#  Role of the companion lensing galaxy in the _CLASS_ gravitational lens
B1152+199

M. Zhang,1,2 Q. Yuan,1,3 J.-Y. Liu,1,3 L. Zhang4
1Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150 Science
1-Street, Urumqi 831001, China
2Key Laboratory for Radio Astronomy, Chinese Academy of Sciences, 2 West
Beijing Road, Nanjing 210008, China
3University of Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049,
China
4College of Big Data and Information Engineering, Guizhou University, Guiyang
550025, China E-mail<EMAIL_ADDRESS>
(10 January 2022)

###### Abstract

We reinvestigate the Cosmic Lens All-Sky Survey (_CLASS_) gravitational lens
B1152+199 using archived Hubble Space Telescope (_HST_) data and Very Long
Baseline Interferometry (_VLBI_) data. A consistent luminosity ratio within
effective radius between the host galaxy and the X-galaxy is measured from
_HST_ tri-band images, which leads to a mass ratio between the two galaxies as
$r_{b}\sim 2$. To determine the role of the X-galaxy in the lens system, we
modelled the dual-lens system with constraints from the _VLBI_ -resolved jet
components and the _HST_ images. The 8.4-GHz global-_VLBI_ data currently
provide the most stringent constraints to the mass model, especially to the
radial power-law slope. The optimized models for this two-image three-
component radio lens favour a steeper-than-isothermal inner slope. The jet
bending in image B was also investigated and it turned out to be rather a
misalignment than a curvature. The goodness of fit indicates that the role of
the X-galaxy is crucial in the lens system if three pairs of resolved jet
components are to be fitted. When we imported the optimal model from radio
constraints to optical modelling with the _HST_ tri-band data, the
optimization kept the consistency of the optimal model and successfully
reproduced the features observed in the _HST_ images. This implies that the
diffuse emission discovered in the _HST_ images is actually a detection of the
secondary lensing effects from the companion lens.

###### keywords:

gravitational lensing: strong – galaxies: jets – galaxies: structure – dark
matter

††pubyear: 2022††pagerange:  Role of the companion lensing galaxy in the
_CLASS_ gravitational lens B1152+199–A.4

##  1 Introduction

Gravitational lensing is an effect of light deflection caused by gravitational
fields due to the distribution of lensing masses. According to the different
lensing strengths, it can be categorized as strong or weak regimes, where the
multiple images or distorted shapes of background galaxies are produced
respectively. The strong gravitational lensing provides a dynamics-independent
method to infer substructures of mass distribution in lensing galaxies,
regardless of whether they are dark or luminous. While the Lambda cold dark
matter (_$\Lambda$ CDM_) cosmological model has enjoyed great success in
predicting large-scale structures in the cosmic microwave background (_CMB_)
observations (Hinshaw et al., 2007), $N$-body simulations in this paradigm
tend to over-predict the amount of substructures in smoothed dark matter halos
(Navarro et al., 1996) that is yet to be confirmed by observational evidence,
especially at galactic scales. Recently, there has been some progress that a
large number of dwarf galaxies in the Milky Way have been detected with the
Sloan Digital Sky Survey (_SDSS_) and the Dark Energy Survey (_DES_) (Bechtol
et al., 2015). The galactic-scale hydrodynamical simulations have also been
done to investigate the subhalo existence (Despali et al., 2016; Despali &
Vegetti, 2017), as well the line-of-sight structures (Gilman et al., 2018;
Despali et al., 2018). The tension between _CDM_ model and observations seems
mitigated with more newfound Milky Way dwarfs. However, the underdetected
substructures in extragalactic early-type galaxies still leave the
discrepancy.

Generally, substructures can be inferred from mass modelling, provided the
discrete lensing mass can be verified. The direct observable phenomena include
the flux-ratio anomalies (Mao & Schneider, 1998; Bradač et al., 2002; Keeton
et al., 2003; Biggs et al., 2004; Keeton et al., 2006) and surface brightness
anomalies (Vegetti et al., 2012), as well as the time-delay anomalies which is
more difficult to detect. The flux-ratio anomaly happens in the event of a
caustic crossing in a quad-image system, which violates the cusp or fold
relations, while the surface brightness anomaly takes place on the arc or
Einstein ring. It should be noticed that there is evidence of galactic-scale
discs may account for the flux-ratio anomalies from both the observational and
the theoretical perspectives (Hsueh et al., 2016, 2017; Hsueh et al., 2018;
Gilman et al., 2018). It is also possible to observe morphological distortions
due the presence of a substructure, i.e. the gravitational millilensing
(Wambsganss & Paczynski, 1992; Metcalf, 2002). Then, indirect inference can be
drawn about the lensing mass, either via a mass or potential reconstruction
with non-parametric methods, such as extended structures (arcs and rings)
reconstruction via Maximum Likelihood (_ML_) methods (Koopmans, 2005, e.g.)
and point-like structure ensemble modelling (Saha & Williams, 1997). There has
been extensive work on non-parametric gravitational potential reconstruction
from lensing (Vegetti & Koopmans, 2009; Vegetti et al., 2010, 2012; Vegetti &
Vogelsberger, 2014; Vegetti et al., 2018; Ritondale et al., 2019).

There have been several luminous cases in which faint dwarf galaxies have been
found, and whereby models were fit to the observables: one is the anomalous
flux-ratio in B2045+265 (McKean et al., 2007), another is the single
perturbing mass (object X) model in MG0414+0534 (Ros et al., 2000; Kochanek,
2006; MacLeod et al., 2013; Stacey & McKean, 2018), and yet another possible
case is MG2016+112 (Koopmans et al., 2002; More et al., 2009; Spingola et al.,
2019b). To be noticed, there are still interesting anomaly cases like
B1422+231 (Patnaik et al., 1992) in which the perturbing object has not been
identified. Additionally, Cohn & Kochanek (2004) have shown that image
multiplicities in lenses were mostly produced by substructure-perturbed
systems, and many previously discovered simple lens system were detected with
satellites (Ros et al., 2000; More et al., 2009; MacLeod et al., 2013;
Spingola et al., 2018).

The gravitational lens B1152+199 was discovered through the _CLASS_ survey
(Myers et al., 1999) and investigated (Rusin et al., 2002; Metcalf, 2002) as a
primary dual-image lens with a possible substructure inference. Higher-
frequency radio interferometric observations have also been studied recently
(Asadi et al., 2020) to testify the bending significance of the jet. In Rusin
et al.’s investigation, there is a faint blob identified as the X galaxy west
to the lens galaxy. Due to the inadequate number of observables, there is a
degeneracy between the mass profile and the perturber mass. Therefore, no
significant effect of the X galaxy on the model could be addressed.

In this paper, we reinvestigate the optical and radio interferometric
observations of B1152+199 to date and present the data analysis and lens
modellings with regard to the impacts of detected arc-shaped diffuse emission
and multi-component images. The role of the satellite galaxy in the lensing
configuration is prudently examined, which was not shown in previous
literature. Section 2 analyses the _HST_ and _VLBI_ data and raises the
question about the diffuse emission and the role of the X-galaxy. Then the
answers to the question are inferred by the mass modelling with both point and
image constraints in Section 3. Section 4 discusses the observation and
modelling results and their implications. Section 5 summarizes our findings
and proposes future investigations.

##  2 Data

###  2.1 _HST_ observations and data reduction

Since the lensing galaxy has not been proved to be radio-loud with
interferometric detections, the optical observations then become very useful
to provide extra constraints on the lens position and mass profile, as well as
to spectroscopically determine the redshifts of images and lenses. To study
the lensing configuration of B1152+199, there were several HST observations
carried out including imaging with the Wide Field and Planetary Camera 2
(_WFPC2_) using $V$-band (F555W) and $I$-band (F814W) filters in 2000 (8248,
PI: Wilkinson), and imaging with the Near Infrared Camera and Multi-Object
Spectrometer (_NICMOS_) using $H$-band (F160W) filter in 2004 (9744, PI:
Kochanek). The lensing galaxy (G) and a satellite galaxy (X) were detected
with _HST/WFPC2_ $I$-band image of this dual-image (A and B) system, and the
redshifts of the lens and the source were measured with the Keck II telescope
as $z_{l}$=0.439 and $z_{s}$=1.019 respectively (Rusin et al., 2002). The CfA-
Arizona Space Telescope LEns Survey of gravitational lenses (_CASTLES_)
(Kochanek et al., 1999) has maintained an up-to-date catalogue of the _HST_
optical lenses with introductory measurements including B1152+199.

