11institutetext:  Instituto de Astrofísica, Pontificia Universidad Católica de
Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile
22institutetext: Sub-Dep. of Astrophysics, Dep. of Physics, University of
Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
33institutetext: INAF - Osservatorio Astronomico di Capodimonte, Via
Moiariello 16, 80131, Naples, Italy 44institutetext: European Southern
Observatory, Karl-Schwarzschild-Straße 2, 85748, Garching, Germany
55institutetext: Universita degli studi dell’Insubria, via Valleggio 11, 22100
Como, Italy 66institutetext: Instituto de Astrofísica de Canarias, Vía Láctea
s/n, E-38205 La Laguna, Tenerife, Spain 77institutetext: Departamento de
Astrofísica, Universidad de La Laguna, E-38205 La Laguna, Tenerife,
88institutetext: European Southern Observatory, Alonso de Cordova 3107,
Vitacura, Casilla 19001, Santiago de Chile, Chile 99institutetext: INAF -
Osservatorio Astronomico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze,
Italy 1010institutetext: School for Physics and Astronomy, Sun Yat-sen
University, Guangzhou 519082, Zhuhai Campus, China 1111institutetext: Max-
Planck-Institut für extraterrestrische Physik, Giessenbachstrasse, 85748
Garching, Germany 1212institutetext: INAF - Osservatorio astronomico di
Padova, Vicolo Osservatorio 5, I-35122 Padova, Italy 1313institutetext: INAF -
Osservatorio Astronomico di Brera, via Brera 28, 20121 Milano, Italy
1414institutetext: Argelander-Institut für Astronomie, Auf dem Hügel 71,
D-53121 - Bonn, Germany 1515institutetext: Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109,
USA

# INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersion

G. D’Ago, Corresponding author<EMAIL_ADDRESS>INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersionINSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersion C. Spiniello INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersionINSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion L. Coccato INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion C. Tortora INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion F. La Barbera
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion M. Arnaboldi
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion D. Bevacqua
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion A. Ferré-Mateu
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersionINSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion A. Gallazzi
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion J. Hartke INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersionINSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion L. K. Hunt INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion I. Martín-Navarro
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion N.R. Napolitano
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion
C. Pulsoni INSPIRE: INvestigating Stellar Population In RElics III. Second
data release (DR2)††thanks: The data described in this paper are publicly
available via the ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion M. Radovich
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion P. Saracco INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion D. Scognamiglio
INSPIRE: INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersionINSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion S. Zibetti INSPIRE:
INvestigating Stellar Population In RElics III. Second data release
(DR2)††thanks: The data described in this paper are publicly available via the
ESO Phase 3 Archive Science
(https://archive.eso.org/scienceportal/home?data_collection=INSPIRE): testing
the systematics on the stellar velocity dispersionINSPIRE: INvestigating
Stellar Population In RElics III. Second data release (DR2)††thanks: The data
described in this paper are publicly available via the ESO Phase 3 Archive
Science (https://archive.eso.org/scienceportal/home?data_collection=INSPIRE):
testing the systematics on the stellar velocity dispersion

(Received November 23, 2022; accepted January 23, 2023)

###### Abstract

Context. The project called INvestigating Stellar Population In RElics
(INSPIRE) is based on VLT/X-Shooter data from the homonymous on-going ESO
Large Program. It targets 52 ultra-compact massive galaxies at $0.1<z<0.5$
with the goal of constraining their kinematics and stellar population
properties in great detail and of analysing their relic nature.

Aims. This is the second INSPIRE data release (DR2), comprising 21 new systems
with observations completed before March 2022. For each system, we release
four one-dimensional (1D) spectra to the ESO Science Archive, one spectrum for
each arm of the X-Shooter spectrograph. They are at their original resolution.
We also release a combined and smoothed spectrum with a full width at half
maximum resolution of 2.51Å. In this paper, we focus on the line-of-sight
velocity distribution, measuring integrated stellar velocity dispersions from
the spectra, and assessing their robustness and the associated uncertainties.

Methods. For each of the 21 new systems, we systematically investigated the
effect of the parameters and set-ups of the full spectral fitting on the
stellar velocity dispersion ($\sigma$) measurements. In particular, we tested
how $\sigma$ changes when several parameters of the fit as well as the
resolution and spectral coverage of the input spectra are varied.

Results. We found that the effect that causes the largest systematic
uncertainties on $\sigma$ is the wavelength range used for the fit, especially
for spectra with a lower signal-to-noise ratio (S/N $\leq 30$). When using
blue wavelengths (UVB arm) one generally underestimates the velocity
dispersion (by $\sim 15$ km/s). The values obtained from the near-IR (NIR) arm
present a larger scatter because the quality of the spectra is lower. We
finally compared our results with those in literature, finding a very good
agreement overall.

Conclusions. Joining results obtained in DR1 with those presented here,
INSPIRE contains 40 ultra-compact massive galaxies, corresponding to 75% of
the whole survey. By plotting these systems in a stellar mass–velocity
dispersion diagram, we identify at least four highly reliable relic candidates
among the new systems. Their velocity dispersion is larger than that of
normal-sized galaxies of similar stellar mass.

###### Key Words.:

Galaxies: evolution – Galaxies: formation – Galaxies: elliptical and
lenticular, cD – Galaxies: kinematics and dynamics – Galaxies: stellar content
– Galaxies: star formation

##  1 Introduction

The most massive and oldest galaxies in the Universe (early-type galaxies,
ETGs) play a fundamental role in the process of structure formation because
they account for more than half of the total mass in the Universe (Blumenthal
et al., 1984). However, the details of their formation and evolution are a
contentious question in present-day extragalactic astrophysics and cosmology.
Recently, a two-phase formation scenario has been proposed to explain their
mass assembly (Naab et al., 2009; Oser et al., 2010, 2012; Hilz et al., 2013;
Rodriguez-Gomez et al., 2016). A few billion years after the Big Bang, intense
and fast star formation episodes ($\tau\sim 100$ Myr, star formation rate
$\geq 1000$ $M_{\odot}\,yr^{-1}$) created ultra-compact and massive objects.
When these galaxies are observed to be passive, they are usually referred to
as red nuggets (Damjanov et al., 2011). Subsequently, a second longer phase,
dominated by accretion, mergers, and gas inflows, drove structural evolution
and size growth, shaping the local giant elliptical galaxies we observe today
(Daddi et al., 2005; van Dokkum et al., 2008; Buitrago et al., 2018). The
stars that formed during the first phase become then the core of today’s ETGs,
occupying their innermost regions (Ferré-Mateu et al., 2019; Pulsoni et al.,
2021; Barbosa et al., 2021). A fraction of these red nuggets could also end up
in the bulges of massive nearby spiral galaxies (de la Rosa et al., 2016;
Costantin et al., 2021).

However, the exact formation mechanism of red nuggets and the physics that
transform them into today’s massive galaxies is far from clear, and these
questions have a significance well beyond the context of the size evolution of
quiescent galaxies. The dense centres of local massive galaxies host the most
massive black holes in the Universe (Kormendy & Ho, 2013; Ferré-Mateu et al.,
2015). Their chemical enrichment history is extremely different from that of
the Milky Way (Worthey, 1994), and their stars may have formed with a bottom-
heavy (i.e. dwarf-rich) stellar initial mass function (IMF; Conroy et al.
2013; Martín-Navarro et al. 2015a; Sarzi et al. 2018; Barbosa et al. 2021).

In local ETGs, the material accreted in the second phase unfortunately
contaminates the in situ (i.e. first phase) component that encodes the
information about high-redshift baryonic processes, affecting its spatial and
orbital distributions. In turn, obtaining spectra of high-$z$ red nuggets with
a signal-to-noise ratio (S/N) that is high enough to perform detailed stellar
population analyses would require prohibitive integration times with current
instrumentation. Because of the stochastic nature of mergers, a small fraction
of red nuggets fortunately survived without experiencing any interactions.
They are now massive compact relic galaxies (Trujillo et al., 2009). Relics
are thus the only objects that allow us to study the physical processes that
shaped the mass assembly of galaxies in the high-$z$ universe with the same
amount of detail achievable in the nearby Universe.

So far, only three relics have been found and were fully characterised in the
local Universe: NGC1277 (Trujillo et al., 2014; Martín-Navarro et al., 2015b;
Beasley et al., 2018), Mrk 1216, and PGC 032873 (Ferré-Mateu et al., 2017).
These three objects all show a very quick (time-scales ¡1 Gyr) high-$z$ star
formation and thus are consistent with having stars with a very old mean mass-
weighted age ($\sim 13$ Gyr, almost as old as the Universe). Moreover, their
morphology, kinematics, and density profiles perfectly resemble those of $z>2$
red nuggets (Ferré-Mateu et al., 2017).

With the effort of enlarging the number of confirmed relics and also detecting
them at higher redshifts, we started the project called INvestigating Stellar
Population In RElics (INSPIRE), which aims at building the first large
catalogue of relics at $0.1<z<0.5$ (Spiniello et al., 2021a, b). The
(relatively $\sim 30$) high S/N and wide wavelength (from UVB to NIR) spectra
from the X-Shooter spectrograph (XSH; Vernet et al. 2011) now allow us to
infer the stellar kinematics and population properties (age, metallicity,
elemental abundance, and low-mass end of the IMF slope) of a sample of ultra-
compact massive galaxies (UCMGs) at $0.1<z<0.5$ with stellar masses
$M_{\star}>6\times 10^{10}M_{\odot}$ and effective radii
$\mathrm{R}_{\mathrm{e}\,}$¡2 kpc.

