# Stellar Dynamical Models for 797 $z\sim 0.8$ Galaxies from LEGA-C

Josha van Houdt  Max-Planck Institut für Astronomie Königstuhl, D-69117,
Heidelberg, Germany  Arjen van der Wel  Sterrenkundig Observatorium,
Universiteit Gent, Krijgslaan 281 S9, 9000 Gent, Belgium  Max-Planck Institut
für Astronomie Königstuhl, D-69117, Heidelberg, Germany  Rachel Bezanson
University of Pittsburgh, Department of Physics and Astronomy, 100 Allen Hall,
3941 O’Hara St, Pittsburgh PA 15260, USA  Marijn Franx  Leiden Observatory,
Leiden University, P.O.Box 9513, NL-2300 AA Leiden, The Netherlands  Francesco
D’Eugenio  Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9,
9000 Gent, Belgium  Ivana Barisic  Max-Planck Institut für Astronomie
Königstuhl, D-69117, Heidelberg, Germany  Eric F. Bell  Department of
Astronomy, University of Michigan, 1085 South University Ave., Ann Arbor, MI
48109, USA  Anna Gallazzi  INAF-Osservatorio Astrofisico di Arcetri, Largo
Enrico Fermi 5, I-50125 Firenze, Italy  Anna de Graaff  Leiden Observatory,
Leiden University, P.O.Box 9513, NL-2300 AA Leiden, The Netherlands  Michael
V. Maseda  Leiden Observatory, Leiden University, P.O.Box 9513, NL-2300 AA
Leiden, The Netherlands  Camilla Pacifici  Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  Jesse van de Sande  Sydney
Institute for Astronomy, School of Physics, A28, The University of Sydney,
NSW, 2006, Australia  ARC Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D)  David Sobral  Department of Physics, Lancaster
University, Lancaster LA1 4YB, UK  Caroline Straatman  Sterrenkundig
Observatorium, Universiteit Gent, Krijgslaan 281 S9, 9000 Gent, Belgium  Po-
Feng Wu  National Astronomical Observatory of Japan, Osawa 2-21-1, Mitaka,
Tokyo 181-8588, Japan

###### Abstract

We present spatially resolved stellar kinematics for 797 $z=0.6-1$ galaxies
selected from the LEGA-C survey and construct axisymmetric Jeans models to
quantify their dynamical mass and degree of rotational support. The survey is
$K_{s}$-band selected, irrespective of color or morphological type, and allows
for a first assessment of the stellar dynamical structure of the general
$L^{*}$ galaxy population at large lookback time. Using light profiles from
Hubble Space Telescope imaging as a tracer, our approach corrects for
observational effects (seeing convolution and slit geometry), and uses well-
informed priors on inclination, anisotropy and a non-luminous mass component.
Tabulated data include total mass estimates in a series of spherical apertures
(1, 5, and 10 kpc; 1$\times$ and 2$\times$$R_{\rm{e}}$), as well as rotational
velocities, velocity dispersions and anisotropy. We show that almost all star-
forming galaxies and $\sim$50% of quiescent galaxies are rotation-dominated,
with deprojected $V/\sigma\sim 1-2$. Revealing the complexity in galaxy
evolution, we find that the most massive star-forming galaxies are among the
most rotation-dominated, and the most massive quiescent galaxies among the
least rotation-dominated galaxies. These measurements set a new benchmark for
studying galaxy evolution, using stellar dynamical structure for galaxies at
large lookback time. Together with the additional information on stellar
population properties from the LEGA-C spectra, the dynamical mass and
$V/\sigma$ measurements presented here create new avenues for studying galaxy
evolution at large lookback time.

galaxies: high-redshift - galaxies: kinematics and dynamics

††software: EAZY (Brammer et al., 2008), JAM (Cappellari, 2008), MGE
(Cappellari, 2002), emcee (Foreman-Mackey et al., 2013), corner.py (Foreman-
Mackey, 2016)

##  1 Introduction

The formation and evolutionary history of galaxies is encoded in the motions
of their stars. The dynamical state of the Milky Way is now being revealed in
ever more detail with Gaia (Gaia Collaboration et al. 2018a, b) and stellar
kinematics of nearby galaxies has been a very active area of research since
the 1970s (e.g., Bertola & Capaccioli 1975; Binney 1978; Kormendy &
Illingworth 1982; Davies et al. 1983). In the present-day Universe, much has
been learned from resolved kinematics measurements made with single slit
observations (e.g. Tully & Fisher 1977; Faber & Jackson 1976; Djorgovski &
Davis 1987; van der Marel 1991). More recently, this has been superseded by
integral field observations, revealing kinematic patterns in two dimensions
(e.g. Emsellem et al. 2004; Krajnović et al. 2011) and qualifying the dynamics
of various types of galaxies (Emsellem et al. 2011; Graham et al. 2018). In
particular, early-type galaxies are found to exhibit very diverse rotational
patterns and are often split into fast rotators and slow rotators. Fast
rotators are consistent with being axisymmetric, oblate rotators, whereas slow
rotators are more dispersion dominated and often have complex dynamical
structures (Emsellem et al. 2007). These observations have in turn sparked
significant theoretical effort (e.g. Jesseit et al. 2009; Bois et al. 2010,
2011; Naab et al. 2014; Lagos et al. 2018a, b; van de Sande et al. 2019), to
understand the large variety in observed kinematic signatures and explain the
varying degrees of ordered motion by investigating the relative importance of
e.g. (minor) merging and accretion.
So far, a clear observational picture has not been produced at higher
redshift. Yet it is highly desirable to understand the structure of high-
redshift galaxies, when they were in the middle of forming the bulk of their
stars and shortly thereafter. Ever following in the footsteps of studies of
the local Universe, at higher redshift we have largely relied on photometry to
guide progress. We have learned that the population of galaxies at $z\sim 1$
is bimodal in color and can be separated into quiescent and star-forming
galaxies based on optical colors (e.g. Labbé et al. 2005; Williams et al.
2009). Like in the local Universe, these groups show distinct structural
properties (e.g. Bell et al. 2012) and correlations (e.g. Franx et al. 2008;
van der Wel et al. 2014a). In particular, projected ellipticities have been
used to study intrinsic shapes of galaxy populations which, for lack of more
direct measures, have been associated with a galaxy’s state of rotational
support (van der Wel et al. 2014b; Chang et al. 2013). Oblate, axisymmetric
galaxies are found to form a larger fraction of the total galaxy population
with increasing redshift, which is interpreted as the increased rotational
support at high redshift.
Kinematic studies of ionized gas have revealed that large, star-forming disks
exist at least as early as $z\sim 2$ (Förster Schreiber et al. 2006; Genzel et
al. 2008), but that those are more pressure-supported than gas disks in the
present-day Universe (e.g., Kassin et al. 2012; Wisnioski et al. 2015) .
Moreover, an increasing fraction of lower mass galaxies have not yet formed
regular, rotating structures (e.g., Förster Schreiber et al. 2009). This
evolutionary trend has been successfully reproduced by hydrodynamical
simulations (Ceverino et al. 2014; Martig et al. 2014; Pillepich et al. 2019).
However, there are limitations to the use of ionized gas as a tracer of
dynamical structure: while to first order the integrated velocity dispersions
of ionized gas and stars agree fairly well (Bezanson et al., 2018a) there is
large scatter for several reasons. First, it reveals only the state of the
youngest parts of galaxies. Second, the spatial distribution of the emission
is different from the total mass distribution: it has a patchy structure,
which makes it a problematic tracer (Varidel et al. 2019). The stellar
component does not suffer from these drawbacks: they are the dominant mass
component in most massive galaxies (within the inner 5-10 kpc for most
galaxies in our sample), and the stellar light itself is a reasonable proxy
for the shape of the mass distribution.
But the necessary data to access the stellar dynamical structure – high-S/N
continuum spectroscopy – is challenging to obtain and have so far been limited
to small samples (van der Marel & van Dokkum 2007; van der Wel & van der Marel
2008; Newman et al. 2015, 2018; Guérou et al. 2017; Toft et al. 2017) that
only include very massive, passive galaxies. Even when these measurements can
be made, the interpretation is complicated by the impact of observing
conditions, such as the slit geometry and beam smearing. To make progress,
detailed dynamical models can help. At the expense of making assumptions about
the underlying galaxies (e.g., axi-symmetry and regular motions) one can
account for the impact of the large slit width, inclination and seeing and
estimate the intrinsic dynamical properties of galaxies.

