11institutetext: Purple Mountain Observatory, Chinese Academy of Sciences, 10
Yuanhua Road, Nanjing 210023, People’s Republic of China
11email<EMAIL_ADDRESS>22institutetext: Université Paris-Saclay, Université
Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France 33institutetext:
Institut de Radioastronomie Millimétrique, 300 Rue de la Piscine, 38406 Saint-
Martin d’Hères, France 44institutetext: National Radio Astronomy Observatory,
800 Bradbury Dr., SE Ste 235, Albuquerque, NM 87106, USA 55institutetext:
LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne
Universités, 75014 Paris 66institutetext: Kavli Institute for the Physics and
Mathematics of the Universe, The University of Tokyo, Kashiwa, 277-8583, Japan
77institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, People's Republic of China 88institutetext: Center
for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan 99institutetext: Department of Astronomy,
School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan 1010institutetext: School of Physics and Astronomy, University
of Southampton, Highfield SO17 1BJ, UK 1111institutetext: Center for
Extragalactic Astronomy, Department of Physics, Durham University, South Road,
Durham DH1 3LE, UK 1212institutetext: National Astronomical Research Institute
of Thailand, Don Kaeo, Mae Rim, Chiang Mai 50180, Thailand 1313institutetext:
Department of Physics, Faculty of Science, Chulalongkorn University, 254
Phayathai Road, Pathumwan, Bangkok 10330, Thailand 1414institutetext: European
Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching bei Munchen,
Germany 1515institutetext: Cosmic Dawn Center (DAWN), Denmark

# Fitting pseudo-S${\rm\acute{e}}$rsic (Spergel) light profiles to galaxies in
interferometric data: the excellence of the $uv$-plane

Qing-Hua Tan 1122 Emanuele Daddi 22 Victor de Souza Magalhães 3344 Carlos
Gómez-Guijarro 22 Jérôme Pety 3355 Boris S. Kalita 667788 David Elbaz 22
Zhaoxuan Liu 668899 Benjamin Magnelli 22 Annagrazia Puglisi Anniversary
Fellow10101111 Wiphu Rujopakarn 12121313 John D. Silverman 668899 Francesco
Valentino 14141515 Shao-Bo Zhang 11

Modern (sub)millimeter interferometers, such as ALMA and NOEMA, offer high
angular resolution and unprecedented sensitivity. This provides the
possibility to characterize the morphology of the gas and dust in distant
galaxies. To assess the capabilities of current softwares in recovering
morphologies and surface brightness profiles in interferometric observations,
we test the performance of the Spergel model for fitting in the $uv$-plane,
which has been recently implemented in the IRAM software GILDAS (uv$\\_$fit).
Spergel profiles provide an alternative to the S${\rm\acute{e}}$rsic profile,
with the advantage of having an analytical Fourier transform, making them
ideal to model visibilities in the $uv$-plane. We provide an approximate
conversion between Spergel index and S${\rm\acute{e}}$rsic index, which
depends on the ratio of the galaxy size to the angular resolution of the data.
We show through extensive simulations that Spergel modeling in the $uv$-plane
is a more reliable method for parameter estimation than modeling in the image-
plane, as it returns parameters that are less affected by systematic biases
and results in a higher effective signal-to-noise ratio (S/N). The better
performance in the $uv$-plane is likely driven by the difficulty of accounting
for correlated signal in interferometric images. Even in the $uv$-plane, the
integrated source flux needs to be at least 50 times larger than the noise per
beam to enable a reasonably good measurement of a Spergel index. We
characterise the performance of Spergel model fitting in detail by showing
that parameters biases are generally low (¡ 10%) and that uncertainties
returned by uv$\\_$fit are reliable within a factor of two. Finally, we
showcase the power of Spergel fitting by re-examining two claims of extended
halos around galaxies from the literature, showing that galaxies and halos can
be successfully fitted simultaneously with a single Spergel model.

###### Key Words.:

Methods: data analysis - Techniques: interferometric - Galaxies: high-redshift
- Submillimeter: galaxies

##  1 Introduction

Galaxy morphologies are closely linked to their formation and evolution (e.g.,
Conselice, 2014). One of the most effective ways to understand the structures
of galaxies and how they evolve over time is by measuring the distribution of
light within them, specifically in relation to their radial distance from the
center (e.g., Tacchella et al., 2015). To describe the morphological types of
galaxies, two commonly used measurements are the half-light radius and the
central concentration of light profiles (e.g., the S${\rm\acute{e}}$rsic
index). These measurements provide insights into the shapes and sizes of
galaxies and help classify them into different morphological categories.

Over the past decade, optical and near-infrared (IR) observations have allowed
for the exploration of the structural properties of the stellar components in
high-redshift star-forming galaxies (e.g., Wuyts et al., 2011; van der Wel et
al., 2014; Shibuya et al., 2015; Tacchella et al., 2018; Cutler et al., 2022;
Chen et al., 2022; Kartaltepe et al., 2023). However, obtaining measurements
of purely star-forming components based on ultraviolet (UV) and optical star
formation rate (SFR) tracers are much more difficult, due to the effects of
dust attenuation. At high redshifts, galaxy morphology can manifest in various
ways, ranging from radial gradients (Nelson et al., 2016) to very complex
asymmetrical distributions (Le Bail et al., 2023). In massive ($M_{*}>10^{10}\
M_{\odot}$) galaxies, most of the rest-frame UV/optical emission is re-emitted
in the far-IR/submillimeter windows (e.g., Pannella et al., 2015; Wang et al.,
2019; Fudamoto et al., 2021; Smail et al., 2021; Xiao et al., 2023; Gómez-
Guijarro et al., 2023), which provide an excellent alternative to obtain
unbiased measurements of the morphology of star formation inside galaxies.

Recent developments in (sub)millimeter interferometers, such as ALMA and
NOEMA, have made it possible to study the distribution of dust and molecular
gas in high-redshift galaxies with high sensitivity (e.g., Barro et al., 2016;
Hodge et al., 2016, 2019; Tadaki et al., 2017; Elbaz et al., 2018; Fujimoto et
al., 2018; Gullberg et al., 2019; Jiménez-Andrade et al., 2019; Puglisi et
al., 2019, 2021; Gómez-Guijarro et al., 2022; Stuber et al., 2023). This
provides new observational constraints to the distribution of gas and dust-
obscured star formation in galaxies. ALMA, with the widest array
configurations, can potentially deliver very high spatial resolution of the
order of a few tens of milliarcseconds (depending on frequency), which is
unparalleled even by HST and JWST standards. ALMA has already revealed the
complex structure of star formation on sub-kiloparsec scales in dusty star-
forming galaxies (e.g., Iono et al., 2016; Gullberg et al., 2018; Hodge et
al., 2019; Rujopakarn et al., 2019). Comparing star formation profiles with
stellar mass profiles and morphology (more in general) is crucial for
understanding the structural evolution of galaxies as a result of star
formation (e.g., Cibinel et al., 2015). To extract the full amount of
information contained in the data, it is also essential to have appropriate
technical tools. For example, ALMA and NOEMA interferometers gather data in
the form of visibilities between antennas in the $uv$-plane, unlike the
optical data that are directly images from CCDs or other electronic detectors.

So far, modeling galaxy morphology profiles in the submillimeter, based on
interferometric data, has typically involved reconstructing images from the
visibilities, followed by featuring S${\rm\acute{e}}$rsic (or other kind of)
profiles fitting. This approach has been necessary due to the lack of an
effective way to fit generic S${\rm\acute{e}}$rsic profiles in the $uv$-plane,
as supported by common software packages dedicated to the calibration and
analysis of radio-interferometric data (e.g., CASA, AIPS, MIRIAD, GILDAS; a
summary of profiles provided by each software package in the visibility
modeling is given by Martí-Vidal et al., 2014).

Visibilities are the immediate output of interferometeric observations that
sample the Fourier transform of the sky brightness distribution and consist of
amplitudes and phases (depending on angles between antenna pairs as projected
on the sky). The amplitudes and phases can be represented as imaginary numbers
in the $uv$-plane, with their $uv$ elongation depending on antenna separation
and frequency. As the Earth rotates, each pair of antennas in the
interferometric array will trace out an elliptical track in the $uv$-plane.
The Fourier transform of the measured visibilities produce a dirty image. The
point-spread function (PSF) of the resulting image, known as dirty beam, has a
complex shape with, even in case of relatively good sampling of the
$uv$-plane, numerous positive and negative sidelobes, extending to large
spatial scales.

As each data element in the $uv$-plane (visibility) affects the data at all
spatial scales via the Fourier transform, the signal as well as the noise in
the resulting images are always strongly correlated, especially on scales
according to the full width at half maximum (FWHM) of the dirty beam. This
correlation can significantly impact the measurements of sources’ structure
(e.g., Condon, 1997; Martí-Vidal et al., 2014; Pavesi et al., 2018; Tsukui et
al., 2023). When analyzing image-based measurements, it has been shown that
ignoring signal correlation in interferometric images can lead to a
significant underestimation of statistical uncertainties and, consequently,
misinterpreted results (Tsukui et al., 2023). However, many recent studies
(e.g., Elbaz et al., 2018; Fujimoto et al., 2018; Hodge et al., 2019; Lang et
al., 2019; Fudamoto et al., 2022) measured submillimeter morphologies in the
image plane using tools such as galfit (Peng et al., 2002, 2010). These tools
are optimised for optical/near-IR observations and, as such, do not account
for the complex noise correlation typical of interferometric observations.

Interferometric images are often cleaned during deconvolution, which replaces
the dirty beam with well-defined PSFs. However, these routines are based on
strong assumptions and are not fully objective. Moreover, they do not account
for the strong correlation between pixels on the scale of the beam. For
example, high-fidelity observations of $z\sim 3$ star-forming galaxies by
Rujopakarn et al. (2019) using ALMA show that the small-scale source
structures of galaxies are affected by both the weighting scheme and the
cleaning (deconvolution) algorithm in imaging procedures.

In contrast, working directly in the $uv$-plane would be preferable because
the measured data points are independent there-in. For example, it has been
suggested that analyzing observations of very compact sources in Fourier space
is more reliable than image-based analyses (Martí-Vidal et al., 2012).
However, the lack of tools that allow general profile fitting in the
$uv$-plane is an obstacle. Typical codes allows for Gaussian fitting, in
addition to the standard PSF fitting, and sometimes exponential profiles
(i.e., the particular case of $n=1$ for the S${\rm\acute{e}}$rsic profile).

One possible way to approximate the standard surface brightness profile (i.e.,
general S${\rm\acute{e}}$rsic) of galaxies is to use linear superpositions of
Gaussians (Hogg & Lang, 2013). However, this approach has limitations. For
example, it is difficult to precisely measure how the light is concentrated in
the center of galaxies, and the comparison to results obtained in
optical/near-IR bands becomes prohibitive. This is because general
S${\rm\acute{e}}$rsic profiles are not analytically transformable into Fourier
space, making it computationally intensive to fit a S${\rm\acute{e}}$rsic
profile to visibilities that requires large numbers of numerical Fourier
transforms while fitting the model to the data.

Quite conveniently, the issue with the lack of analytical Fourier-
transformability of the S${\rm\acute{e}}$rsic profile has already been
addressed in the framework of optical imaging, where it became necessary to
account for spatially and temporally varying PSFs, that requires
computationally intensive convolutions. Spergel (2010) proposed a solution
based on the incomplete Bessel function of the third kind. This function
closely approximates S${\rm\acute{e}}$rsic functions and is commonly known as
the Spergel profile (see Section 2.2 for a description of the profile). The
Spergel profile has been recently also incorporated into the MAPPING procedure
of GILDAS 111http://iram.fr/IRAMFR/GILDAS (Guilloteau & Lucas, 2000). This
allows for the study of galaxies morphology in interferometric observations
with unprecedented detail, enabling comparison to optical studies. The idea is
to model galaxy structures accurately using functions that approximate the
S${\rm\acute{e}}$rsic profile in the $uv$-plane (i.e.,
pseudo-S${\rm\acute{e}}$rsic).