This is the second of the three planned yearly data releases, in which we
analyse the spectra of 21 systems whose observations were completed before
March 2022111The data are publicly available through the ESO Science Archive,
https://archive.eso.org/scienceportal/home?data_collection=INSPIRE.. Here, we
focus on analysing the line-of-sight velocity distribution (LOSVD) of the 21
new systems, and, in particular, on measuring the integrated stellar velocity
distribution ($\sigma$). The latter is often used as proxy for the total mass
of the galaxy, and it has been used in the literature to select relic
candidates (e.g. Saulder et al., 2015). Moreover, in DR1, we found that relics
and especially extreme relics have larger integrated $\sigma$ than non-relics
and normal-sized galaxies of similar stellar masses. Hence, we need to ensure
that based on the quality and wavelength range of our spectra, we can robustly
measure the integrated stellar velocity dispersion values from them. After
demonstrating that the $\sigma$ can be securely inferred from medium-S/N
spectra, this can be used as a selection criterion, together with the
$\mathrm{R}_{\mathrm{e}\,}$ and stellar mass, to pre-select good relic
candidates from on-going and future wide-sky surveys such as the Galaxy
Evolution Survey with the 4-metre Multi-Object Spectrograph Telescope (4MOST;
de Jong et al., 2019). This paper therefore presents a systematic and
quantitative analysis of all the parameters and code set-ups that might bias
and influence the stellar velocity dispersion measurements. In addition, it
presents the first effort to show that the stellar velocity dispersion is not
higher for all compact objects, as was proposed by Saulder et al. (2015), for
example, but only for a sub-sample of them. These probably are the most
reliable relic candidates.

The paper is organised as follows. The sample and current status of the
observations, as well as previous results obtained with INSPIRE, are described
in Sect. 2. The data reduction and analysis, including the extraction of the
1D spectra, the telluric correction on visual (VIS) and NIR arms, and the
combination of the three arms, are described in Sect. 3, and the kinematical
analysis and results are presented in Sect. 4. In Sect. 5 we compare velocity
dispersion values obtained from X-Shooter spectra to those inferred from
GAMA222the Galaxy And Mass Assembly survey spectra for ten objects in common.
In Sect. 6 we present the stellar mass-velocity dispersion plot for DR1 and
the DR2 INSPIRE objects. Finally, we present our conclusions and outline the
future development of INSPIRE in Sect. 7.

Throughout the paper, we assume a standard $\Lambda$-cold dark matter
($\Lambda$CDM) cosmology with $H_{0}=69.6$ km/s Mpc-1,
$\Omega_{\mathrm{vac}}=0.714,$ and $\Omega_{\mathrm{M}}=0.286$ (Bennett et
al., 2014).

##  2 INSPIRE project status and sample

INSPIRE is based on an on-going ESO Large Program (LP, ID: 1104.B-0370, PI: C.
Spiniello) that started in P104 (October 2019) with the aim to
spectroscopically follow up 52 UCMGs at redshift $0.1<z<0.5$ with the XSH
spectrograph (Vernet et al., 2011). The sample was collected through a
dedicated observational effort to find and spectroscopically confirm as many
UCMGs in the Kilo Degree Survey (KiDS; Kuijken, 2011) as possible (de Jong et
al., 2015; Kuijken et al., 2019). The results of this census for UCMGs were
presented in Tortora et al. (2016, 2018), and Scognamiglio et al. (2020),
hereafter T16, T18, and S20. These objects are the perfect candidates to host
very old stars as their $g-i$ broad-band colours are compatible with the
colour of a stellar population with an integrated age $\geq 8$ Gyrs
(considering a solar, super-solar, and sub-solar metallicity; see Fig. 1 in
INSPIRE DR1). They also all have remarkably small sizes
($\mathrm{R}_{\mathrm{e}\,}$$<2$ kpc) and high stellar masses
(M${}_{\star}\,$$>6\times 10^{10}$M${}_{\odot}\,$). The redshift window
covered by the sample, shown in Fig. 1, is $0.1<z<0.5,$ with a peak at $z\sim
0.28$.

Figure 1: Redshift distributions for the INSPIRE targets in DR1 (blue), DR2
(red), and for the final sample (grey).

At the time of writing, 148 of the total 154 hours have been delivered by ESO.
Forty systems were completely observed by the end of the ESO Period 108 (31
March 2022). Of these, 19 have been publicly released as part of the INSPIRE
DR1, whereas the remaining 21 constitute the data release presented here that
is publicly released via the ESO Phase 3 Science
Archive333https://doi.org/10.18727/archive/36.

We briefly describe the survey and observation strategy below. A more detailed
description can be found in DR1. Depending on the $r$-band surface brightness
luminosity ($\langle\mu_{\mathrm{e}}\rangle$) and aperture magnitudes
($\text{mag}_{r}$), both taken from the KiDS survey DR4 catalogue (Kuijken et
al., 2019), the exposure times on target varied from 2810 to 11240 seconds.
This allowed us to obtain an integrated 1D spectrum with an S/N that was good
enough (S/N $\geq 15$ per Å ) to constrain the stellar age, metallicity, and
[Mg/Fe] abundance of the stellar populations and hence confirm the relic
nature of the UCMGs. For each of the targets, structural parameters were
computed from KiDS images in $g,r,i$ bands and stellar masses inferred from
spectral energy distribution (SED) fitting in $ugri$ bands. These numbers were
reported in T18 and S20.

The slit widths that we chose for the UVB, VIS, and NIR arms are $1\arcsec.6$,
$1\arcsec.5,$ and $1\arcsec.5$ respectively. The position angles (P.A.) of the
slit were always oriented along the major axis of the galaxies, taken from T18
or S20. The observations were carried out in nodding mode, with a dithering
scheme consisting of multiple frames shifted by a small amount from the slit
centre to facilitate a proper sky subtraction. Similarly to DR1, the seeing
during the observations ranged between $0\arcsec.85$ to $1\arcsec.2$, with a
median value of $\sim 1\arcsec$.

The feasibility of the method and techniques were extensively tested in
Spiniello et al. (2021a, hereafter INSPIRE Pilot), where the first three
objects of the survey were presented. Then, in Spiniello et al. (2021b,
hereafter INSPIRE DR1), we applied the same routines to 19 systems that were
fully observed until March 2020. We confirmed ten new relic galaxies,
demonstrating that they had formed more than 75% of their stellar mass at
$z>2$, hence extending the number by a factor of 3.3 and pushing the redshift
up to $z\sim 0.5$. The remaining nine systems showed a longer star formation.

An important result initially proposed by Ferré-Mateu et al. (2017) that was
confirmed in DR1 is the degree of relicness: Some of the relics were already
fully assembled in terms of stellar mass soon after the Big Bang (BB) and
before the end of the first formation phase ($z\sim 2$, extreme relics),
whereas other relics formed a (high) fraction of their stars through a
starburst in a very short time at high $z$ but then had subsequent lower-z
star formation events (relics). Hence, the star formation history (SFH) of
relics can be more or less extreme, and this might correlate with other
morphological and stellar characteristics and possibly with the environment in
which they live.

As part of this DR2, we release the 1D NIR spectra of the 19 galaxies that
were presented in DR1 to the ESO Phase 3 Science Archive, together with 1D
UVB-VIS-NIR spectra of the 21 new systems. In particular, we obtain and
release three fully reduced and flux-calibrated 1D extracted spectra per
galaxy, one for each arm of the detector (UVB, VIS, and NIR), at their
original resolutions. In addition, we combine the arms, after smoothing
everything to a common resolution with a full width at half maximum (FWHM) of
$\sim 2.51\AA$ and release the final spectrum as an additional data product.

The INSPIRE DR2 targets along with their coordinates are listed in Table 1. We
list the final exposure time and the P.A. of the slit, as well as photometric
properties ($r$-band magnitudes and surface brightness),
$\mathrm{R}_{\mathrm{e}\,}$ derived as the median of the quantities obtained
from $g$, $r$ and $i$-band KiDS images and stellar masses, from SED fitting in
$ugri$ bands. Finally, in the last column of the table, we report the sample
from which the object was taken. Ten of the systems analysed in T18 or S20
were also independently observed with the AAOmega spectrograph and are part of
the GAMA second data release database (DR2; Liske et al. 2015) or fourth data
release (DR4; Driver et al. 2022). For these ten galaxies, we can directly
compare the kinematic results obtained from the GAMA spectra with those
computed from our spectra with their better resolution and higher S/N XSH (see
Sec. 5).

ID | RA | DEC | Exp.T. | P.A. | magr | $\langle\mu_{\mathrm{e}}\rangle_{r}$ | $\langle\mathrm{R}_{\mathrm{e}}\rangle\,$ | $\langle\mathrm{R}_{\mathrm{e}}\rangle\,$ | M⋆ | SAMPLE
---|---|---|---|---|---|---|---|---|---|---
KiDS | (deg) | (deg) | (sec) | (deg) | (AB) | (AB) | ($\arcsec$) | (kpc) | ($10^{11}$M⊙) |
J0844+0148 | 131.0553886 | +1.8132204 | 11240 |  $-$37.4 | 19.78 | 18.53 | 0.26 | 1.14 | 0.71 | S20/GAMA
J0904-0018 | 136.0518949 | -0.3054848 | 5620 |  $-$96.6 | 19.11 | 18.06 | 0.26 | 1.16 | 1.3 | S20/GAMA
J0909+0147 | 137.3989150 | +1.7880025 | 5620 | 10.3 | 18.68 | 16.05 | 0.30 | 1.05 | 1.05 | T18/GAMA
J0917-0123 | 139.2701850 |  $-$1.3887918 | 11240 |  $-$62.9 | 19.21 | 17.99 | 0.27 | 1.37 | 2.19 | S20
J0920+0126 | 140.1291393 | +1.4431610 | 11240 |  $-$115.6 | 19.52 | 18.82 | 0.33 | 1.51 | 0.98 | S20/GAMA
J1026+0033 | 156.7231818 | +0.5580979 | 5620 |  $-$83.1 | 17.39 | 16.98 | 0.34 | 1.02 | 1.48 | SDSS
J1040+0056 | 160.2152308 | +0.9407580 | 11240 | 53.4 | 19.52 | 18.85 | 0.31 | 1.29 | 0.93 | S20
J1114+0039 | 168.6994335 | +0.6510299 | 8430 | 34.0 | 19.0 | 17.89 | 0.34 | 1.52 | 1.62 | S20
J1128-0153 | 172.0885023 |  $-$1.8890642 | 8430 |  $-$2.9 | 18.56 | 17.94 | 0.35 | 1.27 | 1.30 | T18
J1142+0012 | 175.7023296 | +0.2043419 | 2810 |  $-$84.8 | 17.02 | 17.90 | 0.71 | 1.40 | 0.84 | S20/GAMA
J1154-0016 | 178.6922828 |  $-$0.2779248 | 8430 | 25.4 | 19.52 | 18.28 | 0.22 | 1.06 | 0.64 | T18/GAMA
J1156-0023 | 179.2186145 |  $-$0.3946597 | 5620 | 15.8 | 18.83 | 17.01 | 0.26 | 1.04 | 1.39 | T18/GAMA
J1202+0251 | 180.5132277 | +2.8515452 | 8430 |  $-$70.6 | 19.43 | 18.53 | 0.31 | 1.49 | 0.68 | S20
J1218+0232 | 184.7355807 | +2.5449139 | 5620 | 1.8 | 19.23 | 18.72 | 0.31 | 1.40 | 0.93 | S20
J1228-0153 | 187.0640987 |  $-$1.8989049 | 5620 |  $-$74.1 | 18.85 | 18.57 | 0.36 | 1.61 | 1.15 | S20
J1411+0233 | 212.8336012 | +2.5618381 | 8430 | 37.3 | 18.86 | 17.44 | 0.21 | 1.07 | 1.55 | S20/GAMA
J1436+0007 | 219.0481314 | +0.1217459 | 5620 |  $-$100.6 | 18.27 | 18.27 | 0.39 | 1.40 | 1.15 | S20/GAMA
J2202-3101 | 330.5472803 |  $-$31.0183808 | 8430 |  $-$87.6 | 19.43 | 18.77 | 0.31 | 1.45 | 1.10 | T18
J2204-3112 | 331.2228147 |  $-$31.2002605 | 8430 |  $-$86.1 | 19.32 | 18.74 | 0.35 | 1.39 | 0.90 | T18
J2257-3306 | 344.3966471 |  $-$33.1144449 | 8430 | 3.5 | 19.42 | 17.09 | 0.29 | 1.18 | 0.93 | T18/GAMA
J2356-3332 | 359.1261248 |  $-$33.5334748 | 11240 |  $-$46.5 | 19.81 | 18.37 | 0.22 | 1.06 | 0.98 | T18
Table 1:  INSPIRE DR2 sample. We list from left to right the galaxy ID and
coordinates, the exposure times and the position angles (along the major axis
of the galaxy) of the XSH observations, the aperture magnitudes (MAG_AUTO from
the KiDS DR3 catalogue, corrected for extinction), the surface brightness
luminosity averaged within the $\mathrm{R}_{\mathrm{e}\,}$, both in $r$-band,
$\mathrm{R}_{\mathrm{e}\,}$ in arcseconds and kiloparsec, computed as median
of the quantities obtained from the $g,r,i$ bands, and the stellar masses from
the SED fitting. Finally, in the last column, we list the sample from which
each object was taken. The six objects with a double reference were selected
from T18(or S20), but were then also found in the GAMA DR4.

