: The full lightcurve for the source in 0.1-300 GeV energy
corresponding to the period May 1, 2020 (MJD 58970) - September 30, 2021 (MJD
59500). There are several flaring episodes within this period. Middle:
Lightcurve for the 0.1-1 GeV band. We observe multiple flaring events here as
well, the most prominent being around June 05, 2021 (MJD $\sim$ 59370). Bottom
Lightcurve for the 1-300 GeV band. We observe strong flaring activity only
around June 15, 2021.

We observe that the entire domain of observation is interspersed with several
flaring episodes, but frequent and short-duration flaring events are most
prominent during the $\sim$ 150 day span from February 05, 2021 (MJD 59250) to
July 05, 2021 (MJD 59400).

###  3.1 Estimation of variability

In order to quantify the flux variability, we use the fractional variability
amplitude (Vaughan et al., 2003) defined as

$F_{var}=\sqrt{\frac{\sigma_{XS}^{2}}{\mu^{2}}},$ (4)

where $\sigma_{XS}$ is the excess variance of a time series with
$x_{i}~{}\text{counrs s}^{-1}$ (Nandra et al., 1997) and $\mu$ is the sample
mean. The excess variance is in turn defined as

$\sigma_{XS}^{2}=\sigma^{2}-\frac{1}{N}\sum_{i}^{N}{\sigma_{i}}^{2},$ (5)

where $\sigma$ is the standard deviation of the sample of size N and
$\sigma_{i}$ is the statistical uncertainty of each of the data comprising the
full sample. The normalized excess variance is defined to be
$\sigma_{NXS}={\sigma_{XS}^{2}}/{\mu^{2}}$, and the fractional variability
happens to be the square root of this normalized excess variance. The error
corresponding to the fractional variability is estimated as

$err(F_{var})=\sqrt{\Bigg{(}\sqrt{\frac{1}{2N}}.\frac{\sigma^{2}}{\mu^{2}F_{var}}\Bigg{)}^{2}+\Bigg{(}\sqrt{\frac{\sigma^{2}}{N}}\frac{1}{\mu}\Bigg{)}^{2}},$
(6)

where the meaning of the respective quantities are as explained above (Prince
et al., 2018).

During the initial phase of low flaring activity (between MJD = 58970 and MJD
= 59250), we observe the normalized excess variance in 0.1-300 GeV to be
$0.85$, and the fractional variability to be $0.92\pm 0.07$. However, towards
the final later segment of observation (beyond MJD = 59250) where frequent
flaring activities are noticed, the fractional variability becomes $1.18\pm
0.70$.

The rise and decay times of the flaring events were calculated by fitting the
150 days long $\gamma$-ray lightcurve with sum of exponential profiles of the
form

$f(t)=F_{0}+\sum_{i=1}^{N}2F_{i}\Bigg{(}\exp\Bigg{[}{\frac{t_{0}-t_{i}}{T_{r}}\Bigg{]}}+\exp\Bigg{[}{\frac{t_{i}-t_{0}}{T_{d}}}\Bigg{]}\Bigg{)}^{-1},$
(7)

where $F_{0}$ is the baseline flux, $F_{i}$ is the source flux at time $t_{0}$
representing the approximate amplitude of the flare, $T_{r}$ and $T_{d}$ are
respectively the rise and decay times of the flares. In HEB, we observe
frequent flaring activity, but the photon intensity in this band is at least
one order of magnitude lesser compared to broadband flux (0.1-300 GeV). We
examine the rise and decay time, as well as the flux doubling time-scale only
for the 150 days time window spanning between MJD = 59250 and MJD = 59400 by
fitting the broadband flaring region using a combination of sum of exponential
profiles, as shown in Figure-2. Between MJD = 59265 and MJD = 59270, the
0.1-300 GeV flux rises from ($3.61\pm 1.52$)$\times{10^{-8}}~{}\text{erg
cm}^{-2}\text{s}^{-1}$ to ($2.31\pm 0.57$)$\times{10^{-7}}~{}\text{erg
cm}^{-2}\text{s}^{-1}$, and on MJD = 59275 the flux again drops to ($5.17\pm
2.40$)$\times{10^{-8}}~{}\text{erg cm}^{-2}\text{s}^{-1}$. The computation of
the flux doubling time was undertaken using the formula

$F(t_{2})=F(t_{1})2^{\frac{t_{2}-t_{1}}{\tau_{D}}},$ (8)

where $F_{t1,t2}$ are the fluxes at time instances $t_{1}$ and $t_{2}$
respectively and $\tau_{D}$ is the flux doubling time. From this formula, we
find the flux doubling time for this flare to be 3.22 days. The rise and decay
time obtained from the sum of exponential fits were respectively found to be
2.40 days and 0.55 days respectively. Subsequently, between MJD = 59300 and
MJD = 59350, we do not observe either a fully quiescent state or a full-
fledged flare, and we call this a ‘plateau’ state after (Prince et al., 2017).

Figure 2: The flaring zone between MJD = 59250 and MJD = 59400 is fitted with
sum of exponential profiles. The time-stamp in the X-axis is relative to MJD =
59250.

The average flux from the source is $(8.10\pm 2.10)\times 10^{-8}~{}\text{erg
cm}^{-2}\text{s}^{-1}$, with the flux peaking at $(1.78\pm 0.17)\times
10^{-7}~{}\text{erg cm}^{-2}\text{s}^{-1}$. Subsequent to MJD = 59350,
corresponding to which simultaneous X-ray data is available, we observe twin
sharp flaring events within a span of $\sim$ 25 days. The peak flux of these
individual flaring events were $(6.05\pm 2.24)\times 10^{-7}~{}\text{erg
cm}^{-2}\text{s}^{-1}$ and $(5.04\pm 0.50)\times 10^{-7}~{}\text{erg
cm}^{-2}\text{s}^{-1}$ respectively. The rise and decay times for the first
twin flare appeared to be 1.6 days and 2.9 days respectively, and
corresponding to the second twin flare they turned out to be 0.9 days and 1.3
days respectively. The shortest flux doubling time-scale thus turns out to be
0.9 days, which can be used to constrain the physical size of the emission
region as well. From the shortest flux doubling time-scale of 0.9 days, we
quantify the variability time-scale to be
$t_{\text{var}}=\tau_{D}\times\ln{2}=0.63$ days.

Table 2: Results from spectral analysis of Swift-XRT/UVOT and Fermi-LAT data Instrument | Parameter | Value | Units
---|---|---|---
| Spectral index ($\alpha$) | $-2.17\pm 0.03$ | -
Fermi-LAT | Prefactor ($N_{0}$) | $2.36\pm 0.11$ | $10^{-11}~{}\text{ph cm}^{-2}\text{s}^{-1}\text{MeV}^{-1}$
| Flux ($F_{\text{0.1-300 GeV}}$) | $3.02\pm 0.14$ | $10^{-7}~{}\text{ph cm}^{-2}\text{s}^{-1}$
| $\Gamma_{X}$ | $1.58\pm 0.05$ | -
Swift-XRT | $K$ | $3.74\pm 0.17$ | $10^{-4}~{}\text{ph cm}^{-2}\text{s}^{-1}\text{keV}^{-1}$
| Flux ($F_{\text{0.3-8.0 keV}}$) | $2.43\pm 0.11$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$
| u band flux | $4.22\pm 0.10$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$
| b band flux | $6.50\pm 0.14$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$
Swift-UVOT | v band flux | $8.15\pm 0.22$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$
| w1 band flux | $2.15\pm 0.06$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$
| w2 band flux | $1.03\pm 0.03$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$
| m2 band flux | $1.57\pm 0.04$ | $10^{-12}~{}\text{erg cm}^{-2}\text{s}^{-1}$

###  3.2 Timing Analysis

We have carried out a temporal analysis of the $\gamma-$ ray emission from a
blazar TXS 1700+685 with the Fermi- LAT data from May 1, 2020 to September 30,
2021. In the low-frequency domain, we have detected a sharp peak in the
periodogram at $\sim 0.06~{}\text{days}^{-1}$ which has $\sim 3\sigma$
significance (Figure-3(a)). The significance of the periodogram peak is
estimated by the method proposed by (Emmanoulopoulos et al., 2013) and
simulating 1000 $\gamma$-ray light curve
444https://github.com/samconnolly/DELightcurveSimulation, as elucidated in
(Connolly, 2015). However, we also detect another peak at $\sim
0.62~{}\text{days}^{-1}$ which has marginally $>2\sigma$ detection
significance (Figure-3(b)).

(a) Low frequency periodogram

(b) High frequency periodogram

Figure 3: Low and high frquency periodograms. In low frequency, we observe a
$>3\sigma$ significant peak at $\sim$ 17 days, while in high frequency the
peak is only $\sim 2.5\sigma$ significant.

