# The HI angular momentum-mass relation

Sushma Kurapati1,2, Jayaram N. Chengalur 1 , Marc A.W. Verheijen 3,1
1 National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, PO Box 3, Pune 411007, India
2 The Inter-University Centre for Astronomy and Astrophysics, Post Bag 4,
Ganeshkhind, Pune, 411007, India
3 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700
AV Groningen, Netherlands E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We study the relationship between the Hi  specific angular momentum ($\rm
j_{g}$) and the Hi  mass ($\rm M_{g}$) for a sample of galaxies with well
measured Hi  rotation curves. We find that the relation is well described by
an unbroken power law $\rm j_{g}$$\propto$ $\rm M_{g}$α over the entire mass
range (107 – 1010.5 M⊙), with $\alpha=0.89\pm 0.05$ (scatter 0.18 dex). This
is in reasonable agreement with models which assume that evolutionary
processes maintain Hi  disks in a marginally stable state. The slope we
observe is also significantly different from both the $j\propto M^{2/3}$
relation expected for dark matter haloes from tidal torquing models and the
observed slope of the specific angular momentum-mass relation for the stellar
component of disk galaxies. Our sample includes two Hi–bearing ultra diffuse
galaxies, and we find that their angular momentum follows the same relation as
other galaxies. The only discrepant galaxies in our sample are early–type
galaxies with large rotating Hi  disks which are found to have significantly
higher angular momentum than expected from the power law relation. The Hi
disks of all these early–type galaxies are misaligned or counter-rotating with
respect to the stellar disks, consistent with the gas being recently accreted.
We speculate that late stage wet mergers, as well as cold flows play a
dominant role in determining the kinematics of the baryonic component of
galaxies as suggested by recent numerical simulations.

###### keywords:

dwarf-galaxies: fundamental parameters–galaxies: kinematics and dynamics

††pubyear: 2018††pagerange: The HI angular momentum-mass relation–A

##  1 Introduction

Studies of the correlations between global parameters of galaxies (e.g.
rotation velocity vs. luminosity, or star formation rate vs. stellar mass)
provide critical observational tests of galaxy formation and evolution models.
One specific parameter which is receiving increasing attention is the angular
momentum. This is in part because it is only now that, thanks to large IFU
surveys as well as large Hi surveys, measurements of angular momentum are
becoming available for a sizeable number of galaxies (e.g. Walter et al.,
2008; Hunter et al., 2012; Sánchez et al., 2012; Bryant et al., 2015; Bundy et
al., 2015; Stott et al., 2016). Further, our theoretical understanding of how
galaxies acquire angular momentum is also rapidly evolving as numerical models
become more sophisticated and better able to capture the details of the
physics of the baryonic component of galaxies.

The traditional understanding was that galaxies acquire their angular momentum
via tidal torquing of collapsing proto-halos by neighbouring mass
concentrations. In such models the bulk of the angular momentum gain happens
around the turn around time of the proto-halo and the angular momentum remains
roughly constant after that epoch. The angular momentum of the final collapsed
halo is generally quantified by its ‘spin parameter’
$\lambda_{h}=J_{h}|E_{h}|^{1/2}/(GM_{h}^{5/2})$ (where $J_{h}$ is the total
angular momentum of the halo, $E_{h}$ the total energy, and $M_{h}$ the total
mass). Both analytical calculations (Peebles (1969)), as well as numerical
simulations (e.g. Bett et al. (2007); Rodríguez-Puebla et al. (2016)) find
that the spin parameter is roughly independent of the total mass of the halo.
This leads to a scaling relation between the halo specific angular momentum
$j_{h}$ ($=J_{h}/M_{h}$) and the halo mass, viz. $j_{h}\propto M_{h}^{2/3}$.
If one assumes that the baryonic matter and dark matter have similar
distributions at early times and that galaxies form by cooling and angular
momentum conserving collapse of the baryonic component (Fall (1983); Mo et al.
(1998)) then a similar scaling relation would be expected for the baryonic
component of galaxies, viz. $\rm j_{bar}$ $\propto$ $\rm M_{bar}$2/3. It has
been found that such a scaling relation is indeed obtained for the stellar
component of galaxies, although individual galaxies can deviate significantly
from the relation and the constant of proportionality varies between early and
late type galaxies, as well as with bulge fraction(Fall, 1983; Romanowsky &
Fall, 2012; Fall & Romanowsky, 2018)

This traditional understanding has been challenged by several numerical
simulations (e.g. Desmond et al., 2017; Jiang et al., 2019) of galaxy
formation that include the effects of mergers and coldflows. These simulations
show that the collapsed baryonic component of galaxies does not arise solely
from the cooling and contraction of the initial hot baryonic component of the
proto-halo. Both mergers as well as ‘cold’ flows of baryonic material along
filaments that penetrate deep into the halo (e.g. Kereš et al., 2005, 2009;
Brooks et al., 2009) could result in a significant change in the final
baryonic mass as well as the final angular momentum. The effect on the angular
momentum depends on whether the merger was ‘wet’ (i.e. with a gaseous
progenitor) or dry (i.e. with a gas poor progenitor) as well as the relative
orientation of the spin of the incoming material vis a vis that of the galaxy
(see e.g. Naab et al., 2014; Choi & Yi, 2017; Lagos et al., 2018).
Additionally, the final baryonic mass as well as the angular momentum are
affected by the energy input from stellar winds and supernovae which could
cause an outflow of material from the galaxy. The strength of the outflows
depends on whether the supernovae explosions are clustered or not. Since
clustering of supernovae typically happens only near the centres of galaxies
(i.e. in regions where the material has low specific angular momentum, (sAM))
outflows lead to a preferential loss of low specific angular momentum
material. This results in an increase in the specific angular momentum of the
remaining material in the galaxy (Dutton, 2009; Brook et al., 2011; Sharma et
al., 2012). As a consequence of these processes the angular momentum of the
galaxy is not determined during its formation phase, but rather is affected by
the entire evolutionary path of the galaxy. Both the morphology as well as the
angular momentum appear to be driven more by evolutionary processes and a
galaxy’s merger history than the original spin parameter of the dark matter
halo (see e.g. Stewart et al., 2017; Zjupa & Springel, 2017; Jiang et al.,
2019; Renzini, 2020). For example, simulations show that accretion of low sAM
or counter rotating gas leads to a central compact galaxy ("blue nugget"),
with an outer spheroidal stellar distribution, and possibly an outer counter-
rotating disk, while accretion of co-rotating material typically leads to a
disk type galaxy with gas density stable against rapid gravitational collapse
(see e.g. Zolotov et al., 2015; Jiang et al., 2019; Kretschmer et al., 2020).

Recent observational studies have also begun to indicate that the relationship
between the specific angular momentum and the mass is more nuanced than a
simple $\rm j_{bar}\propto\rm M_{bar}^{2/3}$ scaling. For the stellar $\rm
j_{s}$– $\rm M_{s}$ relation, Posti et al. (2018, 2019) find a unbroken power-
law $\rm j_{s}$$\propto$ $\rm M_{s}$α with $\alpha$ $\sim 0.55\pm 0.02$ for
all the galaxies in the stellar mass range of 107–1011.5 M⊙. On the other hand
for the gas $\rm j_{g}$– $\rm M_{g}$ relation Mancera Piña et al. (2021) have
recently reported a slope of $1.01\pm 0.04$. Earlier studies have also found
that gas rich dwarf galaxies appear to have an excess baryonic specific
angular momentum compared to that expected from the scaling relation for
bulgeless spirals given by Obreschkow & Glazebrook (2014) (e.g. Butler et al.,
2017; Chowdhury & Chengalur, 2017; Kurapati et al., 2018). Kurapati et al.
(2018) also find that the elevation in specific angular momentum occurs for
the galaxies with masses lower than $\sim 10^{9}$ M⊙. The scaling relation
hence appears to depend both on the details of the galaxy morphology, as well
as which component (stars, gas) of the galaxy is under consideration.

