# Normalizing Flows for Domain Adaptation when Identifying $\Lambda$ Hyperon
Events

R. Kelleher11footnotetext: Corresponding author. A. Vossen

###### Abstract

This study focuses on the novel application of a normalizing flow as a method
of domain adaptation. Normalizing flows offer a way to transform data points
between two different distributions. The present study investigates a method
of transforming latent representations of physics data to a normal
distribution and then to a physics distribution again. The final distribution
models a simulated distribution. Following the transformation process, the
data can be classified by a neural network trained on labeled simulation data.
The present study succeeds in training two normalizing flows that can
transform between data (or simulation) and a Gaussian distribution.

##  1 Introduction

In semi-inclusive deep inelastic scattering (SIDIS) experiments [1], electrons
are scattered off of protons to probe the spin structure of the proton.
$\Lambda$ hyperons are produced in the scattering experiment at the CLAS12
detector [2], but have a lifetime too short to detect directly in the
detector. $\Lambda$ hyperons primarily decay into a proton and negatively
charged pion; events where a proton and $\pi^{-}$ are measured in the final
state may have produced a $\Lambda$. However, protons and $\pi^{-}$ may be
produced by other reactions, meaning events where a $\Lambda$ is produced must
be distinguished from those where there are only background processes. A
signal fit can be applied with a peak at the nominal $\Lambda$ mass of 1.1157
GeV. Although this process can work well enough, there is a prominent
background for these events at CLAS12, meaning that the signal fit alone
produces a relatively low signal to background ratio. Improving the signal to
background ratio could help increase confidence in the signal and background
separation of the fit, reducing statistical uncertainty.

One approach to improving signal extraction consists in training a classifier
on kinematic variables related to $\Lambda$ event candidates so that the
classifier can identify which events contained a $\Lambda$ decay. The CLAS12
detector cannot record data for particles that decay before interacting with
the detector volumes. Instead, particles that decay quickly produce decay
products that are measured. These products can help us understand the
processes that occurred during a collision, but do not provide enough
information to definitively identify which decay products resulted from which
parent particles. Thus, measured data cannot be labeled as containing
$\Lambda$ hyperons, meaning they cannot be used for training. Simulated data
can contain all information about events, allowing it to be labeled. Monte-
Carlo event generation (MC) [3] was used to produce SIDIS events that simulate
the physics that occurs in the CLAS12 detector, which is used in the present
study. Training a classifier on this simulation data provides a tool to
identify which simulated events contain a $\Lambda$, but this does not
directly translate to datasets composed of measured events. Differences
between the simulated and measured data causes the classifier to struggle to
accurately classify events from the measured dataset. Previous work has been
done in Ref. [4] where a domain adversarial graph neural network was employed
to improve the classification process and the signal extraction. The present
study builds on the graph neural network (GNN) implementation by adding a
domain adaptation stage in the classification process between the GNN and the
classification network. The domain adaptation is implemented via the use of
two normalizing flow networks that transform the dataset between data and
simulation distributions.

##  2 Normalizing Flows

Normalizing flows are a class of neural networks that transform data points
between a simple base distribution and a complex distribution. This process is
achieved through the layering of many invertible, differentiable mappings.
Generated samples consist in the output of these mappings. Both the density of
the sample under the base distribution and the change of volume originating
from the transformation can be calculated. The product of this density and
volume can be treated as a likelihood, and the negative likelihood can be
thought of as loss [5].

The change of variables formula is central to the normalizing flow
architecture as it enables the computation of the probability density function
(PDF) of a complex distribution. We can start with a base distribution Z for
which we know the PDF (in D dimensions where our input is of dimension D),
then find a function that transforms Z to a distribution X with an unknown but
desired PDF, $p_{X}(x)$. With both the base distribution and transformation,
we can compute the PDF of Z, $p_{Z}(z)$ and sample from it. The transformation
must be bijective, however, to allow for computations of its inverse. To
simplify the task of writing a bijective transformation between X and Z we can
compose many invertible functions together. Because the composition of
multiple invertible functions is itself invertible, we can describe a complex
transformation with many simpler functions.

The functions that describe the transformation from Z to X can be learned
through log-likelihood maximization (or negative log-likelihood minimization).
This likelihood is computed by taking the logarithm of the change of variables
formula:

$\log{p_{X}(x)}=\log{p_{Z}(f(x))}+\log{\Bigl{|}det\Bigl{(}\frac{\partial
f(x)}{\partial x^{T}}\Bigr{)}\Bigr{|}}$ (2.1)

The parameters of Z and $f(x)$ are trained according to this equation.

###  2.1 Architecture

The present study utilizes a flow model based on the RealNVP architecture
introduced by Ref. [6]. The RealNVP architecture implements the transformation
as a composition of coupled functions, where each function scales (with
scaling function $s(x)$) and translates (with translation function $t(x)$) the
input. To improve computation efficiency, the architecture uses coupling
functions where the input vector is split into two segments. The output of the
first segment equal to the input of the first segment; the output of the
second segment is parameterized by the first segment:

$\displaystyle y_{1:d}$ $\displaystyle=x_{1:d}$ (2.2) $\displaystyle
y_{d+1:D}$ $\displaystyle=x_{d+1:D}\cdot exp(s(x_{1:d}))+t(x_{1:d})$

This coupling method improves the computation efficiency for the log-
determinant as the coupling forces the transformation’s jacobian to be
triangular. The determinant of a triangular matrix is the product of its
diagonal entries; because the diagonal entries consist only in the scale
portion of the transformation, and the scaling is exponential, the
exponentials can be summed. Now, the determinant computation consists only of
a summation, which is very favorable for the computation time of the training
process.

##  3 Model Implementation

The flow model implemented for this study utilizes the normalizing-flows
package [7]. This package provides implementations for the layers needed to
create a neural network based on the realNVP architecture. For our study we
used a model with 32 masked affine layers, alternating the mask each layer.
Now we can write a single parametric function (given in Ref. [6]) to replace
eqn. 2.2:

$y=b\cdot x+(1-b)\cdot(x\cdot exp(s(b\cdot))+t(b\cdot x))$ (3.1)

The scale and translation functions used in eqn. 3.1 are implemented as multi-
layer perceptrons (MLPs) using the pytorch python package [8] by Ref. [7].
Because the GNN output had a dimension of 71 we used 71 as the input and
output dimension for the MLPs. We used 2 hidden layers for each MLP, both with
dimensions of 142. The normalizing-flows package provides functions for
calculating negative log-likelihood as explained in section 2.1, as well as an
implementation for back propagating the loss; these functions are used for
training the model.

The models were trained on batches of 100 events output by the GNN. The
training loop is as follows for each step: the batch of events is transformed
using the current model; the likelihood that these samples were drawn from a
gaussian distribution is calculated; the model is updated with gradient
descent. A validation dataset was used to determine when to stop training. A
testing dataset was used at the end of training to ensure the models did not
overfit or diverge.

##  4 Application

###  4.1 Classifier Input

The first application aims to improve the performance of a classifier trained
to identify simulated $\Lambda$ events. We designed an approach where: data
would first be transformed to a normal distribution via a forward pass with a
data-network; second, the normalized data would be passed backwards through a
simulation-network such that it ends up "looking like" the simulation data
that the classifier was trained on. By using an intermediate normal
distribution, we can transform between the data and simulation distributions.
Figure 1 visualizes the process.

Figure 1: The transformation process inserted between latent space extraction
and classification steps.

For this strategy two different networks were trained separately: a data-
network was trained to normalize measured data; a simulation-network was
trained to normalize simulation data. Because the networks represent bijective
functions, they can be reversed to turn normalized samples into simulation-
like samples. The data-network was trained over 6 epochs while the simulation-
network was trained over 11 epochs (due to the size difference in the
datasets).

###  4.2 Distortion Reversal

The second application aims to transform from a distorted dataset to an ideal
dataset. The ideal dataset consists of the transverse momentum $p_{T}$ ,
azimuthal angle $\phi$, and polar angle $\theta$ for both the proton and pion
in every event. The distorted dataset was generated in two different ways: by
drawing a random value from a normal distribution centered at 0 for each
event; by taking a set value for all events. In each case, the distortion
value was added to the proton $p_{T}$. A strategy similar to that in section
4.1 was employed, where we trained two networks: the first was trained to
normalize distorted data; the second was trained to normalize ideal data. The
distorted data was then passed forward through the distorted-network and then
backwards through the ideal-network.