The mass ratio between the lensing galaxy and the X-galaxy is conventionally
derived from the Faber-Jackson relation ($\sigma\propto L^{1/4}$) and the
mass-velocity dispersion relation ($b\propto\sigma^{2}$), together with the
magnitude relation ($F_{1}/F_{2}\simeq 2.512^{\Delta m}$), as done by Rusin et
al. (2002). Since the photometry is essential to speculate the mass scale of
the X-galaxy, we re-measured the _WFPC2_ $V$/$I$-band and _NICMOS_ $H$-band
photometries of both the lens galaxy and the X-galaxy with the drizzled image
data from Hubble Legacy Archive 111 Based on observations made with the
NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive,
which is a collaboration between the Space Telescope Science Institute
(STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA)
and the Canadian Astronomy Data Centre (CADC/NRC/CSA).. Both observations had
a long total exposure time ($V$/$I$-band: 2000 sec, $H$-band: 5311 sec), but
the integration times are not as long as the total the exposure time, since
the _HST_ produces multiple frames for removing cosmic rays, bad pixels and so
that the point spread function (_PSF_) can be better sampled from dithered
observations. We found it is difficult to remove the point spread function
(_PSF_) effect in the image merely by fitting the bright point source with the
convolution kernel. This is in part due to the dynamic range limitations from
having a coarsely sampled _PSF_ , as well as the _PSF_ used will likely not
match the colour of the lensed emission, and therefore, the luminosity
weighted width of the _PSF_ as a function of frequency will be poorly known.
This means there is a mismatch between the _PSF_ and the point-source emission
from quasars.

We composed the _NICMOS_ $H$-band and _WFPC2_ $I$\- and $V$-band images into
the RGB channels and formed a pseudo-colour image as seen in Fig. 1. From the
RGB image, we can notice there are a noticeable amount of emissions around the
lens galaxy and the X-galaxy, especially in the $I$\- and $H$\- band images.
The intrinsic emissions from the galaxies are blended with the spread of the
lensed light from both A and B images. We have utilized the tinytim222The
Hubble Space Telescope PSF generating program by John Krist at STSCI. (Krist,
1993) to generate the _PSF_ of _WFPC2_ and _NICMOS_ , then convolve and
subtracted the models in the _HST_ images retrieved from the Hubble Legacy
Archive (_HLA_). The image-plane model fitting is carried out with the Galfit
devised by Peng et al. (2002, 2010). The foreground galaxies were fitted
against the Sérsic model and the background quasar images were fitted by the
_PSF_ kernel. Obvious the primary image was not representable by a simple
_PSF_ due to saturation, and there is a tendency for the X-galaxy to fit the
spurious emissions spread from the lensed images with a bigger effective
radius. So we masked a significant amount of pixels around the galaxies
including all lensed images and the surrounding diffuse emission to reduce the
_PSF_ effect. The galfit image models and residuals of tri-band data are shown
in Fig. 2. We here list the newly measured photometry and derived mass ratios
in Table 1. Their error estimates are the projected covariances as quoted in
Galfit (Peng et al., 2002). It is noticeable that the new mass ratio between
the host galaxy and the satellite is consistently around 2 across three bands,
rather than ranging broadly from 3 to 5 in the previous measurements. This
leads to different estimations about the X-galaxy’s total mass and conceivably
affects the lens modelling. Though it is arguable whether the optical
luminosity of the lensing galaxy is compromised with uncertainty due to dust
extinction, we note there the positional angle of fitted Sérsic profiles of
the host galaxy and the perturber here in Table 2.

Table 1: Photometry and derived mass ratios between lens and X-galaxy from the _HST_ archived images. Here we take the conventional STmag system for _HST_ observations. | $V$ | $I$ | $H$
---|---|---|---
G mag | 20.88$\pm$0.17 | 20.50$\pm$0.09 | 19.12$\pm$0.02
X mag | 23.25$\pm$0.38 | 22.14$\pm$0.76 | 20.52$\pm$0.14
$b^{\prime}_{G}/b^{\prime}_{X}$ | 2.98$\pm$0.58 | 2.13$\pm$0.74 | 1.90$\pm$0.12
Table 2: Fitted light profile parameters of Sérsic profiles of the galaxies in three _HST_ bands. Values are shown in ($R_{e}$, $n$, $q$, $\phi_{q}$) tuples. The tuples below each parameter tuple shows the quoted covariance errors. Note that the ellipticity $e$ equals to 1-$q$ and the position angles is measured from north to east. | Host galaxy | X-galaxy
---|---|---
$V$ | (0.44”, 0.89, 0.37, -54.24∘) |  (0.17”, 0.62, 0.58, -46.85∘)
|  $\pm$(0.08”, 0.13, 0.03, 1.65∘) |  $\pm$(0.06”, 0.39, 0.11, 12.27∘)
$I$ | (0.35”, 1.39, 0.46, -51.75∘) |  (0.29”, 1.68, 0.63, -52.95∘)
|  $\pm$(0.03”, 0.12, 0.01, 1.64∘) |  $\pm$(0.23”, 1.00, 0.10, 13.48∘)
$H$ | (0.29”, 1.72, 0.53, -51.38∘) |  (0.17”, 1.14, 0.85, -65.60∘)
|  $\pm$(0.01”, 0.05, 0.01, 1.30∘) |  $\pm$(0.03”, 0.25, 0.05, 15.62∘)

Rusin et al. (2002) had already mentioned the detected diffuse emission to the
west of image B in the $I$-band image. To quantitatively estimate whether this
extended structure will play a role in the overall lens modelling, we need
more observational evidence. As we see in _HST_ multiband images, the diffuse
west to image B is barely visible in $V$-band image, when we go to longer
wavelengths like I and H bands, it starts to be noticeable. The diffuse
emission roughly runs along the tangential direction around the X-galaxy which
may form an arc shape. In the $I$-band image, the arc shape is resolved into
three faint blobs which resemble a lensing case of the cusp-crossing quad.
Comparing the cleaned images to the original drizzled images, we can see that
some emissions are restored around the arc region. It could be possible that
the arc-shaped emission is purely produced by surrounding interstellar dust,
however the recent spectroscopic study on the damped Ly$\alpha$ absorption
together with the metallicity estimation from the X-ray observation (Dai et
al., 2020) found no adequate evidence for the dust existence and thus
disfavoured this scheme.

Figure 1: The _RGB_ images of tri-band _HST_ optical observations. The first
three panels from left to right are the _HST/WFPC2_ $V$-band, $I$-band and
_HST/NICMOS_ $H$-band images, which are assigned to the B, G and R channels,
respectively, according to their wavelengths. The last panel is the composed
three-channel _RGB_ image. The hue is set to a black-body profile with a
logarithmic scale.

Figure 2: The image-plane light profile subtraction. From top to bottom, the
rows are the _HST/WFPC2_ $V$-band, $I$-band and _HST/NICMOS_ $H$-band images,
respectively. From left to right, the columns show the _PSF_ models, the
original images, the convolved image models, and the residual images after the
subtraction, respectively. The intensity is plotted on logarithmic scale with
pseudo colours. The pixel ranges are adjusted to show optimal contrast. The
primary and the secondary lensed images, the main lens galaxy, the perturber
galaxy, the diffuse arc and the resolved quad are labelled as A, B, G, X, Arc
and R1, R2, R3 respectively.

###  2.2 _VLBI_ observations and data reduction

Precisely measured observables of lensed images are crucial for lens
modelling, especially when the substructures are to be identified therein. The
Very Long Baseline Interferometry (_VLBI_) is currently the most powerful
method to get highly resolved images for extragalactic radio sources. Since
the light extinction is entirely negligible at centimetre wavelengths, the
positional and flux constraints obtained from radio interferometry can be used
to determine the lens model with more accuracy compared to those in the
optical bands. High-resolution _VLBI_ images thus constrain the lens model the
most and have been utilized to detect deviations from a smooth mass model (Ros
et al., 2000; Bradač et al., 2002; Biggs et al., 2004; Spingola et al., 2018),
galactic central profile (Zhang et al., 2007; Quinn et al., 2016), Hubble
constant (Biggs et al., 2003; Wucknitz, 2004; Wucknitz et al., 2004), jet
proper motion (Spingola et al., 2019b) and new lens discovery survey (Spingola
et al., 2019a).