Here we study the relation between the baryonic specific angular momentum and
mass, as well as that between the Hi specific angular momentum ($\rm j_{g}$ )
and Hi mass ($\rm M_{g}$ ) for a diverse sample of gas rich galaxies including
Hi-bearing ultra-diffuse galaxies (HUDs), normal dwarf galaxies, spiral
galaxies, and early–type galaxies. We discuss the results in terms of the
range of simulations discussed above. This paper is organized as follows. In
§2.1, we describe the sample and observations. In § 3, we determine the gas
angular momentum and gas mass relation. We present a discussion in § 4 and
summarize the main results in § 5.

##  2 Determination of specific angular momentum

###  2.1 Sample

Our sample consists of a wide variety of galaxies including dwarf galaxies, Hi
bearing ultra diffuse galaxies, spiral galaxies, as well as early–type
galaxies. For some galaxies we present fresh measurements. We supplement this
sample with measurements that are available in the literature. We discuss
first the galaxies for which we present fresh measurements.

The Hi bearing ultra-diffuse galaxies (HUDs) were selected from the ‘more
restrictive’ sample of Leisman et al. (2017), with half light radii
r${}_{\mathrm{g},\mathrm{eff}}>1.5$ kpc, ${\mu}_{g,0}>24$ mag arcsec-2, and
g-band Magnitude > -16.8 mag, i.e. more extreme stellar sizes as compared to
their stellar magnitudes. We used new Giant Metrewave Radio Telescope (GMRT)
observations to obtain Hi morphologies and kinematics of 2 HUDs as part of a
pilot study for a larger survey. We derive the rotation curves and Hi surface
brightness profiles for these galaxies using Fully Automated TiRiFiC (FAT;
Kamphuis et al., 2015) by fitting a tilted ring model to the data cube. The
derived rotation curves and surface brightness profiles for ultra–diffuse
galaxies are shown in Figures 16 and 18 respectively.

Spiral galaxies are taken from the samples of Verheijen & Sancisi (2001).
Verheijen & Sancisi (2001) presented results from a synthesis imaging survey
of 43 galaxies in the nearby Ursa Major region using the Westerbork Synthesis
Radio Telescope (WSRT). We require a sample of galaxies where rotation curves
can be reliably derived. Hence, we consider only the galaxies that are well
resolved across the major axis and exclude the galaxies that are in
interacting pairs or that have disturbed velocity fields. 16 of 43 galaxies
remain after the imposition of these criteria. We use K’ band luminosity
profiles (as measured in Verheijen (2001)) to calculate the stellar mass
profiles since the uncertainties in the derivation of the mass to light ratio
are much less in the mid-infrared compared to the optical. We assume a
constant K’ band mass to light ratio of 0.24 to calculate the stellar surface
density profiles (Ponomareva et al., 2018). We use the rotation curves and the
radial Hi surface density profiles from Verheijen (2001) to calculate the
baryonic and gas specific angular momentum.

The early–type galaxies were selected from the ATLAS3D Hi survey of a volume-
limited, complete sample of 166 nearby early-type galaxies (ETGs) brighter
than MK = -21.5 (Serra et al., 2012). In constructing the sub-sample we first
restrict the selection to galaxies with morphologies later than T > -4.0.
Serra et al. (2012) divided the galaxies with detected Hi into four
morphological classes, viz. (i) large discs - most of the Hi rotates regularly
and is distributed in a disc or ring larger than the stellar body of the
galaxy, (ii) small discs - most of the Hi rotates regularly and is distributed
in a small disc confined within the stellar body of the galaxy, (iii)
unsettled - most of the Hi exhibits an unsettled morphology (e.g. tails or
stream of gas) and kinematics, and (iv) clouds - the Hi is found in small,
scattered clouds around the galaxy. In order to calculate the angular momentum
accurately, we require a sample where reliable rotation curves can be derived.
Hence, we have considered only the galaxies from the first class, i.e. those
with large discs that have regular velocity fields. We derived the rotation
curve for these galaxies using FAT or 3DBarolo (Di Teodoro & Fraternali, 2015)
on the final ATLAS3D data products, i.e Hi cubes with a typical angular
resolution of $\sim$ 35 arcsecs. We chose the galaxies that are well resolved
across its major axis ($>$ 6 beams). Our final sample consists of those
galaxies for which reliable rotation curves could be obtained by FAT or
BAROLO. BAROLO was used to derive the rotation curve for only one of the
galaxies (NGC 3941), since the FAT derived rotation curve was unreliable.Two
of the early–type galaxies (NGC3626 and NGC3941) have companions, which were
excluded for the measurements of mass and specific angular momentum. The
derived rotation curves and surface brightness profiles for the final sample
of early–type galaxies are shown in Figures 17 and 19 respectively. In Fig. 6
\- 15, we show (i) the integrated Hi intensity map contours overlaid on the
optical image, (ii) Position-Velocity (PV) diagram taken along the major axis
of the galaxy with the rotation curves overlaid on them. The dashed lines
indicate the systemic velocity and kinematic center. The overplotted violet
triangles represent the rotation curve derived by ’FAT’ (iii) the intensity
weighted first moment of the galaxy, and (iv) velocity field of the best
fitting FAT model for all the HUDs and early-type galaxies. We find that all
the rotation curves broadly follow the PV diagram. We note that PV diagrams
are not fully representative of the rotation curve as they may reflect the
effects of non-circular motion while FAT and other algorithms also take pixels
off the kinematic major axis into account.

As mentioned above, in our analysis we also use the published data from
various earlier studies; the details are given in the relevant sections.

###  2.2 Measuring mass and specific angular momentum

The Hi mass was calculated using the standard formula, MHI= 2.36$\times$ 105
$D^{2}\int$S $dv~{}M_{\odot}$, where $D$ is the distance in Mpc, $S$ is the
flux density in Jy and $dv$ is in km s-1 (Roberts, 1962). The distances for
early-type galaxies are taken from Cappellari et al. (2011), which were
obtained with the surface brightness fluctuation (SBF) method. These distance
estimates have an error of $\sim$ 7 – 10 $\%$. The distance estimates for the
UDGs are taken from Leisman et al. (2017), which were calculated using the
ALFALFA flow model, and the distance uncertainties due to proper motions are
$\lesssim$ 15 $\%$. The distances for the Ursa Major spiral galaxies are taken
from Sorce et al. (2014), which were derived by using the Tully Fisher
relation as an empirical distance estimator which provides a distance accuracy
of $\sim$ 3–10 $\%$. We assume a conservative error of 15 $\%$ on distance
measurements for all the galaxies. The quoted uncertainties on gas mass are
based on the errors on the distance measurements as well as the error on flux
densities. The error on the flux density is dominated by the calibration
uncertainties, which are typically $\sim$ 10%. We do not consider the mass to
light ratio uncertainties while calculating the uncertainties of baryonic
masses for the ultra-diffuse galaxies as they are dominated by the gas mass.
We calculate the uncertainties of the baryonic masses for the UMa spirals
(following (Lelli et al., 2016)) by

$\delta_{M_{\rm bar}}=\sqrt{\delta^{2}_{M_{\rm
g}}+\bigg{(}\gamma_{\ast}\delta_{L_{\ast}}\bigg{)}^{2}+\bigg{(}L\delta_{\gamma_{\ast}}\bigg{)}^{2}+\bigg{(}2M_{\rm
bar}\frac{\delta_{D}}{D}\bigg{)}^{2}}$ (1)

where $\delta_{\rm g}$ and $\delta_{L}$, are the errors on Mg and L due to
uncertainties on total fluxes and $\delta_{\gamma}$ is the uncertainty in mass
to light ratio. We assume $\delta_{\gamma_{\ast}}$ $\sim$ 0.1 dex for K’-band
data (Ponomareva et al., 2018). The specific angular momentum j is computed by
numerically evaluating the integral,