The LEGA-C survey (van der Wel et al. 2016) is the first survey to provide
resolved stellar kinematics for a magnitude-limited sample of galaxies at
$z\sim 0.8$. Bezanson et al. (2018b) presented a first, qualitative look at
the stellar rotation curves and dispersion profiles for a sample of $104$
quiescent galaxies in the LEGA-C survey. It was found that quiescent galaxies
at $z\sim 0.8$, like their local counterparts, show a decrease of rotational
support with increasing mass and that, like in the local population, there is
a subset of massive galaxies seemingly devoid of significant rotation
altogether. But crucially, the LEGA-C galaxies appeared to show significantly
more rotation than local galaxies. However, the interpretation of the observed
kinematics is complicated by the observational setup: the slit has a
considerable width, the seeing is in many cases comparable to the galaxy size,
and the slits are not necessarily oriented along the galaxy’s major axis. An
accurate quantitive description of the dynamical structure of these galaxies
therefore requires dynamical modelling.
In this paper we present, for the first time, a concerted effort to infer the
dynamical masses and dynamical states for a large, representative sample in
the LEGA-C survey of quiescent and star forming galaxies. We achieve this by
using axisymmetric Jeans models. The Jeans method is frequently used to model
galaxies in the local Universe, for instance to constrain mass-to-light ratios
and dark matter fractions (van der Marel 1991; Williams et al. 2009;
Cappellari et al. 2013), or constrain the internal rotation and anisotropy of
galaxies (Satoh 1980; Zhu et al. 2016). Similarly, at $z>0$ they have been
applied to study the dynamical masses (Guérou et al. 2017), initial mass
functions (Newman et al. 2017) and rotation rates (van der Wel & van der Marel
2008; Newman et al. 2018).
In this paper, we present the Jeans models for $797$ galaxies that are part of
the third data release of the LEGA-C survey (van der Wel et al. in prep.). In
Section 2 we introduce the data and the sample selection used in this
analysis. The models are described in Section 3. In Section 4 we show the
results of the fits from the models to the observations, as well as a number
of consistency checks. Some basic results from the derived quantities are
shown in Section 7, after which Section 8 summarises and looks forward to
anticipated follow-up work. Throughout the paper, a standard $\Lambda$CDM
cosmology is assumed with $H_{0}=70$ km s-1, $\Omega_{\Lambda}=0.7$ and
$\Omega_{b}=0.3$.

##  2 Data

###  2.1 LEGA-C Spectroscopy

This work is based on the Large Early Galaxy Astrophysics Census (LEGA-C; van
der Wel et al. 2016). LEGA-C is an ESO Public Spectroscopic Survey with the
VIMOS instrument on the VLT/UT3. The survey targets galaxies at a redshift
between 0.6 and 1.0 and below a redshift-dependent magnitude limit in the $K$
band, acting as a surrogate for a stellar mass selection yet insensitive to
mass-modelling uncertainties. The survey has been completed and has collected
4081 spectra in the COSMOS field. This paper uses the full sample, to be
published in the recent data release of the full sample (van der Wel et al.,
2021).

The integration time is $\sim 20$ hours per mask, resulting in spectra with a
median signal-to-noise of $\sim 19$ Å-1 at a spectral resolution of $R=3500$
(Straatman et al. 2018). Stellar kinematics are extracted with pPXF
(Cappellari & Emsellem 2004; Cappellari 2017), as detailed in Bezanson et al.
(2018b). In short, combinations of two sets of templates, for the stellar
continuum and for the gas (line) emission, are fit to the observed spectra.
The templates are broadened and shifted independently to optimise the fit in
every row of the spectrum with $S/N>2$, giving us the line-of-sight velocity
distribution (LOSVD) at each spatial pixel along the slit. Compared to
Bezanson et al. (2018b), we use the pPXF fit with synthetic templates from C.
Conroy (priv. communication) instead of the empirical models based on the
MILES (Sánchez-Blázquez et al. 2006) templates (Vazdekis et al. 2010). The
Conroy templates offer increased resolution ($\sigma_{\rm{mod}}\sim
12~{}\rm{km~{}s}^{-1}$ compared to $\sigma_{\rm{mod}}=70~{}\rm{km~{}s}^{-1}$
for MILES) – which is ideal compared to the resolution of our spectra
($\sigma_{\rm{instr}}\sim 35~{}\rm{km~{}s}^{-1}$). This high resolution is
essential, as face-on disks in our sample can have stellar velocity
dispersions as low as 40 $\rm{km~{}s}^{-1}$. We only measure the first and
second moment of the LOSVD (the rotation velocity $v_{\star}$ and the observed
velocity dispersion $\sigma_{\star}$, respectively) and do not fit for higher-
order moments.

###  2.2 Ancillary Imaging Data

In addition to the resolved kinematics from LEGA-C, our dynamical models
require a light tracer. We adopt for this purpose the HST/ACS F814W imaging
from COSMOS (Koekemoer et al. 2007), which covers nearly all galaxies in
LEGA-C, and we use this imaging for fitting 2D light profiles with Sérsic
models as described by (van der Wel et al., 2021).

We also incorporate ground-based optical and near-infrared data from DR1 of
the UltraVISTA survey (Muzzin et al. 2013a), the photometric parent sample of
LEGA-C. Using the $EAZY$ (Brammer et al. 2008) photometric redshift code in
combination with the LEGA-C spectroscopic redshifts, we determine restframe
$U-V$ and $V-J$ colours and select quiescent and star-forming galaxies
according to the $UVJ$-criterion in Muzzin et al. (2013b).

###  2.3 Sample Selection for Jeans Modeling

Figure 1: Distribution of redshift (top panel) and integrated stellar velocity
dispersion (bottom panel) for the parent sample of galaxies with regular
morphologies and no flags (see text) in light grey, those with position angles
aligned with the slit to within 45 degrees (darker grey) and the final sample
of 861 spectra that also satisfy the $S/N$ criteria. Figure 2: Axis ratio
distribution distribution of quiescent (top) and star-forming (bottom)
galaxies. The lines indicate the eligible samples (primary targets with
measured (integrated) stellar velocity dispersions and aligned with the slit
to within 45 degrees); the solid histograms show the sub-samples with
sufficient $S/N$ in the spatially resolved spectra for constructing the
dynamical model. The $S/N$ for edge-on, star-forming galaxies suffers from
attenuation, introducing an inclination-dependent bias in our sample of star-
forming galaxies. This bias is absent for quiescent galaxies.  Figure 3:
Spatial extent of the kinematic measurements for $v_{rot}$ and $\sigma_{*}$
for the $860$ kinematic data sets (of 795 unique galaxies) in the final
sample, first in arcseconds and then in multiples of each galaxy’s effective
radius.

The third (and final) LEGA-C release (van der Wel et al., 2021) contains
$3028$ spectra of $K_{s}$-band selected ‘primary’ galaxies in the redshift
range $0.6<z<1$. We construct dynamical models for a subset of these, those 1)
with the major axis is not misaligned with the slit by more than $45^{\circ}$
(1554 left), 2) without a morphology flag (mergers, irregulars, …) or a
spectroscopy flag (significant AGN contributions or severe flux calibration
problems) as defined in the data release paper (1381 left), and 3) with
$S/N>10$ in at least 3 spatial resolution elements (861 left).

Due to the observing strategy and the way in which galaxies are prioritized,
some galaxies are observed in multiple masks. This means we have independent
measurements of the same galaxy under slightly different seeing conditions and
slit alignments. In the sample of 861 we have 797 unique galaxies: there are
$64$ ‘duplicates’ ($7.6\%$ of the sample), which we use for consistency checks
of the method (Sec. 5.2).