####  2.2.1 C-band VLBI observations

There have been several _VLBI_ observations carried out on the target of
B1152+199 in recent years. Hence, we collected and reduced the archived 5-GHz
data from the _VLBA_ 333The Very Long Baseline Array is a radio
interferometric network run by the National Radio Astronomy Observatory (NRAO)
and the _EVN_ 444European VLBI Network is coordinated by the Joint Institute
for VLBI in Europe (JIVE) in order to obtain milli-arcsecond-scale astrometry
of the dual-image lensing systems. The data set includes the _VLBA_
observation (BB133, PI: Biggs) from 2001, the global-_VLBI_ observation
(GJ010, PI: Jackson) from 2003 and the High-Sensitive Array (_HSA_)
observation (BW084, PI: Winn) from 2005. All data were recorded with four
contiguous 8-MHz intermediate frequencies (_IF_) and correlated with a channel
resolution of 0.5 MHz and a time resolution of 2 s. The global-_VLBI_
observation (GJ010) was recorded only with left circular polarization (_LCP_),
while the other two were obtained with dual circular polarization. The
effective on-source integration time of BB133, GJ010 and BW084 was 3.4, 7.7
and 3.5 hours respectively.

To cross-check the observations, we re-reduced the visibility data of the
three epochs to obtain high-dynamic-range images. The data calibration was
carried out with aips555The Astronomical Image Processing System distributed
by NRAO (National Radio Astronomy Observatory).. Standard procedures for data
inspection and flagging, as well as calibration of amplitude, phase and
bandpass, and multi-source splitting, have been applied to the three sets of
VLBI data using appropriate aips tasks. The self-calibration and deconvolution
procedures were carefully used in each case, especially when mapping the jet
in image B. To maintain the highest resolution and suppress extended features,
we cleaned the uniformly weighted data before self-calibration. The three-
epoch cleaned images are shown in Fig. 3. All images are restored to the same
3$\times$1.5 mas2 beam, and the root mean squared (RMS) map noise was cleaned
down to the noise level respectively. The jet in image A has been resolved
into at least two components, while in image B there is only one
distinguishable component. With the highest sensitivity and resolving power of
the _HSA_ , we can see that there is a third jet component just barely
resolved between the outermost and innermost components in image A. There were
up to three jet components resolved in image A, while there was only one
unresolved jet emission protruding out in image B. The discrete jet components
were model-fitted in the image plane with aips task jmfit and the errors were
estimated by the beam size over signal-to-noise ratio, as shown in Table 3.

It is noticeable that the contour lines of the jet in image B show a slight
bending trend, even when the contour lines are raised over 100 ${\rm\mu}$Jy
bm-1. The slight bending impression is mainly guided by outer contours, so it
might come from a non-circular restoring beam and the presence of any
unresolved asymmetric structures. To investigate the fidelity of observing a
bent jet, we use a flip-subtraction method to check the collimation of a jet
(Section A.1). The result shows the jet bending is merely marginal at the
first contour level. Metcalf (2002) has elaborately shown that this subtle
deflection of the jet can be a strong indication of the existence of local
substructures. Undoubtedly, it is possible to add a local compact perturber to
account for the local image morphology without changing the overall lensed
image properties. Recently Asadi et al. (2020) has also confirmed that the jet
bending in image B is non-significant with higher-frequency (22 GHz and 8.4
GHz) _VLBI_ observations.

Figure 3:  Three-epoch 5-GHz _VLBI_ observations of B1152+199. From top to
bottom, the A and B images are of observations BB133 (2001), GJ010 (2003) and
BW084 (2005), respectively. The data have been uniformly weighted. The
restoring beam size is 3$\times$1.5 mas2 at a position angle of $-9^{\circ}$.
Contours in the map are plotted at multiples of -1, 1, 2, 4, 8, 16, 32, 64,
128, 256, 512, 1024, 2048 $\times$ $3\sigma$ where $\sigma$ is the local RMS
noise. The off-source background noise of each observation is 75, 31 and 21
${\rm\mu}$Jy bm-1respectively. The marginally resolved jet components in image
A are labelled from the core as A1, A2, A3, and the unresolved jet in image B
is labelled ad Bjet.

####  2.2.2 X-band global-_VLBI_ observation

The 8.4-GHz global-_VLBI_ observation (GA036, PI: Asadi) of B1152+199 from
2015 was carried out by Asadi et al. (2020) . It was a full-track observation
with 22 telescopes from both _EVN_ and _VLBA_. The effective on-source
integration time of GA036 was around 11 hours. In the original paper of Asadi
et al. (2020), the radio jet in image B was resolved into 3 discrete
components, which match to the jet components in image A. However, the authors
only used the image for morphological comparison between the curvature
translations in the A and B images. To extract the observables from each
component as mass modelling constraints, we re-reduced the data and produced
maps of the A and B images from the archived data. The data came out from the
correlator as two passes of pipeline-calibrated $uv$ data. We found the phase
centers of the two passes offset and used aips task uvfix to rectify it and
combined the two data set together with task dbcon. Multiple rounds of self-
calibrations of solution intervals from two hours down to one minute were
applied to solve the phase error and achieve the highest dynamic range. The
self-calibration task we used is scimg. The uniform weighting leads to a
mottled feature as seen in Asadi et al. (2020), so we use the Briggs weighting
scheme with a robustness parameter equal to 4 (Briggs, 1995). The images were
restored with a 2.1$\times$0.7 mas2 beam and the background noise is cleaned
down to 29 ${\rm\mu}$Jy bm-1. The cleaned images are shown in Fig. 4 and the
observables measured from image-plane model-fitting with a 2-dimensional
Gaussian are shown in Table 3. The cleaning and image-plane model-fitting are
carried out with the aips task imagr and jmfit. We can notice from Fig. 5 that
the jet position angle seems not changed over a decade.

Figure 4:  8.4-GHz _VLBI_ observations of B1152+199. The restoring beam size is 2.1$\times$0.7 mas2 at a position angle of $172^{\circ}$. Contours in the map are plotted at multiples of -1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 $\times$ $3\sigma$ where $\sigma$ is the local RMS noise ($\sim$34 ${\rm\mu}$Jy bm-1). The red circles indicate the fitted image models of individual components. The jet components are sequentially labelled outward from each core as A1, A2, A3, and B1, B2, B3 respectively.  Table 3:  The observable constraints of B1152+199 from the 5-GHz _HSA_ and 8.4-GHz _VLBI_ observations. The components are fitted by a 2-dimensional Gaussian model in the image plane. The flux is measured as the integral intensity of the component. A and B are the core images, A1, A2, A3 and B1, B2, B3 are the model-fitted jet components in Fig. 4, while Bjet denotes the unresolved jet component in Fig. 3.  HSA | Flux | RA offset | DEC offset
---|---|---|---
(5 GHz) | (mJy) | (mas) | (mas)
A | 33.41 $\pm$ 0.20 | 0.00 $\pm$ 0.01 | 0.00 $\pm$ 0.01
A1 | 3.91 $\pm$ 0.41 | 6.51 $\pm$ 0.12 | 0.32 $\pm$ 0.11
A2 | 1.92 $\pm$ 0.56 | 9.72 $\pm$ 0.16 | 0.33 $\pm$ 0.18
A3 | 8.73 $\pm$ 1.15 | 15.82 $\pm$ 0.25 | 0.75 $\pm$ 0.20
B | 11.91 $\pm$ 0.21 | 935.35 $\pm$ 0.01 | -1245.45 $\pm$ 0.02
Bjet | 5.07 $\pm$ 2.01 | 939.21 $\pm$ 0.58 | -1249.62 $\pm$ 0.65
g-VLBI | Flux | RA offset | DEC offset
(8.4 GHz) | (mJy) | (mas) | (mas)
A | 33.95 $\pm$ 0.07 | 0.000 $\pm$ 0.001 | 0.000 $\pm$ 0.001
A1 | 1.24 $\pm$ 0.16 | 8.33 $\pm$ 0.17 | 0.35 $\pm$ 0.11
A2 | 1.14 $\pm$ 0.18 | 11.27 $\pm$ 0.16 | 0.47 $\pm$ 0.19
A3 | 3.63 $\pm$ 0.29 | 17.69 $\pm$ 0.19 | 0.94 $\pm$ 0.07
B | 10.78 $\pm$ 0.07 | 935.590 $\pm$ 0.001 | -1245.620 $\pm$ 0.003
B1 | 0.48 $\pm$ 0.11 | 937.36 $\pm$ 0.10 | -1248.23 $\pm$ 0.18
B2 | 0.58 $\pm$ 0.13 | 938.78 $\pm$ 0.13 | -1248.79 $\pm$ 0.19
B3 | 1.31 $\pm$ 0.16 | 940.54 $\pm$ 0.10 | -1250.66 $\pm$ 0.15

Figure 5:  Jet component positions obtained from 5-GHz _HSA_ and 8.4-GHz
_VLBI_ observations. The blue dots are measured from the 5-GHz data and the
red dots are measured from the 8.4-GHz data. The dots are connected by
polylines to indicate the direction of the jet stream.