$j\equiv\frac{J}{M}=\frac{\int_{0}^{R}dr2\pi r^{2}\Sigma(r)v(r)cos[\delta
i(r)]}{\int_{0}^{R}dr2\pi r\Sigma(r)}$ (2)

where, $\Sigma(r)$ is the azimuthally averaged surface density, v(r) is the
circular velocity at radius r, $\delta$i(r) is the difference in inclination
with respect to the central disc inclination and R is the radius of the gas
disk. This integral is evaluated separately for the stellar, the gaseous, and
the baryonic components of the galaxy. The procedure used for the estimation
of specific angular momentum is similar for the galaxies from this work and
for the galaxies that are taken from the literature. In this work, only the
stellar, Hi, and Helium contributions to the baryonic mass are considered.
Hence, the total baryonic surface density is $\Sigma_{\rm b}$ =
$\Sigma_{\ast}$ \+ 1.35 $\Sigma_{HI}$ , where the 1.35 factor accounts for the
He contribution. No correction was made for molecular hydrogen as well as
ionized gas. The optical deprojected radial surface brightness profiles were
derived by fitting elliptical annuli to the optical images by using the ELLINT
task in the GIPSY package. The specific angular momentum of gas, stars and
baryons was computed by numerically evaluating the integral in Equation (1).
For all quantities, the integral converges within the error bars. The
uncertainties on the specific angular momentum of the various components are
based on the errors on distance measurements, on rotation velocities, surface
brightness values, and inclination values. The uncertainty in jk ($k$ being
stars or gas) is given by:

${\footnotesize\frac{\delta_{j_{k}}}{j_{k}}=\sqrt{\big{(}\frac{\delta_{D}}{D}\big{)}^{2}+\big{(}\frac{\delta_{i}}{tan(i)}\big{)}^{2}+\sum_{i}^{N}\big{(}\frac{\delta_{v_{i}}}{v_{i}}.\frac{J_{i}}{J}\big{)}^{2}+\sum_{i}^{N}\big{(}\frac{\delta_{S_{i}}}{S_{i}}.\frac{J_{i}}{J}\big{)}^{2}}}$
(3)

where $i$ is the inclination ($\delta_{i}$ is its uncertainty). vi is the
rotation velocity at each point on rotation curve ($\delta_{v_{i}}$ is its
uncertainty), Si is the surface brightness value at each point on surface
brightness profile ($\delta_{S_{i}}$ is its uncertainty), Ji is the evaluated
angular momentum in each ring and J is the total angular momentum of galaxy.
The errors on specific angular momentum are often dominated by the errors on
distance measurement.

In Tables 1 and 2, we present the measurements of stellar, gas, and baryonic
mass and specific angular momentum of UMa spiral galaxies and ultra-diffuse
galaxies respectively: The columns are as follows. Column (1) name of the
galaxy; (2) stellar mass ; (3) gas mass; (4) total baryonic mass; (5) stellar
specific angular momentum; (6) gas specific angular momentum and (7) baryonic
specific angular momentum. In Table 3 we show the properties and the
measurements of the gas mass and gas specific angular momentum of the
early–type galaxies. The columns are as follows. Column (1) name of the
galaxy; (2) morphology ; (3) distance in Mpc (4) gas mass; (5) gas specific
angular momentum and (6) Notes on galaxies.

###  2.3 Convergence criteria

We show the cumulative specific angular momentum profiles for 2 HUDs (both
$\rm j_{bar}$and $\rm j_{g}$profiles), 16 UMa spirals (both $\rm j_{bar}$and
$\rm j_{g}$profiles), and 8 early-type galaxies (only $\rm j_{g}$profiles) in
Figures 20, 21, and 22 respectively. We normalize the specific angular
momentum and radius axes to allow the comparison between the profiles. We also
make a quantitative assessment of the convergence of these profiles. We follow
Posti et al. (2018) to determine whether the specific angular momentum of a
galaxy has converged or not. We select galaxies satisfying the following
criteria:

$\frac{j(<R_{N})-j(R_{N-1})}{j(R_{N})}<0.1\ \ \&\&\ \ \frac{d\ log(j(<R))}{d\
log(R)}\bigg{|}_{R_{N}}<0.5$ (4)

i.e. that the last two points of the j profile differ by less than 10 $\%$ and
that the logarithmic slope of the j profile in the outer-most point is less
than 1/2. We find that all the galaxies except two early-type galaxies
(NGC3626 and NGC6798) follow the convergence criterion.

We also fit a function $\mathcal{P}$ = k. exp(-a/x), (which is zero at
x$\xrightarrow{}$ 0+ becomes constant as x $\xrightarrow{}$ $\infty$ ) to the
last three points of the cumulative j profile. We then, take the ratio $r_{k}$
(k being baryons/gas) of $j_{k}$ at last measured point to the maximum value
that $\mathcal{P}$ would have if extrapolated to infinity. This ratio, (i.e.
the convergence factor), is calculated for all the jbar and jgas profiles and
listed for all the galaxies in Tables 4, 5 and 6. Only 2 of the early-type
galaxies (NGC 3626 and NGC6798) and 1 of the ultra-diffuse galaxies have a
convergence factor $<$ 0.7.

Several angular momentum studies, e.g. Murugeshan et al. (2020), Elson (2017)
do not check for convergence. Disk galaxies show a range of rotation curve
shapes with low mass galaxies typically having rising rotation curves.
However, in general, dwarf galaxies show a large diversity of rotation curve
shapes, even at fixed maximum rotation speed (e.g. Oman et al., 2015).
Restricting the analysis to a set of galaxies that meet a user defined
convergence criterion may lead to a non representative sample, particularly
for the galaxies with lower baryonic masses. For studies where one is
considering the specific angular momentum of the galaxy disk, (as opposed to
making inferences about the halo), it is not clear that a restriction of the
sample to those whose specific angular momentum curves meet a user defined
convergence is essential.

Figure 1: Log $\rm j_{g}$– log $\rm M_{g}$of 2 HUDs (magenta triangles) from
this work, 11 dwarfs residing in Lynx-Cancer void (black circles) from
Kurapati et al. (2018) , 17 field dwarfs (red diamonds from Butler et al.
(2017) and Chowdhury & Chengalur (2017)), 16 spirals residing in the UMa
cluster (plus symbols) from this work, 15 spirals from Obreschkow & Glazebrook
(2014) (pentagons), and 84 galaxies from Mancera Piña et al. (2021) (grey
circles). The cyan, magenta, and yellow symbols (plus symbols and pentagons)
indicate spiral galaxies with bulge fractions, $\beta$ = 0, $\beta$ = 0.1, and
$\beta$ = 0.2 respectively. The green line indicates the best fit for the $\rm
j_{g}$– $\rm M_{g}$relation using the linear regression. The blue line
represents the stability criterion (jHI x 11 km s-1/(G MHI) $\sim$ 1) from
Romeo (2020)

##  3 Results

###  3.1 Specific angular momentum of the Hi disk

As discussed in the introduction, recent simulations indicate that the angular
momentum of galaxies is governed more by evolutionary processes and the
physics of the baryonic material than by the the initial spin of the ‘parent’
dark matter halo. Numerical simulations also support the picture that star
formation occurs in marginally stable gas disks, where inflows provide both
the gas required for the star formation, as well as energy to keep the disks
marginally stable against gravitational instability (e.g. Dekel et al.
(2013)). Models of disk galaxies where the disks are critically stable also
appear to provide a reasonable description of the specific angular momentum-
mass relation of the gas disks of local galaxies (e.g. Zasov, 1974; Zasov &
Zaitseva, 2017; Kurapati et al., 2018; Romeo, 2020). Observational constraints
on models where the gas mass and angular momentum of Hi disks are set by the
requirement that the disk be critically stable have been presented by e.g.
Obreschkow et al. (2016); Zasov & Zaitseva (2017). It would hence be
interesting to look at how well such models fit the Hi specific angular
momentum ($\rm j_{g}$) and the Hi mass ($\rm M_{g}$) for our heterogeneous
sample of galaxies.

