# The Tully-Fisher relation and the Bosma effect

Francesco Sylos Labini,1,2 Giordano De Marzo1,3 Matteo Straccamore,1,3 and
Sébastien Comerón4,5
1Centro Ricerche Enrico Fermi, I-00184, Roma, Italia
2INFN Unitá Roma 1, Dipartimento di Fisica, Universitá di Roma Sapienza,
I-00185 Roma, Italia
3Dipartimento di Fisica, Sapienza, Universitá di Roma, I-00185, Roma, Italia
4Departamento de Astrofísica, Universidad de La Laguna, E-38200, La Laguna,
Tenerife, Spain
5Instituto de Astrofísica de Canarias E-38205, La Laguna, Tenerife, Spain
E-mail<EMAIL_ADDRESS>(FSL)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We show that the rotation curves of 16 nearby disc galaxies in the THINGS
sample and the Milky Way can be described by the NFW halo model and by the
Bosma effect at approximately the same level of accuracy. The latter effect
suggests that the behavior of the rotation curve at large radii is determined
by the rescaled gas component and thus that dark matter and gas distributions
are tightly correlated. By focusing on galaxies with exponential decay in
their gas surface density, we can normalize their rotation curves to match the
exponential thin disc model at large enough radii. This normalization assumes
that the galaxy mass is estimated consistently within this model, assuming a
thin disc structure. We show that this rescaling allows us to derive a new
version of the Tully-Fisher (TF) relation, the Bosma TF relation that nicely
fit the data. In the framework of this model, the connection between the Bosma
Tully-Fisher (TF) relation and the baryonic TF relation can be established by
considering an additional empirical relation between the baryonic mass and the
total mass of the disc, as measured in the data.

###### keywords:

galaxies: general — galaxies: disc — galaxies: haloes — Galaxy: kinematics and
dynamics — galaxies: structure

††pubyear: 2023††pagerange: The Tully-Fisher relation and the Bosma
effect–LABEL:lastpage

##  1 Introduction

The rotation curves of spiral galaxies appear to be roughly flat out to the
outermost observed points, which is in contrast to the nearly Keplerian
decline expected from the luminous disc (see, e.g., Rubin et al. (1980);
Sancisi (1999) and references therein). This discrepancy has led to the
conclusion that spiral galaxies are dominated by a massive dark halo which has
a different nature from the luminous matter. The dark halo is thought to have
an almost spherical mass distribution with an isotropic velocity dispersion,
while the luminous matter forms a rotationally supported disc. This difference
in kinematic and geometrical structure of the disc and the halo is due to the
physical properties of the matter of which they are made of: dissipational
baryonic matter the former, dissipation-less and collision-less non-baryonic
matter the latter.

The halo model is the prevailing physical framework used to interpret
observations of galactic kinematics. This model is able to explain the variety
of rotation curves observed: many galaxies show a flat rotation curve, others
a rising one without a clear tendency toward a flattening, while in some cases
the rotation curve shows a declining behavior. The ability to fit different
rotation curve shapes in the standard halo model (Navarro et al., 1997) is
related to the shape of the dark matter (DM) halo, which decays with a
logarithmic slope ranging from -1 as the radius $r$ approaches zero to -3 for
$r\rightarrow\infty$. In cases where a central cusp is not present, a
generalized version of the halo model can be utilized (Di Cintio et al., 2014;
Read et al., 2016; Ou et al., 2023). This generalized model, allows for the
logarithmic slope to approach 0 as $r$ tends to zero. This modification
provides additional flexibility in capturing the rotation curve behavior,
particularly in the central regions of galaxies where deviations from the NFW
profile may be observed. These characteristics allow the halo model to
successfully fit rotation curves that exhibit various shapes. The standard
halo profile depends on two free parameters, while the generalized halo
profile requires three free parameters. It is worth noting that these
parameters have been found to exhibit strong correlations in numerical
$N$-body calculations. (see, e.g., Macciò et al. (2008); Dutton & Macciò
(2014)).

A different theoretical mass model to interpret disc galaxy kinematics,
proposed long ago (Milgrom, 1983), is based on the Modified Newtonian Dynamics
(MOND) and, interestingly, has passed a number of observational tests (see,
e.g., McGaugh et al. (2016)). This model does not require the introduction of
an additional dark component and only considers the mass of the visible
stellar and gas components. The key feature is that the gravitational force is
assumed to decline more slowly than in Newtonian dynamics, which allows for
the rotation velocity to remain constant without the need for additional mass
beyond the baryonic components.

In this paper, beyond the halo model, we consider a mass model based on
standard Newtonian dynamics. The key difference with the halo model is that we
place DM into the galactic disc. The roots of this model lie in the observed
correlation between DM and the distribution of
$\textrm{H}\scriptstyle\mathrm{I}$ gas, first noted by Bosma (1978, 1981).
This correlation refers to the observation, in a sample of nearby disc
galaxies, that the ratio of the total disc surface density (derived from
rotation curve measurements) and the gas surface density (measured from
$\textrm{H}\scriptstyle\mathrm{I}$ observations) is roughly constant beyond
about one-third of the optical radius. The correlation between DM and gas in
disc galaxies implies that rotation curves at large radii can be rescaled
versions of those derived from the $\textrm{H}\scriptstyle\mathrm{I}$ gas
distribution (Sancisi, 1999; Hoekstra et al., 2001; Hessman & Ziebart, 2011;
Swaters et al., 2012; Sylos Labini et al., 2023b). In this framework, which we
refer to as the Bosma model, at small radii, the total disc surface density is
dominated, completely or partially, by stars, while at large radii it is
dominated by a mass component obtained by rescaling the surface density of the
gas. Since the mass of the gas component is a negligible fraction of the
galaxy mass, the Bosma effect implies that the gas surface density is a proxy
for DM. This means that the distribution of DM follows that of the gas and is
confined to a disc, but is not still observed. Note that the Bosma effect does
not inherently imply that DM is exclusively confined to a disc. It is crucial
to acknowledge that, based on a given radial acceleration profile, such as the
one predicted by the Bosma effect, it is always possible to construct a
spherical distribution of matter that produces the observed acceleration.
However, for the sake of our discussion, we will proceed with the assumption
that DM is restricted to a disc. This assumption allows us to simplify the
modeling and analysis by focusing on a specific configuration where DM is
concentrated within a disc-like structure. While it may not capture all
possible scenarios, this hypothesis provides a useful starting point for
understanding the observed rotational velocities and their relation to the
distribution of DM in galaxies.

In this respect, it is worth recalling that Pfenniger et al. (1994) proposed
the idea that DM could reside in the galactic disc in the form of very cold
molecular hydrogen (i.e., H2). Extensive vast clouds of cold (i.e.,
$T\approx$10K-20K) dust and dark gas, invisible in
$\textrm{H}\scriptstyle\mathrm{I}$ and CO but detected in $\gamma$-rays, were
detected, in the solar neighborhood, by Grenier et al. (2005) who concluded
that dark gas mass in the Milky Way was comparable to the molecular one.
Later, dark gas was found also by the Planck mission (Planck Collaboration et
al., 2011) — see Casandjian et al. (2022) for an recent discussion of cold
dark gas clouds measurements. H2 in less excited and colder regions than those
seen with CO cannot be inferred yet: CO freezes on grains below 20K, so
molecules do not rotate anymore thus emitting rotational lines. Regions colder
than 20K are not rare and for this reason there is room for more undetected H2
in outer gas discs where excitation from stellar radiation decays fast:
unfortunately, a direct detection such clouds in the outer disc of galaxies
appears to be a very hard task from an observational point of view (Combes &
Pfenniger, 1997).

Irrespective of the type of DM in the galactic disc, it has been shown that
the Bosma effect can provide highly accurate rotation curve fits for several
disc galaxies, by using both the observed stellar disc and
$\textrm{H}\scriptstyle\mathrm{I}$ gas as proxies for DM (Hessman & Ziebart,
2011; Swaters et al., 2012). Recent work by Sylos Labini et al. (2023b) has
demonstrated that the rotation curve of the Milky Way can also be accurately
fit using the Bosma effect. Although the presence of correlation does not
necessarily imply causality, its detection may be related to a fundamental
property of disc galaxies — that DM is distributed on the disc, rather than in
a spherical halo around the galaxy, as currently assumed. Therefore, we
believe it is worthwhile to investigate this effect in more detail than has
been previously done.

In this paper, we use 16 galaxies from the The
$\textrm{H}\scriptstyle\mathrm{I}$ Nearby Galaxy Survey (THINGS) data-set
(Walter et al., 2008) and the Milky Way to further examine the Bosma effect.
This sub-sample of the THINGS galaxies is the same used by Hessman & Ziebart
(2011) (with the exception of NGC 3031 that shows signs of strong tidal
interactions). However, unlike previous studies, for a large fraction of the
galaxies in the THINGS sample we use the full gas surface density and mass,
not just the $\textrm{H}\scriptstyle\mathrm{I}$ component. In particular, we
use the measurements of CO inferred H2 by Schruba et al. (2011), which include
11 galaxies in our sample. In addition, for 12 galaxies we use the more
accurate stellar surface density profiles and masses from the stellar density
maps based on the recipes in Querejeta et al. (2015) applied to the Spitzer
Survey of Stellar Structure in Galaxies (S4G — Sheth et al. (2010); Muñoz-
Mateos et al. (2015)) Pipeline 5 dust-corrected 3.5 $\mu$m surface brightness
maps, instead of previous measurements based on the 2MASS survey (De Blok et
al., 2008; Hessman & Ziebart, 2011). To quantitatively evaluate our results,
we fit the Bosma mass model and compare it to both the Navarro-Frenk-White
(NFW) halo mass model (Navarro et al., 1997) and its generalized version (Ou
et al., 2023)

By assuming, consistently with the Bosma model, that the toal mass is confined
on the galactic disk, we shown that rotational curves shown a universal shape
when properly normalized. In particular, when the radius is expressed in units
of the exponential scale-length $R_{\text{d}}$ of the gas (i.e.,
$\textrm{H}\scriptstyle\mathrm{I}$) density profile and the velocity is in
units of the typical system velocity that depends on the total mass and on
$R_{\text{d}}$, we find that the rotation curves of the different galaxies
approximately follow the exponential thin disc approximation (Freeman, 1970).
Given this result, we are able to show that a simple analytical explanation of
the Tully-Fisher relation can be formulated. This is then compared to
observations and related to the baryonic Tully-Fisher (BTF) relation (McGaugh
et al., 2000; McGaugh, 2005).

The structure of the paper is as follows: in Sect.2, the properties of the
sample galaxies are discussed, including their rotation curves, star and gas
surface brightness, and masses. Then in Sect.3, we discuss the fits of the
Bosma and NFW models, and their modifications to the rotation curves (the
results of fitting the individual galaxies are presented in the Appendix),
whereas in Sect.4, the computation of the average rotation curve among the
galaxies in our sample and its comparison to the the exponential thin disc
model is discussed. Furthermore, we show that it is possible to derive a
modified version of the Tully-Fisher (TF) relation in the framework of the
Bosma model and we discusss its relation to the classical TF and BTF relations
(Sect.5). Furthermore we examine the correlation between the observed
acceleration and the acceleration predicted by baryonic matter. We discuss
that this correlation can be naturally understood within the framework of the
Bosma model. Finally in Sect.6, the main conclusions are drawn.

##  2 The sample

The THINGS survey (Walter et al., 2008) is a high spectral and spatial
resolution survey of the $\textrm{H}\scriptstyle\mathrm{I}$ emission of nearby
galaxies that was obtained using the NRAO Very Large Array. The sample has a
velocity resolution of 5.2 km/s or better and an angular resolution of 6"
corresponding to a linear resolution of 0.1/0.5 kpc for galaxies at a distance
of 2/15 Mpc (Walter et al., 2008). This high resolution makes the THINGS
survey a unique sample for the study of galaxy kinematics. The astrophysical
parameters of the galaxies studied in this work can be found in Walter et al.
(2008). We considered the same sample of Hessman & Ziebart (2011) but with the
exception of NGC3031 because its $\textrm{H}\scriptstyle\mathrm{I}$
distribution is affected by strong tidal interactions with neighboring
galaxies and its surface density does not show a simple exponential decay: all
other galaxies show a gas surface brightness exponentially decaying.

The stellar profiles and masses from the stellar density maps were produced
from the dust-corrected 3.6$\mu$m maps in the Pipeline 5 of the S4G (Sheth et
al., 2010; Muñoz-Mateos et al., 2015) using the recipe provided in Querejeta
et al. (2015). We obtained the stellar surface density profile and mass (see
Tab. 1) for galaxies included in this sample. For galaxies not included in
this sample, we used the mass from Hessman & Ziebart (2011) and the stellar
surface density profile from the 2MASS data (De Blok et al., 2008). Note that
in some cases, there are differences between the values of the stellar mass we
used and those reported in Hessman & Ziebart (2011).