##  3 Lens mass modelling

###  3.1 Modelling of _VLBI_ data

A typical softened power-law ellipsoid (_SPLE_) model with elliptical symmetry
can be written as a two-dimensional convergence:

$\kappa(x,y)=\frac{b^{2-\alpha}}{2(s^{2}+x^{2}+y^{2}/q^{2})^{1-\alpha/2}},$
(1)

where $\alpha$ is the power-law index, and $\alpha=1$ for the isothermal case;
$b$ is the mass scaling factor, i.e. the so-called Einstein radius for
circular symmetry; $s$ is the core size (for _SIE_ $s$=0); and $q$ is the
axial ratio. It has been clearly shown from the optical survey from the Sloan
Lens Advanced Camera for Survey (_SLACS_) (Koopmans et al., 2006), the lensing
galaxy most likely has an isothermal mass profile. Besides, the core parameter
is proved to be insensitive to outer image constraints of a dual-image lens
(Zhang et al., 2007). Thus, in our modelling, we fixed $\alpha=1$ and $s=0$ to
check in detail to see how the positional observables gradually constrained
the primary singular isothermal ellipsoid model.

The _HST/WFPC2_ $I$-band and _HST/NICMOS_ $H$-band image have both already
shown the presence of a lensing galaxy as well as the perturber galaxy X. As
such, we defined an object function with complete constraints from
observables, including the discrepancies between the observed and predicted
image positions and fluxes, the galaxy positions and the mass ratio of the
perturber galaxy to the host galaxy, to minimize in the image plane as:

$\displaystyle\chi^{2}=$
$\displaystyle\sum_{i=A,B,A_{1},B_{1},...}\left[\frac{(x^{\prime}_{i}-x_{i})^{2}}{\sigma^{2}_{xi}}+\frac{(y^{\prime}_{i}-y_{i})^{2}}{\sigma^{2}_{yi}}+\frac{(f^{\prime}_{i}-f_{i})^{2}}{\sigma^{2}_{fi}}\right]$
(2)
$\displaystyle+\sum_{g=G,X}\left[\frac{(x^{\prime}_{g}-x_{g})^{2}}{\sigma^{2}_{xg}}+\frac{(y^{\prime}_{g}-y_{g})^{2}}{\sigma^{2}_{yg}}\right]$
$\displaystyle+\frac{(b_{1}/b_{2}-r_{b})^{2}}{\sigma^{2}_{r_{b}}},$

where the observed image positions and fluxes are noted as ($x^{\prime}_{i}$,
$y^{\prime}_{i}$, $f^{\prime}_{i}$), and the observed galaxy positions are
noted as ($x^{\prime}_{g}$, $y^{\prime}_{g}$); all the modelled counterparts
are noted without the prime symbol. The mass ratio derived from photometry is
also used as a constraint, which is reflected in the total $\chi^{2}$. The
ratio between the mass scales of the lensing galaxy ($b_{1}$) and the X-galaxy
($b_{2}$) is noted as $r_{b}$. The denominators are the observational errors,
noted as $\sigma$ with a subscript corresponding to their nominators
respectively. The $\chi^{2}$ in the image plane explicitly defines the
variance between the observational and theoretical data. The constraints from
the multiple jet components are logically complete. So other lensing
properties like image parities, jet curvatures and position angles can be
derived from the alignment of the jet components. We use the direct
observables rather than the derived constraints in our study because the
derived constraints can have degeneracy and introduce ambiguity. For example,
the same positional angle can be formed with a reverse alignment of the
components which have opposite parity. If we consider the number of degrees of
freedom (_NDF_ 666The _NDF_ calculation: $NDF=N_{Cons}-N_{Para}$
Para: $\underbrace{\textrm{$b$ \, ($e$, $\phi_{e}$) \, ($\gamma$,
$\phi_{\gamma}$)}}_{\textrm{$\times N_{lens}$}}$
$\underbrace{\textrm{($x_{s}$, $y_{s}$, $f_{s}$)}}_{\textrm{$\times
N_{src}$}}$
Cons: $\underbrace{\textrm{($x_{i}$, $y_{i}$, $f_{i}$)}}_{\textrm{$\times
N_{img}\times N_{src}$}}$ $\underbrace{\textrm{($x_{g}$,
$y_{g}$)}}_{\textrm{$\times N_{lens}$}}$
$\underbrace{\textrm{$r_{b}$}}_{\textrm{$\times(N_{lens}-1)$}}$
N.B. the lensing galaxy positions are considered as observable constraints in
this paper.=$\nu$), the reduced $\chi^{2}$ ($\bar{\chi}^{2}$=$\chi^{2}/\nu$)
is approaching 1, if the extent of the match between observations and
estimates is in accord with the error variance.

The X-band global-VLBI observation currently provides the highest resolution
of this lens system. As the jet components in both image A and image B were
resolved at 8.4 GHz, it offers very strong positional and flux constraints to
the mass model. The modelling was carried out with the comprehensive modelling
program gravlens a.k.a. lensmodel developed by Keeton (2001). Since the
radially aligned jet components in image B are sensitive to the radial mass
profile, we have tried to optimize the mass model using _SIE_ and _SPLE_ plus
external shear, with and without a second perturber respectively. The
optimized model parameters are shown in Table 4. We can notice that, the
presence of the X-galaxy as a second perturber does give a better fit, while
if the slope of the power-law model is allowed to vary, the fit improves but
not significantly. The optimal power-law profiles indicate the mass
distribution is steeper than isothermal. With the presence of a second
perturber, the profile can be steeper, as the mass distribution is sort of
compensated by the perturber. The critical curves and caustics of the
optimized power-law models are shown in Fig. 6. We can see from the curve
configurations that there are degeneracies between ellipticity, external shear
and the present mass of the X-galaxy. As the goodness of fit indicated, the
X-galaxy is needed to fit the constraints from the three resolved jet
components in the 8.4-GHz _VLBI_ images. The best fit is given by a power-law
model with an _SIS_ perturber, with a reduced $\chi^{2}$ close to unity. The
predicted image and source configurations are shown in Fig. 7. As we can see,
all modelled jet components are aligned linearly very well, except the B1
component has a slight offset from the alignment.

Table 4: The optimized parameters of the isothermal and power-law mass models. _SPLE_ is the softened power-law ellipsoid. _SIS_ is the singular isothermal spheroid. $b^{\prime}$ is the modified mass scale (refer to A.3). $(x,y)$ is the mass profile centroid, while the subscripts $g$ and $X$ denote the host galaxy and the X-galaxy respectively. $e$ is the ellipticity and $\phi_{e}$ is its position angle. $\gamma$ is the external shear and $\phi_{\gamma}$ is its position angle. $\alpha$ is the power-law slope. The _NDF_ is noted as $\nu$.  SIE + $\gamma$ |  |
---|---|---
$b^{\prime}$ = 0.7536 |  |  $\chi^{2}$ = 21.73
$x_{g}$ = 0.5396 |  |  $\nu$ = 10
$y_{g}$ = -0.9645 |  |
$e$ = 0.4596 |  |
$\phi_{e}$ = 84.48 |  |
$\gamma$ = 0.1238 |  |
$\phi_{\gamma}$ = 5.9306 |  |
SIE + SIS + $\gamma$ |  |
$b^{\prime}$ = 0.5516 |  $b^{\prime}_{X}$ = 0.2518 |  $\chi^{2}$ = 13.87
$x_{g}$ = 0.5609 |  $x_{X}$ = -0.0564 |  $\nu$ = 12
$y_{g}$ = -0.9752 |  $y_{X}$ = -1.0284 |
$e$ = 0.4513 |  |
$\phi_{e}$ = 58.64 |  |
$\gamma$ = 0.0826 |  |
$\phi_{\gamma}$ = -45.02 |  |
SPLE + $\gamma$ |  |
$b^{\prime}$ = 0.7530 |  |  $\chi^{2}$ = 11.68
$x_{g}$ = 0.5321 |  |  $\nu$ = 9
$y_{g}$ = -0.9792 |  |
$e$ = 0.5006 |  |
$\phi_{e}$ = -87.53 |  |
$\gamma$ = 0.1012 |  |
$\phi_{\gamma}$ = 20.18 |  |
$\alpha$ = 0.8031 |  |
SPLE + SIS + $\gamma$ |  |
$b^{\prime}$ = 0.6136 |  $b^{\prime}_{X}$ = 0.2328 |  $\chi^{2}$ = 11.26
$x_{g}$ = 0.5538 |  $x_{X}$ = -0.0572 |  $\nu$ = 11
$y_{g}$ = -0.9767 |  $y_{X}$ = -1.0260 |
$e$ = 0.2410 |  |
$\phi_{e}$ = 46.26 |  |
$\gamma$ = 0.1000 |  |
$\phi_{\gamma}$ = -83.44 |  |
$\alpha$ = 0.6716 |  |