We show in Fig. 1 the Hi specfic angular momentum and Hi gas mass relation for
our sample galaxies along with other samples taken from the literature
(details are given in the figure caption). Interestingly, the galaxies show a
smooth $\rm j_{g}$– $\rm M_{g}$ relation across the entire mass range. We
estimate the best fitting linear regression between $\rm j_{g}$ and $\rm
M_{g}$ by using the BCES algorithm (Akritas & Bershady, 1996) that accounts
for errors on both $\rm j_{g}$ and $\rm M_{g}$ as well as for the fact that
these errors are correlated. The best-fit $\rm j_{g}$ -$\rm M_{g}$ relation is
given by, $\rm j_{g}$ = q$\rm M_{g}$α with $\alpha$ = 0.89 $\pm$ 0.05 and
log10 q = -5.42 $\pm$ 0.41. We do not include early–type galaxies while
fitting for the $\rm j_{g}$ – $\rm M_{g}$ relation since these galaxies are
peculiar in that the gas is counter-rotating or misaligned with the stellar
kinematics. We discuss these galaxies further below. The existence of such a
relation between the Hi mass and the Hi specific angular momentum (suggested
earlier also by e.g. Zasov (1974)) is interesting in itself, and we return to
this in Sec. 4. We also note that the slope that we find for the $\rm j_{g}$ –
$\rm M_{g}$ relation is significantly different from that seen for the $\rm
j_{s}$ – $\rm M_{s}$ relation (i.e. $0.55\pm 0.02$) by Posti et al. (2019).
The slope we get is slightly smaller than (but statistically consistent with)
the value of $1.02\pm 0.04$ recently reported by Mancera Piña et al. (2021).
We return to a discussion of this in Sec. 4. Here we focus on how the relation
that we find compares with expectations from the critically stable disk model.

The commonly used criterion for stability is the Toomre Q parameter. Strictly
speaking, the Toomre criterion is for a thin single component disk, while at
least some of the galaxies that we are analysing have significant stellar
masses as well as a non negligible disk thickness. Issues associated with
applying the Toomre-Q criterion to such galaxies have been discussed in detail
in Zasov & Zaitseva (2017) and Romeo (2020). Romeo (2020) in particular takes
a phenomenological approach, and suggests that the different components
(molecular, atomic, stellar) of galaxy disks follow scaling laws derived from
the relation

${j_{i}\hat{\sigma_{i}}\over GM_{i}}\approx 1$

where $\hat{\sigma_{i}}=\bar{\sigma_{i}}/A_{i}$, with $\bar{\sigma_{i}}$ being
the disk averaged velocity dispersion of component $i$ (Hi , H2, or stars),
and $A_{i}$ is the mass averaged Toomre-Q parameter. The value of
$\hat{\sigma_{i}}$ is not derived from fundamental principals, but is instead
determined by a fit to observational data. Romeo (2020) uses data from Leroy
et al. (2008) (which overlaps with the Obreschkow & Glazebrook (2014) sample)
to determine the value for $\hat{\sigma_{i}}$. For the Hi component the
derived value is 11 km/s. We show this empirical relation, log(j) =
log(M)-6.41 (with $\hat{\sigma_{i}}=11$ km/s) in Fig. 1. As can be seen, it
provides a good fit to the data, the majority of which is independent from the
sample used by Romeo (2020). This further supports the idea that the structure
of the Hi disks in galaxies settles to a critical density, as would happen in
models where the star formation rate (and hence mass outflow rate) is driven
by the mass inflow rate.

As discussed in the introduction, simulations also show that spheroidal
stellar distributions with extended counter-rotating gas disks could arise in
systems where there has been a recent accretion of gas whose spin is not
aligned with that of the parent galaxy (e.g. Kretschmer et al., 2020). In the
context of such models it would be interesting to examine the position of
extremely gas rich early–type galaxies in the $\rm j_{g}\ -\rm M_{g}$ plane.
As can be seen from Table 1, for many of the galaxies in our sample the gas is
counter-rotating or misaligned with the stellar kinematics, consistent with
results from simulations. Figure 2 shows the location of early–type galaxies,
as well as all our other sample galaxies in the $\rm j_{g}$ – $\rm M_{g}$
plane. The green solid line indicates the best fit line using the linear
regression for all the galaxies except the early–type galaxies and green
dotted lines indicate 3$\sigma$ vertical scatter. As can be seen, most of the
early–type galaxies lie above the 3$\sigma$ scatter, i.e. their Hi disks have
significantly higher specific angular momentum as compared to other galaxies
with the same Hi mass. A Kolmogorov–Smirnov (KS) test(Fasano & Franceschini,
1987, https://github.com/syrte/ndtest/blob/master/ndtest.py) gives a
probability of 7 $\times$ 10-4 for the early–type galaxies being drawn from
the same distribution as the other galaxies.

Two of these early-type galaxies (NGC3626 and NGC6798, marked with a circle
around the blue open triangles in Fig 2) do not satisfy the convergence
criterion. We note that they do not appear to have discrepant $\rm j_{g}$ as
that of the early-type galaxies with converged specific angular momentum
curves. If the lack of convergence means that these two galaxies have a higher
specific angular momentum than measured, it does not affect the result that
the early–type galaxies in our sample have significantly higher $\rm j_{g}$
than expected from the $\rm j_{g}$\- $\rm M_{g}$ power law. Our observations
hence support the picture in which these gas rich early-type galaxies have
acquired the gas via a recent inflow of high, but misaligned sAM gas. The
elevated specific angular momentum is consistent with the decreased star
formation efficiency of early–type galaxies since higher angular momentum
stabilizes the gas against Jeans instabilities and subsequent star formation
(Toomre, 1964; Obreschkow et al., 2016).

We also show the scaling relation between the atomic gas fraction (fatm) and
the stability parameter (q) for all the galaxies in our sample in Figure 3 for
which these parameters are available. The dotted line shows the relation
expected from Obreschkow et al. (2016). We find that all the galaxies in our
sample largely follow the relation consistently. We do not show early–type
galaxies in this figure since we can not measure their baryonic specific
angular momentum as the gas does not in general co-rotate with the stars for
these galaxies. The dwarf galaxies lie in the saturated region (f${}_{\rm
atm}\sim$ 1). We also show the Hi excess (higher atomic gas fraction) galaxies
from Lutz et al. (2018) for comparison; they follow the relation between the
atomic to baryonic mass fraction and the global stability parameter given by
Obreschkow et al. (2016).

Figure 2:  The location of early–type galaxies with large regular Hi disks
(blue open triangles) and the other galaxies (symbols are the same as in
Figure 1) in the $\rm j_{g}$-$\rm M_{g}$plane. The green solid line indicates
the best fit line using the linear regression for all the galaxies except
early–type galaxies. The green dotted lines indicates 3$\sigma$ scatter around
the relation. The early-type galaxies that do not follow the convergence
criterion are marked with a circle around the blue open triangles. The Hi
disks in gas rich early–type galaxies have high specific angular momentum
compared to other galaxies with similar Hi mass. Figure 3: The atomic gas mass
to baryonic mass fraction versus the global stability parameter q for all the
galaxies in our sample (symbols are the same as in Figure 1). We also show
galaxies from Murugeshan et al. (2020) (cyan stars ($\beta$ = 0) and magenta
stars ($\beta$ = 0.1)) and the Hi excess galaxies from Lutz et al. (2018)
(green squares). These galaxies were not included in the earlier plots for the
$\rm j_{g}$-$\rm M_{g}$relation because $\rm j_{g}$is not available for these
galaxies. We do not show the galaxies from Mancera Piña et al. (2021) as q
values were not presented. The dotted line is the relation predicted by
Obreschkow et al. (2016).