Figure 1 shows the distribution of $z_{spec}$ and $\sigma_{*}$ – the
integrated stellar velocity dispersion – and how the selection criteria affect
these distributions. Figure 2 shows the bias against edge-on star-forming
galaxies: these galaxies have higher dust attenuation and often fall short of
our $S/N$ criterion. This is an important consideration when attempting to
quantify the relative numbers of flat and round star-forming galaxies, and the
evolution thereof. The trend does not exist for the mostly dust-free quiescent
galaxies. For those (and for face-on star-forming galaxies) the $S/N$ can be
too low due to low stellar mass and/or high redshift.

From the single-slit spectra we have one kinematic axis and a spatially
resolved profile for $v_{\star}$ and $\sigma_{\star}$ for typically $\sim
8-12$ spatial pixels, reaching over $2$ effective radii for some of the
galaxies in our sample. Figure 3 shows the range probed for the galaxies in
the sample, in arcsec as well as in units of the galaxy effective radius
$R_{\rm{e}}$.

##  3 Methods

###  3.1 Axisymmetric Jeans models

The dynamical models used in this work are based on the Jeans equations, a set
of stellar hydrodynamic equations relating (average) kinematic properties to
the stellar light and the underlying matter distribution. They are derived by
taking velocity moments of the Collisionless Boltzmann Equation, describing
the evolution of the phase-space density, assuming the system of particles to
be collisionless and in steady-state (e.g. Binney & Tremaine 2008)
In this work we implement this using the Jeans Anisotropic MGE models (JAM;
Cappellari 2008)111The JAM source code is available from http://www-
astro.physics.ox.ac.uk/~mxc/software/. This software uses the computationally
convenient Multi Gaussian Expansion (described below) to model the surface
brightness, the gravitational potential and the point spread function (PSF,
see Sections 3.2 and 3.3). The Jeans equations become solvable by making
observationally-motivated assumptions on the velocity ellipsoid. Via the
anisotropy parameter $\beta_{z}\equiv 1-\langle v_{z}^{2}\rangle/\langle
v_{R}^{2}\rangle$ (in cylindrical coordinates) we can model the flattening of
the velocity ellipsoid in the direction perpendicular to the disk of the
galaxy. The velocity ellipsoid is otherwise assumed to be aligned with the
main cylindrical axes such that cross-terms ($\langle v_{z}v_{R}\rangle$,
$\langle v_{\phi}v_{R}\rangle$, $\langle v_{\phi}v_{z}\rangle$) all vanish.
This leaves us with one independent (possibly spatially varying) second
velocity moment at every position. Given the inclination, this can then be
integrated along the line-of-sight as done in Equation 28 of Cappellari
(2008). This prediction can then be compared to the observed combination of
1st and 2nd order line-of-sight velocity moments
$v_{rms}\equiv\sqrt{v^{2}_{\star}+\sigma_{\star}^{2}}$, allowing us to
constrain the kinematic structure and gravitational potential with only a few
parameters.
$v_{rms}$ allows us to constrain the _total_ rms motion in the galaxy as a sum
of rotation and dispersion, but it gives no information about the relative
importance of each of the components. To distinguish between these we make the
additional assumption that the dispersion in the azimuthal direction is
related to the dispersion in the radial direction such that

$\overline{v_{\phi}}=\kappa\sqrt{\overline{v_{\phi}^{2}}-\overline{v_{R}^{2}}}$
(1)

where the constant rotation parameter $\kappa$ is a fitting parameter (Binney
& Tremaine, 2008; van der Marel & van Dokkum, 2007).

###  3.2 Light and Mass Modelling

An important input of the dynamical models is a detailed parametrisation of
the tracer light (stellar), from which the observed kinematics originates, as
well as the underlying gravitational potential. In our model, the potential is
due to two mass components: the stellar body and dark matter. That latter, in
actuality, represents any component that breaks the mass-follows-light
assumption inherent in the former, but this will be discussed in full below.

Both are described with a set of elliptical Gaussians using the Multi Gaussian
Expansion code (Cappellari 2002)222The MGE software is also available at
http://www-astro.physics.ox.ac.uk/~mxc/software/. Extra Gaussian components
are iteratively added until the fit does not improve significantly. All
Gaussians are assumed to have the same, global PA (no isophotal twists). In
addition, we impose a regularisation to limit the axis ratios around the
global value as long as it does not affect the quality of the fit, which is
most relevant if we add residuals or decompose mass profiles (see below). This
is important since the minimum axis ratio limits the minimum allowed
inclination in the dynamical fit. In some cases (in particular, for Sérsic
profiles that fall off faster than a single Gaussian function, $n<0.5$) we
also allow negative Gaussian components while ensuring that the sum stays
positive.

Figure 4: Deconvolved surface brightness distribution – used as light tracer
input and stellar mass component in our dynamical model – for the galaxy
$1414$. The inset shows the $ACS/F814W$ image, with the VIMOS slit overlaid in
white. The main panel shows observed surface brightness profile in black, as
an average in elliptical annuli following the best-fit GALFIT geometry (PA and
axis ratio). The solid red line is the best-fit Sérsic profile. The dashed
blue line is the multi-Gaussian expansion of the Sérsic profile; individual
Gaussian components are shown in dotted blue. Finally, the dotted vertical
line denotes the effective radius of this galaxy.

####  3.2.1 Light Tracer and Stellar Mass Model

We use the ACS/F814W imaging as the light tracer model. The _HST_ /ACS
instrumental point spread function has a non-negligible size relative to the
galaxy sizes, which is why we use the best-fit Sérsic profiles to describe the
deconvolved surface brightness distributions, following van der Wel et al.
(2012) and van der Wel et al. (2016). These are then subsequently fitted with
the MGE, as illustrated in Figure 4, where we show that this procedure
accurately follows the true light profile over 6 magnitudes in surface
brightness, out to 2.5 effective radii.

We note that for these relatively low-$S/N$ F814W data direct MGE fits
systematically underestimate the fluxes, on average, 20% as inferred from a
comparison with ground-based photometry, where we synthesized F814W photometry
from the SED fits to the ground-based photometry described in Section 2.2. Low
surface brightness light is missed by the MGE, leading to underestimated
effective radii. The Sérsic model fluxes, on the other hand, show no such
offset, justifying the extrapolation of the profiles out to large radii. The
1$\sigma$ scatter for both the MGE and Sérsic fluxes is 15% with respect to
ground-based fluxes.

####  3.2.2 Dark Matter Model

In addition to the luminous matter we also include a dark matter component.
Later, in section 5.2, we will see that the primary role of the dark matter
fit component is to allow us to fit mass to light variations within the inner
10kpc or so of the galaxy, having no predictive power outside of that radius.
Consequently, while we choose a functional form for this component that is
rooted in cosmology, its detailed functional form is not important and does
not significantly influence the results of our overall fits.

We adopt a spherically symmetric NFW-profile (Navarro, Frenk, & White 1997):

$\rho=\frac{\Delta\rho_{crit}}{(r/r_{s})(1+(r/r_{s}))^{2}}$ (2)

where $\rho_{crit}$ is the critical density, $r_{s}$ a characteristic scale
and $\Delta$ the threshold level delimiting the halo, as a matter of
convention often chosen to be $200$. The halo mass $M_{200}$ then corresponds
to the mass inside the radius $r_{200}$ for which the average density is
$200\rho_{crit}$. Furthermore, it is convenient to define the concentration
parameter $c_{200}=r_{200}/r_{s}$. It is well known that there is a
correlation between halo mass and concentration, such that more massive haloes
tend to be less concentrated, though the scatter is large. From the
simulations in Dutton & Macciò (2014) we can use the relation between mass,
concentration and redshift to eliminate one of the two halo parameters:

$\log_{10}\left(c_{200}\right)=a+b\log_{10}\left(M_{200}/[10^{12}h^{-1}M_{\odot}]\right)$
(3)

where the parameters $a$ and $b$ are given (for $\Delta=200$) by

$a=0.520+\left(0.905-0.520\right)e^{-0.617z^{1.21}}$ (4)

and

$b=-0.101+0.026z$ (5)

This allows us to reduce the dark matter component to a 1-parameter model: its
mass $M_{200}$. These models can be easily included in the dynamical fit by
expanding the NFW profile in a series of Gaussians and adding them to the MGE
terms of the stellar matter.