A variety of mechanisms have been proposed to explain the background physical
behavior in blazer $\gamma$-ray emission. These include periodically varying
Doppler beaming from precessing jet (Marscher & Gear, 1985; Camenzind &
Krockenberger, 1992; Abraham, 2000; Caproni et al., 2013), processed related
to the innermost stable orbit of the accretion disc (Broderick & Loeb, 2006;
Pihajoki et al., 2013) and several others.

One possible explanation of the QPO of a blazar involves an accretion disc
hot-spot orbiting close to the innermost stable circular orbit of the central
SMBH (Gupta et al., 2019). These hotspot emissions are quasi-thermal and could
be directly observed in the optical/X-ray band. The optical/X-ray emission
could generate the variation in seed photons and after external Compton
interaction with jet, it can trigger the flux modulation in the $\gamma-$ray
lightcurve (Gupta et al., 2017). However, in this case, the expected period in
the optical/X-ray band of QPO depends on the mass and spin of the central
SMBH. Furthermore, it would not be the same in the gamma-ray band, which would
be Doppler boosted. Due to the lack of optical/X-ray data, we can not comment
on optical/X-ray variability.

Another potential explanation of the QPO in Gamma-ray energy is instability in
the disk/jet system. Magnetically choked accretion flows (MCAFs) in general-
relativistic three-dimensional magnetohydrodynamic simulations are observed to
produce quasi-periodic fluctuations in the energy outflow efficiency of the
relativistic jet close to the black hole (Tchekhovskoy et al., 2011). The
dominant mode of these fluctuations has a period of $\sim 70r_{g}/c$ for the
spin $a=0.9375$. For a central mass of $\sim 10^{8}$ M⊙ has a period of $\sim
1$d when the Schwarzschild radius $r_{s}\approx 3\times 10^{11}$m. Slower-
spinning black holes are expected to have longer periods McKinney et al.
(2012). It is unclear, whether these oscillations can produce an observable
signature in the radio jet or not. However, The different time-scales make it
improbable that this mechanism may explain the QPO seen in the Gamma-ray light
curve in TXS 1700+685.

The quasi periodic flux variation can be induced because of the shock
advancement along the internal helical structure of the jet, or the precession
or twisting configuration of the jet (Camenzind & Krockenberger, 1992; Mohan &
Mangalam, 2015). There could be several underlying mechanisms behind a
precessing jet. The modulation of the jet axis could be induced by the
Lense–Thirring precession of the disk (Fragile & Meier, 2009). It can also be
triggered if the AGN is part of a binary supermassive black hole system
(Ackermann et al., 2015; Sandrinelli et al., 2016b; Sandrinelli et al.,
2016a). However, the oscillation period arising out of such a mechanism
appears to be $\sim$ few years, which has been detected recently in the case
of a few AGN sources (Ackermann et al., 2015; Sandrinelli et al., 2016b;
Sandrinelli et al., 2016a). In our case, we found a periodicity about $\sim$
17 days, which is much lower than the expected value in the case of binary
SMBH scenario.

However, in the scenario of the helical motion of the plasma blob along the
internal helical structure of the jet Mohan & Mangalam (2015), the variation
of the Doppler boosting factor in association with the viewing angle of the
emission region can induce a significant flux modulation in the high energy
domain. Depending on the viewing angle and the boosting factor, the expected
QPO may vary from day to month time scale. Our analysis of the gamma-ray
emission from TXS 1700+685 suggests a similar time scale expected from this
model. The observed modulation of the flux could be the enhanced emission
region moving helically within the curved-jet or curved helical jet itself
Sarkar et al. (2021). As we have a limited idea about the emission mechanism
of blazar and lack of multiwavelength data of this source, it is possible that
the observed modulation in gamma-ray flux is due to some other intrinsic
effect or combinations of these effects.

###  3.3 Auto Correlation Function

We have performed Auto Correlation Function (ACF) for the $\gamma$-ray light
curve (0.3 to 500 GeV) using the discrete correlation function
(DCF)555https://github.com/astronomerdamo/pydcf/blob/master/dcf.py (Edelson &
Krolik, 1988) and $\zeta$-discrete correlation function
($\zeta$-DCF)666https://www.weizmann.ac.il/particle/tal/research-
activities/software (Alexander, 2014) for comparison. Both correlation
functions are followed the same pattern. The most correlated peak is found at
zero time lag which is expected as here we have done the ACF analysis. The 2nd
most significant peak is found $82\pm 5$ days on both sides of the main peak.
The correlation coefficient value of this peak is about 50 for both peaks. In
estimating the significance of the ACF peak, the same procedure as elucidated
in the previous section has been followed. We perform the computation of ACF
over 1000 $\gamma$-ray light curves and estimated the 1$\sigma$, 2$\sigma$ and
3$\sigma$ for each time lag. The $\sigma$ levels (black and gray curves) are
plotted on the correlation function as shown in Figure-4. We have found
82$\pm$5 days peaks are significant up to 3$\sigma$.

Figure 4:  We have also calculated the confidence of the peaks in correlation
function (DCF). Here we found the peak near $\pm 82$ days have a confident
upto $3\sigma$.

###  3.4 Examination of high energy photons

In order to estimate the dominance of high energy photons during the flaring
region, we carried out the Fermi-LAT analysis with $2^{\circ}$ of ROI. We
choose this threshold because the flux of the source was low, the likelihood
analysis could not be performed for most of the time intervals with smaller
ROI. We do not perform the analysis with the ULTRACLEAN class of events
because of the same reason. In Figure-5, we plot the lightcurve corresponding
to the high energy ($>1$ GeV) photons detected within $2^{\circ}$ ROI within
the flaring phase. In (Figure-5), we show the number of such photons with
$>90\%$ probability of detection at successive time intervals.

Figure 5:  Time variation of the number of high energy ($>1$ GeV) photons with
$>90\%$ probability of detection. We detect this number to be peaking on MJD =
59370.

We observe the highest number of high energy photons (total 42) on MJD =
59370, and the highest energy of these photons being 6.74 GeV with the
detection probability 99.80%. The number of such high-energy photons during
the pre-flare plateau region is significantly lower, with the highest number
being 10 on MJD = 59270. The highest energy of the detected photons with high
detection significance corresponding to the entire span of observation
pertaining to our analysis, however, turns out to be 25.41 GeV on MJD = 59275
with a detection probability of 99.97%. Such high energy photons could be
produced as a consequence of the external Compton scattering of the photons
originating from the broad-line region (BLR), or dusty torus by the jets or
synchrotron self-Compton emission.

###  3.5 Gamma-ray Spectral energy distribution during flaring phase

In this section, we describe the key features regarding the $\gamma$-ray
spectral energy distribution (SED). We attempt to approximate the shape of the
SED using different profiles: power-law (PL), log-parabola (LP), and broken
power-law (BPL) as defined in Equations 1-3. The choice of these functional
forms is motivated by the spectral fitting over equispaced logarithmic energy
bins of blazar flares in 100 MeV - 200 GeV range (Ackermann et al., 2010). In
order to determine the quality of the corresponding spectral fits, the
Log(likelihood) values obtained from the unbinned likelihood analysis have
also been examined. In Figure-6, we show the comparison of different profiles.
The spectral index corresponding to the power-law fit turns out to be $2.18\pm
0.03$. However, even by visual inspection the presence of spectral curvature
is detected, which motivates us to examine the alternative phenomenological
models such as logparabola (LP) and broken power law(BPL). The significance of
the spectral curvature is determined via $\text{TS}_{\text{curve}}$, which is
defined as the
$\text{TS}_{\text{curve}}=2(\mathcal{L}(BPL/LP)-\mathcal{L}(PL))$. The
spectral curvature is significant only if $\text{TS}_{\text{curve}}>16$ (Nolan
et al., 2012), where L represents the likelihood of the function. In both of
the cases (LP and BPL) we found this criterion to be satisfied, implying that
spectral curvature is statistically significant. In the case of LP, we find
$\alpha=2.26\pm 0.05$ and $\beta=0.17\pm 0.04$, with the
$TS_{\text{curve}}=18$. On the other hand in the case of BPL fit, we find the
spectral index before the break at 1 GeV to be $1.92\pm 0.07$ and after the
break to be $2.63\pm 0.11$, with $TS_{\text{curve}}=34$. Thus, The BPL photon
index demonstrates the trend of being $<2$ before the break, and after $>2$
after the break. Such a falling spectrum has earlier been observed in the case
of several other FSRQs as well, 3C 454.3 (Abdo et al., 2011), 3C 279 (Paliya
et al., 2015; Prince, 2020).