The $\textrm{H}\scriptstyle\mathrm{I}$ mass and surface density profile were
estimated from the THINGS data set (Walter et al., 2008). Note that the values
of $\textrm{H}\scriptstyle\mathrm{I}$ masses we report in Tab.1 are not
multiplied by 1.39 to correct for He and heavy elements, and thus coincide
with those listed in Tab. 5 of Walter et al. (2008). The H2 mass and surface
density profile are available only for the galaxies included in the sample of
Schruba et al. (2011). For all galaxies except NGC 2366, NGC 3621, NGC 7793,
DDO 154, and IC 2574 for which there are no data, we use the gas surface
density profile defined as the sum of the $\textrm{H}\scriptstyle\mathrm{I}$
and H2 profiles. Note that the H2 distribution may affect the fitting with
models at small radii, but it does not change the behavior at large radii,
which is the regime that matters for the Bosma effect.

The fluxes derived from Pipeline 5 have uncertainties of the order of 10% (see
Appendix A of Querejeta et al. (2015)). As mentioned above, we convert the
flux to mass following the recipe in Querejeta et al. (2015), which has an
accuracy of 0.1 dex. For this reason, quadratically summing these two
uncertainties, we find an error of 30% for the stellar mass. We consider the
same error for the mass of the gas.

In general, the stellar surface brightness of a galaxy may show a simple or
double exponential decay. The latter occurs when the galaxy has a central
bulge. However, the consideration of the bulge implies the inclusion of a new
free parameter in the model. As the bulge is relevant only at very small
radii, we exclude the very inner disc from the fit and thus we avoid adding
another free parameter. The rationale for this choice is that the Bosma effect
primarily pertains to the outer regions of a galaxy. Therefore, considering
the scope of the present work, it is unnecessary to intricately model the
behavior of the inner disc.

Concerning the gas surface brightness profile in the inner disc, this is
generally not exponential. This is true especially for the
$\textrm{H}\scriptstyle\mathrm{I}$ profile, which is roughly flat at small
radii and then has an exponential tail with a characteristic scale-length,
$R_{\text{g}}$, reported in Tab.1. In many cases, the
$\textrm{H}\scriptstyle\mathrm{I}$+H2 profile displays exponential decay from
the very inner regions of the disc. A detailed description of the stellar and
gas surface densities of individual galaxies is reported in the Appendix.

##  3 Mass models

In Section 3.1, we provide a detailed description of the Bosma effect,
focusing on its relevance to the outer regions of a galaxy. Following that, in
Section 3.2, we provide a brief overview of the key elements of the halo
model. We then proceed to discuss the results of the fits in Section 3.3,
specifically considering the four models that we have tested, which include
two halo models and two disc models. For a more comprehensive analysis of each
galaxy, a detailed discussion is provided in the Appendix.

###  3.1 The Bosma effect and the dark matter disc model

The correlation referred to as the "Bosma effect," first noticed by Bosma
(1978, 1981), consists in the fact that the ratio of atomic hydrogen
($\textrm{H}\scriptstyle\mathrm{I}$) gas to total matter surface density as
inferred from the rotation curves appears to be relatively constant in disc
galaxies. This ratio, the Bosma factor, can be defined as the total matter to
$\textrm{H}\scriptstyle\mathrm{I}$ surface densities averaged over the disc,
i.e.,

$\frac{\Sigma_{\text{b}}(R)}{\Sigma_{\text{g}}(R)}\sim\frac{v_{\text{c}}^{2}(R)}{v_{\text{g}}^{2}(R)}\sim\mbox{const.}\gg
1\;$ (1)

where $\Sigma_{\text{b}}$ is the total mass surface density, obtained from the
observed rotation curve $v_{\text{c}}$, $v_{\text{g}}$ is the rotation curve
expected from the observed gas surface density $\Sigma_{\text{g}}$ and $R$ is
the galacto-centric distance. The Bosma factor in Eq. 1, being generally much
larger than unity, can be interpreted as the constant of proportionality
between the centripetal effects of gas and DM, or equivalently as an effective
mass-to-light ratio contribution above that normally attributable to standard
gas components.

This factor has been measured in about hundred galaxies of different Hubble
types from massive early spirals to late dwarf irregulars (Sancisi, 1983;
Carignan, 1985; Carignan & Beaulieu, 1989; Carignan & Puche, 1990; Jobin &
Carignan, 1990; Carignan et al., 1990; Puche et al., 1990; Sancisi, 1999;
Hoekstra et al., 2001). The value of the Bosma factor spans about one order of
magnitude, i.e., ranges between 5 and 50 and more massive galaxies tended to
have larger values.

Hoekstra et al. (2001) analyzed a sample of 24 spiral galaxies with high-
quality rotation curves. They found that, with a few exceptions, the rotation
curves generated by scaling up the centripetal contribution of the
$\textrm{H}\scriptstyle\mathrm{I}$ gas by a constant factor of approximately
10 and not including a spherical DM halo, were comparable in accuracy to those
generated by the NFW halo model. However, they also claimed that it was not
possible to confirm that a real coupling between
$\textrm{H}\scriptstyle\mathrm{I}$ and DM exists due to several imperfections
in their correlation.

Then, Hessman & Ziebart (2011) made a quantitative comparison between the
results of fits with the Bosma effect and of the NFW models. They have pointed
out that the rotation curves of disc galaxies obtained by relying solely on
$\textrm{H}\scriptstyle\mathrm{I}$ scaling are unsatisfactory because the
concentration of $\textrm{H}\scriptstyle\mathrm{I}$ decreases in the central
regions of galaxies. This is because the high-density regions of stars consume
the $\textrm{H}\scriptstyle\mathrm{I}$ gas and convert it into stars or
molecules. To account for this physical effect, the authors suggest using the
stellar discs as proxies, or including both neutral and ionized gas, instead
of only the neutral $\textrm{H}\scriptstyle\mathrm{I}$ component, as the total
gas component is believed to have a closer correlation with the distribution
of DM. Hessman & Ziebart (2011) confirmed the correlation between DM and
$\textrm{H}\scriptstyle\mathrm{I}$ distribution using 17 galaxies from the
THINGS dataset (Walter et al., 2008) and rebutted several arguments against
the effect by Hoekstra et al. (2001). They found fits of similar or even
better quality than those obtained by the standard NFW halo model. This is
indeed the case especially when the additional correlation between the two
parameters of the NFW halo model found in $N$-body simulations (Macciò et al.,
2008) in taken into account.

Swaters et al. (2012) further studied a sample of 43 disc galaxies, ranging
from late-type dwarf galaxies to early-type spirals, by fitting the rotation
curves with mass models based on scaling up the stellar and
$\textrm{H}\scriptstyle\mathrm{I}$ discs. They found that such scaling models
fit the observed rotation curves well in the vast majority of cases, even
though the models have only two or three free parameters (depending on whether
the galaxy has a bulge or not). They also found that these models reproduce
some of the detailed small-scale features of rotation curves such as the
"bumps" and "wiggles", a feature that point toward a real coupling between
$\textrm{H}\scriptstyle\mathrm{I}$ and DM.

The dark matter disc (DMD) model assumes, consistently with the Bosma effect,
that the DM is constrained on the disc and has the same distribution as the
gas, thus the same scale-length of the exponential decay. The model can be
parametrized as

$v_{\text{c}}^{2}=\Upsilon_{\text{s}}v_{\text{s}}^{2}+\Upsilon_{\text{g}}v_{\text{g}}^{2}$
(2)

where $\Upsilon_{\text{s}}$ and $\Upsilon_{\text{g}}$111Note that we used the
pure $\textrm{H}\scriptstyle\mathrm{I}$ mass not multiplied by 1.39: this
factor is generally considered to correct for He and heavy elements. In our
fit this corresponds to a simple rescaling of $\Upsilon_{\text{g}}$. are two
free parameters, while $v_{\text{s}}$ ($v_{\text{g}}$) is the rotation
velocity that the stellar (gas) mass component would induce on a test particle
in the plane of the galaxy if it were placed in isolation without any external
influences. Note that Eq. 2 corresponds to assume that the proportionality
between gas and DM is independent of the galactocentric radius: in view of the
very different physical conditions in the inner disc with respect to those of
the outermost regions of the galaxy we believe this is a strong approximation.
The key feature of the Bosma effect is the rescaling of the
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness at very large radii,
where the stellar surface density becomes negligible. In addition, as
mentioned above, in Eq. 2 we have neglected the bulge in the fits (both in the
case of the Bosma effect and the NFW halo — see below) so to avoid introducing
an additional free parameter in the models. Indeed, the very inner disc of the
galaxy does not represent the main region where the Bosma effect was found.
For this reason, the range of radii where the bulge is present can be
constrained from observations of the stellar surface density and excluded from
fits.

From Eq. 2 it follows that the galaxy mass is

$M_{\text{dmd}}=\Upsilon_{\text{s}}M_{\text{s}}+\Upsilon_{\text{g}}M_{\text{g}}\;,$
(3)

where $M_{\text{s}}$ is the mass of the stellar component and $M_{\text{g}}$
that of the gas component. The mass of the bulge is typically 10% that of the
disc and it is thus inside the our error bars which are of $\sim 30\%$ (see
discussion in Sect.2). In what follows we compute $v_{\text{s}}$ (or
$v_{\text{g}}$) from simulations as an analytical computation of the
centrifugal force for a disc is not straightforward (Feng & Gallo, 2011): we
thus generate a thin disc with the observed surface brightness of the stellar
(or gas) component and same mass and we numerically compute the centripetal
acceleration.

###  3.2 The halo model

The Navarro Frenk and White (NFW) mass model can be written as

$v_{\text{c}}^{2}=v_{\text{s}}^{2}+v_{\text{g}}^{2}+v_{h}^{2}$ (4)

where $v_{h}$ corresponds to the equilibrium velocity in a NFW halo profile
described by (Navarro et al., 1997)

$\rho(r)=\frac{\rho_{0}}{\frac{r}{R_{\text{s}}}\left(1+\frac{r}{R_{\text{s}}}\right)^{2}}\;,$
(5)

where $R_{\text{s}}$ and $\rho_{0}$ are two free parameters. Results can be
expressed in terms of the virial radius $R_{\text{vir}}=cR_{\text{s}}$ and
halo virial mass $M_{\text{vir}}=M(R_{\text{vir}})$, where $c$ is the
concentration parameter (Navarro et al., 1997).

Normally, NFW profiles are characterized by two model parameters
($\rho_{0},R_{\text{s}}$ in Eq. 5), and mass model fits usually consider both
parameters as free (see, e.g., De Blok et al. (2008); Eilers et al. (2019)).
However, it was shown by Macciò et al. (2008) that the $N$-body calculations
which resulted in the NFW profile model clearly show that the concentration
parameter is not an independent parameter but is in fact strongly correlated
with $V_{\text{vir}}$, defined as the rotational velocity at $R_{\text{vir}}$.
Quantitatively they found the following empirical relation

$c_{\text{vir}}\approx 7.80\left(\frac{V_{\text{vir}}}{100\;\;\mbox{km
s}^{-1}}\right)^{-0.294}$ (6)

(derived from Eq. (10) of Macciò et al. (2008) — see Dutton & Macciò (2014)
for a slightly different phenomenological fit). Including this intrinsic
correlation reduces the number of fit parameters by one. In what follows, we
consider the two parameter fit and report the value of one of the two
parameters (we choose the concentration parameter) if the additional
correlation of Eq. 6 is taken into account. This is because a potential
systematic effect of adiabatic compression was not taken into account in the
mass models (Gnedin et al., 2004; Sellwood & McGaugh, 2005) and this process
may help reconcile the parameters found with the Eq. 6 as it has the effect of
raising the effective concentration of the resulting halo above that of the
primordial initial condition to which the mass-concentration relation applies
(McGaugh, 2016). It is worth noticing that when the additional correlation
given by Eq. 6 is considered, the quality of the fit generally worsen as it
was noticed by Hessman & Ziebart (2011).

To establish a benchmark for our analysis, we have adopted a core DM model
that has a compact core thus avoiding the cusp for $r\rightarrow 0$. This is
known to provide a better fit to observed rotation curves compared to cusp
profiles like the NFW profile. In the literature, several functional behaviors
have been considered to account for the interactions between baryons and DM,
such as the models proposed by Di Cintio et al. (2014); Read et al. (2016). In
this study, we utilize the generalized NFW (gNFW) profile, which has been
recently employed for the Milky Way by Ou et al. (2023). Unlike the standard
NFW profile, which exhibits a divergence towards smaller radii, the gNFW
profile introduces an additional free parameter that modulates the inner and
outer asymptotic power law slopes of the NFW profile. This additional
parameter allows for a fully cored density profile, enabling the gNFW profile
to possess a power law slope down to $-3$ at radii larger than the scale
radius. By introducing this additional free parameter, the gNFW profile offers
a more straightforward and manageable framework while still capturing
important features of the density distribution in DM halos. For our analysis,
we compute the circular velocity curve based on the density profile described
by the following equation:

$\rho(r)=\frac{\rho_{0}}{\left(\frac{r}{R_{\text{s}}}\right)^{\beta}\left(1+\frac{r}{R_{\text{s}}}\right)^{3-\beta}}\;,$
(7)

where $\beta$ is the additional free parameter. Note that $\beta=1$ gives the
usual NFW profile of Eq. 5.