We note that, in practice, this implementation means that the dark matter
component always has the same slope (the inner slope from the NFW profile) and
the only variable is the normalization, as dictated by $M_{200}$. The reason
for this is that $r_{s}$ always lies beyond the spatial extent of our
observations. We stress again that we interpret the dark matter component
_not_ as a constraint on the mass of entire dark matter halo, but as a
constraint on deviations from the mass-follows-light assumption, which
physically can be due to dark matter, but also contributions from gas and
stellar $M/L$ variations. Global differences in stellar $M/L$ are accounted
for varying the $M/L$ of the light component, as are gas disks that follow the
same profile as the stellar light. However, if gas or stellar mass profiles
differ from the stellar light profile, then the ‘dark matter’ component will
attempt to account for this. The assumption of a spherical dark matter
distribution is then not correct, but for the enclosed mass estimate this has
a minor effect.

We note that we have to allow for a dark component with a negative mass. This
may be counter-intuitive, since setting a strictly positive prior seems
physically self-evident. The two reasons for allowing a negative mass here are
1) avoid a positive bias in the total mass, and 2) allow for inverted stellar
$M/L$ gradients. The positive bias occurs when the dark component is not
constrained by the data, such that the median in the posterior is always
positive. Allowing for negative masses improves the calculation of the
uncertainties by symmetrizing the posterior around zero if there are no
constraints on the dark component, avoiding the aforementioned bias. A side
effect is that for a small set of galaxies we find a negative total mass (see
Section 6), which should be interpreted as unavoidable noise in the
measurements.

###  3.3 Effective Seeing

We correct for beam smearing on an object-by-object basis. Thus, we wish to
quantify the effective seeing of each galaxy, which reflects a combination of
atmospheric seeing as well as observational and instrumental effects that
broaden the light distribution. We do this by comparing the wavelength-
collapsed LEGA-C spectra (equivalent to a radial light profile of that part of
the galaxy covered by the slit) with a model light curve. The model light
profile is obtained by convolving the ACS/F814W images successively with a
seeing kernel and the slit. Then we rebin this to the $0.205\arcsec$ VIMOS
pixel scale, and integrate the resulting image.

For the seeing kernel we use a Moffat profile (Moffat 1969):

$M(\alpha,\gamma)=A\left(1+\frac{(x-x_{0})^{2}}{\gamma^{2}}\right)^{-\alpha}$
(6)

where $A$ is the normalisation and $\gamma$ and $\alpha$ are the Moffat
parameters we fit for. Given $\alpha$ and $\gamma$, it is straightforward to
show that the full-width-half-maximum (FWHM) is given by
$FWHM=2\gamma\sqrt{2^{1/\alpha}-1}$. Generally, a Moffat profile has been
shown to give a better description of the effect of atmospheric turbulence
than a Gaussian function (Trujillo et al. 2001) because of the wings,
controlled by the $\alpha$ parameter. We allow for a possible gradient in the
background flux (due to e.g. imperfections in the spectroscopic data
reduction) by adding a variable linear component to the Moffat-convolved model
light profile. If $\alpha$ is poorly constrained we fix it at $\alpha=4.765$
(Trujillo et al., 2001) and only fit for $\gamma$. For objects where we did
not find a good fit we use $\alpha=4.765$ and $\gamma$ derived from the mask-
average FWHM and $\alpha$.
It is encouraging that despite atmospheric dispersion, drift, flexure and
hysteresis the true PSF (typically, 0.75$\arcsec$) is in fact better than the
DIMM seeing reported at the telescope (typically, 0.95$\arcsec$, Straatman et
al. 2018). Interestingly, the variation from mask to mask is smaller than the
variation from object to object within a given mask. This suggests that
technical details of the instrument (focus, positional accuracy, alignment and
location within the mask) are more important in determining the outcome of a
spectrum than the variation of atmospheric conditions averaged over 20 hours
of integration (good conditions were required to execute observations).

###  3.4 Systematic Effects in Slit Spectroscopy

In general there are several things to keep in mind when using slit
spectroscopy, in particular at the targeted redshift range where the slit
width of 1$\arcsec$ is comparable to the effective radii of galaxies.

####  3.4.1 Slit Misalignment

During observations the slits are centered on the galaxies as accurately as
possible, but this might still be prone to both random and systematic offsets.
This will lead to an offset in the measured velocity profile, as well as
asymmetries in the $v$ and $\sigma$ profiles if the slit is not aligned with
the galaxy’s major axis, even if the galaxy is perfectly axisymmetric.
Although for a small subset of individual sources this offset can be
determined (see Appendix A), finding constraints systematically for all
sources was not possible. Nevertheless, the sources for which we _can_
constrain the offset suggest the slits are generally well-centered, with
offsets of typically $0$, $1$, or sporadically more pixels, where 1 pixel
corresponds to 0.205 arcsec. This motivates us to use a Gaussian prior on the
slit center $x_{0}$ in the dynamical models, centered at $0$, with a standard
deviation of $1.5$ pixels.

####  3.4.2 Slit Orientation

Figure 5: Illustration of how an artificial velocity gradient arises in the
slit. The black lines illustrate the slit, with dashed lines denoting the
separation between spectral pixels. The orange blob illustrates the galaxy,
the major axis of which is rotated with respect to the direction of the slit.
Because of this, the center of the light of any individual pixel is not in the
middle of the slit, but rather slightly shifted - indicated here with a black
cross. The conversion of this spatial offset to an offset in wavelength and
thus in an offset in derived velocity is indicated at the bottom (even though
this varies slightly from galaxy to galaxy due to different spectral
coverage.)

In deriving the rotational velocity at any particular spatial position, the
light is tacitly assumed to come from a homogeneous light distribution that
fills the entire 1-arcsec slit in the wavelength direction (this is how the
spectral resolution $R$ in instrument manuals is usually defined). However,
this is not true for galaxies that have light disributions that change
substantially in wavelength direction of the slit, which is the case for stars
or distant galaxies. This results in spurious velocity shifts if the major
axis is misalignment with respect to the slit. We derive and subtract this
offset as follows. First, we convolve the F814W MGE with the PSF. For every
pixel, we then find the light-weighted center which we convert into a shift in
wavelength, using the dispersion of $0.6$Å per pixel. Using the median of the
observed spectral coverage of each object, we then convert this into a shift
in velocity. This is illustrated in Figure 5. This artificial offset reaches
up to $20$ km$\cdot$s-1, being highest away from the center and for the
flattest, most misaligned galaxies.

####  3.4.3 Pixel Convolution

To compare the JAM predictions to the observations, we need to convolve the
model grid with a kernel that is a combination of the seeing PSF as well as
the slit. The former is an MGE of the Moffat profile derived in Section 3.3.
The latter is a top-hat function of 1 arcsec in width (the size of the slit in
the wavelength direction).

###  3.5 Inclination and Anisotropy Priors

The observed kinematics depend on the inclination, but the only _a priori_
constraint we have is the observed axis ratio. If we know the distribution
$f(p,q)$ of the _intrinsic_ axis ratios $p\equiv a/b$ and $q\equiv b/c$ of a
given class of objects (e.g. quiescent galaxies in a given redshift range) we
can construct the likelihood of observing the observed axial ratio
$q^{\prime}=a^{\prime}/b^{\prime}$ for a given inclination. This will then
serve as the prior on the inclination in the fits of the Jeans models to the
total rms motion in the galaxy.
The intrinsic axis ratios $p$ and $q$ are related to the observed axis ratio
$q^{\prime}$ through two viewing angles. In the oblate axisymmetric case (with
$p=1$) this becomes a simple relation between the inclination, the observed
axis ratio $q^{\prime}$ and the intrinsic short-to-intermediate ratio $q$:

$q^{\prime}=\cos^{2}{i}+q\sin^{2}{i}$ (7)

We use the methodology described by Chang et al. (2013) to reconstruct the
intrinsic $q$ distribution, assumed to be Gaussian, from an observed set of
axis ratios, as shown in Figure 4. For our sample we find a mean
$\mu_{q}=0.41$ and a scatter $\sigma_{q}=0.18$. This distribution allows us to
derive a prior for the inclination: given an observed, projected axis ratio,
what is the probability distribution for the intrinsic shape and, hence, the
inclination? The mathematical tools for this inversion are presented by van de
Ven & van der Wel (2021).