These results can also serve to constrain the possible origin of the
$\gamma$-ray emission region. If the power-law could approximate the SED well,
it would imply that the emission region is outside the broad-line region
(BLR). However, the presence of a significant spectral curvature implies that
the photons are further reprocessed in the BLR (like photon-photon pair
production), which attenuates the $\gamma$-ray flux leading to a spectral
break (Liu & Bai, 2006). The presence of the significant cut-off at 1 GeV
explains the non-detection of very high energy photons throughout the period
of observation, and also implies that the 25.41 GeV photon detected on MJD =
59275 might have originated from outside of the BLR.

###  3.6 Broadband SED modelling

Modeling the multi-waveband SED turns out to be a powerful tool in
understanding the flaring mechanism associated with the source under
consideration. In our work, we consider a one-zone leptonic model of jet
acceleration based on the stochastic or shock acceleration mechanism
(Ghisellini & Tavecchio, 2009; Tramacere et al., 2011). We use the publicly
available jetset implementation for the purpose of our SED modelling (Massaro
et al., 2006; Tramacere et al., 2009, 2011). In this model, it is assumed that
a spherically symmetric blob of size $R$ filled with relativistic population
of electrons is moving through an entangled magnetic field ($B$). The leptons
in the jet are accelerated to an ultra-relativistic regime via a stochastic or
shock acceleration mechanism inside the jet. The plasma blobs are assumed to
be moving at an angle $\theta$ relative to the observer’s line of sight,
implying that the emission will be beamed by a factor
$\delta=1/[\Gamma(1-\beta\cos{\theta})]$, where $\Gamma$ is the bulk Lorentz
factor of the blob. We assume the relativistic population of electrons within
the blob is satisfying an empirical broken power-law distribution of the form,

$n({\gamma})=\begin{cases}K_{1}\gamma^{p_{1}}&\text{when
$\gamma_{\text{min}}\leq\gamma\leq\gamma_{\text{break}}$}\\\
K_{2}\gamma^{p_{2}}&\text{when
$\gamma_{\text{break}}\leq\gamma\leq\gamma_{\text{max}}$}\\\ \end{cases}$ (9)

where $\gamma_{min},\gamma_{max}$ and $\gamma_{\text{break}}$ are respectively
the minimum, maximum, and the break energies of the relativistic electrons,
and $p_{1}$ and $p_{2}$ are the spectral indices below and above the break.
The normalization constants $K_{1}$ and $K_{2}$ are related to each other by
$K_{2}=K_{1}(\gamma_{\text{break}})^{p_{2}-p_{1}}$.

Figure 6: SED during the flaring phase approximated using different profiles
(power-law, broken power-law, and log-parabola). We observe a statistically
significant break at $\sim$ 1 GeV. Figure 7: Multiwavelength SED suring the
flaring phase modeled in one-zone leptonic framework. The best-fit model
parameters are listed in Table 3.

The first hump of the broadband SED is produced by the synchrotron emission as
a consequence of the interaction of relativistic electrons with the magnetic
field. These synchrotron photons serve as the seed photons for the inverse
Compton process by the same population of relativistic electrons named as
synchrotron self-Compton (SSC), and this SSC leads to the high energy hump.
Further, several other processes which fall in the category of external
Compton (EC) process also contribute to the genesis of this hump. The seed
photons for the EC emission come from
(i) direct emission from the accretion disk,
(ii) reprocessed emission from the infrared region of the surrounding dusty
torus (DT),
(iii) reprocessed emission in the optical/UV domain from the BLR.

We consider the seed photons from BLR, DT as well as the disk for the purpose
of our broadband SED modeling. The thermal emission of the accretion disc is
approximated by the multi-temperature blackbody radiation, such that the
radial profile of temperature is provided as

$T^{4}(r)=\frac{3R_{S}R_{D}}{16\epsilon\pi\sigma
r^{3}}\Bigg{(}1-\sqrt{\frac{3R_{S}}{r}}\Bigg{)}K,$ (10)

where $R_{S}$ is the Schwarzschild radius, $\epsilon$ is the efficiency of
accretion, $\sigma$ is the Stefan-Boltzmann constant. The outer boundary of
the disc is assumed to be $500R_{S}$. In our SED modeling, we let the
parameters regarding the electron distribution as well as the magnetic field
vary, while the inner and outer extent of the BLR ($1.0\times 10^{18}$ cm and
$2.0\times 10^{18}$ cm respectively) as well as the size of the DT ($5.0\times
10^{18}$ cm) are left frozen. The broadband spectral fitting is shown in
Figure-7 and the fitted parameters are provided in Table 3.

Table 3: Broadband SED modeling parameters.

Parameter | Unit | Value
---|---|---
$B^{1}$ | Gauss | 0.12
$\theta^{2}$ | degree | 3.60
$\Gamma^{3}$ | - | 7.08
$\delta^{4}$ | - | 11.78
$\gamma_{\text{min}}^{5}$ | - | 4.87$\times 10^{2}$
$\gamma_{\text{max}}^{6}$ | - | 1.05$\times 10^{4}$
$\gamma_{\text{break}}^{7}$ | - | 4.19$\times 10^{3}$
$\alpha_{1}^{8}$ | - | 3.32
$\alpha_{2}^{9}$ | - | 3.15
$U_{e}^{{}^{\prime}10}$ | $\text{erg cm}^{-3}$ | 0.51
$U_{B}^{{}^{\prime}11}$ | $\text{erg cm}^{-3}$ | $6.09\times 10^{-4}$
$U_{e}^{{}^{\prime}}/U_{B}^{{}^{\prime}12}$ | - | 839.08
$N^{13}$ | $\text{cm}^{-3}$ | $7.41\times 10^{2}$
| Parameters external to the jet |
$L_{d}^{14}$ | $10^{45}~{}\text{erg s}^{-1}$ | 1.73
$T_{DT}^{15}$ | K | 1.28$\times 10^{4}$
$\tau_{DT}^{16}$ | - | 0.93
$R_{DT}^{17}$ | $10^{18}\text{cm}$ | 5.0
$R_{BLR}^{\text{in}~{}18}$ | $10^{18}\text{cm}$ | 1.0
$R_{BLR}^{\text{out}~{}19}$ | $10^{18}\text{cm}$ | 2.0
$\tau_{BLR}^{20}$ | - | 0.24

1 Magnetic field, 2 Viewing angle relative to the line of sight, 3 Bulk
Lorentz factor of the jet, 4 Doppler beaming factor, 5 Minimum Lorentz factor
of the injected electrons, 6 Maximum Lorentz factor of the injected electrons,
7 reak Lorentz factor of the injected electrons, 8 Low energy spectral index,
9 High energy spectral index, 10 Electron energy density in comoving frame, 11
Magnetic field energy density in comoving frame, 12 Equipartition value, 13
Particle density, 14 Disk luminosity, 15 Temperature of the dusty torus, 16
Optical depth of the dusty torus, 17 Distance of the dusty torus, 18 Inner
radius of the BLR, 19 Outer radius of the BLR, 20 Optical depth of the BLR,

##  4 Results and Discussions

In this section, we discuss the implications of our timing and correlation
study, as well as the broadband spectral fitting.

###  4.1 Self-correlation and the plausible effect of gravitational lensing

The gravitational light-bending effect of the emitted radiation of quasars by
the intervening foreground sources has been well recognized in the literature
(Torres et al., 2003). For a gravitationally lensed blazar source, the
$\gamma$-ray lightcurve is expected to show some time-delay, as has already
been found in the case of PKS 1830-211 (Barnacka et al., 2011) and B0218+357
(Cheung et al., 2014). The resolution of Fermi-LAT is not sufficient to
resolve the lensed images located within 1 arc second spatial resolution
(Pramesh Rao & Subrahmanyan, 1988), but the time-lag between different photon
paths detected from the self-correlation of the $\gamma$-ray lightcurves could
be a plausible way to detect such lensing effects (Abdo et al., 2015; Abhir et
al., 2021a).

In our case, we have detected a strong self-correlation ($>3\sigma$ detection
significance) of the 0.1-300 GeV $\gamma$-ray lightcurve at a time-delay of 82
days (Figure-4). We conjecture the presence of such a strong signature of a
time delay to be a consequence of the microlensing effect of the $\gamma$-ray
photons originating from different emission regions. However, conclusive
pieces of evidence can only be gathered from detailed radio monitoring of this
source, since it would then be possible to infer the presence and impact of
such lensing effects using flux ratios (Lovell et al., 1996). Also, since the
emission region corresponding to radio and $\gamma$-ray could be different,
emission size induced chromaticity would be an important indicator to probe
such effects.