###  3.2 Baryonic Specific Angular Momentum

Figure 4: The log $\rm j_{bar}$-log $\rm M_{bar}$plane. The black solid line
is the best fit line for the dwarf galaxies from Kurapati et al. (2018) and
the red dashed line is the best fit line for all the galaxies. The symbols are
the same as in Figure 1. The blue circles represent galaxies (only the
galaxies with reliable rotation curves) from Elson (2017).

Several earlier studies (e.g. Obreschkow & Glazebrook, 2014; Kurapati et al.,
2018; Murugeshan et al., 2020; Mancera Piña et al., 2021) have studied the
correlation between the specific angular momentum and mass of the baryonic
(i.e. gas + stars) component of galaxies. In particular Kurapati et al. (2018)
investigated the relation considering the baryonic mass for a sample of gas
rich dwarf galaxies in a range of environments, using the bulgeless spiral
relation found by Obreschkow & Glazebrook (2014) for comparison. They
concluded that the baryonic specific angular momentum of dwarfs was higher as
compared to the extrapolation of the trend seen for higher mass bulge-less
spirals. The increase in specific angular momentum was found to set in at a
mass threshold of $\sim$ 109 M⊙. However, more recent studies (e.g. Mancera
Piña et al. (2021)) find that the $\rm j_{bar}$-$\rm M_{bar}$relation is well
fit by a single power law.

Fig. 4 shows the relation between the baryonic specific angular momentum and
the baryonic mass for the UMa sample of spiral galaxies from this work, along
with the galaxies in Obreschkow & Glazebrook (2014); Butler et al. (2017);
Chowdhury & Chengalur (2017); Elson (2017); Kurapati et al. (2018); Murugeshan
et al. (2020); Mancera Piña et al. (2021) We do not use the early–type
galaxies from the current work this analysis since, as discussed above for
these galaxies the gas does not in general co-rotate with the stars. If fit a
single power law to our sample (red dashed line in Fig. 4), we obtain a slope
0.55 $\pm$ 0.02 in good agreement with the slope measured by Mancera Piña et
al. (2021). However, all the galaxies having $\rm M_{bar}$$\lesssim$ 108 M⊙
lie significantly below the relation while the galaxies having 108
M${}_{\odot}\lesssim$ $\rm M_{bar}$$\lesssim$ 109 M⊙ are consistently above
the relation. In this context, we note that the Mancera Piña et al. (2021)
sample does not contain galaxies with baryonic mass lower than 108M⊙. This
suggests that there may be some curvature in the the $\rm j_{bar}$–$\rm
M_{bar}$ relation, and it may not be best described by a single unbroken power
law.

In order to test how well the $\rm j_{bar}$– $\rm M_{bar}$relation is
described a single unbroken power law, we estimate the best fit linear
regression between $\rm j_{bar}$and $\rm M_{bar}$independently for two
separate mass ranges, viz. galaxies with $\rm M_{bar}$$>$ 109 M⊙ and for the
galaxies with $\rm M_{bar}$$<$ 109 M⊙. We find significantly different slopes
for the $\rm j_{bar}$– $\rm M_{bar}$relation, with a slope of 0.80 $\pm$ 0.05
for low mass galaxies and a slope of 0.52 $\pm$ 0.02 for high mass galaxies.
This suggests that a single power law is not the best description of the data.
On the other hand, the slopes of $\rm j_{g}$\- $\rm M_{g}$relation match
within the error bars for low mass galaxies ($\sim$ 0.85 $\pm$ 0.05) and high
mass galaxies ($\sim$ 0.93 $\pm$ 0.07), suggesting that a single power law is
an adequate description.

We also use the “Chow test" (Chow, 1960) to test whether the $\rm j_{bar}$–
$\rm M_{bar}$relation is well fit by a single unbroken power law. The test
checks for the statistical significance of postulated change in slope in a
regression. We run the Chow test on both the $\rm j_{bar}$– $\rm M_{bar}$and
$\rm j_{g}$\- $\rm M_{g}$relations (https://github.com/jtloong/chow_test) to
test the significance of a difference in slope between the high and low mass
ends, with the nominal dividing point taken as 109 M⊙. As per the Chow test,
the null hypothesis that the baryonic j-M relation follows a unbroken power
law is rejected at a high significance, with a p value of 5.5 $\times$ 10-5.
This indicates that there is a significant change in slope, with a broken
power law providing a better fit. We note that the test does not independently
identify a sharp break-point; a curved relation with a gradual change in slope
as one moves from lower to higher baryonic masses would also be better fit by
a broken power law rather than a single one. In contrast to this, we find a
fairly large p value (0.2) for the gas j-M relation which indicates that it is
adequately fit by a single power law. We note that the number of galaxies with
low baryonic masses in the sample is still quite modest, and it is important
to increase the sample size at the low mass end. We return to a discussion of
the possible reasons for the change in slope seen in the $\rm j_{bar}\ -\rm
M_{bar}$ relation in the next section.

Finally, we draw attention to one more feature in Fig. 4, the location of the
ultra-diffuse dwarfs. Ultra-diffuse dwarf galaxies (UDGs) are a recently
discovered population of very low surface brightness galaxies, which have
stellar masses of dwarfs, but have sizes as large as L∗ galaxies (van Dokkum
et al., 2015). The majority of the UDGs are red and appear more common in
clusters. However, galaxies with extreme ratios of stellar mass to stellar
scale length were also discovered in Hi surveys of isolated environments (e.g.
Leisman et al. 2017). These galaxies are called Hi–bearing ultra diffuse
galaxies (HUDs), which are bluer and have significant reservoirs of Hi as
compared to the UDGs in the clusters. Several formation scenarios have been
proposed to explain the formation of the UDGs. Some authors propose a scenario
in which UDGs are failed Milky Way like galaxies which lost their gas after
forming the first stars (van Dokkum et al., 2015) and others argue that they
are genuinely dwarf galaxies residing in high spin parameter halos (Amorisco &
Loeb, 2016). These formation scenarios can be tested by measuring the angular
momentum of UDGs and comparing them to the normal dwarf galaxies. Figure 4
shows the baryonic specific angular momentum versus the baryonic mass of
Hi–bearing UDGs (magenta triangles) and normal dwarf galaxies of similar
masses. We find that the specific angular momentum of Hi–bearing UDGs is
similar to other normal dwarf galaxies at a given mass. This is in contrast
with the formation scenarios that predict that UDGs are genuinely dwarf
galaxies residing in high spin haloes. We note that our result is in
contradiction with a recent analysis by Mancera Piña et al. (2021), who find
that ultra diffuse galaxies have higher-than-average stellar specific angular
momentum, but have circular velocities much lower than galaxies with similar
baryonic mass. They assume flat rotation curves and exponential light curves
and approximate the stellar specific angular momentum to be the product of
circular velocity and the stellar disk scale length since they lack the
resolution to integrate angular momentum over the entire disk. This, along
with the very small size of our sample, make it important to confirm these
results with high resolution observations of a larger sample.