##  5 Results

###  5.1 Transformation

The models were successful in learning to normalize their respective inputs,
as shown in Fig. 2. The validation loss appeared to match the training loss
well, suggesting we did not encounter over-fitting. The normalized
distributions appear to follow a normal distribution, however we can notice
that some dimensions have distributions that differ from the expected result.
Some distributions are not centered around 0 and some are skewed.

Figure 2: Histograms are shown for several dimensions of the latent
representation of data before (left) and after normalization (right).

###  5.2 Classification

The classifier output for the simulation, the data, and the transformed data
are shown in Fig. 3. The classifier output for the transformed data matches
that of the simulation much better than that of the data, suggesting that the
domain adaptation helped align the data with the inputs that the classifier
was trained on. Furthermore, the figure of merit ($N_{signal}/\sqrt{N_{tot}}$)
and purity ($N_{sig}/N_{tot}$) for the data and transformed data are shown in
Fig. 3, where $N_{tot}$ represents the total number of events being fit and
$N_{sig}$ represents the number of events that reside inside the fit. The
figure of merit appears flatter when transformed with the flow model. The
flatness of the curve may be desirable as it allows for cuts to be made almost
anywhere on the curve without sacrificing the figure of merit.

Figure 3: The classifier output for the data, simulation, and transformed data
are shown as well (left). The figure of merit and purity are plotted for both
transformed and latent data fits (right).

###  5.3 Distortion Reversal

The flow model did not appear to succeed in the second application’s aim of
reversing the distortion (as discussed in section 4.2). The flow model was
able to recover the broader features of the distribution, primarily the
position of the peak, but failed to reconstruct finer details such as
secondary peaks. There are many possible improvements to this attempt at
distortion reversal that may be able to improve the results. One change that
may be beneficial could be to use conditional flow models. Conditional
normalizing flows allow for modeling of conditional probability densities;
with a conditional normalizing flow, one could model the distribution of the
proton $p_{T}$ conditional on the other variables $\phi$ and $\theta$. This
strategy may capture correlations between kinematic variables better, allowing
for better distortion reversal in future studies.

## References

  * [1] C.A. Aidala, S.D. Bass, D. Hasch and G.K. Mallot, _The spin structure of the nucleon_ , _Rev. Mod. Phys._ 85 (2013) 655.
  * [2] V. Burkert, L. Elouadrhiri, K. Adhikari, S. Adhikari, M. Amaryan, D. Anderson et al., _The clas12 spectrometer at jefferson laboratory_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 959 (2020) 163419.
  * [3] L. Mankiewicz, A. Schäfer and M. Veltri, _Pepsi — a monte carlo generator for polarized leptoproduction_ , _Computer Physics Communications_ 71 (1992) 305.
  * [4] M. McEneaney and A. Vossen, _Domain-adversarial graph neural networks for lambda hyperon identification with clas12_ , _Journal of Instrumentation_ 18 (2023) P06002.
  * [5] I. Kobyzev, S.J. Prince and M.A. Brubaker, _Normalizing flows: An introduction and review of current methods_ , _IEEE Transactions on Pattern Analysis and Machine Intelligence_ 43 (2021) 3964–3979.
  * [6] L. Dinh, J.N. Sohl-Dickstein and S. Bengio, _Density estimation using real nvp_ , _ArXiv_ abs/1605.08803 (2016) .
  * [7] V. Stimper, D. Liu, A. Campbell, V. Berenz, L. Ryll, B. Schölkopf et al., _normflows: A pytorch package for normalizing flows_ , _Journal of Open Source Software_ 8 (2023) 5361.
  * [8] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan et al., _PyTorch: An Imperative Style, High-Performance Deep Learning Library_ , in _Advances in Neural Information Processing Systems 32_ , H. Wallach, H. Larochelle, A. Beygelzimer, F. d’Alché Buc, E. Fox and R. Garnett, eds., pp. 8024–8035, Curran Associates, Inc., 2019, http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf.


# SPARC HSBs, and LSBs, the surface density of dark matter haloes, and MOND.

Antonino Del Popolo11footnotemark: 1<EMAIL_ADDRESS>Dipartimento
di Fisica e Astronomia, University Of Catania, Viale Andrea Doria 6, 95125
Catania  Institute of Astronomy, Russian Academy of Sciences, 119017,
Pyatnitskaya str., 48 , Moscow, Russia

###### Abstract

In this paper, we use SPARC’s HSBs, and LSBs galaxies to verify two issues.
The first one is related to one claim of [1] D09, namely: is the DM surface
density (DMsd) a constant universal quantity, equal to
$\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.15\pm 0.2$, or does it depend on the
baryon surface density of the system? The second one, is based on a MOND
prediction that for HSBs the DMsd is constant, and equal to
$\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.14$, while for LSBs the surface density
is not constant and takes values that are smaller than for HSBs and D09
prediction [2]. We find that HSBs shows a constant DMsd vs magnitude as in
D09, and a constant DMsd vs $\Sigma_{\rm eff}$ as in MOND prediction, for HSBs
with $\Sigma_{\rm eff}>200L_{\odot}/pc^{2}$, and $\Sigma_{\rm
eff}>300L_{\odot}/pc^{2}$. However, the value of the DMsd is larger,
$\Sigma\simeq 2.61$ (in the case of the DMsd-magnitude with $\Sigma_{\rm
eff}>300L_{\odot}/pc^{2}$), and $\Sigma\simeq 2.54$ (in the case of the
surface DMsd-surface brightness with $\Sigma_{\rm eff}>200L_{\odot}/pc^{2}$).
This value slightly depends on the threshold to determine wheter a galaxy is
HSB. In the case of LSBs, for $\Sigma_{\rm eff}<100L_{\odot}/pc^{2}$, and
$\Sigma_{\rm eff}<25L_{\odot}/pc^{2}$, the surface density vs magnitude, for
lower magnitudes, is approximately equal to that predicted by D09, but several
galaxies, for magnitude $M>-17$ have smaller values than those predicted by
D09. The DMsd vs $\Sigma_{\rm eff}$ shows a similar behavior in qualitative,
but not quantitative, agreement with MOND predictions. In summary, in the case
of HSBs both D09 and MOND are in qualitative, but not quantitative, agreement
with the data. In the case of LSBs D09 is mainly in disagreement with the
data, and MOND only in qualitative agreement with them.

###### keywords:

Galaxies; Alternative theory of gravity; galaxies surface density

††journal: Journal of LaTeX Templates

##  1 Introduction

In spite of the fact that the $\Lambda$CDM model, is a very good model in
describing the observations on intermediate scales [3, 4, 5, 6, 7, 8, 9], and
cosmology scales [6, 7, 8], it shows several drawbacks. To start with we
recall the cosmological constant problem [10, 11], and the cosmic coincidence
problem [12]. Apart the mentioned problems several others are present. At
large scale there is the unsolved tensions between the the current value of
the Hubble parameter, $H_{0}$, having different values when measured with
different methods, for example using the CMB, and supernovae and stars in the
relatively recent universe [13]. Moreover, there is another tension between
the data of Planck 2015 and the $\sigma_{8}$ growth rate of perturbations
[14], and with CFHTLenS weak lensing [15] data. In the large-angle
fluctuations in the CMB are present a power hemispherical asymmetry [16, 17,
18, 19, 20, 21], a quadrupole-octupole alignment [22, 23, 24, 25, 26], and a
cold spot [27, 28, 29].

Other problems are present on smaller scales ($\simeq 1-10$ kpcs). Those often
recalled are the ”Cusp/Core” problem, namely the discrepancy between the cuspy
profiles obtained in dark matter (DM) only N-body simulations ([30, 31, 32],
and the observations of dwarf spirals, dwarf spheroidals (dSphs), and Low
Surface Brightness (LSBs) galaxies showing cored profiles ([33, 34, 35, 36,
37, 38, 39, 40, 36, 37, 38, 39, 41]) . The “missing satellite problem”
consists in the discrepancy between the number of satellites or subhaloes
predicted by DM only N-body simulations ([42, 43]), and the number of
subhaloes really observed. Moreover, the subhaloes obtained in simulations are
too dense with respect to those observed around the Milky Way [44, 45]. The
last is dubbed “too-big-to-fail” problem. Finally, we cite the issue of the
location on planes of satellite galaxies of the Milky Way and M31 [46],
difficult to explain. This is dubbed “satellites planes problem”, but recently
[47] showed a solution. The quoted problems have been attacked from different
fronts. Some authors proposed to modify the power spectrum (e.g. [48]), others
proposed to modify either the particles constituting DM ([49, 50, 51, 52]), or
the theory of gravity ([53, 54, 55, 56, 57]). Apart these drastic solutions,
other astrophysical solutions have been proposed. They are based on the role
of baryons in “heating” DM. A well known mechanism is that related to
supernovae feedback ([58, 59, 60, 61, 62]), and another one is related to the
transfer of energy and angular momentum from baryons to DM through dynamical
friction ([63, 64, 65, 38, 66, 40, 67].