###  4.2 Estimation of minimum Doppler factor

The numerical value corresponding to the minimum Doppler factor can be
estimated from the highest energy photon detected during the observation.
Assuming the optical depth $\tau_{\gamma\gamma}(E_{h})$ of the highest energy
photon $E_{h}$ to $\gamma\gamma$ interaction being 1, the expression for the
minimum Doppler factor could be read as

$\delta_{\text{min}}=\Bigg{(}\frac{\sigma_{T}D_{L}^{2}(1+z)^{2}f_{e}E_{h}}{4t_{\text{var}}m_{e}c^{4}}\Bigg{)}^{1/6},$
(11)

where $\sigma_{T}$ is the Thomson cross-section, $D_{L}$ is the luminosity
distance of the source which turns out to be 1547 Mpc with under standard
cosmology parameters, $z(=0.3)$ is the redshift of the source, $f_{e}$ is the
0.3-8.0 keV X-ray flux, $t_{\text{var}}$ is the observed variability time-
scale of the source, $E_{h}$ is the energy of the highest energy photon. In
our case, during the flare we observe the fastest variability time-scale to be
0.63 days, and the highest-energy photon with significant detection
probability to be 6.67 GeV. With these parameters, $\delta_{\text{min}}$ turns
out to be 8.76. Our fitted estimation of the Doppler factor (=11.78) turns out
to be more than this threshold.

###  4.3 Broadband emission during the flaring state

In the broadband SED, each segment indicates some constraints on the allowed
parameter space. In the case of blazars, some important constraints are:

  1. 1.

The synchrotron emission component suffers a sharp decline below $10^{12}$ Hz
because of the synchrotron self-absorption mechanism, in which the photons
interact with the ambient electrons and lose energy. Since the radio data is
above the predicted curve and demonstrates only the emission from the extended
segment from the jet, radio data provides only a weak constraint on the low
energy synchrotron spectrum.

  2. 2.

The optical/UV data is primarily contributed by the synchrotron and the
optical emission from the disk, this segment of the SED provides constraints
on the magnetic field ($B$), particle density ($N$) and the first spectral
index ($\alpha_{1}$).

  3. 3.

The high energy peak contributed by the X-ray/$\gamma$-ray is contributed by
the synchrotron self-Compton (SSC) and external Compton (EC) processes. This
segment is determined by the second spectral slope ($\alpha_{2}$), the
physical distances of BLR and DT ($R_{\text{BLR}}$ and $R_{\text{DT}}$).

  4. 4.

Overall emission is also a function of the bulk Lorentz factor of the jet
($\Gamma$), viewing angle ($\theta$), Doppler beaming factor ($\delta$).

Because of the weak constraint provided by the radio data and given the fact
that in the absence of simultaneous radio observation we are using only
historical data for referencing purposes, the radio segment of the SED is not
well approximated by our model, which is common in Blazar SEDs ( (e.g.,
D’Ammando et al. (2012)). The fitted spectrum is provided in (Figure-5) and
the fitted parameters are listed in Table 3. The lower energy hump is mostly
contributed by the synchrotron component, and the contribution from the disk
emission is found to be sub-dominant. Reharding the high energy hump, we
observe the X-ray segment is well modeled with the SSC component, while the
$\gamma$-ray segment is produced by both SSC and EC components. The generation
of keV photons in synchrotron emission requires the presence of a strong
magnetic field or the injection of a large number of high-energy electron
populations. The equipartition value that we have observed in our case implies
that the bulk of the power is carried away by the electron population, and the
magnetic field carries only a tiny fraction of power. This is reflected from
the low magnetic field ($\sim 0.12$ G) we have obtained from our spectral fit.
Our estimation of the magnetic field is in agreement with the previous
estimation for which the equipartition value is similarly on the higher side
(e.g. in PKS 1424-418; Abhir et al. (2021b) for which the estimated magnetic
field turns out to be $\sim 0.2$ G), and is at variance with low equipartition
value estimation corresponding to stronger magnetic fields (Buson et al.,
2014). The bulk Lorentz factor ($\Gamma$) and the Doppler factor ($\delta$)
are also significantly lower compared to what was obtained in Buson et al.
(2014).

The highest-energy photon obtained during the flaring phase was found to be
$6.67$ GeV. This is below the threshold for which the photons become opaque
(more than $\sim$ 20 GeV) and are reprocessed by the BLR region. Because of
this reason, we included the photons from DT, BLR as well as the disk as seeds
for the EC mechanism. We observe the EC contribution by the BLR region to be
dominant in the $\gamma$-ray segment, and the EC from the DT to be
subdominant. The EC component corresponding to the disk photons seems to be
negligible (turns out to be 4-5 orders of magnitude smaller compared to the
other two components).

###  4.4 Estimation of jet power

We have estimated the approximate power carried by different components of the
jet, as well as the entire jet by the following equation

$P_{\text{jet}}=\pi
R_{e}^{2}\Gamma^{2}c(U_{e}^{\prime}+U_{B}^{\prime}+U_{p}^{\prime}),$ (12)

where the quantities $U_{e}^{\prime},U_{B}^{\prime}$ and $U_{p}^{\prime}$ are
respectively the energy densities corresponding to the electrons, magnetic
field and protons in the co-moving frame of the jet (the un-primed quantities
are in observer’s rest frame while the primed quantities are in co-moving
frame). The power contributed by the leptonic component is represented by

$P_{e}^{\prime}=\frac{3\Gamma^{2}c}{4R_{e}}\int_{E_{\text{min}}}^{E_{\text{max}}}EQ(E)dE,$
(13)

where $Q(E)$ is the injected particle spectrum and $E_{\text{min}}$ and
$E_{\text{max}}$ are the minimum and maximum energies of the electron
calculated by multiplying the rest mass energy of the electron (in MeV) with
$\gamma_{\text{min}}$ and $\gamma_{\text{max}}$ respectively. In order to
calculate the power carried by the protons, we assume the ratio of the number
of electron-positron pairs to the number of protons by 20:1. The power
contributed by the magnetic field, on the other hand, can be calculated using

$P_{B}^{\prime}=R_{e}^{2}\Gamma^{2}c\frac{B^{2}}{8}.$ (14)

where $B$ is the strength of the magnetic field obtained from the broadband
fit.

Using the fitted parameters, we find the jet power contributed by the
electrons and protons to be $1.61\times 10^{43}~{}\text{erg s}^{-1}$ and
$1.06\times 10^{42}~{}\text{erg s}^{-1}$ respectively. The power deposited by
the magnetic field component turns out to be $3.02\times 10^{42}~{}\text{erg
s}^{-1}$, which is subdominant compared to the leptonic component. The total
jet power turns out to be $2.03\times 10^{43}~{}\text{erg s}^{-1}$.

Figure 8: Histogram of the logarithm of three days binned gamma-ray flux. The
green dotted line and the blue dashed line represent the Gaussian and
lognormal PDFs, respectively. Figure 9: The rms-flux scatter plot, with a best
fit line (magenta).

###  4.5 Flux distribution of TXS 1700+685

The flux distribution study can provide insight into the underlying mechanisms
producing the variations in blazars. The flux variability in blazars has been
observed to be stochastic in nature. Typically, the flux distribution is
Gaussian in the case of linear stochastic process, while a lognormal flux
distribution (i.e. a Gaussian distribution in the logarithm of flux) is
expected for a non-linear stochastic process. Several blazars show lognormal
flux distributions using multi-wavelength lightcurves, and at different time-
scales (Giebels & Degrange, 2009; H.E.S.S. Collaboration et al., 2010;
Tluczykont et al., 2010; Kushwaha et al., 2016; Sinha et al., 2018; Khatoon et
al., 2020; Scargle, 2020; Prince et al., 2021). The advent of high sensitive
Fermi-LAT (100 MeV - 300 GeV) enables us to characterize the flux distribution
study in $\gamma$-ray energy. Several blazars seen by Fermi-LAT, show
lognormal behavior in their long-term $\gamma$-ray lightcurves (Ackermann et
al., 2015; Shah et al., 2018; Romoli et al., 2018; Bhatta & Dhital, 2020). The
lognormal variability in blazar is usually explained by assuming the jet
emission arising from non-linear multiplicative processes originated in the
accretion disc. However, minute-scale variability as observed in many blazars
may not be produced by the fluctuations in the disc, and that favors the
variability to produce in the jet itself. The “mini jets-in-a-jet" statistical
model provides a possible explanation where the addition of emission from a
large collection of mini jets can explain the lognormal flux distribution
(Biteau & Giebels, 2012). Furthermore, Sinha et al., 2018 showed that the
lognormal flux distribution can be obtained when the underlying particle
acceleration process is time-dependent. In such cases, a Gaussian perturbation
in the particle acceleration time-scale can give rise to a lognormal flux
distribution and linear flux-rms correlations at high energies.