###  3.3 Results of the fits

Rotation curves of the galaxies in our sample were published by De Blok et al.
(2008); these same rotation curves were measured also by Sylos Labini et al.
(2023a) by using a different method. They have three different behaviors: NGC
925, NGC 2976 and IC 2574 show a growing rotation curve without a clear
tendency for a flattening; NGC 2366, NGC 2403, NGC 3198, NGC 3521, NGC 3621,
NGC 6946, NGC 7331, DDO 154 show a well defined flattening after the inner
velocity increase; finally NGC 2841, NGC 2903, NGC 4736, NGC 5055 and NGC 7793
show a decreasing rotation curve. For this last group of galaxies the decrease
between the maximum velocity and the outermost point corresponds to a change
between 15% (NGC 2841, NGC 2903) and 40% (NGC 4736).

Both the numerical fits of the Bosma model, where we consider
$\Upsilon_{\text{s}}$ and $\Upsilon_{\text{g}}$ as free parameters (see Eq.
2), and the fits with the standard NFW mass model, are similar to those of
Hessman & Ziebart (2011). However, there are some discrepancies deriving from
the high sensitivity to to the details of the stellar and gas mass and surface
density profiles, which, as mentioned above, are generally different. In
addition, in cases where the stellar surface density displays a double
exponential decay corresponding to the presence of a bulge in addition to the
disc, we have excluded the very inner disc from the fit. Given the sensitivity
of the fit to the details of the range of radii considered, this is another
reason for obtaining results that are different from Hessman & Ziebart (2011)
for the precise values of the parameters, but, of course, not for the general
trend.

An intriguing test for the DMD model involves treating the characteristic
length-scale of the exponential decay of the DM disk ($R_{\text{d}}$) as a
free parameter in the fitting process, rather then being equal to that of the
gas surface density profile. This approach allows for an exploration of
whether the value of $R_{\text{d}}$ obtained from the gaseous disk provides
the best match to the data. By doing so, it relaxes the strong assumption that
gas and dark matter are tightly correlated in their distribution. To this end,
we conducted a DMD analysis (that we named modified DMD model, or mDMD) by
considering $\Upsilon_{\text{g}}$ and $R_{\text{d}}$ as the free parameters,
while $\Upsilon_{\text{s}}$ is fixed and equal to one. The results of this
analysis are included in Table 2. Upon examining the new DMD fits, we observe
that, in general, the quality of the fit is relatively poorer compared to the
fits where the free parameters were $\Upsilon_{\text{g}}$ and
$\Upsilon_{\text{s}}$. This is evident from the higher values of the
$\chi^{2}$ statistic obtained in the new DMD fits. This result is attributed
to the poor fit in the inner regions, where the stellar component dominates,
rather than the outer disc.

However, there are a few cases where the fits are comparable, or in some
instances, even better, particularly when $\Upsilon_{\text{s}}$ is close to
unity. Furthermore, we have determined that there is a linear correlation
between $\Upsilon_{\text{g}}$ and $R_{d}$, with a correlation coefficient of
approximately $r\approx 0.7$. This correlation implies that there is some
degree of interdependence between the gas mass-to-light ratio
$\Upsilon_{\text{g}}$ and the scale length of the disc $R_{d}$.

Results of fit to the DMD, mDMD, NFW and gNFW models for individual galaxies
are reported in the Appendix (see, respectively, Tab.1-Tab.4).

In summary, our findings suggest that the DMD model fits generally result in
reduced $\chi^{2}$ values comparable to those obtained by the NFW fits. As
expected, the gNFW profiles tend to provide improved fits compared to the NFW
profiles due to the increased flexibility of the density profile shape. This
improvement is primarily attributed to the additional free parameter in the
gNFW model, enabling better alignment with the observed data. However, it is
important to note that the primary distinction between the two profiles
emerges at small radii, which are often excluded from the fitting procedure to
avoid complications arising from the inclusion of a bulge component.
Consequently, the exclusion of the inner disc region, combined with the
avoidance of an additional free parameter associated with the bulge, may
contribute to the comparable performance of the NFW fits in many cases.

Furthermore, it is worth noting that the suboptimal reduced $\chi^{2}$ values
observed in several cases are often a consequence of irregular features, such
as bumps and wiggles, present in the rotation curves. These non-smooth
characteristics pose challenges to the fitting with a smooth model and can
contribute to the overall discrepancies observed between the model predictions
and the observed data.

Of course, this same problem affect the fits with the DMD model. The Bosma
effect corresponds to the fact that the distribution of
$\textrm{H}\scriptstyle\mathrm{I}$ is a proxy of the global matter surface
density in galaxies. Hessman & Ziebart (2011) have shown for this same sample
of galaxies that it can be a noisy tracer. A close comparison of the gas and
total rotation curves has revealed that in some galaxies, the observed "bumps"
and "wiggles" in the rotation curve can be explained by the Bosma model, where
the rotation curve is more closely tied to the gas density. However, in other
galaxies, there is not such a close correlation. This is because the wiggles
can be created by spiral arm structures, which can convert
$\textrm{H}\scriptstyle\mathrm{I}$ to other molecules, rendering it an
inadequate tracer.

Regarding the DMD model, our analysis suggests that the mass-to-light factor
for the stellar component is peaked at approximately
$\Upsilon_{\text{s}}\approx 1$, indicating a close correspondence between the
stellar mass and the observed light. In a few cases, the mass-to-light factor
exceeds unity, suggesting a higher stellar mass relative to the observed
light. This variation in $\Upsilon_{\text{s}}$ highlights the diversity in the
stellar populations and properties among the analyzed galaxies. Overall, these
observations provide valuable insights into the fitting performance of the DMD
and gNFW models, as well as the characteristics of the stellar mass-to-light
ratios in the analyzed galaxies.

For the NFW and gNFW cases, we report (see Tabs.3-4) the values of the best-
fit concentration parameter $c$, obtained from the fit with two free
parameters. In addition, we also report the value of $c_{\text{vir}}$ computed
from Eq. 6 by inserting the value of the rotational velocity at
$R_{\text{vir}}$ that is obtained from fit with to free parameters. It is
noticed that in all cases these are not consistent with the predictions of
$\Lambda$CDM, which provides the well-defined mass-concentration relation Eq.
6. We thus find that the Galactic halo has a concentration parameter which is
different, and generally higher, than theoretical expectations based on
cosmological simulations (Macciò et al., 2008). High values of the
concentration parameter have also been found by other studies (Bovy et al.,
2012; Deason et al., 2012; Kafle et al., 2014; McMillan, 2017; Monari et al.,
2018; Lin & Li, 2019; Eilers et al., 2019): these are in tension with
theoretical expectations based on cosmological simulations.

##  4 The exponential thin disc model approximation for nearby disc galaxies

The rotation curves of different galaxies can vary significantly, not only due
to differences in their intrinsic masses and sizes but also because of
variations in their shapes. Here, we present a normalization method that
enables us to calculate their average behavior that can be further compared
with a theoretical model. The two main ingredients are the following.

(i) The first is that the mass surface density profile can be approximated by
an exponential function. Indeed, the Bosma surface density profile is

$\Sigma_{\text{b}}(R)=\Upsilon_{\text{g}}\Sigma_{\text{g}}+\Upsilon_{\text{s}}\Sigma_{\text{s}}\;.$
(8)

At sufficiently large radii $\Sigma_{\text{b}}(R)$ is proportional to the gas
surface brightness, i.e.,
$\Sigma_{\text{b}}(r)\approx\Upsilon_{\text{g}}\Sigma_{\text{g}}(r)$. Thus if
$\Sigma_{\text{g}}(r)\sim\exp(-R/R_{\text{g}})$, the full mass density surface
brightness also decreases exponentially with a characteristic scale of
$R_{\text{g}}$ and at large enough radii we have

$\Sigma_{\text{b}}(R)\sim\exp\left(-\frac{R}{R_{\text{d}}}\right)$ (9)

where, in general if the rotation curve is extended enough beyond the optical
radius, we have $R_{\text{d}}\approx R_{\text{g}}$.

Figure 1: The average rotation curve (Av.) of the galaxies in our sample is
obtained by normalizing, for each galaxy, the scale radius and the rotation
curve respectively to the characteristic scale of $R_{\text{g}}$ and to the
characteristic velocity Eq. 10 and then by making the average over all
galaxies in the sample. Grey lines are the rotation curves of the galaxies
reported in the labels. The analytical behavior of the exponential thin disc
model (ETD) is reported as comparison.

(ii) The second ingredient is that a galaxy rotation curve $v_{\text{c}}(r)$
can be well approximated, at large enough radii where Eq. 9 is satisfied, by
the Exponential Thin disc (ETD) behavior (Freeman, 1970):

$v_{\text{c}}^{2}(R)=4\pi
G\Sigma_{\text{b}}^{0}R_{\text{d}}y^{2}[I_{0}(y)K_{0}(y)-I_{1}(y)K_{1}(y)],$
(10)

where $y=R/R_{\text{d}}$ and $I_{n}$ and $K_{n}$ are the modified Bessel
functions and

$\Sigma_{\text{b}}^{0}=\Upsilon_{\text{s}}\Sigma_{\text{s}}^{0}+\Upsilon_{\text{g}}\Sigma_{\text{g}}^{0}\approx\Upsilon_{\text{g}}\Sigma_{\text{g}}^{0}\;,$
(11)

is the amplitude of the total mass surface density (see Eq. 3).

Given the points mentioned above, we derive the average rotation curve of the
galaxies in our sample using the following procedure. First for each galaxy’s
rotation curve, we normalize the scale radius to characteristic lenght-scale
$R_{\text{d}}$ and the observed velocity to the characteristic velocity
defined as

$v_{\text{d}}=\sqrt{\frac{GM_{\text{dmd}}}{R_{\text{d}}}}\;,$ (12)

where $M_{\text{dmd}}$ is the mass of the disc given by Eq. 2. Then, we
determine the average of the so-normalized rotation curves.

Note that at small radii, Eq. 9 is not well satisfied because the stellar
component, which dominates the total mass distribution, has a different scale-
length for its exponential decay (or even exhibits two different exponential
behaviors in cases where there is a bulge) compared to that of the gas. This
variability in scale-lengths makes the re-scaling of the rotation curves of
different galaxies highly variable. Conversely, at larger radii, the agreement
is much stronger. The results are depicted in Fig. 1, where the average
rotation curve aligns well with the ETD model.

##  5 The Tully Fisher relation and the Dark Matter disc

Tully & Fisher (1977) discovered an intriguing observational relationship
between the luminosity of a disc galaxy and its maximum rotational velocity,
denoted by $v_{\text{max}}$. This relationship, known as the Tully-Fisher (TF)
relation, has been widely employed in estimating distances beyond our own
galaxy and has become a cornerstone of observational astrophysics. However,
the underlying theoretical explanation for the TF relation remains elusive, as
the connection between the production of stellar light and rotational velocity
is not straightforward from a physical point of view.

McGaugh et al. (2000) extended the TF relation by correlating the rotation
velocity with the detected baryon mass over five orders of magnitude in
stellar mass and one order of magnitude in velocity, resulting in what is
known as the baryonic TF (BTF) relation. This relation is believed to have a
more direct physical causality link between mass and rotational velocity
through gravitational dynamics than the observational TF relation. The
baryonic mass considered by McGaugh et al. (2000) includes the stellar
component, whose mass is estimated using a plausible stellar mass-to-light
ratio, as well as neutral hydrogen mass measured through 21 cm observations.
By combining the $\textrm{H}\scriptstyle\mathrm{I}$ mass with the stellar
mass, the scatter plot of velocity versus mass for a sample of 243 galaxies
was found to be straighter. This inclusion of
$\textrm{H}\scriptstyle\mathrm{I}$ mass was necessary because low surface
brightness galaxies in the sample can have a very low stellar mass content,
which can even be smaller than the gas mass, as shown studies focusing on the
BTF at the smallest scale (see, e.g., Iorio et al. (2017)). The BTF relation
has the form

$M_{\text{bar}}=(M_{\text{s}}+M_{\text{g}})\propto v_{\text{max}}^{\beta}\;,$
(13)

where $v_{\text{max}}$ is the peak value of the rotation curve and the
exponent was empirically found to be $\beta\approx 4$. As noticed by McGaugh
(2005) the use of a stellar mass-to-light ratio estimated from stellar
population models causes a significant systematic uncertainty in the stellar
mass, which is particularly sensitive to the initial mass function. Depending
on the choice of model, one can derive BTF relations with slopes anywhere in
the range of between 3 and 4. Although the formulation of the BFT relation
expressed by Eq. 13 represents an important step towards understanding its
physical basis, the problem is still unresolved.

Figure 2:  Left panel: the Bosma Tully Fisher relation (see Eq. 16): $\Lambda$
(in $10^{10}M_{\odot}$ kpc-1) vs $v_{\text{max}}$ (in km s-1); the exponential
thin disc (ETD) model of Eq. 17 is reported: note that this is not a fit.
Right Panel: the baryonic Tully Fisher relation $M_{\text{bar}}$ (in units of
$10^{10}M_{\odot}$) vs $v_{\text{max}}$ (in km sec-1). The best fit line L1
has slope $3.2$; for reference we report a line (L2) with slope 4. For
comparison the data by McGaugh (2005) are also shown.