In this context, scaling relations are very helpful to understand complex
phenomena. [68], by fitting the rotation curves of 55 galaxies with a pseudo-
isothermal profile, obtained some relations among the DM halos parameters.

They introduced the quantity $\Sigma=\mu_{0D}=\rho_{0}r_{0}$222$r_{0}$ is the
core radius of the pseudo-isothermal profile, and $\rho_{0}$ its central
density, which behaves as a DMsd. According to their studies, for late type
galaxies, $\Sigma=\mu_{0D}\simeq 100M_{\odot}/pc^{2}$ independently from
galaxy luminosity. This result has been studied and verified by several
authors. [1] (D09), in agreement with [68], found again a quasi-universality
of $\Sigma$, by analysing a set of 1000 galaxies (spirals, dwarfs,
ellipticals, etc). Promptly, [2], showed that in MOND $\Sigma$, in the
Newtonian regime, has a similar behavior to that described by D09, but noticed
that in the case of galaxies having a low surface density $\Sigma$ has a
smaller value. Recently, [69] extended the work of [2] to spiral galaxies
showing that for high baryon DMsd $\Sigma$ is constant in agreement with [2],
and D09, and for low baryon surface density $\Sigma$ decreases.

From the literature is known that the Burkert profile gives good fits to LSBs,
and dwarfs rotation curves ([70, 71, 38]. In the case of giant galaxies, and
ellipticals the previous conclusion is not valid ([72, 73, 74, 72]).

Several authors obtained results in disagreement with those of D09.

By means of a much larger sample of that of D09 [75] showed that dark matter
column density systematically increases with the mass of the halo. [76] showed
that the column density, and the Newtonian acceleration, correlates with
different quantities in agreement with [75], and in disagreement with D09
results.

In early-type galaxies, [77] did not find the existence of a universal DMsd.

[78], in agreement with [77], [76], and [75], found signs of a correlation of
DMsd with $M_{200}$.

[79] found a a correlation between the Newtonian acceleration and the virial
mass $M_{\rm vir}$.

Several correlations between the surface density and a series of quantities
are shown by [67], similarly to what found by [80]. In the last paper the DMsd
was obtained by fitting the rotation curves of the galaxies with a Burkert
profile, and using the Markov Chain Monte Carlo (MCMC) method to infer the
values of the parameters.

In this paper, using SPARC’s HSBs, and LSBs we want to study the behavior of
the DMsd, to see whether it is either constant as claimed by D09, or it
depends on the baryon surface density according to MOND. To this aim, we study
the DMsd vs the magnitude and the baryon surface density in HSBs, and LSBs.
HSBs shows a constant behavior of the DM, while LSBs shows that the behavior
depends from the baryon surface density.

The paper is organized as follows. In Sect. 2, we introduce the SPARC sample
and the analysis that was performed in a previous paper ([80]) to obtain the
DMsd. In Sect. 3, we discuss the results, and Sect. 4 is devoted to
discussion.

##  2 SPARC data set, and data analysis: a summary

In this section, we give a summary of the SPARC data used, and the analysis on
the data, as performed by [80]. SPARC ($\it Spitzer$ Photometry and Accurate
Rotation Curves) dataset [81] is constituted by 175 late-type galaxies with
high quality rotation curves obtained from $\rm HI/H\alpha$ studies, and with
surface photometry at 3.6 $\mu\rm m$, giving the mass-to-light ratio
$\Upsilon_{*}$ conversion factor. The gas mass is provided by 21 cm
observations. Almost all SPARC galaxies have a disc structure, with some
having also bulges. The baryon component is constituted by the disc, bulge,
and gas components. The morphologies present in SPARC goes from SO to Irr.
[80] obtained the DM surface density by fitting the rotation curves with a
Burkert profile

$\rho(r)=\frac{\rho_{0}r_{0}^{3}}{(r+r_{0})(r^{2}+r_{0}^{2})},$ (1)

where $r_{0}$, and $\rho_{0}$ represent the scale radius and the central
density of the halo, respectively.

The total rotational velocity is given by

$\displaystyle V_{\rm tot}^{2}=V_{\rm DM}^{2}+V_{\rm bar}^{2}=V_{\rm
DM}^{2}+\Upsilon_{\rm d}V^{2}_{\rm disc}+\Upsilon_{\rm b}V^{2}_{\rm
bulge}+V^{2}_{\rm gas},$ (2)

where $V_{\rm disc},~{}V_{\rm bulge},~{}V_{\rm gas}$ are the velocities of the
baryonic component, and $V_{\rm DM}$ is that of the dark matter component,
given by

$\displaystyle\frac{V_{\rm
DM}^{2}}{V_{200}^{2}}=\frac{C_{200}}{x}\frac{\ln{(1+x)}+\frac{1}{2}\ln{(1+x^{2})}-\arctan{x}}{\ln{(1+C_{200})}+\frac{1}{2}\ln{(1+C_{200}^{2})}-\arctan{C_{200}}},$
(3)

and the concentration $C_{200}$ and the rotation velocity $V_{200}$ at the
virial radius $r_{200}$ are given by

$\displaystyle C_{200}=r_{200}/r_{0},~{}~{}V_{200}=10C_{200}r_{0}H_{0},$ (4)

where $H_{0}$ is the Hubble constant (chosen to be
$73~{}\mathrm{Km\,s^{-1}\,Mpc^{-1}}$). The mass-to-light ratios for the disc
and bulge component, are given by $\Upsilon_{\rm d}$, and $\Upsilon_{\rm b}$,
respectively. An important point to recall is that the galaxy distance, and
the disc inclination, affect the stellar components and the total observed
rotational velocities, respectively. Then if the galaxy distance is changed
from its value $D$ to $D^{\prime}=D\delta_{\rm D}$333$\delta_{\rm D}$ is a
dimensionless distance factor, also the radius is affected and becomes
$R^{\prime}=R\delta_{\rm D}$, and also the baryonic component velocity which
becomes $V^{\prime}_{\rm k}=V_{\rm k}\sqrt{\delta_{D}}$444‘k’ denotes disc,
bulge, or gas.

A change in disk inclination from $i$ to $i^{\prime}=i\delta_{\rm
i}$555$\delta_{\rm i}$ is a dimensionless inclination factor, produces a
change in the observed rotation curves and their uncertainties, as

$\displaystyle
V_{\mathrm{obs}}^{\prime}=V_{\mathrm{obs}}\frac{\sin(i)}{\sin\left(i^{\prime}\right)},\quad\delta
V_{\mathrm{obs}}^{\prime}=\delta
V_{\mathrm{obs}}\frac{\sin(i)}{\sin\left(i^{\prime}\right)}.$ (5)

The observed rotational curve is fitted with a theoretical curve depending on
several parameters: $V_{200}$, $C_{200}$, $\Upsilon_{\rm d}$, $\Upsilon_{\rm
b}$, $\delta_{\rm D}$ and $\delta_{\rm i}$. The calculation is performed
through a Bayesian analysis. The posterior probability of parameter space is
given by

$\displaystyle P(V_{200},C_{200},\Upsilon_{\rm d},\Upsilon_{\rm b},\delta_{\rm
D},\delta_{\rm i}|\mathrm{SPARC})=\mathcal{L}(V_{200},C_{200},$
$\displaystyle\Upsilon_{\rm d},\Upsilon_{\rm b},\delta_{\rm D},\delta_{\rm
i}|\mathrm{SPARC})P(V_{200},C_{200},\Upsilon_{\rm d},\Upsilon_{\rm
b},\delta_{\rm D},\delta_{\rm i}),$ (6)

where the likelihood is obtained through $\mathcal{L}\sim e^{-\chi^{2}/2}$,
and where $\chi^{2}$ is given by

$\displaystyle\chi^{2}=\sum_{k=1}^{N}\left(\frac{V_{\rm tot}(R^{\prime}_{\rm
k};V_{200},C_{200},\Upsilon_{\rm d},\Upsilon_{\rm b},\delta_{\rm
D})-V^{\prime}_{\rm obs,k}}{\delta V_{\rm obs,k}^{{}^{\prime}}}\right)^{2},$
(7)