While we see in Figure 4 that star-forming galaxies, on average, have flatter
intrinsic shapes than quiescent galaxies (Chang et al., 2013; van der Wel et
al., 2014b), adopting different shape distributions as priors for the
inclination would introduce a systematic difference in rotational properties
for quiescent and star-forming galaxies, even if they were kinematically the
same. This spurious result would be explained, trivially, by the priors and
not by the stellar kinematics. In order to avoid this we adopt the same
intrinsic shape distribution (that is, the same inclination prior) for
different types of galaxies such that any difference in dynamical structure
can be attributed to differences in the observed kinematics. As we will see,
despite not assigning different priors to different types of galaxies, the
model successfully distinguishes between rotating and non-rotating galaxies.

Given the assumed inclination, the anisotropy $\beta_{z}$ can in principle be
well-constrained with 2D kinematics (Cappellari 2008) but the long-slit data
is expected to put weak constraints on the anisotropy. We therefore use a
uniform prior of $-0.5<\beta_{z}<0.5$.

Figure 6: The observed axis ratio distribution is fitted with a population of
intrinsically oblate objects, viewed under random angles. The grey histogram
is the observed axis ratio distribution for the whole primary LEGA-C sample;
the black line is the best-fit oblate model. The red and blue lines are
similar fits, but only to the star forming or quiescent galaxies,
respectively. The joint fit is used in the remainder of the paper to describe
the intrinsic shape distribution of the galaxy population, which, given the
observed projected axis ratio of an individual galaxy, gives a prior
probability distribution for the inclination in the dynamical model. Figure 7:
Covariance of the parameters of the JAM model fit to the observed $v_{rms}$ of
galaxy $1414$ (see Fig. 4 for the light profile and HST image), with the best
fitting dynamical model shown in the top right corner. The priors on the slit
center, $x_{0}$, and the inclination, $i$, are shown in their respective one
dimensional histogram as red lines. Contours indicate the 68%, 90% and 99%
confidence intervals. There are clear degeneracies, most notably between
stellar mass and dark matter mass. Some parameters (inclination and slit
position) are not constrained at all by the data; marginalizing over these
parameters helps to produce realistic uncertainties on the well-constrained
physical parameters of interest: $M_{dyn,<Re}$ and $\kappa$. These are shown
in red because they are derived from fitting parameters, but not fitting
parameters themselves. The top-right panel shows the ‘convolved model’
quantities include aperture and projection effects as well as seeing. The
‘intrinsic model’ only include aperture and projection effects. The red and
blue data points and models represent the line-of-sight velocity $V$ and
velocity dispersion $\sigma$, respectively, and the black data points and
models represent $v_{\rm{rms}}$. The spectroscopic aperture is show in the
inset HST images. The quantities printed in the top-left corner of the panel
refer to quantities presented in Sections 3.1 and 6.

##  4 Fitting Procedure

Following the methods and assumptions described in Sec. 3 we start with a fit
of the line-of-sight second velocity moment of the JAM to the observed
$v_{\rm{rms}}=\sqrt{v^{2}+\sigma^{2}}$. For axisymmetric galaxies that are
centred in the slit the velocity profile is symmetric. Observing such a
galaxy, one would expect observations at $+r$ and $-r$ to be independent
observations of the same value. This is not true if the slit is slightly
offset (see also Section 3.3) which is why we do not symmetrise the data.
However, to ensure that asymmetric data points do not influence the fits too
much, the uncertainties are set to the maximum of the formally measured
uncertainties in $v_{\rm{rms}}$ and half the difference between the
$v_{\rm{rms}}$ values at $+r$ and $-r$. In addition, at all radii a minimum
uncertainty of $5$ km s-1 is imposed to avoid that data points with spuriously
low uncertainties dominate the fits. The fits extend to where we have
measurements of $\sigma$, even though the velocity measurements tend to extend
further out. Finally, we measure and subtract the systematic velocity as a
function of the slit center $x_{0}$ by fitting an isotropic ($\beta_{z}=0$),
stellar-mass only model at every value of $x_{0}$.

Table 1: Free parameters in the fit, with assumed priors Parameter | Prior
---|---
$\beta$ | Flat prior on -0.5<$\beta$<0.5
M/L | Unconstrained
$v_{DM}$ | Flat prior on -600<$v_{dm}$<600 km$\cdot$s-1
Inclination |  $q$-dependent prior, assuming an intrinsic thickness distribution of $\mathcal{N}(0.41,0.18)$ (see section $3.5$)
Slit center, $x_{0}$ | Gaussian, centered on 0 with a $1.5$ pixel standard deviation

The fit is done using the Markov Chain Monte Carlo (MCMC) software _emcee_
333The EMCEE code can we downloaded from
https://emcee.readthedocs.io/en/latest/ (Foreman-Mackey et al. 2013), which
samples the parameters space with an ensemble of ‘walkers’. The five free
parameters in the fit are the stellar mass-to-light ratio ($M/L$), the virial
mass of the dark matter halo parameterized by the circular velocity ($v_{c}$),
the inclination ($i$), the anisotropy $\beta_{z}$ and the slit centering
$x_{0}$. We use 40 walkers, with a burn-in run of 80 iterations. Convergence
was verified by visually examining the parameter values as the fit progresses,
which usually occurs after several hundred iterations.

Figure 8: Observed and best-fit model kinematic profiles for a subset of 12
galaxies. The galaxies are ordered by mass (from left to right) and degree of
rotational support (from bottom to top). The quantities printed in the top-
left corner of each panel refer to quantities presented in Sections 3.1 and 6.
The top-left panel includes a legend of the data points and models. The
‘convolved model’ quantities include aperture and projection effects as well
as seeing. The ‘intrinsic model’ only include aperture and projection effects.
The red and blue data points and models represent the line-of-sight velocity
$V$ and velocity dispersion $\sigma$, respectively, and the black data points
and models represent $v_{\rm{rms}}$, the quadratic combination of $V$ and
$\sigma$. The spectrosopic aperture is show in the inset HST images.

As an example, Figure 7 shows the covariance between the various parameters,
for a galaxy that was chosen to be typical in terms of the observed velocity
dispersion (192 km s-1) and radial extent of the observed kinematics (5 pixels
or 1$\arcsec$ – one-sided). The inclination is unconstrained by the data and
is equal to the prior, as is the slit center $x_{0}$. $\beta_{z}$ is weakly
constrained, as expected. Dark matter mass and stellar mass are degenerate,
but the total mass within $R_{e}$ ($\log M_{dyn,<Re}$) is well constrained.
Note that this is not an independent parameter but computed from $M_{200}$ and
$M/L$. The best-fit models have a median reduced $\chi^{2}$ of $1.03$, so
given the simplicity of the models the fits are more than adequate.

The posterior for the rotation parameter $\kappa$ is shown in Figure 7, but is
not fit simultaneously (see Section 3.1). Instead, the best-fit $\kappa$ is
computed at every step of each Markov chain $(\beta_{z}$, $M/L$, $i$,
$M_{200}$, $x_{0}$), without influencing the overall likelihood. The rotation
parameter $\kappa$ has been used in the past as a computationally convenient
way of disentangling rotational from random motion, and for an isotropic
oblate rotator, $\kappa=1$ by definition. As long as the rotation is not
larger than $v_{rms}$, $\kappa$ can be arbitrarily higher than $1$, which
means that the model contains more rotation than is required to explain the
observed flattening, and significant anisotropy (in the vertical direction)
contributes to the thickness of the system. A maximum $\kappa=\kappa_{max}$
can be defined for which $v_{rms}=v_{rot}$. If the best-fit is found for
$\kappa>\kappa_{max}$, we set $\kappa=\kappa_{max}$. These cases can occur for
various reasons: if the galaxy has an anisotopy that differs from Eq. 1; if
the assumptions of the axisymmetric modelling are violated; if the assumption
of a spatially constant $\kappa$ is a poor one; or if $\kappa$ is an improper
(or too simplistic) parameterisation in the first place.