In order to consider the connection between the Bosma effect and the TF
relation, Pfenniger & Revaz (2005) made a fit of the type

$\log(M_{\text{s}}+cM_{\text{g}})=a+b\log(v_{\text{max}})$ (14)

where $a,b,c$ are considered as free parameters so that the usual BFT relation
corresponds to $c=1$. A value $c>1$ corresponds to a re-scaling of the
$\textrm{H}\scriptstyle\mathrm{I}$ mass in agreement with the Bosma model.
They found at the optimal value $c=2.98$, where the linear fit parameters are
$a=3.11$ and $b=3.36$. Eq. 14 means that the mass of the gas (specifically,
only hydrogen) was considered as proxy of a more massive DM component in
agreement with the prescription of the Bosma effect. The conclusion of
Pfenniger & Revaz (2005) was that the BFT relation is improved when the
$\textrm{H}\scriptstyle\mathrm{I}$ mass is multiplied by a factor of about 3,
but they also noticed that larger factors up to 11-16 still improve the fit
over the original one using only the detected baryons. This result suggests
that a substantial amount of baryons would remain to be found in spirals.

We can now advance beyond the results presented in Pfenniger & Revaz (2005) by
connecting the TF relation to the average rotation curve we have obtained,
taking the Bosma effect into account. Since the average rotation curve of the
galaxies in our sample aligns well with the ETD model, we can establish a
simple relationship between the observed peak velocity $v_{\text{max}}$ in the
rotation curve and the characteristic velocity $v_{\text{d}}$ as given by Eq.
12. Indeed, for the ETD model, we find from Eq. 10 that (Freeman, 1970).

$\frac{v_{\text{max}}}{v_{\text{d}}}=\alpha\approx 0.62\;.$ (15)

Note that, for the case of the ETD model, the peak of the rotation curve
$v_{\text{max}}$ is reached for $R=2.2R_{\text{d}}$. In our sample the
estimation of the Bosma surface density gives $R_{\text{d}}$ and thus we can
determine $v_{\text{max}}=v_{\text{c}}(R=2.2R_{\text{d}})$ from the rotation
curve. In this respect, it should be noted that while the rotation curve is
flat in many cases, there are instances where it may increase or decrease: for
this reason our definition of $v_{\text{max}}$ is unambiguous and can be
applied for all rotation curve shapes.

Considering Eq. 12 and Eq. 15 we find

$\Lambda\equiv\frac{M_{\text{dmd}}}{R_{\text{d}}}=\frac{v_{\text{max}}^{2}}{\alpha^{2}G}\;,$
(16)

that we name the Bosma TF ( BoTF) relation. This can be rewritten as

$\Lambda\approx 6.1\times 10^{-5}v_{\text{max}}^{2}$ (17)

where $\Lambda$ has units $10^{10}M_{\odot}$ kpc-1 and $v_{\text{max}}$ is in
km s-1.

In the left panel of Fig. 2 we plot the BoTF relation (Eq. 16): we note that
the data points are very nicely fitted by Eq. 17, i.e., $\Lambda$ scales as
$v_{\text{max}}^{2}$ with very small scattering. Moreover, the constant of
proportionality between them can be analytically determined and it is simply
$(\alpha^{2}G)^{-1}$.

We emphasize that if the average rotation curve approximately matches with the
ETD model then the relation between $\Lambda$ and $v_{\text{max}}$, i.e. the
BoTF, is entirely justified. However, the BoTF does not contain any
information other than the fact that we have fitted the DMD model to the
rotation curves. Indeed, when the DMD mass is divided by the scale radius, the
only dependence remaining is from $v^{2}(R_{\text{d}})$ (and thus
$v_{\text{max}}$). As a result, $\Lambda$ is a parameter that does not provide
information about the location of the maximum, and hence on the rotation curve
shape. It merely depends on the possibility of the models to match the
’average magnitude’ of the rotational velocity. This is achieved by
construction when we fit the DMD to the observed rotation curves. So, the BoTF
relation should always be present, even if the ETD does not provide a good
overall match to the rotation curve shape.

In brief, since the BoTF is inherent of the ETD model and given that the ETD
model well describes the rotation curves of the galaxies we considered, not
surprisingly we observe the BoTF relation also in the real data.

The right panel of Fig. 2 illustrates the BTF relation (Eq. 13). It is evident
that for the galaxies within our sample, the best-fit exponent of Eq. 13 is
approximately $\beta\approx 3.3$, and a fit with $\beta=4$ remains acceptable.
It’s worth noting that Lelli et al. (2019) demonstrated that the slope of the
BTFR can depend on the choice of the velocity axis. Specifically, for
$v_{max}$, they reported a slope of approximately $\approx 3.52$ (as shown in
their Figure 2 and Table 1), which aligns closely with our findings.

The BoTF relation (Eq. 17) is indirectly connected to the BTF (Eq. 13): if
there is a relationship between $M_{\text{dmd}}$ and $M_{\text{bar}}$, then it
is possible to establish a connection between the BTF and the BoTF. Let’s
delve further into this issue.

Figure 3:  Left panel: $\Lambda$ (in $10^{10}M_{\odot}$ kpc-1) in function of
the baryonic mass (in $10^{10}M_{\odot}$), i.e. sum of mass of the stellar and
gas components $M_{\text{bar}}$. The best fit line L3 ($\Lambda=0.81\times
M_{\text{bar}}^{0.61}$) and the line L4 with slope $\Lambda\propto
M_{\text{bar}}^{0.5}$ are reported. Right panel: the DMD mass $M_{\text{dmd}}$
(in $10^{10}M_{\odot}$) as a function of the baryonic mass $M_{\text{bar}}$
(in $10^{10}M_{\odot}$). The best fit line L5 ($M_{\text{dmd}}=3.5\times
M_{\text{bar}}^{0.9}$) is reported.

Fig.3 (left panel) shows that $\Lambda$ and $M_{\text{bar}}$ are tightly
correlated: this correlation originates from the fact that the DM mass in the
Bosma model $M_{\text{dmd}}$, the baryonic mass $M_{\text{bar}}$, the stellar
mass $M_{\text{s}}$ and the mass of the gaseous component $M_{\text{g}}$ are
linked by Eq. 3. However, we cannot a priori determine the functional relation
between $\Lambda$ and $M_{\text{bar}}$. Empirically, in our sample, we find
that

$\Lambda\propto M_{\text{bar}}^{\gamma}$ (18)

where $\gamma\approx 0.6$; in addition we also find that

$M_{\text{bar}}\propto M_{\text{dmd}}^{0.86}\;,$

i.e., the baryonic and DMD mass are almost linearly correlated (see the right
panel of Fig.3) .

Using that $\Lambda\propto M_{\text{bar}}^{\gamma}$ and Eq. 16, we find that
the BTF relation can be written as

$M_{\text{bar}}\propto\Lambda^{\frac{1}{\gamma}}\approx\left(\frac{v_{\text{max}}^{2}}{\alpha^{2}G}\right)^{\frac{1}{\gamma}}\propto
v_{\text{max}}^{\frac{2}{\gamma}}\;.$ (19)

If we take $\gamma=0.6$, as measured in our sample, the exponent in Eq. 19 is
3.3 as, indeed, we found in our data (see the right panel of Fig.2). However,
as shown in the right panel of Fig. 2, the value $\gamma=0.5$, that is still a
good fit, gives the exponent 4 in Eq. 19 equal to the value reported by
McGaugh (2005). Note that in our sample the maximum velocity spans about one
decade, i.e. from 50 km s-1 to $\approx 300$ km s-1, which is about the same
range in the sample considered by McGaugh (2005) (also reported in the right
panel of Fig.2): the main difference is that in the latter case the scatter is
larger.

###  5.1 Discussion

In summary, we can conclude that the BoTF relation observed in our sample of
galaxies can be attributed to the following factors:

i) The surface density of the gas component follows an exponential decay for
sufficiently large radii.

ii) By rescaling the rotation curve of the gas component by a constant factor,
we obtain a rotation curve that matches the observed data. This phenomenon,
known as the Bosma effect, provides reasonable fits to the rotation curves at
large radii.

As a result of these empirical behaviors, the observed rotation curve can be
well approximated by an ETD model, which establishes a relationship between
$\Lambda$, the ratio between the total mass of the DMD disc and its
characteristic length-scale, and the maximum rotational velocity
($v_{\text{max}}$) as expressed in Eq. 17. Additionally, considering the
empirical correlation between $\Lambda$ and the baryonic mass
($M_{\text{bar}}$) we can establish a scaling argument that connects the BoTF
relation (Eq. 17) to the BTF relation (Eq. 13). While the presence of such a
correlation is expected based on the construction of the model (Eq. 2), the
precise value of the exponent needs to be determined from the data.

Figure 4:  Left panel: the radial acceleration relation, $g_{\text{obs}}$ vs
$g_{\text{bar}}$, for the galaxies of our sample. The data of McGaugh et al.
(2016) (average values and dispersion) are reported as reference (RAR). Right
panel: the radial acceleration relation in the DMD model, i.e.,
$g_{\text{dmd}}$ vs $g_{\text{bar}}$ (see text for details).

###  5.2 The Bosma effect and the radial acceleration relation

The radial acceleration relation (RAR) is an empirical relationship that has
been observed to apply to a broad spectrum of galaxies (McGaugh et al., 2016;
Lelli et al., 2017). It suggests a connection between the visible matter and
the overall gravitational potential within these galaxies. More precisely, the
RAR establishes a connection between the radial acceleration $g_{\text{obs}}$,
which is inferred from the rotation curves of disc galaxies and is
attributable to the total mass of the galaxy, including both visible and dark
matter components

$g_{\text{obs}}=\frac{V_{\text{c}}(R)^{2}}{R}=\frac{\partial\Phi_{\text{tot}}(R)}{\partial
R}\;,$

where $V_{\text{c}}(R)$ is the rotation curve and where $\Phi_{\text{tot}}$ is
the gravitational potential due to the total mass of the galaxy, and the
radial acceleration predicted by the observed distribution of baryons
$g_{\text{bar}}$. Specifically, by observing the distribution of stars and gas
within a galaxy, one can create baryonic mass models. Subsequently, by
considering that galactic discs possess a small but non-zero thickness, we
numerically solve the Poisson equation, allowing the computation of
$\Phi_{\text{bar}}(R)$, which represents the gravitational potential
attributed to the baryonic components. This potential results in the
acceleration experienced due to the cumulative effect of these baryonic
components

$g_{\text{bar}}=\frac{\partial\Phi_{\text{bar}}(R)}{\partial R}\;.$

Even though the mass models for individual galaxies exhibit considerable
diversity, as we have previously discussed for the case of our sample, there
is a robust and consistent correlation between the observed acceleration
$g_{\text{obs}}$ and the acceleration predicted by the baryonic matter
$g_{\text{bar}}$ for all galaxies. It’s important to note that in principle,
there is no inherent guarantee that $g_{\text{obs}}$ should exhibit such a
strong correlation with $g_{\text{bar}}$ when DM dominates the gravitational
dynamics of galaxies (McGaugh et al., 2016; Lelli et al., 2017). The
distribution of DM follows directly from the relation, and can be written
entirely in terms of the baryons:
$g_{\text{dm}}=g_{\text{obs}}-g_{\text{bar}}\;.$ The observed correlation
between $g_{\text{obs}}$ and $g_{\text{bar}}$ implies that dark and baryonic
mass are strongly coupled and suggests that the baryons are the source of the
gravitational potential.

The RAR finds a natural interpretation in the framework of the Bosma effect
and DMD model, as in this case DM follows the distribution of baryons as
traced by the gas component. To show that this is the case, first we have
computed $g_{\text{obs}}$ and $g_{\text{bar}}$ from our data and we have found
that galaxies in our sample follow the RAR (see the left panel of Fig.4) as
measured by McGaugh et al. (2016). Then, we have computed the correlation
between $g_{\text{bar}}$ and the radial acceleration from the best DMD model
$g_{\text{dmd}}$ finding that they are also tightly correlated (see the right
panel of Fig.4). Even in this case, it is not possible to predict in advance
the specific exponent that relates these two radial accelerations: however, we
can infer that such a correlation is expected to naturally arise in the DMD
model, where the distribution of baryonic matter traces that of DM.