In the previous relation, $N$ is the number of data point for each galaxy,
$V^{\prime}_{\rm obs,k}$ and $\delta V^{\prime}_{\rm obs,k}$ are the observed
rotation curve and its uncertainty at the radius $R_{\rm k}$. The total
rotation velocity $V_{tot}$ at the radius $R^{\prime}_{k}$ depends on galactic
parameters $\\{\Upsilon_{\rm d},\Upsilon_{\rm b},\delta_{\rm D}\\}$, and halo
parameters $\\{V_{200},C_{200}\\}$. Since prior probabilities of parameters
are uncorrelated it is given by

$P(V_{200},C_{200},\Upsilon_{\rm d},\Upsilon_{\rm b},\delta_{\rm
D},\delta_{\rm i})=P(V_{200})P(C_{200})P(\Upsilon_{\rm d})P(\Upsilon_{\rm
b})P(\delta_{\rm D})P(\delta_{\rm i}).$ (8)

The priors on galactic parameters are set as in [82]: on $\delta_{\rm D}$ and
$\delta_{\rm i}$ are imposed Gaussian priors around 1 with standard deviations
given by the observational relative errors. On $\Upsilon_{*}$ is used a log-
normal prior around their fiducial values $\Upsilon_{\rm d}=0.5$ and
$\Upsilon_{\rm b}=0.7$ with a standard deviation of 0.1 dex. In the case of
the halo parameters is used a flat prior with
$10<V_{200}<500~{}\mathrm{km\,s^{-1}},1<C_{200}<100$. Maximizing the posterior
probability, one gets the best fitting value. The values of $\rho_{0}r_{0}$,
obtained with the previous method by [80], are in the second column of Table
1, while the values of the luminosity, and the effective surface brightness
(taken from SPARC’s webpage

    ttp://astroweb.cwru.edu/SPARC/SPARC_Lelli2016c.mrt) can be found in column 3, and 4 of Table 1, respectively.
    %
    %\begin{verbatim}
    %

ttp://astroweb.cwru.edu/SPARC/SPARC_Lelli2016c.mrt

As already reported, by means of the Bayesian method, [80] fitted 175 galaxy
rotation curve of the SPARC sample. The fit to three representative galaxies
has been shown in Fig. 1 of [80]. Burkert’s profile gives a good fit to the
galaxies studied. The galaxies studied, except five, have a reduced
$\chi^{2}<10$, and the best fitting values and the reduced $\chi^{2}$ for the
SPARC sample are listed in Table 1 of [80].

##  3 Results

Figure 1: Upper panel: the DMsd in terms of the magnitude, $M$ for HSBs. The
data points with error-bars represent the HSBs with surface brightness larger
than 300 $\rm L_{\odot}/pc^{2}$. The dashed line is the D09 value, and the
solid line the fit to the data. Lower panel: same as the upper panel but for
HSBs with surface brightness larger than 200 $\rm L_{\odot}/pc^{2}$.

Figure 2: Upper panel: the DMsd in terms of the effective surface brightness,
$\Sigma_{\rm eff}$ for HSBs. The data points with error-bars represent the
HSBs with surface brightness larger than 300 $\rm L_{\odot}/pc^{2}$. The
dashed line is the D09 value, the shaded region the $1\sigma$ confidence level
region obtained using MOND, and the solid line the fit to the data. Lower
panel: same as the upper panel but for HSBs with surface brightness larger
than 200 $\rm L_{\odot}/pc^{2}$.  Figure 3: Upper panel: DMsd in terms of the
magnitude, $M$ for LSBs. The data points with error-bars represent the LSBs
with surface brightness smaller than 100 $\rm L_{\odot}/pc^{2}$. The dashed
line is the D09 value, and the solid line the fit to the data. Bottom panel:
the DMsd in terms of the effective surface brightness, $\Sigma_{\rm eff}$ for
LSBs with surface brightness smaller than 100 $\rm L_{\odot}/pc^{2}$. The
dashed line is the D09 value, and the solid line the fit to the data. The
shaded region the $1\sigma$ confidence level region obtained using MOND.
Figure 4: Same as Fig. 3 but for DMsd smaller than 25 $\rm L_{\odot}/pc^{2}$.

As discussed in the abstract, in this paper we have two goals. The first one
is related to one claim of D09, namely that the DMsd is a constant universal
quantity, equal to $\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.15\pm 0.2$. The
second one, a MOND prediction, namely that for HSBs the DMsd is constant, and
equal to $\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.14$, while for LSBs the DMsd is
not constant and has smaller values than for HSBs. At the first order the DMsd
behaves as $\Sigma\simeq\sqrt{\Sigma_{b}\Sigma_{M}}$. In order to answer to
these two questions, we used the SPARC ($\it Spitzer$ Photometry and Accurate
Rotation Curves) sample [81], constituted by 175 late-type galaxies with high
quality rotation curves. We divided the sample in HSBs, and LSBs. In order to
distinguish between these two classes of objects, we recall that the division
between LSBs and HSBs is at 23 $\rm mag/arcsec^{2}$ in B band. Recalling that
in SPARC we have a photometry at 3.6 $\rm\mu m$, an estimate of the threshold
between LSBs and HSBs can be reasonably put at the effective surface
brightness $\Sigma_{\rm eff}=200L_{\odot}/pc^{2}$ (or 100 $M_{\odot}/pc^{2}$
for $M/L=0.5$)666Notice that the surface brightness distribution in SPARC is
continuous even if the definition has a certain degree of arbitrariness.

In the paper, we consider several thresholds. For the HSBs, we use values of
the effective surface brightness, $\Sigma_{\rm eff}$, $>200L_{\odot}/pc^{2}$,
and $>300L_{\odot}/pc^{2}$. For LSBs values of $\Sigma_{\rm
eff}<100L_{\odot}/pc^{2}$, and $\Sigma_{\rm eff}<25L_{\odot}/pc^{2}$. For a
given value of $M/L$, the quoted threshold corresponds to thresholds in mass
surface density $\rm M_{\odot}/pc^{2}$. The results are plotted in Figs. 1-4.
In Fig. 1, we plot the DMsd in terms of the magnitude, $M$. The DMsd of the
HSBs is represented by the data points with error-bars. The data points,
surface density of galaxies and errors, are obtained by [80] (Table A. 1). In
the upper panel we used the galaxies (HSBs) with surface brightness $>300\rm
L_{\odot}/pc^{2}$. The lower panel represents the same quantities as the upper
panel but in this case for HSBs with surface brightness larger than 200 $\rm
L_{\odot}/pc^{2}$. In both panels, the dashed line is the D09 value, and the
solid line the fit to the data. Both plots give the same information: the DMsd
is constant, independent on the magnitude. This is qualitatively in agreement
with D09, but the value of the constant DMsd is not
$\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.15\pm 0.2$, but larger
($\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.61$, for the upper panel). Another
difference between D09 result and ours is that D09 claims that their result is
valid for all kind of galaxies: LSBs, and HSBs. In reality, their conclusions
is based on the analysis of dwarf spheroidal satellites of the Milky Way. They
include only one well studied LSB: NGC 3741. In other terms their claim that
their result can be applied to all galaxies is not correct. The D09 result and
our are in qualitative agreement because they are mainly using HSBs as we are
doing in this first part of the analysis. Then HSBs are characterized by a
constant DMsd. In Fig. 2, we plot the surface density in terms of the
effective surface brightness, $\Sigma_{\rm eff}$. As in Fig. 1, the data
points with error-bars represent the HSBs with surface brightness larger than
300 $\rm L_{\odot}/pc^{2}$ (upper panel), and 200 $\rm L_{\odot}/pc^{2}$
(lower panel). The dashed line is the D09 value, and the solid line the fit to
the data. We also plotted, as a shaded region, the $1\sigma$ confidence level
region obtained using MOND (see Appendix). The plot gives two information: a)
the DMsd is constant and does not depend on $\Sigma_{\rm eff}$. b) MOND
predicts a constant value of the DMsd equal to
$\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.14$ almost identical to that of D09,
qualitatively in agreement with our result, for what concerns the constancy,
but in disagreement with the value that we obtain,
$\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.54$ (upper panel). In Fig. 3, the upper
panel represents the DMsd in terms of the magnitude, $M$. The data points with
error-bars represent the LSBs with surface brightness smaller than 100 $\rm
L_{\odot}/pc^{2}$. The dashed line is the D09 value, and the solid line the
fit to the data. The plot shows that the DMsd decreases with increasing
magnitude (smaller luminosity). However, in the range of magnitudes
$\simeq-20.5$ to $\simeq-17.5$ the surface density, within the errors, is in
agreement with a flat surface density close to that of D09. For smaller
magnitudes several LSBs have a smaller value than that predicted by D09. In
other terms, for $\Sigma_{\rm eff}<100\rm L_{\odot}/pc^{2}$, for decreasing
values of the magnitude the LSBs tend to have smaller values of the surface
density with respect to D09. The bottom panel in Fig. 3, represents the DMsd
in terms of $\Sigma_{\rm eff}$. The behavior for decreasing values of
$\Sigma_{\rm eff}$ is similar to that of the upper panel. Within the errors
limits, and in the range $1.4\preceq\log{\Sigma_{\rm eff}}\preceq 2$, the DMsd
is in agreement with the D09 result, after it has smaller values than D09.
Still in the bottom panel it is also plotted a shaded region which is the
$1\sigma$ confidence level region obtained using MOND. The plot shows that in
the range $1.6\preceq\log{\Sigma_{\rm eff}}\preceq 2$, MOND is in agreement
with D09 prediction, while for smaller values of $\Sigma_{\rm eff}$ it shows a
decline towards smaller values of the surface density, in agreement with the
data. The upper panel of Fig. 4, represents the surface density in terms of
the magnitude, $M$. The data points with error-bars represent the LSBs with
surface brightness smaller than 25 $\rm L_{\odot}/pc^{2}$. The trend with the
magnitude of the DMsd is similar to that shown in Fig. 3, even if the values
of the DMsd are smaller than that of D09 since the most negative magnitudes.
The bottom panel in Fig. 4, represents the DMsd in terms of $\Sigma_{\rm
eff}$. As before the dots with error-bars represent the LSBs galaxies the
shaded region is the $1\sigma$ confidence level region obtained using MOND,
the solid line the fit to the LSBs with surface brightness smaller than 25
$\rm L_{\odot}/pc^{2}$, and the dashed line D09 prediction. Within the errors
limits, and in the range $1.35\preceq\Sigma_{\rm eff}\preceq 1.5$, DMsd is in
agreement with the D09 result, after it has smaller values than D09. The
behavior for decreasing values of $\Sigma_{\rm eff}$ is similar to that of the
upper panel. In this case MOND predicts a surface density smaller than that of
D09 in the entire range of the data. As in Fig. 3, it shows a decline towards
smaller values of the DMsd, in agreement with the data.