##  3 Data reduction and analysis

###  3.1 Data reduction and 1D extraction

As already explained in the INSPIRE DR1, we performed an ad hoc extraction of
the one 1D spectra to take into account the fact that these galaxies are not
spatially resolved and the spectra are dominated by seeing, as the
$\mathrm{R}_{\mathrm{e}\,}$ of all objects in arcseconds (apparent sizes, on
average $\mathrm{R}_{\mathrm{e}\,}$$\sim 0\arcsec.3$) are much smaller than
the median seeing of the observations ($\sim 1\arcsec$ on average). Hence, we
reduced the data using the ESO XSH pipeline (v3.5.3) under the ESO Reflex
Workflow (Freudling et al. 2013, version 2.11.3), only up to the creation of
the 2D spectral frames (one for each arm). Then, we subsequently used our own
Python routines, developed for the INSPIRE Pilot , which were used in INSPIRE
DR1. We cannot use ESO internal data products because they only comprise the
already extracted 1D spectra.

Finally, for the VIS and NIR arms, we corrected all the spectra for telluric
absorption lines using the code molecfit (Smette et al. 2015, version 4.2),
which was run with its interactive ESO Reflex workflow. The telluric
correction was performed with the recipe molecfit_model that fits telluric
absorption features on the telluric standard that is observed in the same
night and with the same instrument set-up as the galaxies. After we determined
the column densities of the various molecules in the spectrum, we constructed
the telluric correction and took the difference in air-mass between the
observations of the telluric standard and the galaxy into account. For this
purpose, we used the recipe molecfit_calctrans.

In previous papers of the INSPIRE series, we have extracted spectra with two
different approaches. On the one hand, we collapsed the whole slit, but
weighted the pixels by their flux (following the optimal extraction approach
described in Naylor 1998). Alternatively, as a second approach, we also
extracted the spectra of each galaxy from an aperture that contained more or
less the same fraction of light for the different objects (R50, containing
$\sim 50$% of the total light, but a mix from inside and outside the real
$\mathrm{R}_{\mathrm{e}\,}$, given the spatial resolution of the data; see
INSPIRE DR1 for more details). The R50 approach is best when the INSPIRE
sample is to be compared with other galaxy samples from the literature because
this is the most comparable aperture, at least in terms of light fraction, to
that extracted at one $\mathrm{R}_{\mathrm{e}\,}$ for normal-size galaxies. In
INSPIRE DR1, we proved that the extraction method does not change the
kinematics and stellar population results, and hence it does not play a role
in the relic confirmation. Therefore, in this DR2, we extracted spectra
following the R50 approach alone.

###  3.2 Arm combination and smoothing

When deriving the stellar population parameters, it is important to use a wide
wavelength range that allows breaking the age-metallicity degeneracy (Worthey,
1994) and thus properly inferring the stellar population parameters. Hence,
the three arms in the INSPIRE data set must be combined. In order to do this,
they must first be brought to the same final resolution. We point out that the
UVB and VIS spectra have the same spatial and spectral sampling (scale =
$0\arcsec.16/\text{px}$, $\delta\lambda\sim 0.156$ Å), while the NIR has lower
resolution (scale = $0\arcsec.25/\text{px}$, $\delta\lambda\sim 0.467$ Å).

First, we computed the redshift of each galaxy and independently identified
the most prominent stellar absorption lines in the different arms. The
redshift values are always consistent between the arms, and in 20 out of 21
cases, they are also consistent with the values reported in T18 and S20
(within 0.0005, the nominal uncertainties on the redshifts). Only for
J1218+0232 do we find a slightly higher redshift from the XSH spectrum with a
higher S/N than that used by T18 and S20 ($\Delta
z=z_{\text{XSH}}-z_{\text{S20}}=0.0352$).

Then, we reported all the spectra for all the systems and in all arms to the
same resolution at fixed FHWM and the same binning. We chose an
FWHM${}_{\mathrm{fin}}=2.51$Å, which is equal to that of the MILES single
stellar population (SSP) models (Vazdekis et al., 2015) that we used for the
kinematic analysis. For the smoothing, we adopted the same spectral
convolution procedure as was successfully employed in the INSPIRE Pilot and in
the INSPIRE DR1, based on the use of a Gaussian function with a variable sigma
(following the prescription of Cappellari 2017).

After smoothing and binning, we finally joined the three arms using data from
the UVB up to $\lambda=5560$Å (observed wavelength), from VIS up to
$\lambda=10100$Å, and from the NIR for redder wavelengths. The arms extend
slightly further (UVB 5595Å and VIS 10240 Å) and overlap by $\sim 300\AA$.
Anyway, we point out that the cut we set for each of the three arms was chosen
in order to avoid extremely noisy wavelengths at the borders. This choice has
no effect on the results presented in the paper because we measured the
kinematics both from the single-arm spectra and from the combined ones.

The redshifts inferred from the combined and smoothed spectra are reported in
the second column of Table 2. They are fully consistent with those computed
from the single-arms spectra. Finally, we brought all the spectra to $z=0$.

###  3.3 Signal-to-noise calculation

To calculate the S/N of the 1D spectra, we followed the same recipe as was
used in DR1. We used the IDL code DER_SNR (Stoehr et al., 2008), which
estimates the S/N directly from the flux, assuming that it is Gaussian
distributed and uncorrelated in wavelength bins spaced two pixels apart.

We obtained three different estimates, one for each arm separately (across the
entire wavelength, which slightly varies from one object to the next because
their redshifts are different), and then, we also computed the arithmetic mean
of the three. These numbers are reported in Table 2, along with the redshifts
inferred from the final combined and smoothed spectra. We recall that the
three independent estimates of the redshift obtained from the three single
spectra are perfectly consistent with the estimate computed from the combined
spectrum. For all the objects, the S/N increases from the UVB to the VIS,
demonstrating the passive nature of the systems, but then decreases again in
the NIR; this is likely due to the noisier nature of the NIR spectra. We did
not compute the S/N on the combined and smoothed spectra because the
convolution causes a noise correlation across different pixels, and hence the
assumption made by the DER_SNR code is no longer valid.

##  4 Analysis of the line-of-sight velocity distribution

To derive the LOSVD, and, in particular, to obtain the integrated stellar
velocity dispersion values, we used the Python-based penalised pixel-fitting
software (pPXF, v8.1.0; Cappellari & Emsellem 2004; Cappellari 2017). We only
used an additive Legendre polynomial (ADEGREE) to correct for the continuum
shape during the fit, but did not use a multiplicative one, nor did we use any
regularisation (see Cappellari 2017 for more details) because in this paper,
we focus on the kinematics without computing stellar population parameters and
obtaining the relic confirmation, which will be presented in a follow-up paper
(Spiniello et al., in prep).

The main purpose of our analysis is to assess the robustness of the inferred
velocity dispersion values that will be used in forthcoming INSPIRE papers as
a proxy for the total mass in the dynamical modelling of UCMGs in general and
relics in particular (Tortora et al., in prep). Moreover, the velocity
dispersion also appears to be a good way to select high-confidence relic
candidates because we found in the INSPIRE DR1 that relics have a higher
velocity dispersion than non-relics with similar sizes and stellar masses.