Modeling the galaxy submillimeter structure using a Spergel profile has been
discussed for selected examples of high-redshift star-forming galaxies (Kalita
et al., 2022; Rujopakarn et al., 2023). However, despite the novelty of the
exercise, there have been no attempts in the literature to systematically
characterise the performances of Spergel modeling of light profiles of
galaxies in the $uv$-plane. This includes the returned fidelity and accuracy
in parameter estimation, required signal-to-noise ratios (S/N) for attempting
complex modeling, and possible degeneracies between parameters. This is
similar to what has been extensively done in the optical images for galfit or
other tools during the last decades (e.g., Moriondo et al., 2000; Pignatelli
et al., 2006; Häussler et al., 2007; Mancini et al., 2010; Hoyos et al., 2011;
Hiemer et al., 2014; Lange et al., 2016; Tortorelli & Mercurio, 2023, and many
others). Additionally, the conversion of Spergel indices to Sersic indices,
which enables the comparison of submillimeter/millimeter measurements of
Spergel profiles to those derived from the classic approach of galaxy light
profile modeling (i.e., galfit; Peng et al., 2002), was only briefly explored
by Spergel (2010).

In this paper, we aim to provide a technical assessment of the use of the
Spergel profile. Our investigation focuses on the robustness of profile
fitting with Spergel models using simulated data with real noise from actual
observations, and compares it with the traditional application of profile
fitting on interferometric reconstructed images. We plan to use these results
to perform Spergel modeling on a large galaxy sample taken from the ALMA
archive in a forthcoming paper (Q. Tan et al. in preparation).

Figure 1: Left: Comparison of surface density profiles for
S${\rm\acute{e}}$rsic (solid lines) and Spergel (dashed lines) functions. The
light profile with S${\rm\acute{e}}$rsic index of 0.5, 1, 2, 4, and 5, and
Spergel index of 0.85, 0.5, -0.3, -0.6, and -0.7, are shown in different
colors. Right: Comparison of the cumulative distributions for
S${\rm\acute{e}}$rsic (solid lines) and Spergel (dashed lines) functions. By
definition, $R_{\rm e}$, contains half of the integrated galaxy light.

The paper is structured as follows. In Section 2, we recall the definition of
the radial profile functions used to constrain the light profile of galaxies
in the image-plane (S${\rm\acute{e}}$rsic) and $uv$-plane (Spergel),
respectively. We compare the two profiles and discuss how their parameters can
be converted from one to the other, for this comparison. Section 3 introduces
the method used to generate the simulated data, with a description of the
uv$\\_$fit algorithm. In Section 4, we present the results of a study that
tested the robustness of profile fitting in the $uv$-plane against model fits
to image data. This includes a comparison of the recovery of the structural
parameters and the accuracy of the measurements. The analysis of the absolute
accuracy of $uv$-plane modeling, the reliability of parameter uncertainties,
and the covariance of fitted parameters are described in Section 5. In Section
6, we discuss the simulation results and explore possible reasons for the
different measurement performances of interferometric data. We also discuss
the implication for the study of galaxy morphology, using an example based on
re-examining previously published ALMA data. The conclusions of the paper are
presented in Section 7.

##  2 S${\rm\acute{e}}$rsic and Spergel radial profile functions

In this section, we first recall the definitions of the S${\rm\acute{e}}$rsic
and Spergel profiles. Then we proceed to a qualitative comparison between them
as their indices vary, emphasizing the role played by the spatial scales
actually observed. Finally, we present empirical recipes to convert Spergel
indices to their equivalent S${\rm\acute{e}}$rsic ones, as well as effective
radii and total fluxes. These recipes will be used throughout the paper to
directly compare Spergel and S${\rm\acute{e}}$rsic fits under realistic noise
conditions from our simulations.

###  2.1 The S${\rm\acute{e}}$rsic profile

As a generalization of the $r^{1/4}$ law, the $r^{1/n}$ profile first proposed
by Sersic (1968) is one of the most common functions used to describe how the
intensity of a galaxy varies with distance from its center. The surface
density (or equivalently surface brightness) of the S${\rm\acute{e}}$rsic
profile can be written as

$\Sigma(R)=\Sigma_{\rm e}{\rm exp}\left[-\kappa\left(\left(\frac{R}{R_{\rm
e}}\right)^{1/n}-1\right)\right],$ (1)

where $R$ is the projected distance to the source center, $R_{\rm e}$ is the
effective radius containing half of the total luminosity, $\Sigma_{\rm e}$ is
the surface brightness at $R_{\rm e}$, and $n$ is the S${\rm\acute{e}}$rsic
index which determines the shape of the light profile (see Fig. 1). The
parameter $\kappa$ is a function of S${\rm\acute{e}}$rsic index and is such
that $\Gamma(2n)=2\gamma(2n,\kappa)$, where $\Gamma$ and $\gamma$ represent
the complete and incomplete gamma functions (Ciotti & Bertin, 1999),
respectively. We will use the terms half-light radius or effective radius
interchangeably to refer to the radius within which half of a galaxy’s
luminosity is contained.

The S${\rm\acute{e}}$rsic index, $n$, determines the degree of curvature of
the profile, with $n=0.5$ giving a Gaussian profile, $n=1$ an exponential disk
profile and $n=4$ generally associated with galaxy bulges. As the index $n$
increases, the core steepens more rapidly for $R<R_{\rm e}$, and the intensity
of the outer wing at $R>R_{\rm e}$ is significantly extended.

However, as mentioned in the introduction, the general S${\rm\acute{e}}$rsic
profile is not analytically transformable in Fourier space for most values of
the parameter $n$, as the parameter $\kappa$ cannot be solved in closed form.
Various techniques have been developed to address this issue when calculations
of the Fourier transform are needed, including numerical integration methods,
approximations, and asymptotic expressions (e.g., Ciotti & Bertin, 1999;
Mazure & Capelato, 2002; Baes & Gentile, 2011). The inability to solve the
S${\rm\acute{e}}$rsic profile analytically in Fourier space creates challenges
in certain scenarios. For example, when performing convolutions, such as
corrections for seeing, or when directly working in Fourier space, like with
interferometric data.

###  2.2 The Spergel profile

Spergel (2010) introduced an alternative to the S${\rm\acute{e}}$rsic model
for galactic luminosity profiles with functional form

$\Sigma_{\nu}(R)=\frac{c_{\nu}^{2}L_{\rm 0}}{R_{\rm
e}^{2}}f_{\nu}\left(\frac{c_{\nu}R}{R_{\rm e}}\right)$ (2)

where
$f_{\nu}(x)=\left(\frac{x}{2}\right)^{\nu}\frac{K_{\nu}(x)}{\Gamma(\nu+1)}$,
$\Gamma$ is the Gamma function, $K_{\nu}$ is a modified spherical Bessel
function of the third kind, $c_{\nu}$ is a constant, $R_{\rm e}$ is the half-
light radius, and $\nu$ is known as Spergel index that controls the relative
peakiness of the core and the relative prominence of the wings (similar to
S${\rm\acute{e}}$rsic $n$), with a theoretical limit of $\nu>-1$.

This family of functions is found to provide a good fit for galaxy light
profiles and resembles the S${\rm\acute{e}}$rsic function over a range of
indices. The Spergel profile at $\nu=0.5$ is identical to an exponential
profile, which is equivalent to a S${\rm\acute{e}}$rsic profile with $n=1$.
However, the two functions do not exactly coincide for different Spergel $\nu$
(see Fig. 1).

The Spergel profile has a significant advantage over the S${\rm\acute{e}}$rsic
profile because it is analytic in both real space and Fourier space (Spergel,
2010), which means that it can be described mathematically using equations,
making it easier to work with and analyze in the $uv$-plane.

###  2.3 Qualitatively relating the S${\rm\acute{e}}$rsic and Spergel profiles

In the left panel of Fig. 1, a comparison is shown between
S${\rm\acute{e}}$rsic and Spergel profiles. This comparison covers a range of
$n$ and $\nu$ indices. All the profiles are normalized at $R_{\rm e}$. Within
a certain range of $\nu$, the Spergel profiles resemble S${\rm\acute{e}}$rsic
profiles in shape such that a relation can be conceived between $\nu$ and $n$.
For example, the Spergel profile at $\nu=-0.6$ and in the radial range near
the effective radius, exhibits similarities to a de Vaucouleurs $n=4$ profile
(see also Spergel, 2010). However, when elongating far from the normalization
point, these profiles start to differ at both the innermost (i.e., $R<0.1\
R_{\rm e}$) and outermost (i.e., $R>5\ R_{\rm e}$) regions, indicating that
converting one index into the other depends on the observed scales. This is
generally the case for all S${\rm\acute{e}}$rsic models with $n>1$, respect to
their first-order matching Spergel profiles with $-1<\nu<0.5$: they display
steeper inner profiles and drop faster at large radii. This is discussed more
further down in this Section.

We also see from Fig. 1 that Spergel models are not meant to reproduce
S${\rm\acute{e}}$rsic models with a flatter shape than $n=1$, i.e., $n<1$. For
example, even choosing $\nu=0.85$ the Spergel model only slightly flattens
compared to the $\nu=0.5$ ($n=1$) case, and it remains far from resembling a
Gaussian model $n=0.5$. Given that fitting Gaussian models in the $uv$-plane
is a straightforward approach, we suggest using this method instead of
attempting Spergel fits with high values of $\nu>1$.

In the right panel of Fig. 1, we show cumulative distributions for the
S${\rm\acute{e}}$rsic and Spergel profiles. When truncated at $10\ R_{\rm e}$,
an exponential profile (S${\rm\acute{e}}$rsic $n=1$ and Spergel $\nu=0.5$)
retains almost all of its flux ($\sim 100\%$). In contrast, a
S${\rm\acute{e}}$rsic profile with $n=4$ retains 96.1$\%$ of its flux, while
99.7$\%$ of the flux is contained within this radius for a Spergel profile
with $\nu=-0.6$. At the small radius, within the inner $0.05\ R_{\rm e}$, an
exponential profile contains 0.3$\%$ of the flux, while $n=4$ contains 3.2$\%$
of the flux. For a Spergel profile with $\nu=-0.6$, 5.4$\%$ of the flux is
contained within this radius. As the index $n$ increases (or decreases for
$\nu$), the differences between the fluxes contained within the radius of the
small and large ends also somewhat increase, while remaining overall
contained. A comparison of the S${\rm\acute{e}}$rsic profile with the Spergel
profile using mathematical simulations is presented in Appendix A (see Fig.
16).

Figure 2: Comparison S${\rm\acute{e}}$rsic and Spergel indexes. The data
(crosses) in the top-panel represent the galfit S${\rm\acute{e}}$rsic indices
measured for sources created with input Spergel model, for a range of $\nu$
values from $-0.7$ to 0.5, and for different source effective radii (expressed
in terms of the FWHM of the synthesized beam, $R_{\rm e}/\theta_{\rm b}$)
ranging from 0.1 to 2.0. The solid curves in the top panel denote the best-fit
empirical relation between Spergel $\nu$ and galfit S${\rm\acute{e}}$rsic $n$,
depending on $R_{\rm e}/\theta_{\rm b}$, as expressed by Eq. (3). The lower
panel displays normalised residuals to the proposed relation.

Figure 3: Similar to Fig. 2, but showing the ratio of effective radii (left)
and total fluxes (right) obtained from fitting S${\rm\acute{e}}$rsic models
simulated galaxies created using Spergel profiles (crosses), plotted as a
function of their Spergel index. The solid curves denote the best-fit
empirical relations as expressed by Eq.(4). The bottom panels show normalised
residuals to the best-fit, which are largely within 10%.