Summarizing the results till now reported, concerning HSBs, the DMsd is
constant in terms of the magnitude (in qualitative agreement with D09) and
$\Sigma_{\rm eff}$ (in qualitative agreement with MOND), but larger than the
prediction of D09, and MOND. In the case of the LSBs, there is a decrease of
the DMsd with the luminosity (in disagreement with D09), and a decrease of
$\Sigma_{\rm eff}$ (in agreement with MOND), and it is usually smaller than
the D09 prediction. MOND, is in qualitative agreement with the data predicts a
decrease of the DMsd with $\Sigma_{\rm eff}$.

##  4 Discussion

By means of the HSBs, and LSBs in SPARC we studied the DMsd of the class of
galaxies in the sample. Our interest was that of understanding whether the DM
surface density is a constant universal quantity equal to
$\log{(\rm\Sigma/M_{\odot}pc^{-2})}=2.15\pm 0.2$, as claimed by D09 or if it
assumes different classes of objects and their baryon surface density. At the
same time, we studied the MOND prediction ([2, 69]), namely a constant DMsd in
HSBs close to that predicted by D09, and a smaller, decreasing DMsd with
decreasing surface brightness (baryon surface density) in the case of LSBs. In
order to study these two issues, we grouped the SPARC sample in HSBs, and
LSBs, and we also considered for these two groups of galaxies several
thresholds in $\Sigma_{\rm eff}$. In the case of HSBs, and for galaxies
characterized by $\Sigma_{\rm eff}>200L_{\odot}/pc^{2}$, and $\Sigma_{\rm
eff}>300L_{\odot}/pc^{2}$, we found that the DMsd vs magnitude is constant, as
in D09, and similarly, we found a constant DMsd vs $\Sigma_{\rm eff}$ as in
MOND prediction. In spite of the fact that the result of our study, in the
case of HSBs is in qualitative agreement with D09, and MOND, the value of the
DMsd is larger than the one predicted by D09, and MOND. In the case of LSBs,
we also used two thresholds: $\Sigma_{\rm eff}<100L_{\odot}/pc^{2}$, and
$\Sigma_{\rm eff}<25L_{\odot}/pc^{2}$. For both thresholds, the DMsd is
decreasing, with increasing value of magnitude (decreasing luminosity), in all
magnitude range. In the case of $\Sigma_{\rm eff}<100L_{\odot}/pc^{2}$, the
DMsd, within the errors, and in in the range of magnitudes $\simeq-20.5$ to
$\simeq-17.5$ is in agreement with a flat surface density close to that of
D09. Going to smaller negative magnitudes several LSBs have a smaller value
than that predicted by D09. In the case of LSBs with surface brightness
smaller than 25 $\rm L_{\odot}/pc^{2}$, the trend with magnitude of the
surface density is similar to that of the previous case ($\Sigma_{\rm
eff}<100L_{\odot}/pc^{2}$), even if the values of the DMsd are smaller than
that of D09 since the most negative magnitudes. Concerning the trend of the DM
surface density in terms of the $\Sigma_{\rm eff}$ for the two thresholds
($\Sigma_{\rm eff}<100L_{\odot}/pc^{2}$, and $\Sigma_{\rm
eff}<25L_{\odot}/pc^{2}$), it is decreasing with decreasing $\Sigma_{\rm
eff}$. In the case $\Sigma_{\rm eff}<100L_{\odot}/pc^{2}$, the DMsd is, in the
errors limits, and in the range $1.4\preceq\log{\Sigma_{\rm eff}}\preceq 2$,
in agreement with the D09 result, after it has smaller values than D09. In the
case $\Sigma_{\rm eff}<25L_{\odot}/pc^{2}$ the DMsd is, in the errors limits,
and in the range $1.35\preceq\Sigma_{\rm eff}\preceq 1.5$, in agreement with
the D09 result, after it has smaller values than D09. Concerning MOND’s
predictions, in the case $\Sigma_{\rm eff}<100L_{\odot}/pc^{2}$, and
$1.6\preceq\log{\Sigma_{\rm eff}}\preceq 2$, MOND is in agreement with D09
prediction, while for smaller values of $\Sigma_{\rm eff}$ it shows a decline
towards smaller values of the DMsd, in agreement with the data. In the case
$\Sigma_{\rm eff}<25L_{\odot}/pc^{2}$, for all values of $\Sigma_{\rm eff}$,
DMsd is smaller than D09 prediction, and shows a decline towards smaller
values of the DMsd, in agreement with data.

In summary, the SPARC’s HSBs tell us that both D09, and MOND correctly predict
a constant DMsd, even if the value predicted is smaller than that obtained
from data. SPARC’s LSBs shows a decrease of the DMsd with luminosity and
surface brightness (or baryon surface density), in agreement with MOND but not
with D09.

Appendix: DMsd in MOND

In this section, we summarize the calculation developed in [69] to obtain the
DMsd in MOND. Using the MOND formulation by [83], the potential $\phi$ is
given by a generalization of Poisson equation

$\nabla[\mu(|\nabla\phi|/a_{0})\nabla\phi]=4\pi G\rho$ (9)

where the baryon density is indicated by $\rho$, $\mu(x)$ is the interpolating
function, and $a_{0}$ is the MOND constant. The difference between the MOND
acceleration field $\nabla\phi$ and the Newtonian one, interpreted from a
Newtonian point of view is explained by the presence of dark matter or as
indicated by Milgrom, by ”phantom matter” having a density [84]

$\rho_{\rm p}=\frac{1}{4\pi G}\Delta\phi-\rho$ (10)

Using Eq. (9), we can write

$\rho_{P}({\bf r\rm})=-\frac{1}{4\pi
Ga_{0}}\frac{\mu^{\prime}}{\mu}\nabla|\nabla\phi|\nabla\phi+\rho({\bf
r})(1/\mu-1).$ (11)

The previous equation can be written as

$\rho_{P}=\rho_{P_{1}}+\rho_{P_{2}}=\frac{-a_{0}}{4\pi G}{\bf
e}\cdot\nabla\mathcal{V}(|\nabla\phi|/a_{0})+(1/\mu-1)\rho.$ (12)

After defining $\mathcal{V}(x)=\int L(x)dx$, $x={\rm g}/a_{0}$,
$L=\frac{\mu^{\prime}}{\mu}x$, and defining a vector $\bf e$ in the direction
of $\nabla\phi$, we also have

$\rho_{P}=\rho_{P_{1}}+\rho_{P_{2}}=\frac{-a_{0}}{4\pi G}{\bf
e}\cdot\nabla\mathcal{V}(|\nabla\phi|/a_{0})+(1/\mu-1)\rho,$ (13)

with

$\rho_{P_{1}}=\frac{-a_{0}}{4\pi G}{\bf
e}\nabla\mathcal{V}(|\nabla\phi|/a_{0}),$ (14) $\rho_{P_{2}}=(1/\mu-1)\rho.$
(15)

The integral of Equation (13) is calculated considering two cases: $x\geq 1$
(acceleration above the MOND universal constant), and $x\leq 1$. Let us
consider the second term (Equation 15), $\rho_{P_{2}}=(1/\mu-1)\rho$. We write
$a_{N}=\frac{g_{N}}{a_{0}}=({\rm g}/a_{0})\mu({\rm g}/a_{0})$, where
$g_{N}={\rm g}\mu({\rm g/a_{0}})$ is the Newtonian acceleration.