In order to assess how solid the inferred $\sigma$ values are, we performed an
extensive number of tests to determine the dependence of the kinematic
measurements upon the input spectra and the various assumptions and parameters
in the pPXF fit: the ADEGREE and the moments of the LOSVD were left as free
parameters, the wavelength range, and the masked pixels. We did not need to
investigate on the effect of changing the stellar templates in the fitting
procedure because this was already done in the INSPIRE Pilot. No detectable
difference in the inferred $\sigma$ was found. Therefore, we only fit with the
EMILES models presented in Vazdekis et al. (2015). We limited ourselves to the
safe range of parameters, following the prescription of the authors of the
models444The safe ranges are defined as follows: ages: $[0.063-17.8]$ Gyr, and
metallicites: $[-2.32,0.22]$, [$\alpha$/Fe]: $[0.0-0.4]$. See
http://research.iac.es/proyecto/miles/pages/ssp-models/safe-ranges.php for
more details.. We used the models with a bimodal stellar IMF with a fixed
slope of $\Gamma=1.3$ and PADOVA00 theoretical isochrones (Girardi et al.,
2000). In a forthcoming publication of the INSPIRE series, we will directly
investigate the non-universality of the IMF (Martín-Navarro et al., 2023). In
this analysis, we study the UVB+VIS+NIR and fit up to 10000 Å (rest frame)
alone, where IMF effects are weaker (since dwarf stars emit more strongly at
redder wavelengths), and where the majority of the narrow, strong stellar
absorption lines are. The three spectra separately in the three arms as well
as the combined spectra are made public for the astronomical community and can
be downloaded directly from the ESO Science Archive. They range from
2300Å(2800) to 18000Å (23000) for the highest (lowest) redshift in the sample.

ID | z | S/N | S/N | S/N | S/N
---|---|---|---|---|---
KiDS | ($\pm 0.0005$) | UVB | VIS | NIR | MEAN
J1142+0012 | 0.1077 | 57.9 | 124.1 | 69.8 | 83.9
J1026+0033 | 0.1743 | 38.9 | 113.6 | 68.3 | 73.6
J0909+0147 | 0.2151 | 20.7 | 75.3 | 45.1 | 47.0
J1228-0153 | 0.2973 | 23.2 | 70.1 | 40.8 | 44.7
J1128-0153 | 0.2217 | 21.1 | 69.2 | 37.0 | 42.4
J1411+0233 | 0.3598 | 24.1 | 73.2 | 26.9 | 41.4
J1436+0007 | 0.221 | 21.1 | 67.2 | 29.0 | 39.1
J1156-0023 | 0.2552 | 22.6 | 60.9 | 31.8 | 38.4
J0920+0126 | 0.3117 | 17.9 | 55.6 | 29.7 | 34.4
J1114+0039 | 0.3004 | 19.5 | 54.0 | 28.4 | 34.0
J2204-3112 | 0.2581 | 14.4 | 54.1 | 24.5 | 31.0
J0917-0123 | 0.3602 | 12.2 | 50.3 | 27.5 | 30.0
J1040+0056 | 0.2716 | 11.5 | 46.7 | 31.4 | 29.9
J1202+0251 | 0.3298 | 14.7 | 45.9 | 25.3 | 28.6
J0844+0148 | 0.2837 | 12.9 | 45.0 | 28.0 | 28.6
J0904-0018 | 0.2989 | 12.6 | 44.3 | 26.6 | 27.8
J1154-0016 | 0.3356 | 16.6 | 42.8 | 23.5 | 27.6
J2257-3306 | 0.2575 | 17.8 | 40.0 | 24.0 | 27.3
J2202-3101 | 0.3185 | 13.1 | 47.6 | 20.5 | 27.1
J1218+0232 | 0.308 | 14.6 | 42.0 | 22.7 | 26.4
J2356-3332 | 0.3389 | 11.5 | 34.2 | 17.7 | 21.1
Table 2: Spectroscopic properties of the INSPIRE DR2 sample. We list from left
to right the ID, the redshift computed from the combined spectra, the three
estimates of the S/N (per Å) from the three arms at their original resolution,
and the arithmetic mean of these three values.

###  4.1 Changing the ADEGREE parameter

According to the pPXF recommendations, when performing a full spectral fitting
to obtain the LOSVD, only an additive Legendre polynomial should be used to
correct for the continuum shape during the kinematic fit. The degree of the
additive polynomial, which is regulated by the ADEGREE keyword in pPXF, might
influence the final result on the integrated stellar velocity dispersion,
especially in case of spectra with a low S/N. In previous INSPIRE papers, we
have tested a range of values for the ADEGREE parameter, always fixing it to
the value that stabilises the results against changes of other parameters
while at the same time minimizing the reduced $\chi^{\prime 2}$ ($\chi^{2}$
divided by the number of good pixels used for the fit.). We decided to use the
same ADEGREE value for all the systems (ADEGREE$\sim 20$) to speed up the
computational time, but risked to over-fit the noise affecting the stellar
continuum in some cases, however. Here, we repeated this test in a more
systematic way, using a broader range of values and more spectra. In
particular, we tested ADEGREE values from 1 to 30 and found that at low
polynomials degrees ($<5$), the inferred velocity dispersion is generally not
stable. Then it reaches a plateau around a degree of 8-15, and in some cases,
it then starts to wiggle again at very high degree values ($>17$). However, as
expected, this plateau falls in a slightly different range of ADEGREE values
for different systems, and this prevents us from choosing the same degree for
all them. In Fig. 2 we indicate the fiducial choice of the ADEGREE value with
a large red dot. This adopted choice for each system lies on the median
$\sigma$ retrieved from the test, and it has two purposes: it represents a
good central guess for the bootstrap routines we present in Sect. 4.4 because
it is far from the region in which $\sigma$ is strongly unstable. It also
allows us to be sensitive to the uncertainties arising from the choice of a
specific ADEGREE, hence assessing the error budget associated with this
parameter. This is because the velocity dispersion values change slightly for
degrees around the fiducial value.

Finally, we note that the S/N does not appear to depend on the input spectrum
(shown in each panel), and the overall variation in $\sigma$ is generally
about 5%. This is shown in the figure as the shaded blue region.

Figure 2: ADEGREE test: Variation in the measured $\sigma$ as a function of
the degree of the additive polynomial. The spectra are ordered from top to
bottom and from left to right in descending S/N (shown in each panel in
parentheses). The larger red dots are the ADEGREE chosen for the fiducial fit,
and the shaded blue region shows a variation on the $\sigma$ from the median
value of $\pm 2.5$%. Figure 3: MOMENTS test: Variation in the measured
$\sigma$ as a function of the moments of the LOSVD that were constrained in
the fit. The objects are in the same order as in Fig. 2, from high (top left)
to low (bottom right) S/N. In each panel, the galaxy ID and the S/N are
plotted in the right bottom corner, the larger red dot is the MOMENT chosen
for the fiducial fit (=4), and the shaded blue region shows a variation on the
$\sigma$ from the median value of $\pm 2.5$%.

###  4.2 Changing the MOMENTS parameter

The pPXF allows us to constrain from two and up to six moments of the Gauss-
Hermite parameterisation (van der Marel & Franx, 1993), using the keyword
MOMENTS. In an ideal case, the higher moments ($h_{3}$,…,$h_{6}$) should be
completely uncorrelated to the velocity (V) and velocity dispersion
($\sigma)$. However, when the LOSVD is not perfectly sampled by the data,
performing the fit with a different number of moments might influence the
resulting $\sigma$ values. Hence, we ran different fits changing the MOMENTS
from two to six in order to test the effect of adopting different choices for
the MOMENTS parameter. We show the resulting $\sigma$ values in Fig. 3, where
galaxies are again ordered according to the mean S/N. During this test, we
used the fiducial value for the ADGREE parameter for each galaxy (i.e. the red
dot in Fig. 2). No correlation with the S/N is found in this case either, and
the $\sigma$ values inferred using different MOMENTS are very close to each
other and in many cases are consistent within the errors. For the majority of
systems, a small increase in $\sigma$ ($\sim 5-10$ km/s) is observed from
MOMENTS = 2 to MOMENTS = 4. Then, in 16 out of 21 cases the result is
virtually unchanged when the number of fitted moments is further increased. In
two (three) cases, $\sigma$ decreases (increases) when the number of moments
in the fit is increased. Given the results of this test, we consider as
fiducial values for the velocity dispersion the values computed with MOMENT=4
for all the 21 systems. In all but two cases, the variation with respect to
the median value is always below 5% (shaded region in the figure).

Figure 4: Fit range test. The plot displays the relative difference in
$\sigma$ between the measurements obtained by fitting the widest and narrowest
wavelength range. For spectra with a lower S/N, the spread is larger and the
variation in the velocity dispersion is in some cases as high as 20%.

###  4.3 Changing the wavelength range of the fitted region

Another effect we tested is the effect of the fitted spectral wavelength on
the inferred $\sigma$ values. For this purpose, we repeated the fit many times
with the fiducial ADEGREE and MOMENT parameters, but each time with a smaller
wavelength range by systematically shifting the blue and the red ends limits
of the fitted window. We started from the widest range, [3000$-$10000] Å and
decreased in steps of 200 Å on each side of the fitting window down to the
smallest range, [6000$-$7000] Å. In this case, we find a dependence on the
S/N: A larger variation in the inferred $\sigma$ values is observed in low-S/N
spectra. This is visible from Fig. 4, where we plot the relative difference in
velocity dispersion between the value obtained by fitting the widest
wavelength range and the value obtained fitting the narrowest range, against
the S/N. For S/N$<35,$ the scatter is larger than 20%
($\Delta\sigma/\sigma_{\text{max}}>0.2$), while it stabilises around that
level or below it for spectra with higher S/N. For the majority of the cases,
higher values of $\sigma$ are found when the 4000 Å break is masked out. This
effect is smaller for high S/N spectra, however, where the results are much
more stable against change in the fitted region.

Figure 5: Random masking test. The objects are in the same order as in Fig. 2,
from high (top left) to low (bottom right) S/N. The figure shows the effect on
the $\sigma$ of masking a window of 200 $\AA$ placed on each of the spectral
lines falling within the fitting limits out from the pPXF fit.

To evaluate the effect of masking a certain line from the fit, we ran another
test in which we repeated the fit on the same spectrum across the fiducial
fitting window (3000$-$10000 Å) 50 times, each time masking one strong
(absorption or emission) stellar feature. We note that frequently, different
lines are less than 200 Å apart. We nevertheless ran the masking test one time
for each line, always centreing the masking window on the corresponding line
(sometimes partially masking nearby lines as well). The result of this test is
shown in Fig. 5 where again we group the spectra according to their mean S/N.
Within each group, the spectra were also ordered from the highest (top) to the
lowest (bottom) S/N. In this case, a mild dependence on the S/N is found
overall, although changes on a single-system level are also visible. In
general, the two regions appear to produce a non-negligible effect on the
$\sigma$ measurements for more than one galaxy: the region between 3800 and
4300 Å, and that around the Mgb and Fe strong absorption line ($\sim
5100$-$5400$ Å). Overall, however, the differences in sigma ($\Delta\sigma$)
are never larger than 10%.