Examples of the observed kinematic profiles and corresponding best-fit models
in Figure 8 are shown for 12 galaxies spanning the full dynamic range of mass
and kinematic structure. The axi-symmetric models are sufficiently flexible to
reproduce the full range of observed kinematic and structural properties: non-
rotating and rotating galaxies, dispersion profiles that peak and dip in the
center, late- and early-type galaxies, face-on and edge-on galaxies, aligned
and mis-aligned galaxies, and compact and extended galaxies.

##  5 Tests and Model Validation

###  5.1 Is a dark component required?

If we leave out the dark component, our model reverts to the commonly used
mass-follows-light approach, where the total mass profile is identical to the
MGE stellar light model, with only the $M/L$ scaling as a free parameter. We
show the comparison in Figure 9. Importantly, the formal uncertainties are
very small for the mass-follows-light model (typically, just 6%). This is not
realistic for total mass estimates of high-redshift galaxies with limited
spatial resolution. The uncertainties for the default model are much larger
and we verify in Section 5.2 that these are realistic based on duplicate
observations. The median differences in mass are modest ($<0.10$ dex) but for
individual galaxies the default model often produces a mass that is two times
higher than the mass-follows-light mass. Moreover, there are strong and
systematic differences in the $\kappa$ parameter that traces the degree of
rotational support. For the mass-follow-light models the round (near face-on)
galaxies have the highest $\kappa$ values. This is counterintuitive and
implies a bias: round galaxies should have similar or lower $\kappa$ values
compared to flat galaxies.

On the basis of the biased $\kappa$ estimates and the unrealistically small
uncertainties for the mass-follows-light models we conclude that the dark
component is an essential ingredient. It allows for the required flexibility
in modeling the observed kinematic data, especially for face-on disks. As
discussed in Section 3.2.2 the dark component does not necessarily reflect
dark matter alone, but accounts for deviations from the mass follows light
assumption due to stellar $M/L$ variations and the contribution from gas as
well.

Figure 9: Model comparison for two different assumptions: with dark matter
(our default model) and without dark matter (mass follows light). Left:
Enclosed mass within the effective radius; Right: rotation parameter $\kappa$,
both color coded with the projected ellipticity. Typical error bars are shown
in the top-left corners. For the majority of galaxies the mass estimates agree
between the two models, but there is a prominent tail for which the mass-
follows-light models produce lower mass estimates. Notably, the uncertainties
for the mass follows light models are unrealistically small ($\sim 6\%$),
presumably due to the lack of flexibility in the model. The clear ellipticity
dependence for $\kappa$ betrays strong projection effects. The mass-follows-
light models produce systematically higher $\kappa$ for near face-on, rotating
galaxies, which implies that those are overestimated and that mass-follows-
light models do not accurately capture the dynamical structure of galaxies in
a manner that is independent of viewing angle.

###  5.2 Duplicate Observations

A number of galaxies have been observed multiple times in LEGA-C as a
consequence of the observing approach. These provide independent kinematic
measurements of the same object, so we can leverage this to test the
robustness of the parameters we derive and obtain an independent estimate of
all uncertainties associated with the spectroscopic data. In the sample with
dynamical models there are 65 such duplicates. Figure 10 compares the
dynamical mass and rotation parameter $\kappa$ for these galaxies. The two
sets of observations scatter around the one-to-one line with a scatter that
corresponds reasonably well with the expectation from the formal (random)
uncertainties. Theses uncertainties, inferred from the posterior
distributions, are on average 50% larger than the scatter for the mass
estimate. We attribute this to the flexibility in our model afforded by a mass
component that does not follow the light distribution. The comparison of
duplicate observations may imply that this conservative approach leads to
overestimated random uncertainties. On the other hand, the light tracer,
inclination prior and other elements in the model are identical for the
duplicates, and any uncertainties associated with those remain hidden in this
test: the duplicate results are only independent in terms of kinematic
measurements, but the light tracer, stellar mass model, inclination and
anisotropy priors are identical. The prudent decision, then, is to use the
uncertainties as inferred from the posterior distributions.

Figure 10: Mass (left) and rotation parameter $\kappa$ comparison for galaxies
with repeat observations. The scatter compares reasonably well with the formal
uncertainties, and imply that the these uncertainties that are inferred from
the posterior distributions are accurate. See text for details.

##  6 Catalog Contents

The dynamical modeling results are made available electronically, and Table 2
shows an excerpt. We provide the total mass in 5 different spherical
apertures: 1 kpc, 5kpc, 10kpc, $R_{\rm{e}}$444$R_{\rm{e}}$ is the semi-major
axis of the ellipse that contains half of the Sérsic model light., and
2$R_{\rm{e}}$; these are spherical apertures, rather than projected,
cylindrical apertures. A value is only included in the table if that aperture
does not amount to a significant extrapolation given the radial extent of the
stellar kinematic data. We allow for a maximum of 20% extrapolation in radial
extent. Given this constraint, the number of objects with mass estimates
within these 5 apertures are 861, 809, 206, 730, and 319, respectively. As one
would expect, for most objects the radial extend of the kinematic measurements
is similar to the effective radius: the median kinematic radial extent-to-
effective radius ratio is 1.45 (see Figure 3).

Please note the presence of a small subset of galaxies with negative mass
estimates. This arises due to the prior on the dark mass component that is
centered at zero to avoid a positive bias in the total mass estimates (see
Section 3.2.2). All negative mass estimates have large uncertainties, so that
a positive mass is never ruled out with more than 95% confidence, as is
expected given the derived uncertainties.

The rotation parameter $\kappa$ is described in Section 3.1. The rotational
velocity $V_{R_{\rm{e}}}$ is calculated at $R_{\rm{e}}$ for the edge-on,
deprojected model. The effective velocity dispersion $\sigma_{R_{\rm{e}}}$ is
calculated for the light-weighted, face-on deprojected model within
$R_{\rm{e}}$. The catalog contains best values and lower and upper error bars.
These are obtained from the posterior distributions, where the best value is
the median and the upper (lower) error bars reflect the difference between the
84th (16th) percentile and the median.

A rotation proxy that is popular in the literature is the projected quantity
$\lambda_{R}$, which is frequently used by local IFU studies (and first
introduced for SAURON in Emsellem et al. 2007). Calculating $\lambda_{R}$
directly from our data is not useful given the large seeing convolution kernel
and single kinematic axis, so instead we could infer it from the model. But,
as shown by Harborne et al. (2019, 2020), even surveys of relatively nearby
galaxies suffer from significant seeing convolution and the results would not
comparable, and it would be misleading to name such a quantity $\lambda_{R}$.
Moreover, given the dependence on the model it makes little sense to leave in
inclination-dependent effects – hence our choice to calculate face- and edge-
on projected quantities.

A direct comparison with present-day galaxies is, at this point, not possible:
the only way to infer evolution in the intrinsic dynamical structure of
galaxies is create the equivalent dynamical models for present-day galaxies.

In our own previous work (Bezanson et al. 2018b) we measured
$(v_{5}/\sigma_{0})$ without correcting for seeing or slit losses – instead we
used IFU data from CALIFA (Sánchez et al. 2012), downgraded to resemble our
high-redshift data, in order to infer redshift evolution. Our models now
provide the intrinsic $(v_{5}/\sigma_{0})$ and we find the effect of
downgrading the CALIFA data agrees very well with the change from intrinsic
(model) $(v_{5}/\sigma_{0})$ and directly observed $(v_{5}/\sigma_{0})$. This
validates the conclusions drawn by Bezanson et al. (2018b) that high-redshift
quiescent galaxies show more rotation than their present-day counterparts,
which we interpreted as direct evidence for merging driving galaxy growth and
angular momentum reduction.