To estimate the uncertainty of optimized parameters, we deploy a Markov chain
Monte Carlo (_MCMC_) realization with 13 adaptive random walks and a maximum
step of 10000 through the multi-parameter space. Uniform priors are used for
input uncertainties of all observables. The multivariate probability
distributions of the lens parameters for the optimal SPLE+SIS+$\gamma$777To
avoid the symbol overload confusion, the external shear and power-law index
use the slant ($\gamma$) and upright ($\upgamma$) Greek letters respectively.
model are shown in Fig. 8. The marginalized posterior distributions of lens
parameters are illustrated as the histograms along the diagonal. Other panels
show the joint posterior densities of all coupled parameters and resemble the
realistic covariances.

Figure 6:  The critical curves and caustics from the power-law mass modelling
results. The left panel is for the power-law ellipsoid only, the right panel
is for the power-law ellipsoid with an _SIS_ X-galaxy perturber. Green lines
are the critical curves, and red lines are the caustics. Black dots and
triangles denote the images and sources respectively.

Figure 7: Predicted image and source positions from the optimal _SPLE+SIS_
model with 8.4-GHz _VLBI_ multi-component constraints. The observed image
positions are denoted as black dots with isotropic error bars, while the
predicted image positions are denoted as red circles. The source positions are
denoted as red triangles. The blue dashed lines indicate the collimated
alignment of the modelled components in the image plane and the source plane.
The error bars have been magnified 4 times for visual convenience. Figure 8:
Lens parameter estimation with _MCMC_ realizations for the optimal _SPLE_ with
the _SIS_ X-galaxy perturber. The histograms lying on the diagonal show the
marginalized posterior distributions of each lens parameter. The two-
dimensional plots show the joint posterior densities of all parameter pairs.
The contours are binned into five intervals. The parameter notations are the
same as those in Table 4.

###  3.2 Modelling of _HST_ data

The photometric measurements showed a significant existence of the X-galaxy.
The derived photometric mass of this companion galaxy is nearly half of the
primary lens’ mass and its impact on the lens system should be checked with
lens modelling. Since the discrete blobs and the arc-shape emission are
detected in the _WFPC2_ $I$-band and _NICMOS_ $H$-band images respectively, we
need the mass modelling to check its consistency with a dual-lens system.

We carried out the modelling with the lenstronomy program, by using the entire
emission from the extended images as constraints. It is a multi-purpose Python
package to model strong gravitational lenses which is presented in Birrer &
Amara (2018) and is based on Birrer et al. (2015). It does light profile
decomposition onto a shapelet basis rather than direct pixelation, which gives
it advantages to be used in describing an arbitrary profile within a limited
parameter space. Since every pixel in the image is helping to constrain the
model, we can investigate a bigger parameter space.

So we did combinatorial modelling trials, by assuming the lens mass model of
the host and the companion X-galaxy as a model pair of any two of the
following mass model pairs: (_SIE_ , _SIS_), (_SIE_ , _SIE_), (_SPEP_ ,
_SPEP_) and (_SPEMD_ , _SPEMD_), where _SIS_ denotes a singular isothermal
spheroid mass, _SIE_ denotes a singular isothermal ellipsoid mass, _SPEP_
denotes a softened power-law potential, and _SPEMD_ denotes a smooth power-law
mass density. The softening core or smoothing scale was set to approach zero
to act as the singular case. All trials were carried out both with external
shear and without it. The central position and ellipticity of a mass profile
and its light profile are set to be consistent during optimization. The power-
law potential and mass density profiles were also tested with varying slopes.
Since the source must have a non-delta light profile to contribute the
extended emissions in lens modelling, in the optimization, the source profiles
were set to Sérsic ellipses. Besides, the lensed images were used as position
constraints but they were assigned Sérsic profiles to optimize the light
modelling. However, we found it difficult to converge to the same optimal
parameters as obtained from the _VLBI_ data modelling with gravlens. This is
because the light profiles of the lenses and source need to be optimized
simultaneously in lenstronomy, the increased parameter space leads to non-
unique solutions especially when the model constraining strength is weak.

As the lensing objects are the same at the radio and optical bands, the same
lens model must reproduce both observations. So we imported the optimal
gravlens model for the _VLBI_ 8.4-GHz observation to lenstronomy, and kept the
optimized parameters confined within a certain threshold, to see how similar
to the _HST_ images that the positions and fluxes of can be reproduced. To
keep the dimensionality of the parameter space low, we assumed there is only
one background source. The active galactic nucleus (AGN) contributes to the
compact emissions in both radio and optical bands, while its host galaxy
contributes to the extended emissions in the optical band. All tri-band _HST_
data were used for simultaneous optimization. The optimized results are shown
in Fig. 9 and 10, the statistics and optimized model parameters are listed in
Table 5 and Table 6. Ideally, the reduced $\chi^{2}$ should be close to unity.
From the reconstructed source light in the image plane, as shown in Fig. 10,
we can see the Einstein ring around the X-galaxy is distorted by the primary
lens and the detected diffuse emission is actually a part of the ring. We can
notice that the optimal lens model slightly overfitted the _HST_ $V$\- and
$I$-band data. The average residuals over three bands at the targeted diffuse
arc area have been reduced to about 0.7$\sigma$. Taking the source position as
a free parameter, we found that the optimized source positions with the radio
and optical observations are slightly offset as shown in Table 7.

Table 5:  The statistics of the optimal modelling with tri-band _HST_ data. $\bar{\chi}^{2}$ is the reduced $\chi^{2}$. $\sigma$ is the standard deviation of the residual noise and the residuals have been normalized by $\sigma$. The arc box means the area in the dashed boxes in Fig. 9.  |  | Residuals ($\sigma$)
---|---|---
| $\bar{\chi}^{2}$ | Full image | Arc box
$V$-band | 0.81 | 0.90 | 0.27
$I$-band | 0.67 | 0.82 | 0.55
$H$-band | 1.02 | 1.01 | 1.25

We can see that with a secondary lens - the X-galaxy, the reconstructed tri-
band images can practically resemble arc-shaped emission at the observed
location. This implies the diffuse emission are highly likely from the lensed
background source rather than intrinsically from the host galaxy, such as a
spiral arm. From Table 2 and 6, we can see the fitted Sérsic index tends to be
non-flattish but not significantly far from a flat index ($n\sim 1$) of a disk
galaxy over a bulge-dominated system. Recently Mao et al. (2017) detected
differential polarization and Faraday rotation in the two lensed images, which
suggests a fairly significant axisymmetric magnetic field in the host galaxy.
So, the possibility for the non-smooth light structure of the host galaxy to
interfere with the lens system has not totally been ruled out.