ID |  $\sigma$ (km/s) |  $\Delta\sigma_{BOOT}$ (km/s)
---|---|---
KIDS | COMB | $\Delta_{noise}$ | $\Delta_{adeg}$ | $\Delta_{mom}$ | $\Delta_{wav}$
J1142+0012 | 129 |  $\pm$6 |  $\pm$6 |  $\pm$6 |  $\pm$6
J1026+0033 | 225 |  $\pm$5 |  $\pm$7 |  $\pm$8 |  $\pm$6
J0909+0147 | 401 |  $\pm$15 |  $\pm$18 |  $\pm$17 |  $\pm$12
J1228-0153 | 191 |  $\pm$6 |  $\pm$8 |  $\pm$7 |  $\pm$7
J1128-0153 | 192 |  $\pm$8 |  $\pm$9 |  $\pm$8 |  $\pm$8
J1411+0233 | 217 |  $\pm$7 |  $\pm$9 |  $\pm$8 |  $\pm$9
J1436+0007 | 193 |  $\pm$9 |  $\pm$10 |  $\pm$10 |  $\pm$10
J1156-0023 | 177 |  $\pm$11 |  $\pm$11 |  $\pm$10 |  $\pm$10
J0920+0126 | 190 |  $\pm$8 |  $\pm$8 |  $\pm$8 |  $\pm$9
J1114+0039 | 181 |  $\pm$10 |  $\pm$13 |  $\pm$12 |  $\pm$10
J2204-3112 | 227 |  $\pm$14 |  $\pm$16 |  $\pm$13 |  $\pm$14
J0917-0123 | 239 |  $\pm$13 |  $\pm$13 |  $\pm$13 |  $\pm$14
J1040+0056 | 240 |  $\pm$14 |  $\pm$18 |  $\pm$16 |  $\pm$14
J1202+0251 | 165 |  $\pm$17 |  $\pm$14 |  $\pm$13 |  $\pm$17
J0844+0148 | 224 |  $\pm$12 |  $\pm$12 |  $\pm$15 |  $\pm$15
J0904-0018 | 205 |  $\pm$11 |  $\pm$12 |  $\pm$11 |  $\pm$13
J1154-0016 | 163 |  $\pm$8 |  $\pm$10 |  $\pm$10 |  $\pm$9
J2257-3306 | 185 |  $\pm$15 |  $\pm$13 |  $\pm$12 |  $\pm$13
J2202-3101 | 221 |  $\pm$13 |  $\pm$12 |  $\pm$12 |  $\pm$10
J1218+0232 | 171 |  $\pm$11 |  $\pm$13 |  $\pm$11 |  $\pm$16
J2356-3332 | 162 |  $\pm$15 |  $\pm$15 |  $\pm$15 |  $\pm$16
Table 3: Results of the kinematics bootstrap analysis for the INSPIRE DR2
sample. We report from left to right the ID, the fiducial $\sigma$ resulting
from the bootstrap analysis on the combined and smoothed spectra
($\sigma_{\text{COMB}}$), and the uncertainties associated with the four
bootstrap routines. Each subsequent column includes the randomisation of the
previous test (see text for more details).

###  4.4 Bootstrap

After assessing the effects of the main fit parameters individually, we set up
different bootstrap routines, always repeating 250 pPXF fits of the same
spectrum, to quantitatively infer uncertainties on the velocity dispersion
values and to combine all the tests we performed so far in a systematic way.
Specifically, we started to randomise each time the flux of every single pixel
according to a Gaussian distribution of the noise around the observed flux and
computed the uncertainties associated with this ($\Delta_{noise}$). Then,
keeping the randomisation of the noise, we also added a random selection of
the ADEGREE within the range 1-30 in the bootstrap, and evaluated the
uncertainties in this case as well ($\Delta_{adeg}$). Subsequently, we ran a
third bootstrap in which we combined the noise randomisation and the ADEGREE
randomisation, and we also randomly changed the moments of the LOSVD
($\Delta_{mom}$), always from two to six. We finally ran another bootstrap
including the three effects described above and a randomisation of the fit
limit ($\Delta_{wav}$), chosen within 2500 pixels from the blue and red end of
the entire wavelength range. We followed the same approach as described in
Sec. 4.3.

All these tests allowed us to control the uncertainties introduced by the
choice of the fit parameters during the fit. The $\Delta\sigma$ that are
listed in Table 3 are always $<20$ km/s. Each $\Delta$ column in the table was
obtained by also including the previous randomisation(s). Thus, the last
column refers to the case where noise, ADEGREE, and MOMENTS were all
randomised. We note that randomizing on more parameters at the same time does
not necessarily correspond to an increase in the associated uncertainty. The
four $\Delta\sigma$ columns of Table 3 all list comparable values,
demonstrating that randomizing on all parameters gives a good indication of
the total uncertainty associated with all the possible parameters of the fit.
Finally, a clear dependence on the S/N of the spectra is found, as shown in
Fig. 6. The uncertainties are $\sim 10$ (4) % for the lowest (highest) S/N
spectra.

ID |  $\sigma$ (km/s) |  $\sigma$ (km/s) |  $\sigma$ (km/s) |  $\sigma$ (km/s) | $\Delta\sigma_{COMB-UVB}$ | $\Delta\sigma_{COMB-VIS}$ | $\Delta\sigma_{COMB-NIR}$
---|---|---|---|---|---|---|---
KIDS | UVB | VIS | NIR | COMB | (km/s) | (km/s) | (km/s)
J1142+0012 | 112$\pm$7 | 147$\pm$5 | 173$\pm$12 | 129$\pm$3 | 17$\pm$10 | -17$\pm$9 | -44$\pm$16
J1026+0033 | 208$\pm$4 | 229$\pm$6 | 221$\pm$13 | 225$\pm$3 | 16$\pm$7 | -4$\pm$9 | 4$\pm$16
J0909+0147 | 365$\pm$8 | 429$\pm$11 | 449$\pm$27 | 401$\pm$7 | 36$\pm$15 | -27$\pm$18 | -47$\pm$33
J1228-0153 | 177$\pm$5 | 188$\pm$6 | 207$\pm$7 | 191$\pm$3 | 15$\pm$8 | 3$\pm$9 | -16$\pm$10
J1128-0153 | 185$\pm$5 | 206$\pm$6 | 166$\pm$9 | 192$\pm$4 | 7$\pm$10 | -13$\pm$11 | 26$\pm$14
J1411+0233 | 197$\pm$5 | 211$\pm$6 | 200$\pm$9 | 217$\pm$3 | 20$\pm$9 | 6$\pm$9 | 17$\pm$12
J1436+0007 | 188$\pm$5 | 193$\pm$6 | 191$\pm$16 | 193$\pm$4 | 5$\pm$10 | 1$\pm$10 | 2$\pm$21
J1156-0023 | 166$\pm$6 | 182$\pm$5 | 239$\pm$17 | 177$\pm$4 | 11$\pm$10 | -5$\pm$9 | -62$\pm$21
J0920+0126 | 168$\pm$7 | 166$\pm$5 | 209$\pm$11 | 190$\pm$6 | 22$\pm$13 | 24$\pm$11 | -19$\pm$17
J1114+0039 | 154$\pm$7 | 191$\pm$8 | 215$\pm$13 | 181$\pm$8 | 26$\pm$15 | -10$\pm$15 | -35$\pm$21
J2204-3112 | 212$\pm$8 | 253$\pm$9 | 153$\pm$12 | 227$\pm$7 | 14$\pm$16 | -27$\pm$17 | 74$\pm$20
J0917-0123 | 225$\pm$11 | 237$\pm$7 | 192$\pm$19 | 239$\pm$6 | 14$\pm$16 | 2$\pm$13 | 47$\pm$24
J1040+0056 | 227$\pm$12 | 226$\pm$9 | 155$\pm$13 | 240$\pm$8 | 13$\pm$20 | 14$\pm$16 | 85$\pm$20
J1202+0251 | 169$\pm$9 | 175$\pm$9 | 176$\pm$10 | 165$\pm$6 | -4$\pm$15 | -10$\pm$15 | -11$\pm$16
J0844+0148 | 195$\pm$9 | 237$\pm$9 | 194$\pm$9 | 224$\pm$6 | 29$\pm$15 | -13$\pm$15 | 30$\pm$15
J0904-0018 | 206$\pm$10 | 192$\pm$9 | 200$\pm$15 | 205$\pm$6 | -1$\pm$16 | 13$\pm$15 | 4$\pm$21
J1154-0016 | 136$\pm$7 | 194$\pm$8 | 94$\pm$9 | 163$\pm$4 | 26$\pm$12 | -31$\pm$12 | 69$\pm$14
J2257-3306 | 170$\pm$6 | 186$\pm$9 | 200$\pm$18 | 185$\pm$6 | 16$\pm$13 | -1$\pm$15 | -15$\pm$25
J2202-3101 | 187$\pm$10 | 227$\pm$7 | 238$\pm$15 | 221$\pm$5 | 33$\pm$16 | -7$\pm$13 | -17$\pm$20
J1218+0232 | 168$\pm$8 | 191$\pm$9 | 170$\pm$9 | 171$\pm$6 | 3$\pm$14 | -20$\pm$15 | 1$\pm$15
J2356-3332 | 148$\pm$9 | 181$\pm$10 | 151$\pm$10 | 162$\pm$6 | 14$\pm$15 | -19$\pm$16 | 12$\pm$16
Table 4: Results of the tests on the input spectra and their resolution.
Systems are ordered from the highest to the lowest S/N. We report from left to
right the ID, the velocity dispersion values computed from the single-arm
spectra at their original resolution and that measured from the combined and
smoothed ones (COMB), and the differences in $\sigma$ between these three
spectra ($\Delta\sigma$). We note that the uncertainties quoted in the table
are just the formal errors produced by pPXF.  Figure 6: Relative variation of
$\sigma$ against the mean S/N of the spectra. The four different symbols show
the values relative to the four different bootstrap routines described in Sec.
4.4, and the points are colour-coded by the S/N of the spectra to which they
refer. The uncertainties are about 10% for the lowest S/N spectra (red) and
$\sim 4$% for the highest S/N spectra (blue). The colour gradient of the data
points allows distinguishing the different systems. Figure 7: Effect of
changing the resolution on the $\sigma$ measurement. First panel (from the
left): Comparison between the velocity dispersion values obtained from the
combined spectrum and those obtained from the three single arms at their
original resolution. The solid black line shows the one-to-one identity
relation. Second panel: Difference between the velocity dispersion computed
from the combined and smooth spectra and those measured from the single-arm
spectra as a function of the mean S/N of each system. The horizontal lines
represent the mean offset found for the three arms and correspond to the
vertical lines in the following panel (we use a different line style for each
different arm). The y-axis is flipped to better compare it with the x-axis in
the third panel. No clear correlation with the S/N is found. Third panel:
Distribution of the $\sigma$ differences between the arms, drawn from the
histograms, assuming a Gaussian profile. A small offset for the UVB (VIS) is
visible, which systematically underestimates (overestimates) the $\sigma$ by
14 km/s ($-8$ km/s) on average. For the NIR, the distribution peaks at around
3 km/s, but a much larger scatter is found. Fourth panel: Relative shift in
$\sigma$ against the $\sigma$ measured from the single-arm spectra.

