Morpho-Kinematics of the molecular gas in a quasar host galaxy at redshift z = 0:654

 Communications in Physics, Vol.31, No. 2 (2021), pp. 149-168
 DOI:10.15625/0868-3166/15599
 MORPHO-KINEMATICS OF THE MOLECULAR GAS IN A QUASAR HOST
 GALAXY AT REDSHIFT z = 0.654
 T. T.THAI1,2, P. TUAN-ANH1,†, P. DARRIULAT1, D. T. HOAI1, P. T. NHUNG1, P. N. DIEP1,
 N. B. NGOC1 AND N. T. PHUONG1
1Vietnam National Space Center, Vietnam Academy of Science and Technology
 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
2Graduate University of Science and Technology
 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
 E-mail: †ptanh@vnsc.org.vn
 Received 14 October 2020
 Accepted for publication 29 December 2020
 Published 27 January 2021
 Abstract. We present a new study of archival ALMA observations of the CO(2-1) line emission
 of the host galaxy of quasar RX J1131 at redshift z=0.654, lensed by a foreground galaxy. A
 simple lens model is shown to well reproduce the optical images obtained by the Hubble Space
 Telescope. Clear evidence for rotation of the gas contained in the galaxy is obtained and a simple
 rotating disc model is shown to give an excellent overall description of the morpho-kinematics of
 the source. The possible presence of a companion galaxy suggested by some previous authors is
 not confirmed. Detailed comparison between model and observations gives evidence for a more
 complex dynamics than implied by the model. Doppler velocity dispersion within the beam size in
 the image plane is found to account for the observed line width.
 Keywords: galaxies: evolution – galaxies: ISM – radio lines: galaxies.
 Classification numbers: 98.65-r.
 I. INTRODUCTION
 I.1. General features
 RX J1131-1231 (simply called RX J1131 in the following), is a distant quasar, at redshift
 zs ∼0.654, corresponding to a distance of ∼1.45 Gpc or a time of ∼7.5 Gyr after the Big Bang,
 about half way from us, and some 4 Gyr later than the time of maximal star formation [1, 2]. At
 such distance, 1 arcsec spans 7.03 kpc. It hosts a Super Massive Black Hole (SMBH) in its centre
 ©2021 Vietnam Academy of Science and Technology
150 T. T. THAI et al.
 8
with a mass of ∼2 10 M; it rotates extremely fast, reaching near half the light velocity [3].
The quasar and its host galaxy are gravitationally lensed by a galaxy in the foreground, at redshift
zL ∼0.295. They are the object of numerous studies, in particular aiming at a better understanding
of the cosmological parameters governing the expansion of the Universe ( [4, 5] and references
therein). Microlensing caused by stars transiting across the line of sight to the quasar has been
used to study the structure of the lens halo ( [6, 7] and references therein).
 Infrared observations obtained by Herschel [10] have measured the spectral energy distribu-
 Fig. 1. Left: dependence on the redshifts of the source (zS in abscissa) and of the lens (zL
 in ordinate) of the ratio between their respective angular diameter distances daS/daL [1,2]).
 The relative size of the lens with respect to the source is proportional to daS/daL. The stars
 show the locations of quasars RX J1131 (black, P18) and RX J0911 (red, [8], [9]). The
 sizes of the host galaxies are compared to the size of the lens caustic in the central and
 right panels respectively.
tion (SED), and archival VLA observations (Program ID: AW741; PI: Wucknitz) analysed by
Leung et al. 2017 [11], referred to as L17 in the following, have shown resolved continuum emis-
sion from the jets and the core of the foreground elliptical galaxy as well as emission toward the
background quasar.
 Thorough analyses of high angular resolution HST optical and NIR images [12–14] and of
Keck Adaptive Optics images [15] have produced a detailed description of the lensing properties
in the neighbourhood of the quasar. They reveal a typical long axis quad configuration [16,17], the
quasar being located within the eastern cusp of the lens caustic. As emission from the lens galaxy
is simultaneously detected, the parameters of the lensing potential can be accurately evaluated.
However, they probe only the vicinity of the cusp of the caustic curve. As the emission of the
quasar host galaxy covers the whole caustic curve and extends even farther out, one cannot take
it as granted that the simple lens model obtained from the study of the quasar images reliably
applies to the whole host galaxy. This is at variance with the gravitational lensing of quasar hosts
that are farther away and cover only part of the caustic in addition to being intrinsically smaller.
The central region of the caustic corresponds to images close to an Einstein ring configuration,
which dominates the picture in the case of RX J1131. We illustrate this feature in Figure 1, which
shows the location of RX J1131 in the plane zL vs zS, lens vs source redshifts, together with that of
other multiple imaged systems [12], and compares it with the case of a typical farther away quasar
host galaxy, RX J0911 [8, 9].
 MORPHO-KINEMATICS OF TH ... The modelled spectra are obtained by de-lensing the images pro-
duced by lensing the model disc source and convolved with the beam. Results are illustrated in
the left panel of Figure 12. Qualitatively, the general trend is well reproduced by the model but
significant differences are observed in the central segments: the data display larger Doppler ve-
locities on the red side and lower Doppler velocities on the blue side than implied by the model.