In this work, we presented the flux distribution study for the source TXS
1700+685, with three days binned gamma-ray flux lightcurve. To select the
statistically significant lightcurve, we considered the flux points for which
flux is greater than 2-$\sigma$ level, such that ${\frac{F}{\Delta{F}}}$ > 2\.
We produced the normalized histogram of the logarithm of flux and fit it with
the Gaussian and lognormal probability density functions (PDFs, Shah et al.
(2018)) (Figure-8). It is found that the lognormal function fits the
distribution significantly better with reduced chi-square
(${\chi^{2}}_{red}){\approx}$ 0.91 for 11 degrees of freedom (dof), than the
Gaussian function (${\chi^{2}}_{red}{\approx}$ 2.29 for 11 dof). The linear
dependence of the mean flux $<F>$ with its excess root mean square (rms)
variability, is an essential feature for the lognormal process. We estimated
the excess variance (Vaughan et al., 2003),
$\sigma_{XS}=\sqrt{\sigma^{2}-\overline{\sigma_{err}^{2}}}$ with $\sigma^{2}$
and $\overline{\sigma_{err}^{2}}$ represent the variance and the average of
the square of the measurement errors, using 50d binned data. The rms-flux plot
is shown in Figure-9, fitted with a linear function of slope 0.57$\pm$0.1. The
Spearman’s rank correlation coefficient ($r_{s}$) and the correlation
probability (p) are obtained between $<F>$ and $\sigma_{XS}$, which results
$r_{s}$ = 0.78 and p = $4.5\times 10^{-3}$, suggesting linear rms-flux
relation. The obtained lognormal feature in the flux distribution and the
linearity between the excess variance to the flux implies the variation in
flux is lognormal.

##  5 Summary

In the present study, we comment on the variability and the spectral
characteristics of the blazar TXS 1700+685 using a two-component external
Compton model along with SSC for the first time. For the broadband spectral
modeling, we focus on the flaring phase for which the simultaneous X-ray
observation is present, and the emission region is found to be within BLR. The
high energy peak of the broadband SED corresponding to the flaring state is
mainly reproduced by the EC process of seed photons originating from the BLR
and DT, and the contribution from the disk is found to be sub-dominant. The
power deposited by the jet is found to be primarily contributed by the
leptonic component, instead of the magnetic field component.

It would be really interesting to undertake the multi-frequency correlation
and spectral investigation of this source over several outbursts in order to
understand the dominance of leptonic/hadronic processes in different phases of
its evolution. It would be interesting to follow up on this source using radio
observation to detect any possible micro-lensing effects and connect it with
self-correlation studies across its evolution. We strongly encourage the
detailed multi-wavelength follow-up of this source in order to uncover such
stimulating effects.

##  6 Acknowledgements

This work made use of data collected by Fermi and the Swift missions, as well
as the archival radio data obtained from SEDbuilder tool. A.B. acknowledges
Jayant Abhir for productive discussions regarding the Fermi analysis, and
Andrea Tramacere for his helpful suggestions regarding multi-wavelength
modelling. DB acknowledges Science and Engineering Research Board - Department
of Science and Technology for Ramanujan Fellowship - SB/S2/ RJN-038/2017. For
this work, PN acknowledges CSIR fellowship and the support from Physical
Research Laboratory which is funded by the Department of Space, India. RP
acknowledges the support by the Polish Funding Agency National Science Centre,
project 2017/26/A/ST9/00756 (MAESTRO 9), and MNiSW grant DIR/WK/2018/12.

##  7 Data Availability

This work has made use of data from Fermi-LAT, Swift-XRT, Swift-UVOT and
archival data from SED builder tool. All the data used are available in the
public domain. Details are provided in Section 2.

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$\omega_{\mathcal{H}}\in\scaleobj{1.5}{\wedge}^{2}H_{1}(S_{g};\mathbb{Q})$ is
the characteristic element for some $\mathcal{H}\in\mathcal{H}(x)$. Then for
any $\gamma\wedge\delta$ with $\gamma,\delta$ primitive and
$\langle\gamma,\delta\rangle=1$, there is an $x\subseteq X_{g}$ and
$\mathcal{H}\in\mathcal{H}(x)$ with $\omega_{\mathcal{H}}=\gamma\wedge\delta$.
The set of such $\gamma\wedge\delta$ spans
$\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}\otimes\mathbb{Q}$ by Lemma 4.4, so the
image of the composition

$H_{1}(X_{g}^{\mathcal{W}};\mathbb{Q})\rightarrow
H_{1}(X_{g};\mathbb{Q})\rightarrow\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}\otimes\mathbb{Q}/Q\omega^{\prime}_{a}.$

contains
$\operatorname{im}(\scaleobj{1.5}{\wedge}^{2}(\mathcal{W}^{\perp}\cap[a]^{\perp})\otimes\mathbb{Q}\rightarrow\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}\otimes\mathbb{Q}/\mathbb{Q}\omega_{a}^{\prime})$,
so the lemma is complete. ∎

We will need another auxiliary lemma about the rational abelianizations of the
Torelli groups of surfaces.

######  Lemma 8.5.

Let $g\geq 4$, and let $\mathcal{T}_{0},\mathcal{T}_{1}$ be two surfaces each
equipped with an embedding $\iota^{i}:\mathcal{T}_{i}\hookrightarrow S_{g}$.
Assume that the following hold:

  * •

each embedding $\iota^{i}$ is clean,

  * •

$S_{g}{\ssearrow}\iota^{i}(\mathcal{T}_{i})$ is connected,

  * •

$g(\mathcal{T}_{i})\geq 3$ for $i=0,1$,

  * •

the pullback $S$ of the maps $\iota^{0}$ and $\iota^{1}$ is a connected,
smooth manifold,

  * •

$S_{g}{\ssearrow}\left(\iota^{0}(\mathcal{T}_{0})\cap\iota^{1}(\mathcal{T}_{1})\right)$
is connected, and

  * •

$g(S)\geq 3$.

The following commuting square

$\textstyle{H_{1}(\mathcal{I}(S,S_{g});\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\mathcal{I}(\mathcal{T}_{1},S_{g});\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}^{1}}$$\textstyle{H_{1}(\mathcal{I}(\mathcal{T}_{2},S_{g});\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}^{2}}$$\textstyle{H_{1}(\mathcal{I}_{g};\mathbb{Q}),}$

is a pullback square.

###### Proof.

For each $\mathcal{T}_{i}$, let $\kappa^{i}$ denote the map
$S\rightarrow\mathcal{T}_{i}$, and let $\iota$ denote the composition
$\iota^{i}\circ\kappa^{i}$. Note that since $S$ is a pullback, we have
$\iota^{1}\circ\kappa^{1}=\iota^{0}\circ\kappa^{0}$, so the definition of
$\iota$ does not depend on $i$. Now, since we have assumed that each embedding
$\iota^{i}$ is clean, we have $\iota^{i}_{*}$ injective by a theorem of Putman
[25, Theorem B]. The map $\iota$ is also clean by hypothesis, and therefore
the pushforward $\iota_{*}$ is injective by the same theorem of Putman, so it
suffices to show that
$\operatorname{im}(\iota_{*})=\operatorname{im}(\iota_{*}^{0})\cap\operatorname{im}(\iota_{*}^{1})$.
Lemma 4.5 says that that
$\operatorname{im}(\mathcal{I}(\iota^{i})_{*})=\operatorname{im}(\scaleobj{1.5}{\wedge}^{3}H_{1}(\mathcal{T}_{i};\mathbb{Q})\rightarrow\scaleobj{1.5}{\wedge}^{3}H_{1}(S_{g};\mathbb{Q}))$
and
$\operatorname{im}(\mathcal{I}(\iota)_{*})=\operatorname{im}(\scaleobj{1.5}{\wedge}^{3}H_{1}(S;\mathbb{Q})\rightarrow\scaleobj{1.5}{\wedge}^{3}H_{1}(S_{g};\mathbb{Q}))$.
Since the functor $\scaleobj{1.5}{\wedge}^{3}$ from $\mathbb{Q}$–vector spaces
to $\mathbb{Q}$–vector spaces sends pullbacks of monomorphisms to pullbacks of
monomorphisms, it is enough to show that

$\operatorname{im}(H_{1}(\mathcal{T}_{0};\mathbb{Q})\rightarrow
H_{1}(S_{g};\mathbb{Q}))\cap\operatorname{im}(H_{1}(\mathcal{T}_{1};\mathbb{Q})\rightarrow
H_{1}(S_{g};\mathbb{Q}))=\operatorname{im}(H_{1}(S;\mathbb{Q})\rightarrow
H_{1}(S_{g};\mathbb{Q})).$

This follows from our hypotheses that $S_{g}{\ssearrow}\iota(S)$ and
$S_{g}{\ssearrow}\iota^{i}(\mathcal{T}_{i})$ are connected, so embedding $S$
and $\mathcal{T}_{i}$ into $S_{g}$ does not introduce any new relations in
$H_{1}(S;\mathbb{Q})$ or $H_{1}(\mathcal{T}_{i};\mathbb{Q})$. ∎

We now prove the following.