###  4.5 Changing the input spectra and their resolution

After assessing how the parameters and set-ups of the pPXF code influence the
velocity dispersion measurements, we performed one more test by changing the
input spectra, both in terms of the considered wavelength band and in terms of
spectral resolution and binning. In particular, we ran the code independently
on four different spectra for each galaxy: the UVB, the VIS, and the NIR at
their original resolution ($R_{\text{UVB}}=3200$, $R_{\text{VIS}}=5000$, and
$R_{\text{NIR}}=4300$), and on the combined and smoothed spectrum described in
Sec.3.2, up to $\lambda=10000$Å. For this final test, we did not run the
bootstrap analysis, did not mask single lines, nor changed the limits of
wavelength range on which the fit is performed, as we already know the
contribution on the uncertainties that this has. We also fixed ADEGREE to the
fiducial value of each system and MOMENTS = 4, given the results of the tests
performed so far.

Overall, the spectral resolution plays a small role, leading to $\sigma$
values consistent within the (statistical) uncertainties. This is expected,
given the relatively high stellar velocity dispersion values covered by our
sample (130-400 km/s), corresponding to a resolution that is well below the
resolution of the XSH data and of the MILES models. The effect of changing the
resolution of the input spectra on the $\sigma$ is shown in the first panel in
Fig. 7, where we plot the $\sigma$ values measured from the combined spectrum
(between 3000 and 10000 Å) on the x-axis and the values obtained from the
single arms on the y-axis. A fair agreement is found between the measurements,
especially for UVB and VIS. The measurements in NIR show a much larger
scatter, as shown in the second panel of this figure. Here we plot the
difference between the $\sigma$ measured from the combined spectrum and that
measured from the single arms versus the mean S/N of the corresponding
spectrum. Independently of the S/N, the difference in the NIR is larger ($>50$
km/s in many cases). The third panel of Fig. 7 shows the histograms of the
$\Delta\sigma$ (Table 4), from which a much larger scatter is visible from the
NIR spectra than from UVB and VIS. This is due to the several spikes, bad
pixels, and residuals in this arm. A small systematic difference between the
values computed from the UVB ($\sim 14$ km/s) and the VIS ($\sim-8$ km/s) is
found as well. Finally, the rightmost panel demonstrates that the $\sigma$
shift is not a relative effect (e.g. 5% of the $\sigma$ value), but rather an
apparently absolute shift of 20% in UVB and VIS at most, but up to 60% in the
NIR (for systems with a lower velocity dispersion and relatively low S/N
spectra).

The values of the stellar velocity dispersion measured from the single arms,
those measured from the combined and smoothed spectra, and the differences
between them are all listed in Table 4. The errors quoted on the
$\Delta\sigma$ are the sum of the errors of the two terms. All the quantities
in the table are rounded to the integer.

The main conclusion that can be drawn from this test is that when different
input spectra are used that cover different wavelength ranges, the inferred
stellar velocity dispersion values can change systematically and by up to 20%
between UVB and VIS and up to 60% when NIR is considered as well (however,
this happens only for the lower S/N spectra). We speculate that the different
(generally higher) values inferred from blue to red might have a physical
origin, especially for non-relics, which might include multiple populations,
possibly with different kinematic properties within the galaxy. Excluding the
region below $4000$Å might exclude the contribution of relatively younger
stars, while focussing on $\lambda>6000$Å might mean that only the oldest and
reddest stars are considered. Finally, in our case, the spectral region at
$\lambda>10000$Å becomes very noisy and does not always allow us to obtain a
very precise constraint on the stellar velocity dispersion.

Hence, in conclusion, the $\sigma$ values computed from the wavelength range
[3000-10000] Å probably provide the most robust estimate for XSH spectra with
a medium or high S/N. We therefore quote them as our fiducial choice. Figure 8
shows the fiducial fit for all the systems, grouped and ordered according to
the S/N of the corresponding spectrum.

Figure 8: Fiducial best fit (red) over-plotted on the combined and smoothed
galaxy spectra (red) for all the systems in DR2 ordered from high to low S/N
(left to right and top to bottom). Noisier regions around 4300-4500 Å and
around 7500-8000 Å show the wavelength at which the different arms were
joined.
*For clarity of the stellar continuum best fit, the peaks of the strong emission lines in the spectrum of J1142+0012 are cut in the plot.

##  5 Comparison with literature measurements

Figure 9: Relative difference between the velocity dispersion measured from
XSH and that measured from GAMA plotted against the $\sigma$ measured from
XSH. The fit was made on the ten objects in common within the same wavelength
range and with the same fit settings. The disagreement is about 20% (shaded
region) or more for galaxies with $\sigma<$170 km/s, and it is much smaller
for galaxies with a higher velocity dispersion.

Ten of the 21 systems analysed in this INSPIRE DR2 have also been targeted by
the GAMA Survey. Generally, the GAMA spectra have a lower or comparable S/N,
but much lower (2x) spectral resolution (R$\sim 1300$) and thus a larger
instrumental dispersion.

We ran pPXF on these ten GAMA spectra and on the ten INSPIRE spectra by
adopting the same fit settings (i.e. the same wavelength range, MOMENTS=4, and
the optimal ADEGREE choice for each galaxy), after correcting for the flux
shift between the blue and red GAMA arms that is due to a bad splicing. The
fitted wavelength range for both data sets is $\sim 3500-7000\AA$.

We note that the two set of spectra are both fully seeing dominated, and hence
the inferred velocity dispersion values should be considered as lower limits
(for more details, see Appendix A of the INSPIRE DR1). For the INSPIRE case,
the R50 apertures are about $\sim 0.5-0.6\arcsec$, and the GAMA spectra are
extracted from a circular aperture of radius=$1\arcsec$. We decided not to
apply any aperture-corrections to the $\sigma$ because the equations derived
in the literature (e.g. Eq. 1 of Cappellari et al. 2006) for normal-sized
galaxies probably cannot be applied to ultra-compact galaxies. However, we
note that bringing the measurements to $\mathrm{R}_{\mathrm{e}\,}$ will
account for a variation by 5% of the $\sigma$ at most.

The velocity dispersion values obtained from the XSH and GAMA spectra are
compared in Fig. 9. We plot the quantity
$\Delta\sigma/\sigma_{\text{XSH}}\equiv(\sigma_{\text{GAMA}}-\sigma_{\text{XSH}})/\sigma_{\text{XSH}}$,
representing the relative disagreement between the two values, which are
listed in Table 5, along with the uncertainties given by pPXF (only random
errors). The agreement is fairly good. The velocity dispersion of only one
system (J1142+0012) is different by more than 1$\sigma$ error. The scatter is
non-negligible for this system, which has the lowest velocity dispersion
($\sigma<150$ km/s) and also the highest S/N. Table 5 is indeed ordered from
the highest to the lowest S/N spectra, highlighting that $\Delta\sigma$ does
not depend on the S/N of INSPIRE spectra.

Except for three systems with the lowest velocity dispersion, all the
$\Delta\sigma/\sigma$ are negatives. This is in line with the fact that the
XSH spectra are integrated over a slightly smaller aperture than the GAMA
spectra.

ID |  $\sigma$ (km/s) |  $\sigma$ (km/s) | $\Delta\sigma_{XSH-GAMA}$
---|---|---|---
KIDS | XSH | GAMA | (km/s)
J1142+0012 | 130$\pm$4 | 155$\pm$12 | 25$\pm$16
J0909+0147 | 403$\pm$7 | 381$\pm$17 | -22$\pm$24
J1411+0233 | 219$\pm$3 | 207$\pm$17 | -13$\pm$20
J1436+0007 | 194$\pm$4 | 186$\pm$11 | -8$\pm$15
J1156-0023 | 168$\pm$5 | 175$\pm$35 | 8$\pm$40
J0920+0126 | 186$\pm$5 | 175$\pm$18 | -12$\pm$22
J0844+0148 | 229$\pm$9 | 185$\pm$49 | -44$\pm$58
J0904-0018 | 200$\pm$7 | 169$\pm$29 | -31$\pm$36
J1154-0016 | 166$\pm$5 | 195$\pm$38 | 29$\pm$43
J2257-3306 | 180$\pm$6 | 168$\pm$23 | -12$\pm$29
Table 5: Comparison with GAMA. Systems are ordered from highest to lowest S/N
. The last column shows the difference between the measurements obtained from
the two spectra, using a similar wavelength range and the same ADEGREE and
MOMENT parameters. The uncertainty on the $\Delta\sigma$ is assumed to be the
sum of the two single uncertainties on the $\sigma$.  Figure 10:  M⋆–$\sigma$
relation for the INSPIRE DR1+DR2 galaxies compared to normal-sized ETGs. Empty
blue squares represent the UCMGs in DR1, and filled blue squares indicate
confirmed relics. Empty red circles show the galaxies in DR2. The black line
draws the median of the stellar mass-velocity dispersion relation for normal
galaxies in KiDS with SDSS DR16 spectroscopy, while the shaded region
highlights the 16th–84th percentile interval. Because the INSPIRE spectra are
seeing dominated, the estimates of the velocity dispersion values must be
considered as lower limits. The black error bar in the legend shows the mean
error we retrieve from the bootstrap, and the arrows associated with the
single INSPIRE objects show the estimated strength of a 7% systematic
correction that takes the seeing effect into account (see Appendix A of the
INSPIRE DR1).