In the case $n=1$, and $\mu=\frac{x}{(1+x^{n})^{1/n}}=\frac{{\rm
g}/a_{0}}{1+{\rm g}/a_{0}}$, multiplying for $\mu$ we have

$\mu=\frac{\mu{\rm g}/a_{0}}{\mu{\rm g}/a_{0}+\mu}=\frac{a_{N}}{a_{N}+\mu},$
(16)

solving with respect to $\mu$ we have

$\mu=-1/2a_{N}+1/2\sqrt{a_{N}^{2}+4a_{N}}.$ (17)

For a double exponential disk, with cylindrical radius $R$ and altitude $z$,
we have

$a{{}_{N}}\simeq\frac{GM}{a_{0}R{{}^{2}}}=\frac{G2\pi\rho{{}_{0}}h{{}_{z}}h{{}_{r}}{{}^{2}}}{a{{}_{0}}R{{}^{2}}}\simeq\frac{\Sigma{{}_{b}}}{\Sigma{{}_{M}}}\frac{h{{}_{r}}{{}^{2}}}{R{{}^{2}}}$
(18)

being $\Sigma_{b}$, the baryonic surface density, approximated by an
integrated constant volume density
($\Sigma_{b}\simeq\int\rho_{0}dz=\rho_{0}h_{z}$).

The MOND interpolating function becomes

$\mu=-1/2\,{\frac{\Sigma_{b}\,{{\it
h_{r}}}^{2}}{\Sigma_{M}\,{R}^{2}}}+1/2\,\sqrt{{\frac{{\Sigma_{b}}^{2}{{\it
h_{r}}}^{4}}{{\Sigma_{M}}^{2}{R}^{4}}}+4\,{\frac{\Sigma_{b}\,{{\it
h_{r}}}^{2}}{\Sigma_{M}\,{R}^{2}}}}.$ (19)

Then

$\frac{1-\mu}{\mu}\int_{-\infty}^{+\infty}\rho
dz=2(1-\mu)x\Sigma_{M}\frac{R^{2}}{h_{r}^{2}}\exp^{-R/h_{r}}.$ (20)

The integral of the first (Eq. 13) term is given by

$\displaystyle\int^{+\infty}_{-\infty}\rho_{P_{1}}dz$ $\displaystyle=$
$\displaystyle 2\int^{\infty}_{0}\rho_{P_{1}}dz=$ (21)
$\displaystyle\Sigma_{M}[\mathcal{V(\infty)}-\mathcal{V}(0)]=$
$\displaystyle\Sigma_{M}\int^{\infty}_{0}L(x)dx=a\Sigma_{M},$

where $\Sigma_{M}=\frac{a_{0}}{2\pi G}$.

Integrating the Burkert profile D09 obtained a surface density given by

$\Sigma_{0}=2\int\rho
dr=2\int\frac{\rho_{0}r_{0}^{3}}{(r+r_{0})(r^{2}+r^{2}_{0})}=\frac{\pi}{2}\rho_{0}r_{0}=\frac{\pi}{2}\Sigma_{c}.$
(22)

If we call $\Sigma_{c}^{*}$ the MOND analog of $\Sigma_{c}$ we have

$\Sigma_{c}^{*}=\frac{2\lambda}{\pi}\Sigma_{M},$ (23)

with $\Sigma_{M}=138\frac{a_{0}}{1.2\times 10^{-8}\rm
cms^{-2}}M_{\odot}/pc^{2}$.

The full integral can be written as

$F=F_{1}+F_{2}=\frac{2}{\pi}\Sigma_{M}\int_{0}^{x}L(x)dx+\frac{2}{\pi}2(1-\mu)x\Sigma_{M}\frac{R^{2}}{h_{r}^{2}}\exp^{-R/h_{r}}.$
(24)

As shown in [69], in the case $x\leq 1$ we have

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\Sigma_{M}\arctan{(\sqrt{\frac{\Sigma_{b}}{\Sigma_{M}}}\frac{h_{r}}{R})}+$
(25) $\displaystyle\frac{2}{\pi}2\frac{R\Sigma_{M}}{{\it
h_{r}}}\left(1+1/2\,{\frac{\Sigma_{b}\,{{\it
h_{r}}}^{2}}{\Sigma_{M}\,{R}^{2}}}-1/2\,\sqrt{{\frac{{\Sigma_{b}}^{2}{{\it
h_{r}}}^{4}}{{\Sigma_{M}}^{2}{R}^{4}}}+4\,{\frac{\Sigma_{b}\,{{\it
h_{r}}}^{2}}{\Sigma_{M}\,{R}^{2}}}}\right)$
$\displaystyle\times\sqrt{{\frac{\Sigma_{b}}{\Sigma_{M}}}}{{\rm
e}^{-{\frac{R}{{\it h_{r}}}}}}.$

At small $x$ the first term in the right hand side tend to zero, and $F$ is
dominated by the second term. At first order the trend of $F$ is
$\sqrt{\Sigma_{b}\Sigma_{M}}$.

In the case $x\geq 1$, we have

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\Sigma_{M}\arctan{(\frac{\Sigma_{b}}{\Sigma_{M}}\frac{h_{r}^{2}}{R^{2}})}+$
(26) $\displaystyle\frac{4}{\pi}\,\left(1+1/2\,{\frac{\Sigma_{b}\,{{\it
h_{r}}}^{2}}{\Sigma_{M}\,{R}^{2}}}-1/2\,\sqrt{{\frac{{\Sigma_{b}}^{2}{{\it
h_{r}}}^{4}}{{\Sigma_{M}}^{2}{R}^{4}}}+4\,{\frac{\Sigma_{b}\,{{\it
h_{r}}}^{2}}{\Sigma_{M}\,{R}^{2}}}}\right)$
$\displaystyle\times\Sigma_{M}\,{{\rm e}^{-{\frac{R}{{\it
h_{r}}}}}}R^{2}/h_{r}^{2}.$

For large $x$, the second term tends to 0, O(1), and the first one is
$\simeq\Sigma_{M}$.

As it comes from the previous results and the plots in [69] there is a double
trend of the surface density. For $R=h_{r}$, and at small
$\Sigma_{b}/\Sigma_{M}$, , the surface density increases as
$\sqrt{\frac{\Sigma_{b}}{\Sigma_{M}}}\Sigma_{M}$. For larger
$\Sigma_{b}/\Sigma_{M}$, the behavior of the surface density flattens till
when, at large $\Sigma_{b}/\Sigma_{M}$ tends to $\Sigma_{M}$.

Acknowledgements

The author thanks Valerio Pirronello for a careful reading of the manuscript.
Financial contribution from the Programma ricerca di Ateneo UNICT 2020-22
linea 2 is graciously acknowl- edged. All authors read and approved the final
manuscript.