######  Lemma 8.6.

Let $L\subseteq[a]^{\perp}$ be a free abelian subgroup with $g(L)=2$ and
$\dim(L\otimes\mathbb{Q})=4$. Let $V=L\otimes\mathbb{Q}$. Let $\mathcal{V}$ be
as in Lemma 8.1. Then the natural map

$\psi:\bigoplus_{\mathcal{V}^{\prime}\subseteq\mathcal{V}:|\mathcal{V}^{\prime}|=4}H_{1}(X_{g}^{V}\cap
X_{g}^{\mathcal{V}^{\prime}};\mathbb{Q})\rightarrow
H_{1}(X_{g}^{V};\mathbb{Q})$

is surjective.

###### Proof.

We begin with the following claim.

_Claim_. The pushforward $H_{1}(X_{g}^{V};\mathbb{Q})\rightarrow
H_{1}(X_{g};\mathbb{Q})$ is an injection, and the image of this map is sent to
$\scaleobj{1.5}{\wedge}^{2}V^{\perp}\cap\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}$
under the isomorphism
$H_{1}(X_{g};\mathbb{Q})\cong\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}/\mathbb{Q}\omega_{\alpha}$
from Lemma 8.4.

_Proof of claim_. Let $S_{2}^{1}\subseteq S_{g}$ be a compact subsurface such
that $\operatorname{im}\left(H_{1}(S_{2}^{1};\mathbb{Z})\hookrightarrow
H_{1}(S_{g};\mathbb{Z})\right)=V$. Let $\mathcal{I}_{g-2}^{1}$ denote the
subgroup of $\mathcal{I}_{g}$ generated by elements that fix $\partial
S_{2}^{1}$ and restrict to the identity on $S_{2}^{1}$. Let
$\mathcal{C}_{[a]}(S_{g},S_{2}^{1})$ denote the subcomplex of
$\mathcal{C}_{[a]}(S_{g})$ generated by curves $c$ such that $c$ is disjoint
from $S_{2}^{1}$. By a result of Kent, Leininger and Schleimer [15, Theorem
7.2], the fibers of the natural map
$\mathcal{C}_{[a]}(S_{g},S_{2}^{1})\rightarrow\mathcal{C}_{[a]}(S_{g-2})$ are
all trees, so $\mathcal{C}_{[a]}(S_{g},S_{2}^{1})$ is homotopy equivalent to
$\mathcal{C}_{[a]}(S_{g-2})$. Then a theorem of the author [21, Theorem A]
says that $\mathcal{C}_{[a]}(S_{g-2})$ is at least 1–acyclic, and thus
$\mathcal{C}_{[a]}(S_{g},S_{2}^{1})$ is at least 1–acyclic as well. Therefore
the equivariant homology spectral sequence
$\mathbb{E}_{p,q}^{r}(\mathcal{I}^{1}_{g-2},\mathcal{C}_{[a]}(S_{g},S_{2}^{1});\mathbb{Q})$
converges to $H_{1}(\mathcal{I}^{1}_{g-2};\mathbb{Q})$. Hence there is a right
exact sequence

$H_{1}(\operatorname{Stab}_{\mathcal{I}_{g-2}^{1}}(a);\mathbb{Q})\rightarrow
H_{1}(\mathcal{I}_{g-2}^{1};\mathbb{Q})\rightarrow
H_{1}(X_{g}^{V};\mathbb{Q})\rightarrow 0.$

Since $g\geq 51$, we have $g\left(S_{g-2}^{1}{\ssearrow}a\right)\geq 3$. Then
the inclusion $S_{g-2}^{1}{\ssearrow}a\hookrightarrow S_{g-2}^{1}$ is a clean
embedding (as in Section 3), so a theorem of Putman [25, Theorem B] says that
the pushforward map
$H_{1}(\operatorname{Stab}_{\mathcal{I}_{g-2}^{1}}(a);\mathbb{Q})\rightarrow
H_{1}(\mathcal{I}_{g-2}^{1};\mathbb{Q})$ is an injection. Hence the above
right exact sequence is in fact exact:

$0\rightarrow
H_{1}(\operatorname{Stab}_{\mathcal{I}_{g-2}^{1}}(a);\mathbb{Q})\rightarrow
H_{1}(\mathcal{I}_{g-2}^{1};\mathbb{Q})\rightarrow
H_{1}(X_{g}^{V};\mathbb{Q})\rightarrow 0.$

Furthemore, there is a morphism of short exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\operatorname{Stab}_{\mathcal{I}_{g-2}^{1}}(a);\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{a}}$$\textstyle{H_{1}(\mathcal{I}_{g-2}^{1};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H_{1}(X_{g}^{V};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\rho}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\operatorname{Stab}_{\mathcal{I}_{g}}(a);\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\mathcal{I}_{g};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(X_{g};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\rho_{a}$ and $\rho$ are the natural pushforward maps. Lemma 8.5 says
that left square is a pullback. Furthermore, each map in this square is an
injection by the aforementioned theorem of Putman, so the map
$\operatorname{cok}(\rho_{a})\rightarrow\operatorname{cok}(\rho)$ is
injective. Furthermore, since $\rho$ is injective, we have $\ker(\rho)=0$.
Then the snake lemma says that there is an exact sequence

$\ker(\rho)\rightarrow\ker(\overline{\rho})\rightarrow\operatorname{cok}(\rho_{a})\rightarrow\operatorname{cok}(\rho).$

The map $\operatorname{cok}(\rho_{a})\rightarrow\operatorname{cok}(\rho)$ is
injective and $\ker(\rho)=0$, so $\ker(\overline{\rho})=0$, and in particular
we that $\overline{\rho}:H_{1}(X_{g}^{V};\mathbb{Q})\rightarrow
H_{1}(X_{g};\mathbb{Q})$ is an injection. The second part of the claim follows
from the fact that for any edge $x\subseteq X_{g}^{V}$, the characteristic
element $\omega_{\mathcal{H}}$ for any $\mathcal{H}\in\mathcal{H}(x)$ lies in
$\scaleobj{1.5}{\wedge}^{2}V^{\perp}\cap\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}\otimes\mathbb{Q}$,
because $V$ is a vector space. This implies that
$\operatorname{im}(H_{1}(X_{g}^{V};\mathbb{Q})\rightarrow
H_{1}(X_{g};\mathbb{Q})$ is contained in
$\operatorname{im}(\scaleobj{1.5}{\wedge}^{2}V^{\perp}\cap\scaleobj{1.5}{\wedge}^{2}[a]^{\perp}\otimes\mathbb{Q}$.
Then the second part of Lemma 8.4 gives the containment in the other
direction, so we have equality.

Given the claim, we now continue with the proof of the lemma. The claim says
that
$H_{1}(X_{g}^{V};\mathbb{Q})\cong\scaleobj{1.5}{\wedge}^{2}[V]^{\perp}\cap[a]^{\perp}\otimes\mathbb{Q}$.
Since $\dim(V)=4$ and the elements of $\mathcal{V}$ have pairwise trivial
algebraic intersection, the set $\operatorname{proj}_{V^{\perp}}(\mathcal{V})$
contains at least six linearly independent elements. Let
$\mathcal{B}\subseteq\mathcal{V}$ be a subset of maximal size with
$\mathcal{B}^{\prime}=\operatorname{proj}_{V^{\perp}}(\mathcal{B})$ linearly
independent. Since we have assumed that each element $v\in\mathcal{V}$ has
trivial algebraic intersection with $[a]$, we have
$\mathcal{B}\subseteq[a]^{\perp}\cap V^{\perp}$. Then since
$\left|\mathcal{B}^{\prime}\right|=6$, Lemma 8.3 applied to the set
$\mathcal{B}^{\prime}$ and the vector space $V^{\perp}\cap[a]^{\perp}$ with
$m=4$ says that the natural map

$\psi:\bigoplus_{\mathcal{V}^{\prime}\subseteq\mathcal{B}^{\prime}:\left|\mathcal{V}^{\prime}\right|=4}\scaleobj{1.5}{\wedge}^{2}(V^{\perp}\cap(\mathcal{V}^{\prime})^{\perp}\cap[a]^{\perp})\otimes\mathbb{Q}\rightarrow\scaleobj{1.5}{\wedge}^{2}(V^{\perp}\cap[a]^{\perp})\otimes\mathbb{Q}$

is a surjection. Then, for any $v\in\mathcal{B}$, we have $v^{\perp}\cap
V^{\perp}=w^{\perp}\cap V^{\perp}$, where $w=\operatorname{proj}_{V}(v)$, so
the natural map