Moreover, in the central segment, the line width predicted by the model is much smaller than that
observed in the data [22]. A natural interpretation of such an effect is disc warping causing an
effective dependence on θ of the sine of the inclination angle in Relation (4). However, including
warping in the model by writing Vz = V(R)cosθ sinϕ with ϕ depending simply on θ and R, gives
only a modest improvement of the match between model and observations. This suggests that a
more complex dynamics than described by the simple model is at stake.
 Beam convolution causes an effective smearing of the disc region contributing to each of the
nine segments, making the radial distribution of the mean Doppler velocity measured in the central
segments less steep and causing an increase of the velocity dispersion. This important result is a
warning: measuring the velocity dispersion requires a careful evaluation of the contribution of
rotation within the finite angular resolution of relevance. Making the angular acceptance smaller
will decrease the direct contribution of rotation, independent from beam convolution, but will
not decrease the contribution resulting from the smearing caused by the beam. This contribution
is important: using the central segment as an example, the line width predicted by the model
increases from 60 km s−1 to nearly 100 km s−1 when accounting for beam convolution, namely a
beam contribution of over 70 km s−1 (both contributions add up in quadrature).
164 T. T. THAI et al.
 Fig. 11. Left panels display the brightness distribution in the source plane obtained by
 de-lensing depletion D (left) and excess E (centre-left) respectively. Right panels display
 2
 the dependence of χ on the factors FD and FE measuring the amplitudes of the depletion
 (centre-right) and excess (right) respectively when the other model parameters are fixed
 at their best-fit values.
 0
 Fig. 12. Dependence of on x (major axis of the projection of the disc on the plane
 of the sky) for the data (black) and the model (red) is shown in the left panel and the
 rotation curve in the right panel together with the L17 (blue curve) and P18 (red crosses)
 results.
 The differences between observed and predicted velocity dispersions may receive contribu-
tions from the intrinsic line width (turbulence), which is absent from the model, or from different
rotation contributions, as could be caused by disc warping. We recall that P18 claim that the obser-
vation of a high dispersion in the vicinity of the quasar demonstrates that the region of enhanced
emission studied in the preceding section is the seat of increased gas turbulence. However, the
differences observed in the central segments of the band used to evaluate the rotation curve do not
consist of a simple broadening of the line, as would be expected from turbulence, but reveal rather
an excess of red-shifted emission.
 Maps of the intensity, mean Doppler velocity and Doppler velocity dispersion in the λ vs ω
plane reveal important differences between model and observations [22]: predicted mean Doppler
velocities tend to be too small in the 90◦<θ<180◦ quadrant and too large in the 270◦<θ<360◦
quadrant of the disc plane; velocity dispersions reach some 120 km s−1 for the model and some
 MORPHO-KINEMATICS OF THE MOLECULAR GAS IN A QUASAR HOST GALAXY ... 165
 Fig. 13. Line profiles in the region |λ| <0.25 and 110◦<ω<250◦. Left: observations
 with cuts at 11.7 µJy (black) or 16 µJy (red); the lines show Gaussian fits with σ=85
 km s−1 and 56 km s−1 respectively. Gaussian fits to the predictions of the disc model
 without (centre) and with (right) beam convolution give σ values of 18 km s−1 and 58
 km s−1 respectively.
160 km s−1 for the observations. We failed to obtain a substantial improvement of the quality of
the match between model and observations by including a simple description of disc warping in
the model: this confirms the need for a more complex dynamics than implied by the simple disc
model. From a number of comparisons between the line widths predicted by the model with and
without beam convolution, the contribution is found to be in the range between 50 and 70 km s−1.
 An independent evaluation is obtained by selecting the central region of the λ vs ω plane
(left column of Figure 10) defined as |λ| <0.25 and |ω − 180◦| <70◦. We segment it in 10×14
 ◦
pixels of 0.05 in λ and 10 in ω. For each pixel (i, j) we evaluate the mean value i j of the
Doppler velocity and show the distribution of Vz−i j in Figure 13. We obtain this way an
evaluation of the line profile. The observed line widths (σ) are 85 and 56 km s−1 for cuts at 11.7
(1.5-σ) and 16 µJy (2-σ) per 50×50×8.417 mas2 km s−1 respectively. As the model includes no
intrinsic line broadening, the line broadening shown by the model is entirely due to the contribution
of rotation. Gaussian fits give σ values of 18 and 58 km s−1 without and with beam convolution
respectively, implying a contribution of ∼50 km s−1 from the smearing effect of the beam size.
The line width measured with the 16 µJy cut does not require any contribution of turbulence
while that measured with an 11.7 µJy cut allows for a contribution reaching ∼60 km s−1: without
a precise understanding of the contribution of noise, it is therefore difficult to obtain a reliable
evaluation of the line broadening caused by turbulence.