##  6 Stellar mass–velocity dispersion relation

In INSPIRE DR1, we found a quantitative difference between relics and non-
relics in the stellar mass–stellar velocity dispersion space. In particular,
$\sigma$ at fixed stellar mass is higher for relics and especially extreme
relics than for non-relics and normal-sized passive galaxies.

We can now reproduce the M⋆–$\sigma$ plot taking advantage of the increased
number statistics that we achieved with DR1+DR2. This plot can potentially
have a strong predictive power if we identify that only relics (and not all
UCMGs in general) are outliers in this relation, having higher velocity
dispersion values.

The M⋆–$\sigma$ relation is plotted in Fig. 10 for the 40 INSPIRE objects we
analysed so far. We recall that the velocity dispersion values measured from
the R50 spectra are underestimated because of the seeing effect, and thus must
be interpreted as a lower limit to the $\sigma$, as indicated by the arrows.
Uncertainties are not drawn for each single system for clarity of the plot,
but a mean error, derived from the bootstrapping procedure, is shown in the
top left corner of the plot. Blue points are DR1 galaxies, and red points are
the new objects that we added with this paper. The solid black line shows the
M⋆–$\sigma$ relation for a sample of normal-sized ETGs from the KIDS survey
for which SDSS DR16 spectroscopy is available. In this case, the velocity
dispersion was corrected for $\mathrm{R}_{\mathrm{e}\,}$. The shaded region
represents the 16th–84th percentile confidence interval. The majority of DR1
confirmed relics (filled squares) generally lie above the median trend for
normal galaxies in KiDS, and only DR1 confirmed relics systematically also lie
above the 84th percentile. We note, nevertheless, that two (non-extreme)
relics confirmed in DR1 do not have $\sigma$ that exceeds expectations
significantly.

At least four red points (DR2 UCMGs) appear to have a higher velocity
dispersion than normal-size galaxies of similar stellar mass. Hence, we
identify them as the most reliable relic candidates from the DR2 sample. We
therefore conclude that we currently cannot confirm the (tentative) result
found in DR1, and that a careful and detailed stellar population analysis
remains the only and best way to confirm the relic nature of ultra-compact
massive galaxies. This will be performed on the 21 new objects in a
forthcoming paper of the INSPIRE series (Spiniello et al., in prep.). With a
larger number statistic of confirmed relics, we will be able to reassess the
situation and finally conclude whether all relics are outliers in the
M⋆–$\sigma$ relation.

##  7 Summary and conclusions

This paper accompanies the INSPIRE second data release (DR2), which is also
released as ESO Phase 3 collection (see Sec. 2). We have reduced and analysed
the X-Shooter spectra of an additional 21 UCMGs, selected from the KiDS Survey
(; ) or from the GAMA Survey, to be good relic candidates. These new 21
spectra are added to those already released in Spiniello et al. (2021b) as
part of the INSPIRE DR1, and bring the total number of objects analysed so far
to 40.

After reducing the data with the standard ESO XSH pipeline, up to the
production of a 2D spectrum, we have extracted the 1D integrated spectra in
each arm, from an aperture containing $\sim 50$% of the total light (R50). We
note, however, that this does not exactly correspond to extracting spectra at
$\mathrm{R}_{\mathrm{e}\,}$ because the light contained in the R50 aperture is
a mixture from inside and outside the real $\mathrm{R}_{\mathrm{e}\,}$ being
the data seeing limited. In DR1, we showed that the kinematics does not depend
on the extraction method. We then corrected the VIS and NIR for telluric
emission using the molecfit code and finally obtained a combined UVB+VIS+NIR
spectrum for each galaxy after smoothing the spectra to the common resolution
of 2.51Å in FWHM. This is the same resolution as for the model spectrum
templates.

For each object, we release three final 1D spectra, one for each arm, at the
original instrumental resolution (R$=3200$, R$=5000$, and R$=4300$ in UVB,
VIS, and NIR, respectively). The fluxes are given in units of erg cm-2 s-1
Å-1, and the wavelength is always measured in air. The combined and smoothed
spectrum is also released as an additional ancillary product, together with
the original spectra. Finally, we release the NIR spectra of the 19 systems
presented in DR1, for which UVB and VIS are already publicly available. In
this case, we provide both the R50 and the OptExt version for completeness
(see DR1 for more information). We add the combined and smoothed spectra as
ancillary files for these DR1 objects as well.

The main result of this paper is an in-depth kinematical analysis that
demonstrates the validity and robustness of the stellar velocity dispersion
measurements and the systematics associated with the derivation of kinematic
features with full spectral fitting approaches. We presented the integrated
$\sigma$ values obtained from the spectra of the 21 UCMGs and carried out a
very detailed quantitative analysis of the statistical and systematic
uncertainties on these values due to the different assumptions, parameters,
and set-ups of the pPXF code, which was used for the full spectral fitting.
For the purpose, we also set up a bootstrap analysis, which allowed us to take
the statistical uncertainties from the observations and the uncertainties
introduced by the specific parameter set-up into account.

The conclusion of this analysis is that because of the fairly high S/N of the
spectra ($20<S/N_{\text{MEAN}}<85$), the stellar velocity dispersion values
are robust and generally precise to the 5% level. However, the wavelength
range used in the fit plays a non-negligible role in inferring the $\sigma$
values: it shifts the inferred values also by $\sim 30$% in the worst cases.
This mainly depends on the S/N of the input spectrum.

In detail, we found that the degree of the additive Legendre polynomial that
we used to correct for the continuum shape affects the inferred values of
$\sigma$ only slightly, unless too low ($<5$) or too high ($>25$) values are
used. The number of moments of the Gauss-Hermite parameterisation of the LOSVD
used in the fit also plays a negligible role on the stellar velocity
dispersion values. We found a mean systematic shift of $\sim 14$ ($\sim-8$)
km/s between the stellar velocity dispersion values extracted from the UVB
(VIS) spectra and those measured from the combined and smoothed spectra. This
shift does not depend strongly on the S/N. The values computed from the NIR
are overall consistent, but show a much larger scatter (and hence a larger
formal error) because of the noisier nature of the spectra at these redder
wavelengths (spikes and sky and telluric residuals) and the absence of strong
and narrow absorption lines. The uncertainties computed from the NIR appear to
be larger for systems with lower $\sigma$. On the other hand, modifying the
wavelength range on which the fit is performed plays the largest role,
changing the inferred $\sigma$ by more than the statistical uncertainties
returned by the fit and by up to 20% in the worst cases. The velocity
dispersion values are systematically higher when a redder wavelength range is
used, within 3000-10000Å. This mostly happens for lower S/N spectra, as shown
in Fig. 4, where the velocity dispersion measurements are significantly larger
when the blue end of the wavelength is excluded.

We finally analysed the GAMA spectra of ten galaxies in common with the
INSPIRE DR2 sample. We repeated the pPXF run on the GAMA and XSH spectra using
an identical configuration for the fit. The velocity dispersion agreed fairly
well with the dispersion measured from the XSH spectra in a similar wavelength
region. Only for systems with the lowest velocity dispersion were the values
inferred from GAMA spectra (with a much lower spectral resolution)
overestimated by $\sim 25\%$.

In conclusion, our work has shown that assessments of stellar kinematics and
measurements of the stellar velocity dispersion are mainly robust to different
assumptions on the fitting parameters. They are more sensitive to changes in
wavelength coverage than previously thought, however.

In the next paper of the series (Spiniello et al., in prep.), we will focus on
the stellar population properties of these 21 UCMGs. Constraining age,
metallicity, and [Mg/Fe], we will be able to confirm a fraction of them as
relics, further increasing the number of spectroscopically confirmed relics in
the low-$z$ Universe. We will therefore test whether the velocity dispersion
can be used as a selection criterion to select the most reliable relic
candidates among UCMGs.

Finally, the third and final INSPIRE data release is foreseen after completion
of all the observations. Twelve additional UCMGs will be targeted, whose
kinematics and stellar populations will be studied.

## Acknowledgements

GD acknowledges support by ANID, BASAL, FB210003. CS is supported by an
‘Hintze Fellowship’ at the Oxford Centre for Astrophysical Surveys, which is
funded through generous support from the Hintze Family Charitable Foundation.
CS, CT, FLB, AG, SZ, and PS acknowledge funding from the INAF PRIN-INAF 2019
program 1.05.01.85.11. AFM has received financial support through the
Postdoctoral Junior Leader Fellowship Programme from ‘La Caixa’ Banking
Foundation (LCF/BQ/LI18/11630007) and from the Severo Ochoa Excellence scheme
of the MCIU (CEX2019-000920-S). DS is a member of the International Max Planck
Research School (IMPRS) for Astronomy and Astrophysics at the Universities of
Bonn and Cologne.

The Geryon cluster at the Centro de Astro–Ingenieria UC was extensively used
for the calculations performed in this paper. BASAL CATA PFB–06 and FB210003,
the Anillo ACT-86, FONDEQUIP AIC–57, and QUIMAL 130008 provided funding for
several improvements to the Geryon cluster.

GAMA is a joint European-Australasian project based around a spectroscopic
campaign using the Anglo-Australian Telescope. The GAMA input catalogue is
based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared
Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained
by a number of independent survey programmes including GALEX MIS, VST KiDS,
VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio
coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and
the participating institutions. The GAMA website is http://www.gama-
survey.org/.

Based on observations made with ESO Telescopes at the La Silla Paranal
Observatory under programme ID 179.A-2004 and ID 177.A-3016.

The authors wish to thank the ESO Archive Science Group for the great support
with the Data Release.

We finally thank the anonymous referee for the helpful comments and
constructive remarks on this manuscript.