Table 2: Stellar Dynamical Parameters

ID1 | ID2 | $r_{\rm{max}}$ | $R_{e}$ | $\log M_{1}$ | $\log M_{5}$ | $\log M_{10}$ | $\log M_{R_{e}}$ | $\log M_{2R_{e}}$ | $\kappa$ | $V_{R_{e}}$ | $\sigma_{R_{e}}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | kpc | kpc | $M_{\odot}$ | $M_{\odot}$ | $M_{\odot}$ | $M_{\odot}$ | $M_{\odot}$ |  | $km~{}s^{-1}$ | $km~{}s^{-1}$
5 | 4792 | 6.9 | 5.0 | $9.99^{+0.08}_{-0.06}$ | $10.88^{+0.27}_{-0.24}$ |  $\cdots$ | $10.88^{+0.27}_{-0.24}$ |  $\cdots$ | $1.10^{+0.07}_{-0.04}$ | $179^{+27}_{-34}$ | $90^{+5}_{-4}$
15 | 6556 | 4.7 | 3.2 | $10.68^{+0.03}_{-0.03}$ | $10.73^{+0.53}_{-0.35}$ |  $\cdots$ | $10.84^{+0.18}_{-0.12}$ |  $\cdots$ | $0.34^{+0.20}_{-0.34}$ | $69^{+7}_{-90}$ | $204^{+13}_{-16}$
16 | 6859 | 4.9 | 7.6 | $10.83^{+0.04}_{-0.03}$ | $11.33^{+0.17}_{-0.16}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $0.36^{+0.15}_{-0.06}$ | $120^{+4}_{-2}$ | $214^{+9}_{-11}$
19 | 7419 | 13.0 | 11.5 | $9.72^{+0.25}_{-0.31}$ | $11.00^{+0.19}_{-0.20}$ | $11.46^{+0.21}_{-0.20}$ | $11.54^{+0.23}_{-0.21}$ |  $\cdots$ | $0.78^{+0.18}_{-0.09}$ | $123^{+7}_{-7}$ | $160^{+13}_{-13}$
26 | 10462 | 4.4 | 3.9 | $9.80^{+0.14}_{-0.12}$ | $10.70^{+0.52}_{-0.39}$ |  $\cdots$ | $10.55^{+0.46}_{-0.36}$ |  $\cdots$ | $0.57^{+4.42}_{-0.57}$ | $80^{+6}_{-19}$ | $58^{+10}_{-12}$
27 | 10902 | 5.9 | 12.0 | $10.22^{+0.07}_{-0.05}$ | $11.02^{+0.25}_{-0.25}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $1.04^{+3.96}_{-1.04}$ | $211^{+34}_{-71}$ | $119^{+11}_{-10}$
38 | 14375 | 6.2 | 2.8 | $10.76^{+0.02}_{-0.02}$ | $11.27^{+0.17}_{-0.16}$ |  $\cdots$ | $11.11^{+0.07}_{-0.08}$ | $11.29^{+0.19}_{-0.18}$ | $0.70^{+0.10}_{-0.06}$ | $241^{+5}_{-6}$ | $208^{+14}_{-31}$
39 | 14729 | 6.3 | 10.4 | $10.36^{+0.06}_{-0.06}$ | $11.21^{+0.16}_{-0.18}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $1.18^{+0.30}_{-0.08}$ | $306^{+43}_{-53}$ | $175^{+16}_{-17}$
62 | 25994 | 4.7 | 3.1 | $10.41^{+0.05}_{-0.05}$ | $11.10^{+0.19}_{-0.23}$ |  $\cdots$ | $10.91^{+0.11}_{-0.16}$ |  $\cdots$ | $1.08^{+0.05}_{-0.12}$ | $249^{+22}_{-49}$ | $145^{+7}_{-7}$
68 | 26283 | 7.8 | 4.2 | $10.51^{+0.04}_{-0.03}$ | $10.95^{+0.33}_{-0.35}$ |  $\cdots$ | $10.91^{+0.26}_{-0.28}$ | $11.01^{+0.71}_{-0.73}$ | $0.13^{+0.04}_{-0.13}$ | $29^{+1}_{-34}$ | $157^{+7}_{-9}$
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

Note. — (1): LEGA-C ID (DR3); (2):Ultra-VISTA ID (Muzzin et al., 2013b); (3):
Extent of stellar kinematic data; (4): Effective radius of the galaxy; (5-9):
Model mass enclosed in spherical apertures; (10): Rotation parameter; (11)
Rotational velocity at $R_{e}$ (edge-on); (12): Velocity dispersion within
$R_{e}$ (face-on). Values and uncertainties are based on 16th, 50th, 84th
percentiles of posterior parameter distributions. The machine-readable table
has 861 entries. We choose the following notation for a negative mass (see
text): $\log M=-10$ means $M=-10^{10}$ (not $M=10^{-10}$).

##  7 Stellar dynamical structure of $z\sim 0.8$ Galaxies

Figure 11: Projected (left) and deprojected (right)
$(V/\sigma)_{R_{\rm{e}}}$vs. projected shape, color coded with Sérsic index
$n$. Flat projected shapes represent a fairly good prediction for a high
degree of rotational support, but for individual galaxies a round shape is a
poor predictor. High-Sérsic index galaxies show a large range in rotational
support, overlapping with the distribution of low-Sérsic index galaxies, but
with an additional population of slowly rotating galaxies.  Figure 12:
Deprojected $(V/\sigma)_{R_{\rm{e}}}$vs. dynamical mass within $R_{e}$ for
quiescent (right) and star-forming galaxies (right), color coded with
deprojected (edge-on) rotation velocity $V$ at $R_{e}$. The correlations are
generally weak, and quiescent galaxies overlap strongly with star-forming
galaxies in terms of $(V/\sigma)_{R_{\rm{e}}}$. Only at the high-mass end
there is a clear difference between the populations, where quiescent galaxies
rotate slowly, and star-forming galaxies are strongly rotation-dominated.

It is beyond the scope of this paper to explore rotational support, angular
momentum and the masses of galaxies in great detail, and in forthcoming papers
we will explore the connection between star-formation history and dynamical
structure, the various mass components of galaxies, and the comparison with
dynamical modeling based on ionized gas kinematics. But to illustrate the
richness of our data we show in Figures 11 and 12 the distribution of galaxy
shapes, $(V/\sigma)_{R_{\rm{e}}}$and dynamical mass $M_{R_{e}}$.

The left-hand panel of Figure 11 is strongly reminiscent of integral field
surveys of present-day galaxies (e.g., Emsellem et al., 2011; van de Sande et
al., 2017; Falcón-Barroso et al., 2019), with a strong correlation between
projected shape and (projected) degree of rotational support. Low Sérsic index
(late-type) galaxies generally have higher $(V/\sigma)_{R_{\rm{e}}}$than high-
Sérsic index (early-type) galaxies, even at fixed projected ellipticity. In
particular, we see a population of rather flat ($\epsilon\sim 0.4$), but non-
rotating high-$n$ galaxies. Such galaxies also exist in the present-day
Universe and can be either anisotropically flattened systems or dynamically
complex galaxies with, for example, counter-rotating disks (Rix et al., 1992).
The spatial resolution of our data is obviously insufficient to distinguish
between the different explanations, but we can conclude that the net rotation
is low compared to the degree of flattening. There also is a small set of data
points at very large $(V/\sigma)_{R_{\rm{e}}}$, away from the general locus:
here the best-fit dynamical model has a high anisotropy (with very small
vertical dispersion), but with large uncertainties (not shown in the Figure,
but included in the catalog).