###  5.3 The Tully Fisher relation in the halo model

It is worth noticing that, in the framework of the NFW halo model, the
interpretation of the BFT relation depends on how dark and baryonic mass are
related. For an halo at virial equilibrium one finds (see, e.g., McGaugh
(2012))

$M_{\text{vir}}=AV_{\text{vir}}^{3}$ (20)

where $A=4.6\times 10^{5}M_{\odot}$ km-3 s3 so that DM halos obey a scaling
relation reminiscent of the TF relation with some important differences from
the observed one: indeed, both mass and velocity refer to the virial radius
$R_{\text{vir}}$ that is much larger than the typical range of observed radii.
In order to map these theoretical quantities into the observed baryonic mass
and circular velocity, we need to introduce some factors which crucially
depend on the fact that $M_{\text{vir}}$ is correlated with $M_{\text{bar}}$.
In our sample, we have observed a significant scatter in the relation between
the baryonic mass ($M_{\text{bar}}$) and the halo mass ($M_{\text{vir}}$).
This finding is consistent with the results obtained by Mancera Piña et al.
(2022) (see Fig. 5)). However, upon excluding two galaxies from the fit, we
have identified a correlation between the baryonic mass and the halo mass,
given by $M_{\text{bar}}\sim M_{\text{vir}}^{\xi}$, where $\xi\approx 1.4$
(see the left panel of Fig. 5). This correlation can be considered as
analogous to Eq. 18 for the case of the BoTF relation. From Eq. 20 we thus
find

$M_{\text{bar}}\sim V_{\text{vir}}^{3\xi}\sim V_{\text{vir}}^{4.3}\;.$ (21)

This behavior is in proximity to the observed trend (see Fig. 2). The right
panel of Figure 5 illustrates the same plot, but in this case, the halo mass
was computed using the gNFW model. The trend observed in this plot is very
similar to the previous one: most of the galaxies exhibit a consistent
relationship between the baryonic mass and the halo mass. However, there are
only two galaxies for which the halo mass differs significantly. These two
galaxies, NGC 925 and NGC 2976, are characterized by a growing rotation curve.
The similarity in trends between the two plots, obtained using different halo
mass calculation methods (NFW and gNFW models), suggests that the overall
relationship between baryonic mass and halo mass remains robust. The slight
discrepancies observed in the halo mass for specific galaxies can be
attributed to the unique characteristics of their rotation curves.

Figure 5:  Left panel: baryonic mass $M_{\text{bar}}$ (in $10^{10}M_{\odot}$)
vs NFW virial halo mass $M_{\text{vir}}$ (in $10^{10}M_{\odot}$). The
reference line L6 has slope 1.43. Right panel: The same but for the gNFW
virial halo mass $M_{\text{vir}}$ (in $10^{10}M_{\odot}$). The gray dots are
the data from Mancera Piña et al. (2022).

###  5.4 The Tully Fisher relation in Modified Newton Dynamics model

As a final remark, we note that MOND (Milgrom, 1983) makes strong a priori
predictions about the BFT relation, which is a consequence of the form of the
force law in MOND. This, for small enough accelerations, deviates from purely
the Newtonian behavior: in the deep modified regime the effective acceleration
$a=\sqrt{a_{0}g_{N}}$ where $g_{N}$ is the Newtonian acceleration calculated
for the observed baryonic mass in the usual way and $a_{0}$ is the
characteristic acceleration at which the MOND regime becomes effective. For
circular motion around a point source, by equating the centripetal
acceleration to this effective force, one obtains

$a_{0}GM_{\text{bar}}=v_{\text{max}}^{4}\;.$ (22)

Note that the normalization is set by the constants of nature $a_{0},G$. As
discussed above in our sample we find a value of the exponent of the BFT
relation that is definitively smaller than 4.

##  6 Conclusion

We have shown that the rotation curves of a group of nearby disc galaxies can
be accurately modeled using the Bosma effect and that it is possible to
rescale the rotation curves of different disc galaxies to the simple
exponential thin disc behavior. This model, referred to as the Dark Matter
Disc model (DMD), posits that DM is confined to a thin disc. The disc’s mass
surface density is dominated by that of the stars at small radii and by that
of the gas rescaled at larger ones. Both the star and gas surface brightness
show approximately exponential decays, but with different characteristic
scale-lengths where the stars have a smaller scale-length than the gas. At
small radii, the circular velocity is proportional to that caused by the
stellar component, with the proportionality constant close to 1 in most cases,
i.e., the rotation curve in the inner disc can be almost entirely explained by
solely luminous matter. Therefore, the inner shape of rotation curves is well
predicted by the distribution of observed baryons (van Albada & Sancisi, 1986;
Palunas & Williams, 2000), implying that galactic discs are maximal in the
sense that they contribute the bulk of the mass at small radii.

At large radii, the circular velocity is proportional to that caused by the
gas, where the proportionality constant can be as large as 10 or more,
indicating that the observed $\textrm{H}\scriptstyle\mathrm{I}$ and gas is
just a proxy for a more massive DM component as assumed by the Bosma effect.

Due to the exponential decay of the gas surface brightness in the galaxies
under consideration, and assuming that the DM is confined to a thin disc due
to the Bosma effect, the rotation curves at large radii for the selected
sample of disc galaxies can be effectively approximated by the behavior of a
thin disc (Freeman, 1970). To achieve this renormalization, the galaxy mass
$M_{\text{dmd}}$ needs to be estimated consistently within the framework of
the exponential thin disc model. It is important to note that this
renormalization procedure is applicable primarily to galaxies with a surface
brightness that decays exponentially. For galaxies whose surface brightness
does not follow an exponential decay, the simple renormalization described
above may not be valid, and the rotation curves of such galaxies are expected
to deviate from the behavior expected for an exponential thin disc.

A global link between the baryonic mass of a galaxy and the amplitude of its
rotation curve has already been established in the baryonic Tully-Fisher (BTF)
relation (McGaugh et al., 2000; McGaugh, 2005). The success of the baryonic
scaling model indicates there is a more local coupling between the
distribution of the baryons and the rotation curve shape. However, the reason
why the BTF relation should be a power law relation between the disc baryonic
mass and the disc rotational velocity remains unclear, because the improvement
found on the TF relation still does not explain its physical origin; it just
points toward an explanation based on the disc baryonic content instead of
either only the stellar or only the detected baryonic content.

In our study, we have demonstrated that the BTF relation can be formulated
within the framework of the Bosma model, which we refer to as the Bosma TF
(BoTF) relation. The BoTF relation arises from the observed exponential thin
disc behavior exhibited by the disc galaxies in our sample. This behavior
manifests as a strong correlation between the total mass $M_{\text{dmd}}$, the
characteristic scale-length of the exponential decay $R_{\text{d}}$, and the
rotation velocity $v_{\text{c}}$. To establish a connection between the BoTF
relation and the more widely known BTF relation, an additional relationship
between the disc DM mass $M_{\text{dmd}}$ and the baryonic mass
$M_{\text{bar}}$ is required. Within the DMD model, the specific functional
form of this relation cannot be determined a priori. However, our analysis of
the data has revealed a tight power-law relationship between these two
quantities.

Given this situation, it is indeed not surprising that the Radial Acceleration
Relation (RAR) finds a natural framework within the context of the Bosma
model. The RAR is an empirical relationship that has been observed to hold for
a wide range of galaxies, as demonstrated by studies such as (McGaugh et al.,
2016; Lelli et al., 2017) . This relation suggests a connection between the
visible matter and the overall gravitational potential in these galaxies. In
the Bosma model, where DM follows the distribution of gas, the RAR can be
readily incorporated. This model naturally accommodates the observed empirical
relationship, as the distribution of DM is tied to the distribution of
baryonic matter, particularly the gas component.

The hypothesis that DM is confined to a disc presents several theoretical
challenges, including the dynamical stability of massive discs and the origin
of their exponential mass distribution. In the standard Cold Dark Matter (CDM)
scenario, the process of structure formation follows a bottom-up hierarchical
pattern, where non-collisional and non-dissipational non-baryonic DM forms a
nearly spherical halo structure with an almost isotropic velocity dispersion.
In this framework, baryonic matter undergoes dissipative collapse in the
gravitational field of the halo structure, leading to the formation of a disc
dominated by rotational motions. This disc is embedded in a DM component,
which is instead dominated by isotropic motions and is nearly spherical in
shape. The prevalence of rotational motion in the disc is a consequence of the
conservation of angular momentum during the collapse process.

The confinement of DM to a disc challenges the conventional framework and
raises questions about the stability and origin of such disc structures. In
response to this challenge, it has been proposed that heavy DM discs can form
through non-collisional and non-dissipational processes if gravitational
collapse follows a top-down scenario. This approach, as discussed in studies
by Benhaiem et al. (2019); Sylos Labini et al. (2020); Sylos Labini & Capuzzo-
Dolcetta (2020), suggests that in a top-down gravitational collapse model, as
initial deviations from spherical symmetry are amplified during the violent
collapse phase, isolated out-of-equilibrium perturbations may give rise to
massive discs in quasi-stable configurations. This formation mechanism
broadens the range of possibilities for the origin of disc structures and
offers an alternative perspective to the bottom-up hierarchical scenario.
Notably, recent work by Peebles (2020) advocates for this kind of monolithic
top-down scenario for galaxy formation. This approach aims to address the
challenges faced by standard CDM-like models in explaining crucial
observational aspects of galaxies. A monolithic top-down collapse could
potentially be facilitated by a sharp cutoff in the matter power spectrum at
small scales, such as the one associated with a warm dark matter initial mass
fluctuation power spectrum.

From an observational standpoint, as mentioned in the introduction, a direct
detection such cold clouds in the outer disc of galaxies appears to be a very
hard task (Combes & Pfenniger, 1997). However, we note that disc galaxies
offer valuable opportunities to impose stringent constraints on the symmetries
of their mass distribution through the study of strong gravitational lensing.
Strong lensing provides a distinct advantage compared to methods relying on
stellar or gas kinematics, as it does not depend on the detection of luminous
tracers within the galaxy under investigation. Instead, lensing directly
probes the mass distribution in regions predominantly influenced by DM (see,
e.g., Sonnenfeld et al. (2020) and references therein). In this regard, future
samples of strong lensing events in disc galaxies, coupled with high-
resolution imaging, will enable the measurement of the relative mass ratios
between the bulge, disc, and DM halo of these galaxies. Furthermore, it will
facilitate the determination of the distribution of the total mass flattening
parameter, which characterizes the overall shape of the mass distribution.
These observations hold the potential to refine our understanding of the mass
distribution within disc galaxies and the differences and/or similarities
between the distributions of the baryonic and DM components.

## Acknowledgments

FSL thanks for useful comments and discussions Roberto Capuzzo-Dolcetta, Žofia
Chrobáková, Frederic Hessman, Michael Joyce, Martin Lopez-Corredoira, Daniel
Pfenniger, Alessandro Sonnefeld and Hai-Feng Wang. We also thank an anonymous
referee for a number of interesting comments and suggestions that have allowed
us to improve the presentation of our results. This work has made use of data
from the The $\textrm{H}\scriptstyle\mathrm{I}$ Nearby Galaxy Survey (THINGS)
data-set available at the web address
https://www2.mpia-hd.mpg.de/THINGS/

##  Data availability

The data used in this paper are from the THINGS survey (Walter et al., 2008).

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##  7 Appendix: Results for individual galaxies