###  2.4 Converting the Spergel index into the equivalent
S${\rm\acute{e}}$rsic index

To empirically derive the conversion (spatial-scale dependent) between Spergel
$\nu$ and the S${\rm\acute{e}}$rsic $n$, we create noise-free222 The model
sources were created with an extremely high S/N to ensure a robust galfit
measurement images of Spergel two-dimensional models, as described in the next
section, and use galfit to measure the corresponding S${\rm\acute{e}}$rsic
index. In Fig. 2, the cross symbols represent the galfit measurements, which
we refer to the intrinsic best value and are labelled as $n_{\rm galfit}$,
compared to the input $\nu$ for different source sizes of $R_{\rm
e}/\theta_{\rm b}$ ranging from 0.1 to 2.0. Here $\theta_{\rm b}$ is defined
as the synthesized circularized beam size, given by $\sqrt{ab}$, where $a$ and
$b$ represent the FWHMs of the major and minor axes of the synthesized beam,
respectively.

As a zero-order check of the procedure, Fig. 2 shows that for a Spergel
profile with $\nu=0.5$, sources with different sizes all return a
S${\rm\acute{e}}$rsic index of $n=1$ in galfit, as expected. This result is
accurate within $3\%$, which represents the inherent maximal precision of this
empirical calibration. We find that $\nu=-0.6$ corresponds to $n=4$ for
R${}_{\rm e}/\theta_{\rm b}\sim 0.9$–1.0, which is close to when the FWHM
scale of the galaxy and of the beam are identical, as expected (Spergel,
2010). However, when R${}_{\rm e}/\theta_{\rm b}<0.5$ an input $\nu=-0.6$
converts to $n\sim 2$–3, while models with $\nu=-0.6$ and R${}_{\rm
e}/\theta_{\rm b}>1$ correspond to steeper $n>4$. The dependence of the
conversion on R${}_{\rm e}/\theta_{\rm b}$ systematically decrease with
increasing the Spergel indices, and nearly vanishes for $\nu>0$ (Fig. 2).

We find that the whole set of measurements can be well described by the form:

$n(\frac{R_{\rm e}}{\theta_{\rm b}},\nu)\sim p_{1}\frac{R_{\rm e}}{\theta_{\rm
b}}{\rm exp}(p_{2}\nu)+p_{3}\nu^{2}+p_{4}\nu+p_{5}$ (3)

for sources with $R_{\rm e}/\theta_{\rm b}$ ranging from 0.1 to 2.0,
respectively. The solid lines in Fig. 2 represent the best-fit relation with
coefficients $p_{1}=0.0249$, $p_{2}=-7.72$, $p_{3}=0.191$, $p_{4}=-0.721$, and
$p_{5}=1.32$. The residuals of the fit indicate that the uncertainties of
S${\rm\acute{e}}$rsic $n$ are largely within 10% for model sources with
$R_{\rm e}/\theta_{\rm b}$ of 0.1–2.0 at $\nu>-0.7$ (see Fig. 2). We confirm
this trend when comparing the Spergel index with the S${\rm\acute{e}}$rsic
index analytically through a mathematical matching between the
S${\rm\acute{e}}$rsic and Spergel functions (see Fig. 17).

This analysis demonstrates that converting a Spergel index to a
S${\rm\acute{e}}$rsic index depends on the ratio between the angular size of
the galaxy and that of the beam of the observations being examined. We have
mapped this relationship over a reasonably large range of this ratio, i.e.,
$R_{\rm e}/\theta_{\rm b}$ of 0.1–2.0. Anticipating results from forthcoming
paper, which focuses on analyzing the morphologies of an ALMA archival sample
of about 100 distant star-forming galaxies in the submillimeter bands (Q. Tan
et al., in preparation), we find that around 82% (93%) sources fall within the
range of $R_{\rm e}/\theta_{\rm b}=0.1-1.0\ (0.1-2.0)$, while approximately
88% (99%) of sources exhibit best-fitting $\nu>-0.7$ ($\nu<1$). The median
$R_{\rm e}/\theta_{\rm b}$ is 0.32 and the semi-interquartal range is 0.2–0.6.
This suggests that the calibration presented in Fig. 2 is representative of
general observations of distant galaxies with ALMA and NOEMA.

###  2.5 Converting half-light radii and total fluxes from Spergel to
S${\rm\acute{e}}$rsic

The differences in the profiles imply that also for sizes (half-light radii)
and total fluxes, a conversion might be required when comparing results based
on Spergel to those from S${\rm\acute{e}}$rsic. Based on our simulations, we
find that both the size and flux density estimated by galfit using a
S${\rm\acute{e}}$rsic profile tend to be larger than when using Spergel, as
the S${\rm\acute{e}}$rsic index becomes larger. Instead, axis ratios and
position angles are unaffected. Fig. 3 shows the ratios of $R_{\rm e}$ and
flux densities measured from fitting S${\rm\acute{e}}$rsic models to simulated
galaxies created as Spergel models, as a function of Spergel index. Both the
ratio of $R_{\rm e}$ and flux density exhibit a similar increasing trend as
the profile get steeper. The ratio becomes larger for better resolved sources
(i.e., larger $R_{\rm e}/\theta_{\rm b}$). For example, in the case of a
large-sized ($R_{\rm e}/\theta_{\rm b}=1$) galaxy with a de Vacouleurs-like
profile of $\nu=-0.6$, the $R_{\rm e}$ estimated by the S${\rm\acute{e}}$rsic
model can be larger by about 30–40%, while the excess flux density is around
15–20%. By comparison, the relative increase in $R_{\rm e}$ and flux density
are less significant for sources with flatter profile or smaller sizes
compared to the beam.

To correct for these systematical biases and thus to enable comparison of
measurements using Spergel and S${\rm\acute{e}}$rsic profiles, we fit the
distribution of the ratio of half-light radius and total flux measured from
S${\rm\acute{e}}$rsic to the Spergel-based input values. To distinguish the
fitted parameters between those obtained from Spergel versus
S${\rm\acute{e}}$rsic profile modeling, we label $R_{\rm e}$ and $S_{\rm sp}$
the half-light radius and total flux measured from Spergel profile fitting,
respectively. For S${\rm\acute{e}}$rsic-based measurements, we use instead
$R_{\rm e,se}$ and $S_{\rm se}$. We find that both ratios of $R_{\rm
e,se}/R_{\rm e}$ and $S_{\rm se}/S_{\rm sp}$ can be accurately described by a
similar form:

$r(\frac{R_{\rm e}}{\theta_{\rm b}},\nu)\sim p_{1}(\frac{R_{\rm
e}}{\theta_{\rm b}})^{2}{\rm exp}(p_{2}\nu+p_{3}\frac{R_{\rm e}}{\theta_{\rm
b}})+p_{4}\nu+p_{5}$ (4)

for sources with $R_{\rm e}/\theta_{\rm b}$ ranging from 0.1 to 2.0. For the
size ratio of $R_{\rm e,se}/R_{\rm e}$, the best-fit gives coefficients
$p_{1}=0.00138$, $p_{2}=-8.96$, $p_{3}=0.260$, $p_{4}=-0.0260$, and
$p_{5}=0.996$, while for the flux ratio of $S_{\rm se}/S_{\rm sp}$, the best-
fit coefficients are $p_{1}=0.00217$, $p_{2}=-7.43$, $p_{3}=0.149$,
$p_{4}=0.00942$, and $p_{5}=1.00$ (see the solid lines in Fig. 3). Analytical
calculations, matching the Spergel profile with S${\rm\acute{e}}$rsic profiles
numerically (see Appendix A), fully confirm the trends encoded in the Eq.(4)
above and in Fig. 3.

We caution that while we believe that our methodology captures the bulk of the
systematic effects in the conversion as encoded in the $R_{\rm e}/\theta_{\rm
b}$ ratio, some further systematics might be expected depending on higher
order terms describing the actual shape of the beam. We derived best fitting
parameters for Eq.(3) and Eq.(4) averaging over three different ALMA array
configurations. By comparing to results from single ALMA array configurations,
we estimate that further systematic uncertainties are small. The details of
the three ALMA array configuration are summarized in Table 1 (see Appendix B).

##  3 Technical aspects: simulating galaxies under realistic noise conditions,
and measuring their properties

###  3.1 Realistic noise map

Each simulated galaxy was created by inserting a model source signal into an
empty dataset with realistic noise. The noise was obtained from real data
using ALMA band 7 observed visibilities from galaxies in a recent survey
(e.g., Puglisi et al., 2019, 2021; Valentino et al., 2020).

To analyze the data, the calibrated ALMA visibilities for several targets were
exported from CASA using the exportuvfits. The exported data was then
converted to $uv$-tables using the GILDAS fits$\\_$to$\\_$uvt task. The four
spectral windows were combined using the uv$\\_$continuum and uv$\\_$merge
tasks. Prior to introducing the model source into the $uv$-plane, any detected
sources were subtracted from the visibility data using the best-fitting
Spergel model to the visibilities. This produced a residual data set which,
upon inspection, did not reveal any further source.

Fig. 4 (left) shows the map of the residual $uv$ data for a case in our
simulations: the primary beam field of view (FOV) is 18′′, and the FWHM of the
synthesized beam using natural weighting is $\theta_{\rm
b}=1.0^{\prime\prime}$. The right panel of Fig. 4 shows the pixel
distribution, which can be fitted well with a Gaussian profile. This indicates
that the data is mostly noise without any other significant features. The
noise level measured from the Gaussian fit is 40 $\mu$Jy beam-1. We refer to
the rms noise derived in this way as $\sigma_{\rm b}$ in the following
sections. This is an objective characterization of the noise in the data,
independent on source properties.

We emphasize that each empty galaxy map can be used for a large number of
independent simulations. This can be achieved by placing simulated signal at
random positions within the primary beam. Considering that half-synthesized-
beam offsets produce an independent noise realization, the number of such
independent realizations are of order 1000 for each empty ALMA band-7 dataset.

Figure 4: Left: One of the empty datasets used for our simulations, after
subtracting a source (which was close to the phase center) from the
visibilities. The dashed white circle represents the primary beam, i.e., the
FOV of the data set. Right: Histogram of pixel values within the primary beam.
The red line represents the best-fitting Gaussian model. The fit suggests that
the noise data is at least approximately Gaussian.

###  3.2 Source models

The model sources were created using GILDAS through the MAPPING task
uv$\\_$fit. We generate elliptical Spergel model sources by fixing their seven
free parameters: centroid position, flux density, effective radius $R_{\rm
e}$, minor-to-major axis ratio, position angle (PA), and Spergel index
$\nu$333In practice this is done by fitting to the empty data a model with all
parameters fixed and negative flux. The residual image will then have the
desired model added (and positive).. The flux density varies over a range with
step sizes of factors of two. This range corresponds to a total flux density
that is normalized by the noise, varying from 25 to 400, which is represented
by the ratio of integrated flux density to the pixel rms noise
(Stot/$\sigma_{\rm b}$). We define S/N as the ratio of the integrated flux
density to the noise per beam Stot/$\sigma_{\rm b}$: the advantage of this
choice lays in its model-independence and reliance on very basic properties of
the source and of the noise. Note that for extended sources, the S/N defined
in this way is obviously higher than the S/N eventually recovered for the
integrated flux density coming from a full Spergel/S${\rm\acute{e}}$rsic
profile.