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Table 1:  In the table, the first column gives the name of the galaxy, the second one the best fitting values of the DM surface density (obtained in [80]), and the relative errors, the third column the luminosity at 3.6 $\rm\mu m$, and the relative errors given in the SPARC’s webpage, and the fourth column the effective surface brightness given in SPARC’s webpage. Name | $\log{(\rho_{0}r_{0})}(M_{\odot}/pc^{3})$ |  L[3.6 $\mu m$] $(10^{9}L_{\odot})$ | $\Sigma_{eff}(L_{\odot}/pc^{2})$
---|---|---|---
UGC02487 |  2.33 $\pm$0.23 |  489.955$\pm$4.061 | 818.14
UGC02885 |  2.24$\pm$0.11 |  403.525$\pm$4.088 | 421.53
NGC6195 |  2.18$\pm$0.1 |  391.076$\pm$6.123 | 686.8
UGC11455 |  3.01$\pm$0.11 |  374.322$\pm$3.792 | 571.26
NGC5371 |  2.13$\pm$0.09 |  340.393$\pm$1.881 | 560.83
NGC2955 |  2.57$\pm$0.11 |  319.422$\pm$4.413 | 974.61
NGC0801 |  1.82$\pm$0.06 |  312.57$\pm$3.455 | 818.14
ESO563-G021 |  3.28$\pm$0.13 |  311.177$\pm$2.579 | 429.37
UGC09133 |  1.92$\pm$0.06 |  282.926$\pm$2.345 | 1284.78
UGC02953 |  3.64$\pm$0.06 |  259.518$\pm$0.717 | 1544.65
NGC7331 |  2.4$\pm$0.06 |  250.631$\pm$0.693 | 2470.76
NGC3992 |  2.82$\pm$0.18 |  226.932$\pm$0.836 | 360.44
NGC6674 |  1.69$\pm$0.07 |  214.654$\pm$1.977 | 560.83
NGC5985 |  3.37$\pm$0.15 |  208.728$\pm$1.538 | 288.95
NGC2841 |  2.25$\pm$0.04 |  188.121$\pm$0.52 | 983.63
IC4202 |  3.55$\pm$0.06 |  179.749$\pm$3.311 | 391.59
NGC5005 |  2.67$\pm$0.21 |  178.72$\pm$0.494 | 1789.91
NGC5907 |  2.93$\pm$0.04 |  175.425$\pm$0.646 | 449.6
UGC05253 |  2.77$\pm$0.09 |  171.582$\pm$0.79 | 1488.78
NGC5055 |  2.22$\pm$0.06 |  152.922$\pm$0.282 | 1383.02
NGC2998 |  1.94$\pm$0.09 |  150.902$\pm$2.085 | 479.55
UGC11914 |  2.95$\pm$0.08 |  150.028$\pm$0.553 | 2425.66
NGC3953 |  2.79$\pm$0.45 |  141.301$\pm$0.521 | 587.26
UGC12506 |  2.9$\pm$0.07 |  139.571$\pm$3.214 | 144.82
NGC0891 |  2.82$\pm$0.06 |  138.34$\pm$0.255 | 1617.45
UGC06614 |  2.22$\pm$0.17 |  124.35$\pm$2.52 | 1461.61
UGC02916 |  2.68$\pm$0.08 |  124.153$\pm$1.83 | 2425.66
UGC03205 |  3.06$\pm$0.09 |  113.642$\pm$1.361 | 587.26
NGC5033 |  2.65$\pm$0.09 |  110.509$\pm$0.407 | 2036.24
NGC4088 |  2.21$\pm$0.15 |  107.286$\pm$0.494 | 457.96
NGC4157 |  2.12$\pm$0.1 |  105.62$\pm$0.486 | 848.85
UGC03546 |  2.57$\pm$0.11 |  101.336$\pm$0.747 | 2403.42
UGC06787 |  1.9$\pm$0.06 |  98.256$\pm$0.543 | 1891.6
NGC4051 |  2.44$\pm$0.37 |  95.268$\pm$0.439 | 319.76
NGC4217 |  2.98$\pm$0.16 |  85.299$\pm$0.393 | 483.98
NGC3521 |  2.35$\pm$0.1 |  84.836$\pm$0.156 | 2232.7
NGC2903 |  2.82$\pm$0.2 |  81.863$\pm$0.151 | 632.16
NGC2683 |  2.67$\pm$0.19 |  80.415$\pm$0.222 | 1139.8
NGC4013 |  2.03$\pm$0.05 |  79.094$\pm$0.364 | 746.16
NGC7814 |  2.71$\pm$0.06 |  74.529$\pm$0.343 | 2709.13
UGC06786 |  2.43$\pm$0.09 |  73.407$\pm$0.676 | 992.73
NGC3877 |  2.88$\pm$0.11 |  72.535$\pm$0.401 | 598.18
NGC0289 |  1.84$\pm$0.12 |  72.065$\pm$0.465 | 1587.93
NGC1090 |  2.82$\pm$0.22 |  72.045$\pm$0.796 | 283.68
NGC3726 |  1.96$\pm$0.12 |  70.234$\pm$0.388 | 198.08
UGC09037 |  2.33$\pm$0.1 |  68.614$\pm$1.769 | 334.83
NGC6946 |  2.31$\pm$0.16 |  66.173$\pm$0.122 | 571.26
NGC4100 |  2.79$\pm$0.11 |  59.394$\pm$0.328 | 388
NGC3893 |  2.68$\pm$0.16 |  58.525$\pm$0.377 | 1573.37
UGC06973 |  2.98$\pm$0.11 |  53.87$\pm$0.347 | 3317.65
ESO079-G014 |  2.56$\pm$0.15 |  51.733$\pm$0.524 | 157.34
UGC08699 |  2.17$\pm$0.09 |  50.302$\pm$0.695 | 2191.95
NGC4138 |  2.82$\pm$0.29 |  44.111$\pm$0.284 | 1909.11
NGC3198 |  2.15$\pm$0.06 |  38.279$\pm$0.212 | 178.99
NGC3949 |  2.86$\pm$0.23 |  38.067$\pm$0.28 | 1058.84
NGC6015 |  1.73$\pm$0.1 |  32.129$\pm$0.237 | 331.76
NGC3917 |  2.44$\pm$0.1 |  21.966$\pm$0.202 | 112.94
NGC4085 |  2.55$\pm$0.15 |  21.724$\pm$0.2 | 856.7
NGC4389 |  3.03$\pm$0.22 |  21.328$\pm$0.216 | 207.41
NGC4559 |  2.05$\pm$0.15 |  19.377$\pm$0.107 | 211.27
NGC3769 |  2.25$\pm$0.12 |  18.679$\pm$0.189 | 626.37
NGC4010 |  2.27$\pm$0.11 |  17.193$\pm$0.19 | 14.75
NGC3972 |  2.48$\pm$0.12 |  14.353$\pm$0.172 | 164.75
UGC03580 |  2.33$\pm$0.07 |  13.266$\pm$0.195 | 614.94
NGC6503 |  2.26$\pm$0.03 |  12.845$\pm$0.059 | 774.16
UGC11557 |  2.36$\pm$0.53 |  12.101$\pm$0.212 | 106.86
UGC00128 |  2.18$\pm$0.1 |  12.02$\pm$0.565 | 20.36
F579-V1 |  2.46$\pm$0.4 |  11.848$\pm$0.742 | 56.6
NGC4183 |  2.38$\pm$0.18 |  10.838$\pm$0.15 | 86.46
F571-8 |  2.79$\pm$0.14 |  10.164$\pm$0.412 | 825.71
NGC2403 |  2.34$\pm$0.04 |  10.041$\pm$0.028 | 341.06
UGC06930 |  2.29$\pm$0.25 |  8.932$\pm$0.14 | 73.93
F568-3 |  2.31$\pm$0.29 |  8.346$\pm$0.592 | 23.81
UGC01230 |  2.5$\pm$0.54 |  7.62$\pm$0.379 | 28.63
NGC0247 |  2.06$\pm$0.18 |  7.332$\pm$0.027 | 33.79
NGC7793 |  2.01$\pm$0.16 |  7.05$\pm$0.026 | 233.79
UGC06917 |  2.33$\pm$0.1 |  6.832$\pm$0.12 | 53.07
NGC1003 |  1.7$\pm$0.09 |  6.82$\pm$0.075 | 140.87
F574-1 |  2.31$\pm$0.12 |  6.537$\pm$0.596 | 29.98
F568-1 |  2.54$\pm$0.21 |  6.252$\pm$0.564 | 20.36