##  4 Discussion

As we saw above both the $\rm j_{g}$ – $\rm M_{g}$ relation (this work and
Mancera Piña et al. (2021)) as well as the $\rm j_{s}$ – $\rm M_{s}$ relation
(Posti et al., 2018) follow an unbroken power law over the mass range of 107 –
1010.5 M⊙. However their slopes are very different, viz. with slopes of
$0.89\pm 0.05$ (this work) and $0.55\pm 0.02$ (Posti et al., 2018)
respectively. In contrast to these relations, the $\rm j_{bar}$ \- $\rm
M_{bar}$ relation is better fit by a broken power law, with gas rich low mass
galaxies having higher specific angular momentum compared to the extrapolated
trend from high mass galaxies. We test the scenario that this increase in
baryonic specific angular momentum essentially reflects the different relative
contributions of the stellar and gas angular momenta to the total baryonic
angular momentum following from the different slopes of stellar and gas j–M
relations.

We calculate the general expected trend of $\rm j_{bar}$ with $\rm M_{bar}$ by
using the observed Mg \- Ms and $\rm j_{g}$\- $\rm M_{g}$ and js – Ms
relations as well as the relation between MHI and Ms from Maddox et al.
(2015). The predicted $\rm j_{bar}$ \- $\rm M_{bar}$ curve from these
relations is shown in Fig. 5, where we have included the contribution of
helium to the gas mass by multiplying MHI by a factor of 1.35. For the
predicted relation we have used the median Hi mass values and their 1$\sigma$
uncertainties derived for galaxies in the Maddox et al. (2015) sample for
which SDSS spectra are available. The blue solid curve indicates the expected
$\rm j_{bar}$\- $\rm M_{bar}$relation and the blue shaded region indicates the
1$\sigma$ scatter. The relation obtained if use the relation from Maddox et
al. (2015) derived from the sample that includes galaxies for which no SDSS
spectrum is available matches the blue curve within the 1$\sigma$ scatter. As
can be seen, the model curve is in broad qualitative agreement with the data
showing a change in slope around $10^{9}\rm M_{\odot}$, although the $\rm
j_{bar}$ values in this mass range are somewhat larger than the observed
points. We note that the actual gas fractions of the sample being used would
shift the location of the points in the $\rm j_{bar}$–$\rm M_{bar}$ plane.
However, the broad qualitative agreement between this simple model and the
data supports the idea that the different slopes obtained at the low and high
mass ends of the $\rm j_{bar}\ -\rm M_{bar}$ relation is a consequence of the
changing contribution of the stars and gas to the total angular momentum at
the high and low $\rm M_{bar}$ ends.

Figure 5: The log $\rm j_{bar}$-log $\rm M_{bar}$plane. The black dotted line
is the best fit line for the dwarf galaxies from Kurapati et al. (2018) and
the red dashed line is the best fit line for all the galaxies. The symbols are
the same as in previous figure. The model is shown as blue shaded region

It is also interesting to ask how the power law relations between the specific
angular momentum and the mass fit in with the other known scaling relations?
To address this question we consider a simple model galaxy where the stars and
gas have exponential surface brightness profiles with scale lengths rs and rg
as well as a flat rotation curve with V(r) = Vc. This gives j $\propto$ Vcr.
From this expression we calculate the expected slope ($\alpha_{\rm s}$) for
the $\rm j_{s}$-$\rm M_{s}$ relation by using the stellar mass size relation,
rs $\propto$ M${}_{\rm s}^{0.28}$ (Bell & de Jong, 2001) and the stellar TF
relation, Vc $\propto$ M${}_{\rm s}^{0.31}$ (Bloom et al., 2017), which gives
$\alpha_{\rm s}$ $\sim$ 0.28+0.31 = 0.59 which compares well with the observed
slope of $\sim$ 0.55 $\pm$ 0.02. For the gas rich galaxies we use the baryonic
TF relation, V${}_{\rm c}\propto$ M${}_{\rm bar}^{0.33}$ (Ponomareva et al.,
2018) and assume that the baryons are dominated by gas for the dwarf galaxies,
which gives V${}_{\rm c}\propto$ M${}_{\rm g}^{0.33}$. Observations suggest
that the Hi mass and radius are related as r${}_{\rm g}\propto$ M${}_{\rm
g}^{0.51}$ (Wang et al., 2016). This also arises from simulations if we assume
a exponential Hi disk, with the inner radius and outer radius being set by
molecular transition and observational threshold for detecting Hi (Obreschkow
et al., 2009). The combination of baryonic TF relation and the gas mass-size
relation gives the slope ($\alpha_{\rm g}$) for the $\rm j_{g}$ – $\rm M_{g}$
relation, $\alpha_{\rm g}$ $\sim$ 0.51+0.33 = 0.84, in excellent agreement
with what we observe. For less gas rich galaxies, if we combine the stellar TF
relation Vc $\propto$ M${}_{\rm s}^{0.31}$ from Bloom et al. (2017) and the
gas fraction relation $\rm M_{g}$/ $\rm M_{s}$= $\rm M_{s}$-0.53 from Romeo
(2020), we find V${}_{\rm c}\propto$ M${}_{\rm g}^{0.66}$. and $\alpha_{\rm
g}$ $\sim$ 0.66+0.51 = 1.17 in reasonable agreement with what we observe. We
cannot determine the predicted values of the intercepts, since they depend on
the ratio of the size of the HI disk to its scale length, which is not well
measured.

##  5 Summary and Conclusions

We have examined the relation between the Hi  mass and the Hi  specific
angular momentum for a sample of galaxies with well measured rotation curves
that spans a range of morphology. Combining this with data from the
literature, we find that the relation is well fit by an unbroken power law
over the mass range $10^{7}-10^{10.5}$ $\rm M_{\odot}$. The scatter in this
relation is small, and this supports earlier suggestions that the relationship
between the angular momentum and mass is a fundamental scaling relation for
galaxies. The slope that we measure for the power law $0.89\pm 0.05$ is also
significantly different from that expected for dark matter haloes from tidal
torquing models which predict $j\propto M^{2/3}$, as well as the observed
slope of $0.55\pm 0.02$ for the stellar component of disk galaxies. The slope
we get for the $\rm j_{g}-\rm M_{g}$ relation is slightly smaller than (but
statistically consistent with) the value of $1.02\pm 0.04$ recently reported
by Mancera Piña et al. (2021). Our sample includes two Hi  bearing ultra-
diffuse galaxies, and we find that they lie along the relationship defined by
the other galaxies in our sample. The only discrepant galaxies are early–type
galaxies with large well rotating disks. These are found to have excess
angular momentum compared to that predicted by the relation. In all of these
galaxies, the Hi  disk also appears to be kinematically distinct from the
stellar disk, indicating that it represents recently acquired material.
Overall, our observations leads us to suggest that the kinematics of the
baryonic component of gas rich galaxies is more affected by cold flows and
late stage mergers than by the spin parameter of the ‘parent’ dark matter halo
as suggested by recent numerical simulations. We also find that the $\rm
j_{bar}-\rm M_{bar}$ relation is better fit by a broken power law than a
single power law, indicating that there is some curvature in this relation. We
suggest that the change in slope of the $\rm j_{bar}-\rm M_{bar}$ relation
arises from the changing gas to stellar mass ratio with mass as well as the
different slopes of the $\rm j_{s}-\rm M_{s}$ and $\rm j_{g}-\rm M_{g}$
relations.

## Acknowledgements

This paper is based in part on observations taken with the GMRT. We thank the
staff of the GMRT who made these observations possible. The GMRT is run by the
National Centre for Radio Astrophysics of the Tata Institute of Fundamental
Research. This work was supported by the DAE grant 12-R&D-TFR-5.02-0700. MV
acknowledges support by the Netherlands Foundation for Scientific Research
(NWO) through VICI grant 016.130.338 and thanks the NCRA staff for their
hospitality. We are grateful for insightful comments from one of the anonymous
reviewers that helped improve this paper.