Galaxy | $R_{\text{d}}$ | $M_{\text{s}}$ | $M_{\text{g}}$ | $\Upsilon_{\text{s}}$ | $\Upsilon_{\text{g}}$ | $M_{\text{bar}}$ | $M_{\text{dmd}}$ | $\chi^{2}$ | $v_{\text{max}}$
---|---|---|---|---|---|---|---|---|---
NGC 9252,3 | 6.0$\pm$1 | 0.72 | 0.5 | 1.0 | 6.4 | 1.22 | 4.0 (0.73,3.2) | 2.7 | 117$\pm$7
NGC 23663 | 1.8$\pm$0.1 | 0.018 | 0.07 | 5.0 | 3.6 | 0.088 | 0.35 (0.1,0.25) | 0.3 | 55$\pm$10
NGC 24031,2 | 4.5$\pm$0.1 | 0.33 | 0.35 | 2.9 | 12.0 | 0.68 | 5.2 (1.0,4.2) | 1.1 | 130$\pm$5
NGC 28411,2 | 10.0$\pm$0.5 | 11.0 | 1.1 | 1.3 | 46.6 | 12.1 | 65.6 (14.3,51.3) | 1.9 | 300$\pm$8
NGC 29031,2 | 7.0$\pm$0.5 | 4.0 | 0.9 | 1.5 | 18.5 | 4.9 | 16.6 (6.0,10.6) | 2.6 | 200$\pm$8
NGC 29761,2 | 1.0$\pm$0.1 | 0.2 | 0.02 | 0.8 | 7.3 | 0.22 | 0.30 (0.15,0.15) | 1.4 | 75$\pm$3
NGC 31981,2 | 10.3$\pm$0.5 | 1.8 | 1.4 | 1.8 | 8.1 | 3.2 | 14.5 (3.2, 11.3) | 0.7 | 151$\pm$5
NGC 35211,2 | 6.0$\pm$0.5 | 9.0 | 1.6 | 1.2 | 4.8 | 10.6 | 18.7 (11,7.7) | 0.9 | 230$\pm$10
NGC 36213 | 9.0$\pm$1 | 1.4 | 1.3 | 1.5 | 8.0 | 2.7 | 12.5 (2.1,10.4) | 0.6 | 140$\pm$10
NGC 47361,2 | 2.5$\pm$0.5 | 2.5 | 0.14 | 0.24 | 16.4 | 2.6 | 2.9 (0.6, 2.3) | 0.9 | 158$\pm$5
NGC 50551,2 | 15.0$\pm$2 | 6.0 | 1.2 | 1.1 | 22 | 7.3 | 33 (6.6, 26.4) | 1.2 | 200$\pm$10
NGC 69462 | 5.5$\pm$0.5 | 4.4 | 1.4 | 1.4 | 5.5 | 5.8 | 13.9 (6.2, 7.7) | 1.7 | 202$\pm$5
NGC 73312 | 6.5$\pm$0.2 | 12.0 | 1.7 | 1.1 | 7.7 | 13.7 | 26.3 (13.2, 13.1) | 0.2 | 248$\pm$10
NGC 77931 | 2.0$\pm$0.1 | 0.41 | 0.09 | 1.5 | 9.6 | 0.5 | 1.5 (0.6, 0.9) | 0.3 | 116$\pm$5
DDO 1543 | 3.1$\pm$0.1 | 0.002 | 0.036 | 1.0 | 12.6 | 0.038 | 0.46 (0.002,0,46) | 1.2 | 48$\pm$4
IC 25743 | 4.5$\pm$1.0 | 0.074 | 0.15 | 2.7 | 3.7 | 0.22 | 0.75 (0.2,0.55) | 0.6 | 65$\pm$8
MW2 | 9.0$\pm$0.2 | 8.0 | 0.5 | 1.2 | 26.1 | 8.5 | 22.3 (9.2,13.1) | 1.1 | 229$\pm$5
Table 1: The index 1 is for the galaxies for which the data from the Pipeline 5 of S4G are available providing both the stellar surface brightness profile and the stellar mass; the index 2 when the H2 surface density profile from Schruba et al. (2011) is known ; the index 3 when the stellar surface brightness is from the 2MASS survey deBlok et al. (2008) and the stellar mass from Hessman & Ziebart (2011). The best-fit values of $\Upsilon_{\text{s}}$, $\Upsilon_{\text{g}}$ and $M_{\text{s}},M_{\text{g}},M_{\text{bar}},M_{\text{dmd}}$ (in units of $10^{10}M_{\odot}$) are reported; the contributions related to the stellar $\Upsilon_{\text{s}}M_{\text{s}}$ and gas $\Upsilon_{\text{g}}M_{\text{g}}$ components are shown in parenthesis. For the case of the Milky Way (MW) we refer the reader to Sylos Labini et al. (2023b) for further details. Galaxy | $R_{\text{d}}$ | $\Upsilon_{\text{g}}$ | $\chi^{2}$
---|---|---|---
NGC 925 | 25.5 | 90 | 1.9
NGC 2366 | 3.0 | 7.9 | 0.2
NGC 2403 | 5.1 | 14.9 | 2.2
NGC 2841 | 13.0 | 52.5 | 2.5
NGC 2903 | 8.7 | 16.7 | 0.7
NGC 2976 | 16.2 | 1080 | 1.5
NGC 3198 | 10.4 | 9.3 | 0.9
NGC 3521 | 5.2 | 5.4 | 3.2
NGC 3621 | 10.0 | 8.7 | 0.5
NGC 4736 | 18.6 | 56 | 0.7
NGC 5055 | 42.6 | 69 | 5.3
NGC 6946 | 5.3 | 5.4 | 2.2
NGC 7331 | 8.7 | 9.8 | 0.3
NGC 7793 | 3.6 | 24.4 | 1.4
DDO 154 | 3.3 | 12.4 | 0.1
IC 2574 | 24.7 | 178 | 0.6
MW | 7.4 | 24 | 2.0
Table 2:  Dark Matter disc fits obtained by considering free parameters the gas length-scale $R_{d}$ (in kpc) and the DMD mass, i.e. $M_{\text{DMD}}=\Upsilon_{\text{g}}M_{g}+M_{s}$ where $M_{g},M_{s}$ are reported in Tab.1. In the last column it is reported the value of reduced $\chi^{2}$ obtained from the fit. Galaxy | $R_{\text{vir}}$ | $M_{\text{vir}}$ | $c$ | $c_{\text{vir}}$ | $\chi^{2}$
---|---|---|---|---|---
NGC 925 | 240 | 155 | 2.4 | 7.1 | 4.8
NGC 2366 | 99 | 11 | 5.5 | 9.0 | 0.72
NGC 2403 | 138 | 30 | 15.0 | 8.0 | 0.43
NGC 2841 | 275 | 233 | 16.0 | 6.6 | 0.4
NGC 2903 | 185 | 71 | 13.3 | 7.5 | 0.3
NGC 2976 | 118 | 18 | 2.0 | 10.2 | 1.7
NGC 3198 | 160 | 46 | 10.1 | 8.0 | 0.3
NGC 3521 | 9.6 | 4.5 | 2.0 | 7.9 | 0.5
NGC 3621 | 186 | 72 | 6.8 | 7.8 | 0.1
NGC 4736 | 25 | 0.17 | 19.0 | 13.2 | 1.1
NGC 5055 | 211 | 107 | 6.5 | 7.5 | 0.5
NGC 6946 | 161 | 47 | 15 | 7.7 | 0.8
NGC 7331 | 200 | 90 | 11.8 | 7.4 | 0.1
NGC 7793 | 208 | 102 | 5.6 | 7.6 | 2.6
DDO154 | 87 | 7.5 | 4.5 | 9.9 | 0.5
IC2574 | 89 | 8.0 | 4.5 | 9.9 | 2.0
MW | 178 | 63 | 14.3 | 7.5 | 1.7
Table 3:  The best-fit values of the virial radius $R_{\text{vir}}$ (in kpc) and the mass $M_{\text{vir}}$ (in units of $10^{10}M_{\odot}$) of the NFW halo, together with the best-fit concentration parameter $c$ and the expected concentration parameter $c_{\text{vir}}$ from Eq. 6 are reported.  Galaxy | $R_{\text{vir}}$ | $M_{\text{vir}}$ | $\beta$ | $c$ | $c_{\text{vir}}$ | $\chi^{2}$
---|---|---|---|---|---|---
NGC 925 | 484 | 1274 | 0.4 | 4.9 | 5.6 | 2.2
NGC 2366 | 120 | 19 | 0.9 | 5.4 | 8.5 | 0.71
NGC 2403 | 139 | 30 | 0.9 | 16.8 | 8.0 | 0.40
NGC 2841 | 270 | 223 | 0.9 | 18.0 | 6.5 | 0.4
NGC 2903 | 180 | 66 | 0.7 | 18.0 | 7.4 | 0.3
NGC 2976 | 486 | 1300 | 0.5 | 4.9 | 5.8 | 1.7
NGC 3198 | 156 | 43 | 0.8 | 13.0 | 7.9 | 0.3
NGC 3521 | 9.6 | 4.5 | 1.0 | 2.0 | 7.9 | 0.5
NGC 3621 | 167 | 52 | 0.7 | 11.3 | 7.8 | 0.1
NGC 4736 | 53 | 1.6 | 0.1 | 10.5 | 10.9 | 1.0
NGC 5055 | 184 | 71 | 0.2 | 16.8 | 7.3 | 0.5
NGC 6946 | 160 | 46 | 1.0 | 15.0 | 7.7 | 0.8
NGC 7331 | 200 | 90 | 1.0 | 11.8 | 7.4 | 0.1
NGC 7793 | 129 | 24 | 0.4 | 17.8 | 5.6 | 2.0
DDO154 | 60 | 2.4 | 0.2 | 17.1 | 10.2 | 0.2
IC2574 | 103 | 12.2 | 0.1 | 10.5 | 9.0 | 0.8
MW | 175 | 60 | 0.8 | 17.8 | 7.4 | 0.8
Table 4:  The best-fit values of the virial radius $R_{\text{vir}}$ (in kpc)
and the mass $M_{\text{vir}}$ (in units of $10^{10}M_{\odot}$) of the gNFW
halo (see Eq. 7), together with the best-fit concentration parameter $c$ and
the expected concentration parameter $c_{\text{vir}}$ from Eq. 6 are reported.

(a) NGC 925

(b) NGC 925

(c) NGC 2366

(d) NGC 2366

(e) NGC 2403

(f) NGC 2403

Figure 6: For each galaxy we show the surface brightness profile of the
stellar and gas component and the rotation curve

(a) NGC 2841

(b) NGC 2841

(c) NGC 2903

(d) NGC 2903

(e) NGC 2976

(f) NGC 2976

Figure 7: As Fig.6

(a) NGC 3198

(b) NGC 3198

(c) NGC 3521

(d) NGC 3521

(e) NGC 3621

(f) NGC 3621

Figure 8: As Fig.6.

(a) NGC 4736

(b) NGC 4736

(c) NGC 5055

(d) NGC 5055

(e) NGC 6946

(f) NGC 6946

Figure 9: As Fig.6.

(a) NGC 7331

(b) NGC 7331

(c) NGC 7793

(d) NGC 7793

(e) DDO 154

(f) DDO 154

Figure 10: As Fig.6.

(a) IC 2574

(b) IC 2574

Figure 11: As Fig.6.

(a) MW

(b) MW

(c) MW

Figure 12:  Data for the Milky Way. Left panel: stellar density on the disc
(units are number of stars kpc-3). Center panel:
$\textrm{H}\scriptstyle\mathrm{I}$ and $\textrm{H}\scriptstyle\mathrm{I}$+H2
surface density. Right panel: rotation curve. The three lines are respectively
the best fits with the NFW and DMD mass models and the best exponential thin
disc (ETD) approximation: note that in this later case the small radii
behavior is not well fitted because of the it neglects the bulge.

###  7.1 NGC 925

The surface brightness profile of the stellar component (2MASS data — see De
Blok et al. (2008)) displays an exponential decay while the surface brightness
profiles of both $\textrm{H}\scriptstyle\mathrm{I}$ and
$\textrm{H}\scriptstyle\mathrm{I}$ +H2 are almost flat for $R<10$ kpc. The
contribution to the gas mass of the H2 component is marginal as
$M_{\text{$\textrm{H}\scriptstyle\mathrm{I}${}}}=0.46\times 10^{10}M_{\odot}$
(Walter et al., 2008) and $M_{\text{H}_{2}}=0.04\times 10^{10}M_{\odot}$
(Schruba et al., 2011). The Bosma brightness profile shows an approximate
exponential decay with $R_{\text{d}}\approx 6$ kpc from very small scale,
where the stellar component dominates at small radii and the gas component in
the outer regions. The maximum radius up to which the rotation curve is
measured is about $2.5R_{\text{d}}$; at that distance $v_{\text{c}}(R)$ is
still rising. The very inner disc rotation curve, i.e., $R<3$ kpc, can be
accounted for by the stellar component alone (where we used the same mass
$M_{\text{s}}=0.72\times 10^{10}M_{\odot}$ as Hessman & Ziebart (2011)). On
larger radii, the DMD requires $\approx$ 8 times the
$\textrm{H}\scriptstyle\mathrm{I}$ +H2 mass to fit the rotation curve. The
total DMD mass is about 3.5 times larger than the visibile one. The NFW model
has a virial mass approximately 120 times larger than the baryonic one;
correspondingly, the velocity at the virial radius $V_{\text{vir}}\approx 160$
km/s is larger than the observed maximum rotation velocity $\approx 120$ km/s.
This problematic situation arises when fitting a linearly rising rotation
curve with an NFW model (McGaugh & de Blok, 1998), leading to a concentration
parameter of $c\approx 2.5$ that is larger than the expected value according
to Eq. 6 of $c_{\text{vir}}\approx 6.7$.

In our analysis, we have found that both the DMD model and its modified
version (mDMD), where the free parameters $\Upsilon_{\text{g}}$ (disc mass-to-
light ratio) and $R_{\text{d}}$ (disc scale length) are considered, yield
similar reduced $\chi^{2}$ values. This implies that both models provide
reasonable fits to the data, and there is no significant preference for one
over the other based on the goodness of fit alone.

We note that the gNFW (generalized NFW) model generally provides a better
agreement with the data compared to the NFW model. This is because the gNFW
model offers more flexibility at small radii, allowing it to capture the
detailed features that characterize the observed rotation curve.

Although the obtained reduced $\chi^{2}$ values are relatively high, around
$\approx 2$, this is not uncommon in studies of rotation curves. The bumps and
wiggles present in the rotation curve data can introduce some level of
uncertainty and make achieving a perfect fit challenging. Therefore, the
higher reduced $\chi^{2}$ values are primarily a reflection of the intrinsic
complexities in the rotation curve data rather than a fundamental limitation
of the models themselves.