The simulation process involves setting the size of the sources in units of
$\theta_{\rm b}$, i.e., $R_{\rm e}/\theta_{\rm b}$ = 0.1, 0.2, 0.4, and 0.7,
to represent very compact, small, intermediate, and large-sized (relative to
the beam) sources, respectively. For each set of Monte Carlo (MC) sources that
share a fixed effective radius and axis ratio $q$ ($\equiv b/a$), the Spergel
model sources vary in both their flux density (thus, S/N) and Spergel index
with values of $\nu=$ 0.5, $-0.3$, $-0.5$, $-0.6$, and $-0.7$. The position
angle remain constant. To mimic real observations, we add the model sources to
the realistic noise data that is derived from observed visibilities, to
produce a simulated data set (see Fig. 4 for an example).

###  3.3 Visibility model-fitting with Spergel profile

Fitting elliptical Spergel models to the simulated dataset is performed using
the task uv$\\_$fit. All the seven fitted parameters are allowed to vary. As
typically done within MAPPING’s uv$\\_$fit in GILDAS, we generate a range of
initial guesses for each fitted parameter, which are built into a
N-dimensional list of combinations of these guesses. This approach helps to
explore a wider range of possible solutions and thus identify the best fit
model and return a well-sampled range for uncertainties. We test the fit by
setting the starting range parameters of initial guesses within a factor of 2
centered on the input mock values and the number of start parameters (e.g., 3
guesses for each parameter). We found that for simulated sources (where model
parameters are known a priori), the results using single initial guesses
identical to the real parameters are not significantly different from when the
code is run using multiple initial guesses for each parameter. However, using
uv$\\_$fit with a large range of initial guesses is critical for real
observations, where the source’s properties are not known beforehand.

Figure 5: Example of a simulated source generated with a Spergel profile of
$S_{\rm tot}/\sigma_{\rm b}=$50, $R_{\rm e}/\theta_{\rm b}$=0.4, axis ratio of
$q$ = 0.75, position angle of PA = 30∘, and Spergel index of $\nu=-0.6$. From
left to right: the dirty image (top-left), dirty beam (PSF used for
convolution in galfit; bottom-left), best-fit source models convolved with the
dirty beam (middle), and residuals after subtracting the model source (right).
The model source and the model-subtracted residual shown in the top and bottom
rows were derived from the uv$\\_$fit and galfit fits, respectively. Each
image cutout is $9^{\prime\prime}\times 9^{\prime\prime}$. The contours start
from 2$\sigma$ and increase in steps of 4$\sigma$. White crosses mark the
best-fit source position obtained from uv$\\_$fit.

###  3.4 Image model-fitting with S${\rm\acute{e}}$rsic profile

Each $uv$-plane simulation is imaged and then fitted with a single-component
S${\rm\acute{e}}$rsic profile using galfit (Peng et al., 2002). The galfit run
is performed on the dirty maps, which are created by Fourier transforming
visibilities without cleaning. The (full) dirty beam is used as the galfit
PSF. The known parameters of the mock sources are used as initial guesses for
galfit. All parameters are left free without constraints in the fitting. The
initial guess of the S${\rm\acute{e}}$rsic $n$ is calculated by converting the
input $\nu$ using Eq. (3). The S${\rm\acute{e}}$rsic fits also provide
measurements of seven free parameters: central position, total magnitude,
effective radius, S${\rm\acute{e}}$rsic index, axis ratio, and PA. In some
cases, galfit may fail to provide accurate measurements. To ensure the
validity of our comparisons, we remove measurements in both the image- and
$uv-$plane for sources with output parameters marked as problematic in galfit.
All this procedure is extremely favorably biased towards positively amplifying
the performances of galfit.

Fig. 5 illustrates a typical case of a simulated source generated with a
Spergel profile and with $S_{\rm tot}/\sigma_{\rm b}$ ratio of 50.
Additionally, it shows the best-fit models derived from uv$\\_$fit and galfit,
respectively. Both methods provide good constraints to the simulated source at
this S/N ratio, as no significant component is visible in the residual map.

Figure 6: Results from simulations of a single model source. The distribution
of recovered parameters, from uv$\\_$fit to the visibilities (panels labelled
with uvfit) and galfit in the image plane (panels labelled with galfit),
allows us to measure their respective accuracy in recovering the intrinsic
known parameters (dashed vertical lines), including the flux density and
structural parameters ($R_{\rm e}$, $q$, and $n$). We present two examples
using a Spergel profile as input source: $R_{\rm e}/\theta_{\rm b}$ = 0.4, $q$
= 0.75, PA = 30∘, $\nu=-0.6$ (top-two panels), and $R_{\rm e}/\theta_{\rm b}$
= 0.2, $q$ = 0.6, PA = 30∘, $\nu=0.5$ (bottom-two panels). The distribution of
recovered parameter is color-coded by flux density ($S_{\rm tot}/\sigma_{\rm
b}=$25, 100, and 400, indicated by different colors in the inset panel). The
S${\rm\acute{e}}$rsic indices shown in the uvfit panels are obtained by
converting the best-fit Spergel indices, based on Eq. (3). For all parameters,
we have kept into account the conversion from Spergel-based fits to
S${\rm\acute{e}}$rsic-based fits discussed in Section 2 (see Eqs.(3) and (4)),
to remove any underlying systematics coming from the difference in the
profiles.

###  3.5 Details of the UV_FIT implementation in GILDAS

The uv_fit command uses the SLATEC/DNLS1E implementation of the Levenberg-
Marquardt algorithm (see Press et al., 1992, for an intuitive presentation) to
minimize the reduced $\chi^{2}$ of this non-linear least-square problem. This
algorithm only requires to deliver a routine that computes the complex
function and its partial derivatives with respect to the different fitted
parameters. Appendix C delivers the equations for the elliptical Spergel
profile and its partial derivatives. Once the minimum of the least-square
problem is found, the routine SLATEC/DCOV is called to compute the covariance
matrix at this minimum. The diagonal elements of this covariance matrix are
the $\pm 1\sigma$ uncertainties on each fitted parameters.

In estimation theory, the Fisher matrix, $\boldsymbol{I}_{F}$, quantifies the
amount of information in the least-square problem. When the noise on the
measurements (the visibilities) is well modeled by an uncorrelated centered
white Gaussian random variable of standard deviation $\sigma$, the computation
of the Fisher matrix reduces to (Stoica & Moses, 2005)

$\forall(i,j)\quad\left[\boldsymbol{I}_{F}\right]_{ij}=\sum_{k=1}^{n_{\mathrm{visi}}{}}\frac{1}{\sigma_{k}^{2}}\frac{\partial
V_{k}}{\partial\varphi_{i}}\frac{\partial V_{k}}{\partial\varphi_{j}},$ (5)

where $[\boldsymbol{I}_{F}]_{ij}$ stands for the term $(i,j)$ of the Fisher
matrix, $\sigma_{k}$ and $V_{k}$ are the noise and the fitted visibility
function for visibility $k$, and $(\varphi_{i})$ is the vector of fitted
parameters. In our case,
$(\varphi_{i})=(x_{0},y_{0},L_{0},R_{\mathrm{maj}},R_{\mathrm{min}},\mathrm{\phi},\nu)$,
i.e., the central position of the Spergel profile as an offset with respect to
the phase center, its luminosity, its major and minor half-light radius, its
position angle, and its index. The Cramer-Rao Bound (CRB) for each fitted
parameter, ${\cal B}(\varphi_{i})$, is defined as the $i$th diagonal element
of the inverse of the Fisher matrix

${\cal B}(\varphi_{i})=\left[\boldsymbol{I}_{F}^{-1}\right]_{ii}.$ (6)

The CRB is the reference precision of the least-square problem. Indeed, the
variance of any unbiased estimator of the parameter $i$ will always be larger
than the associated CRB (Garthwaite et al., 1995), or

$\mathrm{var}(\varphi_{i})\geq{\cal B}(\varphi_{i}).$ (7)

In other words, an efficient fitting algorithm will deliver variances for the
estimated parameters, which reach the associated CRB values. In practice,
sufficiently large signal-to-noise ratios are required to ensure that the
$\chi^{2}$ minimization converges towards the actual solution. Additional
explanations and an example of application to the fit of CO(1-0) profiles in
the local inter-stellar medium can be found in Roueff et al. (2021).

##  4 Analysis: Comparison of $uv$-plane and image plane performances

In this section, we compare the parameter estimates obtained by fitting
Spergel profile in the $uv$-plane and S${\rm\acute{e}}$rsic profile in the
image-plane to the same data, and thus comparatively assess their performances
for the recovery of all the key parameters. Additionally, we investigate the
reliability of the uncertainties returned by the fitting codes. For all
parameters, the systematic terms that arise due to the intrinsic differences
in the profiles (Section 2, see Eqs.(3) and (4)) are always included in the
comparison.

###  4.1 Comparison of structural parameter measurements

In Fig. 6, we present the results of our simulations, using only two models of
galaxies as typical examples, for clarity. The input mock source was chosen
with $R_{\rm e}/\theta_{\rm b}$ of 0.4 (0.2), axis ratio $q$ of 0.75 (0.6),
and Spergel index $\nu$ of $-0.6$ (0.5). The distribution of recovered fitted
parameter is shown in the top-two (bottom-two) panels. The $\nu=0.5$ case is
particularly useful as it provides perfect coincidence with the Spergel and
Sersic ($n=1$) models.

We extracted the distribution of the structural parameters obtained by fitting
general Spergel profiles (panels labelled with uvfit) and
S${\rm\acute{e}}$rsic profiles (panels labelled with galfit) with all
parameters free, respectively (Fig. 6). The distribution of recovered best
fitting values is shown for flux density, size, axis ratio, and
S${\rm\acute{e}}$rsic index, where the true values are shown by vertical
lines. The dispersion of recovered values can be used as a gauge of the
measurement uncertainty. The average difference between recovered and true
values probes any measurement biases and accuracy. In all cases, the scatter
of the distributions decreases as the S/N of the simulated sources increases,
as expected. Similarly, any systematic bias decreases with S/N, both in the
image plane and in the $uv$-plane.

These two examples demonstrate that, when comparing fits in the $uv$-plane to
measurements of each structural parameter obtained from profile fitting in the
image-plane with galfit, it is clear that the latter exhibit larger scatter
and have larger uncertainties. Also, there is evidence for systematic biases
that are more pronounced for image-plane fitting with galfit, especially at
low S/N and clearly visible for both the low and high S${\rm\acute{e}}$rsic
cases (albeit larger for the latter).

The systematic relative deviations of all key parameters of the fit (biases)
for a larger variety of input parameters are shown in Fig. 7, which again
shows how the S${\rm\acute{e}}$rsic fits gets increasingly biased at low S/N
much more rapidly than uv-plane fits. As the biases vanishes at the highest
S/N even for the S${\rm\acute{e}}$rsic case, we conclude that these are not
due to the previously discussed systematic differences in the profiles.

Fig. 8 compares the measurement uncertainty between $uv$-plane and image-
plane. To enhance the readability of the figure, we plotted only the
measurements obtained from fitting model sources with a size of $R_{\rm
e}/\theta_{\rm b}$ = 0.2 and 0.7, and flux density of $S_{\rm tot}/\sigma_{\rm
b}$ = 25, 100, and 400 in Figs. 7 and 8. The following sections provide
detailed results on the recovery bias of individual key parameters. Then, the
focus shifts to the comparative uncertainty in parameters estimates. Again, we
emphasize how the case $\nu=0.5$/$n=1$ is included, where the
S${\rm\acute{e}}$rsic and Spergel models are identical, and it behaves fully
similar to the other $\nu$/$n$ cases, demonstrating that the results are not
driven by small differences between models at higher $n$.