Table 6:  The optimized parameters of lens mass, lens light and source light profiles with _HST_ image constraints. $(e_{1},e_{2})$ are the orthogonal components of the eccentricity related to the ellipticity and its position angles as: $e_{1}=(1-q)\cos 2\phi_{q}/(1+q)$, $e_{2}=(1-q)\sin 2\phi_{q}/(1+q)$. $(\gamma_{1},\gamma_{2})$ are the orthogonal components of the external shear related to its position angles as: $\gamma_{1}=\gamma\cos 2\phi_{\gamma}$, $\gamma_{2}=\gamma\sin 2\phi_{\gamma}$. The length unit of $R_{e}$ and $(x,y)$ is in arcsec and $Amp$ is the amplitude parameter in normalized counts. The coordinates’ origin is at the image center. $\upgamma$ is the power-law index.  Mass | $\theta_{E}$ | $e_{1}$ | $e_{2}$ | $x$ | $y$ | $\upgamma$ |
---|---|---|---|---|---|---|---
G | 0.630 | 0.182 | -0.291 | 0.337 | -0.360 | 2.328 |
X | 0.233 |  |  | -0.299 | -0.480 |  |
| $\gamma_{1}$ | $\gamma_{2}$ |  |  |  |  |
$\gamma$ | -0.048 | -0.040 |  |  |  |  |
Light | $Amp$ | $R_{e}$ | $n$ | $e_{1}$ | $e_{2}$ | $x$ | $y$
G | 211.070 | 0.630 | 2.503 | 0.182 | -0.291 | 0.337 | -0.360
X | 53.138 | 0.716 | 1.530 |  |  | -0.299 | -0.480
S | 4.464e4 | 0.047 | 1.095 | -0.274 | -0.209 | -0.074 | -0.082
Table 7:  The predicted background source position. The coordinates’ origin is at the A image core. The errors are projected covariance estimated from _MCMC_ trials.  | Radio (VLBI) | Optical (HST)
---|---|---
$\Delta$RA (′′) | 0.184$\pm$0.020 | 0.180$\pm$0.006
$\Delta$DEC (′′) | -0.789$\pm$0.012 | -0.709$\pm$0.009

Figure 9:  The optimal models of the image, sources, lensing convergence and
magnification. The top three rows are for the _HST_ $V$-band, $I$-band and
$H$-band data, respectively. The bottom two rows show the source, mass and
light models. In the reconstructed image panel, the red stars, green pluses
and yellow crosses denote the observed image positions, the modelled lens
positions and source positions respectively. In the reconstructed source
panel, the blue stars denote the source light centroid positions. The dashed
boxes indicates the area used for arc residual statistics.

Figure 10:  The optimal light models and convolved images of the lenses and
lensed source. The top row shows the light models of the lens and lensed
source. The bottom three rows are convolved images for the _HST_ $V$-band,
$I$-band and $H$-band data, respectively.

###  3.3 Comparison between models for optical and radio datasets

Ideally, the optical models for optical and radio data should converge to the
same point in the parameter space. However, this only applies when there is a
unique minimization of the model on the same dimensionality of the parameter
space and the constraining strength of the observables. Our combinatorial
trials have shown the chance to get a consistent model independently from
radio and optical data is low.

Fitting to the light profiles in _HST_ images utilizes information from all
pixels. Besides the lens mass model, the light profiles of the lens and the
source need to be constrained at the same time. The increased observable
constraints are competing with the expanded parameter space then. Due to the
degeneracy between model parameters, the optimization will face more local
minima when the parameter space increases. Advanced algorithms like particle
swarm optimization (PSO) hardly mitigate this. Intuitively, different image
features imply different optimal models. Unlike the radio emissions, in a
galaxy lens, the optical emissions are easily affected by many factors, like
the blending between foreground and background emissions, the dust absorptions
and even the instrumental deficiency. As we have noticed, the diffuse emission
looks different in the _HST_ $I$-band and $H$-band images. Compared with the
arc-shape emission in the $H$-band image, the discrete blobs in the $I$-band
image give an impression of a quad-image configuration. If we fit each band
individually, it is easy to reach the modelling inconsistency. Provided with
multi-band data, it is better to optimize the model over the whole bands
simultaneously to avoid the discrepancy between individual bands.
Nevertheless, the different image features at different bands still need to be
explained, such as, by noise or resolution limits. The different appearances
of the diffuse emission in _HST_ images are probably due to the
instrumentation effect, as the $I$-band image has a lower signal-to-noise
ratio but a higher resolution than the $H$-band image.

Unlike the _HST_ image modelling, the modelling of _VLBI_ data utilize
accurate positional and flux constraints from the resolved jet components in
radio images. Given only the mass model of the lens to be constrained, the
_VLBI_ data can help to determine the mass model parameters more precisely
with the image-plane optimization, especially with the multi-image multi-
component lens examples, like B1152+199. Since the lens model is the same, a
practical approach is to use the high-resolution radio observables to
constrain the lens mass model first and feed the optimal model to the optical
optimization to get the optimal light models of the lenses and source second.

The parametric modelling has shown that the X-galaxy does improve the
modelling, given the tight constraints from the 8.4-GHz global _VLBI_
observation. If the pow-law profile is let to vary, the optimal mass model of
the primary lens tends to have a steeper-than-_SIE_ slope. When the X-galaxy
perturber is added, its mass profile becomes even steeper to compensate for
the mass profile extension due to the presence of the X-galaxy. This is
consistent with the previous investigation done by Rusin et al. (2002) and
Auger (2008).

##  4 Discussions

Here we propose some topics on the lenses, images, sources, mass and light
models with regard to the observations and lens modelling, that need to be
discussed as follows.

###  4.1 Contribution of the X-galaxy

The arc-shaped emission detected in _HST/NICMOS_ $H$-band data conjured up an
image of secondarily lensed arc in the lens system. Its convergence centre
coincides with the location of the surrounded X-galaxy. Our detailed mass
modelling with both _HST_ $V$-, $I$\- and $H$-band observations have shown
that the X-galaxy does contribute to the optical morphology in the lensed
optical images. That means the diffuse emission in _HST_ images is actually
the detection of the lensing effects of the companion galaxy.

As we can see from the optical modelling, the diffuse arc is mainly
contributed by the extended structure from the host galaxy of the source. So
there is no direct lensing effect of the X-galaxy detected in the radio
observation. The lensed images are well resolved and serval discrete sub-
components can be identified in the 8.4-GHz global _VLBI_ observation. So the
observables from those three resolved jet components of this dual-image lens
add up to the constraining strength and make it possible to find the subtle
contribution of the X-galaxy in fitting to the constraints. Our parametric
mass modelling shows that the X-galaxy is required to achieve an optimal
fitting to all the observables, especially when handling the misalignment of
the jet components in image B. If the radial profile is left as a free
parameter, the optimal mass model of the primary lens turns to have a steeper
profile as the presented _SIS_ perturber can contribute part of the extended
mass distribution in the whole lens system. This could be explained merely as
a modelling degeneracy if the direct lensing effects of companion lens have
not been detected. In other words, the up-to-date optical and radio
observations testify the role of the X-galaxy in this lens system is not
ignorable.

###  4.2 Evidence for a bent jet

B1152+199 is a particularly controversial lensing system, which first provided
evidence for substructure lensing because of a bending jet in one of the
lensed images and recently was found to not have such change in the jet
direction. The possible bent radio jet in one of the lensed images was seen as
evidence for a local perturbation in the mass model by CDM substructure
(Metcalf, 2002). This has motivated a number of recent observations like Asadi
et al. (2020). The 8.4-GHz global _VLBI_ observation has resolved the sub-
components in the lensed images with a position accuracy down to about 0.1
mas, as seen in Fig. 4 and 7. We may notice that the sub-components line up
very well in each image, except that there is a very slight misalignment for
component B1 in image B. The orthogonal distance between B1 and the
collimation line is about 0.5 mas. The optimized lens model also reflects this
small offset in its predicted image positions. However, it is arguable whether
this small offset is enough to depict a bent jet. As asserted by Asadi et al.
(2020), it exhibits no apparent milliarcsecond-scale curvature. Nevertheless,
our modelling has shown that the lens model can easily tackle this slight
misalignment within the given measurement errors, as long as the X-galaxy
perturber is considered in the mass model, i.e., no jet curvature or extra
substructure is needed for that. This is different from Metcalf (2002)’s
approach. We have provided a toy model to produce an evidently bent jet with
an SIE+SIS+$\gamma$ lensing configuration in Section A.2.

###  4.3 Source structures

As the diffuse emission is considered as a secondary lensed feature, it is
natural to speculate there is a secondary lensed source, as either independent
or attached to the primary one by a certain structure. Assuming that the
background source is an AGN abiding in an elliptical host galaxy, our optical
modelling shows that the extended structure of the host galaxy spread over the
caustics and forms the lensed arc. The radio observations see the radio
emissions from the AGN and the optical observations see the optical emissions
from both the AGN and the host galaxy.

There are also other possibilities, such as that the compact radio quasar just
has an extended companion galaxy to contribute to the diffuse lensed features
with itself not being so extended. We should point out that, either the case
of an extended host galaxy or the case of an extended companion, is merely a
projection along the line of sight. The source components could be in
different source planes but coincidentally align along the line of sight, as a
discovered case by Gavazzi et al. (2008). We show the modelling of a two-
source case in Section A.4.