###  7.2 NGC 2366

The stellar surface brightness (2MASS data — see De Blok et al. (2008)),
exhibits a single exponential decay for $R>1$ kpc. The
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness displays a constant
behavior at small radii, then it shows approximately an exponential decay with
the same characteristic scale-length of the stellar component. No information
of the H2 surface brightness is available for this galaxy. The Bosma
brightness profile shows an exponential decay with $R_{\text{d}}\approx 1.9$
kpc. The total baryonic mass is dominated by the
$\textrm{H}\scriptstyle\mathrm{I}$ component (Walter et al., 2008) that is
about four times larger than the stellar mass (Hessman & Ziebart, 2011). The
maximum radius up to which the rotation curve is measured is three time
$R_{\text{d}}$. The rotation curve shows a flattening for $R>2$ kpc. Fitting
the rotation curve with the DMD model we find that $\Upsilon_{\text{s}}=5.8$
and $\Upsilon_{\text{g}}=3.2$, so that we obtain that the total mass in this
model is about 3.6 times larger than the baryonic one. A NFW model requires a
halo with a virial mass 34 times larger than the baryonic mass. The
concentration parameter as calculated from Eq. 6 is $c_{\text{vir}}\approx
8.7$ and it is different from the best-fit value $c\approx 5.5$. The DMD and
mDMD models give similar reduced $\chi^{2}\approx 0.2$ value, and the same
occurs for the NFW and gNFW models with $\chi^{2}\approx 0.7$ .

###  7.3 NGC 2403

The stellar surface brightness (S4G data) shows a double exponential decay,
signaling the presence of a bulge at small radii whose mass is estimated to be
about 10% that of the disc (Hessman & Ziebart, 2011). The
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness is close to constant in
the very inner disc and then decays exponentially. The
$\textrm{H}\scriptstyle\mathrm{I}$ +H$2$ surface brightness displays an
exponential decay with a characteristic scale-length that is approximately
three times larger than the stellar one $R_{\text{s}}$. The Bosma brightness
profile shows an approximate exponential decay with $R_{\text{d}}\approx 5$
kpc. The maximum radius up to which the rotation curve is measured is about
three times $R_{\text{d}}$. The stellar mass is similar to the
$\textrm{H}\scriptstyle\mathrm{I}$ mass (Walter et al., 2008) while the H$2$
contribution (Schruba et al., 2011) represents only a small fraction of the
total gas content. The inner disc requires three times the mass of the stellar
component to fit the rotation curve. The DMD mass is eight times larger than
the sum of the stellar and gas mass, while the NFW halo virial mass is almost
45 times larger. In this case, we find from Eq. 6 that $c_{\text{vir}}\approx
7.9$, which is about half the best-fit value of $c\approx 15.0$. In this case,
the best fit for the DMD model yields a reduced $\chi^{2}$ value of 1.1, which
is half that of the mDMD model. Both the NFW and gNFW models result in smaller
$\chi^{2}$ values of 0.4..

###  7.4 NGC 2841

The stellar surface brightness (S4G data) shows the typical bulge- disc
transition at small radii, i.e. $\sim 3$ kpc; on large radii it shows an
exponential decay a characteristic scale-length $R\approx 3.4$ kpc, which is
about three times smaller than the $\textrm{H}\scriptstyle\mathrm{I}$
exponential scale-length. As for other galaxies, we neglect the contribution
of the bulge (whose mass is about 20% that of the disc Hessman & Ziebart
(2011)) and consider the range for $R>4$ kpc for a fitting with the mass
models. The H2 mass represents 20% of the total gas content (Schruba et al.,
2011), and gives a non-negligible contribution to the gas surface brightness
only at small radii. The $\textrm{H}\scriptstyle\mathrm{I}$ +H2 surface
brightness shows an approximate exponential decay with characteristic scale
$R_{\text{g}}=10.6$ kpc. The Bosma brightness profile is dominated by the
rescaled gas component at all radii but the very inner ones where the bulge
dominates and thus it has the same characteristic scale of the
$\textrm{H}\scriptstyle\mathrm{I}$ component. The maximum radius up to which
the rotation curve is measured is 4.5 times $R_{\text{d}}$: this rotation
curve is one of those that extends most in terms of $R/R_{\text{d}}$. The peak
of the rotation curve is at $\approx 10$ kpc and the velocity is $\approx 320$
km/s while in the outermost regions, $R>40$ kpc the rotation curve decreases
of about 15%, reaching $\approx 270$ km/s: this is thus an example of a slowly
decaying rotation curve. To explain the behavior of the rotation at large
radii a substantial amount of DM is required. Indeed, we find that
$\Upsilon_{\text{s}}\approx 1.3$ and $\Upsilon_{\text{g}}\approx 47$ so that
the main mass contribution in this model is given by the rescaled gas
component. The total mass for the DMD case is 5.4 times larger than the
baryonic mass, while the virial mass of the NFW halo is about 18 times larger.
From Eq. 6, we find that $c_{\text{vir}}\approx 6.5$, while the best-fit value
of $c\approx 16.7$.

The DMD and mDMD models yield similar reduced $\chi^{2}$ values of
approximately 2, while the NFW and gNFW models produce smaller values, around
$\chi^{2}\approx 0.4$.

###  7.5 NGC 2903

Because this galaxy has a bulge, the stellar surface brightness (S4G data)
shows a double exponential decay, with $R_{\text{d}}=0.25$ kpc for $R<1$ kpc
and $R_{\text{d}}=2.5$ kpc for $R>4$ kpc. The mass of the bulge is about 10%
of that of the disc (Hessman & Ziebart, 2011) and we exclude it from the fit.
The the $\textrm{H}\scriptstyle\mathrm{I}$ surface brightness profile is close
to flat for $R<10$ kpc, whereas the $\textrm{H}\scriptstyle\mathrm{I}$ +H2
surface brightness profile shows a double exponential decay, with H$2$ being
dominant for $R<7$ kpc, having a characteristic scale of $R_{\text{g}}\approx
2.5$ kpc, while $R_{\text{g}}\approx 8$ kpc for $R>10$ kpc. We performed the
fit with the $\textrm{H}\scriptstyle\mathrm{I}$ profile alone as the H2 gives
a very high contribution at small radii where the bulge dominates. The Bosma
brightness profile shows an approximate exponential decay with
$R_{\text{d}}\approx 7$ kpc for $R>10$ kpc. As for the case of NGC 2841, also
this galaxy shows a slowly decaying rotation curve in the observed range of
distances, passing from 215 km/s at 5 kpc to 185 km/s at 30 kpc, i.e. a
decrease of about 15%. The maximum radius up to which the rotation curve is
measured is three times $R_{\text{d}}$. The sum of the stellar and gas
components is sufficient to explain the inner disc rotation curve while in the
outer parts, the best fit with a DMD model gives $\Upsilon_{\text{g}}\approx
18$. The DMD mass is less than 4 times larger than the baryonic mass, while
the NFW virial mass is about 13 times larger. From Eq. 6, we find
$c_{\text{vir}}\approx 7.4$, which is different from the best-fit value of
$c\approx 13.2$.

The mDMD model obtains a best-fit value for $R_{\text{g}}$ that is twice the
one measured in the gas surface density profile. This higher value of
$R_{\text{g}}$ enables a superior fit for the mDMD model compared to the DMD
model, resulting in a $\chi^{2}$ that is reduced by one-third. In contrast,
both the NFW and gNFW models yield smaller values, approximately
$\chi^{2}\approx 0.3$.

###  7.6 NGC 2976

The stellar surface brightness profile (S4G data) shows a double exponential
decay with $R_{\text{d}}=1.2$ kpc for $R<1$ kpc and $R_{\text{d}}=0.7$ kpc for
$R>1$ kpc. This is not indicative of a typical bulge, as the characteristic
scale-length in the inner part is only slightly larger than in the outer part.
The $\textrm{H}\scriptstyle\mathrm{I}$ +H2 surface brightness profile shows
some distinct features, including a flat behavior at small radii ($R<1.3$
kpc), an exponential decay with $R_{\text{d}}=0.6$ kpc for $1<R<4$ kpc, and
$R_{\text{d}}=3$ kpc for $R>4$ kpc. However, the rotation curve is determined
only up to $R=3$ kpc, so the last regime is not relevant. The Bosma brightness
profile shows an approximate exponential decay with $R_{\text{d}}\approx 1$
kpc and the maximum radius up to which the rotation curve is measured is three
times $R_{\text{d}}$. This galaxy shows signs of strong tidal interactions in
its $\textrm{H}\scriptstyle\mathrm{I}$ distribution (Bigiel & Blitz, 2012).
The velocity profile is almost entirely accounted for by the stars, and the
DMD is only 1.5 larger than the baryonic matter component. However, the NFW
halo virial mass is 80 times larger than the baryonic matter component. The
value of $c_{\text{vir}}\approx 8.3$ is twice the best-fit value of $c\approx
2.7$, suggesting that Eq. 6 is not satisfied.

The four models produce similar reduced $\chi^{2}$ values, approximately
around 1.7. These elevated values can be attributed to the prominent bumps
present in the rotation curve of this galaxy.

###  7.7 NGC 3198

Even this galaxy presents a stellar surface brightness profile (S4G data) with
a bulge- disc structure, i.e. a double exponential behavior: at small radii
($R<1.5$ kpc), it decays exponentially with a scale-length of
$R_{\text{s}}=0.7$ kpc, and then at larger radii it shows a slope with
$R_{\text{s}}=3.4$ kpc. The mass of the bulge is 20% that of disc: as for the
other galaxies we have excluded the very inner disc from the fit. The
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness is close to constant at
small radii and then shows an exponential decay with a scale-length of
$R_{\text{g}}\approx 12$ kpc. The contribution of H2 (Schruba et al., 2011) is
small and limited to very small radii. The $\textrm{H}\scriptstyle\mathrm{I}$
+H$2$ surface brightness has an exponential decay from the very inner disc
with the same characteristic scale-length as the
$\textrm{H}\scriptstyle\mathrm{I}$ . The Bosma brightness profile shows an
approximate exponential decay with $R_{\text{d}}\approx 10$ kpc, i.e. that of
the gas component as the stellar component dominates only in the very inner
disc. As the maximum radius up to which the rotation curve is measured is
about 40 kpc, this is four times $R_{\text{d}}$. In the DMD case, the stellar
component accounts for about half of the observed rotation curve up to 10 kpc
as for the best fit model $\Upsilon_{\text{s}}=1.8$. At larger radii the
rescaled gas component, with $\Upsilon_{\text{g}}=8.1$ dominates the mass
distribution and the total DMD mass is about five times larger than the
baryonic component. In the case of the NFW halo the best fit virial mass is
about 13 times larger than the baryonic component. The best fit concentration
parameter is again different from the phenomenological one: Eq. 6 gives
$c_{\text{vir}}\approx 10.3$, while $c\approx 7.6$.

The DMD and mDMD models yield nearly identical reduced $\chi^{2}$ values,
while the same is true for the NFW and gNFW models.

###  7.8 NGC 3521

The stellar surface brightness profile (S4G data) exhibits a double
exponential behavior. At small radii (for $R<1$ kpc), it decays with a scale-
length of $R_{\text{s}}=1.4$ kpc, and at larger radii has a scale-length of
$R_{\text{s}}=4$ kpc. The $\textrm{H}\scriptstyle\mathrm{I}$ surface
brightness is nearly constant at small radii, and for $R>10$ kpc, it exhibits
an exponential decay with a scale-length of $R_{\text{g}}\approx 7$ kpc. The
contribution of H2 (Schruba et al., 2011) is limited to within 10 kpc. The
Bosma brightness profile shows an approximate exponential decay with
$R_{\text{d}}\approx 5$ kpc so that the maximum radius up to which the
rotation curve is measured is five times $R_{\text{d}}$. The stellar component
is almost enough to account for the observed rotation curve up to 15 kpc. The
DMD mass model requires $\Upsilon_{\text{g}}=3.1$, while
$\Upsilon_{\text{s}}=1.5$: in this case the total mass only 1.8 times larger
than the baryonic one. For the NFW mass model, the halo virial mass is about
two times larger than the baryonic mass. The best fit concentration parameter,
$c\approx 35$, is smaller than the valued predicted by Eq. 6 gives
$c_{\text{vir}}\approx 8$.

The DMD, NFW, and gNFW models all result in similar reduced $\chi^{2}$ values.
However, in this instance, the mDMD model’s best fit yields a high value of
$\chi^{2}=3.2$. This is attributed to the mDMD model’s inability to accurately
fit the innermost portion of the rotation curve.

###  7.9 NGC 3621

The stellar surface brightness profile (2MASS data — see De Blok et al.
(2008)) has a single exponential behavior with a scale-length of
$R_{\text{s}}=$ 2.3 kpc. The $\textrm{H}\scriptstyle\mathrm{I}$ surface
brightness shows a depletion at small radii, i.e. $R<3$ kpc and then an
exponential decay with a scale-length of $R_{\text{g}}\approx$ 10 kpc. No data
are available for the H2 surface density profile. The Bosma brightness profile
shows an approximate exponential decay with $R_{\text{d}}\approx 10$ kpc and
the maximum radius up to which the rotation curve is measured is twice
$R_{\text{d}}$. The stellar and gas masses are very similar, i.e.
$M_{\text{s}}\approx M_{\text{g}}\approx 1.4\times 10^{10}M_{\odot}$ (Hessman
& Ziebart, 2011). The stellar component is sufficient to account for the
observed rotation curve at very small radii, i.e., $R<5$ kpc. The best fit of
the DMD mass model requires $\Upsilon_{\text{s}}\approx 1.5$ whereas the gas
component is about 9 times larger. The total DMD mass is about 5 times larger
than the baryon mass. In the the NFW mass model, the halo virial mass is
estimated to be 26 times the visible component. In this case the best-fit
value of the concentration parameter, $c\approx 6.8$, is very close to the
value predicted by Eq. 6, i.e. $c_{\text{vir}}\approx 7.3$.