Figure 7: Relative difference between the input and measured parameters, flux
density, effective radius, axis ratio, and S${\rm\acute{e}}$rsic index (from
left to right), obtained from Spergel fits to the $uv$-plane (top panels) and
S${\rm\acute{e}}$rsic fits in the image-plane using galfit (bottom panels) in
our simulations, plotted as a function of light concentration (i.e.,
S${\rm\acute{e}}$rsic index) for various $S_{\rm tot}/\sigma_{\rm b}$ and
source sizes. Sources with a size of $R_{\rm e}/\theta_{\rm b}=$ 0.2 and 0.7
are marked by diamond and triangle respectively, while the inset panel of
colorbar show the different input flux density. The error bars denote the
interquartile range of the distribution divided by the square root of the
number of simulations.

####  4.1.1 Flux measurements

The left panels of Fig. 7 show the median relative difference between the
recovered and input flux densities, given by $(S_{\rm out}-S_{\rm in})/S_{\rm
in}$, plotted against S${\rm\acute{e}}$rsic $n$ (converted from input $\nu$
using Eq. (3)). For both methods, there is a positive bias in recovering the
flux density, leading to an overestimate of the flux. The magnitude of this
bias depends on both the S/N (Stot/$\sigma_{\rm b}$) and the source extension
($R_{\rm e}/\theta_{\rm b}$).

For image-plane fitting with galfit our simulations show that even relatively
compact sources with Stot/$\sigma_{\rm b}$ of 25 exhibit systematic errors of
$>20$% in the recovered flux densities. Large-sized sources with $R_{\rm
e}/\theta_{\rm b}=0.7$ can be boosted by approximately $60\%$ of flux density
at S/N$\sim 25$. In contrast, the systematic biases of flux densities measured
from $uv$-fitting are much smaller, with a mean value of $\sim$5% at the
faintest S/N$\sim 25$.

####  4.1.2 Size measurements

In terms of size estimates, the $uv$-plane method shows significantly smaller
systematic offsets than image-plane measurements (see the second column panels
of Fig. 7). Both methods have higher relative biases on $R_{\rm e}$ than flux
density. In general, both methods tend to overestimate $R_{\rm e}$ for sources
with low $S_{\rm tot}/\sigma_{\rm b}$.

In the image-plane, for a compact, faint (Stot/$\sigma_{\rm b}=25$) source
with disk-like profile, the $R_{\rm e}$ can be overestimated by up to 70%. On
the other hand, the visibility-based $R_{\rm e}$ is on average overestimated
by only about 15% for the same model source.

####  4.1.3 Axis ratio measurements

In Fig. 7 (third column panels), the accuracy of recovering the axis ratio $q$
is compared by fitting in the image-plane and $uv$-plane, respectively. Both
image-based and visibility-based measurements tend to systematically
underestimate the recovered $q$ in most cases, but again the bias is much
stronger in the image-plane. In addition, the accuracy of $q$ estimates is
influenced by the size of galaxies, with the highest bias occurring for the
most compact sources.

####  4.1.4 S${\rm\acute{e}}$rsic index measurements

For image-based measurements, the S${\rm\acute{e}}$rsic $n$ estimate is often
biased towards a lower value in most cases (Fig. 7, fourth column). There is a
significant increase in systematic offsets of measured S${\rm\acute{e}}$rsic
$n$ as light concentration increases. In other words, the difference in
estimating $n$ becomes large at low S/N and when the galaxy profile is steep.
For the faintest source (i.e., Stot/$\sigma_{\rm b}=25$) with a de
Vaucouleurs-like profile in the simulation, the S${\rm\acute{e}}$rsic $n$
estimates can be underestimated by about 70%. The underestimation goes down to
30% when the source becomes less centrally concentrated with a disk-like
profile.

While the visibility-based concentration index shows systematic offsets at low
S/N, it is still much less biased than the image-based one. The bulk of the
error on $n$ estimates obtained from the $uv$-plane is limited to within about
20$\%$ or less. It should be noted that for very compact sources at the lowest
S/N, the estimates of $n$ tend to be higher than the true value. Furthermore,
the systematic error in the fit is greater for sources with an exponential
profile than those with a steeper profile.

Figure 8: Similar to Fig. 7, but we plot the ratio of the scatter of the
measurements derived from Spergel fits in the $uv$-plane to that derived from
S${\rm\acute{e}}$rsic fits in the image plane for fitted parameters. From left
to right, we show flux density, effective radius, axis ratio, and
S${\rm\acute{e}}$rsic index, as a function of light concentration (i.e.,
S${\rm\acute{e}}$rsic index).

###  4.2 Comparison of the relative measurement uncertainty

Apart from systematic biases, it is also important to evaluate the scatter of
the measurements, both in the uv-plane and in the image plane, to verify if
any of the two returns better measurements with lower scatter.

In order to evaluate such relative measurement uncertainty, we use the scatter
of the distribution of recovered values, evaluated as the median absolute
deviation (MAD) of the data around the true value (i.e., input mock value)
converted to $\sigma$ using $\sigma=$1.48$\times$MAD, which is what expected
for a Normal distribution. The use of MAD is preferred in order to to be less
dependent on outliers while capturing the bulk of the spread in the sample. By
defining the deviation with respect to the true value, the measurement bias is
also taken into account when evaluating the overall uncertainty in the
measurements. Fig. 8 compares the scatter of measurements obtained from
$uv$-fitting with those obtained in the image-plane for each key structural
parameter.

Again, we find that the random uncertainties in all structure parameter
recoveries are systematically larger for image-based estimates than those
obtained by visibility analysis. The median scatter ratio of measurements
obtained from $uv$-fitting to that obtained from image-plane for the recovered
flux densities is 0.5, while for the recovered effective radius, axis ratio,
and S${\rm\acute{e}}$rsic $n$, the median scatter ratios are 0.5, 0.6, and
0.5, respectively. We did not find a significant correlation between the
scatter ratio and the S/N of data. These findings are in line with a study
that compared the performance of stacking data in the $uv$-plane and image-
plane, which found that $uv$-stacking resulted in significantly improved
accuracy of size estimates, with typical errors less than half compared to
image-stacking (Lindroos et al., 2015). We emphasize that the small residual
systematics shown in Figs. 2 and 3, inherent in the conversion of Spergel-
based parameters of our simulations to S${\rm\acute{e}}$rsic parameters, only
account for a negligible amount of the excess noise coming from galfit (and
have no effect at all for the $n=1$ case).

In conclusion, we find that measurements in the image-plane are not only more
subject to bias, but also returning parameters that are more substantially
affected by noise, compared to those in the $uv$-plane.

##  5 Analysis: absolute accuracy of $uv$-plane modeling and reliability of
uncertainties

In this section, we focus on the performances of the Spergel model fitting in
GILDAS uv$\\_$fit. We analyze its accuracy in retrieving intrinsic galaxy
morphological parameters and verify the reliability of parameter errors
returned by the code. In addition, we evaluate the presence of correlations
between fitted parameters in the presence of noise.

Figure 9: The relative accuracy of the $uv$-based parameter estimates, flux
density, effective radius, axis ratio, and S${\rm\acute{e}}$rsic index (from
left to right), given by $\sigma$(para)/para, where $\sigma$(para) is the
uncertainties of the parameter estimates distribution and calculated as
1.48$\times$MAD, as a function of S/N of flux density. The top panels show the
results derived from fits with an elliptical Gaussian (orange) and circular
Gaussian (green) model, while the results from a Spergel model $uv$-fit with
input $R_{\rm e}/\theta_{\rm b}$ of 0.1, 0.2, 0.4, and 0.7 are shown in panels
from second to fifth rows, respectively. Here we only display results for the
cases with a Spergel index of 0.5 and $-0.7$ in our simulation. For other
cases with a Spergel index between $\nu=0.5$ and $-0.7$, the results are found
to be within or close to those shown in panels between the second and fifth
rows. The dashed vertical lines represent the threshold of $S_{\rm
tot}/\sigma_{\rm b}$ of 50.

###  5.1 S/N recoverable for fitted parameters in the $uv$-plane

We check the achievable accuracy for the parameter estimates from Spergel fits
in the uv-plane by computing the ratio of the uncertainties of the parameter
estimates to the input mock value, given by $\sigma$(para)/para. The
uncertainties of the parameter estimates, $\sigma$(para), can be evaluated as
1.48$\times$MAD, as discussed in Section 4.2.

In Fig. 9, the accuracy of parameter estimates is shown to vary with the S/N
of the data. The parameters being estimated are flux density, size, axis
ratio, and S${\rm\acute{e}}$rsic index. To enhance readability of the figure,
only Spergel models with an exponential ($\nu=0.5$) profile and steep profile
with $\nu=-0.7$ (close to a de Vaucouleurs profile) were considered. These
models are shown in the panels between the second and fifth rows. It is worth
noting that for other cases where the Spergel index is assumed to be between
$\nu=0.5$ and $-0.7$, the measurements are found to be either within or close
to the values obtained from the above two cases. Therefore, the results
presented in Fig. 9 can be considered representative of the Spergel model.

Fig. 9 shows that the uncertainties in measuring galaxy shape parameters, such
as size and Spergel index, are significantly larger than those for flux
density. The most difficult parameter to estimate is the Spergel index. The
uncertainty in estimating the Spergel index can be as high as $70\%$ for model
sources with a Stot/$\sigma_{\rm b}$ of 25 or lower, suggesting that the
measurement of Spergel index can be highly uncertain. As the Stot/$\sigma_{\rm
b}$ increases to 50, the accuracy of Spergel index estimates significantly
improves, with a median value of $\sigma$($n$)/$n$ of 0.36, which we consider
as the bare minimum to define a meaningful estimate. At Stot/$\sigma_{\rm b}$
of 50, the uncertainties of the flux density, size, and axis ratio are
significantly smaller, with median values of 9%, 18%, and 22% of estimates,
respectively. To obtain a meaningful and reliable profile fitting result with
the Spergel model in $uv$-plane, our simulations strongly suggest that a
Stot/$\sigma_{\rm b}$ of at least 50 should be required.

In addition, we find that the accuracy of both the flux density and size is
lower for simulated data with a steep profile compared to the model data with
a flat profile. On average, the accuracy of flux densities and sizes for the
galaxy with an exponential profile ($\nu=0.5$) is 50$\%$ higher than those of
the galaxy with a de Vaucouleurs-like profile ($\nu=-0.7$). However, it is
important to note that the measurements of Spergel $\nu$ tend to be less
accurate for smaller sources ($R_{\rm e}/\theta_{\rm b}\leqslant$0.2) compared
with larger sources in the data. We have also found that both the flux
densities and sizes of the small-sized sources are more accurately measured,
with a typical factor of about 40%. The differences in the accuracy of
parameter estimates imply that all the fitted parameters are interrelated.

Figure 10: Comparison between the errors obtained from the scatter of the
distribution of parameter estimates and the errors estimated by uv$\\_$fit at
each fit in our simulation, as a function of the S/N of flux density. The
fitted parameters, from left to right, are flux density, effective radius,
axis ratio, and Spergel index. The scatter of the parameter estimates
distribution is calculated as $\sigma$ = 1.48$\times$MAD. Simulated sources
with a size of $R_{\rm e}/\theta_{\rm b}=$ 0.2 (diamond) and 0.7 (triangle)
are presented and color-coded by the Spergel $\nu$. The black bars represent
the median value of the ratio between the error in each fitted parameter
obtained from simulations and the median of the error measured by uv$\\_$fit
at each flux S/N bin for the simulated sources in this work.

###  5.2 Reliability of parameter uncertainties from UVFIT

In Fig. 10, we compare the average value of parameter errors in simulated
galaxies to the posterior scatter of the recovered distributions to explore
their reliability. We find that in most cases, the uncertainties of parameters
measured from Spergel profile fitting are underestimated, although typically
within a factor of 2. The underestimation is generally less important for
steep profiles ($\nu=-0.7$) than for disks ($\nu=0.5$) and for extended versus
compact galaxies. At each flux S/N bin, the ratio between the error obtained
from simulations and the median of the error measured by uv$\\_$fit for the
whole set of simulated sources is in the range of 1.4–1.9, 2.0–3.2, 1.9–2.5,
and 1.4–1.6 (see the black bars in Fig. 10), with a mean value of 1.6, 2.5,
2.1, and 1.5 for the parameter estimates of flux density, effective radius,
axis ratio, and Spergel index, respectively.