The light centroid of the background source is roughly 80 mas offset from the
position of the radio core as seen in Table 7. This corresponds to a distance
scale of $\sim$625 pc in the source plane888A flat $\Lambda$CDM cosmological
model is used ad hoc as $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$ and
$H_{0}=72$ kms-1Mpc-1.. It is not uncommon to find radio-optical offsets in
AGNs (Orosz & Frey, 2013), even in the (_CLASS_) lens samples (Skipper &
Browne, 2018; Spingola & Barnacka, 2020). In the gravitational lens modelling,
the source position is self-constrained by ray-tracing image positions back
from the image plane to the source plane. The positional observables in the
image plane build up a self-relative reference frame. Our lens modelling is
proven to be an independent method to measure the relative offset without
resorting to an absolute celestial reference frame. Since the offset between
the radio and optical core positions of AGNs may indicate a profound
evolutionary scenario of the central black hole and the dark matter halo in
the host galaxy, the gravitational lensing thus becomes an effective research
tool on this topic.

It needs to be mentioned that in the optimized two-source configuration, the
optimized source light profile becomes flatter, while the X-galaxy light
profile becomes steeper; the radio-optical offset of the primary source is
then lowered to $\sim$60 mas. Since the secondary source mainly contributes to
the extended emissions along the Einstein ring which blends with the emissions
from the lens and the bright images of the lensed AGN, its presence does not
violate the image number theorem. We can notice from Fig. 10 that, one extra
central image is blended with the X-galaxy.

###  4.4 Inner power-law slope

The $\Lambda$CDM halo density profile, as applied to clusters, is shallower
than isothermal near the centre and steeper than isothermal near the virial
radius (Navarro et al., 2004). While down to the galactic scale, the
statistical results from the Sloan _SLACS_ survey (Koopmans et al., 2006)
shows that the inner regions of lens galaxies exhibit isothermal density
profiles. The optimal lens model obtained from the 8.4-GHz _VLBI_ data
modelling has inferred a steeper-than-isothermal inner slope of the primary
lens galaxy, which is also verified by the optical modelling with tri-band
_HST_ data. Its inconsistency with the statistical results implies the lens
galaxy is at a different stage of evolution. It has been found that the lens
system with a companion is more likely to have steeper density profiles
(Auger, 2008). From the lens modelling point of view, in a nearly-isothermal
lens system, if a companion is included in the modelling, the optimization
will steepen the inner slope of the primary lens to compensate for the added
extended mass profile by the companion. This modelling degeneracy can be
broken when the extra lensing effects due to the companion lens are detected.
The detection of the diffuse arc in the lens B1152+199 has put the claim of a
steeper inner slope on a solid foothold. Dobke et al. (2007) used N-body
simulation to demonstrate that the steepening of the density profile is due to
the temporary interactions within galaxy groups and will return to
isothermality eventually.

###  4.5 Quad, arc or ring

Though the current modelling restores the optical observation to the greatest
extent, we still notice that there are subtle differences between the observed
diffuse features and the reconstructed emissions. The source light
reconstruction in the image plane does show a fractured ring. After the
convolution with the _PSF_ s, only the diffuse arc remains visible since other
parts of emissions are totally buried in the glows of the two quasar images.
As limited by the sensitivities and resolutions of current observations, there
is no compact core resolved from the diffuse emission, so it is hard to tell
whether the lensing configuration is a quad, arc or ring. Nevertheless, we can
notice that the shape of the fractured ring does resemble a quad
configuration, which verified our naive speculation from the _HST_ $I$-band
image. The only difference is that the model resembles a fold-crossing rather
than a cusp-crossing case. As the arc is composed of two extended images (R2,
R3) and other two images (including R1) (refer to Fig. 2) are shifted close
the primarily lensed images A and B, due to the presence of the primary lens.

If the diffuse arc can be resolved into discrete images, it will definitely
add up to the positional observables and help to constrain the lens model to
another level. The high-sensitivity infrared observation with the upcoming
James Webb Space Telescope (_JWST_) can definitely have a deeper insight into
this lens system.

##  5 Conclusions

In this paper, we reinvestigated the archived _HST/WFPC2_ $V/I$-band and
_HST/NICMOS_ $H$-band data as well as the 5-GHz and 8.4-GHz _VLBI_ data of the
target lens _CLASS_ B1152+199. Our re-analysis and lens modelling of the
archival data, has shown that improved observations are important for
astronomers to understand this lens system.

The resolved radio sub-components in lensed images give more direct
constraints on the parametric lens model and the image-plane optimization
gives more precise estimations on the model parameters. Currently, the 8.4-GHz
global-_VLBI_ observation now offers the highest resolution of the discrete
jet components in the A and B images. The collimation of the jet components in
image B dismisses the previous debate over the jet bending (Asadi et al.,
2020). Actually, the jet bending can be proved non-significant with lower
resolution data from 5-GHz observation, as shown in Section A.1. Nevertheless,
a slight misalignment within 0.5 mas of a B jet component can still be
detected. Meanwhile, the model fitting of resolved components gives the
strongest up-to-date constraints on the macro mass model. Especially, the
radial mass profile may be inferred from the radially aligned jet components.
So we tested the observables with _SIE_ and _SPLE_ models plus external shear
both with and without an _SIS_ perturber. The optimal models tell us that the
companion X-galaxy is needed for better optimization and a steeper-than-
isothermal mass profile is thus preferred.

The photometric measurements from tri-band images give a consistent luminosity
ratio between the host galaxy and the X-galaxy, which is different from the
previous study by Rusin et al. (2002). This leads to a different estimation of
the mass ratio between two galaxies, which affects the mass model
considerably. Our attention was brought to the diffuse emission that was
detected in the _HST_ $H$-band and $I$-band images around the X-galaxy. A
careful inspection has revealed that the diffuse arc-shaped emission in the
$H$-band image is resolved into three faint blobs in the $I$-band image. We
utilized the shapelet modelling program lenstronomy to investigate the
possibility of the diffuse emission as a secondary lensed feature by the
presence of the X-galaxy. Optical modelling is very useful to reconstruct the
morphology of the observed image. However, as there are light profiles of the
lens and the source to be constrained at the same time, the expanded parameter
space makes its determination of the mass model less effective. So we imported
the optimal lens model from the radio modelling to the optical modelling as a
reference model to keep the optimizations consistent. The optimization with
optical data shows that a power-law ellipsoid with external shear, plus an
isothermal spheroid companion, can reproduce the optical features observed in
_HST_ images.

An under-constrained lens system leads to parameter degeneracy, so that a
subtle observed effect may be explained by many possibilities. It is risky to
draw conclusions without considering all the components detected in the lens
system. The X-galaxy and diffuse optical emissions were detected from the
first discovery of the B1152+199 lens (Rusin et al., 2002). However, it is
until the sub-components are fully resolved by the 8.4-GHz globally _VLBI_
observation that the X-galaxy can finally find its role in this lens system.
So a high-sensitivity full-track global _VLBI_ follow-up observation is still
the most powerful tool to probe the mass distribution of those radio lenses.

Though it is favoured by current mass modelling that the dubious diffuse
emission around the X-galaxy in _HST_ images are lensed features from the
source, the diffuse emission may still play an important role in further
determining the light and mass profiles of the lens and source when a higher-
sensitivity-and-resolution observation is achieved. Hence future
investigations of this specific lens system with more powerful or technically
advanced ground-based or space-based optical telescopes are needed. As shown
in the work of Lagattuta et al. (2012), the Keck adaptive optics can provide
better constraints than the _HST_. Proved by the mass modelling, the X-galaxy,
as a massive companion satellite of the host galaxy, does play a consequential
role in the overall mass model. This kind of companion galaxies are not rare
in _CLASS_ lenses, such identified samples as B1608+656 (Myers et al., 1995),
B2108+213 (McKean et al., 2005) and B1359+154 (Myers et al., 1999). As a
matter of fact, we don’t even know the redshift of the X-galaxy (Rusin et al.,
2002), not even mention the redshift of the diffuse emission. The
spectrographs of very faint galaxies ($\sim$30 mag) have been successfully
taken by the _HST_ slitless grism and Keck-I Multi-Object Spectrometer For
Infra-Red Exploration (_MOSFIRE_) (Schmidt et al., 2014). Even the detection
of the arc-blob emission in another _HST_ filter will give a crude first-order
inference on the photometric redshift with the colours. Yet the redshifts of
the primary lensing galaxy and the X-galaxy have not been confirmed the same
until the present.