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##  Appendix A Data

In this appendix, we present the data and the kinematic analysis of 2 Hi–
bearing ultra diffuse galaxies and 8 early–type galaxies. In Fig. 6 \- 15, we
show (i) the integrated Hi intensity map contours overlaid on the optical
image, (ii) Position-velocity diagram taken along the major axis of the galaxy
with the rotation curves overlaid on them. The dashed lines indicate the
systemic velocity and kinematic center. The overplotted violet triangles
represent the rotation curve derived by ’FAT’ (iii) the intensity weighted
first moment of the galaxy, and (iv) velocity field of the best fitting FAT
model for all the HUDs and early-type galaxies.

Figure 6: Hi data and kinematics for the galaxy AGC121790. Top left: Hi
distribution of the galaxy overlaid on the SDSS g–band data,Top right:
Position velocity diagram (outer contour corresponds to 2$\sigma$) with the
rotation curves overlaid on them. Bottom left: velocity field of data, and
Bottom right:velocity field of the best fitting fat model. velocity contours
run from 2910 to 2946 km s-1 with a spacing of 6 km s-1. Figure 7: Hi data and
kinematics for the galaxy AGC749401. Top left: Hi distribution of the galaxy
overlaid on the SDSS g–band data,Top right: Position velocity diagram (outer
contour corresponds to 2$\sigma$)with the rotation curves overlaid on them.
Bottom left: velocity field of data, and Bottom right:velocity field of the
best fitting fat model. velocity contours run from 2712 to 2754 km s-1 with a
spacing of 6 km s-1.  Figure 8: Hi data and kinematics for the galaxy NGC3626.
Top left: Hi distribution of the galaxy overlaid on the SDSS g–band data,Top
right: Position velocity diagram (outer contour corresponds to 2$\sigma$) with
the rotation curves overlaid on them. Bottom left: velocity field of data, and
Bottom right:velocity field of the best fitting fat model. Velocity contours
run from 1300 to 1660 km s-1 with a spacing of 20 km s-1. Figure 9: Hi data
and kinematics for the galaxy NGC4262. Top left: Hi distribution of the galaxy
overlaid on the SDSS g–band data,Top right: Position velocity diagram (outer
contour corresponds to 2$\sigma$) with the rotation curves overlaid on them.
Bottom left: velocity field of data, and Bottom right:velocity field of the
best fitting fat model. Velocity contours run from 1145 to 1545 km s-1 with a
spacing of 25 km s-1. Figure 10: Hi data and kinematics for the galaxy
NGC4278. Top left: Hi distribution of the galaxy overlaid on the SDSS g–band
data,Top right: Position velocity diagram (outer contour corresponds to
2$\sigma$) with the rotation curves overlaid on them. Bottom left: velocity
field of data, and Bottom right:velocity field of the best fitting fat model.
Velocity contours run from 425 to 800 km s-1 with a spacing of 25 km s-1.
Figure 11: Hi data and kinematics for the galaxy NGC5582. Top left: Hi
distribution of the galaxy overlaid on the SDSS g–band data,Top right:
Position velocity diagram (outer contour corresponds to 2$\sigma$) with the
rotation curves overlaid on them. Bottom left: velocity field of data, and
Bottom right:velocity field of the best fitting fat model. Velocity contours
run from 1225 to 1600 km s-1 with a spacing of 25 km s-1.  Figure 12: Hi data
and kinematics for the galaxy NGC6798. Top left: Hi distribution of the galaxy
overlaid on the SDSS g–band data,Top right: Position velocity diagram (outer
contour corresponds to 2$\sigma$) with the rotation curves overlaid on them.
Bottom left: velocity field of data, and Bottom right:velocity field of the
best fitting fat model. Velocity contours run from 2200 to 2500 km s-1 with a
spacing of 15 km s-1. Figure 13: Hi data and kinematics for the galaxy
NGC3941. Top left: Hi distribution of the galaxy overlaid on the SDSS g–band
data,Top right: Position velocity diagram (outer contour corresponds to
2$\sigma$) with the rotation curves overlaid on them. Bottom left: velocity
field of data, and Bottom right:velocity field of the best fitting fat model.
Velocity contours run from 800 to 1050 km s-1 with a spacing of 25 km s-1.
Figure 14: Hi data and kinematics for the galaxy UGC6176. Top left: Hi
distribution of the galaxy overlaid on the SDSS g–band data,Top right:
Position velocity diagram (outer contour corresponds to 2$\sigma$) with the
rotation curves overlaid on them. Bottom left: velocity field of data, and
Bottom right:velocity field of the best fitting fat model. Velocity contours
run from 2560 to 2800 km s-1 with a spacing of 15 km s-1.  Figure 15: Hi data
and kinematics for the galaxy UGC9519. Top left: Hi distribution of the galaxy
overlaid on the SDSS g–band data,Top right: Position velocity diagram (outer
contour corresponds to 2$\sigma$) with the rotation curves overlaid on them.
Bottom left: velocity field of data, and Bottom right:velocity field of the
best fitting fat model. Velocity contours run from 1560 to 1680 km s-1 with a
spacing of 20 km s-1.

Figure 16: Rotation curves of ultra-diffuse galaxies, AGC121790 and AGC749401.

Figure 17: Rotation curves of early–type galaxies. Error bars on NGC4278 are taken to be the velocity resolution of the data cube (13 km s-1, which is greater than median of FAT errors of all galaxies) since FAT error bars were unreliable. Table 1: Measured values of specific angular momentum and mass (stars, gas, and baryons) for ultra–diffuse galaxies in this study  Galaxy |  dist |  Ms |  Mg |  Mbar |  js |  jg |  jbar
---|---|---|---|---|---|---|---
|  (Mpc) |  (log10 M⊙) |  (log10 M⊙) |  (log10 M⊙) |  (log10 kpc km/s) |  (log10 kpc km/s) |  (log10 kpc km/s)
AGC749401 |  41.8 |  7.31 $\pm$ 0.131 |  8.66 $\pm$ 0.137 |  8.68 $\pm$ 0.131 |  2.12 $\pm$ 0.110 |  2.39 $\pm$ 0.093 |  2.39 $\pm$ 0.092
AGC121790 |  37.5 |  6.40 $\pm$ 0.131 |  8.89 $\pm$ 0.137 |  8.89 $\pm$ 0.136 |  2.00 $\pm$ 0.081 |  2.46 $\pm$ 0.074 |  2.46 $\pm$ 0.075