Figure 11: Top: corner plots showing the covariances between the free
parameters modeled in Spergel profile fitting. The example model source has a
flux density of $S_{\rm tot}/\sigma_{\rm b}=100$, a size of $R_{\rm
e}/\theta_{\rm b}=0.2$, an axis ratio of $q$=0.6, and a Spergel index of
$\nu=-0.5$. The shaded density histograms show the two-parameter distributions
with Spearman rank correlation coefficient and p-value marked in each panel.
The one-dimensional histograms at the top of each column represent the
individual parameter distributions, annotated with the median values. The
boundaries of the 25th and 75th percentiles of the distribution are plotted as
dashed lines, while blue vertical lines show the true values. Bottom:
Spearman’s rank correlation coefficients of the two-parameter pairs as a
function of source size. The points are colour-coded by the input flux
density.

###  5.3 Covariance of Spergel model fitted parameters

In this section, we examine the covariance between fitted parameters estimated
from uv$\\_$fit using a Spergel model. The top panels in Fig. 11 show the
correlations between the fitted parameters, which are flux density $S_{\rm
tot}/\sigma_{\rm b}$, effective radius $R_{\rm e}/\theta_{\rm b}$, axis ratio
$q$, and Spergel index $\nu$, for a simulated dataset with an input $S_{\rm
tot}/\sigma_{\rm b}$ of 100, $R_{\rm e}/\theta_{\rm b}$ of 0.2, $q$ of 0.6,
and $\nu$ of $-0.5$, respectively. In this case, the Spearman correlation
coefficient, which is calculated by dividing the covariance by the intrinsic
scatter of each parameter, show weak correlations ($|\rho|\lesssim 0.3$)
between fitted parameters of size, axis ratio, and $\nu$, while moderate
correlations ($|\rho|\sim 0.5$) are found between the flux density and both
$\nu$ and size.

The bottom part of Fig. 11 summarizes the pairwise correlation coefficients
calculated for all the data sets in our simulation. Generally, we find that
the correlation between variables becomes more prominent as the S/N increases,
except for the correlation between flux density and size. For sources that are
significantly more compact than the beam ($R_{\rm e}/\theta_{\rm b}<0.2$), the
correlation between the flux density and measured source size is weaker when
the source is detected with a high S/N. We did not find any significant
correlation between q and other parameters. This indicates that the measured
axis ratios are almost independent of these parameters.

There is a positive correlation between flux density and S${\rm\acute{e}}$rsic
$n$ (anti-correlated with Spergel $\nu$), indicating that sources with higher
measured S${\rm\acute{e}}$rsic $n$ tend to have a larger flux density. In
addition, a strong positive correlation is seen between flux density and size,
except for very compact sources with $R_{\rm e}/\theta_{\rm b}$ of 0.1, where
the correlation is relatively weak. This means that sources for which sizes
are overestmated have a tendency to be also boosted in flux density.

##  6 Discussion

The results presented in the previous sections demonstrate that studying
galaxies morphologies in the $uv$-plane leads to better performance than
imaging the data and then using galfit to study morphologies in the image-
plane. This approach offers exciting possibilities for studying morphologies
of galaxies in SFR tracers. However, there are several issues that merit
discussion.

###  6.1 On the differences between Spergel and S${\rm\acute{e}}$rsic profiles

A comparison between the Spergel and S${\rm\acute{e}}$rsic profiles shows that
the Spergel model has a steeper core and faster declining wings compared to
the S${\rm\acute{e}}$rsic model (see Fig. 1 and Fig. 16), except for the case
of Spergel $\nu=$0.5 which is equivalent to an exponential profile
(S${\rm\acute{e}}$rsic $n$=1). The question of whether Spergel or
S${\rm\acute{e}}$rsic models provide better fits to actual galaxies remains
open and requires future investigation. The current available data quality may
not be sufficient to determine a clear preference between the two functional
forms, in absolute terms. For the time being and with the typical data
available, we deem the two functional forms as equivalent.

Converting a Spergel index to a S${\rm\acute{e}}$rsic one (as well as $R_{\rm
e}$ and total flux) requires knowledge of the intrinsic size of the galaxy and
the angular resolution of the observations. In the case of this study, the
angular resolution is determined by the synthesized beam of the
interferometric data. We believe this requirement also applies implicitly to
optical observations. When a S${\rm\acute{e}}$rsic index is derived from
optical data, it applies to the range of scales actually observed in the data.
It may not apply beyond the observed scales by definition. It is possible that
a somewhat different Spergel/S${\rm\acute{e}}$rsic index might be recovered
when re-observing the same galaxy with a much different surface brightness
sensitivity.

It is also relevant to question whether the differences in the profiles at the
outer and inner ranges could affect the systematic biases of the parameter
estimates, particular for a steep profile. Along these lines, considering that
we have simulated Spergel models and then fitted them with galfit in the
image-plane, one might wonder if the under-performance of galfit in the image-
plane could be simply related to the discussed differences between Spergel and
S${\rm\acute{e}}$rsic models.

As already mentioned through-out the description of results in the previous
section, we believe that the argument can be dismissed, but it’s worth
recalling the key evidences here. Based on the bottom panels of Fig.7, the
simulations with the highest S/N (400 in this case) show that any average bias
in S${\rm\acute{e}}$rsic modeling is vanishing or strongly reduced, as
expected by construction (Eq.(3)), showing that any residual systematics
beyond our conversions do not have a measurable impact. The systematic
differences in the recovered profile parameters ($R_{\rm e}$ and
S${\rm\acute{e}}$rsic $n$, crucially) are in fact strongly S/N-dependent, and
therefore mostly an effect of noise, rather than arising from structural
differences. Additionally, Fig.8 shows that the excess noise in parameter
recovery from image-plane fitting is not a function of S/N, while the
systematic deviations between the profiles are expected to become more evident
at the highest S/N. Also importantly, Fig.8 shows that an entirely consistent
situation is seen for the $n=1$ case, which is fully identical in shape to a
Spergel $\nu$ of 0.5.

Fig.12 extends further this point by showing the performance comparison of
uv$\\_$fit versus galfit in the case of Gaussian sources and PSF models. In
this case, we directly simulated these shapes in the $uv$-plane, which are not
Spergel models, and there is no shape difference with respect to the model
fitted by galfit. However, the overall behaviour is qualitatively the same as
in Figs.7 and 8. There are stronger biases in the image plane (top panels),
and the effective noise is also much higher there (lower panels).

Figure 12: Top: relative difference between recovered and input parameters of
flux density (left), size (middle), and axis ratio (right), which were
obtained from fitting with an elliptical Gaussian (orange), a circular
Gaussian (green), and a point (blue) source model using uv$\\_$fit (open
symbols) and galfit (solid symbols). Bottom: similar to the top panels, but we
plot the ratio of measurement uncertainty derived from galfit in the image-
plane to that obtained from uv$\\_$fit in the $uv$-plane. The measurement
uncertainties are estimated as $\sigma=$ 1.48$\times$MAD.

###  6.2 On the origin of the poor performances in the image plane with GALFIT
on interferometric data

It has been already shown that ignoring noise correlation in interferometric
images can lead to a significant underestimation of the statistical
uncertainty of the results (e.g., Tsukui et al., 2023). To test this idea
further, we carried out aperture photometry as an alternative to galfit for
point-source simulations. Aperture photometry is one of techniques used to
measure flux density in astronomical observations, which is also used in
interferometric images (e.g., Lang et al., 2019; Gómez-Guijarro et al., 2022).
It should be noted that, for extended sources, there is an added complication
of correcting for flux losses, and this technique has the limitation of not
allowing to estimate morphological parameters. However, it is well-defined for
flux density measurements of point sources.

Figure 13 shows the systematic bias and the measurement uncertainty on the
recovery of the flux density for point sources using different fitting
methodologies, i.e., uv$\\_$fit with a Point source model, galfit with a PSF
profile that is identical to the dirty beam of the simulated data, and
aperture photometry. The latter is performed with an aperture radius equal to
the circularized radius of beam size and at the position returned by galfit
PSF fitting. It is clear that the flux density estimates obtained through
aperture photometry are fully consistent to those measured in the $uv$-plane,
with a similar level of accuracy in the estimates. Both of these methods
exhibit smaller systematic errors and scatters when compared to fitting with a
PSF model using galfit. In a way, aperture photometry measurements being
fairly basic and raw just are not fooled by false coherence in the signal
induced by correlation, and return the full S/N performances in the limiting
case of point-like emission (Fig.13-right).

###  6.3 Warnings against cleaning dataset in view of morphological
measurements

It is common practice to perform deconvolution of low-frequency radio
interferometric data to remove strong sidelobes from bright sources in very
large fields, a process known as cleaning. We emphasize that in ALMA and NOEMA
observations the field of view is tiny compared to the radio and the source
density is basically always manageable, so that cleaning can generally be
safely avoided. Cleaning, as any deconvolution approach, is a model-dependent
process which in most implementations arbitrarily interpolates the observed
source with a superposition of point sources, whose side-lobes are then
subtracted from the data using the dirty beam. Due to its arbitrariness and
its approximation of real sources as multiple point-sources, it is clearly not
ideal for analysis of galaxies morphological profiles. We finally emphasize
that, obviously, the cleaning process would not remove correlations in the
signal in the image plane, which are due to Fourier transforming the
visibilities.

Figure 13: Simulation results of the flux density bias (left) and the relative
uncertainty of measurement on recovery of flux density (right) for point
sources using uv$\\_$fit (red), galfit (black), and aperture photometry (blue)
in the image-plane fitting, as a function of the flux S/N. The error bars in
the left panel represent the standard error on the mean. The dashed line in
the right panel is a 1:-1 line. Figure 14: Distribution of the ratio of the
size obtained from a Spergel fit to that from a Gaussian fit. The dashed line
shows the median of the distribution, while the dotted lines represent the
16th and 84th percentiles of the distribution, respectively.

Figure 15: Examples of ALMA images of CO(5-4) and [CII] emission from high-z
star-forming galaxies PACS-787 (top) and HZ7 (bottom). Left: the ALMA dirty
image. Right: the residual map with the primary disk component modeled by a
single Spergel profile subtracted. The image size is $3.5\arcsec\times
3.5\arcsec$. The synthesized beam ($0.15\arcsec\times 0.14\arcsec$ and
$0.36\arcsec\times 0.35\arcsec$ for PACS-787 and HZ7, respectively) is
presented at the bottom left in the left panels. Contours start from $\pm
2\sigma$ and increase by a factor of 1.5.

###  6.4 The cost of using Spergel: recommendations

To investigate the morphologies of galaxies observed in interferometric
images, Spergel modeling directly in the $uv$-plane should be the preferred
approach. However, a high S/N ratio of at least 50, and ideally much more,
must be required to perform a Spergel fit that meaningfully constrain the
Spergel index, as we have previously commented based on Fig.9. This
requirement is similar to optical observations of galaxies and their galfit
modeling, where a high S/N of order 100 is needed to attempt derivation of a
S${\rm\acute{e}}$rsic index (van der Wel et al., 2014; Magnelli et al., 2023).

These considerations should be extended further to include other parameters of
interest, such as the size or even the flux density. These parameters are
easier to derive and require less S/N compared to a
Spergel/S${\rm\acute{e}}$rsic index. However, there is a tension between
extracting a meaningful measure from the data and ensuring an unbiased
measure. It is important to determine the best approach in balancing these
factors.