## References

  * Barbosa et al. (2021) Barbosa, C. E., Spiniello, C., Arnaboldi, M., et al. 2021, A&A, 649, A93
  * Beasley et al. (2018) Beasley, M. A., Trujillo, I., Leaman, R., & Montes, M. 2018, Nature, 555, 483
  * Bennett et al. (2014) Bennett, C. L., Larson, D., Weiland, J. L., & Hinshaw, G. 2014, ApJ, 794, 135
  * Blumenthal et al. (1984) Blumenthal, G. R., Faber, S. M., Primack, J. R., & Rees, M. J. 1984, Nature, 311, 517
  * Buitrago et al. (2018) Buitrago, F., Ferreras, I., Kelvin, L. S., et al. 2018, ArXiv e-prints [arXiv:1807.02534]
  * Cappellari (2017) Cappellari, M. 2017, MNRAS, 466, 798
  * Cappellari et al. (2006) Cappellari, M., Bacon, R., Bureau, M., et al. 2006, MNRAS, 366, 1126
  * Cappellari & Emsellem (2004) Cappellari, M. & Emsellem, E. 2004, PASP, 116, 138
  * Conroy et al. (2013) Conroy, C., Dutton, A. A., Graves, G. J., Mendel, J. T., & van Dokkum, P. G. 2013, ApJ, 776, L26
  * Costantin et al. (2021) Costantin, L., Pérez-González, P. G., Méndez-Abreu, J., et al. 2021, ApJ, 913, 125
  * Daddi et al. (2005) Daddi, E., Renzini, A., Pirzkal, N., et al. 2005, ApJ, 626, 680
  * Damjanov et al. (2011) Damjanov, I., Abraham, R. G., Glazebrook, K., et al. 2011, ApJ, 739, L44
  * de Jong et al. (2015) de Jong, J. T. A., Verdoes Kleijn, G. A., Boxhoorn, D. R., et al. 2015, A&A, 582, A62
  * de Jong et al. (2019) de Jong, R. S., Agertz, O., Berbel, A. A., et al. 2019, The Messenger, 175, 3
  * de la Rosa et al. (2016) de la Rosa, I. G., La Barbera, F., Ferreras, I., et al. 2016, MNRAS, 457, 1916
  * Driver et al. (2022) Driver, S. P., Bellstedt, S., Robotham, A. S. G., et al. 2022, MNRAS, 513, 439
  * Ferré-Mateu et al. (2019) Ferré-Mateu, A., Forbes, D. A., McDermid, R. M., Romanowsky, A. J., & Brodie, J. P. 2019, ApJ, 878, 129
  * Ferré-Mateu et al. (2015) Ferré-Mateu, A., Mezcua, M., Trujillo, I., Balcells, M., & van den Bosch, R. C. E. 2015, ApJ, 808, 79
  * Ferré-Mateu et al. (2017) Ferré-Mateu, A., Trujillo, I., Martín-Navarro, I., et al. 2017, MNRAS, 467, 1929
  * Freudling et al. (2013) Freudling, W., Romaniello, M., Bramich, D. M., et al. 2013, A&A, 559, A96
  * Girardi et al. (2000) Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, A&AS, 141, 371
  * Hilz et al. (2013) Hilz, M., Naab, T., & Ostriker, J. P. 2013, MNRAS, 429, 2924
  * Kormendy & Ho (2013) Kormendy, J. & Ho, L. C. 2013, ARA&A, 51, 511
  * Kuijken (2011) Kuijken, K. 2011, The Messenger, 146, 8
  * Kuijken et al. (2019) Kuijken, K., Heymans, C., Dvornik, A., et al. 2019, A&A, 625, A2
  * Liske et al. (2015) Liske, J., Baldry, I. K., Driver, S. P., et al. 2015, MNRAS, 452, 2087
  * Martín-Navarro et al. (2015a) Martín-Navarro, I., Barbera, F. L., Vazdekis, A., Falcón-Barroso, J., & Ferreras, I. 2015a, MNRAS, 447, 1033
  * Martín-Navarro et al. (2015b) Martín-Navarro, I., La Barbera, F., Vazdekis, A., et al. 2015b, MNRAS, 451, 1081
  * Martín-Navarro et al. (2023) Martín-Navarro, I., Spiniello, C., Tortora, C., et al. 2023, MNRAS, submitted
  * Naab et al. (2009) Naab, T., Johansson, P. H., & Ostriker, J. P. 2009, ApJ, 699, L178
  * Naylor (1998) Naylor, T. 1998, MNRAS, 296, 339
  * Oser et al. (2012) Oser, L., Naab, T., Ostriker, J. P., & Johansson, P. H. 2012, ApJ, 744, 63
  * Oser et al. (2010) Oser, L., Ostriker, J. P., Naab, T., Johansson, P. H., & Burkert, A. 2010, ApJ, 725, 2312
  * Pulsoni et al. (2021) Pulsoni, C., Gerhard, O., Arnaboldi, M., et al. 2021, A&A, 647, A95
  * Rodriguez-Gomez et al. (2016) Rodriguez-Gomez, V., Pillepich, A., Sales, L. V., et al. 2016, MNRAS, 458, 2371
  * Sarzi et al. (2018) Sarzi, M., Spiniello, C., La Barbera, F., Krajnović, D., & van den Bosch, R. 2018, MNRAS, 478, 4084
  * Saulder et al. (2015) Saulder, C., van den Bosch, R. C. E., & Mieske, S. 2015, A&A, 578, A134
  * Scognamiglio et al. (2020) Scognamiglio, D., Tortora, C., Spavone, M., et al. 2020, The Astrophysical Journal, 893, 4
  * Smette et al. (2015) Smette, A., Sana, H., Noll, S., et al. 2015, A&A, 576, A77
  * Spiniello et al. (2021a) Spiniello, C., Tortora, C., D’Ago, G., et al. 2021a, A&A, 646, A28
  * Spiniello et al. (2021b) Spiniello, C., Tortora, C., D’Ago, G., et al. 2021b, A&A, 654, A136
  * Stoehr et al. (2008) Stoehr, F., White, R., Smith, M., et al. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 394, Astronomical Data Analysis Software and Systems XVII, ed. R. W. Argyle, P. S. Bunclark, & J. R. Lewis, 505
  * Tortora et al. (2016) Tortora, C., La Barbera, F., Napolitano, N. R., et al. 2016, MNRAS, 457, 2845
  * Tortora et al. (2018) Tortora, C., Napolitano, N. R., Spavone, M., et al. 2018, MNRAS, 481, 4728
  * Trujillo et al. (2009) Trujillo, I., Cenarro, A. J., de Lorenzo-Cáceres, A., et al. 2009, ApJ, 692, L118
  * Trujillo et al. (2014) Trujillo, I., Ferré-Mateu, A., Balcells, M., Vazdekis, A., & Sánchez-Blázquez, P. 2014, ApJ, 780, L20
  * van der Marel & Franx (1993) van der Marel, R. P. & Franx, M. 1993, ApJ, 407, 525
  * van Dokkum et al. (2008) van Dokkum, P. G., Franx, M., Kriek, M., et al. 2008, ApJ, 677, L5
  * Vazdekis et al. (2015) Vazdekis, A., Coelho, P., Cassisi, S., et al. 2015, MNRAS, 449, 1177
  * Vernet et al. (2011) Vernet, J., Dekker, H., D’Odorico, S., et al. 2011, A&A, 536, A105
  * Worthey (1994) Worthey, G. 1994, ApJS, 95, 107


introduce selection bias. Meanwhile, to investigate the effect of the
neighborhood effect, we randomly block off some user rows and item columns to
get the mask matrix. Specifically, we let
$m_{u}\sim\operatorname{Bern}(p_{u}),\forall u\in\mathcal{U}$,
$m_{i}\sim\operatorname{Bern}(p_{i}),\forall i\in\mathcal{I}$ and
$m_{u,i}=m_{u}\cdot m_{i}$, where $\operatorname{Bern}(\cdot)$ denotes the
Bernoulli distribution and $p_{u}$ and $p_{i}$ are the mask ratio for user and
item. We set $p_{u}$ and $p_{i}$ equal to 1 in RQ1, and set
$p_{u}=(|\mathcal{U}|-n_{u})/{|\mathcal{U}|}$ and
$p_{i}=(|\mathcal{I}|-n_{i})/{|\mathcal{I}|}$ in RQ2, where $|\mathcal{U}|$
and $|\mathcal{I}|$ are the total user and item numbers, and $n_{u}$ and
$n_{i}$ are the user and item mask numbers, respectively. We tune the
$n_{u}\in\\{50,150,250,350\\}$ and let
$n_{i}=n_{u}\cdot|\mathcal{I}|/|\mathcal{U}|$. Then we obtain the propensities
with $p_{u,i}\leftarrow p_{u,i}\cdot
m_{u,i}=p\alpha^{\max\left(0,4-r_{u,i}\right)}\cdot m_{u,i}$. For different
mask numbers, we adjust $p$ for unmasked user-item pairs to ensure the total
observed sample is 5% of the entire matrix [Schnabel et al., 2016]. The
different $n_{u}$ and $n_{i}$ correspond to the different strengths of the
neighborhood effect. Next, we follow Wang et al. [2019], Guo et al. [2021] to
add a uniform distributed variable to introduce noise to obtain the estimated
propensities
$\frac{1}{\hat{p}_{u,i}}=\frac{\beta}{{p}_{u,i}}+\frac{1-\beta}{{p}_{o}}$,
where $\beta$ is from a uniform distribution ${U}(0,1)$ and
${p}_{o}=\frac{1}{|\mathcal{D}|}\sum_{(u,i)\in\mathcal{D}}o_{u,i}$.

##  Appendix F Real-World Experiment Details

Dataset. We verify the effectiveness of the proposed estimators on three real-
world datasets. Coat 333https://www.cs.cornell.edu/~schnabts/mnar/ contains
6,960 MNAR ratings and 4,640 missing-at-random (MAR) ratings. Both MNAR and
MAR ratings are from 290 users and 300 items. Yahoo!
R3444http://webscope.sandbox.yahoo.com/ contains 311,704 MNAR ratings and
54,000 MAR ratings. The MNAR ratings are from 15,400 users and 1,000 items,
and the MAR ratings are from the first 5,400 users and 1,000 items. For both
datasets, ratings are binarized to 1 if $r_{u,i}\geq 3$, and 0 otherwise.
KuaiRec555https://github.com/chongminggao/KuaiRec [Gao et al., 2022] is a
public large-scale industrial dataset, which contains 4,676,570 video watching
ratio records from 1,411 users for 3,327 videos. The records less than 2 are
set to 0 and otherwise are set to 1.

Experimental Details. All the experiments are implemented on PyTorch with Adam
as the optimizer. For all experiments, we use NVIDIA GeForce RTX 3090 as the
computing resource. We tune the learning rate in $\\{0.005,0.01,0.05,0.1\\}$
and weight decay in $[1e-6,1e-2]$. We tune bandwidth value in
$\\{40,45,50,55,60\\}$ for Coat, $\\{1000,1500,2000,2500,3000\\}$ for Yahoo!
R3 and $\\{100,150,200,250,300\\}$ for KuaiRec.