UGC06983 |  2.39$\pm$0.1 |  5.298$\pm$0.102 | 53.56
UGC05986 |  2.43$\pm$0.15 |  4.695$\pm$0.048 | 76.71
NGC0055 |  1.94$\pm$0.04 |  4.628$\pm$0.013 | 54.55
ESO116-G012 |  2.33$\pm$0.16 |  4.292$\pm$0.071 | 90.54
UGC07323 |  2.01$\pm$0.25 |  4.109$\pm$0.042 | 61.49
UGC05005 |  1.84$\pm$0.31 |  4.1$\pm$0.283 | 25.63
F561-1 |  1.9$\pm$0.59 |  4.077$\pm$0.327 | 22.12
NGC0024 |  2.8$\pm$0.35 |  3.889$\pm$0.036 | 151.65
F568-V1 |  2.53$\pm$0.28 |  3.825$\pm$0.384 | 31.39
UGC06628 |  2.16$\pm$0.6 |  3.739$\pm$0.076 | 34.74
UGC02455 |  3.63$\pm$0.31 |  3.649$\pm$0.034 | 254
UGC07089 |  1.85$\pm$0.2 |  3.585$\pm$0.089 | 37.4
UGC05999 |  2.24$\pm$0.54 |  3.384$\pm$0.231 | 22.95
NGC2976 |  2.46$\pm$0.26 |  3.371$\pm$0.019 | 308.2
UGC05750 |  1.8$\pm$0.21 |  3.336$\pm$0.264 | 6.87
NGC0100 |  2.13$\pm$0.2 |  3.232$\pm$0.063 | 64.99
UGC00634 |  2.08$\pm$0.28 |  2.989$\pm$0.146 | 26.11
F563-V2 |  2.62$\pm$0.4 |  2.986$\pm$0.267 | 23.6
NGC5585 |  2$\pm$0.09 |  2.943$\pm$0.033 | 90.54
NGC0300 |  2.11$\pm$0.23 |  2.922$\pm$0.008 | 147.51
UGC06923 |  2.17$\pm$0.18 |  2.89$\pm$0.077 | 166.28
F574-2 |  0.74$\pm$0.5 |  2.877$\pm$0.384 | 10.89
UGC07125 |  1.77$\pm$0.17 |  2.712$\pm$0.08 | 28.11
UGC07524 |  2.04$\pm$0.07 |  2.436$\pm$0.025 | 29.71
UGC06399 |  2.21$\pm$0.09 |  2.296$\pm$0.072 | 30.82
UGC07151 |  2.28$\pm$0.09 |  2.284$\pm$0.025 | 76.71
F567-2 |  1.87$\pm$0.66 |  2.134$\pm$0.305 | 11.5
UGC04325 |  2.67$\pm$0.17 |  2.026$\pm$0.035 | 41
UGC00191 |  2.21$\pm$0.39 |  2.004$\pm$0.063 | 50.21
F563-1 |  2.38$\pm$0.25 |  1.903$\pm$0.17 | 14.22
F571-V1 |  1.97$\pm$0.42 |  1.849$\pm$0.267 | 15.3
UGC07261 |  2.44$\pm$0.47 |  1.753$\pm$0.048 | 39.16
UGC10310 |  2.21$\pm$0.31 |  1.741$\pm$0.053 | 28.37
UGC02259 |  2.51$\pm$0.17 |  1.725$\pm$0.038 | 47.08
F583-4 |  2.01$\pm$0.25 |  1.715$\pm$0.185 | 24.94
UGC12732 |  1.73$\pm$0.2 |  1.667$\pm$0.048 | 26.84
UGC06818 |  2.15$\pm$0.22 |  1.588$\pm$0.057 | 56.08
UGC04499 |  2.1$\pm$0.21 |  1.552$\pm$0.043 | 33.79
F563-V1 |  1$\pm$0.41 |  1.54$\pm$0.165 | 9.75
UGC06667 |  2.26$\pm$0.07 |  1.397$\pm$0.066 | 18.06
UGC02023 |  2.73$\pm$0.6 |  1.308$\pm$0.033 | 27.59
UGC04278 |  2.08$\pm$0.15 |  1.307$\pm$0.026 | 34.11
UGC12632 |  2.04$\pm$0.18 |  1.301$\pm$0.03 | 13.09
UGC08286 |  2.46$\pm$0.03 |  1.255$\pm$0.018 | 39.52
UGC07399 |  2.6$\pm$0.16 |  1.156$\pm$0.024 | 113.98
NGC4214 |  1.9$\pm$0.22 |  1.141$\pm$0.008 | 363.77
UGC05414 |  1.98$\pm$0.24 |  1.123$\pm$0.028 | 32.87
UGC08490 |  2.31$\pm$0.14 |  1.017$\pm$0.012 | 124.98
IC2574 |  1.69$\pm$0.12 |  1.016$\pm$0.012 | 15.88
UGC06446 |  2.29$\pm$0.16 |  0.988$\pm$0.032 | 37.05
F583-1 |  2.13$\pm$0.15 |  0.986$\pm$0.093 | 11.19
UGC11820 |  1.3$\pm$0.18 |  0.97$\pm$0.047 | 20.18
UGC07690 |  2.45$\pm$0.35 |  0.858$\pm$0.018 | 184.01
UGC04305 |  2.31$\pm$0.44 |  0.736$\pm$0.007 | 76.71
NGC2915 |  2.46$\pm$0.07 |  0.641$\pm$0.008 | 347.4
UGC05716 |  1.62$\pm$0.12 |  0.588$\pm$0.042 | 27.59
UGC05829 |  1.77$\pm$0.4 |  0.564$\pm$0.019 | 10.59
F565-V2 |  2.06$\pm$0.18 |  0.559$\pm$0.098 | 6.93
DDO161 |  1.65$\pm$0.14 |  0.548$\pm$0.015 | 20.74
DDO170 |  1.99$\pm$0.2 |  0.543$\pm$0.03 | 9.39
NGC1705 |  2.71$\pm$0.11 |  0.533$\pm$0.01 | 347.4
UGC05721 |  2.63$\pm$0.16 |  0.531$\pm$0.011 | 233.79
UGC08837 |  2.5$\pm$0.24 |  0.501$\pm$0.015 | 15.73
UGC07603 |  2.32$\pm$0.19 |  0.376$\pm$0.008 | 81.81
UGC00891 |  1.89$\pm$0.18 |  0.374$\pm$0.017 | 19.09
UGC01281 |  1.99$\pm$0.04 |  0.353$\pm$0.009 | 13.83
UGC09992 |  2.03$\pm$0.52 |  0.336$\pm$0.017 | 20.18
D512-2 |  1.79$\pm$0.3 |  0.325$\pm$0.022 | 9.22
UGC00731 |  2.06$\pm$0.11 |  0.323$\pm$0.019 | 25.63
UGC08550 |  2.14$\pm$0.17 |  0.289$\pm$0.009 | 45.38
UGC07608 |  2.18$\pm$0.47 |  0.264$\pm$0.012 | 16.48
NGC4068 |  2.57$\pm$0.32 |  0.236$\pm$0.005 | 30.26
NGC2366 |  1.9$\pm$0.06 |  0.236$\pm$0.005 | 31.98
UGC05918 |  1.95$\pm$0.21 |  0.233$\pm$0.011 | 5.36
D631-7 |  1.89$\pm$0.06 |  0.196$\pm$0.009 | 20.93
NGC3109 |  2$\pm$0.04 |  0.194$\pm$0.002 | 11.4
UGCA281 |  2.03$\pm$0.2 |  0.194$\pm$0.007 | 12.5
DDO168 |  2.12$\pm$0.08 |  0.191$\pm$0.005 | 18.23
DDO064 |  2.01$\pm$0.28 |  0.157$\pm$0.007 | 17.41
PGC51017 |  1.64$\pm$0.39 |  0.155$\pm$0.014 | 15.03
UGCA442 |  1.96$\pm$0.08 |  0.14$\pm$0.005 | 7.6
UGC07866 |  1.77$\pm$0.37 |  0.124$\pm$0.004 | 21.92
UGC07232 |  3.21$\pm$0.3 |  0.113$\pm$0.002 | 83.34
UGC07559 |  1.65$\pm$0.37 |  0.109$\pm$0.004 | 17.9
NGC6789 |  2.87$\pm$0.32 |  0.1$\pm$0.003 | 59.27
KK98-251 |  1.52$\pm$0.32 |  0.085$\pm$0.007 | 7.89
UGC05764 |  2.36$\pm$0.15 |  0.085$\pm$0.006 | 9.31
CamB |  2.55$\pm$0.28 |  0.075$\pm$0.003 | 7.89
ESO444-G084 |  2.27$\pm$0.22 |  0.071$\pm$0.003 | 19.81
DDO154 |  1.86$\pm$0.04 |  0.053$\pm$0.002 | 19.99
UGC07577 |  2.18$\pm$0.41 |  0.045$\pm$0.002 | 11.94
D564-8 |  1.46$\pm$0.25 |  0.033$\pm$0.004 | 10.11
NGC3741 |  1.82$\pm$0.04 |  0.028$\pm$0.001 | 42.94
UGC04483 |  1.84$\pm$0.2 |  0.013$\pm$0.001 | 29.71
UGCA444 |  1.83$\pm$0.05 |  0.012$\pm$0 | 11.94