The top-panel of Fig.9 provides information on the uncertainties in flux
density, size, and axis ratio for Gaussian models, as a function of their S/N
ratios. By comparing this information to the bottom panels, one can make a
decision between a less complex (Gaussian) fit and a more complex (Spergel)
fit to a given dataset. For example, Gaussian fits can provide size
uncertainties of about 20% down to an S/N ratio ($S_{\rm tot}/\sigma_{\rm b}$)
of 20, whereas Spergel fits do not offer the same level of accuracy at lower
S/N ratios. Similar considerations apply to flux density and axis ratio. In
addition, there are no significant differences in the accuracy and uncertainty
estimates between circular and elliptical Gaussian models (see Fig. 9 and Fig.
12). This demonstrates that the increase in degrees of freedom in Spergel
models leads to greater uncertainty in the measured parameters, including flux
density and size.

Figure 14 shows the distribution of the ratio of the size obtained from a
Spergel fit to that from a Gaussian fit, measured for a sample of about 100
high-$z$ star-forming galaxies observed with the ALMA (Q. Tan et al., in
preparation). The median ratio between the $R_{\rm e}$ size obtained from a
Spergel fit and the FWHM size obtained from a Gaussian fit is
$0.47_{-0.08}^{+0.10}$, where uncertainties correspond to the 16th and 84th
percentiles of the distribution. This implies that in most cases, the FWHM
size measured from a Gaussian model can be used as a good approximate of the
effective radius using the relation $R_{\rm e}$=FWHM/2. However, it is worth
noting that almost all of the outliers with $R_{\rm e}$/FWHM ratio far from
the median value in Fig. 14 are sources measured with large
S${\rm\acute{e}}$rsic index $n$ ($n>2$; Q. Tan et al., in preparation). This
suggests that the difference in size measured from Gaussian fit and the
Spergel fit could be significant when the source has a large
S${\rm\acute{e}}$rsic index.

Finally, to achieve a global optimal solution for the Spergel fit, we
recommend utilizing the fitting results obtained from Gaussian or exponential
fits as prior knowledge for the initial guesses of each structure parameter.
This provides good starting guesses, which helps the solver in achieving
convergence and in providing a reasonable level of uncertainty. Such
preliminary information will also inform the user about the actual merit of
proceeding to a more demanding full-Spergel fit.

###  6.5 The power of Spergel fitting: a test case exemplification

Fitting Spergel models to interferometric data can be complex and requires
deep high S/N data. However, it can also potentially brings powerful insights
and enable investigations into new scientific questions.

We aim to demonstrate the potential of our approach by re-examining two cases
of published ALMA observations of distant sources that were claimed to contain
giant halos surrounding the central galaxies (PACS-787 from Silverman et al.
(2018), including several coauthors of this paper, and HZ7 from Lambert et al.
(2023)). These two specific examples are taken from a growing body of results
that report the existence of large halos, often observed in [CII]$\lambda
158\mu$m and other tracers (e.g., Ginolfi et al., 2017; Fujimoto et al., 2019,
2020; Pizzati et al., 2020; Cicone et al., 2021; Herrera-Camus et al., 2021;
Jones et al., 2023; Li et al., 2023; Posses et al., 2023; Scholtz et al.,
2023, etc). Extended halos around galaxies are generally interpreted as
evidence for accretion, outflows, tidal stripping, or other phenomena
affecting distant galaxies. This presents a relevant opportunity for further
constraining these processes. However, it is worth considering whether these
halos are genuine different structures from the galaxies or simply the outer
scale extension of high S${\rm\acute{e}}$rsic-index profiles. In many cases,
simple Gaussian fitting was attempted for the central galaxies, and high
S${\rm\acute{e}}$rsic index profiles are well known to display large halos
(e.g., Mancini et al., 2010).

We have downloaded the data for both datasets as described below. In ALMA
Cycle 6, HZ7 was observed with 80 minutes of on-source integration time in the
C43-4 configuration with 47/45 12m antennas in band 7 (Project 2018.1.01359.S;
PI: M. Aravena). The [CII] emission at 303.93GHz falls into one of four SPWs
with a native channel width of 15.625 MHz. PACS-787 was observed with high
(C40-6, 43 12m antennas with a maximum baseline of 1.1 km) and low (C40-1, 42
12m antennas with a maximum baseline of 278.9 m) resolution configuration in
Band 6 with 19.7 and 10.2 minutes of on-source integration time in ALMA Cycle
4 (Project 2016.1.01426.S; PI: J. Silverman). The CO(5-4) emission at
228.17GHz falls into a native channel width of 3.91 MHz in high-resolution
observation and 15.62 MHz in low-resolution observation. The calibration
targets for the three observations above are J1058+0133 for bandpass,
pointing, and flux, and J0948+0022 for phase.

The left panels of Fig. 15 show the dirty images of PACS-787 and HZ7, where
emission on large scales of several arcsec (tens of kpc) can be readily seen.
We modeled the emission in the $uv$-space with simple Spergel profiles. For
PACS-787, which contains two galaxies in the process of merging, we used two
Spergel components (one for each galaxy), while for HZ7, we used a single
Spergel. The Spergel fitting in both cases is able to fully account for the
emission from the galaxies, simultaneously reproducing the inner emission and
the outer halos. The residuals are clean, and no further components are
needed, as shown in the right panels of Fig.15. For the case of PACS-787, the
two Spergel component have $\nu=-0.36,-0.14$ and $R_{\rm e}/\theta_{\rm b}\sim
0.7$ for both. For HZ7, we find $\nu=-0.42$ and $R_{\rm e}/\theta_{\rm b}\sim
1.3$. Based on Eq.(3) and Fig.2, this corresponds to S${\rm\acute{e}}$rsic
$n\sim 1.5$–3, which is well above the Gaussian approximation ($n=0.5$) and
also above the exponential case ($n=1$), although not as steep as a de
Vacouleurs profile ($n=4$).

We have shown that the full emission in these two systems can be fit with a
single component model, which raises questions about the interpretation of the
outer emission as a halo. Although high S${\rm\acute{e}}$rsic values indicate
the presence of both a central component and a profile extending further out
than a disk model with both smoothly connected, there is no apparent solution
of continuity between the inner parts and the outer halos. This is similar to
the inner versus outer parts of elliptical galaxies, and therefore suggestions
of different physical origins for them are less substantiated.

We emphasize that we have only re-evaluated two cases of halos from the
literature (out of many more existing) to exemplify the power of fitting more
complex Spergel models to ALMA data. It is beyond the scope of this work to
re-analyse all similar observations from the ALMA archive. However, we
obviously anticipate that in other cases, the halos might disappear once
fitted with a Spergel model. This is not only because of the analysis
presented here but also based on one of the key results from our forthcoming
companion paper (Q. Tan et al. in preparation), which suggests that $n>1$
models are required for most ALMA observations of distant galaxies.

##  7 Conclusions

The Spergel’s Bessel-function-based luminosity profile is a good approximation
to S${\rm\acute{e}}$rsic profile and has the significant advantage of being
analytic with a simple Fourier transform. We have performed a thorough
analysis of the new Spergel fits method for visibilities in the $uv$-plane,
comparing it to the S${\rm\acute{e}}$rsic fits for imaged data. Our study aims
to assess the effectiveness of the Spergel model fitting based on visibility
to galaxy light profiles. We have also tested the robustness of fitting in the
$uv$-plane by using simpler forms of point and Gaussian profiles. The main
findings of our study are:

  1. 1.

The conversion of Spergel $\nu$ into S${\rm\acute{e}}$rsic $n$ can be closely
approximated by a two-variable function $n(\frac{R_{\rm e}}{\theta_{\rm
b}},\nu)\sim 0.0249\frac{R_{\rm e}}{\theta_{\rm b}}{\rm
exp}(-7.72\nu)+0.191\nu^{2}-0.721\nu+1.32$ in the ($\frac{R_{\rm
e}}{\theta_{\rm b}},\nu$)-plane. In most cases, the differences between the
best-fit value and the one measured from simulated data for
S${\rm\acute{e}}$rsic $n$ are found to be within 10%.

  2. 2.

When wishing to compare results from Spergel to S${\rm\acute{e}}$rsic fitting
regarding sizes and total fluxes, similar conversions need to be applied. We
find that both the size and total flux estimated by galfit using a
S${\rm\acute{e}}$rsic profile tend to be larger than when using Spergel, as
the profile becomes steeper than an exponential profile, while the fitted
parameters of axis ratios and position angles are unaffected. The variation of
both the ratio of $R_{\rm e,se}/R_{\rm e}$ and $S_{\rm se}/S_{\rm sp}$ can be
described by a similar two-variable function of $r(\frac{R_{\rm
e}}{\theta_{\rm b}},\nu)\sim p_{1}(\frac{R_{\rm e}}{\theta_{\rm b}})^{2}{\rm
exp}(p_{2}\nu+p_{3}\frac{R_{\rm e}}{\theta_{\rm b}})+p_{4}\nu+p_{5}$. The
best-fit coefficients for the size ratio of $R_{\rm e,se}/R_{\rm e}$ are
$p_{1}=0.00138$, $p_{2}=-8.96$, $p_{3}=0.260$, $p_{4}=-0.0260$, and
$p_{5}=0.996$, and for the flux ratio of $S_{\rm se}/S_{\rm sp}$ are
$p_{1}=0.00217$, $p_{2}=-7.43$, $p_{3}=0.149$, $p_{4}=0.00942$, and
$p_{5}=1.00$, respectively.

  3. 3.

Our MC simulations have shown that fitting directly in the $uv$-plane (rather
than imaging the dataset and fitting in the image plane) leads to more
consistent and reliable results. The accuracy of fitted structure parameter
estimates obtained from $uv$-plane fits using a Spergel profile is
significantly higher, with smaller systematic errors and scatters on the
recovery of parameters. In comparison, image-based measurements obtained from
galfit using a S${\rm\acute{e}}$rsic model tend to have higher systematic
biases and larger uncertainties (worse parameter accuracy by a factor of two).

  4. 4.

We have verified the reliability of the parameter uncertainties returned by
GILDAS uv$\\_$fit modeling. The parameter uncertainties are generally somewhat
underestimated, but still correct to better than a factor of two.

  5. 5.

We recommend to attempt full-flagged Spergel profile fitting only to sources
detected with a $S_{\rm tot}/\sigma_{\rm b}$ of at least 50. This is needed
for minimal accuracy and reliability of the S${\rm\acute{e}}$rsic index
(converted from the Spergel index) estimates in the $uv$-plane. The
corresponding median value of $\sigma(n)/n$ is 0.36, which we deemed as the
minimum threshold for a meaningful and accurate estimate of
S${\rm\acute{e}}$rsic $n$.

  6. 6.

The total flux and size estimates obtained from Spergel fitting show larger
uncertainties at fixed S/N compared to Gaussian and point profile functions,
which have fewer degrees of freedom. For Spergel profile fitting, the
uncertainties in measuring galaxy shape parameters were found to be
significantly higher than those in measuring flux density. The least accurate
constraint, requiring the deepest data, is the Spergel index.

  7. 7.

As a test case, we re-analysed literature claims for discovery of extended
halos surrounding distant galaxies. We find that single Spergel models without
any extra added halo can fully explain these observations.

High-quality interferometric data, such as now routinely obtained from ALMA
and NOEMA, allow us to study the morphology of distant galaxies in their
submillimeter band emission. This emission primarily arises from thermal dust
and molecular gas, which are closely related to star formation. Fitting a
Spergel model in the visibility plane is the preferred method for modeling
such emission. This should open-up a new window of investigation to further
our general understanding of the evolution of galaxies, in coming years.