$\psi:\bigoplus_{\mathcal{V}^{\prime}\subseteq\mathcal{B}:\left|\mathcal{V}^{\prime}\right|=4}\scaleobj{1.5}{\wedge}^{2}(V^{\perp}\cap(\mathcal{V}^{\prime})^{\perp})\otimes\mathbb{Q}\rightarrow\scaleobj{1.5}{\wedge}^{2}(V^{\perp}\cap[a]^{\perp})\otimes\mathbb{Q}$

is a surjection. We have
$H_{1}(X_{g}^{V})\cong\scaleobj{1.5}{\wedge}^{2}(V^{\perp}\cap[a]^{\perp})\otimes\mathbb{Q}$
by the claim. Furthermore, by Lemma 8.4, we have
$\operatorname{im}(H_{1}(X_{g}^{V}\cap X_{g}^{v});\mathbb{Q})\rightarrow
H_{1}(X_{g};\mathbb{Q}))\supseteq\scaleobj{1.5}{\wedge}^{2}(V^{\perp}\cap
v^{\perp})$ for any $v\in\mathcal{B}$. Then by substituting in each of these
homology groups in the previous equality, we see that the natural map

$\psi:\bigoplus_{\mathcal{V}^{\prime}\subseteq\mathcal{B}:\left|\mathcal{V}^{\prime}\right|=4}H_{1}(X_{g}^{V}\cap
X_{g}^{\mathcal{V}^{\prime}};\mathbb{Q})\rightarrow
H_{1}(X_{g}^{V};\mathbb{Q})$

is surjective, so the lemma is complete. ∎

We are now almost ready to conclude Section 8.1. We will first prove the
following auxiliary result, and then we will prove Lemma 8.1.

######  Lemma 8.7.

Let $(x,f)\in\bigoplus_{x\in X_{g}^{(1)}}A_{x}$ be a class. There is a linear
combination in $\mathbb{E}_{1,1}^{2}$ given by

$(x,f)=\sum_{i=1}^{m}(x_{i},f_{i})$

such that, for each $1\leq i\leq m$, each $x_{i}$ has a representative
$\widehat{x}_{i}\subseteq\mathcal{C}_{[a]}(S_{g})$ and $f_{i}$ has a
representative
$F_{i}\in\operatorname{Stab}_{\mathcal{I}_{g}}(\widehat{x}_{i})$. These
representatives have the property that there is a subsurface
$\mathcal{T}_{i}\subseteq S_{g}$ with $\mathcal{T}_{i}\cong S_{2}^{1}$,
$F_{i}$ supported on $\mathcal{T}_{i}$, and $\widehat{x}_{i}$ disjoint from
$\mathcal{T}_{i}$.

###### Proof.

Let $\widehat{x}\subseteq\mathcal{C}_{[a]}(S_{g})$ be a representative for
$x$. Let $S^{\prime},S^{\prime\prime}$ be the connected components of
$S_{g}{\ssearrow}\widehat{x}$. We have a surjection
$\mathcal{I}(S^{\prime},S_{g})\times\mathcal{I}(S^{\prime\prime},S_{g})\rightarrow\operatorname{Stab}_{\mathcal{I}_{g}}(\widehat{x})$,
and hence by the Künneth formula we have a surjection

$H_{1}(\mathcal{I}(S^{\prime},S_{g});\mathbb{Q})\oplus
H_{1}(\mathcal{I}(S^{\prime\prime},S_{g});\mathbb{Q})\rightarrow
H_{1}(\operatorname{Stab}_{\mathcal{I}_{g}}(\widehat{x});\mathbb{Q}).$

Hence we may assume that, without loss of generality, $f$ is represented by a
mapping class $F\in\mathcal{I}(S^{\prime},S_{g})$. Furthermore, we may assume
that $g(S^{\prime})\geq 4$. Indeed, if $g(S^{\prime})\leq 3$, then $g\geq
51\geq 7$ implies that there is a 2–cell
$\widehat{\sigma}\subseteq\mathcal{C}_{[a]}(S_{g})$ such that
$\widehat{x}\subseteq\widehat{\sigma}$, and such that for
$\widehat{y},\widehat{z}\subseteq\widehat{\sigma}$ the other two edges, the
connected component of $S_{g}{\ssearrow}\widehat{y}$ that contains
$S^{\prime}$ has genus at least 3, and similarly for $\widehat{z}$. Then since
$g(S^{\prime})\geq 4\geq 3$, Lemma 4.3 says that
$\mathcal{I}(S^{\prime},S_{g})$ is generated by bounding pair maps. Then
bounding pair maps supported on separating curves vanish in
$H_{1}(\mathcal{I}_{g};\mathbb{Q})$ since Dehn twists along separating curves
vanish under $\tau_{g}$. Then Lemma 4.5 says that if
$\iota:S^{\prime}\rightarrow S_{g}$ is the inclusion map, the pushforward
$\mathcal{I}(\iota)_{*}$ in $H_{1}$ is injective, so bounding pair maps
supported on separating curves are trivial in
$H_{1}(\mathcal{I}(S^{\prime},S_{g});\mathbb{Q})$. Hence the class $f$ is a
linear combination of classes represented by bounding pair maps supported on
nonseparating curves contained in $S^{\prime}$, so we may assume that $f$ has
a representative $T_{c,c^{\prime}}\in\mathcal{I}(S^{\prime},S_{g})$ for $c\cup
c^{\prime}$ a bounding pair with $[c]\neq 0$. We now have two cases.

_Case 1: $[c]\neq[a]$_. We first show that we can assume that no connected
component of $S^{\prime}{\ssearrow}(c\cup c^{\prime})$ has genus zero. Suppose
otherwise, so one connected component of $S^{\prime}{\ssearrow}(c\cup
c^{\prime})$ is a subsurface $P$ with $P\cong S_{0}^{4}$. Since we have
assumed that $g(S^{\prime})\geq 4\geq 3$, there is another curve
$c^{\prime\prime}$ disjoint from $c\cup c^{\prime}$ and not equal to $c$ or
$c^{\prime\prime}$ with $[c^{\prime\prime}]=[c]$ and such that both connected
components of $S^{\prime}{\ssearrow}(c\cup c^{\prime\prime})$ have positive
genus. Now, we have
$T_{c,c^{\prime}}=T_{c,c^{\prime\prime}}T_{c^{\prime\prime},c^{\prime}}$, and
both $c\cup c^{\prime\prime}$ and $c^{\prime\prime}\cup c^{\prime}$ satisfied
the desired condition on the genera of connected components. Now, assuming
that both connected components of $S^{\prime}{\ssearrow}(c\cup c^{\prime})$
have positive genus, we can rewrite $T_{c,c^{\prime}}$ as a product of
bounding pair maps $T_{c_{0},c_{1}}T_{c_{1},c_{2}}\ldots T_{c_{n-1},c_{n}}$
such that:

  * •

$c_{0}=c$, $c_{n}=c^{\prime}$, and

  * •

at least one connected component of $S^{\prime}{\ssearrow}(c_{i}\cup c_{i+1})$
has genus one.

Hence we may assume without loss of generality that $c\cup c^{\prime}$ is
supported on a surface of genus one with two boundary components. Then there
is a surface $\mathcal{T}$ that contains $c\cup c^{\prime}$ and is disjoint
from $\widehat{x}$ such that $\mathcal{T}\cong S_{2}^{1}$, as desired.

_Case 2: $[c]=[a]$_. We will reduce to Case 1, by showing that
$[T_{c,c^{\prime}}]\in H_{1}(\mathcal{I}(S^{\prime},S_{g});\mathbb{Q})$ is a
sum $[T_{d,d^{\prime}}]+[T_{e,e^{\prime}}]$ with $d\cup d^{\prime}$ and $e\cup
e^{\prime}$ bounding pairs such that $[d],[e]\neq[a]$. As in Case 1, we may
assume that the connected component of $S^{\prime}{\ssearrow}(c\cup
c^{\prime})$ that does not contain $\widehat{x}$ has genus one. Then by Lemma
4.5, we have
$H_{1}(\mathcal{I}(S^{\prime},S_{g});\mathbb{Q})\cong\scaleobj{1.5}{\wedge}^{3}H_{1}(S^{\prime};\mathbb{Q})$.
By Lemma 4.2, if the connected component $S^{\prime\prime}$ of
$S^{\prime}{\ssearrow}(c\cup c^{\prime})$ that does not contain $\widehat{x}$
has $a,b\in H_{1}(S^{\prime\prime};\mathbb{Z})$ a pair of primitive elements
with $\langle a,b\rangle=1$, then the image of $T_{c,c^{\prime}}$ in
$\scaleobj{1.5}{\wedge}^{3}H_{1}(S^{\prime};\mathbb{Q})$ is
$[a]\wedge[b]\wedge[c]$. Now, choose nonzero primitive $[d],[e]\in
H_{1}(S^{\prime};\mathbb{Z})$ such that $[d]+[e]=[c]$ and
$[d],[e]\in[a]^{\perp}\cap[b]^{\perp}$. Then we take $T_{d,d^{\prime}}$ and
$T_{e,e^{\prime}}$ bounding pair maps in $\mathcal{I}(S^{\prime},S_{g})$ such
that $[T_{d,d^{\prime}}]=[a]\wedge[b]\wedge[d]$ and
$[T_{e,e^{\prime}}]=[a]\wedge[b]\wedge[e]$. ∎

We are now ready to conclude Section 8.1.