Figure 18: Hi surface brightness as a function of radius for the ultra-diffuse galaxies AGC121790, and AGC749401 respectively. Table 2: Measured values of specific angular momentum and mass (stars, gas, and baryons) for UMa spiral galaxies in this study Galaxy |  dist |  Ms |  Mg |  Mbar |  js |  jg |  jbar
---|---|---|---|---|---|---|---
|  (Mpc) |  (log10 M⊙) |  (log10 M⊙) |  (log10 M⊙) |  (log10 kpc km/s) |  (log10 kpc km/s) |  (log10 kpc km/s)
NGC3877 |  15.27 |  10.6 $\pm$ 0.170 |  9.14 $\pm$ 0.137 |  10.62 $\pm$ 0.164 |  2.87 $\pm$ 0.065 |  3.04 $\pm$ 0.142 |  2.88 $\pm$ 0.066
NGC3917 |  17.14 |  10.13 $\pm$ 0.170 |  9.34 $\pm$ 0.137 |  10.19 $\pm$ 0.147 |  2.75 $\pm$ 0.065 |  3.07 $\pm$ 0.169 |  2.81 $\pm$ 0.077
NGC3953 |  18.82 |  10.73 $\pm$ 0.170 |  9.62 $\pm$ 0.137 |  10.76 $\pm$ 0.158 |  3.14 $\pm$ 0.067 |  3.46 $\pm$ 0.120 |  3.17 $\pm$ 0.068
NGC3972 |  20.04 |  10.10 $\pm$ 0.170 |  9.31 $\pm$ 0.137 |  10.16 $\pm$ 0.147 |  2.70 $\pm$ 0.066 |  2.98 $\pm$ 0.132 |  2.75 $\pm$ 0.071
NGC3992 |  27.04 |  10.91 $\pm$ 0.170 |  10.23 $\pm$ 0.137 |  11.0 $\pm$ 0.143 |  3.36 $\pm$ 0.068 |  3.84 $\pm$ 0.068 |  3.49 $\pm$ 0.068
NGC4013 |  19.23 |  10.81 $\pm$ 0.170 |  9.65 $\pm$ 0.137 |  10.84 $\pm$ 0.159 |  2.93 $\pm$ 0.065 |  3.42 $\pm$ 0.065 |  2.99 $\pm$ 0.065
NGC4100 |  20.04 |  10.62 $\pm$ 0.170 |  9.74 $\pm$ 0.137 |  10.67 $\pm$ 0.151 |  3.01 $\pm$ 0.066 |  3.39 $\pm$ 0.156 |  3.07 $\pm$ 0.074
NGC4157 |  16.29 |  10.87 $\pm$ 0.170 |  9.96 $\pm$ 0.137 |  10.92 $\pm$ 0.152 |  2.92 $\pm$ 0.065 |  3.40 $\pm$ 0.097 |  3.01 $\pm$ 0.068
NGC4183 |  17.21 |  10.12 $\pm$ 0.170 |  9.65 $\pm$ 0.137 |  10.25 $\pm$ 0.132 |  2.76 $\pm$ 0.066 |  3.06 $\pm$ 0.137 |  2.85 $\pm$ 0.082
NGC4217 |  18.19 |  10.75 $\pm$ 0.170 |  9.53 $\pm$ 0.137 |  10.78 $\pm$ 0.161 |  2.97 $\pm$ 0.065 |  3.27 $\pm$ 0.114 |  2.99 $\pm$ 0.066
UGC6399 |  22.08 |  9.48 $\pm$ 0.171 |  9.19 $\pm$ 0.137 |  9.66 $\pm$ 0.122 |  2.54 $\pm$ 0.069 |  2.79 $\pm$ 0.141 |  2.64 $\pm$ 0.091
UGC6446 |  18.03 |  8.83 $\pm$ 0.172 |  9.62 $\pm$ 0.137 |  9.69 $\pm$ 0.121 |  2.29 $\pm$ 0.088 |  2.88 $\pm$ 0.100 |  2.83 $\pm$ 0.112
UGC6667 |  18.45 |  9.80 $\pm$ 0.170 |  9.06 $\pm$ 0.137 |  9.87 $\pm$ 0.145 |  2.67 $\pm$ 0.065 |  2.70 $\pm$ 0.125 |  2.67 $\pm$ 0.068
UGC6917 |  20.41 |  9.42 $\pm$ 0.171 |  9.54 $\pm$ 0.137 |  9.78 $\pm$ 0.107 |  2.67 $\pm$ 0.073 |  2.99 $\pm$ 0.117 |  2.88 $\pm$ 0.101
UGC6983 |  21.28 |  9.29 $\pm$ 0.170 |  9.74 $\pm$ 0.137 |  9.87 $\pm$ 0.111 |  2.62 $\pm$ 0.071 |  3.08 $\pm$ 0.095 |  3.00 $\pm$ 0.091
UGC7089 |  11.50 |  9.06 $\pm$ 0.171 |  8.80 $\pm$ 0.137 |  9.25 $\pm$ 0.120 |  2.19 $\pm$ 0.067 |  2.54 $\pm$ 0.066 |  2.34 $\pm$ 0.067
Table 3: Measured values of gas mass and gas specific angular momentum of 6 S0 and 2 elliptical galaxies in this study Galaxy |  Morp |  dist |  Mg |  jg |  Notes
---|---|---|---|---|---
|  |  (Mpc) |  (log10 M⊙) |  (log10 kpc km s-1) |
NGC3626 |  S0 |  19.5 |  9.19 $\pm$ 0.137 |  3.60 $\pm$ 0.073 |  C
NGC4262 |  S0 |  15.4 |  8.84 $\pm$ 0.137 |  3.27 $\pm$ 0.086 |  R
NGC4278 |  E |  15.6 |  8.90 $\pm$ 0.137 |  3.64 $\pm$ 0.091 |  M, L
NGC5582 |  E |  27.7 |  9.78 $\pm$ 0.137 |  3.85 $\pm$ 0.074 |  R
NGC3941 |  S0 |  11.9 |  8.86 $\pm$ 0.137 |  3.15 $\pm$ 0.073 |  C, R
NGC6798 |  S0 |  37.5 |  9.64 $\pm$ 0.137 |  3.67 $\pm$ 0.077 |  C
UGC6176 |  S0 |  40.1 |  9.21 $\pm$ 0.137 |  3.21 $\pm$ 0.068 |  W
UGC9519 |  S0 |  27.6 |  9.40 $\pm$ 0.137 |  3.02 $\pm$ 0.081 |  M
Notes on Hi morphology and kinematics: |
C = Hi counter-rotating relative to the stellar kinematics; |
L= lopsided Hi morphology; M= Hi misaligned relative to |
the stellar kinematics; R= ring; W= warp |
Table 4: Convergence factors of 16 UMa spirals for $\rm j_{bar}$and $\rm j_{g}$profiles.  Galaxy |  rbar |  rg
---|---|---
N3877 |  0.90 |  0.90
N3917 |  0.94 |  0.86
N3953 |  0.98 |  0.90
N3972 |  0.94 |  0.82
N3992 |  0.97 |  0.95
N4013 |  0.97 |  0.88
N4100 |  0.96 |  0.90
N4157 |  0.95 |  0.88
N4183 |  0.91 |  0.82
N4217 |  0.97 |  0.83
U6399 |  0.86 |  0.78
U6446 |  0.94 |  0.94
U6667 |  0.66 |  0.73
U6917 |  0.89 |  0.88
U6983 |  0.94 |  0.94
U7089 |  0.82 |  0.86
Table 5: Convergence factors of 2 ultra-diffuse galaxies for $\rm j_{bar}$and $\rm j_{g}$profiles. Galaxy |  rbar |  rg
---|---|---
AGC121790 |  0.74 |  0.74
AGC749401 |  0.61 |  0.60
Table 6: Convergence factors of 8 early-type galaxies for $\rm j_{g}$profiles.  Galaxy |  rg
---|---
NGC4262 |  0.80
NGC4278 |  0.71
NGC3626 |  0.61
NGC3941 |  0.89
NGC5582 |  0.85
NGC6798 |  0.43
UGC9519 |  0.74
UGC6176 |  0.85

Figure 19: Hi surface brightness as a function of radius for the early-type
galaxies NGC3626, NGC4262, NGC4278, NGC5582, NGC6798, NGC3941, UGC6176, and
UGC9519 respectively.

Figure 20: The cumulative jbar and jg profiles of 16 UMa spirals. We normalize
the axes to allow the comparison between the profiles. The one to one relation
is shown with a blue solid line.

Figure 21:  The cumulative jbar and jg profiles of 2 ultra-diffuse galaxies.
We normalize the axes to allow the comparison between the profiles. The one to
one relation is shown with a blue solid line. Figure 22: The cumulative jg
profiles of of 8 early-type galaxies. We normalize the axes to allow the
comparison between the profiles. The one to one relation is shown with a blue
solid line.