###### Acknowledgements.

We thank the anonymous referee for constructive suggestions to improve the
paper. We are grateful to the IRAM director, Karl Schuster, for the
implementation of the Spergel models as part of the GILDAS software package.
QT and ED would like to dedicate this paper to the memory of late Prof. Yu
Gao, who offered kind support for this work. R.I.P. We wish to thank Daizhong
Liu for useful discussions. This work has been partly funded by China
Scholarship Council. QT acknowledges support from the NSFC (grant Nos.
12033004, 12003070, 11803090). CGG acknowledges support from CNES. JPe
acknowledges support by the French Agence Nationale de la Recherche through
the DAOISM grant ANR-21-CE31-0010 and by the Programme National “Physique et
Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP, co-funded
by CEA and CNES. A.P. acknowledges support by an Anniversary Fellowship at
University of Southampton and by STFC through grants ST/T000244/1 and
ST/P000541/1. This work made use of the following Python libraries:
Astropy(Astropy Collaboration et al., 2022), Numpy(Harris et al., 2020),
Matplotlib(Hunter, 2007), and Emcee(Foreman-Mackey et al., 2013).

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##  Appendix A Matching Spergel profiles with Sersic profiles through
mathematical simulations numerically

Within a valid range of $\nu$ ($-0.85\leqslant\nu\leqslant 4$; see Spergel
2010), the matching between Spergel and S${\rm\acute{e}}$rsic profiles is
derived by minimizing the sum of the difference between the Spergel and
S${\rm\acute{e}}$rsic functions over a radial range (see Fig. 1 and Fig. 16),
i.e., $\Sigma({\rm log}((I/I_{\rm e})_{\rm Spergel})-{\rm log}((I/I_{\rm
e})_{\rm S\acute{e}rsic}))^{2}$. We use the emcee package (Foreman-Mackey et
al. 2013) in Python to perform Markov chain Monte Carlo (MCMC) search for the
best-fit.

To match with the S${\rm\acute{e}}$rsic index measured from galfit (see
Section 2.4), we varied the radial range used for analytic matching and found
that the best-fit is given by limiting the radius between about 0.01
$\theta_{\rm b}$ and 2.0 $\theta_{\rm b}$ for the source sizes (in units of
$\theta_{\rm b}$, $R_{\rm e}/\theta_{\rm b}$) ranging between 0.1 and 2.0.
Fig. 17 summarizes the output returned from S${\rm\acute{e}}$rsic model versus
true parameters of the profile fits by setting the intensity, $R_{\rm e}$, and
S${\rm\acute{e}}$rsic index $n$ as free parameters in the MCMC sampler. We
find that the relationship between the S${\rm\acute{e}}$rsic output and
Spergel input parameters exhibit similar trends as seen in Fig. 2 and Fig. 3.

Figure 16: Left: surface density profiles for Spergel function (black)
compared to the best-fit one with a S${\rm\acute{e}}$rsic function (red)
through mathematical simulations. Right: comparison of the integrated surface
density profiles for the S${\rm\acute{e}}$rsic and Spergel functions shown in
the left panel. The values of $R_{\rm e,in}$ and $I_{\rm e,in}$ are held fixed
and represent the input parameters in the Spergel profile.

Figure 17: Comparison between Spergel index $\nu$ and S${\rm\acute{e}}$rsic
index $n$ (left), the ratio of $R_{\rm e}$ (middle) and total flux (right)
obtained from profile fitting using a S${\rm\acute{e}}$rsic function to the
input value in the Spergel model as a function of Spergel index $\nu$.

##  Appendix B Details of the three ALMA configurations used to generate
simulation data

To generate simulation data, three different ALMA array configurations data
were used in this work. Table 1 lists the details of each configuration,
including the major and minor size of the beam, the PA of the beam, FOV, and
the number of antennas.

Table 1: Details of the three ALMA array configurations Name | $\theta_{\rm MAJ}$ | $\theta_{\rm MIN}$ | PA | FOV | NANT
---|---|---|---|---|---
| (arcsec) | (arcsec) | (degree) | (arcsec) |
Config-A | 0.210 | 0.194 | 34.5 | 18.0 | 40
Config-B | 0.59 | 0.52 | 83.9 | 18.0 | 41
Config-C | 1.10 | 0.85 | 109.3 | 18.8 | 46

##  Appendix C Details on the elliptical Spergel profile

###  C.1 Definition and Fourier Transform

The circular Spergel profile ($\Sigma_{(\nu,0,0)}^{\mathrm{circ}}$) located at
the phase center $(0,0)$ is written in Equation (2) as

$\Sigma_{(\nu,0,0)}^{\mathrm{circ}}(\theta)=\frac{c_{\nu}^{2}L_{0}}{R_{0}^{2}}\,f_{\nu}\left(\frac{c_{\nu}\theta}{R_{0}}\right),$
(8)

where $L_{0}$ is the total luminosity, $R_{0}$ is the half light radius,
$c_{\nu}$ is a tabulated function of $\nu$ and

$f_{\nu}(x)=\left(\frac{x}{2}\right)^{\nu}\frac{K_{\nu}(x)}{\Gamma(\nu+1)},$
(9)

in which $\Gamma$ is the Gamma function, and $K_{\nu}(x)$ is the modified
spherical Bessel function of the third kind. Spergel (2010) tabulates the
values of $c_{\nu}$ every $0.05$ for a range of $\nu$ from $-0.90$ to $0.85$.
We approximated it with the following approximation

$c_{\nu}=\alpha+\beta\nu+\gamma\log(2+\nu)$ (10)

with

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-0.403713,$
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle-0.228101,$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle+2.400961.$

Figure 18 compares the tabulated values with our analytical approximation. It
results in relative error of up to 10% for $\nu<-0.6$ and less than 1% for
$\nu>-0.6$.

Figure 18: Comparison between tabulated values of the $c_{\nu}$ function with
an analytical approximation.

We use the radio-astronomy convention to define the conjugate coordinates of
the angular coordinates $(\theta_{l},\theta_{m})$ relative to the projection
center of the image as $(u,v)$ with

$u\,\theta_{l}=\lambda,\quad\mbox{and}\quad v\,\theta_{m}=\lambda,$ (11)

where $\lambda$ is the wavelength of the observed line. The
$(\theta_{l},\theta_{m})$ and $(u,v)$ coordinates are expressed in radian and
meter, respectively. In the $uv-$plane, the Fourier transform of the circular
Spergel profile $\widetilde{\Sigma}_{(\nu,0,0)}^{\mathrm{circ}}(u,v)$ can be
written as

$\widetilde{\Sigma}_{(\nu,0,0)}^{\mathrm{circ}}(u,v)=L_{0}\,\left[1+\left(2\pi\,\frac{R_{0}}{c_{\nu}}\right)^{2}\left(u^{2}+v^{2}\right)\right]^{-(1+\nu)}$
(12)

Noting $(R_{\mathrm{maj}},R_{\mathrm{min}})$ the major and minor half light
radii, and $\phi$ the position angle of the elliptical Spergel profile, we
yield the following generalization

$\widetilde{\Sigma}_{(\nu,0,0)}^{\mathrm{elli}}(u,v)=L_{0}\,\left[1+\frac{r_{\mathrm{maj}}^{2}u_{\mathrm{rot}}^{2}+r_{\mathrm{min}}^{2}v_{\mathrm{rot}}^{2}}{c_{\nu}^{2}}\right]^{-(1+\nu)}$
(13)

where $(u_{\mathrm{rot}},v_{\mathrm{rot}})$ are the coordinates of the
$uv-$plane rotated by $-\phi$ in order to bring the major axis along the
$u_{\mathrm{rot}}$ axis, i.e.,

$\displaystyle u_{\mathrm{rot}}$ $\displaystyle=$ $\displaystyle
u\sin{\mathrm{\phi}}+v\cos{\mathrm{\phi}},$ (14) $\displaystyle
v_{\mathrm{rot}}$ $\displaystyle=$ $\displaystyle
u\cos{\mathrm{\phi}}-v\sin{\mathrm{\phi}},$ (15)

and $(r_{\mathrm{maj}},r_{\mathrm{min}})$ are the reduced major and minor half
light radii, i.e.,

$r_{\mathrm{maj}}=2\pi\,R_{\mathrm{maj}}\quad\mbox{and}\quad
r_{\mathrm{min}}=2\pi\,R_{\mathrm{min}}.$ (16)

An additional phase term appears when the Spergel profile is centered at an
offset $(\theta_{l0},\theta_{m0})$ with respect to the phase center.

$\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}(u,v)=\widetilde{\Sigma}_{(\nu,0,0)}^{\mathrm{elli}}(u,v)\,\exp\left[2i\,\pi\left(u\,\theta_{l0}+v\,\theta_{m0}\right)\right]$
(17)

###  C.2 Partial derivatives

In order to differenciate the elliptical Spergel profile, we first rewrite it
as

$\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}(u,v)=L_{0}\,\left[g(u,v)\right]^{-(1+\nu)}\,p(u,v)$
(18)

with

$g(u,v)=1+\frac{r_{\mathrm{maj}}^{2}u_{\mathrm{rot}}^{2}+r_{\mathrm{min}}^{2}v_{\mathrm{rot}}^{2}}{c_{\nu}^{2}},$
(19)

and

$p(u,v)=\exp\left[2i\,\pi\left(u\,\theta_{l0}+v\,\theta_{m0}\right)\right].$
(20)

The derivative with respect to the luminosity is

$\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial
L_{0}}(u,v)=\left[g(u,v)\right]^{-(1+\nu)}\,p(u,v).$ (21)

The derivatives with respect to the offset from the phase center are

$\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial\theta_{l0}}(u,v)=2i\,\pi\,u\,\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}(u,v),$
(22)

and

$\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial\theta_{m0}}(u,v)=2i\,\pi\,v\,\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}(u,v).$
(23)

The derivatives with respect to the major and minor half-light radius are

$\displaystyle\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial
R_{\mathrm{maj}}}(u,v)$ (24) $\displaystyle=$ $\displaystyle-
L_{0}\left(2\pi\,\frac{2\,r_{\mathrm{maj}}\,u_{\mathrm{rot}}^{2}}{c_{\nu}^{2}}\right)$
$\displaystyle\times(\nu+1)\,\left[g(u,v)\right]^{-(\nu+2)}\,p(u,v),$

and

$\displaystyle\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial
R_{\mathrm{min}}}(u,v)$ (25) $\displaystyle=$ $\displaystyle-
L_{0}\left(2\pi\,\frac{2\,r_{\mathrm{min}}\,v_{\mathrm{rot}}^{2}}{c_{\nu}^{2}}\right)$
$\displaystyle\times(\nu+1)\,\left[g(u,v)\right]^{-(\nu+2)}\,p(u,v).$

The derivative with respect to the position angle is

$\displaystyle\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial\mathrm{\phi}}(u,v)$
(26) $\displaystyle=$ $\displaystyle-
L_{0}\left[\frac{2(r_{\mathrm{maj}}^{2}-r_{\mathrm{min}}^{2})\,u_{\mathrm{rot}}\,v_{\mathrm{rot}}}{c_{\nu}^{2}}\right]$
$\displaystyle\times(\nu+1)\,\left[g(u,v)\right]^{-(\nu+2)}\,p(u,v).$

Finally, the derivative with respect to the Spergel index is

$\displaystyle\frac{\partial\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}}{\partial\nu}(u,v)$
$\displaystyle=$
$\displaystyle\widetilde{\Sigma}_{(\nu,\theta_{l0},\theta_{m0})}^{\mathrm{elli}}(u,v)\times$
$\displaystyle\left[(\nu+1)\frac{2(r_{\mathrm{maj}}^{2}u_{\mathrm{rot}}^{2}+r_{\mathrm{min}}^{2}v_{\mathrm{rot}}^{2})}{g\,c_{\nu}^{3}}\left(\beta+\frac{\gamma}{\nu+2}\right)-\log{g}\right].$