###### Proof of Lemma 8.1.

Let $(x,f)\in\bigoplus_{x\in X_{g}^{(1)}}A_{x}$ be a class, where $x\subseteq
X_{g}$ is an edge and $f\in A_{x}$. Let $\widehat{x}$ be a lift of $x$ to
$\mathcal{C}_{[a]}(S_{g})$ such that $a$ is a vertex of $\widehat{x}$. We will
show that the image of $(x,f)$ in $\mathbb{E}_{1,1}^{2}$ is contained in
$\operatorname{im}(\xi)$. By Lemma 8.7, it suffices to prove the result in the
case that $f$ is represented by a bounding pair map
$F\in\operatorname{Stab}_{\mathcal{I}_{g}}(\widehat{x})$ and that there is an
inclusion $\iota:S_{2}^{1}\hookrightarrow S_{g}$ such that $\widehat{x}$ is
disjoint from $\operatorname{im}(\iota)$ and $F$ is supported on
$\operatorname{im}(\iota)$. Let $V=\operatorname{im}(\iota_{*})$. By Lemma
8.6, there is a linear combination in $H_{1}(X_{g}^{V};\mathbb{Q})$ given by

$[x]=\sum_{i=0}^{n}\lambda_{i}[x_{i}]$

such that each $x_{i}$ is contained in $X_{g}^{\mathcal{V}^{\prime}}\cap
X_{g}^{V}$ for some $\mathcal{V}^{\prime}\subseteq\mathcal{V}$ with
$\left|\mathcal{V}^{\prime}\right|=4$. Since each $x_{i}\subseteq X_{g}^{V}$,
we have $f\in A_{x_{i}}$ for each $x_{i}$, and thus there is a linear
combination in $\mathbb{E}_{1,1}^{2}$ given by

$(x,f)=\sum_{i=0}^{n}(x_{i},f).$

Hence it is enough to prove the result in the case that $x\in
X_{g}^{\mathcal{V}^{\prime}}$ for some
$\mathcal{V}^{\prime}\subseteq\mathcal{V}$ with
$\left|\mathcal{V}^{\prime}\right|=4$. Let
$\mathcal{H}(x)=\\{\mathcal{H}_{0}^{x},\mathcal{H}_{1}^{x}\\}$ and, without
loss of generality, assume $V\subseteq\mathcal{H}_{0}^{x}$. Now, if
$\mathcal{V}^{\prime}\not\subseteq\mathcal{H}_{0}^{x}$ for $i=\\{0,1\\}$, we
have $(x,f)\in\mathbb{E}_{1,1}^{2,v}$ for any $v\in\mathcal{V}^{\prime}$.
Otherwise, the vector space $\mathcal{H}_{0}^{x}\otimes\mathbb{Q}$ and the set
$\mathcal{V}^{\prime}$ satisfy the hypotheses of Lemma 4.8. Hence there is a
linear combination

$f=\sum_{j=1}^{m}\lambda_{j}f_{j}$

where each $f_{j}\in\scaleobj{1.5}{\wedge}^{3}(\mathcal{H}_{0}^{x}\cap
v_{i}^{\perp})\otimes\mathbb{Q}$ for some $v_{i}\in\mathcal{V}^{\prime}$.
Hence in $\mathbb{E}_{1,1}^{2}$, we have

$(x,f)=\sum_{j=1}^{m}\lambda_{j}(x,f_{j})$

with each $(x,f_{j})\in\mathbb{E}_{1,1}^{2,v}$ for some
$v\in\mathcal{V}^{\prime}\subseteq\mathcal{V}$. Therefore we have
$(x,f)\in\operatorname{im}(\xi)$ as desired. ∎

###  8.2. The proof of Theorem B

We begin by showing that $\mathbb{E}_{1,1}^{2}$ is finite dimensional.

###### Proof of Proposition 1.2.

Let $\mathcal{V}\subseteq[a]^{\perp}$ be a set of 9 elements such that
$\mathcal{V}$ can be represented by the connected components of a
nonseparating multicurve $M\subseteq S_{g}{\ssearrow}a.$ Let
$G=\operatorname{im}(\operatorname{Mod}(S_{g}{\ssearrow}a)\rightarrow\operatorname{Sp}(2g,\mathbb{Z}))=\operatorname{Sp}\left([a]^{\perp},\mathbb{Z}\right)$
and let $\mathcal{F}=\\{T_{v}:v\in\mathcal{V}\\}$. We will show that

$(G,\mathcal{F},\mathbb{E}_{1,1}^{2})$

is a representation of transvective type. Hypotheses (1) and (4) of the
definition of representation of transvective type follow from Lemma 2.2.
Hypothesis (2) is the content of Lemma 7.1 and hypothesis (3) is the content
of Lemma 8.1. Hence the proposition follows by Proposition 2.1. ∎

We are now ready to complete Section 8.

###### Proof of Theorem B.

Let $g\geq 51$ and $\mathbb{E}_{*,*}^{*}$ denote the equivariant homology
spectral sequence for the action of $\mathcal{I}_{g}$ on $C_{[a]}(S_{g})$. By
a result of the author [21, Theorem A], $\mathbb{E}_{p,q}^{r}$ converges to
$H_{2}(\mathcal{I}_{g};\mathbb{Q})$ for $p+q=2$. Hence it suffices to show
that $\mathbb{E}_{2,0}^{2}$ and $\mathbb{E}_{1,1}^{2}$ are finite dimensional.
The vector space $\mathbb{E}_{2,0}^{2}$ is finite dimensional by Proposition
1.1 and $\mathbb{E}_{1,1}^{2}$ is finite dimensional by Proposition 1.2. ∎

##  9\. The proof of Theorem A

We now prove Theorem A. We will begin by proving Lemma 9.1, which is an
alternate proof of a corollary of a result of Kassabov and Putman result [14,
Theorem A]. We will then use Theorem B along with Proposition 2.1 to prove
Theorem A.

######  Lemma 9.1.

Let $g\geq 3$. The vector space
$H_{0}(\operatorname{Sp}(2g,\mathbb{Z});H_{2}(\mathcal{I}_{g};\mathbb{Q}))$ is
finite dimensional.

###### Proof.

Let $\mathbb{E}_{p,q}^{r}$ be the Leray–Serre spectral sequence associated to
the short exact sequence

$1\rightarrow\mathcal{I}_{g}\rightarrow\operatorname{Mod}(S_{g})\rightarrow\operatorname{Sp}(2g,\mathbb{Z})\rightarrow
1.$

The vector space $H_{2}(\operatorname{Mod}(S_{g});\mathbb{Q})$ is finite
dimensional [8, Section 5.4]. Then since $H_{1}(\mathcal{I}_{g};\mathbb{Q})$
is finite dimensional [12], the modules
$\mathbb{E}_{p,q}^{2}=H_{p}(\operatorname{Sp}(2g,\mathbb{Z});H_{q}(\mathcal{I}_{g};\mathbb{Q}))$
are all finite dimensional for $q\leq 1$ [26, Corollary 3]. Hence both the
image and the kernel of the pushforward map

$H_{0}(\operatorname{Sp}(2g,\mathbb{Z});H_{2}(\mathcal{I}_{g};\mathbb{Q}))\rightarrow
H_{2}(\operatorname{Mod}(S_{g});\mathbb{Q})$

are finite dimensional, so
$H_{0}(\operatorname{Sp}(2g,\mathbb{Z});H_{2}(\mathcal{I}_{g};\mathbb{Q}))$ is
finite dimensional. ∎

We now prove the main result of the paper.

###### Proof of Theorem A.

Let $T_{[a]}$ denote transvection along $[a]$. We will show that

$\left(\operatorname{Sp}(2g,\mathbb{Z}),T_{[a]},H_{2}(\mathcal{I}_{g};\mathbb{Q})\right)$

is a representation of transvective type. Hypothesis (1) follows from Lemma
2.2. Hypothesis (2) follows from Lemma 9.1. Hypothesis (3) is the content of
Theorem B. Hypothesis (4) also follows from Lemma 2.2. Therefore, by
Proposition 2.1 we have that $H_{2}(\mathcal{I}_{g};\mathbb{Q})$ is finite
dimensional. ∎

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