 In summary, a simple rotating disc model gives a global picture that describes well the
general trend of the observed kinematics. However, such a model fails to reproduce quantitatively
the details of the Doppler velocity distribution: it reveals a more complex dynamics. Attempts to
describe the mismatch as the result of a simple disc warping have failed. An important result of
our analysis is that the smearing in the image plane caused by the beam size contributes between
50 and 70 km s−1 to the dispersion of the Doppler velocity. It combines in quadrature with the
contribution of rotation within the angular acceptance being probed by the pixel size of relevance.
Taken together, these effects make it difficult to evaluate reliably the contribution of turbulence to
the line width. But they prevent claiming that such contribution is important: within experimental
166 T. T. THAI et al.
uncertainties the line width can be accounted for by the dispersion of rotation velocities within the
beam size.
V. SUMMARY AND CONCLUSIONS
 The present study of the emission of the CO(2-1) molecular line by the host galaxy of quasar
RX J1131 uses ALMA observations of unprecedented quality; they have been previously analysed
in much detail by their proponents, [18], P18. We have shown that the HST images of the quasar
and of the lens galaxy are very well reproduced by a simple lensing potential, sphere+external
shear. We obtained parameters that are in agreement with the results of previous authors within
errors. We discussed the uncertainties attached to the model parameters and their correlations and
explained them by remarking that these results probe only the environment of the eastern cusp of
the caustic. At this stage, the validity of the model at larger distances, in the region covered by the
emission of the host galaxy, cannot be taken as granted. But we demonstrated its validity from our
analysis of the ALMA observations of the CO(2-1) line emission. We reduced the raw data and
produced cleaned images, at variance with P18 who perform their analysis in the uv plane. We
obtained morpho-kinematics properties of the gas that are in good agreement with the results of
P18. We used two different methods for de-lensing these images, which produce source brightness
distributions in agreement with each other as well as with the P18 results. We have shown that the
general morphology of the image is dictated by the properties of the lens and confined within a
band bracketing the critical curve. This leaves in practice no freedom to conceive lensing potentials
that would be significantly different from those that we and other authors have been using. It
guarantees the robustness of the results obtained by P18 within observational uncertainties.
 We used polar coordinates better adapted to this geometry than Cartesians to reveal very
clearly the evidence for rotation. We looked for possible deviations from the predictions of a
simple rotating disc model, in particular for the possible presence of a companion as was predicted
by L17 using Plateau de Bure observations of much less good angular resolution than the ALMA
observations; but our analysis did not support such a presence. We paid special attention to the red-
most velocity interval, which displays a particularly complex morphology. We found evidence for
enhanced emission revealing a significant deviation from the prediction of the simple disc model.
However the associated hot spot in the source plane overlaps the caustic, implying important
uncertainties on its de-lensed properties.
 We discussed the rotation curve in some detail and argued that it probably rises faster than
assumed by P18 in the vicinity of the quasar; we noted that the angular resolution is such that
the dispersion of the rotation velocity within the beam size in the image plane causes an important
effective broadening of the line width, which we evaluated in the range between 50 and 70 km s−1 .
This is sufficient to account for the observed line width when applying a ∼2-σ cut on the data.
However, uncertainties attached to the effect of noise prevent from giving a reliable evaluation of
the contribution of turbulence. This is at variance with the conclusion of P18 who attribute the high
Doppler velocity dispersion observed in the vicinity of the AGN to gas turbulence exclusively.
 P18 obtained from their analysis a number of results concerning the physical properties of
the galaxy, which they compared with those of other similar galaxies. The nature of our study
prevents us from adding much to their conclusions. We simply remark that their estimate of
 11
the total dynamical mass enclosed within 5 kpc, (1.46±0.31)×10 M, may be affected by the
steepness of the rotation curve: at fixed radius, the dynamical mass is proportional to the square of
 MORPHO-KINEMATICS OF THE MOLECULAR GAS IN A QUASAR HOST GALAXY ... 167
the rotation velocity. The analysis presented in the preceding sections implies therefore a possibly
higher value of the gas mass, obtained by P18 and accordingly affect the value of the CO−H2
conversion factor.
ACKNOWLEDGMENT
 We express our deep gratitude to Professors Frederic Courbin and Matus Rybak who kindly
provided us with copies of data files summarizing the results of the P18 analysis. This paper uses
archival ALMA data from project 2013.1.01207.S (PI: Paraficz Danuta). ALMA is a partnership
of ESO (representing its member states), NSF (USA), NINS (Japan), NRC(Canada), NSC/ASIAA
(Taiwan), and KASI (South Korea), in cooperation with Chile. The Joint ALMA Observatory is
operated by ESO, AUI/NRAO and NAOJ. We are deeply indebted to the ALMA partnership,
whose open access policy means invaluable support and encouragement for Vietnamese astro-
physics. Financial support from the World Laboratory and VNSC is gratefully acknowledged.
This research is funded by the Vietnam National Foundation for Science and Technology Devel-
opment (NAFOSTED) under grant number 103.99-2019.368.
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