## Acknowledgements

MZ is grateful to the anonymous referee for the constructive and informative
suggestions which helped to improve the paper. We also thank Simon Birrier for
his practical insider advice about the lenstronomy code’s behaviours. This
work was supported by the National Science Foundation of China (12173078 and
11773062) and the West Light Foundation of Chinese Academy of Sciences
(2017-XBQNXZ-A-008). The _VLBA_ is an instrument of the National Radio
Astronomy Observatory, a facility of the National Science Foundation operated
by Associated Universities, Inc. The European _VLBI_ Network is a joint
facility of independent European, African, Asian, and North American radio
astronomy institutes. The scientific results from data presented in this
publication are derived from the following EVN project codes: BB133, GJ010,
BW084 and GA036. This research used observations made with the NASA/ESA Hubble
Space Telescope, obtained from the data archive at the Space Telescope Science
Institute. STScI is operated by the Association of Universities for Research
in Astronomy, Inc. under NASA contract NAS 5-26555.

## Data availability

The data underlying this article are available in the Hubble Legacy Archive at
https://hla.stsci.edu and the NRAO Science Data Archive at
https://archive.nrao.edu.

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##  Appendix A Extra materials

###  A.1 Jet flip-subtraction

For collimated jet emission, if we flip the image along the collimation axis
and subtract mirrored image from the original one, we will find the flux along
the collimation axis cancelled. However, if a jet is really bent, its flux
cannot be cancelled by flip-subtraction no matter how the collimation axis is
chosen. The systematics of the residuals with mirrored parities should be
detected for a flip-subtracted bent structure, see Fig. 11. So, rather than
using an elliptical beam, we chose a circular beam to restore image B to check
for any unbiased shape perception (Fig. 12). The radius of the circular beam
is chosen as the minor axis of the original elliptical beam calculated from
$uv$ coverage. Along the major axis direction, the image is thus slightly
super-resolved, which may help to trace unresolved components. The collimating
axis was determined by the peak positions of the image components. We can see
that, when restored with a circular beam, the jet in image B from _VLBA_ and
global-_VLBI_ observations looks less bent than that restored with the
elliptical beam shown in Fig. 3. Though a slightly noticeable bend can be seen
from the contour connection between the core and the jet component in the
_HSA_ observation, after flip-subtraction, the marginal jet bending is only at
a $3\sigma$ detection level.

Figure 11:  Flip-subtraction of a bent jet. The solid lines are the bent jet
and dashed lines are the negative mirror image of the jet after flip-
subtraction. A, B and C show the effects of choosing different flipping axes.
A real bent jet cannot be cancelled along any flipping axis.  Figure 12:
Three-epoch _VLBI_ observations and flip-subtractions of image B restored with
a circular beam of 1.5$\times$1.5 mas2 . From top to bottom, the A and B
images correspond with observations BB133 (2001), GJ010 (2003) and BW084
(2005), respectively. The data have been uniformly weighted. Contours in the
map are plotted in multiples of -1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,
1024, 2048 $\times$ $3\sigma$, where $\sigma$ is the local RMS noise ($\sim
100\mu$Jy). The images in the middle column are flipped along the jet
direction, while the image in the right column is the residual image after the
subtraction of the original image from the flipped image.

###  A.2 Toy model to produce a bent jet

Here present a toy model to show that with the companion galaxy as a secondary
lens, the macro lens model can produce a bent jet without any extra
substructure local to the jet. The exaggerated error ellipses are used to give
a greater relaxation for the lens model to move around the image positions. We
can see from the middle panel in Fig. 13, merely an SIE+SIS+$\gamma$ model is
adequate enough to produce a bent jet if the lensing strength of the companion
is emphasized. The diffuse emission as a lensing effect can be treated as a
partial Einstein ring, an arc or a quad-image case.

Figure 13:  Predicted image and source positions from the optimal model with
multi-component and quad-image constraints. The observed image positions are
denoted as blue error ellipses, while the predicted image positions are
denoted as green squares. The source positions are denoted as red triangles.
The S’ denotes the secondary source.

###  A.3 Parameter translation between gravlens and lenstronomy

We summarize here the parameter conventions of the two modelling software
packages and the conversion between them to facilitate cross-check and clarify
the use of symbols. The gravlens Alpha model’s parameter set is {
$b^{\prime}$, ($e$, $\phi_{e}$), ($\gamma$, $\phi_{\gamma}$), $\alpha$ },
while the lenstronomy _SPEMD_ model’s parameter set is accordingly {
$\theta_{E}$, ($e_{1}$, $e_{2}$), ($\gamma_{1}$, $\gamma_{2}$), $\upgamma$ }.

Convergence:

$\displaystyle\kappa(x,y)$
$\displaystyle=\frac{1}{2}{b^{2-\alpha}\over{(x^{2}+y^{2}/q^{2})^{(2-\alpha)/2}}}$
(3) $\displaystyle=\frac{1}{2}{b^{\prime
2-\alpha}\over{[(1-\epsilon)x^{2}+(1+\epsilon)y^{2}]^{(2-\alpha)/2}}}$ (4)
$\displaystyle=\frac{3-\upgamma}{2}{\theta_{E}^{\upgamma-1}\over(qx^{2}+y^{2}/q)^{(\upgamma-1)/2}}$
(5)

Mass scale (1-dimensional):

$\displaystyle\frac{b^{\prime}}{b}=\sqrt{\frac{2q^{2}}{1+q^{2}}}$ (6)
$\displaystyle\frac{\theta_{E}}{b^{\prime}}=\sqrt{\frac{1+q^{2}}{2q}}\frac{1}{\alpha^{1/(2-\alpha)}}$
(7) $\displaystyle\frac{\theta_{E}}{b}=\frac{\sqrt{q}}{\alpha^{1/(2-\alpha)}}$
(8)

Ellipticity:

$\displaystyle q^{2}=(1-\epsilon)/(1+\epsilon)$ (9) $\displaystyle e=1-q$ (10)
$\displaystyle e_{1}=\frac{1-q}{1+q}\cos{2\phi_{e}},\quad
e_{2}=\frac{1-q}{1+q}\sin{2\phi_{e}}$ (11)

External shear:

$\gamma_{1}=\gamma\cos{2\phi_{\gamma}},\quad\gamma_{2}=\gamma\sin{2\phi_{\gamma}}$
(12)

Power-law index:

$\upgamma=3-\alpha$ (13)

###  A.4 Model case with two background sources

Here we present an example model with two lensed background sources which is
optimized with tri-band _HST_ images (see Fig 14). Since the secondary source
also contributes to forming the Einstein ring, the primary source needs not to
be so extended as shown in Fig. 9 and 10. The optimized model parameters are
shown in Table 8

Figure 14:  The optimal models for the two-source configuration. The top three rows are for the _HST_ $V$-band, $I$-band and $H$-band data, respectively. The bottom two rows show the source, mass and light models. In the reconstructed image panels, the red stars, green pluses and yellow crosses denote the observed image positions, the modelled lens positions and source positions respectively. In the reconstructed source panel, the blue stars denote the source light centroid positions.  Table 8:  The optimized parameters of lens mass, lens light and source light profiles with _HST_ image constraints for a two-source case. The symbol notations are the same as in Table 6.  Mass | $\theta_{E}$ | $e_{1}$ | $e_{2}$ | $x$ | $y$ | $\upgamma$ |
---|---|---|---|---|---|---|---
G | 0.627 | 0.112 | -0.086 | 0.310 | -0.393 | 2.328 |
X | 0.233 |  |  | -0.302 | -0.391 |  |
| $\gamma_{1}$ | $\gamma_{2}$ |  |  |  |  |
$\gamma$ | -0.094 | -0.007 |  |  |  |  |
Light | $Amp$ | $R_{e}$ | $n$ | $e_{1}$ | $e_{2}$ | $x$ | $y$
G | 178.803 | 0.698 | 2.111 | 0.112 | -0.086 | 0.310 | -0.393
X | 79.702 | 0.279 | 2.161 |  |  | -0.302 | -0.391
S1 | 1.434e5 | 0.042 | 0.712 | -0.258 | -0.309 | -0.070 | -0.100
S2 | 240.880 | 0.057 | 0.592 | -0.090 | 0.203 | 0.060 | -0.204