All models yield reduced $\chi^{2}$ values below 1, indicating excellent fits.

###  7.10 NGC 4736

The stellar component surface brightness (S4G data) exhibits a complex
behavior that can be modeled as three exponential decays. The first is
characterized by $R_{\text{s}}=0.2$ for radii less than 1 kpc, followed by
$R_{\text{s}}=1.0$ for 1 kpc $<R<6$ kpc, and $R_{\text{s}}=5.0$ for $R>5$ kpc.
We fit the models only for $R>1$ kpc, ignoring the bulge. Despite this, the
profile of the stellar surface brightness still displays a unique behavior at
small radii, and the rotation curve also exhibits a very clear decline. The
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness is nearly constant,
while the $\textrm{H}\scriptstyle\mathrm{I}$ +H$2$ (data from (Schruba et al.,
2011)) surface brightness shows approximately a double exponential behavior
with $R_{\text{d}}=1.0$ for $R<5$ kpc and $R_{\text{d}}=10.0$ for $R>5$ kpc.
The Bosma brightness profile does not follow a simple exponential decay in the
same range of radii where the rotation curve was measured and can fitted by a
rough exponential decay with $R_{\text{d}}\approx 2.2$ kpc but with a large
modulation about this behavior both at small and a large radii. The maximum
radius up to which the rotation curve is measured is 4 times $R_{\text{d}}$.
This is one of the most extended rotation curves in the sample. The peak of
the rotation curve is at $\approx 1$ kpc and the velocity is $\approx 195$
km/s while in the outermost regions, $R\approx 9$ kpc the rotation curve is
decreased of about 40%, reaching $\approx 120$ km/s. The stellar component
alone can take into account the observed rotation curve at small radii: this
is the reason why the best fit with te Bosma model gives
$\Upsilon_{\text{s}}\approx 3$. In this respect we note that the stellar mass
we sued was $M_{\text{s}}=4\times 10^{10}M_{\odot}$ which is three times
larger the value reported by Hessman & Ziebart (2011) (excluding the bulge’s
mass). The DMD mass is only 10% larger than the baryonic mass. In the NFW mass
model the halo virial represents a small fraction of the baryonic mass,
showing that the stellar component is sufficient to take into account almost
the whole mass budget. Finally the best-fit value concentration parameter
$c\approx 13$, larger than $c_{\text{vir}}\approx 13$. predicted from Eq. 6.

The DMD, mDMD, NFW, and gNFW models all yield similar reduced $\chi^{2}$
values, approximately around 1. This relatively high value is attributed to
the presence of multiple bumps and wiggles in the rotation curve.

###  7.11 NGC 5055

The stellar surface brightness profile (S4G data) has a double exponential
behavior, typical signature of the bulge, with $R_{\text{s}}=0.5$ for $R<3$
kpc and $R_{\text{s}}=5.0$ for $R_{\text{d}}>3$ kpc. The
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness has a plateau at small
radii. The $\textrm{H}\scriptstyle\mathrm{I}$ +H$2$ (data from (Schruba et
al., 2011)) surface brightness shows a double exponential behavior with
$R_{\text{g}}=5.0$ for $R<20$ kpc and $R_{\text{g}}\approx 20$ for $R>22$ kpc.
We performed the fit with the $\textrm{H}\scriptstyle\mathrm{I}$ profile alone
as the H2 gives a very high contribution at small radii where the bulge
dominates. The Bosma brightness profile shows an approximate exponential decay
with $R_{\text{d}}\approx 20$ kpc and the maximum radius up to which the
rotation curve is measured is twice $R_{\text{d}}$. The peak of the rotation
curve is at $\approx 10$ kpc and the velocity is $\approx 210$ km/s while in
the outermost regions, $R>40$ kpc the rotation curve is decreased of about
20%, reaching $\approx 165$ km/s. The sum of the stellar and gas components is
almost sufficient to take into account the inner disc behavior (we avoid to
fit the bulge as for the other cases whose mass is about 10% of that of the
disc Hessman & Ziebart (2011)). The total DMD mass is about four times larger
than the visibile mass while the NFW virial mass is 16 larger than the
visibile mass. The best-fit concentration parameter is $c\approx 2.7$ that is
close to $c_{\text{vir}}\approx 6.6$ from Eq. 6.

The DMD, NFW, and gNFW models all result in similar reduced $\chi^{2}$ values,
each smaller than 1. In contrast, the mDMD model obtains a value of
$\chi^{2}=5.3$ due to the gas component’s inability to accurately fit the
inner region of the rotation curve.

###  7.12 NGC 6946

The stellar surface brightness profile (2MASS data — see De Blok et al.
(2008))has a double exponential behavior with $R_{\text{s}}=0.6$ for $R<1.5$
kpc and $R_{\text{s}}=2.8$ for $R_{\text{d}}>2$ kpc: this is an indication for
the presence of a bulge whose mass is estimated to be 0.27 $\times
10^{10}M_{\odot}$ while the mass of the disc is 4.2 $\times 10^{10}M_{\odot}$
(Hessman & Ziebart, 2011). We avoid to include the bulge in the fit and we
consider only $R>2$ kpc. The $\textrm{H}\scriptstyle\mathrm{I}$ +H2 surface
brightness profile (data from (Schruba et al., 2011)) shows approximately an
exponential behavior with $R_{\text{g}}\approx 4$, while the
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness has a plateau at small
radii. The Bosma brightness profile shows an approximate exponential decay
with $R_{\text{d}}\approx 4$ kpc and the maximum radius up to which the
rotation curve is measured is five times $R_{\text{d}}$. We performed the fit
with the $\textrm{H}\scriptstyle\mathrm{I}$ profile alone as the H2 gives a
very high contribution at small radii where the bulge dominates. The rotation
curve shows and increase at small radii and the a flattening. The stellar plus
gas component are sufficient to take into account the rotation curve in the
inner disc. At larger radii the gas mass must be scaled by a factor
$\Upsilon_{\text{g}}\approx 7$. The total DMD mass is about 1.8 times larger
than the baryonic mass whereas the NFW virial mass is respectively 6 times
larger than the baryonic mass. The best-fit concentration parameter is
$c\approx 12$ and is larger than $c_{\text{vir}}\approx 7$ predicted by Eq. 6.

Both the DMD and mDMD yield small values of reduced $\chi^{2}$ values,
approximately around 2. Conversely, the NFW and gNFW models produce reduced
$\chi^{2}$ values of around 1.

###  7.13 NGC 7331

The stellar surface brightness profile (2MASS data — see De Blok et al.
(2008)) shows a double exponential decay signaling the presence of a bulge
with $R_{\text{s}}=2$ kpc at small radii and $R_{\text{s}}\approx 6$ kpc at
large ones. The $\textrm{H}\scriptstyle\mathrm{I}$ surface brightness profile
is flat at small radii and then it shows an exponential behavior with
$R_{\text{g}}\approx 7$; the $\textrm{H}\scriptstyle\mathrm{I}$ +H2 surface
brightness exponentially decays with the same characteristic scale from small
scales. The Bosma brightness profile shows an approximate exponential decay
with $R_{\text{d}}\approx 7$ kpc and the maximum radius up to which the
rotation curve is measured is three times $R_{\text{d}}$. We avoid to fit the
bulge as for the other cases whose mass is about 10% of that of the disc
(Hessman & Ziebart, 2011). The sum of the stellar and gas components is
sufficient takes into account the observed rotation curve at small radii, i.e.
$R<5$ kpc. The total DMD mass is about 2 times larger than the visibile mass
while the NFW virial mass is about 7 times larger than the visibile mass. The
best-fit concentration parameter is $c\approx 11.9$ that is close to
$c_{\text{vir}}\approx 7.1$ from Eq. 6.

All models produce reduced $\chi^{2}$ values that are small, around 0.2.

###  7.14 NGC 7793

The stellar surface brightness (S4G data) shows a simple exponential decay
with $R_{\text{s}}=1.2$ whereas the $\textrm{H}\scriptstyle\mathrm{I}$ surface
brightness profile is flat at small radii and then decays exponentially with a
characteristic scale-length close to the stellar one $R_{\text{s}}\approx 2$.
No data are available for the H2 distribution. The Bosma brightness profile
shows an approximate exponential decay with $R_{\text{d}}\approx 2$ kpc and
the maximum radius up to which the rotation curve is measured is four times
$R_{\text{d}}$. The peak of the rotation curve is at $\approx 4.5$ kpc and the
velocity is $\approx 116$ km/s while in the outermost regions, $R>7$ kpc the
rotation curve is decreased of about 17%, reaching $\approx 95$ km/s. The
inner disc rotation curve, i.e., for $R<3$ kpc, can be explained by the
stellar component solely. The total DMD mass is about 3 times larger than the
baryonic mass. The halo virial mass is 200 times larger than the baryonic
mass: this problematic situation occurs when one fits a linearly rising
rotation curve with a NFW model (McGaugh & de Blok, 1998). The concentration
parameter value: $c_{\text{vir}}\approx 7.0$ from Eq. 6 is larger than the
best-fit value $c\approx 5.6$.

In this particular case, the best-fit model is the DMD, which results in a
$\chi^{2}$ value of 0.3, whereas the other models yield $\chi^{2}$ values
greater than 1.

###  7.15 DDO 154

The stellar surface brightness profile (2MASS data — see De Blok et al.
(2008)) shows decay at small radii and the it flattens although being very
noisy: the stellar component is however a negligible component of the whole
mass budget as the mass of the gas component is almost 20 times larger.
Instead, the $\textrm{H}\scriptstyle\mathrm{I}$ surface brightness shows an
exponential decay with a scale-length $R_{\text{s}}=3.1$. No data are
available for the H2 distribution. The Bosma brightness profile shows an
approximate exponential decay with $R_{\text{d}}\approx 3.1$ kpc and the
maximum radius up to which the rotation curve is measured is almost three
times $R_{\text{d}}$. In the DMD mass model the total mass is about 12 times
larger than the baryonic mass. In the NFW mass model the halo virial mass is
instead about 190 times larger than the baryonic mass: even in this case the
best-fit concentration parameter is smaller than the values given by from Eq.
6, i.e., $c\approx 4.5$ instead of $c_{\text{vir}}=9.1$.

The mDMD model produces a $\chi^{2}$ value of approximately 0.1, similar to
the NFW and gNFW models, while the DMD model yields a $\chi^{2}$ of 1.2.

###  7.16 IC 2574

The stellar surface brightness profile (2MASS data — see De Blok et al.
(2008)) has an exponential behavior with $R_{\text{s}}=2.9$ while the
$\textrm{H}\scriptstyle\mathrm{I}$ surface brightness is constant at small
radii, i.e. $R<7$ kpc, and then it shows a sharp decay. No data are available
for the H2 distribution. The Bosma brightness profile shows an approximate
exponential decay with $R_{\text{d}}\approx 4$ kpc: however, especially at
large radii, the exponential approximation works badly because in that range
of radii the surface brightness is dominated by the rescaled
$\textrm{H}\scriptstyle\mathrm{I}$ one. The maximum radius up to which the
rotation curve is measured is twice times $R_{\text{d}}$. Both fits with the
DMD and the NFW work well. The observed $\textrm{H}\scriptstyle\mathrm{I}$
mass is twice the stellar mass and the total DMD mass is about 3 times larger
than the baryonic one. The NFW virial mass is about190 times larger than the
baryonic mass. Finally the best-fit concentration factor $c\approx 4.5$, is
smaller than the value obtained from Eq. 6, i.e., $c_{\text{vir}}=9.0$.

Both the DMD and mDMD models result in small values of reduced $\chi^{2}$,
approximately around 0.6. In contrast, the NFW model yields $\chi^{2}=2$, and
the gNFW model yields 0.8.

###  7.17 Milky Way

Sylos Labini et al. (2023b) made a DMD fit to the rotation curve of the MW
measured by the latest data releases of the Gaia mission by Eilers et al.
(2019); Wang et al. (2023). They have obtained that, by fixing
$\Upsilon_{\text{s}}=1$ the gas amplification is about
$\Upsilon_{\text{g}}=16$. The sum of the stellar and gas components have a
more complicated behavior than a simple exponential function. However, a
exponential decay with $R_{\text{d}}=5.7$ kpc well fits with rotation curve if
the mass is $M_{\text{d}}=17\times 10^{10}M_{\odot}$. Note that the best fit
values of Sylos Labini et al. (2023b) give a local density of $\sim 0.3$ Gev
cm-3 and for the Bosma case 13,2 Gev-3.

All models produce reduced $\chi^{2}$ values that are between 1 and 2.