The crowding in the lower-left part of the figure (round, non-rotating
galaxies) is largely due to projection effects. Therefore, in the right-hand
panel we show the deprojected $(V/\sigma)_{R_{\rm{e}}}$(as explained in
Section 6). Even for nearly round galaxies with $\epsilon\sim 0.1$ our
modeling successfully recovered the degree of rotational support, as indicated
by the lack of a correlation between $(V/\sigma)_{R_{\rm{e}}}$and $\epsilon$
for low-$n$ galaxies. This population must consist of galaxies with a fairly
homogeneous dynamical structure, with $(V/\sigma)_{R_{\rm{e}}}$$\sim 1-2$, and
for which the scatter in projected $(V/\sigma)_{R_{\rm{e}}}$is mostly
explained by viewing angle (inclination). The high-$n$ galaxies are less
homogeneous in dynamical structure. Like in the present-day Universe, there is
a population of intrinsically round, slowly-rotating galaxies, mixed with a
population that is fast rotating, and similar in dynamical structure as the
low-$n$ population. We note that our models successfully distinguish between
rotation- and dispersion-dominated galaxies; our inclination prior, based on
an intrinsically oblate shape, did not preclude this.

It is well documented that Sérsic index correlates well with star-forming
activity at all redshifts (Bell et al., 2012), and it is informative to split
the population into quiescent and star-forming galaxies on the basis of their
rest-frame $U-V$ and $V-J$ colors, computed as explained in our 2nd data
release paper (Straatman et al., 2018). In Figure 12 we show the
$(V/\sigma)_{R_{\rm{e}}}$distribution as a function of dynamical mass
$M_{R_{e}}$ for quiescent and star-forming galaxies. For the quiescent
galaxies we see that the most massive galaxies, as in the present-day
Universe, are the most likely to be slowly rotating, while for the star-
forming population we see that the most massive galaxies are the most rotation
dominated. This immediately implies different evolutionary pathways for the
most massive galaxies, depending on their star-formation history.

The trend for star-forming galaxies may at first sight seem counter-intuitive,
given that the most massive star-forming galaxies are more bulge-dominated
than low-mass star-forming galaxies (which we see for our sample by an
increase in Sérsic index). Yet these more bulge-dominated galaxies have higher
$(V/\sigma)_{R_{\rm{e}}}$. But these trends are not mutually exclusive:
indeed, $\sigma$ increases with mass, signifying the presence of more massive
bulges, but $V$ increases faster with mass than $\sigma$, signifying the
presence of a large, extended disk. Interestingly, however, the galaxies with
the highest rotation speeds are quiescent: these are compact objects with high
velocity dispersions ($>200$ km s-1), yet flattened and rotation dominated,
with $V_{R_{e}}\sim 350$ km s-1, reminiscent of what has been referred to as
‘fossil’ galaxies in the present-day Universe (e.g., van den Bosch et al.,
2012; Ferré-Mateu et al., 2012).

If we then consider the interesting set of low-mass star-forming galaxies with
little rotation, visual inspection of the HST images and the rotation curves
suggest that this can occur for several reasons: some of these galaxies are
somewhat irregular systems for which the assumption of axisymmetry is likely
invalid, but the assessment of low $(V/\sigma)_{R_{\rm{e}}}$ still accurate;
some galaxies are truly non-rotating systems and appear to be young,
spheroidal galaxies that just fall short of being classified as quiescent; for
other objects the stellar light distribution obviously does not follow the
stellar mass distribution, and bright, blue star-forming regions or spiral
arms affect the analysis of the HST light profiles, and thereby the geometry
of the adopted axisymmetric system in the dynamical model.

##  8 Summary & Conclusions

We present axisymmetric Jeans models for 797 z$=0.6-1$ galaxies with spatially
resolved stellar kinematic measurements from the LEGA-C survey (van der Wel et
al. 2016; Straatman et al. 2018; van der Wel et al. 2021). The $K_{s}$-band
selected parent sample of over 3000 galaxies makes no selection on color or
morphology, and we select the subset objects that satisfy additional criteria
($S/N$, slit alignment, and regular morphology), but is otherwise unbiased
except for a bias against edge-on disks due to attenuation. This is by far the
largest sample of galaxies with resolved stellar kinematics at this redshift
to date, by a factor 30, and the only sample comprising star-forming / late-
type galaxies, as previous samples were focused solely on passive galaxies.
The application of Jeans models is important because unlike galaxies in the
present-day Universe, high-redshift galaxies are at most a few arcseconds in
extent - not much lager than both the width of the slits ($1$”) and the
ground-based seeing ($\sim 0.8$”). Our primary concern is to correct the
observed rotation and velocity dispersion profiles (Bezanson et al. 2018b) for
the convolution by slit and seeing, and an unknown inclination angle.

We use _HST_ $F814W$ imaging to describe the tracer light. The underlying
gravitational potential is modelled with a mass-follows light stellar
component plus a NFW dark matter halo. Other free parameters are the
inclination, the velocity anisotropy, and the positioning of the galaxy within
the slit; these three parameters are not well constrained by our data, but we
assess their impact on rotation and mass estimates by marginalizing over well-
informed prior distributions. We provide a full treatment of the seeing and
slit geometry on a galaxy-by-galaxy basis, and the JAM methodology is adapted
to accept non-square pixels (Section 3.4). We derive (total) dynamical masses
and proxies for the total, _intrinsic_ rotational support of galaxies from
these models (Table 2).

This data product provides the community with the tools to study the masses
and kinematic properties of a large sample of galaxies, which was not possible
before at large look-back time. For the first time we show the
$(V/\sigma)_{R_{\rm{e}}}$distribution of galaxies at large look-back time
(Fig. 11) and the complexity in evolutionary pathways revealed by the large
variety in dynamical structure seen even for galaxies with the same
(dynamical) mass (Fig. 12).

These results are timely because cosmological hydrodynamical simulations have
now reached the resolution to predict the stellar dynamical structure for
large, representative samples. Further analysis of the models and the related
data products will improve our understanding of these galaxies even further,
and this will be published in a number of follow-up projects in the near
future.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement 683184). Based on observations made with ESO Telescopes at
the La Silla or Paranal Observatories under programmes ID 194.A2005 and ID
1100.A-0949 (the LEGA-C Public Spectroscopic Survey). JvdS acknowledges
support of an Australian Research Council Discovery Early Career Research
Award (project number DE200100461) funded by the Australian Government.

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##  Appendix A Slit Mis-Centering

For most purposes, the slit can be assumed to be centered on the center of the
galaxy. This is how the masks are designed, with accurate coordinates
available from _HST_ astrometry. Furthermore, given the angular sizes of
galaxies compared to the slit width, even with a $1$ pixel offset in either
direction most light would still be captured. Even so, the kinematics will be
affected by an offset, leading to (small) asymmetries and offsets in the
observed velocity and dispersion profiles.

Here we ask the question whether it is possible to constrain the slit position
by comparing the observed LEGA-C light profile with the expected light profile
from a rebinned, convolved _HST_ image with a slit placed at different
offsets. For single, symmetric sources there are too many degeneracies to make
this feasible, but galaxies with asymmetric profiles, or galaxies with a close
neighbour that happens to fall within the slit, make this a viable option
(Figure 14). Owing to their relative position, the flux ratio between the two
sources is a function of the assumed offset (perpendicular to the slit, in the
wavelength direction), which constrains the observed offset.

Unfortunately, such sources are scarce, with on average $\sim 5$ useful
examples per mask. This is insufficient for a robust determination, as there
is no guarantee that galaxies within the same mask or even the same quadrant
will share the same offset. However, given the ensemble of offsets across the
entire survey the majority suggest offsets of $1-2$ pixels around $0$, with
incidental suggestions of bigger offsets, justifying the assumed prior in
Section 3.

Figure 13: Example of a galaxy with a close neighbour in the slit, visible and
fittable in the light profile. In red-dashed, the normalized light profile
obtained from collapsing the spectra in the wavelength direction. In
successive greytones the lightprofile obtained by convolving the $HST/ACS$
image with $0.9$ Moffat seeing profile, rebinning, and placing a $5$ pixel
slit (with pixels of $0.205$ arcseconds) with varying offsets. The inset shows
the $HST/ACS$ image (North up, East left). The x-axis shows the distance from
the bottom (South side) of the slit, in arcsec.